TPTP Problem File: SLH0495^1.p
View Solutions
- 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 : Hales_Jewett/0002_Hales_Jewett/prob_00435_017062__5653822_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1349 ( 707 unt; 77 typ; 0 def)
% Number of atoms : 3231 (1088 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 8852 ( 384 ~; 56 |; 126 &;7028 @)
% ( 0 <=>;1258 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 361 ( 361 >; 0 *; 0 +; 0 <<)
% Number of symbols : 74 ( 71 usr; 11 con; 0-4 aty)
% Number of variables : 3133 ( 267 ^;2830 !; 36 ?;3133 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:39:20.841
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (71)
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
fun_upd_nat_set_nat: ( nat > set_nat ) > nat > set_nat > nat > set_nat ).
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_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
sgn_sgn_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
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_Hales__Jewett_Ois__line,type,
hales_is_line: ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ois__subspace,type,
hales_is_subspace: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > $o ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
inf_inf_int: int > int > int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_It__Nat__Onat_J,type,
semila1667268886620078168et_nat: ( set_nat > set_nat > set_nat ) > set_nat > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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__Num__Onum_J,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
zero_n2687167440665602831ol_nat: $o > 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__Num__Onum,type,
collect_num: ( num > $o ) > set_num ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
set_ord_lessThan_num: num > set_num ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_v_B____,type,
b: nat > set_nat ).
thf(sy_v_L,type,
l: nat > nat > nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_t,type,
t: nat ).
% Relevant facts (1264)
thf(fact_0_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% assms(1)
thf(fact_1_assms_I2_J,axiom,
ord_less_nat @ one_one_nat @ t ).
% assms(2)
thf(fact_2_assms_I3_J,axiom,
hales_is_line @ l @ n @ t ).
% assms(3)
thf(fact_3_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_4_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_5_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_6_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_7_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_8_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_9_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_10_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_11_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_12_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_13_lessThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_14_lessThan__iff,axiom,
! [I: num,K: num] :
( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
= ( ord_less_num @ I @ K ) ) ).
% lessThan_iff
thf(fact_15_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_16_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_17_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_18_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_19_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_20_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_21_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_22_less__imp__neq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_23_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_24_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_25_order_Oasym,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ~ ( ord_less_num @ B @ A2 ) ) ).
% order.asym
thf(fact_26_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_27_ord__eq__less__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_28_ord__eq__less__trans,axiom,
! [A2: num,B: num,C: num] :
( ( A2 = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_29_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_30_ord__less__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_31_ord__less__eq__trans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_32_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X3 )
=> ( P @ Y3 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_33_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_34_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_35_antisym__conv3,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_num @ Y @ X )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_36_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_37_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_38_linorder__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_39_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_40_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_41_dual__order_Oasym,axiom,
! [B: num,A2: num] :
( ( ord_less_num @ B @ A2 )
=> ~ ( ord_less_num @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_42_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_43_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_44_dual__order_Oirrefl,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_45_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_46_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: 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 @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_47_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: 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 @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_48_linorder__less__wlog,axiom,
! [P: num > num > $o,A2: num,B: num] :
( ! [A3: num,B2: num] :
( ( ord_less_num @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: num] : ( P @ A3 @ A3 )
=> ( ! [A3: num,B2: num] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_49_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_50_order_Ostrict__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_51_order_Ostrict__trans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_52_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_53_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_54_not__less__iff__gr__or__eq,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ( ord_less_num @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_55_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_56_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_57_dual__order_Ostrict__trans,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_num @ B @ A2 )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_58_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_59_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_60_order_Ostrict__implies__not__eq,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_61_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_62_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_63_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_64_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A2: num] :
( ( ord_less_num @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_65_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_66_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_67_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_68_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_69_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_70_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_71_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_72_linorder__neqE,axiom,
! [X: num,Y: num] :
( ( X != Y )
=> ( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_73_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_74_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_75_order__less__asym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_asym
thf(fact_76_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_78_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_79_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_80_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_81_linorder__neq__iff,axiom,
! [X: num,Y: num] :
( ( X != Y )
= ( ( ord_less_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_82_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_83_order__less__asym_H,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_84_order__less__asym_H,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ~ ( ord_less_num @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_85_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_86_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_87_order__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_88_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_89_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_90_ord__eq__less__subst,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_91_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_92_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_93_ord__eq__less__subst,axiom,
! [A2: num,F: int > num,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_94_ord__eq__less__subst,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_95_ord__eq__less__subst,axiom,
! [A2: int,F: num > int,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_96_ord__eq__less__subst,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_97_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_98_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_99_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_100_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_101_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_102_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_103_ord__less__eq__subst,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_104_ord__less__eq__subst,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_105_ord__less__eq__subst,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_106_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_107_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_108_order__less__irrefl,axiom,
! [X: num] :
~ ( ord_less_num @ X @ X ) ).
% order_less_irrefl
thf(fact_109_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_110_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_111_order__less__subst1,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_112_order__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_113_order__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_114_order__less__subst1,axiom,
! [A2: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_115_order__less__subst1,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_116_order__less__subst1,axiom,
! [A2: num,F: int > num,B: int,C: int] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_117_order__less__subst1,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_118_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_119_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_120_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_121_order__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_122_order__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_123_order__less__subst2,axiom,
! [A2: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_124_order__less__subst2,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_125_order__less__subst2,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_126_order__less__subst2,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_127_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_128_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_129_order__less__not__sym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_130_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_131_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_132_order__less__imp__triv,axiom,
! [X: num,Y: num,P: $o] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_133_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_134_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_135_linorder__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
| ( X = Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_136_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_137_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_138_order__less__imp__not__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_139_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_140_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_141_order__less__imp__not__eq2,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_142_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_143_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_144_order__less__imp__not__less,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_145_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_146_lessThan__strict__subset__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_147_lessThan__strict__subset__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_set_num @ ( set_ord_lessThan_num @ M2 ) @ ( set_ord_lessThan_num @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_148_lessThan__strict__subset__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_149_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_150_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_151_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_152_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_153_line__points__in__cube__unfolded,axiom,
! [L: nat > nat > nat,N: nat,T: nat,S: nat,J: nat] :
( ( hales_is_line @ L @ N @ T )
=> ( ( ord_less_nat @ S @ T )
=> ( ( ord_less_nat @ J @ N )
=> ( member_nat @ ( L @ S @ J ) @ ( set_ord_lessThan_nat @ T ) ) ) ) ) ).
% line_points_in_cube_unfolded
thf(fact_154_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_155_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_156_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_157_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_158_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_159_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_160_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_161_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_162_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_163_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_164__092_060open_062_123i_O_Ai_A_060_An_A_092_060and_062_A_I_092_060forall_062s_060t_O_AL_As_Ai_A_061_As_J_125_A_092_060noteq_062_A_123_125_092_060close_062,axiom,
( ( collect_nat
@ ^ [I2: nat] :
( ( ord_less_nat @ I2 @ n )
& ! [S2: nat] :
( ( ord_less_nat @ S2 @ t )
=> ( ( l @ S2 @ I2 )
= S2 ) ) ) )
!= bot_bot_set_nat ) ).
% \<open>{i. i < n \<and> (\<forall>s<t. L s i = s)} \<noteq> {}\<close>
thf(fact_165_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_166_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_167_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_168_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_169_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_170_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_171_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_172_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_173_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_174_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_175_B__def,axiom,
( b
= ( fun_upd_nat_set_nat
@ ( fun_upd_nat_set_nat
@ ^ [I2: nat] : bot_bot_set_nat
@ zero_zero_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( ord_less_nat @ I2 @ n )
& ! [S2: nat] :
( ( ord_less_nat @ S2 @ t )
=> ( ( l @ S2 @ I2 )
= S2 ) ) ) ) )
@ one_one_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( ord_less_nat @ I2 @ n )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ t )
=> ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ t )
=> ( ( l @ X2 @ I2 )
= ( l @ Y4 @ I2 ) ) ) ) ) ) ) ) ).
