TPTP Problem File: SLH0408^1.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 : Query_Optimization/0004_Selectivities/prob_00076_002627__14903942_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1389 ( 569 unt; 120 typ; 0 def)
% Number of atoms : 3350 (1204 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10155 ( 378 ~; 73 |; 150 &;8049 @)
% ( 0 <=>;1505 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 393 ( 393 >; 0 *; 0 +; 0 <<)
% Number of symbols : 106 ( 103 usr; 23 con; 0-3 aty)
% Number of variables : 3480 ( 141 ^;3209 !; 130 ?;3480 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:00:51.473
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Real__Oreal_J,type,
multiset_real: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Num__Onum_J,type,
multiset_num: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
multiset_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (103)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
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__Multiset__Omultiset_Itf__a_J,type,
minus_3765977307040488491iset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
zero_z7348594199698428585et_nat: multiset_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Num__Onum_J,type,
zero_z8056838136647266291et_num: multiset_num ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Real__Oreal_J,type,
zero_z8811559133707751557t_real: multiset_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
zero_zero_multiset_a: multiset_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
if_list_a: $o > list_a > list_a > list_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Omember_001tf__a,type,
member_a: list_a > a > $o ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Onths_001tf__a,type,
nths_a: list_a > set_nat > list_a ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Oremdups__adj_001tf__a,type,
remdups_adj_a: list_a > list_a ).
thf(sy_c_List_Oremove1_001tf__a,type,
remove1_a: a > list_a > list_a ).
thf(sy_c_List_Oreplicate_001tf__a,type,
replicate_a: nat > a > list_a ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Orotate_001tf__a,type,
rotate_a: nat > list_a > list_a ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_Multiset_Oadd__mset_001t__Nat__Onat,type,
add_mset_nat: nat > multiset_nat > multiset_nat ).
thf(sy_c_Multiset_Oadd__mset_001t__Num__Onum,type,
add_mset_num: num > multiset_num > multiset_num ).
thf(sy_c_Multiset_Oadd__mset_001t__Real__Oreal,type,
add_mset_real: real > multiset_real > multiset_real ).
thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
add_mset_a: a > multiset_a > multiset_a ).
thf(sy_c_Multiset_Omset_001tf__a,type,
mset_a: list_a > multiset_a ).
thf(sy_c_Multiset_Oreplicate__mset_001tf__a,type,
replicate_mset_a: nat > a > multiset_a ).
thf(sy_c_Multiset_Oset__mset_001t__Nat__Onat,type,
set_mset_nat: multiset_nat > set_nat ).
thf(sy_c_Multiset_Oset__mset_001t__Num__Onum,type,
set_mset_num: multiset_num > set_num ).
thf(sy_c_Multiset_Oset__mset_001t__Real__Oreal,type,
set_mset_real: multiset_real > set_real ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
size_size_multiset_a: multiset_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Selectivities_Olist__sel__aux_001tf__a,type,
list_sel_aux_a: ( a > a > real ) > a > list_a > real ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
comm_s629917340098488124ar_nat: char > nat ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__a,type,
member_a2: a > set_a > $o ).
thf(sy_v_f,type,
f: a > a > real ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y_H____,type,
y: a ).
thf(sy_v_ya____,type,
ya: list_a ).
thf(sy_v_ys____,type,
ys: list_a ).
thf(sy_v_z_H____,type,
z: a ).
thf(sy_v_za____,type,
za: list_a ).
thf(sy_v_zs____,type,
zs: list_a ).
% Relevant facts (1263)
thf(fact_0__092_060open_062y_H_A_092_060in_062_D_Amset_Ay_092_060close_062,axiom,
member_a2 @ y @ ( set_mset_a @ ( mset_a @ ya ) ) ).
% \<open>y' \<in># mset y\<close>
thf(fact_1__092_060open_062z_H_A_092_060in_062_D_Amset_Ay_092_060close_062,axiom,
member_a2 @ z @ ( set_mset_a @ ( mset_a @ ya ) ) ).
% \<open>z' \<in># mset y\<close>
thf(fact_2_False,axiom,
y != z ).
% False
thf(fact_3_Suc_Oprems,axiom,
( ( mset_a @ ya )
= ( mset_a @ za ) ) ).
% Suc.prems
thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062y_H_Ays_O_Ay_A_061_Ay_H_A_D_Ays_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Y: a,Ys: list_a] :
( ya
!= ( cons_a @ Y @ Ys ) ) ).
% \<open>\<And>thesis. (\<And>y' ys. y = y' # ys \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062z_H_Azs_O_Az_A_061_Az_H_A_D_Azs_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Z: a,Zs: list_a] :
( za
!= ( cons_a @ Z @ Zs ) ) ).
% \<open>\<And>thesis. (\<And>z' zs. z = z' # zs \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_y__def,axiom,
( ya
= ( cons_a @ y @ ys ) ) ).
% y_def
thf(fact_7_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_8_z__def,axiom,
( za
= ( cons_a @ z @ zs ) ) ).
% z_def
thf(fact_9_ex__mset,axiom,
! [X: multiset_a] :
? [Xs: list_a] :
( ( mset_a @ Xs )
= X ) ).
% ex_mset
thf(fact_10_not__Cons__self2,axiom,
! [X2: a,Xs2: list_a] :
( ( cons_a @ X2 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_11_Suc_Ohyps_I2_J,axiom,
( ( suc @ n )
= ( size_size_list_a @ ya ) ) ).
% Suc.hyps(2)
thf(fact_12_mset_Osimps_I2_J,axiom,
! [A: a,X2: list_a] :
( ( mset_a @ ( cons_a @ A @ X2 ) )
= ( add_mset_a @ A @ ( mset_a @ X2 ) ) ) ).
% mset.simps(2)
thf(fact_13_member__rec_I1_J,axiom,
! [X2: a,Xs2: list_a,Y2: a] :
( ( member_a @ ( cons_a @ X2 @ Xs2 ) @ Y2 )
= ( ( X2 = Y2 )
| ( member_a @ Xs2 @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_14_ShiftD,axiom,
! [Kl: list_a,Kl2: set_list_a,K: a] :
( ( member_list_a @ Kl @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) )
=> ( member_list_a @ ( cons_a @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_15__092_060open_062length_Azs_A_061_An_092_060close_062,axiom,
( ( size_size_list_a @ zs )
= n ) ).
% \<open>length zs = n\<close>
thf(fact_16_mset__add,axiom,
! [A: a,A2: multiset_a] :
( ( member_a2 @ A @ ( set_mset_a @ A2 ) )
=> ~ ! [B: multiset_a] :
( A2
!= ( add_mset_a @ A @ B ) ) ) ).
% mset_add
thf(fact_17_Suc__length__conv,axiom,
! [N: nat,Xs2: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs2 ) )
= ( ? [Y3: a,Ys2: list_a] :
( ( Xs2
= ( cons_a @ Y3 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_18_length__Suc__conv,axiom,
! [Xs2: list_a,N: nat] :
( ( ( size_size_list_a @ Xs2 )
= ( suc @ N ) )
= ( ? [Y3: a,Ys2: list_a] :
( ( Xs2
= ( cons_a @ Y3 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_19_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_a] :
( ( size_size_list_a @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_20_neq__if__length__neq,axiom,
! [Xs2: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs2 )
!= ( size_size_list_a @ Ys3 ) )
=> ( Xs2 != Ys3 ) ) ).
% neq_if_length_neq
thf(fact_21_multi__member__split,axiom,
! [X2: a,M: multiset_a] :
( ( member_a2 @ X2 @ ( set_mset_a @ M ) )
=> ? [A3: multiset_a] :
( M
= ( add_mset_a @ X2 @ A3 ) ) ) ).
% multi_member_split
thf(fact_22_insert__noteq__member,axiom,
! [B2: a,B3: multiset_a,C: a,C2: multiset_a] :
( ( ( add_mset_a @ B2 @ B3 )
= ( add_mset_a @ C @ C2 ) )
=> ( ( B2 != C )
=> ( member_a2 @ C @ ( set_mset_a @ B3 ) ) ) ) ).
% insert_noteq_member
thf(fact_23_union__single__eq__member,axiom,
! [X2: a,M: multiset_a,N2: multiset_a] :
( ( ( add_mset_a @ X2 @ M )
= N2 )
=> ( member_a2 @ X2 @ ( set_mset_a @ N2 ) ) ) ).
% union_single_eq_member
thf(fact_24_length__Cons,axiom,
! [X2: a,Xs2: list_a] :
( ( size_size_list_a @ ( cons_a @ X2 @ Xs2 ) )
= ( suc @ ( size_size_list_a @ Xs2 ) ) ) ).
% length_Cons
thf(fact_25_mset__eq__length,axiom,
! [Xs2: list_a,Ys3: list_a] :
( ( ( mset_a @ Xs2 )
= ( mset_a @ Ys3 ) )
=> ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) ) ) ).
% mset_eq_length
thf(fact_26_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_27_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_28_Suc_Ohyps_I1_J,axiom,
! [Y2: list_a,Z2: list_a] :
( ( n
= ( size_size_list_a @ Y2 ) )
=> ( ( ( mset_a @ Y2 )
= ( mset_a @ Z2 ) )
=> ( ( list_sel_aux_a @ f @ x @ Y2 )
= ( list_sel_aux_a @ f @ x @ Z2 ) ) ) ) ).
% Suc.hyps(1)
thf(fact_29_Succ__Shift,axiom,
! [Kl2: set_list_a,K: a,Kl: list_a] :
( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) @ Kl )
= ( bNF_Greatest_Succ_a @ Kl2 @ ( cons_a @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_30_gen__length__code_I2_J,axiom,
! [N: nat,X2: a,Xs2: list_a] :
( ( gen_length_a @ N @ ( cons_a @ X2 @ Xs2 ) )
= ( gen_length_a @ ( suc @ N ) @ Xs2 ) ) ).
% gen_length_code(2)
thf(fact_31_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_32_size__neq__size__imp__neq,axiom,
! [X2: char,Y2: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_33_size__neq__size__imp__neq,axiom,
! [X2: num,Y2: num] :
( ( ( size_size_num @ X2 )
!= ( size_size_num @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_34__092_060open_0620_A_060_Alength_Az_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ za ) ).
% \<open>0 < length z\<close>
thf(fact_35_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_36_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_37_Suc__le__length__iff,axiom,
! [N: nat,Xs2: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs2 ) )
= ( ? [X3: a,Ys2: list_a] :
( ( Xs2
= ( cons_a @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_38_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a2 @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_39_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a2 @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_40__092_060open_0620_A_060_Alength_Ay_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ ya ) ).
% \<open>0 < length y\<close>
thf(fact_41_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_42_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_43_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_44_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_45_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_46_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_47_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_48_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_49_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_50_size__mset,axiom,
! [Xs2: list_a] :
( ( size_size_multiset_a @ ( mset_a @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) ) ).
% size_mset
thf(fact_51_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_52_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_53_in__replicate__mset,axiom,
! [X2: a,N: nat,Y2: a] :
( ( member_a2 @ X2 @ ( set_mset_a @ ( replicate_mset_a @ N @ Y2 ) ) )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( X2 = Y2 ) ) ) ).
