TPTP Problem File: SLH0400^1.p
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%------------------------------------------------------------------------------
% 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 : Digit_Expansions/0000_Bits_Digits/prob_00613_024577__5610974_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1217 ( 454 unt; 55 typ; 0 def)
% Number of atoms : 3519 (1078 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10933 ( 310 ~; 78 |; 245 &;8625 @)
% ( 0 <=>;1675 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 664 ( 664 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 50 usr; 7 con; 0-4 aty)
% Number of variables : 3520 ( 283 ^;3170 !; 67 ?;3520 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:20:50.834
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (50)
thf(sy_c_Bits__Digits_Onth__digit,type,
bits_nth_digit: nat > nat > nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_M_Eo_J,type,
finite_finite_nat_o: set_nat_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite6739761609112101331et_nat: set_set_set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > 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_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $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__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
ord_less_set_nat_o2: set_nat_o > set_nat_o > $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__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le152980574450754630et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $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__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le6029213668185085951_nat_o: set_nat_o > set_nat_o > $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__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001_062_It__Nat__Onat_M_Eo_J,type,
set_or99350221437691188_nat_o: ( nat > $o ) > ( nat > $o ) > set_nat_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or9137876137106135879et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_member_001_062_It__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > set_nat_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_v_b,type,
b: nat ).
thf(sy_v_f,type,
f: nat > nat ).
thf(sy_v_qa____,type,
qa: nat ).
thf(sy_v_ta____,type,
ta: nat ).
% Relevant facts (1158)
thf(fact_0__092_060open_062t_A_060_ASuc_Aq_092_060close_062,axiom,
ord_less_nat @ ta @ ( suc @ qa ) ).
% \<open>t < Suc q\<close>
thf(fact_1_bound,axiom,
! [K: nat] : ( ord_less_nat @ ( f @ K ) @ b ) ).
% bound
thf(fact_2__092_060open_062t_A_092_060noteq_062_ASuc_Aq_092_060close_062,axiom,
( ta
!= ( suc @ qa ) ) ).
% \<open>t \<noteq> Suc q\<close>
thf(fact_3_True,axiom,
ord_less_eq_nat @ ta @ ( suc @ qa ) ).
% True
thf(fact_4_series__bound,axiom,
( ord_less_nat
@ ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( f @ K2 ) @ ( power_power_nat @ b @ K2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ qa ) )
@ ( power_power_nat @ b @ ( suc @ qa ) ) ) ).
% series_bound
thf(fact_5_Suc,axiom,
! [T: nat] :
( ( ( ord_less_eq_nat @ T @ qa )
=> ( ( bits_nth_digit
@ ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( f @ K2 ) @ ( power_power_nat @ b @ K2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ qa ) )
@ T
@ b )
= ( f @ T ) ) )
& ( ~ ( ord_less_eq_nat @ T @ qa )
=> ( ( bits_nth_digit
@ ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( f @ K2 ) @ ( power_power_nat @ b @ K2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ qa ) )
@ T
@ b )
= zero_zero_nat ) ) ) ).
% Suc
thf(fact_6_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_7_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_8_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_9_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_10_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_11_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_12_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_13_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A2 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_14_aux2__digit__gen__sum__repr,axiom,
! [N: nat,B: nat,C: nat,R: nat,A: nat] :
( ( ord_less_nat @ N @ ( power_power_nat @ B @ C ) )
=> ( ( ord_less_nat @ R @ C )
=> ( ( bits_nth_digit @ ( plus_plus_nat @ ( times_times_nat @ A @ ( power_power_nat @ B @ C ) ) @ N ) @ R @ B )
= ( bits_nth_digit @ N @ R @ B ) ) ) ) ).
% aux2_digit_gen_sum_repr
thf(fact_15_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_16_sum__shift__lb__Suc0__0,axiom,
! [F: nat > nat,K3: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K3 ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K3 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_17_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_18_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_19_atLeastatMost__subset__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat,D: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or9137876137106135879et_nat @ A @ B ) @ ( set_or9137876137106135879et_nat @ C @ D ) )
= ( ~ ( ord_le6893508408891458716et_nat @ A @ B )
| ( ( ord_le6893508408891458716et_nat @ C @ A )
& ( ord_le6893508408891458716et_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_20_atLeastatMost__subset__iff,axiom,
! [A: nat > $o,B: nat > $o,C: nat > $o,D: nat > $o] :
( ( ord_le6029213668185085951_nat_o @ ( set_or99350221437691188_nat_o @ A @ B ) @ ( set_or99350221437691188_nat_o @ C @ D ) )
= ( ~ ( ord_less_eq_nat_o @ A @ B )
| ( ( ord_less_eq_nat_o @ C @ A )
& ( ord_less_eq_nat_o @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_21_atLeastatMost__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_22_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_23_atLeastAtMost__iff,axiom,
! [I: set_set_nat,L: set_set_nat,U: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or9137876137106135879et_nat @ L @ U ) )
= ( ( ord_le6893508408891458716et_nat @ L @ I )
& ( ord_le6893508408891458716et_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_24_atLeastAtMost__iff,axiom,
! [I: nat > $o,L: nat > $o,U: nat > $o] :
( ( member_nat_o @ I @ ( set_or99350221437691188_nat_o @ L @ U ) )
= ( ( ord_less_eq_nat_o @ L @ I )
& ( ord_less_eq_nat_o @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_25_atLeastAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_26_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_27_Icc__eq__Icc,axiom,
! [L: set_set_nat,H: set_set_nat,L2: set_set_nat,H2: set_set_nat] :
( ( ( set_or9137876137106135879et_nat @ L @ H )
= ( set_or9137876137106135879et_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le6893508408891458716et_nat @ L @ H )
& ~ ( ord_le6893508408891458716et_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_28_Icc__eq__Icc,axiom,
! [L: nat > $o,H: nat > $o,L2: nat > $o,H2: nat > $o] :
( ( ( set_or99350221437691188_nat_o @ L @ H )
= ( set_or99350221437691188_nat_o @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat_o @ L @ H )
& ~ ( ord_less_eq_nat_o @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_29_Icc__eq__Icc,axiom,
! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L @ H )
= ( set_or4548717258645045905et_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_nat @ L @ H )
& ~ ( ord_less_eq_set_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_30_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_31_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_32_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_33_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_34_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_35_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_36_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_37_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_38_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_39_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_40_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_41_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_42_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_43_nat__add__left__cancel__less,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_44_mult__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ( times_times_nat @ M @ K3 )
= ( times_times_nat @ N @ K3 ) )
= ( ( M = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_45_mult__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K3 @ M )
= ( times_times_nat @ K3 @ N ) )
= ( ( M = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_46_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_47_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_48_nat__add__left__cancel__le,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_49_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_50_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_51_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_52_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_53_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_57_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_58_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_59_mult__less__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_60_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_61_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_62_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_63_mult__le__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_64_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_65_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_66_lift__Suc__mono__le,axiom,
! [F: nat > set_set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le6893508408891458716et_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_67_lift__Suc__mono__le,axiom,
! [F: nat > nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat_o @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat_o @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_68_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_69_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_70_lift__Suc__antimono__le,axiom,
! [F: nat > set_set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le6893508408891458716et_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_71_lift__Suc__antimono__le,axiom,
! [F: nat > nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat_o @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat_o @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_72_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_73_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_74_atLeastatMost__psubset__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat,D: set_set_nat] :
( ( ord_le152980574450754630et_nat @ ( set_or9137876137106135879et_nat @ A @ B ) @ ( set_or9137876137106135879et_nat @ C @ D ) )
= ( ( ~ ( ord_le6893508408891458716et_nat @ A @ B )
| ( ( ord_le6893508408891458716et_nat @ C @ A )
& ( ord_le6893508408891458716et_nat @ B @ D )
& ( ( ord_less_set_set_nat @ C @ A )
| ( ord_less_set_set_nat @ B @ D ) ) ) )
& ( ord_le6893508408891458716et_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_75_atLeastatMost__psubset__iff,axiom,
! [A: nat > $o,B: nat > $o,C: nat > $o,D: nat > $o] :
( ( ord_less_set_nat_o2 @ ( set_or99350221437691188_nat_o @ A @ B ) @ ( set_or99350221437691188_nat_o @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat_o @ A @ B )
| ( ( ord_less_eq_nat_o @ C @ A )
& ( ord_less_eq_nat_o @ B @ D )
& ( ( ord_less_nat_o @ C @ A )
| ( ord_less_nat_o @ B @ D ) ) ) )
& ( ord_less_eq_nat_o @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_76_atLeastatMost__psubset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D )
& ( ( ord_less_set_nat @ C @ A )
| ( ord_less_set_nat @ B @ D ) ) ) )
& ( ord_less_eq_set_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_77_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_78_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_79_le__trans,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K3 )
=> ( ord_less_eq_nat @ I @ K3 ) ) ) ).
% le_trans
thf(fact_80_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_81_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_82_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_83_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_84_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_85_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_86_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_87_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_88_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_89_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_90_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_91_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_92_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
| ( M2 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_93_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_94_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_95_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K3: nat,B: nat] :
( ( P @ K3 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_96_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_97_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_98_obtain__smallest,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ( P @ K4 )
& ! [A3: nat] :
( ( ord_less_nat @ A3 @ K4 )
=> ~ ( P @ A3 ) ) ) ) ).