% B_def
thf(fact_176_psubsetD,axiom,
! [A: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_177_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_178_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_179_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_int @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_180_lessThan__def,axiom,
( set_ord_lessThan_num
= ( ^ [U: num] :
( collect_num
@ ^ [X2: num] : ( ord_less_num @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_181_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_182_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_183_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_184_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_185_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_186_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_187_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_188_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_189_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_190_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_191_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_192_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_193_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_194_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_195_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_196_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_197_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_198_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_199_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_200_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_201_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_202_L2,axiom,
( ( inf_inf_set_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( ord_less_nat @ I2 @ n )
& ! [S2: nat] :
( ( ord_less_nat @ S2 @ t )
=> ( ( l @ S2 @ I2 )
= S2 ) ) ) )
@ ( collect_nat
@ ^ [I2: nat] :
( ( ord_less_nat @ I2 @ n )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ t )
=> ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ t )
=> ( ( l @ X2 @ I2 )
= ( l @ Y4 @ I2 ) ) ) ) ) ) )
= bot_bot_set_nat ) ).
% L2
thf(fact_203_L1,axiom,
( ( sup_sup_set_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( ord_less_nat @ I2 @ n )
& ! [S2: nat] :
( ( ord_less_nat @ S2 @ t )
=> ( ( l @ S2 @ I2 )
= S2 ) ) ) )
@ ( collect_nat
@ ^ [I2: nat] :
( ( ord_less_nat @ I2 @ n )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ t )
=> ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ t )
=> ( ( l @ X2 @ I2 )
= ( l @ Y4 @ I2 ) ) ) ) ) ) )
= ( set_ord_lessThan_nat @ n ) ) ).
% L1
thf(fact_204_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_205_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_206_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_207_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A4: set_nat] : ( A4 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_208_dim0__subspace__ex,axiom,
! [T: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_is_subspace @ S3 @ zero_zero_nat @ N @ T ) ) ).
% dim0_subspace_ex
thf(fact_209_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_210_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_211_sgn__pos,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( sgn_sgn_int @ A2 )
= one_one_int ) ) ).
% sgn_pos
thf(fact_212_fun__upd__same,axiom,
! [F: nat > set_nat,X: nat,Y: set_nat] :
( ( fun_upd_nat_set_nat @ F @ X @ Y @ X )
= Y ) ).
% fun_upd_same
thf(fact_213_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_214_Int__iff,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_215_IntI,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) ) ) ) ).
% IntI
thf(fact_216_Un__iff,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B3 ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_217_UnCI,axiom,
! [C: nat,B3: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B3 )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B3 ) ) ) ).
% UnCI
thf(fact_218_sgn__sgn,axiom,
! [A2: int] :
( ( sgn_sgn_int @ ( sgn_sgn_int @ A2 ) )
= ( sgn_sgn_int @ A2 ) ) ).
% sgn_sgn
thf(fact_219_Un__empty,axiom,
! [A: set_nat,B3: set_nat] :
( ( ( sup_sup_set_nat @ A @ B3 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B3 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_220_sgn__0,axiom,
( ( sgn_sgn_int @ zero_zero_int )
= zero_zero_int ) ).
% sgn_0
thf(fact_221_Un__Int__eq_I1_J,axiom,
! [S4: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S4 @ T2 ) @ S4 )
= S4 ) ).
% Un_Int_eq(1)
thf(fact_222_Un__Int__eq_I2_J,axiom,
! [S4: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S4 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_223_Un__Int__eq_I3_J,axiom,
! [S4: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ S4 @ ( sup_sup_set_nat @ S4 @ T2 ) )
= S4 ) ).
% Un_Int_eq(3)
thf(fact_224_Un__Int__eq_I4_J,axiom,
! [T2: set_nat,S4: set_nat] :
( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S4 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_225_Int__Un__eq_I1_J,axiom,
! [S4: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S4 @ T2 ) @ S4 )
= S4 ) ).
% Int_Un_eq(1)
thf(fact_226_Int__Un__eq_I2_J,axiom,
! [S4: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S4 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_227_Int__Un__eq_I3_J,axiom,
! [S4: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ S4 @ ( inf_inf_set_nat @ S4 @ T2 ) )
= S4 ) ).
% Int_Un_eq(3)
thf(fact_228_Int__Un__eq_I4_J,axiom,
! [T2: set_nat,S4: set_nat] :
( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S4 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_229_sgn__1,axiom,
( ( sgn_sgn_int @ one_one_int )
= one_one_int ) ).
% sgn_1
thf(fact_230_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_231_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_232_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_233_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_234_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_235_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_236_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_237_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_238_sgn__less,axiom,
! [A2: int] :
( ( ord_less_int @ ( sgn_sgn_int @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% sgn_less
thf(fact_239_sgn__greater,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% sgn_greater
thf(fact_240_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_241_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_242_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_243_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_244_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_245_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_246_Int__left__commute,axiom,
! [A: set_nat,B3: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B3 @ C2 ) )
= ( inf_inf_set_nat @ B3 @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_247_Un__left__commute,axiom,
! [A: set_nat,B3: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B3 @ C2 ) )
= ( sup_sup_set_nat @ B3 @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_248_Un__Int__distrib2,axiom,
! [B3: set_nat,C2: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B3 @ C2 ) @ A )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B3 @ A ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_249_Int__left__absorb,axiom,
! [A: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B3 ) )
= ( inf_inf_set_nat @ A @ B3 ) ) ).
% Int_left_absorb
thf(fact_250_Int__Un__distrib2,axiom,
! [B3: set_nat,C2: set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B3 @ C2 ) @ A )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B3 @ A ) @ ( inf_inf_set_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_251_Un__left__absorb,axiom,
! [A: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B3 ) )
= ( sup_sup_set_nat @ A @ B3 ) ) ).
% Un_left_absorb
thf(fact_252_Un__Int__distrib,axiom,
! [A: set_nat,B3: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B3 @ C2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B3 ) @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_253_Int__Un__distrib,axiom,
! [A: set_nat,B3: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B3 @ C2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B3 ) @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_254_Un__Int__crazy,axiom,
! [A: set_nat,B3: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B3 ) @ ( inf_inf_set_nat @ B3 @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B3 ) @ ( sup_sup_set_nat @ B3 @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_255_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_256_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_257_Int__absorb,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_258_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_259_Int__assoc,axiom,
! [A: set_nat,B3: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B3 ) @ C2 )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_260_Un__assoc,axiom,
! [A: set_nat,B3: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B3 ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B3 @ C2 ) ) ) ).
% Un_assoc
thf(fact_261_ball__Un,axiom,
! [A: set_nat,B3: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B3 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_262_bex__Un,axiom,
! [A: set_nat,B3: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B3 ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: nat] :
( ( member_nat @ X2 @ B3 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_263_IntD2,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) )
=> ( member_nat @ C @ B3 ) ) ).
% IntD2
thf(fact_264_IntD1,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_265_UnI2,axiom,
! [C: nat,B3: set_nat,A: set_nat] :
( ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B3 ) ) ) ).
% UnI2
thf(fact_266_UnI1,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B3 ) ) ) ).
% UnI1
thf(fact_267_IntE,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B3 ) ) ) ).
% IntE
thf(fact_268_UnE,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B3 ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B3 ) ) ) ).