% in_replicate_mset
thf(fact_54_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_55_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_56_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_57_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_58_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_59_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_60_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_61_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_62_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_63_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_64_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_65_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_66_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_67_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_68_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_69_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_70_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_71_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_72_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_73_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_74_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_75_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_76_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N5: nat] : ( ord_less_eq_nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_77_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_78_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_79_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_80_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J2: nat] :
( ( M2
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_81_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_82_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_83_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_84_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_85_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_86_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_87_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N5: nat] :
( ( ord_less_nat @ M3 @ N5 )
| ( M3 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_88_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_89_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_90_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_91_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_92_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_93_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_94_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_95_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_96_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_97_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_98_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_99_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
& ( M3 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_100_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_101_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_102_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_103_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_104_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_105_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_106_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_107_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_108_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_109_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_110_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_111_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_112_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_113_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_114_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_115_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I4 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I4 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_116_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_117_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_118_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_119_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_120_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_121_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_122_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_123_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_124_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_125_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_126_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_127_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_128_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_129_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_130_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y4: nat,Z3: nat] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N4: nat] : ( R @ N4 @ ( suc @ N4 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_131_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_132_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_133_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_134_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_135_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_136_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M4: nat] :
( M7
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_137_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_138_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_139_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_140_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_141_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_142_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_143_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_144_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_145_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P @ X4 @ Y4 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_146_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_147_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_148_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_149_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_150_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_151_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_152_length__code,axiom,
( size_size_list_a
= ( gen_length_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_153_length__induct,axiom,
! [P: list_a > $o,Xs2: list_a] :
( ! [Xs: list_a] :
( ! [Ys4: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys4 ) @ ( size_size_list_a @ Xs ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_154_size__eq__Suc__imp__elem,axiom,
! [M: multiset_a,N: nat] :
( ( ( size_size_multiset_a @ M )
= ( suc @ N ) )
=> ? [A4: a] : ( member_a2 @ A4 @ ( set_mset_a @ M ) ) ) ).
% size_eq_Suc_imp_elem
thf(fact_155_impossible__Cons,axiom,
! [Xs2: list_a,Ys3: list_a,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ ( size_size_list_a @ Ys3 ) )
=> ( Xs2
!= ( cons_a @ X2 @ Ys3 ) ) ) ).
% impossible_Cons
thf(fact_156_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_157_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_158_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_159_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_160_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_161_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_162_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_163_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_164_order__refl,axiom,
! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).
% order_refl
thf(fact_165_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_166_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N4: nat] :
( ~ ( P @ N4 )
& ( P @ ( suc @ N4 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_167_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_168_multiset__nonemptyE,axiom,
! [A2: multiset_a] :
( ( A2 != zero_zero_multiset_a )
=> ~ ! [X4: a] :
~ ( member_a2 @ X4 @ ( set_mset_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_169_multi__member__last,axiom,
! [X2: a] : ( member_a2 @ X2 @ ( set_mset_a @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) ) ).
% multi_member_last
thf(fact_170_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_171_nle__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_172_nle__le,axiom,
! [A: num,B2: num] :
( ( ~ ( ord_less_eq_num @ A @ B2 ) )
= ( ( ord_less_eq_num @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_173_le__cases3,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_174_le__cases3,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ( ord_less_eq_real @ X2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_175_le__cases3,axiom,
! [X2: num,Y2: num,Z2: num] :
( ( ( ord_less_eq_num @ X2 @ Y2 )
=> ~ ( ord_less_eq_num @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_num @ Y2 @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_num @ X2 @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_num @ Z2 @ Y2 )
=> ~ ( ord_less_eq_num @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_num @ Y2 @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_num @ Z2 @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_176_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_177_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_178_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
= ( ^ [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
& ( ord_less_eq_num @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_179_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_180_ord__eq__le__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_181_ord__eq__le__trans,axiom,
! [A: num,B2: num,C: num] :
( ( A = B2 )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_182_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_183_ord__le__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_184_ord__le__eq__trans,axiom,
! [A: num,B2: num,C: num] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_185_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_186_order__antisym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_187_order__antisym,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_188_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_189_order_Otrans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_190_order_Otrans,axiom,
! [A: num,B2: num,C: num] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_191_order__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_192_order__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_193_order__trans,axiom,
! [X2: num,Y2: num,Z2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ Z2 )
=> ( ord_less_eq_num @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_194_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_195_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B2: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_196_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B2: num] :
( ! [A4: num,B4: num] :
( ( ord_less_eq_num @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: num,B4: num] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_197_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_198_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_199_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
= ( ^ [A5: num,B5: num] :
( ( ord_less_eq_num @ B5 @ A5 )
& ( ord_less_eq_num @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_200_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_201_dual__order_Oantisym,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_202_dual__order_Oantisym,axiom,
! [B2: num,A: num] :
( ( ord_less_eq_num @ B2 @ A )
=> ( ( ord_less_eq_num @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_203_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_204_dual__order_Otrans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_205_dual__order_Otrans,axiom,
! [B2: num,A: num,C: num] :
( ( ord_less_eq_num @ B2 @ A )
=> ( ( ord_less_eq_num @ C @ B2 )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_206_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_207_antisym,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_208_antisym,axiom,
! [A: num,B2: num] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( ord_less_eq_num @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_209_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_210_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_211_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
= ( ^ [A5: num,B5: num] :
( ( ord_less_eq_num @ A5 @ B5 )
& ( ord_less_eq_num @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_212_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_213_order__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_214_order__subst1,axiom,
! [A: nat,F: num > nat,B2: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_215_order__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_216_order__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_217_order__subst1,axiom,
! [A: real,F: num > real,B2: num,C: num] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_218_order__subst1,axiom,
! [A: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_219_order__subst1,axiom,
! [A: num,F: real > num,B2: real,C: real] :
( ( ord_less_eq_num @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_220_order__subst1,axiom,
! [A: num,F: num > num,B2: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_221_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_222_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_223_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_224_order__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_225_order__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_226_order__subst2,axiom,
! [A: real,B2: real,F: real > num,C: num] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_227_order__subst2,axiom,
! [A: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_228_order__subst2,axiom,
! [A: num,B2: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_229_order__subst2,axiom,
! [A: num,B2: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_230_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_231_order__eq__refl,axiom,
! [X2: real,Y2: real] :
( ( X2 = Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_232_order__eq__refl,axiom,
! [X2: num,Y2: num] :
( ( X2 = Y2 )
=> ( ord_less_eq_num @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_233_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_234_linorder__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_235_linorder__linear,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
| ( ord_less_eq_num @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_236_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_237_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_238_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_239_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_240_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_241_ord__eq__le__subst,axiom,
! [A: num,F: real > num,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_242_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B2: num,C: num] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_243_ord__eq__le__subst,axiom,
! [A: real,F: num > real,B2: num,C: num] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_244_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B2: num,C: num] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_245_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_246_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_247_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_248_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_249_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_250_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > num,C: num] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_251_ord__le__eq__subst,axiom,
! [A: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_252_ord__le__eq__subst,axiom,
! [A: num,B2: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_253_ord__le__eq__subst,axiom,
! [A: num,B2: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_254_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_255_linorder__le__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_256_linorder__le__cases,axiom,
! [X2: num,Y2: num] :
( ~ ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_257_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_258_order__antisym__conv,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq_real @ Y2 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_259_order__antisym__conv,axiom,
! [Y2: num,X2: num] :
( ( ord_less_eq_num @ Y2 @ X2 )
=> ( ( ord_less_eq_num @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_260_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_261_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_262_lt__ex,axiom,
! [X2: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).
% lt_ex
thf(fact_263_gt__ex,axiom,
! [X2: nat] :
? [X_12: nat] : ( ord_less_nat @ X2 @ X_12 ) ).
% gt_ex
thf(fact_264_gt__ex,axiom,
! [X2: real] :
? [X_12: real] : ( ord_less_real @ X2 @ X_12 ) ).
% gt_ex
thf(fact_265_dense,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ? [Z3: real] :
( ( ord_less_real @ X2 @ Z3 )
& ( ord_less_real @ Z3 @ Y2 ) ) ) ).
% dense
thf(fact_266_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_267_less__imp__neq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_268_less__imp__neq,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_269_order_Oasym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order.asym
thf(fact_270_order_Oasym,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ~ ( ord_less_real @ B2 @ A ) ) ).
% order.asym
thf(fact_271_order_Oasym,axiom,
! [A: num,B2: num] :
( ( ord_less_num @ A @ B2 )
=> ~ ( ord_less_num @ B2 @ A ) ) ).
% order.asym
thf(fact_272_ord__eq__less__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_273_ord__eq__less__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_274_ord__eq__less__trans,axiom,
! [A: num,B2: num,C: num] :
( ( A = B2 )
=> ( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_275_ord__less__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_276_ord__less__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_277_ord__less__eq__trans,axiom,
! [A: num,B2: num,C: num] :
( ( ord_less_num @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_278_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_279_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_280_antisym__conv3,axiom,
! [Y2: real,X2: real] :
( ~ ( ord_less_real @ Y2 @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_281_antisym__conv3,axiom,
! [Y2: num,X2: num] :
( ~ ( ord_less_num @ Y2 @ X2 )
=> ( ( ~ ( ord_less_num @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_282_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_283_linorder__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_284_linorder__cases,axiom,
! [X2: num,Y2: num] :
( ~ ( ord_less_num @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_num @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_285_dual__order_Oasym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ~ ( ord_less_nat @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_286_dual__order_Oasym,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ~ ( ord_less_real @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_287_dual__order_Oasym,axiom,
! [B2: num,A: num] :
( ( ord_less_num @ B2 @ A )
=> ~ ( ord_less_num @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_288_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_289_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_290_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_291_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N5: nat] :
( ( P3 @ N5 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_292_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_293_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B2: real] :
( ! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_294_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B2: num] :
( ! [A4: num,B4: num] :
( ( ord_less_num @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: num] : ( P @ A4 @ A4 )
=> ( ! [A4: num,B4: num] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_295_order_Ostrict__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_296_order_Ostrict__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_297_order_Ostrict__trans,axiom,
! [A: num,B2: num,C: num] :
( ( ord_less_num @ A @ B2 )
=> ( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_298_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_299_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_300_not__less__iff__gr__or__eq,axiom,
! [X2: num,Y2: num] :
( ( ~ ( ord_less_num @ X2 @ Y2 ) )
= ( ( ord_less_num @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_301_dual__order_Ostrict__trans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_302_dual__order_Ostrict__trans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_303_dual__order_Ostrict__trans,axiom,
! [B2: num,A: num,C: num] :
( ( ord_less_num @ B2 @ A )
=> ( ( ord_less_num @ C @ B2 )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_304_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_305_order_Ostrict__implies__not__eq,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_306_order_Ostrict__implies__not__eq,axiom,
! [A: num,B2: num] :
( ( ord_less_num @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_307_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_308_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_309_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: num,A: num] :
( ( ord_less_num @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_310_linorder__neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_311_linorder__neqE,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_312_linorder__neqE,axiom,