% obtain_smallest
thf(fact_99_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_100_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_101_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_102_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_103_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_104_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_105_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_106_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_107_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_108_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_109_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K4 )
=> ~ ( P @ I3 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_110_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K3: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K3 ) @ ( F @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_111_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z: nat] :
( ( R2 @ X4 @ Y3 )
=> ( ( R2 @ Y3 @ Z )
=> ( R2 @ X4 @ Z ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_112_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_113_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_114_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_115_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_116_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_117_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M5: nat] :
( M6
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_118_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_119_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_120_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_121_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_122_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_123_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_124_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_125_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_126_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_127_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_128_add__le__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ K3 ) ) ) ).
% add_le_mono1
thf(fact_129_add__le__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_130_le__Suc__ex,axiom,
! [K3: nat,L: nat] :
( ( ord_less_eq_nat @ K3 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K3 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_131_add__leD2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N )
=> ( ord_less_eq_nat @ K3 @ N ) ) ).
% add_leD2
thf(fact_132_add__leD1,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_133_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_134_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_135_add__leE,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K3 @ N ) ) ) ).
% add_leE
thf(fact_136_mult__le__mono2,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J ) ) ) ).
% mult_le_mono2
thf(fact_137_mult__le__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ).
% mult_le_mono1
thf(fact_138_mult__le__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_139_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_140_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_141_sum__mono,axiom,
! [K5: set_set_nat,F: set_nat > nat,G: set_nat > nat] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups8294997508430121362at_nat @ F @ K5 ) @ ( groups8294997508430121362at_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_142_sum__mono,axiom,
! [K5: set_nat,F: nat > nat,G: nat > nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K5 ) @ ( groups3542108847815614940at_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_143_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_set_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_144_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_set_nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_set_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_set_set_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_145_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_146_lift__Suc__mono__less,axiom,
! [F: nat > set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_set_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_147_lift__Suc__mono__less,axiom,
! [F: nat > set_set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_set_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_set_set_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_148_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_149_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_150_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K4 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_151_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_152_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_153_sum__nonpos,axiom,
! [A2: set_set_nat,F: set_nat > nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_154_sum__nonpos,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_155_sum__nonneg,axiom,
! [A2: set_set_nat,F: set_nat > nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_156_sum__nonneg,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_157_Suc__mult__le__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K3 ) @ M ) @ ( times_times_nat @ ( suc @ K3 ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_158_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_159_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_160_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_161_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K4: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K4 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K4 )
=> ( P @ I2 @ K4 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_162_less__trans__Suc,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K3 )
=> ( ord_less_nat @ ( suc @ I ) @ K3 ) ) ) ).
% less_trans_Suc
thf(fact_163_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_164_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_165_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_166_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_167_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_168_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_169_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_170_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_171_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_172_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_173_Suc__lessE,axiom,
! [I: nat,K3: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K3 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K3
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_174_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_175_Nat_OlessE,axiom,
! [I: nat,K3: nat] :
( ( ord_less_nat @ I @ K3 )
=> ( ( K3
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K3
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_176_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_177_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_178_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_179_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_180_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_181_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_182_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_183_less__add__eq__less,axiom,
! [K3: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K3 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K3 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_184_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_185_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_186_add__less__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ K3 ) ) ) ).
% add_less_mono1
thf(fact_187_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_188_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_189_add__less__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K3 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_190_add__lessD1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K3 )
=> ( ord_less_nat @ I @ K3 ) ) ).
% add_lessD1
thf(fact_191_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_192_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_193_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_194_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_195_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_196_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_197_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_198_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_199_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_200_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ M )
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_201_sum_Oreindex__bij__witness,axiom,
! [S2: set_set_nat,I: nat > set_nat,J: set_nat > nat,T2: set_nat,H: nat > nat,G: set_nat > nat] :
( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ S2 )
=> ( member_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_set_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8294997508430121362at_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_202_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: set_nat > nat,J: nat > set_nat,T2: set_set_nat,H: set_nat > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( member_set_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ T2 )
=> ( member_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups8294997508430121362at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_203_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: nat > nat,J: nat > nat,T2: set_nat,H: nat > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( member_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_204_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K3: nat > set_nat,A2: set_set_nat,H: set_nat > nat,Gamma: nat > nat,Phi: set_nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ( ( member_set_nat @ ( K3 @ Y3 ) @ A2 )
& ( ( H @ ( K3 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ( member_nat @ ( H @ X4 ) @ B3 )
& ( ( K3 @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8294997508430121362at_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_205_sum_Oeq__general__inverses,axiom,
! [B3: set_set_nat,K3: set_nat > nat,A2: set_nat,H: nat > set_nat,Gamma: set_nat > nat,Phi: nat > nat] :
( ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ B3 )
=> ( ( member_nat @ ( K3 @ Y3 ) @ A2 )
& ( ( H @ ( K3 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_set_nat @ ( H @ X4 ) @ B3 )
& ( ( K3 @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups8294997508430121362at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_206_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K3: nat > nat,A2: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ( ( member_nat @ ( K3 @ Y3 ) @ A2 )
& ( ( H @ ( K3 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ ( H @ X4 ) @ B3 )
& ( ( K3 @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_207_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_set_nat,H: set_nat > nat,Gamma: nat > nat,Phi: set_nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
& ( ( H @ X5 )
= Y3 )
& ! [Ya: set_nat] :
( ( ( member_set_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ( member_nat @ ( H @ X4 ) @ B3 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8294997508430121362at_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_208_sum_Oeq__general,axiom,
! [B3: set_set_nat,A2: set_nat,H: nat > set_nat,Gamma: set_nat > nat,Phi: nat > nat] :
( ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ B3 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ( H @ X5 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_set_nat @ ( H @ X4 ) @ B3 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups8294997508430121362at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_209_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ( H @ X5 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ ( H @ X4 ) @ B3 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_210_sum_Ocong,axiom,
! [A2: set_nat,B3: set_nat,G: nat > nat,H: nat > nat] :
( ( A2 = B3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= ( groups3542108847815614940at_nat @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_211_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_212_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_213_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_214_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_215_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_216_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_217_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_218_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
? [K2: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_219_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_220_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_221_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_222_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_223_Suc__mult__less__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K3 ) @ M ) @ ( times_times_nat @ ( suc @ K3 ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_224_mult__less__mono2,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_225_mult__less__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ) ).
% mult_less_mono1
thf(fact_226_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_227_sum_Oswap,axiom,
! [G: nat > nat > nat,B3: set_nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( G @ I4 ) @ B3 )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ I4 @ J3 )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_228_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_229_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_230_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_231_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_232_power__gt__expt,axiom,
! [N: nat,K3: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K3 @ ( power_power_nat @ N @ K3 ) ) ) ).
% power_gt_expt
thf(fact_233_sum__SucD,axiom,
! [F: nat > nat,A2: set_nat,N: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( suc @ N ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).
% sum_SucD
thf(fact_234_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_235_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_nat > nat,A2: set_set_nat] :
( ( ( groups8294997508430121362at_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A4: set_nat] :
( ( member_set_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_236_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A2: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_237_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_238_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_239_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_240_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_241_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_242_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_243_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_244_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_245_zero__induct,axiom,
! [P: nat > $o,K3: nat] :
( ( P @ K3 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_246_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_247_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_248_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_249_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_250_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_251_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_252_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_253_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_254_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_255_nat__arith_Osuc1,axiom,
! [A2: nat,K3: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K3 @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K3 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_256_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_257_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_258_Suc__mult__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K3 ) @ M )
= ( times_times_nat @ ( suc @ K3 ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_259_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_260_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_261_add__mult__distrib2,axiom,
! [K3: nat,M: nat,N: nat] :
( ( times_times_nat @ K3 @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_262_add__mult__distrib,axiom,
! [M: nat,N: nat,K3: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K3 )
= ( plus_plus_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) ) ) ).
% add_mult_distrib
thf(fact_263_sum__product,axiom,
! [F: nat > nat,A2: set_nat,G: nat > nat,B3: set_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] :
( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( F @ I4 ) @ ( G @ J3 ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_264_sum__distrib__right,axiom,
! [F: nat > nat,A2: set_nat,R: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R )
= ( groups3542108847815614940at_nat
@ ^ [N4: nat] : ( times_times_nat @ ( F @ N4 ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_265_sum__distrib__left,axiom,
! [R: nat,F: nat > nat,A2: set_nat] :
( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [N4: nat] : ( times_times_nat @ R @ ( F @ N4 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_266_sum_Odistrib,axiom,
! [G: nat > nat,H: nat > nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( plus_plus_nat @ ( G @ X3 ) @ ( H @ X3 ) )
@ A2 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_267_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_268_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_269_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_270_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_271_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_272_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_273_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ ( suc @ X4 ) @ A2 )
=> ( ( F @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_274_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_275_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M: nat,K3: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K3 ) @ ( plus_plus_nat @ N @ K3 ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_276_nat__mult__le__cancel__disj,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_277_nat__mult__less__cancel__disj,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_278_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_279_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_280_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_281_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_282_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_283_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_284_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_285_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_286_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_287_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_288_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_289_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_290_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_291_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_292_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_293_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_294_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_295_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_296_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_297_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_298_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_299_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_300_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_301_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_302_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_303_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_304_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_305_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_306_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_307_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_308_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( I = J )
& ( K3 = L ) )
=> ( ( plus_plus_nat @ I @ K3 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_309_group__cancel_Oadd1,axiom,
! [A2: nat,K3: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K3 @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_310_group__cancel_Oadd2,axiom,
! [B3: nat,K3: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K3 @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_311_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_312_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_313_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_314_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_315_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_316_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_317_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_318_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_319_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_320_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_321_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K3 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_322_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_323_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_324_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_325_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_326_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_327_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_328_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_329_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_330_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_331_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_332_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_333_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_334_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_335_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K3 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_336_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_337_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_338_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_339_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_340_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_341_nat__mult__eq__cancel__disj,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K3 @ M )
= ( times_times_nat @ K3 @ N ) )
= ( ( K3 = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_342_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K3: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K3 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K3 ) ) ).