% UnE
thf(fact_269_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_270_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A4 )
| ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_271_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_272_Int__Collect,axiom,
! [X: nat,A: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_273_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_274_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_275_Un__empty__left,axiom,
! [B3: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_276_disjoint__iff__not__equal,axiom,
! [A: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A @ B3 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ B3 )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_277_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_278_Int__empty__left,axiom,
! [B3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B3 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_279_disjoint__iff,axiom,
! [A: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A @ B3 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_280_Int__emptyI,axiom,
! [A: set_nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B3 ) )
=> ( ( inf_inf_set_nat @ A @ B3 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_281_sgn__0__0,axiom,
! [A2: int] :
( ( ( sgn_sgn_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% sgn_0_0
thf(fact_282_sgn__eq__0__iff,axiom,
! [A2: int] :
( ( ( sgn_sgn_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% sgn_eq_0_iff
thf(fact_283_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_284_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_285_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_286_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_287_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_288_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_289_is__line__elim__t__1,axiom,
! [L: nat > nat > nat,N: nat,T: nat] :
( ( hales_is_line @ L @ N @ T )
=> ( ( T = one_one_nat )
=> ~ ! [B_0: set_nat,B_1: set_nat] :
~ ( ( ( sup_sup_set_nat @ B_0 @ B_1 )
= ( set_ord_lessThan_nat @ N ) )
& ( ( inf_inf_set_nat @ B_0 @ B_1 )
= bot_bot_set_nat )
& ( B_0 != bot_bot_set_nat )
& ! [X5: nat] :
( ( member_nat @ X5 @ B_1 )
=> ! [Xa: nat] :
( ( ord_less_nat @ Xa @ T )
=> ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ T )
=> ( ( L @ Xa @ X5 )
= ( L @ Y3 @ X5 ) ) ) ) )
& ! [X5: nat] :
( ( member_nat @ X5 @ B_0 )
=> ! [S5: nat] :
( ( ord_less_nat @ S5 @ T )
=> ( ( L @ S5 @ X5 )
= S5 ) ) ) ) ) ) ).
% is_line_elim_t_1
thf(fact_290_sgn__1__pos,axiom,
! [A2: int] :
( ( ( sgn_sgn_int @ A2 )
= one_one_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% sgn_1_pos
thf(fact_291_inf__sup__absorb,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_292_sup__inf__absorb,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_293_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_294_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B ) )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_295_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_296_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_297_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_298_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_299_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_300_inf__right__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_301_inf_Oright__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ B )
= ( inf_inf_set_nat @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_302_inf__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_303_inf_Oleft__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_304_inf__idem,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_305_inf_Oidem,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_306_sup_Oright__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ B )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_307_sup__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_308_sup_Oleft__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_309_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_310_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_311_fun__upd__apply,axiom,
( fun_upd_nat_set_nat
= ( ^ [F2: nat > set_nat,X2: nat,Y4: set_nat,Z2: nat] : ( if_set_nat @ ( Z2 = X2 ) @ Y4 @ ( F2 @ Z2 ) ) ) ) ).
% fun_upd_apply
thf(fact_312_fun__upd__triv,axiom,
! [F: nat > set_nat,X: nat] :
( ( fun_upd_nat_set_nat @ F @ X @ ( F @ X ) )
= F ) ).
% fun_upd_triv
thf(fact_313_fun__upd__upd,axiom,
! [F: nat > set_nat,X: nat,Y: set_nat,Z: set_nat] :
( ( fun_upd_nat_set_nat @ ( fun_upd_nat_set_nat @ F @ X @ Y ) @ X @ Z )
= ( fun_upd_nat_set_nat @ F @ X @ Z ) ) ).
% fun_upd_upd
thf(fact_314_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_315_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_316_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_317_inf__Int__eq,axiom,
! [R: set_nat,S4: set_nat] :
( ( inf_inf_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S4 ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( inf_inf_set_nat @ R @ S4 ) ) ) ) ).
% inf_Int_eq
thf(fact_318_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_319_sup__Un__eq,axiom,
! [R: set_nat,S4: set_nat] :
( ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S4 ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( sup_sup_set_nat @ R @ S4 ) ) ) ) ).
% sup_Un_eq
thf(fact_320_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_321_inf__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_322_inf_Oleft__commute,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A2 @ C ) )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_323_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X2 ) ) ) ).
% inf_commute
thf(fact_324_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( inf_inf_set_nat @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_325_inf__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_326_inf_Oassoc,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_327_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_328_inf__sup__aci_I2_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_329_inf__sup__aci_I3_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_330_inf__sup__aci_I4_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_331_sup__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_332_sup_Oleft__commute,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_333_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X2 ) ) ) ).
% sup_commute
thf(fact_334_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_335_sup__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_336_sup_Oassoc,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_337_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_338_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_339_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_340_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_341_fun__upd__idem__iff,axiom,
! [F: nat > set_nat,X: nat,Y: set_nat] :
( ( ( fun_upd_nat_set_nat @ F @ X @ Y )
= F )
= ( ( F @ X )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_342_fun__upd__twist,axiom,
! [A2: nat,C: nat,M2: nat > set_nat,B: set_nat,D: set_nat] :
( ( A2 != C )
=> ( ( fun_upd_nat_set_nat @ ( fun_upd_nat_set_nat @ M2 @ A2 @ B ) @ C @ D )
= ( fun_upd_nat_set_nat @ ( fun_upd_nat_set_nat @ M2 @ C @ D ) @ A2 @ B ) ) ) ).
% fun_upd_twist
thf(fact_343_fun__upd__other,axiom,
! [Z: nat,X: nat,F: nat > set_nat,Y: set_nat] :
( ( Z != X )
=> ( ( fun_upd_nat_set_nat @ F @ X @ Y @ Z )
= ( F @ Z ) ) ) ).
% fun_upd_other
thf(fact_344_fun__upd__idem,axiom,
! [F: nat > set_nat,X: nat,Y: set_nat] :
( ( ( F @ X )
= Y )
=> ( ( fun_upd_nat_set_nat @ F @ X @ Y )
= F ) ) ).
% fun_upd_idem
thf(fact_345_fun__upd__eqD,axiom,
! [F: nat > set_nat,X: nat,Y: set_nat,G: nat > set_nat,Z: set_nat] :
( ( ( fun_upd_nat_set_nat @ F @ X @ Y )
= ( fun_upd_nat_set_nat @ G @ X @ Z ) )
=> ( Y = Z ) ) ).
% fun_upd_eqD
thf(fact_346_fun__upd__def,axiom,
( fun_upd_nat_set_nat
= ( ^ [F2: nat > set_nat,A5: nat,B5: set_nat,X2: nat] : ( if_set_nat @ ( X2 = A5 ) @ B5 @ ( F2 @ X2 ) ) ) ) ).
% fun_upd_def
thf(fact_347_inf_Ostrict__coboundedI2,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_348_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_349_inf_Ostrict__coboundedI2,axiom,
! [B: int,C: int,A2: int] :
( ( ord_less_int @ B @ C )
=> ( ord_less_int @ ( inf_inf_int @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_350_inf_Ostrict__coboundedI1,axiom,
! [A2: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_351_inf_Ostrict__coboundedI1,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ A2 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_352_inf_Ostrict__coboundedI1,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ A2 @ C )
=> ( ord_less_int @ ( inf_inf_int @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_353_inf_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( A5
= ( inf_inf_set_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_354_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( A5
= ( inf_inf_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_355_inf_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( A5
= ( inf_inf_int @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_356_inf_Ostrict__boundedE,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_set_nat @ A2 @ B )
=> ~ ( ord_less_set_nat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_357_inf_Ostrict__boundedE,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_358_inf_Ostrict__boundedE,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ ( inf_inf_int @ B @ C ) )
=> ~ ( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_359_inf_Oabsorb4,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B )
= B ) ) ).
% inf.absorb4
thf(fact_360_inf_Oabsorb4,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb4
thf(fact_361_inf_Oabsorb4,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( inf_inf_int @ A2 @ B )
= B ) ) ).
% inf.absorb4
thf(fact_362_inf_Oabsorb3,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( inf_inf_set_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb3
thf(fact_363_inf_Oabsorb3,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb3
thf(fact_364_inf_Oabsorb3,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( inf_inf_int @ A2 @ B )
= A2 ) ) ).
% inf.absorb3
thf(fact_365_less__infI2,axiom,
! [B: set_nat,X: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ X )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ X ) ) ).
% less_infI2
thf(fact_366_less__infI2,axiom,
! [B: nat,X: nat,A2: nat] :
( ( ord_less_nat @ B @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).
% less_infI2
thf(fact_367_less__infI2,axiom,
! [B: int,X: int,A2: int] :
( ( ord_less_int @ B @ X )
=> ( ord_less_int @ ( inf_inf_int @ A2 @ B ) @ X ) ) ).