! [X2: num,Y2: num] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_num @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_313_order__less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_314_order__less__asym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_315_order__less__asym,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_316_linorder__neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_317_linorder__neq__iff,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
= ( ( ord_less_real @ X2 @ Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_318_linorder__neq__iff,axiom,
! [X2: num,Y2: num] :
( ( X2 != Y2 )
= ( ( ord_less_num @ X2 @ Y2 )
| ( ord_less_num @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_319_order__less__asym_H,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_320_order__less__asym_H,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ~ ( ord_less_real @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_321_order__less__asym_H,axiom,
! [A: num,B2: num] :
( ( ord_less_num @ A @ B2 )
=> ~ ( ord_less_num @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_322_order__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_323_order__less__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_324_order__less__trans,axiom,
! [X2: num,Y2: num,Z2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ( ord_less_num @ Y2 @ Z2 )
=> ( ord_less_num @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_325_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_326_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_327_ord__eq__less__subst,axiom,
! [A: num,F: nat > num,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_328_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_329_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_330_ord__eq__less__subst,axiom,
! [A: num,F: real > num,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_331_ord__eq__less__subst,axiom,
! [A: nat,F: num > nat,B2: num,C: num] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_332_ord__eq__less__subst,axiom,
! [A: real,F: num > real,B2: num,C: num] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_333_ord__eq__less__subst,axiom,
! [A: num,F: num > num,B2: num,C: num] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_334_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_335_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_336_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_337_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_338_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_339_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > num,C: num] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_340_ord__less__eq__subst,axiom,
! [A: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_341_ord__less__eq__subst,axiom,
! [A: num,B2: num,F: num > real,C: real] :
( ( ord_less_num @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_342_ord__less__eq__subst,axiom,
! [A: num,B2: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_343_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_344_order__less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_345_order__less__irrefl,axiom,
! [X2: num] :
~ ( ord_less_num @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_346_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_347_order__less__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_348_order__less__subst1,axiom,
! [A: nat,F: num > nat,B2: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_349_order__less__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_350_order__less__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_351_order__less__subst1,axiom,
! [A: real,F: num > real,B2: num,C: num] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_352_order__less__subst1,axiom,
! [A: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_353_order__less__subst1,axiom,
! [A: num,F: real > num,B2: real,C: real] :
( ( ord_less_num @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_354_order__less__subst1,axiom,
! [A: num,F: num > num,B2: num,C: num] :
( ( ord_less_num @ A @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_355_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_356_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_357_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_358_order__less__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_359_order__less__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_360_order__less__subst2,axiom,
! [A: real,B2: real,F: real > num,C: num] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_361_order__less__subst2,axiom,
! [A: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_362_order__less__subst2,axiom,
! [A: num,B2: num,F: num > real,C: real] :
( ( ord_less_num @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_363_order__less__subst2,axiom,
! [A: num,B2: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_364_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_365_order__less__not__sym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_366_order__less__not__sym,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_367_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_368_order__less__imp__triv,axiom,
! [X2: real,Y2: real,P: $o] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_369_order__less__imp__triv,axiom,
! [X2: num,Y2: num,P: $o] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ( ord_less_num @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_370_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_371_linorder__less__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_372_linorder__less__linear,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_num @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_373_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_374_order__less__imp__not__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_375_order__less__imp__not__eq,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_376_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_377_order__less__imp__not__eq2,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_378_order__less__imp__not__eq2,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_379_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_380_order__less__imp__not__less,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_381_order__less__imp__not__less,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_382_multiset__induct__max,axiom,
! [P: multiset_nat > $o,M: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X4: nat,M8: multiset_nat] :
( ( P @ M8 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M8 ) )
=> ( ord_less_eq_nat @ Xa @ X4 ) )
=> ( P @ ( add_mset_nat @ X4 @ M8 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_max
thf(fact_383_multiset__induct__max,axiom,
! [P: multiset_real > $o,M: multiset_real] :
( ( P @ zero_z8811559133707751557t_real )
=> ( ! [X4: real,M8: multiset_real] :
( ( P @ M8 )
=> ( ! [Xa: real] :
( ( member_real @ Xa @ ( set_mset_real @ M8 ) )
=> ( ord_less_eq_real @ Xa @ X4 ) )
=> ( P @ ( add_mset_real @ X4 @ M8 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_max
thf(fact_384_multiset__induct__max,axiom,
! [P: multiset_num > $o,M: multiset_num] :
( ( P @ zero_z8056838136647266291et_num )
=> ( ! [X4: num,M8: multiset_num] :
( ( P @ M8 )
=> ( ! [Xa: num] :
( ( member_num @ Xa @ ( set_mset_num @ M8 ) )
=> ( ord_less_eq_num @ Xa @ X4 ) )
=> ( P @ ( add_mset_num @ X4 @ M8 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_max
thf(fact_385_multiset__induct__min,axiom,
! [P: multiset_nat > $o,M: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X4: nat,M8: multiset_nat] :
( ( P @ M8 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M8 ) )
=> ( ord_less_eq_nat @ X4 @ Xa ) )
=> ( P @ ( add_mset_nat @ X4 @ M8 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_min
thf(fact_386_multiset__induct__min,axiom,
! [P: multiset_real > $o,M: multiset_real] :
( ( P @ zero_z8811559133707751557t_real )
=> ( ! [X4: real,M8: multiset_real] :
( ( P @ M8 )
=> ( ! [Xa: real] :
( ( member_real @ Xa @ ( set_mset_real @ M8 ) )
=> ( ord_less_eq_real @ X4 @ Xa ) )
=> ( P @ ( add_mset_real @ X4 @ M8 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_min
thf(fact_387_multiset__induct__min,axiom,
! [P: multiset_num > $o,M: multiset_num] :
( ( P @ zero_z8056838136647266291et_num )
=> ( ! [X4: num,M8: multiset_num] :
( ( P @ M8 )
=> ( ! [Xa: num] :
( ( member_num @ Xa @ ( set_mset_num @ M8 ) )
=> ( ord_less_eq_num @ X4 @ Xa ) )
=> ( P @ ( add_mset_num @ X4 @ M8 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_min
thf(fact_388_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_389_leD,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_390_leD,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq_real @ Y2 @ X2 )
=> ~ ( ord_less_real @ X2 @ Y2 ) ) ).
% leD
thf(fact_391_leD,axiom,
! [Y2: num,X2: num] :
( ( ord_less_eq_num @ Y2 @ X2 )
=> ~ ( ord_less_num @ X2 @ Y2 ) ) ).
% leD
thf(fact_392_leI,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% leI
thf(fact_393_leI,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% leI
thf(fact_394_leI,axiom,
! [X2: num,Y2: num] :
( ~ ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X2 ) ) ).
% leI
thf(fact_395_nless__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_396_nless__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_real @ A @ B2 ) )
= ( ~ ( ord_less_eq_real @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_397_nless__le,axiom,
! [A: num,B2: num] :
( ( ~ ( ord_less_num @ A @ B2 ) )
= ( ~ ( ord_less_eq_num @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_398_antisym__conv1,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_399_antisym__conv1,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_400_antisym__conv1,axiom,
! [X2: num,Y2: num] :
( ~ ( ord_less_num @ X2 @ Y2 )
=> ( ( ord_less_eq_num @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_401_antisym__conv2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_402_antisym__conv2,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_403_antisym__conv2,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ( ~ ( ord_less_num @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_404_dense__ge,axiom,
! [Z2: real,Y2: real] :
( ! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_eq_real @ Y2 @ X4 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_ge
thf(fact_405_dense__le,axiom,
! [Y2: real,Z2: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ X4 @ Z2 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_le
thf(fact_406_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_407_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_408_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
& ~ ( ord_less_eq_num @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_409_not__le__imp__less,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_410_not__le__imp__less,axiom,
! [Y2: real,X2: real] :
( ~ ( ord_less_eq_real @ Y2 @ X2 )
=> ( ord_less_real @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_411_not__le__imp__less,axiom,
! [Y2: num,X2: num] :
( ~ ( ord_less_eq_num @ Y2 @ X2 )
=> ( ord_less_num @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_412_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_413_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_real @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_414_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A5: num,B5: num] :
( ( ord_less_num @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_415_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_416_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_417_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A5: num,B5: num] :
( ( ord_less_eq_num @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_418_order_Ostrict__trans1,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_419_order_Ostrict__trans1,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_420_order_Ostrict__trans1,axiom,
! [A: num,B2: num,C: num] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_421_order_Ostrict__trans2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_422_order_Ostrict__trans2,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_423_order_Ostrict__trans2,axiom,
! [A: num,B2: num,C: num] :
( ( ord_less_num @ A @ B2 )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_424_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_425_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ~ ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_426_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A5: num,B5: num] :
( ( ord_less_eq_num @ A5 @ B5 )
& ~ ( ord_less_eq_num @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_427_dense__ge__bounded,axiom,
! [Z2: real,X2: real,Y2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X2 )
=> ( ord_less_eq_real @ Y2 @ W ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_428_dense__le__bounded,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ! [W: real] :
( ( ord_less_real @ X2 @ W )
=> ( ( ord_less_real @ W @ Y2 )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_429_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_430_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A5: real] :
( ( ord_less_real @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_431_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B5: num,A5: num] :
( ( ord_less_num @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_432_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_433_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B5: real,A5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_434_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B5: num,A5: num] :
( ( ord_less_eq_num @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_435_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_436_dual__order_Ostrict__trans1,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_437_dual__order_Ostrict__trans1,axiom,
! [B2: num,A: num,C: num] :
( ( ord_less_eq_num @ B2 @ A )
=> ( ( ord_less_num @ C @ B2 )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_438_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_439_dual__order_Ostrict__trans2,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_440_dual__order_Ostrict__trans2,axiom,
! [B2: num,A: num,C: num] :
( ( ord_less_num @ B2 @ A )
=> ( ( ord_less_eq_num @ C @ B2 )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_441_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_442_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B5: real,A5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_443_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B5: num,A5: num] :
( ( ord_less_eq_num @ B5 @ A5 )
& ~ ( ord_less_eq_num @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_444_order_Ostrict__implies__order,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_445_order_Ostrict__implies__order,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_446_order_Ostrict__implies__order,axiom,
! [A: num,B2: num] :
( ( ord_less_num @ A @ B2 )
=> ( ord_less_eq_num @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_447_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_448_dual__order_Ostrict__implies__order,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ( ord_less_eq_real @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_449_dual__order_Ostrict__implies__order,axiom,
! [B2: num,A: num] :
( ( ord_less_num @ B2 @ A )
=> ( ord_less_eq_num @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_450_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_451_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_452_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_453_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_454_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_455_order__less__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_456_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_457_linorder__not__le,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y2 ) )
= ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_458_linorder__not__le,axiom,
! [X2: num,Y2: num] :
( ( ~ ( ord_less_eq_num @ X2 @ Y2 ) )
= ( ord_less_num @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_459_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_460_linorder__not__less,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_461_linorder__not__less,axiom,
! [X2: num,Y2: num] :
( ( ~ ( ord_less_num @ X2 @ Y2 ) )
= ( ord_less_eq_num @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_462_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_463_order__less__imp__le,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_464_order__less__imp__le,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_465_order__le__neq__trans,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_466_order__le__neq__trans,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_467_order__le__neq__trans,axiom,
! [A: num,B2: num] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_num @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_468_order__neq__le__trans,axiom,
! [A: nat,B2: nat] :
( ( A != B2 )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_469_order__neq__le__trans,axiom,
! [A: real,B2: real] :
( ( A != B2 )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_470_order__neq__le__trans,axiom,
! [A: num,B2: num] :
( ( A != B2 )
=> ( ( ord_less_eq_num @ A @ B2 )
=> ( ord_less_num @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_471_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_472_order__le__less__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_473_order__le__less__trans,axiom,
! [X2: num,Y2: num,Z2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ( ord_less_num @ Y2 @ Z2 )
=> ( ord_less_num @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_474_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_475_order__less__le__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_476_order__less__le__trans,axiom,
! [X2: num,Y2: num,Z2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ Z2 )
=> ( ord_less_num @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_477_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_478_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_479_order__le__less__subst1,axiom,
! [A: nat,F: num > nat,B2: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_480_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_481_order__le__less__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_482_order__le__less__subst1,axiom,
! [A: real,F: num > real,B2: num,C: num] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_483_order__le__less__subst1,axiom,
! [A: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_484_order__le__less__subst1,axiom,
! [A: num,F: real > num,B2: real,C: real] :
( ( ord_less_eq_num @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_485_order__le__less__subst1,axiom,
! [A: num,F: num > num,B2: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_486_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_487_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_488_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_489_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_490_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_491_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > num,C: num] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_492_order__le__less__subst2,axiom,
! [A: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_493_order__le__less__subst2,axiom,
! [A: num,B2: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_494_order__le__less__subst2,axiom,
! [A: num,B2: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_495_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_496_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_497_order__less__le__subst1,axiom,
! [A: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_498_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_499_order__less__le__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_500_order__less__le__subst1,axiom,
! [A: num,F: real > num,B2: real,C: real] :
( ( ord_less_num @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_501_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B2: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_502_order__less__le__subst1,axiom,
! [A: real,F: num > real,B2: num,C: num] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_503_order__less__le__subst1,axiom,
! [A: num,F: num > num,B2: num,C: num] :
( ( ord_less_num @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_504_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_505_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_506_order__less__le__subst2,axiom,
! [A: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_507_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_508_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_509_order__less__le__subst2,axiom,
! [A: num,B2: num,F: num > real,C: real] :
( ( ord_less_num @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_510_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_511_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > num,C: num] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_512_order__less__le__subst2,axiom,
! [A: num,B2: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_513_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_514_linorder__le__less__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_515_linorder__le__less__linear,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
| ( ord_less_num @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_516_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_517_order__le__imp__less__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_518_order__le__imp__less__or__eq,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ( ord_less_num @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_519_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_520_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_521_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_522_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_523_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_524_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_525_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_526_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_527_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X6 ) ) ).