% left_add_mult_distrib
thf(fact_343_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_344_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_345_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_346_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_347_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_348_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_349_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_350_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_351_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_352_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_353_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_354_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_355_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_356_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_357_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_358_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_359_nat__mult__eq__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ( times_times_nat @ K3 @ M )
= ( times_times_nat @ K3 @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_360_nat__mult__less__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_361_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_362_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_363_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_364_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_365_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_366_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_367_nat__mult__le__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_368_psubsetI,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_set_nat @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_369_psubsetI,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_nat @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_370_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_371_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_372_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_373_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_374_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_375_subset__antisym,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_376_subset__antisym,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_377_subsetI,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( member_set_nat @ X4 @ B3 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_378_subsetI,axiom,
! [A2: set_nat,B3: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B3 ) )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_379_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_380_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_381_set__eq__subset,axiom,
( ( ^ [Y5: set_set_nat,Z2: set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [A6: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A6 @ B5 )
& ( ord_le6893508408891458716et_nat @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_382_set__eq__subset,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A6: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_383_subset__trans,axiom,
! [A2: set_set_nat,B3: set_set_nat,C4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ C4 )
=> ( ord_le6893508408891458716et_nat @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_384_subset__trans,axiom,
! [A2: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C4 )
=> ( ord_less_eq_set_nat @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_385_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_386_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_387_subset__refl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_388_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_389_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B5: set_set_nat] :
! [T3: set_nat] :
( ( member_set_nat @ T3 @ A6 )
=> ( member_set_nat @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_390_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A6 )
=> ( member_nat @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_391_equalityD2,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( A2 = B3 )
=> ( ord_le6893508408891458716et_nat @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_392_equalityD2,axiom,
! [A2: set_nat,B3: set_nat] :
( ( A2 = B3 )
=> ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_393_equalityD1,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( A2 = B3 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_394_equalityD1,axiom,
! [A2: set_nat,B3: set_nat] :
( ( A2 = B3 )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_395_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B5: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A6 )
=> ( member_set_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_396_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_397_equalityE,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( A2 = B3 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ~ ( ord_le6893508408891458716et_nat @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_398_equalityE,axiom,
! [A2: set_nat,B3: set_nat] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ~ ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_399_subsetD,axiom,
! [A2: set_set_nat,B3: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_400_subsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_401_in__mono,axiom,
! [A2: set_set_nat,B3: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_402_in__mono,axiom,
! [A2: set_nat,B3: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_403_psubset__trans,axiom,
! [A2: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C4 )
=> ( ord_less_set_nat @ A2 @ C4 ) ) ) ).
% psubset_trans
thf(fact_404_psubset__trans,axiom,
! [A2: set_set_nat,B3: set_set_nat,C4: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B3 )
=> ( ( ord_less_set_set_nat @ B3 @ C4 )
=> ( ord_less_set_set_nat @ A2 @ C4 ) ) ) ).
% psubset_trans
thf(fact_405_psubsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_406_psubsetD,axiom,
! [A2: set_set_nat,B3: set_set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A2 @ B3 )
=> ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_407_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B5: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_408_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_409_Collect__subset,axiom,
! [A2: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_410_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_411_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
( ord_less_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_412_less__set__def,axiom,
( ord_less_set_set_nat
= ( ^ [A6: set_set_nat,B5: set_set_nat] :
( ord_less_set_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_413_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_414_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_415_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_416_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_417_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_418_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_419_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_420_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_421_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_422_subset__iff__psubset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B5: set_set_nat] :
( ( ord_less_set_set_nat @ A6 @ B5 )
| ( A6 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_423_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A6 @ B5 )
| ( A6 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_424_subset__psubset__trans,axiom,
! [A2: set_set_nat,B3: set_set_nat,C4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( ord_less_set_set_nat @ B3 @ C4 )
=> ( ord_less_set_set_nat @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_425_subset__psubset__trans,axiom,
! [A2: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C4 )
=> ( ord_less_set_nat @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_426_subset__not__subset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A6: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A6 @ B5 )
& ~ ( ord_le6893508408891458716et_nat @ B5 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_427_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_428_psubset__subset__trans,axiom,
! [A2: set_set_nat,B3: set_set_nat,C4: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ C4 )
=> ( ord_less_set_set_nat @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_429_psubset__subset__trans,axiom,
! [A2: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C4 )
=> ( ord_less_set_nat @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_430_psubset__imp__subset,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B3 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_431_psubset__imp__subset,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_432_psubset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A6: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A6 @ B5 )
& ( A6 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_433_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B5 )
& ( A6 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_434_psubsetE,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B3 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ord_le6893508408891458716et_nat @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_435_psubsetE,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_436_lambda__zero,axiom,
( ( ^ [H3: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_437_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_438_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_439_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_440_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_441_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_442_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_443_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_444_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_445_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_446_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_447_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_448_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_449_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_450_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_451_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_452_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_453_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_454_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_455_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_456_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_457_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_458_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_459_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_460_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_461_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_462_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_463_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_464__092_060open_062_092_060lbrakk_062_I_092_060Sum_062k_A_061_A0_O_Oq_O_Af_Ak_A_K_Ab_A_094_Ak_J_A_060_Ab_A_K_Ab_A_094_Aq_059_A_092_060And_062t_O_A_I_092_060Sum_062k_A_061_A0_O_Oq_O_Af_Ak_A_K_Ab_A_094_Ak_J_Adiv_Ab_A_094_At_Amod_Ab_A_061_A_Iif_At_A_092_060le_062_Aq_Athen_Af_At_Aelse_A0_J_059_At_A_061_ASuc_Aq_059_A_092_060And_062num_Ak_Abase_O_Anth__digit_Anum_Ak_Abase_A_061_Anum_Adiv_Abase_A_094_Ak_Amod_Abase_092_060rbrakk_062_A_092_060Longrightarrow_062_A_I_I_092_060Sum_062k_A_061_A0_O_Oq_O_Af_Ak_A_K_Ab_A_094_Ak_J_A_L_Af_A_ISuc_Aq_J_A_K_A_Ib_A_K_Ab_A_094_Aq_J_J_Adiv_A_Ib_A_K_Ab_A_094_Aq_J_Amod_Ab_A_061_Af_A_ISuc_Aq_J_092_060close_062,axiom,
( ( ord_less_nat
@ ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( f @ K2 ) @ ( power_power_nat @ b @ K2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ qa ) )
@ ( times_times_nat @ b @ ( power_power_nat @ b @ qa ) ) )
=> ( ! [T4: nat] :
( ( ( ord_less_eq_nat @ T4 @ qa )
=> ( ( modulo_modulo_nat
@ ( divide_divide_nat
@ ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( f @ K2 ) @ ( power_power_nat @ b @ K2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ qa ) )
@ ( power_power_nat @ b @ T4 ) )
@ b )
= ( f @ T4 ) ) )
& ( ~ ( ord_less_eq_nat @ T4 @ qa )
=> ( ( modulo_modulo_nat
@ ( divide_divide_nat
@ ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( f @ K2 ) @ ( power_power_nat @ b @ K2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ qa ) )
@ ( power_power_nat @ b @ T4 ) )
@ b )
= zero_zero_nat ) ) )
=> ( ( ta
= ( suc @ qa ) )
=> ( ! [Num: nat,K4: nat,Base: nat] :
( ( bits_nth_digit @ Num @ K4 @ Base )
= ( modulo_modulo_nat @ ( divide_divide_nat @ Num @ ( power_power_nat @ Base @ K4 ) ) @ Base ) )
=> ( ( modulo_modulo_nat
@ ( divide_divide_nat
@ ( plus_plus_nat
@ ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( f @ K2 ) @ ( power_power_nat @ b @ K2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ qa ) )
@ ( times_times_nat @ ( f @ ( suc @ qa ) ) @ ( times_times_nat @ b @ ( power_power_nat @ b @ qa ) ) ) )
@ ( times_times_nat @ b @ ( power_power_nat @ b @ qa ) ) )
@ b )
= ( f @ ( suc @ qa ) ) ) ) ) ) ) ).
% \<open>\<lbrakk>(\<Sum>k = 0..q. f k * b ^ k) < b * b ^ q; \<And>t. (\<Sum>k = 0..q. f k * b ^ k) div b ^ t mod b = (if t \<le> q then f t else 0); t = Suc q; \<And>num k base. nth_digit num k base = num div base ^ k mod base\<rbrakk> \<Longrightarrow> ((\<Sum>k = 0..q. f k * b ^ k) + f (Suc q) * (b * b ^ q)) div (b * b ^ q) mod b = f (Suc q)\<close>
thf(fact_465_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_466_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_467_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_468_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_469_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_470_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_471_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_472_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_473_cancel__div__mod__rules_I2_J,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
= ( plus_plus_nat @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_474_cancel__div__mod__rules_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
= ( plus_plus_nat @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_475_mod__div__decomp,axiom,
! [A: nat,B: nat] :
( A
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_476_div__mult__mod__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_477_mod__div__mult__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_478_mod__mult__div__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_479_mult__div__mod__eq,axiom,
! [B: nat,A: nat] :
( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_480_nth__digit__def,axiom,
( bits_nth_digit
= ( ^ [Num2: nat,K2: nat,Base2: nat] : ( modulo_modulo_nat @ ( divide_divide_nat @ Num2 @ ( power_power_nat @ Base2 @ K2 ) ) @ Base2 ) ) ) ).