% less_infI2
thf(fact_368_less__infI1,axiom,
! [A2: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ X )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ X ) ) ).
% less_infI1
thf(fact_369_less__infI1,axiom,
! [A2: nat,X: nat,B: nat] :
( ( ord_less_nat @ A2 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).
% less_infI1
thf(fact_370_less__infI1,axiom,
! [A2: int,X: int,B: int] :
( ( ord_less_int @ A2 @ X )
=> ( ord_less_int @ ( inf_inf_int @ A2 @ B ) @ X ) ) ).
% less_infI1
thf(fact_371_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_372_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_373_sup_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A2: int] :
( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_374_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ C @ A2 )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_375_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_376_sup_Ostrict__coboundedI1,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ C @ A2 )
=> ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_377_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( A5
= ( sup_sup_set_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_378_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( A5
= ( sup_sup_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_379_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B5: int,A5: int] :
( ( A5
= ( sup_sup_int @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_380_sup_Ostrict__boundedE,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_set_nat @ B @ A2 )
=> ~ ( ord_less_set_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_381_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_382_sup_Ostrict__boundedE,axiom,
! [B: int,C: int,A2: int] :
( ( ord_less_int @ ( sup_sup_int @ B @ C ) @ A2 )
=> ~ ( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_383_sup_Oabsorb4,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_384_sup_Oabsorb4,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_385_sup_Oabsorb4,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( sup_sup_int @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_386_sup_Oabsorb3,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_387_sup_Oabsorb3,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_388_sup_Oabsorb3,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( sup_sup_int @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_389_less__supI2,axiom,
! [X: set_nat,B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ X @ B )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_390_less__supI2,axiom,
! [X: nat,B: nat,A2: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_391_less__supI2,axiom,
! [X: int,B: int,A2: int] :
( ( ord_less_int @ X @ B )
=> ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_392_less__supI1,axiom,
! [X: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ X @ A2 )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_393_less__supI1,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_nat @ X @ A2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_394_less__supI1,axiom,
! [X: int,A2: int,B: int] :
( ( ord_less_int @ X @ A2 )
=> ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_395_sup__inf__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_396_sup__inf__distrib1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_397_inf__sup__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_398_inf__sup__distrib1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_399_distrib__imp2,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ! [X3: set_nat,Y2: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z3 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ ( sup_sup_set_nat @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_400_distrib__imp1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ! [X3: set_nat,Y2: set_nat,Z3: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z3 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ ( inf_inf_set_nat @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_401_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_402_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_403_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_404_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_405_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_406_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_407_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_408_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_409_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_410_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_411_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_412_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B: set_nat] :
( ( A
= ( inf_inf_set_nat @ K @ A2 ) )
=> ( ( inf_inf_set_nat @ A @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_413_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_nat,K: set_nat,B: set_nat,A2: set_nat] :
( ( B3
= ( inf_inf_set_nat @ K @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B3 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_414_boolean__algebra__cancel_Osup1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B: set_nat] :
( ( A
= ( sup_sup_set_nat @ K @ A2 ) )
=> ( ( sup_sup_set_nat @ A @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_415_boolean__algebra__cancel_Osup2,axiom,
! [B3: set_nat,K: set_nat,B: set_nat,A2: set_nat] :
( ( B3
= ( sup_sup_set_nat @ K @ B ) )
=> ( ( sup_sup_set_nat @ A2 @ B3 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_416_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_417_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_less_as_int
thf(fact_418_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_419_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_420_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_421_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_422_sgn__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( sgn_sgn_int @ A2 )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% sgn_neg
thf(fact_423_add_Oinverse__inverse,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_424_neg__equal__iff__equal,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_425_neg__equal__zero,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= A2 )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_426_equal__neg__zero,axiom,
! [A2: int] :
( ( A2
= ( uminus_uminus_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_427_neg__equal__0__iff__equal,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_428_neg__0__equal__iff__equal,axiom,
! [A2: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A2 ) )
= ( zero_zero_int = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_429_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_430_neg__less__iff__less,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_431_Compl__disjoint2,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
= bot_bot_set_nat ) ).
% Compl_disjoint2
thf(fact_432_Compl__disjoint,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= bot_bot_set_nat ) ).
% Compl_disjoint
thf(fact_433_sgn__minus,axiom,
! [A2: int] :
( ( sgn_sgn_int @ ( uminus_uminus_int @ A2 ) )
= ( uminus_uminus_int @ ( sgn_sgn_int @ A2 ) ) ) ).
% sgn_minus
thf(fact_434_neg__less__0__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_435_neg__0__less__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_436_neg__less__pos,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_pos
thf(fact_437_less__neg__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_438_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_right
thf(fact_439_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_left
thf(fact_440_inf__compl__bot__right,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_right
thf(fact_441_inf__compl__bot__left2,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left2
thf(fact_442_inf__compl__bot__left1,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( inf_inf_set_nat @ X @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left1
thf(fact_443_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_nat,Y: set_nat] :
( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_444_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_nat,Y: set_nat] :
( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_445_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_446_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_447_verit__negate__coefficient_I2_J,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_448_equation__minus__iff,axiom,
! [A2: int,B: int] :
( ( A2
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_449_minus__equation__iff,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= B )
= ( ( uminus_uminus_int @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_450_minus__less__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_451_less__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% less_minus_iff
thf(fact_452_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_453_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_454_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_455_Collect__imp__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_456_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_457_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_458_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_459_inf__cancel__left2,axiom,
! [X: set_nat,A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A2 ) @ ( inf_inf_set_nat @ X @ B ) )
= bot_bot_set_nat ) ).
% inf_cancel_left2
thf(fact_460_inf__cancel__left1,axiom,
! [X: set_nat,A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ A2 ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B ) )
= bot_bot_set_nat ) ).
% inf_cancel_left1
thf(fact_461_sgn__not__eq__imp,axiom,
! [B: int,A2: int] :
( ( ( sgn_sgn_int @ B )
!= ( sgn_sgn_int @ A2 ) )
=> ( ( ( sgn_sgn_int @ A2 )
!= zero_zero_int )
=> ( ( ( sgn_sgn_int @ B )
!= zero_zero_int )
=> ( ( sgn_sgn_int @ A2 )
= ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_462_Compl__Un,axiom,
! [A: set_nat,B3: set_nat] :
( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A @ B3 ) )
= ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B3 ) ) ) ).
% Compl_Un
thf(fact_463_Compl__Int,axiom,
! [A: set_nat,B3: set_nat] :
( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A @ B3 ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B3 ) ) ) ).
% Compl_Int
thf(fact_464_sgn__minus__1,axiom,
( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% sgn_minus_1
thf(fact_465_zsgn__def,axiom,
( sgn_sgn_int
= ( ^ [I2: int] : ( if_int @ ( I2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zsgn_def
thf(fact_466_not__int__zless__negative,axiom,
! [N: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_467_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_468_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_469_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_470_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_471_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_472_verit__comp__simplify1_I1_J,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_473_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A5: nat,B5: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_474_int__if,axiom,
! [P: $o,A2: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_475_sgn__1__neg,axiom,
! [A2: int] :
( ( ( sgn_sgn_int @ A2 )
= ( uminus_uminus_int @ one_one_int ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% sgn_1_neg
thf(fact_476_sgn__if,axiom,
( sgn_sgn_int
= ( ^ [X2: int] : ( if_int @ ( X2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% sgn_if
thf(fact_477_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_478_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_479_sgn__of__nat,axiom,
! [N: nat] :
( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% sgn_of_nat
thf(fact_480_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_481_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_482_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_483_nat__of__bool,axiom,
! [P: $o] :
( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
= ( zero_n2687167440665602831ol_nat @ P ) ) ).
% nat_of_bool
thf(fact_484_ComplI,axiom,
! [C: nat,A: set_nat] :
( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_485_Compl__iff,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_486_of__bool__eq_I1_J,axiom,
( ( zero_n2684676970156552555ol_int @ $false )
= zero_zero_int ) ).