% minf(8)
thf(fact_528_minf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ~ ( ord_less_eq_num @ T @ X6 ) ) ).
% minf(8)
thf(fact_529_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_530_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ord_less_eq_real @ X6 @ T ) ) ).
% minf(6)
thf(fact_531_minf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ( ord_less_eq_num @ X6 @ T ) ) ).
% minf(6)
thf(fact_532_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_533_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ord_less_eq_real @ T @ X6 ) ) ).
% pinf(8)
thf(fact_534_pinf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ( ord_less_eq_num @ T @ X6 ) ) ).
% pinf(8)
thf(fact_535_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_536_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ~ ( ord_less_eq_real @ X6 @ T ) ) ).
% pinf(6)
thf(fact_537_pinf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ~ ( ord_less_eq_num @ X6 @ T ) ) ).
% pinf(6)
thf(fact_538_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_539_verit__comp__simplify1_I3_J,axiom,
! [B6: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
= ( ord_less_real @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_540_verit__comp__simplify1_I3_J,axiom,
! [B6: num,A6: num] :
( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
= ( ord_less_num @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_541_complete__interval,axiom,
! [A: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A @ C4 )
& ( ord_less_eq_nat @ C4 @ B2 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C4 ) )
=> ( P @ X6 ) )
& ! [D: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_542_complete__interval,axiom,
! [A: real,B2: real,P: real > $o] :
( ( ord_less_real @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C4: real] :
( ( ord_less_eq_real @ A @ C4 )
& ( ord_less_eq_real @ C4 @ B2 )
& ! [X6: real] :
( ( ( ord_less_eq_real @ A @ X6 )
& ( ord_less_real @ X6 @ C4 ) )
=> ( P @ X6 ) )
& ! [D: real] :
( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_real @ D @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_543_verit__la__disequality,axiom,
! [A: nat,B2: nat] :
( ( A = B2 )
| ~ ( ord_less_eq_nat @ A @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_544_verit__la__disequality,axiom,
! [A: real,B2: real] :
( ( A = B2 )
| ~ ( ord_less_eq_real @ A @ B2 )
| ~ ( ord_less_eq_real @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_545_verit__la__disequality,axiom,
! [A: num,B2: num] :
( ( A = B2 )
| ~ ( ord_less_eq_num @ A @ B2 )
| ~ ( ord_less_eq_num @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_546_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_547_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_548_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_549_ex__gt__or__lt,axiom,
! [A: real] :
? [B4: real] :
( ( ord_less_real @ A @ B4 )
| ( ord_less_real @ B4 @ A ) ) ).
% ex_gt_or_lt
thf(fact_550_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_551_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_552_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_553_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_554_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_555_pinf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_556_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_557_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_558_pinf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_559_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_560_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_561_pinf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_562_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_563_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_564_pinf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_565_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_566_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ~ ( ord_less_real @ X6 @ T ) ) ).
% pinf(5)
thf(fact_567_pinf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ~ ( ord_less_num @ X6 @ T ) ) ).
% pinf(5)
thf(fact_568_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_569_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ord_less_real @ T @ X6 ) ) ).
% pinf(7)
thf(fact_570_pinf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ( ord_less_num @ T @ X6 ) ) ).
% pinf(7)
thf(fact_571_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_572_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_573_minf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_574_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_575_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_576_minf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_577_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_578_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_579_minf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_580_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_581_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_582_minf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_583_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_584_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ord_less_real @ X6 @ T ) ) ).
% minf(5)
thf(fact_585_minf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ( ord_less_num @ X6 @ T ) ) ).
% minf(5)
thf(fact_586_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_587_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ~ ( ord_less_real @ T @ X6 ) ) ).
% minf(7)
thf(fact_588_minf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ~ ( ord_less_num @ T @ X6 ) ) ).
% minf(7)
thf(fact_589_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_590_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_591_mset__single__iff,axiom,
! [Xs2: list_a,X2: a] :
( ( ( mset_a @ Xs2 )
= ( add_mset_a @ X2 @ zero_zero_multiset_a ) )
= ( Xs2
= ( cons_a @ X2 @ nil_a ) ) ) ).
% mset_single_iff
thf(fact_592_mset__single__iff__right,axiom,
! [X2: a,Xs2: list_a] :
( ( ( add_mset_a @ X2 @ zero_zero_multiset_a )
= ( mset_a @ Xs2 ) )
= ( Xs2
= ( cons_a @ X2 @ nil_a ) ) ) ).
% mset_single_iff_right
thf(fact_593_size__Diff2__less,axiom,
! [X2: a,M: multiset_a,Y2: a] :
( ( member_a2 @ X2 @ ( set_mset_a @ M ) )
=> ( ( member_a2 @ Y2 @ ( set_mset_a @ M ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) @ ( add_mset_a @ Y2 @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M ) ) ) ) ).
% size_Diff2_less
thf(fact_594_size__Diff1__less,axiom,
! [X2: a,M: multiset_a] :
( ( member_a2 @ X2 @ ( set_mset_a @ M ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M ) ) ) ).
% size_Diff1_less
thf(fact_595_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_596_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_597_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_598_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_599_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_600_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_601_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_602_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_603_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_604_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_605_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_606_diff__ge__0__iff__ge,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
= ( ord_less_eq_int @ B2 @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_607_diff__ge__0__iff__ge,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
= ( ord_less_eq_real @ B2 @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_608_diff__gt__0__iff__gt,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
= ( ord_less_int @ B2 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_609_diff__gt__0__iff__gt,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
= ( ord_less_real @ B2 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_610_length__0__conv,axiom,
! [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_a ) ) ).
% length_0_conv
thf(fact_611_mset__zero__iff,axiom,
! [X2: list_a] :
( ( ( mset_a @ X2 )
= zero_zero_multiset_a )
= ( X2 = nil_a ) ) ).
% mset_zero_iff
thf(fact_612_mset__zero__iff__right,axiom,
! [X2: list_a] :
( ( zero_zero_multiset_a
= ( mset_a @ X2 ) )
= ( X2 = nil_a ) ) ).
% mset_zero_iff_right
thf(fact_613_diff__add__mset__swap,axiom,
! [B2: a,A2: multiset_a,M: multiset_a] :
( ~ ( member_a2 @ B2 @ ( set_mset_a @ A2 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ B2 @ M ) @ A2 )
= ( add_mset_a @ B2 @ ( minus_3765977307040488491iset_a @ M @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_614_length__greater__0__conv,axiom,
! [Xs2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs2 ) )
= ( Xs2 != nil_a ) ) ).
% length_greater_0_conv
thf(fact_615_insert__DiffM,axiom,
! [X2: a,M: multiset_a] :
( ( member_a2 @ X2 @ ( set_mset_a @ M ) )
=> ( ( add_mset_a @ X2 @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_616_diff__union__swap2,axiom,
! [Y2: a,M: multiset_a,X2: a] :
( ( member_a2 @ Y2 @ ( set_mset_a @ M ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X2 @ M ) @ ( add_mset_a @ Y2 @ zero_zero_multiset_a ) )
= ( add_mset_a @ X2 @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ Y2 @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_617_transpose_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X2
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X4: a,Xs: list_a,Xss: list_list_a] :
( X2
!= ( cons_list_a @ ( cons_a @ X4 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_618_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_619_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 )
= ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_620_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 )
= ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_621_diff__eq__diff__eq,axiom,
! [A: real,B2: real,C: real,D3: real] :
( ( ( minus_minus_real @ A @ B2 )
= ( minus_minus_real @ C @ D3 ) )
=> ( ( A = B2 )
= ( C = D3 ) ) ) ).
% diff_eq_diff_eq
thf(fact_622_diff__eq__diff__eq,axiom,
! [A: int,B2: int,C: int,D3: int] :
( ( ( minus_minus_int @ A @ B2 )
= ( minus_minus_int @ C @ D3 ) )
=> ( ( A = B2 )
= ( C = D3 ) ) ) ).
% diff_eq_diff_eq
thf(fact_623_diff__mono,axiom,
! [A: int,B2: int,D3: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ D3 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D3 ) ) ) ) ).
% diff_mono
thf(fact_624_diff__mono,axiom,
! [A: real,B2: real,D3: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ D3 @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D3 ) ) ) ) ).
% diff_mono
thf(fact_625_diff__left__mono,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_626_diff__left__mono,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_627_diff__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_628_diff__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_629_diff__eq__diff__less__eq,axiom,
! [A: int,B2: int,C: int,D3: int] :
( ( ( minus_minus_int @ A @ B2 )
= ( minus_minus_int @ C @ D3 ) )
=> ( ( ord_less_eq_int @ A @ B2 )
= ( ord_less_eq_int @ C @ D3 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_630_diff__eq__diff__less__eq,axiom,
! [A: real,B2: real,C: real,D3: real] :
( ( ( minus_minus_real @ A @ B2 )
= ( minus_minus_real @ C @ D3 ) )
=> ( ( ord_less_eq_real @ A @ B2 )
= ( ord_less_eq_real @ C @ D3 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_631_eq__iff__diff__eq__0,axiom,
( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
= ( ^ [A5: real,B5: real] :
( ( minus_minus_real @ A5 @ B5 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_632_eq__iff__diff__eq__0,axiom,
( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
= ( ^ [A5: int,B5: int] :
( ( minus_minus_int @ A5 @ B5 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_633_diff__strict__mono,axiom,
! [A: int,B2: int,D3: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ D3 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D3 ) ) ) ) ).
% diff_strict_mono
thf(fact_634_diff__strict__mono,axiom,
! [A: real,B2: real,D3: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ D3 @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D3 ) ) ) ) ).
% diff_strict_mono
thf(fact_635_diff__eq__diff__less,axiom,
! [A: int,B2: int,C: int,D3: int] :
( ( ( minus_minus_int @ A @ B2 )
= ( minus_minus_int @ C @ D3 ) )
=> ( ( ord_less_int @ A @ B2 )
= ( ord_less_int @ C @ D3 ) ) ) ).
% diff_eq_diff_less
thf(fact_636_diff__eq__diff__less,axiom,
! [A: real,B2: real,C: real,D3: real] :
( ( ( minus_minus_real @ A @ B2 )
= ( minus_minus_real @ C @ D3 ) )
=> ( ( ord_less_real @ A @ B2 )
= ( ord_less_real @ C @ D3 ) ) ) ).
% diff_eq_diff_less
thf(fact_637_diff__strict__left__mono,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_638_diff__strict__left__mono,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_639_diff__strict__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_640_diff__strict__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_641_in__diffD,axiom,
! [A: a,M: multiset_a,N2: multiset_a] :
( ( member_a2 @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N2 ) ) )
=> ( member_a2 @ A @ ( set_mset_a @ M ) ) ) ).
