TPTP Problem File: SLH0811^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Digit_Expansions/0000_Bits_Digits/prob_00378_014745__5544074_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1342 ( 673 unt; 70 typ; 0 def)
% Number of atoms : 3473 (1601 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 11962 ( 304 ~; 83 |; 273 &;9935 @)
% ( 0 <=>;1367 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 554 ( 554 >; 0 *; 0 +; 0 <<)
% Number of symbols : 69 ( 66 usr; 14 con; 0-3 aty)
% Number of variables : 3623 ( 260 ^;3252 !; 111 ?;3623 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:20:31.501
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (66)
thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
gbinomial_int: int > nat > int ).
thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
gbinomial_nat: nat > nat > nat ).
thf(sy_c_Bits__Digits_Onth__digit,type,
bits_nth_digit: nat > nat > nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
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_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Nat__Onat,type,
groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
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_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_M_Eo_J,type,
ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
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_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_v_a____,type,
a: nat ).
thf(sy_v_b,type,
b: nat ).
thf(sy_v_c,type,
c: nat ).
thf(sy_v_d____,type,
d: nat ).
thf(sy_v_e____,type,
e: nat ).
thf(sy_v_f____,type,
f: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_r,type,
r: nat ).
% Relevant facts (1266)
thf(fact_0_assms_I2_J,axiom,
ord_less_nat @ r @ c ).
% assms(2)
thf(fact_1_assms_I1_J,axiom,
ord_less_nat @ n @ ( power_power_nat @ b @ c ) ).
% assms(1)
thf(fact_2_assms_I3_J,axiom,
ord_less_nat @ one_one_nat @ b ).
% assms(3)
thf(fact_3_f__def,axiom,
( f
= ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_digit @ n @ K @ b ) @ ( power_power_nat @ b @ K ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ r ) ) ) ).
% f_def
thf(fact_4__092_060open_062_I_092_060Sum_062k_A_061_A0_O_O_060Suc_Ar_O_Anth__digit_An_Ak_Ab_A_K_Ab_A_094_Ak_J_A_060_Ab_A_094_ASuc_Ar_092_060close_062,axiom,
( ord_less_nat
@ ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_digit @ n @ K @ b ) @ ( power_power_nat @ b @ K ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ r ) ) )
@ ( power_power_nat @ b @ ( suc @ r ) ) ) ).
% \<open>(\<Sum>k = 0..<Suc r. nth_digit n k b * b ^ k) < b ^ Suc r\<close>
thf(fact_5_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_6_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_7_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_8_mult__less__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_9_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_10_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_11_a__def,axiom,
( a
= ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_digit @ n @ K @ b ) @ ( power_power_nat @ b @ K ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ r ) ) ) ) ).
% a_def
thf(fact_12_sum_Oneutral__const,axiom,
! [A2: set_int] :
( ( groups4541462559716669496nt_nat
@ ^ [Uu: int] : zero_zero_nat
@ A2 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_13_sum_Oneutral__const,axiom,
! [A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [Uu: int] : zero_zero_int
@ A2 )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_14_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [Uu: nat] : zero_zero_int
@ A2 )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_15_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A2 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_16_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_17_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_18_mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_19_mult__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_20_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_21_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_22_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_23_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_24_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_25_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_26_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_27_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_28_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_29_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_30_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_31_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_32_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_33_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_34__092_060open_062a_A_060_Ab_A_094_ASuc_Ar_092_060close_062,axiom,
ord_less_nat @ a @ ( power_power_nat @ b @ ( suc @ r ) ) ).
% \<open>a < b ^ Suc r\<close>
thf(fact_35_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_36_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_37_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_38_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_39_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_40_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_41_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_42_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_43_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_44_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_45_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_46_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_47_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_48_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_49_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_50_d__def,axiom,
( d
= ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_digit @ n @ K @ b ) @ ( power_power_nat @ b @ K ) )
@ ( set_or4665077453230672383an_nat @ ( suc @ r ) @ c ) ) ) ).
% d_def
thf(fact_51_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_52_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_53_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_54_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_55_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_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_60_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_61_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_62_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_63_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_64_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_65_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_66_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_67_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_68_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_69_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_70_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_71_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_72_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_73_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_74_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_75_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_76_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,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_77_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_78_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_79_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_80_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M3: nat] :
( ( M
= ( suc @ M3 ) )
& ( ord_less_nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_81_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_82_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_83_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_84_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_85_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_86_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_87_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_88_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_89_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_90_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_91_Suc__mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M )
= ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_92_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_93_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_94_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_95_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_96_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_97_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_98_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_99_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_set_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_100_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_set_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_101_lift__Suc__mono__less__iff,axiom,
! [F: nat > int > $o,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int_o @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int_o @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_102_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_103_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_104_lift__Suc__mono__less,axiom,
! [F: nat > set_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_set_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_105_lift__Suc__mono__less,axiom,
! [F: nat > set_int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_set_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_106_lift__Suc__mono__less,axiom,
! [F: nat > int > $o,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_int_o @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_int_o @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_107_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_108_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_109_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_110_power__Suc2,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_111_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_112_power__Suc,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_Suc
thf(fact_113_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_114_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_115_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_116_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_117_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_118_Suc__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_119_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > int,G: nat > int] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ ( suc @ X4 ) @ A2 )
=> ( ( F @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups3539618377306564664at_int @ F @ A2 )
= ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_120_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_121_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_122_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_123_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_124_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_125_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_126_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_127_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_128_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_129_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_130_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_131_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_132_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_133_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_134_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_135_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_136_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_137_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_138_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_139_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_140_power__gt__expt,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K2 @ ( power_power_nat @ N @ K2 ) ) ) ).
% power_gt_expt
thf(fact_141_sum__SucD,axiom,
! [F: int > nat,A2: set_int,N: nat] :
( ( ( groups4541462559716669496nt_nat @ F @ A2 )
= ( suc @ N ) )
=> ? [X4: int] :
( ( member_int @ X4 @ A2 )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).