% nth_digit_def
thf(fact_481_nat__mult__div__cancel__disj,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ( K3 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= zero_zero_nat ) )
& ( ( K3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_482_nat__mult__div__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_483_aux2__digit__wise__gen__equiv,axiom,
! [K3: nat,A: nat,B: nat] :
( ! [R3: nat] :
( ( ord_less_nat @ R3 @ K3 )
=> ( ( bits_nth_digit @ A @ R3 @ B )
= zero_zero_nat ) )
=> ( ( modulo_modulo_nat @ A @ ( power_power_nat @ B @ K3 ) )
= zero_zero_nat ) ) ).
% aux2_digit_wise_gen_equiv
thf(fact_484_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_485_Suc__mod__mult__self1,axiom,
! [M: nat,K3: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K3 @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_486_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K3: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K3 ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_487_Suc__mod__mult__self3,axiom,
! [K3: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K3 @ N ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_488_Suc__mod__mult__self4,axiom,
! [N: nat,K3: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K3 ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_489_mod__mod__trivial,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_490_mod__add__self1,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self1
thf(fact_491_mod__add__self2,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self2
thf(fact_492_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_493_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_494_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_495_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_496_mod__mult__self2__is__0,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_497_mod__mult__self1__is__0,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_498_mod__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self4
thf(fact_499_mod__mult__self3,axiom,
! [C: nat,B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self3
thf(fact_500_mod__mult__self2,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self2
thf(fact_501_mod__mult__self1,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self1
thf(fact_502_mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_div_trivial
thf(fact_503_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_504_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_505_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_506_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_507_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_508_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_509_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_510_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_511_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_512_mod__mult__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_513_mod__mult__cong,axiom,
! [A: nat,C: nat,A7: nat,B: nat,B6: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A7 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B6 @ C ) )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A7 @ B6 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_514_mod__mult__mult2,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_515_mult__mod__right,axiom,
! [C: nat,A: nat,B: nat] :
( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
= ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_516_mod__mult__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_517_mod__mult__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_518_mod__add__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_519_mod__add__cong,axiom,
! [A: nat,C: nat,A7: nat,B: nat,B6: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A7 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B6 @ C ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A7 @ B6 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_520_mod__add__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_521_mod__add__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_522_power__mod,axiom,
! [A: nat,B: nat,N: nat] :
( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
= ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% power_mod
thf(fact_523_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_524_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_525_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% mod_less_eq_dividend
thf(fact_526_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_527_div__le__mono,axiom,
! [M: nat,N: nat,K3: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K3 ) @ ( divide_divide_nat @ N @ K3 ) ) ) ).
% div_le_mono
thf(fact_528_div__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q3 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) ).
% div_mult2_eq
thf(fact_529_mod__sum__eq,axiom,
! [F: nat > nat,A: nat,A2: set_nat] :
( ( modulo_modulo_nat
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).
% mod_sum_eq
thf(fact_530_mod__eq__self__iff__div__eq__0,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= A )
= ( ( divide_divide_nat @ A @ B )
= zero_zero_nat ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_531_div__add1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_532_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_533_mod__induct,axiom,
! [P: nat > $o,N: nat,P2: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P2 )
=> ( ( ord_less_nat @ M @ P2 )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ P2 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P2 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_534_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_535_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_536_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_537_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_538_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_539_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_540_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_541_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_542_div__mult1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_543_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_544_div__le__mono2,axiom,
! [M: nat,N: nat,K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K3 @ N ) @ ( divide_divide_nat @ K3 @ M ) ) ) ) ).
% div_le_mono2
thf(fact_545_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_546_div__less__iff__less__mult,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_547_mod__eq__nat2E,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q3 )
= ( modulo_modulo_nat @ N @ Q3 ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ~ ! [S3: nat] :
( N
!= ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_548_mod__eq__nat1E,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q3 )
= ( modulo_modulo_nat @ N @ Q3 ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ~ ! [S3: nat] :
( M
!= ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_549_div__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( suc @ ( divide_divide_nat @ M @ N ) ) ) )
& ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
!= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% div_Suc
thf(fact_550_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q3 ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_551_split__mod,axiom,
! [Q: nat > $o,M: nat,N: nat] :
( ( Q @ ( modulo_modulo_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( Q @ M ) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) ) )
=> ( Q @ J3 ) ) ) ) ) ).
% split_mod
thf(fact_552_div__nat__eqI,axiom,
! [N: nat,Q3: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_553_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_554_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_555_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_556_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_div
thf(fact_557_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_558_bits__mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% bits_mod_div_trivial
thf(fact_559_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_560_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_561_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_562_div__mod__decomp,axiom,
! [A2: nat,N: nat] :
( A2
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% div_mod_decomp
thf(fact_563_div__less__mono,axiom,
! [A2: nat,B3: nat,N: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A2 @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B3 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B3 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_564_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
=> ( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo_nat @ M5 @ N3 ) )
=> ( P @ M5 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_565_verit__comp__simplify1_I2_J,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_566_verit__comp__simplify1_I2_J,axiom,
! [A: nat > $o] : ( ord_less_eq_nat_o @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_567_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_568_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_569_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_570_verit__comp__simplify1_I1_J,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_571_verit__comp__simplify1_I1_J,axiom,
! [A: set_set_nat] :
~ ( ord_less_set_set_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_572_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_573_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A7 ) )
= ( ord_less_nat @ A7 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_574_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_575_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B2: nat] :
( ( P @ A4 @ B2 )
= ( P @ B2 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B2: nat] :
( ( P @ A4 @ B2 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B2 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_576_predicate1I,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_nat_o @ P @ Q ) ) ).
% predicate1I
thf(fact_577_less__fun__def,axiom,
( ord_less_nat_o
= ( ^ [F2: nat > $o,G2: nat > $o] :
( ( ord_less_eq_nat_o @ F2 @ G2 )
& ~ ( ord_less_eq_nat_o @ G2 @ F2 ) ) ) ) ).
% less_fun_def
thf(fact_578_dual__order_Orefl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% dual_order.refl
thf(fact_579_dual__order_Orefl,axiom,
! [A: nat > $o] : ( ord_less_eq_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_580_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_581_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_582_order__refl,axiom,
! [X: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ X ) ).
% order_refl
thf(fact_583_order__refl,axiom,
! [X: nat > $o] : ( ord_less_eq_nat_o @ X @ X ) ).
% order_refl
thf(fact_584_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_585_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_586_order__antisym__conv,axiom,
! [Y: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_587_order__antisym__conv,axiom,
! [Y: nat > $o,X: nat > $o] :
( ( ord_less_eq_nat_o @ Y @ X )
=> ( ( ord_less_eq_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_588_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_589_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_590_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_591_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_592_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_593_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_594_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_595_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_596_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_597_ord__le__eq__subst,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_598_ord__le__eq__subst,axiom,
! [A: nat > $o,B: nat > $o,F: ( nat > $o ) > nat,C: nat] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_599_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_600_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_601_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_602_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_603_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_604_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_605_ord__eq__le__subst,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_606_ord__eq__le__subst,axiom,
! [A: nat > $o,F: nat > nat > $o,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_607_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_608_ord__eq__le__subst,axiom,
! [A: nat,F: ( nat > $o ) > nat,B: nat > $o,C: nat > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat_o @ B @ C )
=> ( ! [X4: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_609_ord__eq__le__subst,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_610_ord__eq__le__subst,axiom,
! [A: nat > $o,F: set_nat > nat > $o,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_611_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_612_order__eq__refl,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( X = Y )
=> ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_613_order__eq__refl,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( X = Y )
=> ( ord_less_eq_nat_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_614_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_615_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_616_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_617_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_618_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_619_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_620_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_621_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat_o @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_622_order__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_623_order__subst2,axiom,
! [A: nat > $o,B: nat > $o,F: ( nat > $o ) > nat,C: nat] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_624_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_625_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat_o @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_626_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_627_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_628_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_629_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_630_order__subst1,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_631_order__subst1,axiom,
! [A: nat,F: ( nat > $o ) > nat,B: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat_o @ B @ C )
=> ( ! [X4: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_632_order__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_633_order__subst1,axiom,
! [A: nat > $o,F: nat > nat > $o,B: nat,C: nat] :
( ( ord_less_eq_nat_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_634_order__subst1,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_635_order__subst1,axiom,
! [A: set_nat,F: ( nat > $o ) > set_nat,B: nat > $o,C: nat > $o] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat_o @ B @ C )
=> ( ! [X4: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_636_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_nat,Z2: set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_637_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat > $o,Z2: nat > $o] : ( Y5 = Z2 ) )
= ( ^ [A5: nat > $o,B4: nat > $o] :
( ( ord_less_eq_nat_o @ A5 @ B4 )
& ( ord_less_eq_nat_o @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_638_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_639_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_640_antisym,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_641_antisym,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ord_less_eq_nat_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_642_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_643_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_644_dual__order_Otrans,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_le6893508408891458716et_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_645_dual__order_Otrans,axiom,
! [B: nat > $o,A: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ B @ A )
=> ( ( ord_less_eq_nat_o @ C @ B )
=> ( ord_less_eq_nat_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_646_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_647_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_648_dual__order_Oantisym,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_649_dual__order_Oantisym,axiom,
! [B: nat > $o,A: nat > $o] :
( ( ord_less_eq_nat_o @ B @ A )
=> ( ( ord_less_eq_nat_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_650_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_651_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_652_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_nat,Z2: set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A5 )
& ( ord_le6893508408891458716et_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_653_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat > $o,Z2: nat > $o] : ( Y5 = Z2 ) )
= ( ^ [A5: nat > $o,B4: nat > $o] :
( ( ord_less_eq_nat_o @ B4 @ A5 )
& ( ord_less_eq_nat_o @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_654_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_655_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_656_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: nat,B2: nat] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_657_order__trans,axiom,
! [X: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Z3 )
=> ( ord_le6893508408891458716et_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_658_order__trans,axiom,
! [X: nat > $o,Y: nat > $o,Z3: nat > $o] :
( ( ord_less_eq_nat_o @ X @ Y )
=> ( ( ord_less_eq_nat_o @ Y @ Z3 )
=> ( ord_less_eq_nat_o @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_659_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_660_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z3 )
=> ( ord_less_eq_set_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_661_order_Otrans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_662_order_Otrans,axiom,
! [A: nat > $o,B: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ord_less_eq_nat_o @ B @ C )
=> ( ord_less_eq_nat_o @ A @ C ) ) ) ).