% in_diffD
thf(fact_642_list__nonempty__induct,axiom,
! [Xs2: list_a,P: list_a > $o] :
( ( Xs2 != nil_a )
=> ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs: list_a] :
( ( Xs != nil_a )
=> ( ( P @ Xs )
=> ( P @ ( cons_a @ X4 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_643_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs2: list_a,Ys3: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X4: a,Xs: list_a] : ( P @ ( cons_a @ X4 @ Xs ) @ nil_a )
=> ( ! [Y4: a,Ys: list_a] : ( P @ nil_a @ ( cons_a @ Y4 @ Ys ) )
=> ( ! [X4: a,Xs: list_a,Y4: a,Ys: list_a] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y4 @ Ys ) ) )
=> ( P @ Xs2 @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_644_neq__Nil__conv,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
= ( ? [Y3: a,Ys2: list_a] :
( Xs2
= ( cons_a @ Y3 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_645_remdups__adj_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X4: a] :
( X2
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y4: a,Xs: list_a] :
( X2
!= ( cons_a @ X4 @ ( cons_a @ Y4 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_646_list_Oexhaust,axiom,
! [Y2: list_a] :
( ( Y2 != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y2
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_647_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_648_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_649_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_650_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B5: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B5 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_651_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_652_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A5: real,B5: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B5 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_653_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_654_list__induct2,axiom,
! [Xs2: list_a,Ys3: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X4: a,Xs: list_a,Y4: a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y4 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys3 ) ) ) ) ).
% list_induct2
thf(fact_655_list__induct3,axiom,
! [Xs2: list_a,Ys3: list_a,Zs2: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: a,Xs: list_a,Y4: a,Ys: list_a,Z3: a,Zs: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ Xs @ Ys @ Zs )
=> ( P @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y4 @ Ys ) @ ( cons_a @ Z3 @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys3 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_656_list__induct4,axiom,
! [Xs2: list_a,Ys3: list_a,Zs2: list_a,Ws: list_a,P: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: a,Xs: list_a,Y4: a,Ys: list_a,Z3: a,Zs: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs @ Ws2 )
=> ( P @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y4 @ Ys ) @ ( cons_a @ Z3 @ Zs ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys3 @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_657_mset_Osimps_I1_J,axiom,
( ( mset_a @ nil_a )
= zero_zero_multiset_a ) ).
% mset.simps(1)
thf(fact_658_diff__single__trivial,axiom,
! [X2: a,M: multiset_a] :
( ~ ( member_a2 @ X2 @ ( set_mset_a @ M ) )
=> ( ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) )
= M ) ) ).
% diff_single_trivial
thf(fact_659_diff__single__eq__union,axiom,
! [X2: a,M: multiset_a,N2: multiset_a] :
( ( member_a2 @ X2 @ ( set_mset_a @ M ) )
=> ( ( ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) )
= N2 )
= ( M
= ( add_mset_a @ X2 @ N2 ) ) ) ) ).
% diff_single_eq_union
thf(fact_660_multi__drop__mem__not__eq,axiom,
! [C: a,B3: multiset_a] :
( ( member_a2 @ C @ ( set_mset_a @ B3 ) )
=> ( ( minus_3765977307040488491iset_a @ B3 @ ( add_mset_a @ C @ zero_zero_multiset_a ) )
!= B3 ) ) ).
% multi_drop_mem_not_eq
thf(fact_661_add__mset__remove__trivial__If,axiom,
! [A: a,N2: multiset_a] :
( ( ( member_a2 @ A @ ( set_mset_a @ N2 ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N2 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= N2 ) )
& ( ~ ( member_a2 @ A @ ( set_mset_a @ N2 ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N2 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( add_mset_a @ A @ N2 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_662_add__mset__remove__trivial__eq,axiom,
! [N2: multiset_a,A: a] :
( ( N2
= ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N2 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
= ( member_a2 @ A @ ( set_mset_a @ N2 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_663_multiset__add__sub__el__shuffle,axiom,
! [C: a,B3: multiset_a,B2: a] :
( ( member_a2 @ C @ ( set_mset_a @ B3 ) )
=> ( ( B2 != C )
=> ( ( add_mset_a @ B2 @ ( minus_3765977307040488491iset_a @ B3 @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B2 @ B3 ) @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_664_more__than__one__mset__mset__diff,axiom,
! [A: a,M: multiset_a] :
( ( member_a2 @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( set_mset_a @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_665_empty__Shift,axiom,
! [Kl2: set_list_a,K: a] :
( ( member_list_a @ nil_a @ Kl2 )
=> ( ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_666_size__Suc__Diff1,axiom,
! [X2: a,M: multiset_a] :
( ( member_a2 @ X2 @ ( set_mset_a @ M ) )
=> ( ( suc @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) ) )
= ( size_size_multiset_a @ M ) ) ) ).
% size_Suc_Diff1
thf(fact_667_mset__remove1,axiom,
! [A: a,Xs2: list_a] :
( ( mset_a @ ( remove1_a @ A @ Xs2 ) )
= ( minus_3765977307040488491iset_a @ ( mset_a @ Xs2 ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ).
% mset_remove1
thf(fact_668_remdups__adj__length__ge1,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_a @ ( remdups_adj_a @ Xs2 ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_669_insert__Nil,axiom,
! [X2: a] :
( ( insert_a @ X2 @ nil_a )
= ( cons_a @ X2 @ nil_a ) ) ).
% insert_Nil
thf(fact_670_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_671_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_672_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_673_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_674_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_675_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_676_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_677_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_678_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_679_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_680_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_681_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_682_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_683_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_684_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_685_le__diff__iff_H,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_686_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_687_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_688_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_689_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_690_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_691_remdups__adj_Osimps_I3_J,axiom,
! [X2: a,Y2: a,Xs2: list_a] :
( ( ( X2 = Y2 )
=> ( ( remdups_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs2 ) ) )
= ( remdups_adj_a @ ( cons_a @ X2 @ Xs2 ) ) ) )
& ( ( X2 != Y2 )
=> ( ( remdups_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs2 ) ) )
= ( cons_a @ X2 @ ( remdups_adj_a @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_692_remove1_Osimps_I2_J,axiom,
! [X2: a,Y2: a,Xs2: list_a] :
( ( ( X2 = Y2 )
=> ( ( remove1_a @ X2 @ ( cons_a @ Y2 @ Xs2 ) )
= Xs2 ) )
& ( ( X2 != Y2 )
=> ( ( remove1_a @ X2 @ ( cons_a @ Y2 @ Xs2 ) )
= ( cons_a @ Y2 @ ( remove1_a @ X2 @ Xs2 ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_693_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_694_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_695_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_696_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_697_diff__less__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_698_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_699_remdups__adj_Osimps_I2_J,axiom,
! [X2: a] :
( ( remdups_adj_a @ ( cons_a @ X2 @ nil_a ) )
= ( cons_a @ X2 @ nil_a ) ) ).
% remdups_adj.simps(2)
thf(fact_700_remdups__adj_Oelims,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( remdups_adj_a @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> ( Y2 != nil_a ) )
=> ( ! [X4: a] :
( ( X2
= ( cons_a @ X4 @ nil_a ) )
=> ( Y2
!= ( cons_a @ X4 @ nil_a ) ) )
=> ~ ! [X4: a,Y4: a,Xs: list_a] :
( ( X2
= ( cons_a @ X4 @ ( cons_a @ Y4 @ Xs ) ) )
=> ~ ( ( ( X4 = Y4 )
=> ( Y2
= ( remdups_adj_a @ ( cons_a @ X4 @ Xs ) ) ) )
& ( ( X4 != Y4 )
=> ( Y2
= ( cons_a @ X4 @ ( remdups_adj_a @ ( cons_a @ Y4 @ Xs ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_701_remdups__adj__length,axiom,
! [Xs2: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( remdups_adj_a @ Xs2 ) ) @ ( size_size_list_a @ Xs2 ) ) ).
% remdups_adj_length
thf(fact_702_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_703_size__Diff__singleton,axiom,
! [X2: a,M: multiset_a] :
( ( member_a2 @ X2 @ ( set_mset_a @ M ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_704_size__Diff__singleton__if,axiom,
! [X2: a,A2: multiset_a] :
( ( ( member_a2 @ X2 @ ( set_mset_a @ A2 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A2 @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_a2 @ X2 @ ( set_mset_a @ A2 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A2 @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) )
= ( size_size_multiset_a @ A2 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_705_nths__singleton,axiom,
! [A2: set_nat,X2: a] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_a @ ( cons_a @ X2 @ nil_a ) @ A2 )
= ( cons_a @ X2 @ nil_a ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_a @ ( cons_a @ X2 @ nil_a ) @ A2 )
= nil_a ) ) ) ).
% nths_singleton
thf(fact_706_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_707_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_708_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_709_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_710_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_711_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_712_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_713_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_714_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_715_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_716_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_717_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_718_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_719_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_720_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_721_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_722_nths__all,axiom,
! [Xs2: list_a,I5: set_nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs2 ) )
=> ( member_nat @ I4 @ I5 ) )
=> ( ( nths_a @ Xs2 @ I5 )
= Xs2 ) ) ).
% nths_all
thf(fact_723_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_724_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_725_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_726_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_727_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_728_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_729_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_730_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_731_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_732_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_733_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_734_list__sel__aux_Osimps_I1_J,axiom,
! [Sel: a > a > real,X2: a] :
( ( list_sel_aux_a @ Sel @ X2 @ nil_a )
= one_one_real ) ).
% list_sel_aux.simps(1)
thf(fact_735_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_736_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_737_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_738_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_739_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_740_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_741_nth__Cons__pos,axiom,
! [N: nat,X2: a,Xs2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ N )
= ( nth_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_742_rotate1__length01,axiom,
! [Xs2: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat )
=> ( ( rotate1_a @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_743_Cons__replicate__eq,axiom,
! [X2: a,Xs2: list_a,N: nat,Y2: a] :
( ( ( cons_a @ X2 @ Xs2 )
= ( replicate_a @ N @ Y2 ) )
= ( ( X2 = Y2 )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs2
= ( replicate_a @ ( minus_minus_nat @ N @ one_one_nat ) @ X2 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_744_length__replicate,axiom,
! [N: nat,X2: a] :
( ( size_size_list_a @ ( replicate_a @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_745_length__rotate1,axiom,
! [Xs2: list_a] :
( ( size_size_list_a @ ( rotate1_a @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) ) ).
% length_rotate1
thf(fact_746_nth__Cons__0,axiom,
! [X2: a,Xs2: list_a] :
( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_747_nth__Cons__Suc,axiom,
! [X2: a,Xs2: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ ( suc @ N ) )
= ( nth_a @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_748_nth__equalityI,axiom,
! [Xs2: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ Xs2 @ I4 )
= ( nth_a @ Ys3 @ I4 ) ) )
=> ( Xs2 = Ys3 ) ) ) ).
% nth_equalityI
thf(fact_749_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X7: a] : ( P @ I2 @ X7 ) ) )
= ( ? [Xs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_a @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_750_list__eq__iff__nth__eq,axiom,
( ( ^ [Y6: list_a,Z4: list_a] : ( Y6 = Z4 ) )
= ( ^ [Xs3: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs3 ) )
=> ( ( nth_a @ Xs3 @ I2 )
= ( nth_a @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_751_replicate__Suc,axiom,
! [N: nat,X2: a] :
( ( replicate_a @ ( suc @ N ) @ X2 )
= ( cons_a @ X2 @ ( replicate_a @ N @ X2 ) ) ) ).
% replicate_Suc
thf(fact_752_nth__Cons_H,axiom,
! [N: nat,X2: a,Xs2: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ N )
= ( nth_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_753_remdups__adj__adjacent,axiom,
! [I: nat,Xs2: list_a] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_a @ ( remdups_adj_a @ Xs2 ) ) )
=> ( ( nth_a @ ( remdups_adj_a @ Xs2 ) @ I )
!= ( nth_a @ ( remdups_adj_a @ Xs2 ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_754_nth__mem__mset,axiom,
! [I: nat,Ls: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( member_a2 @ ( nth_a @ Ls @ I ) @ ( set_mset_a @ ( mset_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_755_remdups__adj__replicate,axiom,
! [N: nat,X2: a] :
( ( ( N = zero_zero_nat )
=> ( ( remdups_adj_a @ ( replicate_a @ N @ X2 ) )
= nil_a ) )
& ( ( N != zero_zero_nat )
=> ( ( remdups_adj_a @ ( replicate_a @ N @ X2 ) )
= ( cons_a @ X2 @ nil_a ) ) ) ) ).