% sum_SucD
thf(fact_142_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_143_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_144_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_145_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_146_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_147_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_148_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_149_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_150_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_151_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_152_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_153_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_154_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_155_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_156_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_157_sum_Oreindex__bij__witness,axiom,
! [S2: set_int,I: int > int,J: int > int,T2: set_int,H: int > nat,G: int > nat] :
( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( member_int @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T2 )
=> ( member_int @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S2 )
= ( groups4541462559716669496nt_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_158_sum_Oreindex__bij__witness,axiom,
! [S2: set_int,I: int > int,J: int > int,T2: set_int,H: int > int,G: int > int] :
( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( member_int @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T2 )
=> ( member_int @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4538972089207619220nt_int @ G @ S2 )
= ( groups4538972089207619220nt_int @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_159_sum_Oreindex__bij__witness,axiom,
! [S2: set_int,I: nat > int,J: int > nat,T2: set_nat,H: nat > int,G: int > int] :
( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_int @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4538972089207619220nt_int @ G @ S2 )
= ( groups3539618377306564664at_int @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_160_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: int > nat,J: nat > int,T2: set_int,H: int > int,G: nat > int] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( member_int @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T2 )
=> ( member_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S2 )
= ( groups4538972089207619220nt_int @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_161_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: nat > nat,J: nat > nat,T2: set_nat,H: nat > int,G: nat > int] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S2 )
= ( groups3539618377306564664at_int @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_162_sum_Oreindex__bij__witness,axiom,
! [S2: set_int,I: nat > int,J: int > nat,T2: set_nat,H: nat > nat,G: int > nat] :
( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_int @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_163_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: int > nat,J: nat > int,T2: set_int,H: int > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( member_int @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T2 )
=> ( member_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups4541462559716669496nt_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_164_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: nat > nat,J: nat > nat,T2: set_nat,H: nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_165_sum_Oeq__general__inverses,axiom,
! [B3: set_int,K2: int > int,A2: set_int,H: int > int,Gamma: int > nat,Phi: int > nat] :
( ! [Y3: int] :
( ( member_int @ Y3 @ B3 )
=> ( ( member_int @ ( K2 @ Y3 ) @ A2 )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ( member_int @ ( H @ X4 ) @ B3 )
& ( ( K2 @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups4541462559716669496nt_nat @ Phi @ A2 )
= ( groups4541462559716669496nt_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_166_sum_Oeq__general__inverses,axiom,
! [B3: set_int,K2: int > int,A2: set_int,H: int > int,Gamma: int > int,Phi: int > int] :
( ! [Y3: int] :
( ( member_int @ Y3 @ B3 )
=> ( ( member_int @ ( K2 @ Y3 ) @ A2 )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ( member_int @ ( H @ X4 ) @ B3 )
& ( ( K2 @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups4538972089207619220nt_int @ Phi @ A2 )
= ( groups4538972089207619220nt_int @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_167_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K2: nat > int,A2: set_int,H: int > nat,Gamma: nat > int,Phi: int > int] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ( ( member_int @ ( K2 @ Y3 ) @ A2 )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ( member_nat @ ( H @ X4 ) @ B3 )
& ( ( K2 @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups4538972089207619220nt_int @ Phi @ A2 )
= ( groups3539618377306564664at_int @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_168_sum_Oeq__general__inverses,axiom,
! [B3: set_int,K2: int > nat,A2: set_nat,H: nat > int,Gamma: int > int,Phi: nat > int] :
( ! [Y3: int] :
( ( member_int @ Y3 @ B3 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_int @ ( H @ X4 ) @ B3 )
& ( ( K2 @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A2 )
= ( groups4538972089207619220nt_int @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_169_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K2: nat > nat,A2: set_nat,H: nat > nat,Gamma: nat > int,Phi: nat > int] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ ( H @ X4 ) @ B3 )
& ( ( K2 @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A2 )
= ( groups3539618377306564664at_int @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_170_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K2: nat > int,A2: set_int,H: int > nat,Gamma: nat > nat,Phi: int > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ( ( member_int @ ( K2 @ Y3 ) @ A2 )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ( member_nat @ ( H @ X4 ) @ B3 )
& ( ( K2 @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups4541462559716669496nt_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_171_sum_Oeq__general__inverses,axiom,
! [B3: set_int,K2: int > nat,A2: set_nat,H: nat > int,Gamma: int > nat,Phi: nat > nat] :
( ! [Y3: int] :
( ( member_int @ Y3 @ B3 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_int @ ( H @ X4 ) @ B3 )
& ( ( K2 @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups4541462559716669496nt_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_172_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K2: nat > nat,A2: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ ( H @ X4 ) @ B3 )
& ( ( K2 @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_173_sum_Oeq__general,axiom,
! [B3: set_int,A2: set_int,H: int > int,Gamma: int > nat,Phi: int > nat] :
( ! [Y3: int] :
( ( member_int @ Y3 @ B3 )
=> ? [X5: int] :
( ( member_int @ X5 @ A2 )
& ( ( H @ X5 )
= Y3 )
& ! [Ya: int] :
( ( ( member_int @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ( member_int @ ( H @ X4 ) @ B3 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups4541462559716669496nt_nat @ Phi @ A2 )
= ( groups4541462559716669496nt_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_174_sum_Oeq__general,axiom,
! [B3: set_int,A2: set_int,H: int > int,Gamma: int > int,Phi: int > int] :
( ! [Y3: int] :
( ( member_int @ Y3 @ B3 )
=> ? [X5: int] :
( ( member_int @ X5 @ A2 )
& ( ( H @ X5 )
= Y3 )
& ! [Ya: int] :
( ( ( member_int @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ( member_int @ ( H @ X4 ) @ B3 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups4538972089207619220nt_int @ Phi @ A2 )
= ( groups4538972089207619220nt_int @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_175_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_int,H: int > nat,Gamma: nat > int,Phi: int > int] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ? [X5: int] :
( ( member_int @ X5 @ A2 )
& ( ( H @ X5 )
= Y3 )
& ! [Ya: int] :
( ( ( member_int @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ( member_nat @ ( H @ X4 ) @ B3 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups4538972089207619220nt_int @ Phi @ A2 )
= ( groups3539618377306564664at_int @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_176_sum_Oeq__general,axiom,
! [B3: set_int,A2: set_nat,H: nat > int,Gamma: int > int,Phi: nat > int] :
( ! [Y3: int] :
( ( member_int @ 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_int @ ( H @ X4 ) @ B3 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A2 )
= ( groups4538972089207619220nt_int @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_177_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_nat,H: nat > nat,Gamma: nat > int,Phi: nat > int] :
( ! [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 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A2 )
= ( groups3539618377306564664at_int @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_178_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_int,H: int > nat,Gamma: nat > nat,Phi: int > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ? [X5: int] :
( ( member_int @ X5 @ A2 )
& ( ( H @ X5 )
= Y3 )
& ! [Ya: int] :
( ( ( member_int @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ( member_nat @ ( H @ X4 ) @ B3 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups4541462559716669496nt_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_179_sum_Oeq__general,axiom,
! [B3: set_int,A2: set_nat,H: nat > int,Gamma: int > nat,Phi: nat > nat] :
( ! [Y3: int] :
( ( member_int @ 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_int @ ( H @ X4 ) @ B3 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups4541462559716669496nt_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_180_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_181_sum_Ocong,axiom,
! [A2: set_int,B3: set_int,G: int > nat,H: int > nat] :
( ( A2 = B3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ B3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ A2 )
= ( groups4541462559716669496nt_nat @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_182_sum_Ocong,axiom,
! [A2: set_int,B3: set_int,G: int > int,H: int > int] :
( ( A2 = B3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ B3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups4538972089207619220nt_int @ G @ A2 )
= ( groups4538972089207619220nt_int @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_183_sum_Ocong,axiom,
! [A2: set_nat,B3: set_nat,G: nat > int,H: nat > int] :
( ( A2 = B3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ A2 )
= ( groups3539618377306564664at_int @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_184_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_185_sum_Oswap,axiom,
! [G: nat > int > nat,B3: set_int,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( groups4541462559716669496nt_nat @ ( G @ I3 ) @ B3 )
@ A2 )
= ( groups4541462559716669496nt_nat
@ ^ [J3: int] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ I3 @ J3 )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_186_sum_Oswap,axiom,
! [G: int > nat > nat,B3: set_nat,A2: set_int] :
( ( groups4541462559716669496nt_nat
@ ^ [I3: int] : ( groups3542108847815614940at_nat @ ( G @ I3 ) @ B3 )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups4541462559716669496nt_nat
@ ^ [I3: int] : ( G @ I3 @ J3 )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_187_sum_Oswap,axiom,
! [G: int > int > nat,B3: set_int,A2: set_int] :
( ( groups4541462559716669496nt_nat
@ ^ [I3: int] : ( groups4541462559716669496nt_nat @ ( G @ I3 ) @ B3 )
@ A2 )
= ( groups4541462559716669496nt_nat
@ ^ [J3: int] :
( groups4541462559716669496nt_nat
@ ^ [I3: int] : ( G @ I3 @ J3 )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_188_sum_Oswap,axiom,
! [G: int > int > int,B3: set_int,A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( groups4538972089207619220nt_int @ ( G @ I3 ) @ B3 )
@ A2 )
= ( groups4538972089207619220nt_int
@ ^ [J3: int] :
( groups4538972089207619220nt_int
@ ^ [I3: int] : ( G @ I3 @ J3 )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_189_sum_Oswap,axiom,
! [G: int > nat > int,B3: set_nat,A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( groups3539618377306564664at_int @ ( G @ I3 ) @ B3 )
@ A2 )
= ( groups3539618377306564664at_int
@ ^ [J3: nat] :
( groups4538972089207619220nt_int
@ ^ [I3: int] : ( G @ I3 @ J3 )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_190_sum_Oswap,axiom,
! [G: nat > int > int,B3: set_int,A2: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( groups4538972089207619220nt_int @ ( G @ I3 ) @ B3 )
@ A2 )
= ( groups4538972089207619220nt_int
@ ^ [J3: int] :
( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ I3 @ J3 )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_191_sum_Oswap,axiom,