% of_bool_eq(1)
thf(fact_487_of__bool__eq_I1_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $false )
= zero_zero_nat ) ).
% of_bool_eq(1)
thf(fact_488_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= zero_zero_int )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_489_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= zero_zero_nat )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_490_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
= ( ~ P
& Q ) ) ).
% of_bool_less_iff
thf(fact_491_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
= ( ~ P
& Q ) ) ).
% of_bool_less_iff
thf(fact_492_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= one_one_int )
= P ) ).
% of_bool_eq_1_iff
thf(fact_493_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= one_one_nat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_494_of__bool__eq_I2_J,axiom,
( ( zero_n2684676970156552555ol_int @ $true )
= one_one_int ) ).
% of_bool_eq(2)
thf(fact_495_of__bool__eq_I2_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $true )
= one_one_nat ) ).
% of_bool_eq(2)
thf(fact_496_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2684676970156552555ol_int @ P ) ) ).
% of_nat_of_bool
thf(fact_497_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2687167440665602831ol_nat @ P ) ) ).
% of_nat_of_bool
thf(fact_498_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_499_of__int__of__bool,axiom,
! [P: $o] :
( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
= ( zero_n2684676970156552555ol_int @ P ) ) ).
% of_int_of_bool
thf(fact_500_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_501_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_502_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_503_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_504_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_505_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_506_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_507_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_508_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_509_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_510_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= one_one_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_511_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_minus
thf(fact_512_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_513_zless__nat__conj,axiom,
! [W: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W @ Z ) ) ) ).
% zless_nat_conj
thf(fact_514_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_515_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_516_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_517_Collect__neg__eq,axiom,
! [P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
~ ( P @ X2 ) )
= ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_518_Compl__eq,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A4: set_nat] :
( collect_nat
@ ^ [X2: nat] :
~ ( member_nat @ X2 @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_519_ComplD,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C @ A ) ) ).
% ComplD
thf(fact_520_uminus__set__def,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A4: set_nat] :
( collect_nat
@ ( uminus_uminus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_521_of__bool__eq__iff,axiom,
! [P4: $o,Q2: $o] :
( ( ( zero_n2684676970156552555ol_int @ P4 )
= ( zero_n2684676970156552555ol_int @ Q2 ) )
= ( P4 = Q2 ) ) ).
% of_bool_eq_iff
thf(fact_522_of__bool__eq__iff,axiom,
! [P4: $o,Q2: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P4 )
= ( zero_n2687167440665602831ol_nat @ Q2 ) )
= ( P4 = Q2 ) ) ).
% of_bool_eq_iff
thf(fact_523_split__of__bool__asm,axiom,
! [P: int > $o,P4: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ one_one_int ) )
| ( ~ P4
& ~ ( P @ zero_zero_int ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_524_split__of__bool__asm,axiom,
! [P: nat > $o,P4: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ one_one_nat ) )
| ( ~ P4
& ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_525_split__of__bool,axiom,
! [P: int > $o,P4: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
= ( ( P4
=> ( P @ one_one_int ) )
& ( ~ P4
=> ( P @ zero_zero_int ) ) ) ) ).
% split_of_bool
thf(fact_526_split__of__bool,axiom,
! [P: nat > $o,P4: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
= ( ( P4
=> ( P @ one_one_nat ) )
& ( ~ P4
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_of_bool
thf(fact_527_of__bool__def,axiom,
( zero_n2684676970156552555ol_int
= ( ^ [P5: $o] : ( if_int @ P5 @ one_one_int @ zero_zero_int ) ) ) ).
% of_bool_def
thf(fact_528_of__bool__def,axiom,
( zero_n2687167440665602831ol_nat
= ( ^ [P5: $o] : ( if_nat @ P5 @ one_one_nat @ zero_zero_nat ) ) ) ).
% of_bool_def
thf(fact_529_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_530_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_531_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K2: int] : ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K2 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_532_nat__mono__iff,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_mono_iff
thf(fact_533_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_534_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_pos
thf(fact_535_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_536_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N2: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N2 ) )
=> ( P @ N2 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_537_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_538_nat__less__iff,axiom,
! [W: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M2 )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_539_of__nat__nat,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= ( ring_1_of_int_int @ Z ) ) ) ).
% of_nat_nat
thf(fact_540_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_541_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_542_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_543_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_544_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_545_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_546_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_547_dual__order_Orefl,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_548_numeral__eq__iff,axiom,
! [M2: num,N: num] :
( ( ( numeral_numeral_int @ M2 )
= ( numeral_numeral_int @ N ) )
= ( M2 = N ) ) ).
% numeral_eq_iff
thf(fact_549_numeral__eq__iff,axiom,
! [M2: num,N: num] :
( ( ( numeral_numeral_nat @ M2 )
= ( numeral_numeral_nat @ N ) )
= ( M2 = N ) ) ).
% numeral_eq_iff
thf(fact_550_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_551_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_552_int__eq__iff__numeral,axiom,
! [M2: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( numeral_numeral_int @ V ) )
= ( M2
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_553_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% nat_numeral
thf(fact_554_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_555_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_556_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_557_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_558_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_559_neg__le__iff__le,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_560_inf_Obounded__iff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) )
= ( ( ord_less_eq_set_nat @ A2 @ B )
& ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_561_inf_Obounded__iff,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B @ C ) )
= ( ( ord_less_eq_int @ A2 @ B )
& ( ord_less_eq_int @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_562_inf_Obounded__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_563_le__inf__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_564_le__inf__iff,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) )
= ( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_565_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_566_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_567_sup_Obounded__iff,axiom,
! [B: int,C: int,A2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A2 )
= ( ( ord_less_eq_int @ B @ A2 )
& ( ord_less_eq_int @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_568_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_569_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_nat @ X @ Z )
& ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_570_le__sup__iff,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z )
= ( ( ord_less_eq_int @ X @ Z )
& ( ord_less_eq_int @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_571_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_572_neg__numeral__eq__iff,axiom,
! [M2: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M2 = N ) ) ).
% neg_numeral_eq_iff
thf(fact_573_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_574_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_575_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ N ) ) ).
% of_nat_numeral
thf(fact_576_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% of_nat_numeral
thf(fact_577_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_578_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_579_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_580_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_581_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_582_lessThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_583_lessThan__subset__iff,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
= ( ord_less_eq_num @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_584_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_585_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ( ring_1_of_int_int @ Z )
= ( numeral_numeral_int @ N ) )
= ( Z
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_586_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ K ) ) ).
% of_int_numeral
thf(fact_587_neg__less__eq__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_588_less__eq__neg__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_589_neg__le__0__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_590_neg__0__le__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_591_neg__numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M2 ) ) ).
% neg_numeral_le_iff
thf(fact_592_neg__numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M2 ) ) ).
% neg_numeral_less_iff
thf(fact_593_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_594_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_595_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_596_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= zero_zero_nat ) ).
% nat_neg_numeral
thf(fact_597_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_598_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_599_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_600_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_le_iff
thf(fact_601_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_602_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_less_iff
thf(fact_603_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_604_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_605_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_606_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_607_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_608_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_609_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_610_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_611_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_612_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_613_neg__numeral__le__numeral,axiom,
! [M2: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_614_not__numeral__le__neg__numeral,axiom,
! [M2: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_615_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_616_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_617_neg__numeral__le__one,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_618_neg__one__le__numeral,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).