% remdups_adj_replicate
thf(fact_756_remdups__adj__singleton,axiom,
! [Xs2: list_a,X2: a] :
( ( ( remdups_adj_a @ Xs2 )
= ( cons_a @ X2 @ nil_a ) )
=> ( Xs2
= ( replicate_a @ ( size_size_list_a @ Xs2 ) @ X2 ) ) ) ).
% remdups_adj_singleton
thf(fact_757_nth__non__equal__first__eq,axiom,
! [X2: a,Y2: a,Xs2: list_a,N: nat] :
( ( X2 != Y2 )
=> ( ( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ N )
= Y2 )
= ( ( ( nth_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_758_mset__update,axiom,
! [I: nat,Ls: list_a,V: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( ( mset_a @ ( list_update_a @ Ls @ I @ V ) )
= ( add_mset_a @ V @ ( minus_3765977307040488491iset_a @ ( mset_a @ Ls ) @ ( add_mset_a @ ( nth_a @ Ls @ I ) @ zero_zero_multiset_a ) ) ) ) ) ).
% mset_update
thf(fact_759_remdups__adj__singleton__iff,axiom,
! [Xs2: list_a] :
( ( ( size_size_list_a @ ( remdups_adj_a @ Xs2 ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs2 != nil_a )
& ( Xs2
= ( replicate_a @ ( size_size_list_a @ Xs2 ) @ ( hd_a @ Xs2 ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_760_last__conv__nth,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( last_a @ Xs2 )
= ( nth_a @ Xs2 @ ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_761_length__list__update,axiom,
! [Xs2: list_a,I: nat,X2: a] :
( ( size_size_list_a @ ( list_update_a @ Xs2 @ I @ X2 ) )
= ( size_size_list_a @ Xs2 ) ) ).
% length_list_update
thf(fact_762_list__update__beyond,axiom,
! [Xs2: list_a,I: nat,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ I )
=> ( ( list_update_a @ Xs2 @ I @ X2 )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_763_nth__list__update__eq,axiom,
! [I: nat,Xs2: list_a,X2: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I @ X2 ) @ I )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_764_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_765_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_766_last__list__update,axiom,
! [Xs2: list_a,K: nat,X2: a] :
( ( Xs2 != nil_a )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs2 @ K @ X2 ) )
= X2 ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs2 @ K @ X2 ) )
= ( last_a @ Xs2 ) ) ) ) ) ).
% last_list_update
thf(fact_767_list__update__code_I2_J,axiom,
! [X2: a,Xs2: list_a,Y2: a] :
( ( list_update_a @ ( cons_a @ X2 @ Xs2 ) @ zero_zero_nat @ Y2 )
= ( cons_a @ Y2 @ Xs2 ) ) ).
% list_update_code(2)
thf(fact_768_list__update__code_I3_J,axiom,
! [X2: a,Xs2: list_a,I: nat,Y2: a] :
( ( list_update_a @ ( cons_a @ X2 @ Xs2 ) @ ( suc @ I ) @ Y2 )
= ( cons_a @ X2 @ ( list_update_a @ Xs2 @ I @ Y2 ) ) ) ).
% list_update_code(3)
thf(fact_769_last_Osimps,axiom,
! [Xs2: list_a,X2: a] :
( ( ( Xs2 = nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs2 ) )
= X2 ) )
& ( ( Xs2 != nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs2 ) )
= ( last_a @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_770_last__ConsL,axiom,
! [Xs2: list_a,X2: a] :
( ( Xs2 = nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs2 ) )
= X2 ) ) ).
% last_ConsL
thf(fact_771_last__ConsR,axiom,
! [Xs2: list_a,X2: a] :
( ( Xs2 != nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs2 ) )
= ( last_a @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_772_nth__list__update,axiom,
! [I: nat,Xs2: list_a,J: nat,X2: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( ( ( I = J )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I @ X2 ) @ J )
= X2 ) )
& ( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I @ X2 ) @ J )
= ( nth_a @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_773_list__update__same__conv,axiom,
! [I: nat,Xs2: list_a,X2: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( ( ( list_update_a @ Xs2 @ I @ X2 )
= Xs2 )
= ( ( nth_a @ Xs2 @ I )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_774_mset__swap,axiom,
! [I: nat,Ls: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Ls ) )
=> ( ( mset_a @ ( list_update_a @ ( list_update_a @ Ls @ J @ ( nth_a @ Ls @ I ) ) @ I @ ( nth_a @ Ls @ J ) ) )
= ( mset_a @ Ls ) ) ) ) ).
% mset_swap
thf(fact_775_Bolzano,axiom,
! [A: real,B2: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ! [A4: real,B4: real,C4: real] :
( ( P @ A4 @ B4 )
=> ( ( P @ B4 @ C4 )
=> ( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C4 )
=> ( P @ A4 @ C4 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B2 )
=> ? [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
& ! [A4: real,B4: real] :
( ( ( ord_less_eq_real @ A4 @ X4 )
& ( ord_less_eq_real @ X4 @ B4 )
& ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D ) )
=> ( P @ A4 @ B4 ) ) ) ) )
=> ( P @ A @ B2 ) ) ) ) ).
% Bolzano
thf(fact_776_nth__Cons__numeral,axiom,
! [X2: a,Xs2: list_a,V: num] :
( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ ( numeral_numeral_nat @ V ) )
= ( nth_a @ Xs2 @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_777_nth__rotate1,axiom,
! [N: nat,Xs2: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ ( rotate1_a @ Xs2 ) @ N )
= ( nth_a @ Xs2 @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs2 ) ) ) ) ) ).
% nth_rotate1
thf(fact_778_set__swap,axiom,
! [I: nat,Xs2: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs2 ) )
=> ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs2 @ I @ ( nth_a @ Xs2 @ J ) ) @ J @ ( nth_a @ Xs2 @ I ) ) )
= ( set_a2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_779_numeral__eq__iff,axiom,
! [M2: num,N: num] :
( ( ( numeral_numeral_nat @ M2 )
= ( numeral_numeral_nat @ N ) )
= ( M2 = N ) ) ).
% numeral_eq_iff
thf(fact_780_numeral__eq__iff,axiom,
! [M2: num,N: num] :
( ( ( numeral_numeral_int @ M2 )
= ( numeral_numeral_int @ N ) )
= ( M2 = N ) ) ).
% numeral_eq_iff
thf(fact_781_numeral__eq__iff,axiom,
! [M2: num,N: num] :
( ( ( numeral_numeral_real @ M2 )
= ( numeral_numeral_real @ N ) )
= ( M2 = N ) ) ).
% numeral_eq_iff
thf(fact_782_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_783_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_784_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_785_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_786_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_787_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_788_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_789_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_790_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_791_in__set__remove1,axiom,
! [A: a,B2: a,Xs2: list_a] :
( ( A != B2 )
=> ( ( member_a2 @ A @ ( set_a2 @ ( remove1_a @ B2 @ Xs2 ) ) )
= ( member_a2 @ A @ ( set_a2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_792_in__set__insert,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( insert_a @ X2 @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_793_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_794_in__set__replicate,axiom,
! [X2: a,N: nat,Y2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ ( replicate_a @ N @ Y2 ) ) )
= ( ( X2 = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_795_set__mset__mset,axiom,
! [Xs2: list_a] :
( ( set_mset_a @ ( mset_a @ Xs2 ) )
= ( set_a2 @ Xs2 ) ) ).
% set_mset_mset
thf(fact_796_not__in__set__insert,axiom,
! [X2: a,Xs2: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( insert_a @ X2 @ Xs2 )
= ( cons_a @ X2 @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_797_set__update__subsetI,axiom,
! [Xs2: list_a,A2: set_a,X2: a,I: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ A2 )
=> ( ( member_a2 @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs2 @ I @ X2 ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_798_subset__code_I1_J,axiom,
! [Xs2: list_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ B3 )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs2 ) )
=> ( member_a2 @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_799_set__subset__Cons,axiom,
! [Xs2: list_a,X2: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ ( cons_a @ X2 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_800_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_801_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_802_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_803_in__set__nthsD,axiom,
! [X2: a,Xs2: list_a,I5: set_nat] :
( ( member_a2 @ X2 @ ( set_a2 @ ( nths_a @ Xs2 @ I5 ) ) )
=> ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) ) ) ).
% in_set_nthsD
thf(fact_804_notin__set__nthsI,axiom,
! [X2: a,Xs2: list_a,I5: set_nat] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
=> ~ ( member_a2 @ X2 @ ( set_a2 @ ( nths_a @ Xs2 @ I5 ) ) ) ) ).
% notin_set_nthsI
thf(fact_805_list_Oset__intros_I2_J,axiom,
! [Y2: a,X22: list_a,X21: a] :
( ( member_a2 @ Y2 @ ( set_a2 @ X22 ) )
=> ( member_a2 @ Y2 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_806_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a2 @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_807_list_Oset__cases,axiom,
! [E2: a,A: list_a] :
( ( member_a2 @ E2 @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E2 @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E2 @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_808_set__ConsD,axiom,
! [Y2: a,X2: a,Xs2: list_a] :
( ( member_a2 @ Y2 @ ( set_a2 @ ( cons_a @ X2 @ Xs2 ) ) )
=> ( ( Y2 = X2 )
| ( member_a2 @ Y2 @ ( set_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_809_remove1__idem,axiom,
! [X2: a,Xs2: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( remove1_a @ X2 @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_810_notin__set__remove1,axiom,
! [X2: a,Xs2: list_a,Y2: a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
=> ~ ( member_a2 @ X2 @ ( set_a2 @ ( remove1_a @ Y2 @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_811_mset__eq__setD,axiom,
! [Xs2: list_a,Ys3: list_a] :
( ( ( mset_a @ Xs2 )
= ( mset_a @ Ys3 ) )
=> ( ( set_a2 @ Xs2 )
= ( set_a2 @ Ys3 ) ) ) ).
% mset_eq_setD
thf(fact_812_in__set__member,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
= ( member_a @ Xs2 @ X2 ) ) ).
% in_set_member
thf(fact_813_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_814_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_815_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_816_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_817_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_818_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_819_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_820_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_821_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_822_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_823_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_824_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_825_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_826_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_827_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% one_le_numeral
thf(fact_828_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_829_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_830_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_831_replicate__length__same,axiom,
! [Xs2: list_a,X2: a] :
( ! [X4: a] :
( ( member_a2 @ X4 @ ( set_a2 @ Xs2 ) )
=> ( X4 = X2 ) )
=> ( ( replicate_a @ ( size_size_list_a @ Xs2 ) @ X2 )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_832_replicate__eqI,axiom,
! [Xs2: list_a,N: nat,X2: a] :
( ( ( size_size_list_a @ Xs2 )
= N )
=> ( ! [Y4: a] :
( ( member_a2 @ Y4 @ ( set_a2 @ Xs2 ) )
=> ( Y4 = X2 ) )
=> ( Xs2
= ( replicate_a @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_833_hd__in__set,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( member_a2 @ ( hd_a @ Xs2 ) @ ( set_a2 @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_834_list_Oset__sel_I1_J,axiom,
! [A: list_a] :
( ( A != nil_a )
=> ( member_a2 @ ( hd_a @ A ) @ ( set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_835_in__multiset__in__set,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a2 @ X2 @ ( set_mset_a @ ( mset_a @ Xs2 ) ) )
= ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) ) ) ).