! [G: nat > nat > int,B3: set_nat,A2: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( groups3539618377306564664at_int @ ( G @ I3 ) @ B3 )
@ A2 )
= ( groups3539618377306564664at_int
@ ^ [J3: nat] :
( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ I3 @ J3 )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_192_sum_Oswap,axiom,
! [G: nat > nat > nat,B3: set_nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( G @ I3 ) @ B3 )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ I3 @ J3 )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_193_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_194_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_195_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_196_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A2: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_197_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > nat,A2: set_int] :
( ( ( groups4541462559716669496nt_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_198_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > int,A2: set_int] :
( ( ( groups4538972089207619220nt_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_199_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A2: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_200_sum_Oneutral,axiom,
! [A2: set_int,G: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups4541462559716669496nt_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_201_sum_Oneutral,axiom,
! [A2: set_int,G: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups4538972089207619220nt_int @ G @ A2 )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_202_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G @ A2 )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_203_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_204_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_205_power__commuting__commutes,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_206_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_207_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_208_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_209_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_210_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_211_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_212_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_213_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_214_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_215_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_216_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_217_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_218_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_219_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_220_power__mult,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_221_sum__product,axiom,
! [F: nat > nat,A2: set_nat,G: int > nat,B3: set_int] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] :
( groups4541462559716669496nt_nat
@ ^ [J3: int] : ( times_times_nat @ ( F @ I3 ) @ ( G @ J3 ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_222_sum__product,axiom,
! [F: int > nat,A2: set_int,G: nat > nat,B3: set_nat] :
( ( times_times_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
= ( groups4541462559716669496nt_nat
@ ^ [I3: int] :
( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( F @ I3 ) @ ( G @ J3 ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_223_sum__product,axiom,
! [F: int > nat,A2: set_int,G: int > nat,B3: set_int] :
( ( times_times_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) )
= ( groups4541462559716669496nt_nat
@ ^ [I3: int] :
( groups4541462559716669496nt_nat
@ ^ [J3: int] : ( times_times_nat @ ( F @ I3 ) @ ( G @ J3 ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_224_sum__product,axiom,
! [F: int > int,A2: set_int,G: int > int,B3: set_int] :
( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B3 ) )
= ( groups4538972089207619220nt_int
@ ^ [I3: int] :
( groups4538972089207619220nt_int
@ ^ [J3: int] : ( times_times_int @ ( F @ I3 ) @ ( G @ J3 ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_225_sum__product,axiom,
! [F: int > int,A2: set_int,G: nat > int,B3: set_nat] :
( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ B3 ) )
= ( groups4538972089207619220nt_int
@ ^ [I3: int] :
( groups3539618377306564664at_int
@ ^ [J3: nat] : ( times_times_int @ ( F @ I3 ) @ ( G @ J3 ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_226_sum__product,axiom,
! [F: nat > int,A2: set_nat,G: int > int,B3: set_int] :
( ( times_times_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B3 ) )
= ( groups3539618377306564664at_int
@ ^ [I3: nat] :
( groups4538972089207619220nt_int
@ ^ [J3: int] : ( times_times_int @ ( F @ I3 ) @ ( G @ J3 ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_227_sum__product,axiom,
! [F: nat > int,A2: set_nat,G: nat > int,B3: set_nat] :
( ( times_times_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ B3 ) )
= ( groups3539618377306564664at_int
@ ^ [I3: nat] :
( groups3539618377306564664at_int
@ ^ [J3: nat] : ( times_times_int @ ( F @ I3 ) @ ( G @ J3 ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_228_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
@ ^ [I3: nat] :
( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( F @ I3 ) @ ( G @ J3 ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_229_sum__distrib__right,axiom,
! [F: nat > nat,A2: set_nat,R: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R )
= ( groups3542108847815614940at_nat
@ ^ [N5: nat] : ( times_times_nat @ ( F @ N5 ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_230_sum__distrib__left,axiom,
! [R: nat,F: nat > nat,A2: set_nat] :
( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [N5: nat] : ( times_times_nat @ R @ ( F @ N5 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_231_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_232_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_233_nat__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_234_nat__mult__eq__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_235_mult__less__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_236_mult__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_237_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_238_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_239_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_240_e__def,axiom,
( e
= ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_digit @ n @ K @ b ) @ ( power_power_nat @ b @ ( minus_minus_nat @ K @ ( suc @ r ) ) ) )
@ ( set_or4665077453230672383an_nat @ ( suc @ r ) @ c ) ) ) ).
% e_def
thf(fact_241_ab,axiom,
( a
= ( plus_plus_nat @ ( times_times_nat @ ( bits_nth_digit @ n @ r @ b ) @ ( power_power_nat @ b @ r ) ) @ f ) ) ).
% ab
thf(fact_242_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_243_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_244_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_245_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_246_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > int,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_247_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > nat,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_248_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_249_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_250_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_251_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_252_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_253_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_254_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_255_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_256_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_257_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_258_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_259_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_260_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_261_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_262_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_263_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_264_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_265_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_266_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_267_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_268_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_269_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_270_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_271_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_272_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_273_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_274_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_275_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_276_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_277_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_278_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_279_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_280_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_281_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_282_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_283_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_284_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_285_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_286_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_287_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_288_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_289_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_290_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_291_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_292_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_293_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_294_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_295_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_296_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_297_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_298_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_299_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_300_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_301_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_302_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_303_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_304_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_305_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_306_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_307_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_308_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_309_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_310_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_311_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_312_ar,axiom,
( ( bits_nth_digit @ ( plus_plus_nat @ a @ d ) @ r @ b )
= ( bits_nth_digit @ a @ r @ b ) ) ).
% ar
thf(fact_313_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_314_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_315_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_316_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_317_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_318_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_319_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_320_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_321_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_322_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_323_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_324_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_325_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_326_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_327_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_328_d2r,axiom,
( d
= ( times_times_nat @ ( power_power_nat @ b @ ( suc @ r ) ) @ e ) ) ).
% d2r
thf(fact_329_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_330_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_331__092_060open_062d_A_061_A_I_092_060Sum_062k_A_061_ASuc_Ar_O_O_060c_O_Ab_A_094_ASuc_Ar_A_K_A_Inth__digit_An_Ak_Ab_A_K_Ab_A_094_A_Ik_A_N_ASuc_Ar_J_J_J_092_060close_062,axiom,
( d
= ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( power_power_nat @ b @ ( suc @ r ) ) @ ( times_times_nat @ ( bits_nth_digit @ n @ K @ b ) @ ( power_power_nat @ b @ ( minus_minus_nat @ K @ ( suc @ r ) ) ) ) )
@ ( set_or4665077453230672383an_nat @ ( suc @ r ) @ c ) ) ) ).
% \<open>d = (\<Sum>k = Suc r..<c. b ^ Suc r * (nth_digit n k b * b ^ (k - Suc r)))\<close>
thf(fact_332_ad,axiom,
( ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_digit @ n @ K @ b ) @ ( power_power_nat @ b @ K ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ c ) )
= ( plus_plus_nat @ a @ d ) ) ).