% neg_one_le_numeral
thf(fact_619_neg__numeral__le__neg__one,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_620_not__numeral__le__neg__one,axiom,
! [M2: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_621_not__one__le__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% not_one_le_neg_numeral
thf(fact_622_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_623_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_624_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_625_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_626_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_627_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_628_nle__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_629_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_630_nle__le,axiom,
! [A2: num,B: num] :
( ( ~ ( ord_less_eq_num @ A2 @ B ) )
= ( ( ord_less_eq_num @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_631_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_632_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_633_le__cases3,axiom,
! [X: num,Y: num,Z: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z ) )
=> ( ( ( ord_less_eq_num @ X @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y ) )
=> ( ( ( ord_less_eq_num @ Z @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z )
=> ~ ( ord_less_eq_num @ Z @ X ) )
=> ~ ( ( ord_less_eq_num @ Z @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_634_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_635_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_636_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
= ( ^ [X2: num,Y4: num] :
( ( ord_less_eq_num @ X2 @ Y4 )
& ( ord_less_eq_num @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_637_ord__eq__le__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_638_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_639_ord__eq__le__trans,axiom,
! [A2: num,B: num,C: num] :
( ( A2 = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_640_ord__le__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_641_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_642_ord__le__eq__trans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_643_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_644_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_645_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_646_order_Otrans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_647_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_648_order_Otrans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% order.trans
thf(fact_649_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_650_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_651_order__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_eq_num @ X @ Z ) ) ) ).
% order_trans
thf(fact_652_linorder__wlog,axiom,
! [P: int > int > $o,A2: 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 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_653_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: 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 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_654_linorder__wlog,axiom,
! [P: num > num > $o,A2: num,B: num] :
( ! [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: num,B2: num] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_655_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_656_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_657_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
= ( ^ [A5: num,B5: num] :
( ( ord_less_eq_num @ B5 @ A5 )
& ( ord_less_eq_num @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_658_dual__order_Oantisym,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_659_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_660_dual__order_Oantisym,axiom,
! [B: num,A2: num] :
( ( ord_less_eq_num @ B @ A2 )
=> ( ( ord_less_eq_num @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_661_dual__order_Otrans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_662_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_663_dual__order_Otrans,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_eq_num @ B @ A2 )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_664_antisym,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_665_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_666_antisym,axiom,
! [A2: num,B: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_num @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_667_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_668_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_669_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
= ( ^ [A5: num,B5: num] :
( ( ord_less_eq_num @ A5 @ B5 )
& ( ord_less_eq_num @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_670_order__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_671_order__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_672_order__subst1,axiom,
! [A2: int,F: num > int,B: num,C: num] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_673_order__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_674_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_675_order__subst1,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_676_order__subst1,axiom,
! [A2: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_677_order__subst1,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_678_order__subst1,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_679_order__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_680_order__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_681_order__subst2,axiom,
! [A2: int,B: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_682_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_683_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_684_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_685_order__subst2,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_686_order__subst2,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_687_order__subst2,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_688_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_689_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_690_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_691_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_692_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_693_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_694_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_695_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_696_ord__eq__le__subst,axiom,
! [A2: num,F: int > num,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_697_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_698_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_699_ord__eq__le__subst,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_700_ord__eq__le__subst,axiom,
! [A2: int,F: num > int,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_701_ord__eq__le__subst,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_702_ord__eq__le__subst,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_703_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_704_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_705_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_706_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_707_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_708_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_709_ord__le__eq__subst,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_710_ord__le__eq__subst,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_711_ord__le__eq__subst,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_712_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_713_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_714_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_715_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_716_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_717_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_718_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_719_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_720_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_721_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_722_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_723_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_724_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_725_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_726_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_727_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_728_int__ops_I3_J,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% int_ops(3)
thf(fact_729_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_730_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_731_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_732_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_733_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_734_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_735_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_736_verit__comp__simplify1_I3_J,axiom,
! [B6: num,A6: num] :
( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
= ( ord_less_num @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_737_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_738_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_739_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_740_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_741_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_742_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_743_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_744_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_745_order__less__le__subst2,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_746_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_747_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_748_order__less__le__subst2,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_749_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_750_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_751_order__less__le__subst2,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_752_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_753_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_754_order__less__le__subst1,axiom,
! [A2: num,F: int > num,B: int,C: int] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_755_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_756_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_757_order__less__le__subst1,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_758_order__less__le__subst1,axiom,
! [A2: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_759_order__less__le__subst1,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_760_order__less__le__subst1,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_761_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_762_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_763_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_764_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_765_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_766_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_767_order__le__less__subst2,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_768_order__le__less__subst2,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_769_order__le__less__subst2,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_770_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_771_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_772_order__le__less__subst1,axiom,
! [A2: int,F: num > int,B: num,C: num] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_773_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_774_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_775_order__le__less__subst1,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_776_order__le__less__subst1,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_777_order__le__less__subst1,axiom,
! [A2: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_778_order__le__less__subst1,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_779_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_780_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_781_order__less__le__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_782_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_783_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_784_order__le__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_785_order__neq__le__trans,axiom,
! [A2: int,B: int] :
( ( A2 != B )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_786_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_787_order__neq__le__trans,axiom,
! [A2: num,B: num] :
( ( A2 != B )
=> ( ( ord_less_eq_num @ A2 @ B )
=> ( ord_less_num @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_788_order__le__neq__trans,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_789_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_790_order__le__neq__trans,axiom,
! [A2: num,B: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_num @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_791_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_792_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_793_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_794_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_795_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_796_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_797_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_798_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_799_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_800_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_801_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_802_order__less__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y4: num] :
( ( ord_less_eq_num @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_803_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_804_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_805_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X2: num,Y4: num] :
( ( ord_less_num @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_806_dual__order_Ostrict__implies__order,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_807_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_808_dual__order_Ostrict__implies__order,axiom,
! [B: num,A2: num] :
( ( ord_less_num @ B @ A2 )
=> ( ord_less_eq_num @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_809_order_Ostrict__implies__order,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_810_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_811_order_Ostrict__implies__order,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ( ord_less_eq_num @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_812_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_813_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_814_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B5: num,A5: num] :
( ( ord_less_eq_num @ B5 @ A5 )
& ~ ( ord_less_eq_num @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_815_dual__order_Ostrict__trans2,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_816_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_817_dual__order_Ostrict__trans2,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_num @ B @ A2 )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_818_dual__order_Ostrict__trans1,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_819_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_820_dual__order_Ostrict__trans1,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_eq_num @ B @ A2 )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_821_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_822_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_823_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B5: num,A5: num] :
( ( ord_less_eq_num @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_824_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_int @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_825_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_826_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B5: num,A5: num] :
( ( ord_less_num @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_827_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ~ ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_828_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_829_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A5: num,B5: num] :
( ( ord_less_eq_num @ A5 @ B5 )
& ~ ( ord_less_eq_num @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_830_order_Ostrict__trans2,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_831_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_832_order_Ostrict__trans2,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_833_order_Ostrict__trans1,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_834_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_835_order_Ostrict__trans1,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_836_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_837_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_838_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A5: num,B5: num] :
( ( ord_less_eq_num @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_839_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_840_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_841_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A5: num,B5: num] :