% in_multiset_in_set
thf(fact_836_last__in__set,axiom,
! [As: list_a] :
( ( As != nil_a )
=> ( member_a2 @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_837_Cons__in__subseqsD,axiom,
! [Y2: a,Ys3: list_a,Xs2: list_a] :
( ( member_list_a @ ( cons_a @ Y2 @ Ys3 ) @ ( set_list_a2 @ ( subseqs_a @ Xs2 ) ) )
=> ( member_list_a @ Ys3 @ ( set_list_a2 @ ( subseqs_a @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_838_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X3: a,Xs3: list_a] : ( if_list_a @ ( member_a2 @ X3 @ ( set_a2 @ Xs3 ) ) @ Xs3 @ ( cons_a @ X3 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_839_in__set__product__lists__length,axiom,
! [Xs2: list_a,Xss2: list_list_a] :
( ( member_list_a @ Xs2 @ ( set_list_a2 @ ( product_lists_a @ Xss2 ) ) )
=> ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_840_length__n__lists__elem,axiom,
! [Ys3: list_a,N: nat,Xs2: list_a] :
( ( member_list_a @ Ys3 @ ( set_list_a2 @ ( n_lists_a @ N @ Xs2 ) ) )
=> ( ( size_size_list_a @ Ys3 )
= N ) ) ).
% length_n_lists_elem
thf(fact_841_length__pos__if__in__set,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_842_all__set__conv__all__nth,axiom,
! [Xs2: list_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs2 ) )
=> ( P @ X3 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( P @ ( nth_a @ Xs2 @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_843_all__nth__imp__all__set,axiom,
! [Xs2: list_a,P: a > $o,X2: a] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs2 ) )
=> ( P @ ( nth_a @ Xs2 @ I4 ) ) )
=> ( ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_844_in__set__conv__nth,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
& ( ( nth_a @ Xs2 @ I2 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_845_list__ball__nth,axiom,
! [N: nat,Xs2: list_a,P: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ! [X4: a] :
( ( member_a2 @ X4 @ ( set_a2 @ Xs2 ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_a @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_846_nth__mem,axiom,
! [N: nat,Xs2: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( member_a2 @ ( nth_a @ Xs2 @ N ) @ ( set_a2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_847_set__update__memI,axiom,
! [N: nat,Xs2: list_a,X2: a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( member_a2 @ X2 @ ( set_a2 @ ( list_update_a @ Xs2 @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_848_length__remove1,axiom,
! [X2: a,Xs2: list_a] :
( ( ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( size_size_list_a @ ( remove1_a @ X2 @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) ) )
& ( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( size_size_list_a @ ( remove1_a @ X2 @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) ) ) ) ).
% length_remove1
thf(fact_849_nth__equal__first__eq,axiom,
! [X2: a,Xs2: list_a,N: nat] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_850_mod__by__Suc__0,axiom,
! [M2: nat] :
( ( modulo_modulo_nat @ M2 @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_851_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_852_mod__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( modulo_modulo_nat @ M2 @ N )
= M2 ) ) ).
% mod_less
thf(fact_853_minus__mod__self2,axiom,
! [A: int,B2: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% minus_mod_self2
thf(fact_854_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_855_mod__diff__eq,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% mod_diff_eq
thf(fact_856_mod__diff__cong,axiom,
! [A: int,C: int,A6: int,B2: int,B6: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A6 @ C ) )
=> ( ( ( modulo_modulo_int @ B2 @ C )
= ( modulo_modulo_int @ B6 @ C ) )
=> ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A6 @ B6 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_857_mod__diff__left__eq,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% mod_diff_left_eq
thf(fact_858_mod__diff__right__eq,axiom,
! [A: int,B2: int,C: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% mod_diff_right_eq
thf(fact_859_mod__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% mod_Suc_eq
thf(fact_860_mod__Suc__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_861_mod__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ M2 ) ).
% mod_less_eq_dividend
thf(fact_862_mod__Suc,axiom,
! [M2: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_863_mod__less__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_864_mod__induct,axiom,
! [P: nat > $o,N: nat,P5: nat,M2: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P5 )
=> ( ( ord_less_nat @ M2 @ P5 )
=> ( ! [N4: nat] :
( ( ord_less_nat @ N4 @ P5 )
=> ( ( P @ N4 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N4 ) @ P5 ) ) ) )
=> ( P @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_865_mod__Suc__le__divisor,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_866_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M3: nat,N5: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N5 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N5 ) @ N5 ) ) ) ) ).
% mod_if
thf(fact_867_le__mod__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( modulo_modulo_nat @ M2 @ N )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_868_mod__le__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_869_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
=> ( ! [M4: nat,N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 @ ( modulo_modulo_nat @ M4 @ N4 ) )
=> ( P @ M4 @ N4 ) ) )
=> ( P @ M2 @ N ) ) ) ).
% gcd_nat_induct
thf(fact_870_the__elem__set,axiom,
! [X2: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% the_elem_set
thf(fact_871_psubset__imp__ex__mem,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ? [B4: a] : ( member_a2 @ B4 @ ( minus_minus_set_a @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_872_psubsetD,axiom,
! [A2: set_a,B3: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ( member_a2 @ C @ A2 )
=> ( member_a2 @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_873_hd__rotate__conv__nth,axiom,
! [Xs2: list_a,N: nat] :
( ( Xs2 != nil_a )
=> ( ( hd_a @ ( rotate_a @ N @ Xs2 ) )
= ( nth_a @ Xs2 @ ( modulo_modulo_nat @ N @ ( size_size_list_a @ Xs2 ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_874_card__set__1__iff__replicate,axiom,
! [Xs2: list_a] :
( ( ( finite_card_a @ ( set_a2 @ Xs2 ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs2 != nil_a )
& ? [X3: a] :
( Xs2
= ( replicate_a @ ( size_size_list_a @ Xs2 ) @ X3 ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_875_DiffI,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a2 @ C @ A2 )
=> ( ~ ( member_a2 @ C @ B3 )
=> ( member_a2 @ C @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_876_Diff__iff,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( ( member_a2 @ C @ A2 )
& ~ ( member_a2 @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_877_length__rotate,axiom,
! [N: nat,Xs2: list_a] :
( ( size_size_list_a @ ( rotate_a @ N @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) ) ).
% length_rotate
thf(fact_878_rotate__length01,axiom,
! [Xs2: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat )
=> ( ( rotate_a @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_length01
thf(fact_879_rotate__id,axiom,
! [N: nat,Xs2: list_a] :
( ( ( modulo_modulo_nat @ N @ ( size_size_list_a @ Xs2 ) )
= zero_zero_nat )
=> ( ( rotate_a @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_id
thf(fact_880_DiffE,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ~ ( ( member_a2 @ C @ A2 )
=> ( member_a2 @ C @ B3 ) ) ) ).
% DiffE
thf(fact_881_DiffD1,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ( member_a2 @ C @ A2 ) ) ).
% DiffD1
thf(fact_882_DiffD2,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ~ ( member_a2 @ C @ B3 ) ) ).
% DiffD2
thf(fact_883_rotate__conv__mod,axiom,
( rotate_a
= ( ^ [N5: nat,Xs3: list_a] : ( rotate_a @ ( modulo_modulo_nat @ N5 @ ( size_size_list_a @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_884_card__length,axiom,
! [Xs2: list_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( set_a2 @ Xs2 ) ) @ ( size_size_list_a @ Xs2 ) ) ).
% card_length
thf(fact_885_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_886_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_887_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_888_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_889_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_890_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_891_nth__rotate,axiom,
! [N: nat,Xs2: list_a,M2: nat] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ ( rotate_a @ M2 @ Xs2 ) @ N )
= ( nth_a @ Xs2 @ ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ N ) @ ( size_size_list_a @ Xs2 ) ) ) ) ) ).
% nth_rotate
thf(fact_892_add__right__cancel,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_893_add__right__cancel,axiom,
! [B2: int,A: int,C: int] :
( ( ( plus_plus_int @ B2 @ A )
= ( plus_plus_int @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_894_add__left__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_895_add__left__cancel,axiom,
! [A: int,B2: int,C: int] :
( ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_896_add__le__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_eq_int @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_897_add__le__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_898_add__le__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
= ( ord_less_eq_real @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_899_add__le__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_eq_int @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_900_add__le__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_901_add__le__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
= ( ord_less_eq_real @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_902_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_903_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_904_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_905_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_906_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_907_add__cancel__left__left,axiom,
! [B2: int,A: int] :
( ( ( plus_plus_int @ B2 @ A )
= A )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_908_add__cancel__left__left,axiom,
! [B2: nat,A: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_909_add__cancel__left__left,axiom,
! [B2: real,A: real] :
( ( ( plus_plus_real @ B2 @ A )
= A )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_910_add__cancel__left__right,axiom,
! [A: int,B2: int] :
( ( ( plus_plus_int @ A @ B2 )
= A )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_911_add__cancel__left__right,axiom,
! [A: nat,B2: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_912_add__cancel__left__right,axiom,
! [A: real,B2: real] :
( ( ( plus_plus_real @ A @ B2 )
= A )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_913_add__cancel__right__left,axiom,
! [A: int,B2: int] :
( ( A
= ( plus_plus_int @ B2 @ A ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_914_add__cancel__right__left,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ B2 @ A ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_915_add__cancel__right__left,axiom,
! [A: real,B2: real] :
( ( A
= ( plus_plus_real @ B2 @ A ) )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_916_add__cancel__right__right,axiom,
! [A: int,B2: int] :
( ( A
= ( plus_plus_int @ A @ B2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_917_add__cancel__right__right,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ A @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_918_add__cancel__right__right,axiom,
! [A: real,B2: real] :
( ( A
= ( plus_plus_real @ A @ B2 ) )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_919_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_920_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y2 ) )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_921_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_922_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_923_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_924_add__less__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_int @ A @ B2 ) ) ).
% add_less_cancel_left
thf(fact_925_add__less__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_left
thf(fact_926_add__less__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
= ( ord_less_real @ A @ B2 ) ) ).
% add_less_cancel_left
thf(fact_927_add__less__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_int @ A @ B2 ) ) ).
% add_less_cancel_right
thf(fact_928_add__less__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_right
thf(fact_929_add__less__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
= ( ord_less_real @ A @ B2 ) ) ).
% add_less_cancel_right
thf(fact_930_add__numeral__left,axiom,
! [V: num,W2: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_931_add__numeral__left,axiom,
! [V: num,W2: num,Z2: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_932_add__numeral__left,axiom,
! [V: num,W2: num,Z2: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_933_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_934_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_935_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_936_add__diff__cancel,axiom,
! [A: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel
thf(fact_937_add__diff__cancel,axiom,
! [A: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel
thf(fact_938_diff__add__cancel,axiom,
! [A: real,B2: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
= A ) ).
% diff_add_cancel
thf(fact_939_diff__add__cancel,axiom,
! [A: int,B2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
= A ) ).
% diff_add_cancel
thf(fact_940_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_941_add__diff__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
= ( minus_minus_real @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_942_add__diff__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
= ( minus_minus_int @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_943_add__diff__cancel__left_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_944_add__diff__cancel__left_H,axiom,
! [A: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_945_add__diff__cancel__left_H,axiom,
! [A: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_946_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_947_add__diff__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
= ( minus_minus_real @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_948_add__diff__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_949_add__diff__cancel__right_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_950_add__diff__cancel__right_H,axiom,
! [A: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_951_add__diff__cancel__right_H,axiom,
! [A: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_952_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_953_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_954_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_955_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_956_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_957_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_958_add__le__same__cancel1,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_959_add__le__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_960_add__le__same__cancel1,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_961_add__le__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_962_add__le__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_963_add__le__same__cancel2,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_964_le__add__same__cancel1,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_965_le__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_966_le__add__same__cancel1,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B2 ) )
= ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_967_le__add__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B2 @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_968_le__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_969_le__add__same__cancel2,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B2 @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_970_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_971_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_972_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_973_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_974_add__less__same__cancel1,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_975_add__less__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_976_add__less__same__cancel1,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_977_add__less__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_978_add__less__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_979_add__less__same__cancel2,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_980_less__add__same__cancel1,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_981_less__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_982_less__add__same__cancel1,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B2 ) )
= ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_983_less__add__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B2 @ A ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_984_less__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_985_less__add__same__cancel2,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B2 @ A ) )
= ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_986_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_987_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_988_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_989_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_990_le__add__diff__inverse2,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_991_le__add__diff__inverse2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B2 ) @ B2 )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_992_le__add__diff__inverse2,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_993_le__add__diff__inverse,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_994_le__add__diff__inverse,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_995_le__add__diff__inverse,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_996_diff__add__zero,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_997_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_998_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_999_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_1000_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_1001_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_1002_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_1003_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1004_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1005_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1006_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1007_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1008_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1009_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1010_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1011_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1012_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1013_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1014_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1015_diff__diff__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_1016_diff__diff__eq,axiom,
! [A: real,B2: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_1017_diff__diff__eq,axiom,
! [A: int,B2: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_1018_add__diff__add,axiom,
! [A: real,C: real,B2: real,D3: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D3 ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ ( minus_minus_real @ C @ D3 ) ) ) ).