% ad
thf(fact_333_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_334_sum_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > int] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_335_sum_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_336_Nat_Odiff__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_337_diff__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_338_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_339_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_340_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_341_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_342_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_343_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_344_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_345_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_346_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_347_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_348_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_349_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_350_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_351_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_352_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_353_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_354_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_355_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_356_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_357_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_358_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_359_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_360_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_361_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_362_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_363_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_364_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_365_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_366_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_367_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_368_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_369_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_370_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_371_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_372_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_373_group__cancel_Osub1,axiom,
! [A2: int,K2: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K2 @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_374_group__cancel_Oadd2,axiom,
! [B3: nat,K2: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_375_group__cancel_Oadd2,axiom,
! [B3: int,K2: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K2 @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_376_group__cancel_Oadd1,axiom,
! [A2: nat,K2: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_377_group__cancel_Oadd1,axiom,
! [A2: int,K2: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_378_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_379_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_int @ I @ K2 )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_380_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_381_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_382_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_383_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_384_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_385_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( minus_minus_int @ A4 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_386_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_387_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_388_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_389_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_390_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_391_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_392_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_393_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_394_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_395_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_396_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_397_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_398_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_399_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_400_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_401_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_402_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_403_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_404_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_405_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_406_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K2 = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_407_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_408_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_409_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_410_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_411_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_412_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_413_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_414_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_415_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_416_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_417_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_418_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_419_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_420_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_421_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_422_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_423_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_424_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_425_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_426_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_427_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_428_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_429_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_430_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_431_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_432_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_433_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_434_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_435_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_436_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_437_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_438_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_439_nat__arith_Osuc1,axiom,
! [A2: nat,K2: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_440_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_441_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_442_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_443_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_444_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_445_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_446_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_447_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_448_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_449_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_450_diff__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_451_diff__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_452_add__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_453_add__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_454_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_455_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_456_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_457_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_458_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_459_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_460_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_461_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_462_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_463_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_464_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_465_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_466_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_467_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_468_sum_OatLeastLessThan__rev,axiom,
! [G: nat > nat,N: nat,M: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N @ M ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ ( suc @ I3 ) ) )
@ ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_469_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N5: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N5 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% add_eq_if
thf(fact_470_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N5: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N5 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% mult_eq_if
thf(fact_471_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_472_sum_Oshift__bounds__nat__ivl,axiom,
! [G: nat > nat,M: nat,K2: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K2 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_473_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_474_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_475_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_476_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_477_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_478_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_479_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_480_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_481_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_482_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_483_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_484_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_485_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N5: nat] :
? [K: nat] :
( N5
= ( suc @ ( plus_plus_nat @ M4 @ K ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_486_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_487_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_488_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_489_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_490_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_491_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_492_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_493_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_494_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_495_sum__power__add,axiom,
! [X: int,M: nat,I4: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I3 ) )
@ I4 )
= ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I4 ) ) ) ).
% sum_power_add
thf(fact_496_sum_OatLeast0__lessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_497_sum_OatLeast0__lessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_498_sum_OatLeast__Suc__lessThan,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_nat @ M @ N )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_499_sum_OatLeast__Suc__lessThan,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_500_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_501_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_502_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_503_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_504_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_505_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_506_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_507_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_508_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_509_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_510_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_511_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_512_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_513_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_514_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_515_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_516_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_517_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_518_power__eq__if,axiom,
( power_power_nat
= ( ^ [P2: nat,M4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P2 @ ( power_power_nat @ P2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_519_power__eq__if,axiom,
( power_power_int
= ( ^ [P2: int,M4: nat] : ( if_int @ ( M4 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P2 @ ( power_power_int @ P2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_520_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_521_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_522_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_523_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_524_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_525_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_526_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_527_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_528_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_529_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_530_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_531_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_532_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_533_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_534_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_535_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_536_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_537_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_538_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_539_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_540_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_541_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_542_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_543_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_544_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_545_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_546_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_547_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_548_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_549_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_550_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_551_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_552_lambda__zero,axiom,
( ( ^ [H3: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_553_lambda__zero,axiom,
( ( ^ [H3: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_554_lambda__one,axiom,
( ( ^ [X3: nat] : X3 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_555_lambda__one,axiom,
( ( ^ [X3: int] : X3 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_556_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_557_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_558_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_559_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_560_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_561_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_562_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_563_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_564_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_565_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_566_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_567_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_568_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_569_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_570_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_571_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_572_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_573_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_574_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_575_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_576_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_577_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_578_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_579_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_580_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_581_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_582_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_583_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_584_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_585_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_586_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_587_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_588_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_589_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_590_sum_Oshift__bounds__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_591__092_060open_062_I_092_060Sum_062k_060c_O_Anth__digit_An_Ak_Ab_A_K_Ab_A_094_Ak_J_A_061_Aa_A_L_Ab_A_094_ASuc_Ar_A_K_Ae_092_060close_062,axiom,
( ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_digit @ n @ K @ b ) @ ( power_power_nat @ b @ K ) )
@ ( set_ord_lessThan_nat @ c ) )
= ( plus_plus_nat @ a @ ( times_times_nat @ ( power_power_nat @ b @ ( suc @ r ) ) @ e ) ) ) ).
% \<open>(\<Sum>k<c. nth_digit n k b * b ^ k) = a + b ^ Suc r * e\<close>
thf(fact_592_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_593_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_594_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M4: nat] :
( ( ord_less_nat @ M4 @ N )
& ( P @ M4 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_595_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( P @ M4 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_596_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_597_add__scale__eq__noteq,axiom,
! [R: int,A: int,B: int,C: int,D: int] :
( ( R != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_598_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_599_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_600_mult__less__iff1,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_601_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_602_lessThan__iff,axiom,
! [I: int,K2: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K2 ) )
= ( ord_less_int @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_603_lessThan__iff,axiom,
! [I: nat,K2: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K2 ) )
= ( ord_less_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_604_lessThan__minus__lessThan,axiom,
! [N: nat,M: nat] :
( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( set_ord_lessThan_nat @ M ) )
= ( set_or4665077453230672383an_nat @ M @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_605_lessThan__minus__lessThan,axiom,
! [N: int,M: int] :
( ( minus_minus_set_int @ ( set_ord_lessThan_int @ N ) @ ( set_ord_lessThan_int @ M ) )
= ( set_or4662586982721622107an_int @ M @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_606_sum_OlessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_607_sum_OlessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_608_lessThan__strict__subset__iff,axiom,
! [M: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_609_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_610_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X3: int] : ( ord_less_int @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_611_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_612_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_613_sum_Onat__diff__reindex,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_614_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_615_sum_OlessThan__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_616_sum_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_617_sum__lessThan__telescope_H,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N5: nat] : ( minus_minus_int @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_618_sum__lessThan__telescope,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N5: nat] : ( minus_minus_int @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_619_sum_Onat__group,axiom,
! [G: nat > nat,K2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [M4: nat] : ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M4 @ K2 ) @ ( plus_plus_nat @ ( times_times_nat @ M4 @ K2 ) @ K2 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N @ K2 ) ) ) ) ).