% order.trans
thf(fact_663_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_664_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_665_order__antisym,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_666_order__antisym,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X @ Y )
=> ( ( ord_less_eq_nat_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_667_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_668_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_669_ord__le__eq__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_670_ord__le__eq__trans,axiom,
! [A: nat > $o,B: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_671_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_672_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_673_ord__eq__le__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( A = B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_674_ord__eq__le__trans,axiom,
! [A: nat > $o,B: nat > $o,C: nat > $o] :
( ( A = B )
=> ( ( ord_less_eq_nat_o @ B @ C )
=> ( ord_less_eq_nat_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_675_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_676_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_677_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_nat,Z2: set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [X3: set_set_nat,Y6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y6 )
& ( ord_le6893508408891458716et_nat @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_678_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat > $o,Z2: nat > $o] : ( Y5 = Z2 ) )
= ( ^ [X3: nat > $o,Y6: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y6 )
& ( ord_less_eq_nat_o @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_679_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_680_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [X3: set_nat,Y6: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y6 )
& ( ord_less_eq_set_nat @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_681_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_682_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_683_order__less__imp__not__less,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_684_order__less__imp__not__less,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_less_set_set_nat @ X @ Y )
=> ~ ( ord_less_set_set_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_685_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_686_order__less__imp__not__eq2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_687_order__less__imp__not__eq2,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_less_set_set_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_688_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_689_order__less__imp__not__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_690_order__less__imp__not__eq,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_less_set_set_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_691_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_692_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_693_order__less__imp__triv,axiom,
! [X: set_nat,Y: set_nat,P: $o] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_694_order__less__imp__triv,axiom,
! [X: set_set_nat,Y: set_set_nat,P: $o] :
( ( ord_less_set_set_nat @ X @ Y )
=> ( ( ord_less_set_set_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_695_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_696_order__less__not__sym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_697_order__less__not__sym,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_less_set_set_nat @ X @ Y )
=> ~ ( ord_less_set_set_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_698_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_699_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_700_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_701_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_702_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_703_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_704_order__less__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_705_order__less__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_706_order__less__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_707_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_708_order__less__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_709_order__less__subst1,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_710_order__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_711_order__less__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_712_order__less__subst1,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_713_order__less__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_less_set_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_714_order__less__subst1,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_715_order__less__subst1,axiom,
! [A: set_set_nat,F: set_set_nat > set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_716_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_717_order__less__irrefl,axiom,
! [X: set_nat] :
~ ( ord_less_set_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_718_order__less__irrefl,axiom,
! [X: set_set_nat] :
~ ( ord_less_set_set_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_719_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_720_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_721_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_722_ord__less__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_723_ord__less__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_724_ord__less__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_725_ord__less__eq__subst,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_726_ord__less__eq__subst,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_727_ord__less__eq__subst,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_728_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_729_ord__eq__less__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_730_ord__eq__less__subst,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_731_ord__eq__less__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_732_ord__eq__less__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_733_ord__eq__less__subst,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_734_ord__eq__less__subst,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_735_ord__eq__less__subst,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C: set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_736_ord__eq__less__subst,axiom,
! [A: set_set_nat,F: set_set_nat > set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_737_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_738_order__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z3 )
=> ( ord_less_set_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_739_order__less__trans,axiom,
! [X: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( ord_less_set_set_nat @ X @ Y )
=> ( ( ord_less_set_set_nat @ Y @ Z3 )
=> ( ord_less_set_set_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_740_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_741_order__less__asym_H,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ord_less_set_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_742_order__less__asym_H,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ~ ( ord_less_set_set_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_743_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_744_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_745_order__less__asym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_746_order__less__asym,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_less_set_set_nat @ X @ Y )
=> ~ ( ord_less_set_set_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_747_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_748_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_749_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_750_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_751_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_752_order_Ostrict__implies__not__eq,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_753_order_Ostrict__implies__not__eq,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_754_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_755_dual__order_Ostrict__trans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_756_dual__order_Ostrict__trans,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ( ord_less_set_set_nat @ C @ B )
=> ( ord_less_set_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_757_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_758_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_759_order_Ostrict__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_760_order_Ostrict__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_761_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_762_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B2: nat] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_763_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N4: nat] :
( ( P4 @ N4 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N4 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_764_dual__order_Oirrefl,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_765_dual__order_Oirrefl,axiom,
! [A: set_set_nat] :
~ ( ord_less_set_set_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_766_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_767_dual__order_Oasym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ~ ( ord_less_set_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_768_dual__order_Oasym,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ~ ( ord_less_set_set_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_769_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_770_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_771_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_772_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X4 )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_773_ord__less__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_774_ord__less__eq__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_775_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_776_ord__eq__less__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_777_ord__eq__less__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( A = B )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_778_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_779_order_Oasym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ord_less_set_nat @ B @ A ) ) ).
% order.asym
thf(fact_780_order_Oasym,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ~ ( ord_less_set_set_nat @ B @ A ) ) ).
% order.asym
thf(fact_781_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_782_less__imp__neq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_783_less__imp__neq,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_less_set_set_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_784_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_785_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_786_rev__predicate1D,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ( ord_less_eq_nat_o @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_787_predicate1D,axiom,
! [P: nat > $o,Q: nat > $o,X: nat] :
( ( ord_less_eq_nat_o @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_788_order__le__imp__less__or__eq,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_less_set_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_789_order__le__imp__less__or__eq,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X @ Y )
=> ( ( ord_less_nat_o @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_790_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_791_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_792_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_793_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_794_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_795_order__less__le__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_796_order__less__le__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_797_order__less__le__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_798_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_799_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat > $o,C: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat_o @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_800_order__less__le__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_801_order__less__le__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_802_order__less__le__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat > $o,C: nat > $o] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_nat_o @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_803_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_804_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_805_order__less__le__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_806_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_807_order__less__le__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_less_set_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_808_order__less__le__subst1,axiom,
! [A: nat > $o,F: nat > nat > $o,B: nat,C: nat] :
( ( ord_less_nat_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_809_order__less__le__subst1,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_810_order__less__le__subst1,axiom,
! [A: nat,F: ( nat > $o ) > nat,B: nat > $o,C: nat > $o] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat_o @ B @ C )
=> ( ! [X4: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_811_order__less__le__subst1,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_812_order__less__le__subst1,axiom,
! [A: nat > $o,F: set_nat > nat > $o,B: set_nat,C: set_nat] :