( ( ord_less_num @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_842_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_843_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_844_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_845_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_846_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_847_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y4: num] :
( ( ord_less_eq_num @ X2 @ Y4 )
& ~ ( ord_less_eq_num @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_848_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_849_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_850_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_851_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_852_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_853_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_854_nless__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_int @ A2 @ B ) )
= ( ~ ( ord_less_eq_int @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_855_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_856_nless__le,axiom,
! [A2: num,B: num] :
( ( ~ ( ord_less_num @ A2 @ B ) )
= ( ~ ( ord_less_eq_num @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_857_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_858_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_859_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_860_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_861_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_862_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_863_neg__numeral__neq__numeral,axiom,
! [M2: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_864_numeral__neq__neg__numeral,axiom,
! [M2: num,N: num] :
( ( numeral_numeral_int @ M2 )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_865_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_866_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_867_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_868_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_869_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_870_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_871_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_872_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_873_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_874_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_875_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_876_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_877_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_878_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_879_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I3 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_880_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_881_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_882_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_883_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M5: nat] :
( ( M2
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_884_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_885_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_886_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_887_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_888_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_889_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_890_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_891_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_892_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_893_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_894_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_895_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_896_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_897_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_898_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_899_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_900_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_901_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_902_le__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_minus_iff
thf(fact_903_minus__le__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_904_le__imp__neg__le,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_905_inf_OcoboundedI2,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_906_inf_OcoboundedI2,axiom,
! [B: int,C: int,A2: int] :
( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_907_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_908_inf_OcoboundedI1,axiom,
! [A2: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_909_inf_OcoboundedI1,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ A2 @ C )
=> ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_910_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_911_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( inf_inf_set_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_912_inf_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( ( inf_inf_int @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_913_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_914_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_915_inf_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( ( inf_inf_int @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_916_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_917_inf_Ocobounded2,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_918_inf_Ocobounded2,axiom,
! [A2: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_919_inf_Ocobounded2,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_920_inf_Ocobounded1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_921_inf_Ocobounded1,axiom,
! [A2: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_922_inf_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_923_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( A5
= ( inf_inf_set_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_924_inf_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( A5
= ( inf_inf_int @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_925_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( A5
= ( inf_inf_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_926_inf__greatest,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Z )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_927_inf__greatest,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Z )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_928_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_929_inf_OboundedI,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_930_inf_OboundedI,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ A2 @ C )
=> ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_931_inf_OboundedI,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_932_inf_OboundedE,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_933_inf_OboundedE,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B @ C ) )
=> ~ ( ( ord_less_eq_int @ A2 @ B )
=> ~ ( ord_less_eq_int @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_934_inf_OboundedE,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_935_inf__absorb2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( inf_inf_set_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_936_inf__absorb2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( inf_inf_int @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_937_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_938_inf__absorb1,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( inf_inf_set_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_939_inf__absorb1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( inf_inf_int @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_940_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_941_inf_Oabsorb2,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_942_inf_Oabsorb2,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( inf_inf_int @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_943_inf_Oabsorb2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_944_inf_Oabsorb1,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( inf_inf_set_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_945_inf_Oabsorb1,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( inf_inf_int @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_946_inf_Oabsorb1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_947_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_948_le__iff__inf,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y4: int] :
( ( inf_inf_int @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_949_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( inf_inf_nat @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_950_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Z3 )
=> ( ord_less_eq_set_nat @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_951_inf__unique,axiom,
! [F: int > int > int,X: int,Y: int] :
( ! [X3: int,Y2: int] : ( ord_less_eq_int @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: int,Y2: int] : ( ord_less_eq_int @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ord_less_eq_int @ X3 @ Z3 )
=> ( ord_less_eq_int @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_int @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_952_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_953_inf_OorderI,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_954_inf_OorderI,axiom,
! [A2: int,B: int] :
( ( A2
= ( inf_inf_int @ A2 @ B ) )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% inf.orderI
thf(fact_955_inf_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_956_inf_OorderE,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2
= ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_957_inf_OorderE,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( A2
= ( inf_inf_int @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_958_inf_OorderE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( A2
= ( inf_inf_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_959_le__infI2,axiom,
! [B: set_nat,X: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ X ) ) ).
% le_infI2
thf(fact_960_le__infI2,axiom,
! [B: int,X: int,A2: int] :
( ( ord_less_eq_int @ B @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B ) @ X ) ) ).
% le_infI2
thf(fact_961_le__infI2,axiom,
! [B: nat,X: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).
% le_infI2
thf(fact_962_le__infI1,axiom,
! [A2: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ X ) ) ).
% le_infI1
thf(fact_963_le__infI1,axiom,
! [A2: int,X: int,B: int] :
( ( ord_less_eq_int @ A2 @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B ) @ X ) ) ).
% le_infI1
thf(fact_964_le__infI1,axiom,
! [A2: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).
% le_infI1
thf(fact_965_inf__mono,axiom,
! [A2: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_966_inf__mono,axiom,
! [A2: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ A2 @ C )
=> ( ( ord_less_eq_int @ B @ D )
=> ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B ) @ ( inf_inf_int @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_967_inf__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_968_le__infI,axiom,
! [X: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ B )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_969_le__infI,axiom,
! [X: int,A2: int,B: int] :
( ( ord_less_eq_int @ X @ A2 )
=> ( ( ord_less_eq_int @ X @ B )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_970_le__infI,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_971_le__infE,axiom,
! [X: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_set_nat @ X @ A2 )
=> ~ ( ord_less_eq_set_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_972_le__infE,axiom,
! [X: int,A2: int,B: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ A2 @ B ) )
=> ~ ( ( ord_less_eq_int @ X @ A2 )
=> ~ ( ord_less_eq_int @ X @ B ) ) ) ).
% le_infE
thf(fact_973_le__infE,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_974_inf__le2,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_975_inf__le2,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_976_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_977_inf__le1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_978_inf__le1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_979_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_980_inf__sup__ord_I1_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_981_inf__sup__ord_I1_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_982_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_983_inf__sup__ord_I2_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_984_inf__sup__ord_I2_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_985_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_986_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_987_sup_OcoboundedI2,axiom,
! [C: int,B: int,A2: int] :
( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_988_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_989_sup_OcoboundedI1,axiom,
! [C: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_990_sup_OcoboundedI1,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ C @ A2 )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_991_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_992_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_993_sup_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( ( sup_sup_int @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_994_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_995_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_996_sup_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( ( sup_sup_int @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_997_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_998_sup_Ocobounded2,axiom,
! [B: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_999_sup_Ocobounded2,axiom,
! [B: int,A2: int] : ( ord_less_eq_int @ B @ ( sup_sup_int @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_1000_sup_Ocobounded2,axiom,
! [B: nat,A2: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_1001_sup_Ocobounded1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_1002_sup_Ocobounded1,axiom,
! [A2: int,B: int] : ( ord_less_eq_int @ A2 @ ( sup_sup_int @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_1003_sup_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_1004_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( A5
= ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_1005_sup_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( A5
= ( sup_sup_int @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_1006_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_1007_sup_OboundedI,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1008_sup_OboundedI,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ A2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1009_sup_OboundedI,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1010_sup_OboundedE,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B @ A2 )
=> ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1011_sup_OboundedE,axiom,
! [B: int,C: int,A2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_int @ B @ A2 )
=> ~ ( ord_less_eq_int @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1012_sup_OboundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1013_sup__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( sup_sup_set_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1014_sup__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( sup_sup_int @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1015_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1016_sup__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( sup_sup_set_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1017_sup__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( sup_sup_int @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1018_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1019_sup_Oabsorb2,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1020_sup_Oabsorb2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( sup_sup_int @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1021_sup_Oabsorb2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1022_sup_Oabsorb1,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_1023_sup_Oabsorb1,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( sup_sup_int @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_1024_sup_Oabsorb1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_1025_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X3 )
=> ( ord_less_eq_set_nat @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1026_sup__unique,axiom,
! [F: int > int > int,X: int,Y: int] :
( ! [X3: int,Y2: int] : ( ord_less_eq_int @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: int,Y2: int] : ( ord_less_eq_int @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ Y2 @ X3 )
=> ( ( ord_less_eq_int @ Z3 @ X3 )
=> ( ord_less_eq_int @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_int @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1027_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1028_sup_OorderI,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B ) )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_1029_sup_OorderI,axiom,
! [A2: int,B: int] :
( ( A2
= ( sup_sup_int @ A2 @ B ) )
=> ( ord_less_eq_int @ B @ A2 ) ) ).