% sum.nat_group
thf(fact_620_one__diff__power__eq,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_621_power__diff__1__eq,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ ( power_power_int @ X @ N ) @ one_one_int )
= ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_622_diff__power__eq__sum,axiom,
! [X: int,N: nat,Y: int] :
( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N ) ) @ ( power_power_int @ Y @ ( suc @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [P2: nat] : ( times_times_int @ ( power_power_int @ X @ P2 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ P2 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_623_power__diff__sumr2,axiom,
! [X: int,N: nat,Y: int] :
( ( minus_minus_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_624_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_625_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_626_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_627_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_628_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_629_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_630_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_631_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_632_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_633_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_634_one__diff__power__eq_H,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_635_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( P @ A3 @ B2 )
= ( P @ B2 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B2: nat] :
( ( P @ A3 @ B2 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B2 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_636_inf__period_I1_J,axiom,
! [P: int > $o,D3: int,Q: int > $o] :
( ! [X4: int,K3: int] :
( ( P @ X4 )
= ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ! [X4: int,K3: int] :
( ( Q @ X4 )
= ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
& ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_637_inf__period_I2_J,axiom,
! [P: int > $o,D3: int,Q: int > $o] :
( ! [X4: int,K3: int] :
( ( P @ X4 )
= ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ! [X4: int,K3: int] :
( ( Q @ X4 )
= ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
| ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_638_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_639_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_640_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_641_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_642_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_643_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_644_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_645_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_646_minf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_647_minf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_648_minf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_649_minf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_650_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_651_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_652_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_653_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_654_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_655_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_656_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_657_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_658_pinf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_659_pinf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_660_pinf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_661_pinf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_662_Diff__iff,axiom,
! [C: int,A2: set_int,B3: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
= ( ( member_int @ C @ A2 )
& ~ ( member_int @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_663_DiffI,axiom,
! [C: int,A2: set_int,B3: set_int] :
( ( member_int @ C @ A2 )
=> ( ~ ( member_int @ C @ B3 )
=> ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_664_psubset__imp__ex__mem,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_set_int @ A2 @ B3 )
=> ? [B2: int] : ( member_int @ B2 @ ( minus_minus_set_int @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_665_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_666_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_667_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_668_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_669_DiffE,axiom,
! [C: int,A2: set_int,B3: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
=> ~ ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B3 ) ) ) ).
% DiffE
thf(fact_670_DiffD1,axiom,
! [C: int,A2: set_int,B3: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
=> ( member_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_671_DiffD2,axiom,
! [C: int,A2: set_int,B3: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
=> ~ ( member_int @ C @ B3 ) ) ).
% DiffD2
thf(fact_672_psubsetD,axiom,
! [A2: set_int,B3: set_int,C: int] :
( ( ord_less_set_int @ A2 @ B3 )
=> ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_673_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A5 )
& ~ ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_674_set__diff__eq,axiom,
( minus_minus_set_int
= ( ^ [A5: set_int,B5: set_int] :
( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A5 )
& ~ ( member_int @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_675_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A5 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_676_minus__set__def,axiom,
( minus_minus_set_int
= ( ^ [A5: set_int,B5: set_int] :
( collect_int
@ ( minus_minus_int_o
@ ^ [X3: int] : ( member_int @ X3 @ A5 )
@ ^ [X3: int] : ( member_int @ X3 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_677_less__set__def,axiom,
( ord_less_set_int
= ( ^ [A5: set_int,B5: set_int] :
( ord_less_int_o
@ ^ [X3: int] : ( member_int @ X3 @ A5 )
@ ^ [X3: int] : ( member_int @ X3 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_678_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_679_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_680_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_681_finite__lessThan,axiom,
! [K2: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K2 ) ) ).
% finite_lessThan
thf(fact_682_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_683_sum_Oinfinite,axiom,
! [A2: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_684_sum_Oinfinite,axiom,
! [A2: set_nat,G: nat > int] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G @ A2 )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_685_sum_Oinfinite,axiom,
! [A2: set_int,G: int > int] :
( ~ ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G @ A2 )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_686_sum_Oinfinite,axiom,
! [A2: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_687_sum__eq__0__iff,axiom,
! [F2: set_int,F: int > nat] :
( ( finite_finite_int @ F2 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X3: int] :
( ( member_int @ X3 @ F2 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_688_sum__eq__0__iff,axiom,
! [F2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ F2 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_689_sum_Odelta,axiom,
! [S2: set_int,A: int,B: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K: int] : ( if_nat @ ( K = A ) @ ( B @ K ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K: int] : ( if_nat @ ( K = A ) @ ( B @ K ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_690_sum_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K: nat] : ( if_int @ ( K = A ) @ ( B @ K ) @ zero_zero_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K: nat] : ( if_int @ ( K = A ) @ ( B @ K ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_691_sum_Odelta,axiom,
! [S2: set_int,A: int,B: int > int] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [K: int] : ( if_int @ ( K = A ) @ ( B @ K ) @ zero_zero_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [K: int] : ( if_int @ ( K = A ) @ ( B @ K ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_692_sum_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( if_nat @ ( K = A ) @ ( B @ K ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( if_nat @ ( K = A ) @ ( B @ K ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_693_sum_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K: int] : ( if_nat @ ( A = K ) @ ( B @ K ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K: int] : ( if_nat @ ( A = K ) @ ( B @ K ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_694_sum_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K: nat] : ( if_int @ ( A = K ) @ ( B @ K ) @ zero_zero_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K: nat] : ( if_int @ ( A = K ) @ ( B @ K ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_695_sum_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > int] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [K: int] : ( if_int @ ( A = K ) @ ( B @ K ) @ zero_zero_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [K: int] : ( if_int @ ( A = K ) @ ( B @ K ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_696_sum_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( if_nat @ ( A = K ) @ ( B @ K ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( if_nat @ ( A = K ) @ ( B @ K ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_697_infinite__Iio,axiom,
! [A: int] :
~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).
% infinite_Iio
thf(fact_698_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ N4 )
=> ( ord_less_nat @ X4 @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_699_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ord_less_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_700_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( P @ K )
& ( ord_less_nat @ K @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_701_sum_Oswap__restrict,axiom,
! [A2: set_int,B3: set_nat,G: int > nat > nat,R2: int > nat > $o] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X3: int] :
( groups3542108847815614940at_nat @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B3 )
& ( R2 @ X3 @ Y5 ) ) ) )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [Y5: nat] :
( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( G @ X3 @ Y5 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( R2 @ X3 @ Y5 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_702_sum_Oswap__restrict,axiom,
! [A2: set_nat,B3: set_int,G: nat > int > nat,R2: nat > int > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] :
( groups4541462559716669496nt_nat @ ( G @ X3 )
@ ( collect_int
@ ^ [Y5: int] :
( ( member_int @ Y5 @ B3 )
& ( R2 @ X3 @ Y5 ) ) ) )
@ A2 )
= ( groups4541462559716669496nt_nat
@ ^ [Y5: int] :
( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( G @ X3 @ Y5 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( R2 @ X3 @ Y5 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_703_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
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B3 )
& ( R2 @ X3 @ Y5 ) ) ) )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [Y5: nat] :
( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( G @ X3 @ Y5 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( R2 @ X3 @ Y5 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_704_sum_Ofinite__Collect__op,axiom,
! [I4: set_nat,X: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( plus_plus_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_705_sum_Ofinite__Collect__op,axiom,
! [I4: set_int,X: int > nat,Y: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( plus_plus_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_706_sum_Ofinite__Collect__op,axiom,
! [I4: set_nat,X: nat > int,Y: nat > int] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_zero_int ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( plus_plus_int @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_707_sum_Ofinite__Collect__op,axiom,
! [I4: set_int,X: int > int,Y: int > int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_zero_int ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( plus_plus_int @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_708_prod_Ofinite__Collect__op,axiom,
! [I4: set_nat,X: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( times_times_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_nat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_709_prod_Ofinite__Collect__op,axiom,