( ( ord_less_nat_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_813_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_814_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_815_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_816_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_817_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_818_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat_o @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat_o @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_819_order__le__less__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_820_order__le__less__subst2,axiom,
! [A: nat > $o,B: nat > $o,F: ( nat > $o ) > nat,C: nat] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_821_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_822_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat_o @ ( F @ B ) @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat_o @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_823_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_824_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_825_order__le__less__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_826_order__le__less__subst1,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_827_order__le__less__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_828_order__le__less__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_829_order__le__less__subst1,axiom,
! [A: nat > $o,F: nat > nat > $o,B: nat,C: nat] :
( ( ord_less_eq_nat_o @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat_o @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_830_order__le__less__subst1,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_831_order__le__less__subst1,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_set_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_832_order__le__less__subst1,axiom,
! [A: nat > $o,F: set_nat > nat > $o,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat_o @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_nat_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat_o @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_833_order__less__le__trans,axiom,
! [X: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( ord_less_set_set_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Z3 )
=> ( ord_less_set_set_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_834_order__less__le__trans,axiom,
! [X: nat > $o,Y: nat > $o,Z3: nat > $o] :
( ( ord_less_nat_o @ X @ Y )
=> ( ( ord_less_eq_nat_o @ Y @ Z3 )
=> ( ord_less_nat_o @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_835_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_836_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z3 )
=> ( ord_less_set_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_837_order__le__less__trans,axiom,
! [X: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_less_set_set_nat @ Y @ Z3 )
=> ( ord_less_set_set_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_838_order__le__less__trans,axiom,
! [X: nat > $o,Y: nat > $o,Z3: nat > $o] :
( ( ord_less_eq_nat_o @ X @ Y )
=> ( ( ord_less_nat_o @ Y @ Z3 )
=> ( ord_less_nat_o @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_839_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_840_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z3 )
=> ( ord_less_set_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_841_order__neq__le__trans,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A != B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_842_order__neq__le__trans,axiom,
! [A: nat > $o,B: nat > $o] :
( ( A != B )
=> ( ( ord_less_eq_nat_o @ A @ B )
=> ( ord_less_nat_o @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_843_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_844_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_845_order__le__neq__trans,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_846_order__le__neq__trans,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat_o @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_847_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_848_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_849_order__less__imp__le,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_less_set_set_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_850_order__less__imp__le,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( ord_less_nat_o @ X @ Y )
=> ( ord_less_eq_nat_o @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_851_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_852_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_853_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_854_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_855_order__less__le,axiom,
( ord_less_set_set_nat
= ( ^ [X3: set_set_nat,Y6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y6 )
& ( X3 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_856_order__less__le,axiom,
( ord_less_nat_o
= ( ^ [X3: nat > $o,Y6: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y6 )
& ( X3 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_857_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ( X3 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_858_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y6: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y6 )
& ( X3 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_859_order__le__less,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y6: set_set_nat] :
( ( ord_less_set_set_nat @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_860_order__le__less,axiom,
( ord_less_eq_nat_o
= ( ^ [X3: nat > $o,Y6: nat > $o] :
( ( ord_less_nat_o @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_861_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_nat @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_862_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y6: set_nat] :
( ( ord_less_set_nat @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_863_dual__order_Ostrict__implies__order,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_864_dual__order_Ostrict__implies__order,axiom,
! [B: nat > $o,A: nat > $o] :
( ( ord_less_nat_o @ B @ A )
=> ( ord_less_eq_nat_o @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_865_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_866_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_867_order_Ostrict__implies__order,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_868_order_Ostrict__implies__order,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ord_less_nat_o @ A @ B )
=> ( ord_less_eq_nat_o @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_869_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_870_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_871_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [B4: set_set_nat,A5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A5 )
& ~ ( ord_le6893508408891458716et_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_872_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat_o
= ( ^ [B4: nat > $o,A5: nat > $o] :
( ( ord_less_eq_nat_o @ B4 @ A5 )
& ~ ( ord_less_eq_nat_o @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_873_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_874_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A5 )
& ~ ( ord_less_eq_set_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_875_dual__order_Ostrict__trans2,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_less_set_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_876_dual__order_Ostrict__trans2,axiom,
! [B: nat > $o,A: nat > $o,C: nat > $o] :
( ( ord_less_nat_o @ B @ A )
=> ( ( ord_less_eq_nat_o @ C @ B )
=> ( ord_less_nat_o @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_877_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_878_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_879_dual__order_Ostrict__trans1,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_less_set_set_nat @ C @ B )
=> ( ord_less_set_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_880_dual__order_Ostrict__trans1,axiom,
! [B: nat > $o,A: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ B @ A )
=> ( ( ord_less_nat_o @ C @ B )
=> ( ord_less_nat_o @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_881_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_882_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_883_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [B4: set_set_nat,A5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_884_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat_o
= ( ^ [B4: nat > $o,A5: nat > $o] :
( ( ord_less_eq_nat_o @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_885_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_886_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_887_dual__order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A5: set_set_nat] :
( ( ord_less_set_set_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_888_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat_o
= ( ^ [B4: nat > $o,A5: nat > $o] :
( ( ord_less_nat_o @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_889_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_890_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A5: set_nat] :
( ( ord_less_set_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_891_order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [A5: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B4 )
& ~ ( ord_le6893508408891458716et_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_892_order_Ostrict__iff__not,axiom,
( ord_less_nat_o
= ( ^ [A5: nat > $o,B4: nat > $o] :
( ( ord_less_eq_nat_o @ A5 @ B4 )
& ~ ( ord_less_eq_nat_o @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_893_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_894_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_895_order_Ostrict__trans2,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_896_order_Ostrict__trans2,axiom,
! [A: nat > $o,B: nat > $o,C: nat > $o] :
( ( ord_less_nat_o @ A @ B )
=> ( ( ord_less_eq_nat_o @ B @ C )
=> ( ord_less_nat_o @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_897_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_898_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_899_order_Ostrict__trans1,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_900_order_Ostrict__trans1,axiom,
! [A: nat > $o,B: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ord_less_nat_o @ B @ C )
=> ( ord_less_nat_o @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_901_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_902_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_903_order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [A5: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_904_order_Ostrict__iff__order,axiom,
( ord_less_nat_o
= ( ^ [A5: nat > $o,B4: nat > $o] :
( ( ord_less_eq_nat_o @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_905_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_906_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_907_order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B4: set_set_nat] :
( ( ord_less_set_set_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_908_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat_o
= ( ^ [A5: nat > $o,B4: nat > $o] :
( ( ord_less_nat_o @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_909_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_910_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_911_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_912_less__le__not__le,axiom,
( ord_less_set_set_nat
= ( ^ [X3: set_set_nat,Y6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y6 )
& ~ ( ord_le6893508408891458716et_nat @ Y6 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_913_less__le__not__le,axiom,
( ord_less_nat_o
= ( ^ [X3: nat > $o,Y6: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y6 )
& ~ ( ord_less_eq_nat_o @ Y6 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_914_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_915_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y6: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y6 )
& ~ ( ord_less_eq_set_nat @ Y6 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_916_antisym__conv2,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_917_antisym__conv2,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X @ Y )
=> ( ( ~ ( ord_less_nat_o @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_918_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_919_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_920_antisym__conv1,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ~ ( ord_less_set_set_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_921_antisym__conv1,axiom,
! [X: nat > $o,Y: nat > $o] :
( ~ ( ord_less_nat_o @ X @ Y )
=> ( ( ord_less_eq_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_922_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_923_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_924_nless__le,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ~ ( ord_less_set_set_nat @ A @ B ) )
= ( ~ ( ord_le6893508408891458716et_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_925_nless__le,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ~ ( ord_less_nat_o @ A @ B ) )
= ( ~ ( ord_less_eq_nat_o @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_926_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_927_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_928_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_929_leD,axiom,
! [Y: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ~ ( ord_less_set_set_nat @ X @ Y ) ) ).
% leD
thf(fact_930_leD,axiom,
! [Y: nat > $o,X: nat > $o] :
( ( ord_less_eq_nat_o @ Y @ X )
=> ~ ( ord_less_nat_o @ X @ Y ) ) ).
% leD
thf(fact_931_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_932_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_933_le__fun__def,axiom,
( ord_less_eq_nat_o
= ( ^ [F2: nat > $o,G2: nat > $o] :
! [X3: nat] : ( ord_less_eq_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_934_le__funI,axiom,
! [F: nat > $o,G: nat > $o] :
( ! [X4: nat] : ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_nat_o @ F @ G ) ) ).