% sup.orderI
thf(fact_1030_sup_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_1031_sup_OorderE,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_1032_sup_OorderE,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( A2
= ( sup_sup_int @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_1033_sup_OorderE,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_1034_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_1035_le__iff__sup,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y4: int] :
( ( sup_sup_int @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_1036_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( sup_sup_nat @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_1037_sup__least,axiom,
! [Y: set_nat,X: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ Z @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_1038_sup__least,axiom,
! [Y: int,X: int,Z: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ Z @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_1039_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_1040_sup__mono,axiom,
! [A2: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1041_sup__mono,axiom,
! [A2: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ A2 @ C )
=> ( ( ord_less_eq_int @ B @ D )
=> ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B ) @ ( sup_sup_int @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1042_sup__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1043_sup_Omono,axiom,
! [C: set_nat,A2: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1044_sup_Omono,axiom,
! [C: int,A2: int,D: int,B: int] :
( ( ord_less_eq_int @ C @ A2 )
=> ( ( ord_less_eq_int @ D @ B )
=> ( ord_less_eq_int @ ( sup_sup_int @ C @ D ) @ ( sup_sup_int @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1045_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1046_le__supI2,axiom,
! [X: set_nat,B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X @ B )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1047_le__supI2,axiom,
! [X: int,B: int,A2: int] :
( ( ord_less_eq_int @ X @ B )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1048_le__supI2,axiom,
! [X: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1049_le__supI1,axiom,
! [X: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1050_le__supI1,axiom,
! [X: int,A2: int,B: int] :
( ( ord_less_eq_int @ X @ A2 )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1051_le__supI1,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1052_sup__ge2,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_1053_sup__ge2,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge2
thf(fact_1054_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_1055_sup__ge1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_1056_sup__ge1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge1
thf(fact_1057_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_1058_le__supI,axiom,
! [A2: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X )
=> ( ( ord_less_eq_set_nat @ B @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1059_le__supI,axiom,
! [A2: int,X: int,B: int] :
( ( ord_less_eq_int @ A2 @ X )
=> ( ( ord_less_eq_int @ B @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1060_le__supI,axiom,
! [A2: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1061_le__supE,axiom,
! [A2: set_nat,B: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X )
=> ~ ( ord_less_eq_set_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_1062_le__supE,axiom,
! [A2: int,B: int,X: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_int @ A2 @ X )
=> ~ ( ord_less_eq_int @ B @ X ) ) ) ).
% le_supE
thf(fact_1063_le__supE,axiom,
! [A2: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_1064_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1065_inf__sup__ord_I3_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1066_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1067_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1068_inf__sup__ord_I4_J,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1069_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1070_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1071_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_nonneg
thf(fact_1072_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_1073_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1074_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_1075_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_1076_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_1077_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_1078_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_1079_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_1080_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_1081_not__numeral__less__neg__numeral,axiom,
! [M2: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_1082_neg__numeral__less__numeral,axiom,
! [M2: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_1083_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1084_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1085_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1086_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1087_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1088_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1089_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_1090_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_1091_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_1092_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_1093_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_1094_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_1095_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_1096_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_1097_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1098_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1099_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1100_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1101_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1102_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1103_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1104_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_1105_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_1106_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1107_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1108_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1109_distrib__sup__le,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1110_distrib__sup__le,axiom,
! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z ) ) @ ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1111_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1112_distrib__inf__le,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) @ ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1113_distrib__inf__le,axiom,
! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z ) ) @ ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1114_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1115_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_1116_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_1117_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_1118_int__cases,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_1119_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% of_bool_less_eq_one
thf(fact_1120_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% of_bool_less_eq_one
thf(fact_1121_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1122_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1123_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_1124_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_1125_eq__nat__nat__iff,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z5 ) )
= ( Z = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1126_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1127_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_1128_not__neg__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1129_not__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1130_not__numeral__less__neg__one,axiom,
! [M2: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_1131_neg__one__less__numeral,axiom,
! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).
% neg_one_less_numeral
thf(fact_1132_neg__numeral__less__one,axiom,
! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_1133_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_1134_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_1135_inf__shunt,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% inf_shunt
thf(fact_1136_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1137_shunt1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ Z ) ) ) ).
% shunt1
thf(fact_1138_shunt2,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) @ Z )
= ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% shunt2
thf(fact_1139_sup__neg__inf,axiom,
! [P4: set_nat,Q2: set_nat,R2: set_nat] :
( ( ord_less_eq_set_nat @ P4 @ ( sup_sup_set_nat @ Q2 @ R2 ) )
= ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ P4 @ ( uminus5710092332889474511et_nat @ Q2 ) ) @ R2 ) ) ).
% sup_neg_inf
thf(fact_1140_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1141_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1142_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1143_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1144_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_1145_int__eq__iff,axiom,
! [M2: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z )
= ( ( M2
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_1146_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1147_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1148_nat__eq__iff,axiom,
! [W: int,M2: nat] :
( ( ( nat2 @ W )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1149_nat__eq__iff2,axiom,
! [M2: nat,W: int] :
( ( M2
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1150_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_1151_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila1667268886620078168et_nat @ sup_sup_set_nat @ bot_bot_set_nat
@ ^ [X2: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ X2 )
@ ^ [X2: set_nat,Y4: set_nat] : ( ord_less_set_nat @ Y4 @ X2 ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_1152_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P6: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P6 ) ) ) ) ).
% imp_le_cong
thf(fact_1153_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P6: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P6 ) ) ) ) ).
% conj_le_cong
thf(fact_1154_complete__interval,axiom,
! [A2: int,B: int,P: int > $o] :
( ( ord_less_int @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C3: int] :
( ( ord_less_eq_int @ A2 @ C3 )
& ( ord_less_eq_int @ C3 @ B )
& ! [X5: int] :
( ( ( ord_less_eq_int @ A2 @ X5 )
& ( ord_less_int @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D2: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A2 @ X3 )
& ( ord_less_int @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1155_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A2 @ X5 )
& ( ord_less_nat @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D2: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1156_subsetI,axiom,
! [A: set_nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B3 ) )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% subsetI
thf(fact_1157_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_1158_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_1159_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1160_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1161_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M6: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M6 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1162_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_1163_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1164_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1165_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1166_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M4: nat] :
( M7
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1167_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1168_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_1169_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1170_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_1171_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_1172_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1173_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1174_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1175_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1176_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1177_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1178_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1179_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1180_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1181_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1182_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1183_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1184_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1185_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1186_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1187_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1188_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1189_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1190_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1191_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1192_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_1193_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1194_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1195_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1196_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1197_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1198_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1199_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_1200_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1201_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1202_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1203_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1204_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_1205_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1206_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1207_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X2: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X2 )
@ ^ [X2: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X2 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1208_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1209_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_1210_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1211_semiring__norm_I75_J,axiom,
! [M2: num] :
~ ( ord_less_num @ M2 @ one ) ).
% semiring_norm(75)
thf(fact_1212_semiring__norm_I78_J,axiom,
! [M2: num,N: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% semiring_norm(78)
thf(fact_1213_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_1214_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_1215_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_1216_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_1217_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_1218_nat__2,axiom,
( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% nat_2
thf(fact_1219_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_1220_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1221_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1222_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1223_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1224_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1225_semiring__norm_I80_J,axiom,
! [M2: num,N: num] :
( ( ord_less_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% semiring_norm(80)
thf(fact_1226_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_1227_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1228_semiring__norm_I81_J,axiom,
! [M2: num,N: num] :
( ( ord_less_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% semiring_norm(81)
thf(fact_1229_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1230_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_1231_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_1232_semiring__norm_I79_J,axiom,
! [M2: num,N: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% semiring_norm(79)
thf(fact_1233_semiring__norm_I74_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% semiring_norm(74)
thf(fact_1234_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X32: num] :
( Y
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_1235_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_1236_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1237_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1238_nat__int__add,axiom,
! [A2: nat,B: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) )
= ( plus_plus_nat @ A2 @ B ) ) ).
% nat_int_add
thf(fact_1239_zadd__int__left,axiom,
! [M2: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1240_int__ops_I5_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1241_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_1242_plus__int__code_I2_J,axiom,
! [L2: int] :
( ( plus_plus_int @ zero_zero_int @ L2 )
= L2 ) ).
% plus_int_code(2)
thf(fact_1243_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1244_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1245_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1246_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1247_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_1248_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1249_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z2: int] :
? [N2: nat] :
( Z2
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1250_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1251_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1252_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1253_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1254_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1255_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1256_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1257_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1258_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1259_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1260_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1261_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1262_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
? [K2: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1263_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% less_imp_Suc_add
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [X3: nat] :
( ( member_nat @ X3 @ ( set_ord_lessThan_nat @ one_one_nat ) )
=> ( ( b @ X3 )
!= bot_bot_set_nat ) ) ).
%------------------------------------------------------------------------------