! [I4: set_int,X: int > nat,Y: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_nat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_nat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( times_times_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_nat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_710_prod_Ofinite__Collect__op,axiom,
! [I4: set_nat,X: nat > int,Y: nat > int] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_int ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_int ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( times_times_int @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_int ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_711_prod_Ofinite__Collect__op,axiom,
! [I4: set_int,X: int > int,Y: int > int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_int ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_int ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( times_times_int @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_int ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_712_sum_Ointer__filter,axiom,
! [A2: set_int,G: int > nat,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( if_nat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_713_sum_Ointer__filter,axiom,
! [A2: set_nat,G: nat > int,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( if_int @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_int )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_714_sum_Ointer__filter,axiom,
! [A2: set_int,G: int > int,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( if_int @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_int )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_715_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_716_sum_Orelated,axiom,
! [R2: nat > nat > $o,S2: set_int,H: int > nat,G: int > 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_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R2 @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R2 @ ( groups4541462559716669496nt_nat @ H @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_717_sum_Orelated,axiom,
! [R2: int > int > $o,S2: set_nat,H: nat > int,G: nat > int] :
( ( R2 @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X22: int,Y22: int] :
( ( ( R2 @ X1 @ X22 )
& ( R2 @ Y1 @ Y22 ) )
=> ( R2 @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R2 @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R2 @ ( groups3539618377306564664at_int @ H @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_718_sum_Orelated,axiom,
! [R2: int > int > $o,S2: set_int,H: int > int,G: int > int] :
( ( R2 @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X22: int,Y22: int] :
( ( ( R2 @ X1 @ X22 )
& ( R2 @ Y1 @ Y22 ) )
=> ( R2 @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R2 @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R2 @ ( groups4538972089207619220nt_int @ H @ S2 ) @ ( groups4538972089207619220nt_int @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_719_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_720_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_int,T3: set_int,S2: set_int,I: int > int,J: int > int,T2: set_int,G: int > nat,H: int > nat] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T2 @ T3 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( member_int @ ( I @ B2 ) @ ( minus_minus_set_int @ S2 @ S3 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S3 )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T3 )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S2 )
= ( groups4541462559716669496nt_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_721_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_nat,T3: set_nat,S2: set_nat,I: nat > nat,J: nat > nat,T2: set_nat,G: nat > int,H: nat > int] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T2 @ T3 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( member_nat @ ( I @ B2 ) @ ( minus_minus_set_nat @ S2 @ S3 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S3 )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T3 )
=> ( ( H @ B2 )
= zero_zero_int ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S2 )
= ( groups3539618377306564664at_int @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_722_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_nat,T3: set_int,S2: set_nat,I: int > nat,J: nat > int,T2: set_int,G: nat > int,H: int > int] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T2 @ T3 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( member_nat @ ( I @ B2 ) @ ( minus_minus_set_nat @ S2 @ S3 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S3 )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T3 )
=> ( ( H @ B2 )
= zero_zero_int ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S2 )
= ( groups4538972089207619220nt_int @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_723_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_int,T3: set_nat,S2: set_int,I: nat > int,J: int > nat,T2: set_nat,G: int > int,H: nat > int] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T2 @ T3 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( member_int @ ( I @ B2 ) @ ( minus_minus_set_int @ S2 @ S3 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S3 )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T3 )
=> ( ( H @ B2 )
= zero_zero_int ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4538972089207619220nt_int @ G @ S2 )
= ( groups3539618377306564664at_int @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_724_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_int,T3: set_int,S2: set_int,I: int > int,J: int > int,T2: set_int,G: int > int,H: int > int] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T2 @ T3 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( member_int @ ( I @ B2 ) @ ( minus_minus_set_int @ S2 @ S3 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S3 )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T3 )
=> ( ( H @ B2 )
= zero_zero_int ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4538972089207619220nt_int @ G @ S2 )
= ( groups4538972089207619220nt_int @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_725_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_int,T3: set_nat,S2: set_int,I: nat > int,J: int > nat,T2: set_nat,G: int > nat,H: nat > nat] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T2 @ T3 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( member_int @ ( I @ B2 ) @ ( minus_minus_set_int @ S2 @ S3 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S3 )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T3 )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_726_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_nat,T3: set_int,S2: set_nat,I: int > nat,J: nat > int,T2: set_int,G: nat > nat,H: int > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T2 @ T3 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( member_nat @ ( I @ B2 ) @ ( minus_minus_set_nat @ S2 @ S3 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S3 )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T3 )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups4541462559716669496nt_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_727_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_nat,T3: set_nat,S2: set_nat,I: nat > nat,J: nat > nat,T2: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T2 @ T3 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( member_nat @ ( I @ B2 ) @ ( minus_minus_set_nat @ S2 @ S3 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S3 )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T3 )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_728_sum__eq__Suc0__iff,axiom,
! [A2: set_int,F: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ( F @ X3 )
= ( suc @ zero_zero_nat ) )
& ! [Y5: int] :
( ( member_int @ Y5 @ A2 )
=> ( ( X3 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_729_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 ) )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ A2 )
=> ( ( X3 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_730_sum__eq__1__iff,axiom,
! [A2: set_int,F: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
= one_one_nat )
= ( ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y5: int] :
( ( member_int @ Y5 @ A2 )
=> ( ( X3 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_731_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 )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ A2 )
=> ( ( X3 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_732_sum_Osetdiff__irrelevant,axiom,
! [A2: set_int,G: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X3: int] :
( ( G @ X3 )
= zero_zero_nat ) ) ) )
= ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_733_sum_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( G @ X3 )
= zero_zero_int ) ) ) )
= ( groups3539618377306564664at_int @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_734_sum_Osetdiff__irrelevant,axiom,
! [A2: set_int,G: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X3: int] :
( ( G @ X3 )
= zero_zero_int ) ) ) )
= ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_735_sum_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( G @ X3 )
= zero_zero_nat ) ) ) )
= ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_736_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_737_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N5: nat] : ( ord_less_nat @ N5 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_738_finite__Diff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% finite_Diff
thf(fact_739_finite__Diff,axiom,
! [A2: set_int,B3: set_int] :
( ( finite_finite_int @ A2 )
=> ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ).
% finite_Diff
thf(fact_740_finite__Diff2,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_741_finite__Diff2,axiom,
! [B3: set_int,A2: set_int] :
( ( finite_finite_int @ B3 )
=> ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) )
= ( finite_finite_int @ A2 ) ) ) ).
% finite_Diff2
thf(fact_742_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_743_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_744_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_745_finite__Collect__conjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_746_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_747_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_1: int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_748_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
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_749_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B3: set_int,R2: nat > int > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_750_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B3: set_nat,R2: int > nat > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A2 )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_751_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B3: set_int,R2: int > int > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B3 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A2 )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_752_Diff__infinite__finite,axiom,
! [T2: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_753_Diff__infinite__finite,axiom,
! [T2: set_int,S2: set_int] :
( ( finite_finite_int @ T2 )
=> ( ~ ( finite_finite_int @ S2 )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_754_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B6: set_nat] :
( ( ord_less_set_nat @ B6 @ A6 )
=> ( P @ B6 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_755_finite__psubset__induct,axiom,
! [A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ A2 )
=> ( ! [A6: set_int] :
( ( finite_finite_int @ A6 )
=> ( ! [B6: set_int] :
( ( ord_less_set_int @ B6 @ A6 )
=> ( P @ B6 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_756_sum__gp__basic,axiom,
! [X: int,N: nat] :
( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_757_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_758_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_759_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_760_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_761_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_762_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_763_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_764_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_765_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_766_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_767_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_768_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_769_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_770_finite__atMost,axiom,
! [K2: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K2 ) ) ).