% le_funI
thf(fact_935_le__funE,axiom,
! [F: nat > $o,G: nat > $o,X: nat] :
( ( ord_less_eq_nat_o @ F @ G )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_936_le__funD,axiom,
! [F: nat > $o,G: nat > $o,X: nat] :
( ( ord_less_eq_nat_o @ F @ G )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_937_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_938_pred__subset__eq,axiom,
! [R2: set_set_nat,S2: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ R2 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ S2 ) )
= ( ord_le6893508408891458716et_nat @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_939_pred__subset__eq,axiom,
! [R2: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R2 )
@ ^ [X3: nat] : ( member_nat @ X3 @ S2 ) )
= ( ord_less_eq_set_nat @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_940_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_941_sum__eq__0__iff,axiom,
! [F3: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( ( groups8294997508430121362at_nat @ F @ F3 )
= zero_zero_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ F3 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_942_sum__eq__0__iff,axiom,
! [F3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F3 )
= zero_zero_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ F3 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_943_sum_Oinfinite,axiom,
! [A2: set_set_nat,G: set_nat > nat] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups8294997508430121362at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_944_sum_Oinfinite,axiom,
! [A2: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_945_sum_Odelta_H,axiom,
! [S2: set_set_nat,A: set_nat,B: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S2 )
=> ( ( ( member_set_nat @ A @ S2 )
=> ( ( groups8294997508430121362at_nat
@ ^ [K2: set_nat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_set_nat @ A @ S2 )
=> ( ( groups8294997508430121362at_nat
@ ^ [K2: set_nat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_946_sum_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_947_sum_Odelta,axiom,
! [S2: set_set_nat,A: set_nat,B: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S2 )
=> ( ( ( member_set_nat @ A @ S2 )
=> ( ( groups8294997508430121362at_nat
@ ^ [K2: set_nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_set_nat @ A @ S2 )
=> ( ( groups8294997508430121362at_nat
@ ^ [K2: set_nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_948_sum_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_949_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_950_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_951_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_952_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ N6 )
=> ( ord_less_nat @ X4 @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_953_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_954_sum_Oswap__restrict,axiom,
! [A2: set_set_nat,B3: set_nat,G: set_nat > nat > nat,R2: set_nat > nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ( groups8294997508430121362at_nat
@ ^ [X3: set_nat] :
( groups3542108847815614940at_nat @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y6: nat] :
( ( member_nat @ Y6 @ B3 )
& ( R2 @ X3 @ Y6 ) ) ) )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [Y6: nat] :
( groups8294997508430121362at_nat
@ ^ [X3: set_nat] : ( G @ X3 @ Y6 )
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( R2 @ X3 @ Y6 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_955_sum_Oswap__restrict,axiom,
! [A2: set_nat,B3: set_set_nat,G: nat > set_nat > nat,R2: nat > set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B3 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] :
( groups8294997508430121362at_nat @ ( G @ X3 )
@ ( collect_set_nat
@ ^ [Y6: set_nat] :
( ( member_set_nat @ Y6 @ B3 )
& ( R2 @ X3 @ Y6 ) ) ) )
@ A2 )
= ( groups8294997508430121362at_nat
@ ^ [Y6: set_nat] :
( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( G @ X3 @ Y6 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( R2 @ X3 @ Y6 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_956_sum_Oswap__restrict,axiom,
! [A2: set_nat,B3: set_nat,G: nat > nat > nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] :
( groups3542108847815614940at_nat @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y6: nat] :
( ( member_nat @ Y6 @ B3 )
& ( R2 @ X3 @ Y6 ) ) ) )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [Y6: nat] :
( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( G @ X3 @ Y6 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( R2 @ X3 @ Y6 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_957_sum__mono__inv,axiom,
! [F: set_nat > nat,I5: set_set_nat,G: set_nat > nat,I: set_nat] :
( ( ( groups8294997508430121362at_nat @ F @ I5 )
= ( groups8294997508430121362at_nat @ G @ I5 ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_set_nat @ I @ I5 )
=> ( ( finite1152437895449049373et_nat @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_958_sum__mono__inv,axiom,
! [F: nat > nat,I5: set_nat,G: nat > nat,I: nat] :
( ( ( groups3542108847815614940at_nat @ F @ I5 )
= ( groups3542108847815614940at_nat @ G @ I5 ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_nat @ I @ I5 )
=> ( ( finite_finite_nat @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_959_subset__eq__atLeast0__atMost__finite,axiom,
! [N6: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_960_sum_Ofinite__Collect__op,axiom,
! [I5: set_set_nat,X: set_nat > nat,Y: set_nat > nat] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [I4: set_nat] :
( ( member_set_nat @ I4 @ I5 )
& ( ( X @ I4 )
!= zero_zero_nat ) ) ) )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [I4: set_nat] :
( ( member_set_nat @ I4 @ I5 )
& ( ( Y @ I4 )
!= zero_zero_nat ) ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [I4: set_nat] :
( ( member_set_nat @ I4 @ I5 )
& ( ( plus_plus_nat @ ( X @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_961_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( X @ I4 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( Y @ I4 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( plus_plus_nat @ ( X @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_962_sum_Ointer__filter,axiom,
! [A2: set_set_nat,G: set_nat > nat,P: set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups8294997508430121362at_nat @ G
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups8294997508430121362at_nat
@ ^ [X3: set_nat] : ( if_nat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_963_sum_Ointer__filter,axiom,
! [A2: set_nat,G: nat > nat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( if_nat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_964_sum__nonneg__eq__0__iff,axiom,
! [A2: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( ( groups8294997508430121362at_nat @ F @ A2 )
= zero_zero_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_965_sum__nonneg__eq__0__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ A2 )
= zero_zero_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_966_sum__le__included,axiom,
! [S: set_set_nat,T: set_set_nat,G: set_nat > nat,I: set_nat > set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( finite1152437895449049373et_nat @ T )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X4 ) ) )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ S )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups8294997508430121362at_nat @ F @ S ) @ ( groups8294997508430121362at_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_967_sum__le__included,axiom,
! [S: set_set_nat,T: set_nat,G: nat > nat,I: nat > set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X4 ) ) )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups8294997508430121362at_nat @ F @ S ) @ ( groups3542108847815614940at_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_968_sum__le__included,axiom,
! [S: set_nat,T: set_set_nat,G: set_nat > nat,I: set_nat > nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite1152437895449049373et_nat @ T )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S ) @ ( groups8294997508430121362at_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_969_sum__le__included,axiom,
! [S: set_nat,T: set_nat,G: nat > nat,I: nat > nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S ) @ ( groups3542108847815614940at_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_970_sum__strict__mono__ex1,axiom,
! [A2: set_set_nat,F: set_nat > nat,G: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
& ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_971_sum__strict__mono__ex1,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_972_sum_Orelated,axiom,
! [R2: nat > nat > $o,S2: set_set_nat,H: set_nat > nat,G: set_nat > nat] :
( ( R2 @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
( ( ( R2 @ X1 @ X22 )
& ( R2 @ Y1 @ Y22 ) )
=> ( R2 @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
=> ( ( finite1152437895449049373et_nat @ S2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ S2 )
=> ( R2 @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R2 @ ( groups8294997508430121362at_nat @ H @ S2 ) @ ( groups8294997508430121362at_nat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_973_sum_Orelated,axiom,
! [R2: nat > nat > $o,S2: set_nat,H: nat > nat,G: nat > nat] :
( ( R2 @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
( ( ( R2 @ X1 @ X22 )
& ( R2 @ Y1 @ Y22 ) )
=> ( R2 @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R2 @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R2 @ ( groups3542108847815614940at_nat @ H @ S2 ) @ ( groups3542108847815614940at_nat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_974_sum__eq__Suc0__iff,axiom,
! [A2: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( groups8294997508430121362at_nat @ F @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( ( F @ X3 )
= ( suc @ zero_zero_nat ) )
& ! [Y6: set_nat] :
( ( member_set_nat @ Y6 @ A2 )
=> ( ( X3 != Y6 )
=> ( ( F @ Y6 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_975_sum__eq__Suc0__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ( F @ X3 )
= ( suc @ zero_zero_nat ) )
& ! [Y6: nat] :
( ( member_nat @ Y6 @ A2 )
=> ( ( X3 != Y6 )
=> ( ( F @ Y6 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_976_sum__nonneg__leq__bound,axiom,
! [S: set_set_nat,F: set_nat > nat,B3: nat,I: set_nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups8294997508430121362at_nat @ F @ S )
= B3 )
=> ( ( member_set_nat @ I @ S )
=> ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_977_sum__nonneg__leq__bound,axiom,
! [S: set_nat,F: nat > nat,B3: nat,I: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S )
= B3 )
=> ( ( member_nat @ I @ S )
=> ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_978_sum__nonneg__0,axiom,
! [S: set_set_nat,F: set_nat > nat,I: set_nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups8294997508430121362at_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_set_nat @ I @ S )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_979_sum__nonneg__0,axiom,
! [S: set_nat,F: nat > nat,I: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_nat @ I @ S )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_980_sum__pos2,axiom,
! [I5: set_set_nat,I: set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ I5 )
=> ( ( member_set_nat @ I @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups8294997508430121362at_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_981_sum__pos2,axiom,
! [I5: set_nat,I: nat,F: nat > nat] :
( ( finite_finite_nat @ I5 )
=> ( ( member_nat @ I @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_982_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_983_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_984_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z3: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z3 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z3 ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z3 ) ) ) ).
% crossproduct_eq
thf(fact_985_finite__Collect__le__nat,axiom,
! [K3: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K3 ) ) ) ).
% finite_Collect_le_nat
thf(fact_986_finite__Collect__less__nat,axiom,
! [K3: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K3 ) ) ) ).
% finite_Collect_less_nat
thf(fact_987_finite__Collect__disjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
& ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_988_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_989_finite__Collect__conjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
| ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_990_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_991_finite__Collect__subsets,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [B5: set_set_nat] : ( ord_le6893508408891458716et_nat @ B5 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_992_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_993_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B3: set_set_nat,R2: nat > set_nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B3 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A2 )
& ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_994_pigeonhole__infinite__rel,axiom,
! [A2: set_set_nat,B3: set_nat,R2: set_nat > nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
& ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_995_pigeonhole__infinite__rel,axiom,
! [A2: set_set_nat,B3: set_set_nat,R2: set_nat > set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B3 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B3 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B3 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
& ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_996_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B3: set_nat,R2: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A2 )
& ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_997_not__finite__existsD,axiom,
! [P: set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_998_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_999_finite__has__maximal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
& ( ord_le6893508408891458716et_nat @ A @ X4 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1000_finite__has__maximal2,axiom,
! [A2: set_nat_o,A: nat > $o] :
( ( finite_finite_nat_o @ A2 )
=> ( ( member_nat_o @ A @ A2 )
=> ? [X4: nat > $o] :
( ( member_nat_o @ X4 @ A2 )
& ( ord_less_eq_nat_o @ A @ X4 )
& ! [Xa: nat > $o] :
( ( member_nat_o @ Xa @ A2 )
=> ( ( ord_less_eq_nat_o @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1001_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_nat @ A @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1002_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ A @ X4 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1003_finite__has__minimal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
& ( ord_le6893508408891458716et_nat @ X4 @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1004_finite__has__minimal2,axiom,
! [A2: set_nat_o,A: nat > $o] :
( ( finite_finite_nat_o @ A2 )
=> ( ( member_nat_o @ A @ A2 )
=> ? [X4: nat > $o] :
( ( member_nat_o @ X4 @ A2 )
& ( ord_less_eq_nat_o @ X4 @ A )
& ! [Xa: nat > $o] :
( ( member_nat_o @ Xa @ A2 )
=> ( ( ord_less_eq_nat_o @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1005_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1006_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ X4 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1007_rev__finite__subset,axiom,
! [B3: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B3 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1008_rev__finite__subset,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1009_infinite__super,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S2 )
=> ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_1010_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_1011_finite__subset,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( finite1152437895449049373et_nat @ B3 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_1012_finite__subset,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_1013_finite__psubset__induct,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [A8: set_set_nat] :
( ( finite1152437895449049373et_nat @ A8 )
=> ( ! [B7: set_set_nat] :
( ( ord_less_set_set_nat @ B7 @ A8 )
=> ( P @ B7 ) )
=> ( P @ A8 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_1014_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A8 )
=> ( P @ B7 ) )
=> ( P @ A8 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_1015_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K4: nat] :
( ( ord_less_nat @ N @ K4 )
=> ( P @ K4 ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K4 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K4 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1016_sum__atLeastAtMost__code,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A5: nat] : ( plus_plus_nat @ ( F @ A5 ) )
@ A
@ B
@ zero_zero_nat ) ) ).