% finite_atMost
thf(fact_771_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_772_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_773_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_774_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_775_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_776_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_777_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_778_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_779_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_780_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_781_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_782_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_783_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_784_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_785_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_786_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_787_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_788_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_789_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_790_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_791_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_792_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_793_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_794_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_795_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_796_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_797_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_798_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_799_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_800_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_801_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_802_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( semiri1316708129612266289at_nat @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_803_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_804_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_805_sum_OatMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_806_sum_OatMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_807_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_808_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_809_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_810_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_811_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_812_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_813_infinite__Iic,axiom,
! [A: int] :
~ ( finite_finite_int @ ( set_ord_atMost_int @ A ) ) ).
% infinite_Iic
thf(fact_814_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_815_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_816_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_817_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_818_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_819_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_820_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_821_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_822_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_823_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_824_not__Iic__eq__Icc,axiom,
! [H2: int,L: int,H: int] :
( ( set_ord_atMost_int @ H2 )
!= ( set_or1266510415728281911st_int @ L @ H ) ) ).
% not_Iic_eq_Icc
thf(fact_825_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_826_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z5: int] :
? [N5: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N5 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_827_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_828_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_829_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_830_sum_Onested__swap_H,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nested_swap'
thf(fact_831_int__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A2 ) ) ).
% int_sum
thf(fact_832_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_833_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_834_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_835_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_836_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_837_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_838_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_839_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_840_lessThan__Suc__atMost,axiom,
! [K2: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
= ( set_ord_atMost_nat @ K2 ) ) ).
% lessThan_Suc_atMost
thf(fact_841_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K2: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_842_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_843_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_844_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_845_nat__mult__div__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( K2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= zero_zero_nat ) )
& ( ( K2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_846_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
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_847_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M: nat,K2: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K2 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_848_nat__mult__div__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_849_sum_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N: nat,M: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_850_sum__shift__lb__Suc0__0,axiom,
! [F: nat > int,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_851_sum__shift__lb__Suc0__0,axiom,
! [F: nat > nat,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_852_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_853_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_854_sum_OatMost__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_855_sum_OatMost__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_856_sum__telescope,axiom,
! [F: nat > int,I: nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% sum_telescope
thf(fact_857_sum_Onested__swap,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% sum.nested_swap
thf(fact_858_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( G @ ( suc @ K ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_859_sum_Ohead__if,axiom,
! [N: nat,M: nat,G: nat > int] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.head_if
thf(fact_860_sum_Ohead__if,axiom,
! [N: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.head_if
thf(fact_861_sum_OatMost__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_862_sum_OatMost__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_863_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [G: nat > nat,N: nat,M: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N @ M ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_864_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_865_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_866_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_867_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_868_finite__interval__int4,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_int @ A @ I3 )
& ( ord_less_int @ I3 @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_869_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_870_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_871_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_872_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_873_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_874_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_875_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_876_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_877_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_878_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_879_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_880_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_881_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_882_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_883_int__ops_I8_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_884_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_885_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_886_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_887_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_888_aset_I7_J,axiom,
! [D3: int,A2: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% aset(7)
thf(fact_889_aset_I5_J,axiom,
! [D3: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ T @ A2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% aset(5)
thf(fact_890_aset_I4_J,axiom,
! [D3: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ T @ A2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( X5 != T )
=> ( ( plus_plus_int @ X5 @ D3 )
!= T ) ) ) ) ) ).
% aset(4)
thf(fact_891_aset_I3_J,axiom,
! [D3: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( X5 = T )
=> ( ( plus_plus_int @ X5 @ D3 )
= T ) ) ) ) ) ).
% aset(3)
thf(fact_892_aset_I2_J,axiom,
! [D3: int,A2: set_int,P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb2 @ Xa ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
| ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_893_aset_I1_J,axiom,
! [D3: int,A2: set_int,P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb2 @ Xa ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
& ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_894_bset_I7_J,axiom,
! [D3: int,T: int,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ T @ B3 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% bset(7)
thf(fact_895_bset_I5_J,axiom,
! [D3: int,B3: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% bset(5)
thf(fact_896_bset_I4_J,axiom,
! [D3: int,T: int,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ T @ B3 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( X5 != T )
=> ( ( minus_minus_int @ X5 @ D3 )
!= T ) ) ) ) ) ).
% bset(4)
thf(fact_897_bset_I3_J,axiom,
! [D3: int,T: int,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( X5 = T )
=> ( ( minus_minus_int @ X5 @ D3 )
= T ) ) ) ) ) ).
% bset(3)
thf(fact_898_bset_I2_J,axiom,
! [D3: int,B3: set_int,P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb2 @ Xa ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
| ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_899_bset_I1_J,axiom,
! [D3: int,B3: set_int,P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb2 @ Xa ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
& ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_900_cpmi,axiom,
! [D3: int,P: int > $o,P3: int > $o,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
=> ( ! [X4: int,K3: int] :
( ( P3 @ X4 )
= ( P3 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ( P3 @ X3 ) )
| ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ? [Y5: int] :
( ( member_int @ Y5 @ B3 )
& ( P @ ( plus_plus_int @ Y5 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_901_cppi,axiom,
! [D3: int,P: int > $o,P3: int > $o,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
=> ( ! [X4: int,K3: int] :
( ( P3 @ X4 )
= ( P3 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ( P3 @ X3 ) )
| ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ? [Y5: int] :
( ( member_int @ Y5 @ A2 )
& ( P @ ( minus_minus_int @ Y5 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_902_periodic__finite__ex,axiom,
! [D: int,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K3: int] :
( ( P @ X4 )
= ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
& ( P @ X3 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_903_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M2: nat,N2: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_904_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_905_div__mult2__eq,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q4 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q4 ) ) ).
% div_mult2_eq
thf(fact_906_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_907_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_908_div__less__iff__less__mult,axiom,
! [Q4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q4 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_909_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_910_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_911_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_912_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_913_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_914_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_915_int__power__div__base,axiom,
! [M: nat,K2: int] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( divide_divide_int @ ( power_power_int @ K2 @ M ) @ K2 )
= ( power_power_int @ K2 @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_916_div__if,axiom,
( divide_divide_nat
= ( ^ [M4: nat,N5: nat] :
( if_nat
@ ( ( ord_less_nat @ M4 @ N5 )
| ( N5 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M4 @ N5 ) @ N5 ) ) ) ) ) ).
% div_if
thf(fact_917_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_918_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_919_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 )
=> ! [I3: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_div
thf(fact_920_sum__bounds__lt__plus1,axiom,
! [F: nat > nat,Mm: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( F @ ( suc @ K ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_921_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_922_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_923_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_924_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_925_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_926_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_927_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).
% finite_atLeastAtMost_int
thf(fact_928_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_929_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N5: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
@ N5
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_930_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N5: nat] :
( semiri8420488043553186161ux_int
@ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
@ N5
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_931_div__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% div_mult2_eq'
thf(fact_932_div__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% div_mult2_eq'
thf(fact_933_gbinomial__0_I2_J,axiom,
! [K2: nat] :
( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K2 ) )
= zero_zero_nat ) ).
% gbinomial_0(2)
thf(fact_934_gbinomial__0_I2_J,axiom,
! [K2: nat] :
( ( gbinomial_int @ zero_zero_int @ ( suc @ K2 ) )
= zero_zero_int ) ).
% gbinomial_0(2)
thf(fact_935_gbinomial__0_I1_J,axiom,
! [A: nat] :
( ( gbinomial_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% gbinomial_0(1)
thf(fact_936_gbinomial__0_I1_J,axiom,
! [A: int] :
( ( gbinomial_int @ A @ zero_zero_nat )
= one_one_int ) ).
% gbinomial_0(1)
thf(fact_937_sum_Ozero__middle,axiom,
! [P4: nat,K2: nat,G: nat > int,H: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K2 @ P4 )
=> ( ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K2 ) @ zero_zero_int @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_938_sum_Ozero__middle,axiom,
! [P4: nat,K2: nat,G: nat > nat,H: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K2 @ P4 )
=> ( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K2 ) @ zero_zero_nat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_939_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_940_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_941_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_942_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_943_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_944_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_945_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_946_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_947_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_948_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_949_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_950_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_951_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_952_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_953_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_954_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_955_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_956_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_957_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_958_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_959_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_960_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_961_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_962_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_963_Icc__eq__Icc,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or1266510415728281911st_int @ L @ H )
= ( set_or1266510415728281911st_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_int @ L @ H )
& ~ ( ord_less_eq_int @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_964_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_965_atLeastAtMost__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_eq_int @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_966_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_967_atLeastatMost__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_968_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_969_ivl__subset,axiom,
! [I: int,J: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J @ I )
| ( ( ord_less_eq_int @ M @ I )
& ( ord_less_eq_int @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_970_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_971_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_972_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_973_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_974_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_975_atMost__iff,axiom,
! [I: int,K2: int] :
( ( member_int @ I @ ( set_ord_atMost_int @ K2 ) )
= ( ord_less_eq_int @ I @ K2 ) ) ).
% atMost_iff
thf(fact_976_atMost__iff,axiom,
! [I: nat,K2: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K2 ) )
= ( ord_less_eq_nat @ I @ K2 ) ) ).
% atMost_iff
thf(fact_977_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_978_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_979_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_980_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N5: nat] : ( ord_less_eq_nat @ N5 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_981_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_982_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_983_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_984_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_985_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_986_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_987_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_988_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_989_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_990_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_991_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_992_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_993_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_994_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_995_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_996_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_997_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_998_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_999_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_1000_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_1001_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_1002_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_1003_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_1004_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_1005_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_1006_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_1007_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1008_mult__minus1__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1_right
thf(fact_1009_mult__minus1,axiom,
! [Z2: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1
thf(fact_1010_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_1011_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_1012_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1013_atLeastLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1014_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_1015_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_1016_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_1017_ivl__diff,axiom,
! [I: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1018_ivl__diff,axiom,
! [I: int,N: int,M: int] :
( ( ord_less_eq_int @ I @ N )
=> ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N ) )
= ( set_or4662586982721622107an_int @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1019_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_1020_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_1021_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1022_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1023_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1024_Icc__subset__Iic__iff,axiom,
! [L: nat,H: nat,H2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H2 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_1025_Icc__subset__Iic__iff,axiom,
! [L: int,H: int,H2: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H ) @ ( set_ord_atMost_int @ H2 ) )
= ( ~ ( ord_less_eq_int @ L @ H )
| ( ord_less_eq_int @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_1026_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1027_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_1028_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_1029_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_1030_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_1031_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1032_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1033_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_1034_left__minus__one__mult__self,axiom,
! [N: nat,A: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_1035_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_1036_mult__le__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1037_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1038_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1039_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1040_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_1041_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1042_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_1043_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1044_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_1045_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1046_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1047_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1048_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1049_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1050_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_1051_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K2 ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1052_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_1053_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_1054_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_1055_int__cases,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_1056_sum__mono__inv,axiom,
! [F: int > nat,I4: set_int,G: int > nat,I: int] :
( ( ( groups4541462559716669496nt_nat @ F @ I4 )
= ( groups4541462559716669496nt_nat @ G @ I4 ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_int @ I @ I4 )
=> ( ( finite_finite_int @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1057_sum__mono__inv,axiom,
! [F: nat > nat,I4: set_nat,G: nat > nat,I: nat] :
( ( ( groups3542108847815614940at_nat @ F @ I4 )
= ( groups3542108847815614940at_nat @ G @ I4 ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_nat @ I @ I4 )
=> ( ( finite_finite_nat @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1058_int__cases2,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_1059_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_1060_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_1061_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A7: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A7 ) )
= ( ord_less_int @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1062_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1063_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1064_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1065_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1066_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1067_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1068_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1069_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1070_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1071_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_1072_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_1073_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_1074_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
| ( M4 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1075_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_1076_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
& ( M4 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_1077_obtain__smallest,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ( P @ K3 )
& ! [A8: nat] :
( ( ord_less_nat @ A8 @ K3 )
=> ~ ( P @ A8 ) ) ) ) ).
% obtain_smallest
thf(fact_1078_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_1079_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_1080_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_1081_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_1082_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1083_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1084_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1085_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1086_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1087_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1088_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1089_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_1090_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_1091_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_1092_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_1093_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_1094_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1095_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1096_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1097_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_1098_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1099_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1100_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1101_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_1102_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_1103_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N5: nat] :
? [K: nat] :
( N5
= ( plus_plus_nat @ M4 @ K ) ) ) ) ).
% nat_le_iff_add
thf(fact_1104_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1105_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1106_mult__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1107_mult__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1108_mult__le__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_1109_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1110_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_1111_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_1112_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1113_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1114_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_1115_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1116_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_1117_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_1118_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1119_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_1120_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_1121_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1122_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_1123_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1124_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_1125_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1126_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1127_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1128_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1129_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1130_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1131_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1132_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1133_group__cancel_Oneg1,axiom,
! [A2: int,K2: int,A: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_1134_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_1135_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_1136_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1137_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_1138_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_1139_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1140_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1141_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1142_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1143_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_1144_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_1145_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_1146_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_1147_sum__mono,axiom,
! [K5: set_int,F: int > nat,G: int > nat] :
( ! [I2: int] :
( ( member_int @ I2 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_1148_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_1149_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X3: int] : ( ord_less_eq_int @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_1150_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_1151_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1152_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1153_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_1154_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1155_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_1156_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1157_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M2: nat] :
( M6
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_1158_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_1159_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1160_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_1161_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1162_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1163_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,Z3: nat] :
( ( R2 @ X4 @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X4 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1164_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1165_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_1166_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1167_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_1168_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_1169_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1170_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1171_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1172_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_1173_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_1174_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_1175_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_1176_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_1177_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M7 ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1178_sum__nonpos,axiom,
! [A2: set_int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_1179_sum__nonpos,axiom,
! [A2: set_int,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_1180_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_1181_sum__nonneg,axiom,
! [A2: set_int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_1182_sum__nonneg,axiom,
! [A2: set_int,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_1183_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_1184_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1185_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1186_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1187_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_1188_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1189_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_1190_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_1191_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_1192_add__le__imp__le__diff,axiom,
! [I: int,K2: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_1193_add__le__imp__le__diff,axiom,
! [I: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_1194_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1195_add__le__add__imp__diff__le,axiom,
! [I: int,K2: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1196_add__le__add__imp__diff__le,axiom,
! [I: nat,K2: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1197_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1198_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1199_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1200_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1201_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1202_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1203_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1204_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1205_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1206_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1207_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1208_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1209_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1210_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1211_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1212_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_1213_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1214_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_1215_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1216_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_1217_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1218_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_1219_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1220_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1221_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1222_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_1223_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1224_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_1225_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1226_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_1227_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1228_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_1229_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1230_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_1231_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1232_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_1233_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1234_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_1235_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1236_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_1237_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1238_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_1239_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1240_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1241_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_1242_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1243_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_1244_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1245_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_1246_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1247_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_1248_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_1249_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_1250_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_1251_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1252_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_1253_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1254_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1255_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_1256_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1257_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1258_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_1259_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_1260_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1261_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_1262_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1263_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_1264_group__cancel_Osub2,axiom,
! [B3: int,K2: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K2 @ B ) )
=> ( ( minus_minus_int @ A @ B3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_1265_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% diff_conv_add_uminus
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_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,
( ord_less_nat
@ ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( times_times_nat @ ( bits_nth_digit @ n @ K @ b ) @ ( power_power_nat @ b @ K ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ r ) )
@ ( power_power_nat @ b @ r ) ) ).
%------------------------------------------------------------------------------