% sum_atLeastAtMost_code
thf(fact_1017_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1018_verit__le__mono__div,axiom,
! [A2: nat,B3: nat,N: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B3 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B3 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1019_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1020_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1021_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_1022_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1023_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_1024_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_1025_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1026_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1027_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1028_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1029_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_1030_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_1031_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1032_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1033_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1034_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1035_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1036_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_1037_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_1038_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1039_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_1040_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1041_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1042_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1043_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1044_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1045_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1046_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1047_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1048_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1049_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1050_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1051_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1052_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_1053_lambda__one,axiom,
( ( ^ [X3: nat] : X3 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_1054_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1055_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_1056_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_1057_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1058_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1059_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1060_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1061_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1062_digit__wise__gen__equiv,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( X = Y )
= ( ! [K2: nat] :
( ( bits_nth_digit @ X @ K2 @ B )
= ( bits_nth_digit @ Y @ K2 @ B ) ) ) ) ) ).
% digit_wise_gen_equiv
thf(fact_1063_prod_Ofinite__Collect__op,axiom,
! [I5: set_set_nat,X: set_nat > nat,Y: set_nat > nat] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [I4: set_nat] :
( ( member_set_nat @ I4 @ I5 )
& ( ( X @ I4 )
!= one_one_nat ) ) ) )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [I4: set_nat] :
( ( member_set_nat @ I4 @ I5 )
& ( ( Y @ I4 )
!= one_one_nat ) ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [I4: set_nat] :
( ( member_set_nat @ I4 @ I5 )
& ( ( times_times_nat @ ( X @ I4 ) @ ( Y @ I4 ) )
!= one_one_nat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_1064_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( X @ I4 )
!= one_one_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( Y @ I4 )
!= one_one_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( times_times_nat @ ( X @ I4 ) @ ( Y @ I4 ) )
!= one_one_nat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_1065_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1066_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1067_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1068_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1069_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_1070_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_1071_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1072_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1073_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1074_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1075_power__increasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_1076_power__strict__increasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_1077_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1078_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_1079_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1080_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1081_sum__eq__1__iff,axiom,
! [A2: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( groups8294997508430121362at_nat @ F @ A2 )
= one_one_nat )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y6: set_nat] :
( ( member_set_nat @ Y6 @ A2 )
=> ( ( X3 != Y6 )
=> ( ( F @ Y6 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_1082_sum__eq__1__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A2 )
= one_one_nat )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y6: nat] :
( ( member_nat @ Y6 @ A2 )
=> ( ( X3 != Y6 )
=> ( ( F @ Y6 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_1083_aux1__digit__wise__gen__equiv,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ! [K2: nat] :
( ( bits_nth_digit @ A @ K2 @ B )
= zero_zero_nat ) )
= ( A = zero_zero_nat ) ) ) ).
% aux1_digit_wise_gen_equiv
thf(fact_1084_aux0__digit__wise__gen__equiv,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ? [K4: nat] :
( ( bits_nth_digit @ A @ K4 @ B )
!= zero_zero_nat ) ) ) ).
% aux0_digit_wise_gen_equiv
thf(fact_1085_fold__atLeastAtMost__nat_Oelims,axiom,
! [X: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb @ Xc )
= Y )
=> ( ( ( ord_less_nat @ Xb @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less_nat @ Xb @ Xa2 )
=> ( Y
= ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_1086_fold__atLeastAtMost__nat_Osimps,axiom,
( set_fo2584398358068434914at_nat
= ( ^ [F2: nat > nat > nat,A5: nat,B4: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A5 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B4 @ ( F2 @ A5 @ Acc ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_1087_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1088_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_1089_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_1090_power__decreasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1091_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1092_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1093_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1094_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_1095_digit__shift__preserves__digits,axiom,
! [B: nat,Y: nat,T: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( bits_nth_digit @ ( times_times_nat @ B @ Y ) @ ( suc @ T ) @ B )
= ( bits_nth_digit @ Y @ T @ B ) ) ) ).
% digit_shift_preserves_digits
thf(fact_1096_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_1097_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > nat,P2: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_1098_aux__gen__b__factor,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ one_one_nat @ B )
=> ? [K4: nat,C2: nat] :
( ( A
= ( times_times_nat @ ( power_power_nat @ B @ K4 ) @ C2 ) )
& ( ( modulo_modulo_nat @ C2 @ B )
!= zero_zero_nat ) ) ) ) ).
% aux_gen_b_factor
thf(fact_1099_digit__shift__inserts__zero__least__siginificant__digit,axiom,
! [T: nat,B: nat,Y: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( ord_less_nat @ one_one_nat @ B )
=> ( ( bits_nth_digit @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ B @ Y ) ) @ T @ B )
= ( bits_nth_digit @ ( times_times_nat @ B @ Y ) @ T @ B ) ) ) ) ).
% digit_shift_inserts_zero_least_siginificant_digit
thf(fact_1100_general__digit__base,axiom,
! [T22: nat,T1: nat,B: nat,A: nat] :
( ( ord_less_nat @ T22 @ T1 )
=> ( ( ord_less_nat @ one_one_nat @ B )
=> ( ( bits_nth_digit @ ( times_times_nat @ A @ ( power_power_nat @ B @ T1 ) ) @ T22 @ B )
= zero_zero_nat ) ) ) ).
% general_digit_base
thf(fact_1101_aux3__digit__gen__sum__repr,axiom,
! [D: nat,B: nat,R: nat,A: nat] :
( ( ord_less_nat @ D @ ( power_power_nat @ B @ R ) )
=> ( ( ord_less_nat @ one_one_nat @ B )
=> ( ( bits_nth_digit @ ( plus_plus_nat @ ( times_times_nat @ A @ ( power_power_nat @ B @ R ) ) @ D ) @ R @ B )
= ( bits_nth_digit @ ( times_times_nat @ A @ ( power_power_nat @ B @ R ) ) @ R @ B ) ) ) ) ).
% aux3_digit_gen_sum_repr
thf(fact_1102_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1103_f__le__bound,axiom,
! [K3: nat] : ( ord_less_eq_nat @ ( f @ K3 ) @ ( minus_minus_nat @ b @ one_one_nat ) ) ).
% f_le_bound
thf(fact_1104_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1105_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1106_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1107_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1108_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1109_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1110_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1111_Suc__diff__diff,axiom,
! [M: nat,N: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K3 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K3 ) ) ).
% Suc_diff_diff
thf(fact_1112_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1113_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1114_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1115_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1116_diff__diff__left,axiom,
! [I: nat,J: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K3 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K3 ) ) ) ).
% diff_diff_left
thf(fact_1117_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1118_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1119_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1120_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1121_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1122_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1123_Nat_Odiff__diff__right,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K3 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1124_Nat_Oadd__diff__assoc2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K3 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K3 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1125_Nat_Oadd__diff__assoc,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K3 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1126_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1127_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1128_diff__Suc__diff__eq1,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K3 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K3 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1129_diff__Suc__diff__eq2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K3 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K3 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1130_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1131_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1132_sum__diff__nat,axiom,
! [B3: set_set_nat,A2: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
= ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ F @ B3 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_1133_sum__diff__nat,axiom,
! [B3: set_nat,A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B3 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_1134_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1135_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1136_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1137_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1138_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1139_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1140_less__diff__conv,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ J ) ) ).
% less_diff_conv
thf(fact_1141_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N4 ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N4 ) @ N4 ) ) ) ) ).
% mod_if
thf(fact_1142_diff__commute,axiom,
! [I: nat,J: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K3 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K3 ) @ J ) ) ).
% diff_commute
thf(fact_1143_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_1144_diff__mult__distrib2,axiom,
! [K3: nat,M: nat,N: nat] :
( ( times_times_nat @ K3 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1145_diff__mult__distrib,axiom,
! [M: nat,N: nat,K3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K3 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) ) ) ).
% diff_mult_distrib
thf(fact_1146_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1147_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1148_diff__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K3 ) @ ( plus_plus_nat @ N @ K3 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1149_Nat_Odiff__cancel,axiom,
! [K3: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1150_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1151_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1152_zero__induct__lemma,axiom,
! [P: nat > $o,K3: nat,I: nat] :
( ( P @ K3 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K3 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1153_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1154_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1155_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1156_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1157_less__imp__diff__less,axiom,
! [J: nat,K3: nat,N: nat] :
( ( ord_less_nat @ J @ K3 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K3 ) ) ).
% less_imp_diff_less
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [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 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( bits_nth_digit
@ ( plus_plus_nat @ ( times_times_nat @ ( f @ ( suc @ qa ) ) @ ( power_power_nat @ b @ ( suc @ qa ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( f @ K2 ) @ ( power_power_nat @ b @ K2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ qa ) ) )
@ ta
@ b )
= ( bits_nth_digit
@ ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( f @ K2 ) @ ( power_power_nat @ b @ K2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ qa ) )
@ ta
@ b ) ) ).
%------------------------------------------------------------------------------