% add_diff_add
thf(fact_1019_add__diff__add,axiom,
! [A: int,C: int,B2: int,D3: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D3 ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ ( minus_minus_int @ C @ D3 ) ) ) ).
% add_diff_add
thf(fact_1020_add__implies__diff,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A )
=> ( C
= ( minus_minus_nat @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_1021_add__implies__diff,axiom,
! [C: real,B2: real,A: real] :
( ( ( plus_plus_real @ C @ B2 )
= A )
=> ( C
= ( minus_minus_real @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_1022_add__implies__diff,axiom,
! [C: int,B2: int,A: int] :
( ( ( plus_plus_int @ C @ B2 )
= A )
=> ( C
= ( minus_minus_int @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_1023_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B2: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1024_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B2: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1025_diff__add__eq,axiom,
! [A: real,B2: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% diff_add_eq
thf(fact_1026_diff__add__eq,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).
% diff_add_eq
thf(fact_1027_diff__diff__eq2,axiom,
! [A: real,B2: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B2 @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% diff_diff_eq2
thf(fact_1028_diff__diff__eq2,axiom,
! [A: int,B2: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).
% diff_diff_eq2
thf(fact_1029_add__diff__eq,axiom,
! [A: real,B2: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B2 @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% add_diff_eq
thf(fact_1030_add__diff__eq,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% add_diff_eq
thf(fact_1031_eq__diff__eq,axiom,
! [A: real,C: real,B2: real] :
( ( A
= ( minus_minus_real @ C @ B2 ) )
= ( ( plus_plus_real @ A @ B2 )
= C ) ) ).
% eq_diff_eq
thf(fact_1032_eq__diff__eq,axiom,
! [A: int,C: int,B2: int] :
( ( A
= ( minus_minus_int @ C @ B2 ) )
= ( ( plus_plus_int @ A @ B2 )
= C ) ) ).
% eq_diff_eq
thf(fact_1033_diff__eq__eq,axiom,
! [A: real,B2: real,C: real] :
( ( ( minus_minus_real @ A @ B2 )
= C )
= ( A
= ( plus_plus_real @ C @ B2 ) ) ) ).
% diff_eq_eq
thf(fact_1034_diff__eq__eq,axiom,
! [A: int,B2: int,C: int] :
( ( ( minus_minus_int @ A @ B2 )
= C )
= ( A
= ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_eq_eq
thf(fact_1035_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B2: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B2 )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% group_cancel.sub1
thf(fact_1036_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B2: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B2 )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B2 ) ) ) ) ).
% group_cancel.sub1
thf(fact_1037_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1038_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_1039_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1040_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1041_add__le__imp__le__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_eq_int @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1042_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1043_add__le__imp__le__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1044_add__le__imp__le__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_eq_int @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1045_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1046_add__le__imp__le__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1047_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
? [C3: nat] :
( B5
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_1048_add__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_1049_add__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_1050_add__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_1051_less__eqE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ~ ! [C4: nat] :
( B2
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_1052_add__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_1053_add__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_1054_add__left__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_1055_add__mono,axiom,
! [A: int,B2: int,C: int,D3: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D3 ) ) ) ) ).
% add_mono
thf(fact_1056_add__mono,axiom,
! [A: nat,B2: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D3 ) ) ) ) ).
% add_mono
thf(fact_1057_add__mono,axiom,
! [A: real,B2: real,C: real,D3: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ D3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D3 ) ) ) ) ).
% add_mono
thf(fact_1058_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1059_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1060_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1061_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1062_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1063_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1064_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1065_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1066_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1067_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N5: nat] :
? [K3: nat] :
( N5
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1068_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1069_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1070_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1071_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1072_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1073_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1074_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1075_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1076_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1077_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1078_le__num__One__iff,axiom,
! [X2: num] :
( ( ord_less_eq_num @ X2 @ one )
= ( X2 = one ) ) ).
% le_num_One_iff
thf(fact_1079_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_1080_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1081_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1082_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_1083_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_1084_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1085_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_1086_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1087_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_1088_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1089_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1090_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1091_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1092_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_1093_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1094_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_1095_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1096_is__num__normalize_I1_J,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_1097_add__right__imp__eq,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_1098_add__right__imp__eq,axiom,
! [B2: int,A: int,C: int] :
( ( ( plus_plus_int @ B2 @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_1099_add__left__imp__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_1100_add__left__imp__eq,axiom,
! [A: int,B2: int,C: int] :
( ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_1101_add_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_1102_add_Oleft__commute,axiom,
! [B2: int,A: int,C: int] :
( ( plus_plus_int @ B2 @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_1103_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B5: nat] : ( plus_plus_nat @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_1104_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B5: int] : ( plus_plus_int @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_1105_add_Oright__cancel,axiom,
! [B2: int,A: int,C: int] :
( ( ( plus_plus_int @ B2 @ A )
= ( plus_plus_int @ C @ A ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_1106_add_Oleft__cancel,axiom,
! [A: int,B2: int,C: int] :
( ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ A @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_1107_add_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_1108_add_Oassoc,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_1109_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B2: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_1110_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B2: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B2 ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_1111_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_1112_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B2: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_1113_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1114_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1115_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1116_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1117_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1118_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1119_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1120_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1121_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1122_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1123_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1124_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1125_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1126_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1127_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1128_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1129_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1130_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1131_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1132_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1133_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1134_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1135_add__strict__mono,axiom,
! [A: int,B2: int,C: int,D3: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D3 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_1136_add__strict__mono,axiom,
! [A: nat,B2: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_1137_add__strict__mono,axiom,
! [A: real,B2: real,C: real,D3: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ C @ D3 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_1138_add__strict__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_1139_add__strict__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_1140_add__strict__left__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_1141_add__strict__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1142_add__strict__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1143_add__strict__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1144_add__less__imp__less__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_int @ A @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_1145_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_1146_add__less__imp__less__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
=> ( ord_less_real @ A @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_1147_add__less__imp__less__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_int @ A @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_1148_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_1149_add__less__imp__less__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
=> ( ord_less_real @ A @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_1150_add__decreasing,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_1151_add__decreasing,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_1152_add__decreasing,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_1153_add__increasing,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1154_add__increasing,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1155_add__increasing,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1156_add__decreasing2,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_1157_add__decreasing2,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_1158_add__decreasing2,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_1159_add__increasing2,axiom,
! [C: int,B2: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B2 @ A )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1160_add__increasing2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1161_add__increasing2,axiom,
! [C: real,B2: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B2 @ A )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1162_add__nonneg__nonneg,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1163_add__nonneg__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1164_add__nonneg__nonneg,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1165_add__nonpos__nonpos,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1166_add__nonpos__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1167_add__nonpos__nonpos,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_1168_add__nonneg__eq__0__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ( plus_plus_int @ X2 @ Y2 )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1169_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1170_add__nonneg__eq__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( plus_plus_real @ X2 @ Y2 )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1171_add__nonpos__eq__0__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X2 @ Y2 )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1172_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1173_add__nonpos__eq__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X2 @ Y2 )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1174_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1175_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1176_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1177_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1178_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1179_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1180_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N5: nat] :
? [K3: nat] :
( N5
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1181_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1182_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1183_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1184_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1185_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1186_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1187_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1188_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1189_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1190_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1191_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1192_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1193_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1194_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1195_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1196_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1197_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B2 @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1198_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A @ B2 ) )
= ( ( ( ord_less_nat @ A @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B2 @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1199_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1200_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1201_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N5: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N5 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% add_eq_if
thf(fact_1202_Suc__0__mod__numeral_I1_J,axiom,
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ one ) )
= zero_zero_nat ) ).
% Suc_0_mod_numeral(1)
thf(fact_1203_semiring__norm_I75_J,axiom,
! [M2: num] :
~ ( ord_less_num @ M2 @ one ) ).
% semiring_norm(75)
thf(fact_1204_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_1205_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1206_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1207_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M: nat] :
( ( P @ X2 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1208_mod2__gr__0,axiom,
! [M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% mod2_gr_0
thf(fact_1209_verit__eq__simplify_I8_J,axiom,
! [X23: num,Y23: num] :
( ( ( bit0 @ X23 )
= ( bit0 @ Y23 ) )
= ( X23 = Y23 ) ) ).
% verit_eq_simplify(8)
thf(fact_1210_semiring__norm_I78_J,axiom,
! [M2: num,N: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% semiring_norm(78)
thf(fact_1211_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_1212_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_1213_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_1214_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_1215_mod2__Suc__Suc,axiom,
! [M2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% mod2_Suc_Suc
thf(fact_1216_not__mod2__eq__Suc__0__eq__0,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= ( suc @ zero_zero_nat ) )
= ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_1217_add__self__mod__2,axiom,
! [M2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% add_self_mod_2
thf(fact_1218_Suc__0__mod__numeral_I2_J,axiom,
! [N: num] :
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) )
= one_one_nat ) ).
% Suc_0_mod_numeral(2)
thf(fact_1219_verit__eq__simplify_I10_J,axiom,
! [X23: num] :
( one
!= ( bit0 @ X23 ) ) ).
% verit_eq_simplify(10)
thf(fact_1220_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_1221_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1222_num_Osize_I5_J,axiom,
! [X23: num] :
( ( size_size_num @ ( bit0 @ X23 ) )
= ( plus_plus_nat @ ( size_size_num @ X23 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_1223_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_1224_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_1225_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1226_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_1227_num_Osize__gen_I2_J,axiom,
! [X23: num] :
( ( size_num @ ( bit0 @ X23 ) )
= ( plus_plus_nat @ ( size_num @ X23 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(2)
thf(fact_1228_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_1229_nat__of__char__less__256,axiom,
! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% nat_of_char_less_256
thf(fact_1230_even__diff__nat,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).
% even_diff_nat
thf(fact_1231_nat__dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd_nat @ M2 @ one_one_nat )
= ( M2 = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1232_dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd_nat @ M2 @ ( suc @ zero_zero_nat ) )
= ( M2
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_1233_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_1234_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% even_Suc
thf(fact_1235_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_1236_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_1237_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M2: nat,Q4: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( ( modulo_modulo_nat @ M2 @ Q4 )
= ( modulo_modulo_nat @ N @ Q4 ) )
= ( dvd_dvd_nat @ Q4 @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_1238_less__eq__dvd__minus,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( dvd_dvd_nat @ M2 @ N )
= ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1239_dvd__diffD1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
=> ( ( dvd_dvd_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1240_dvd__diffD,axiom,
! [K: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_nat @ K @ M2 ) ) ) ) ).
% dvd_diffD
thf(fact_1241_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_1242_mod__greater__zero__iff__not__dvd,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ N ) )
= ( ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_1243_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1244_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1245_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_1246_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_1247_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_1248_dvd__antisym,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ M2 @ N )
=> ( ( dvd_dvd_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% dvd_antisym
thf(fact_1249_dvd__diff__nat,axiom,
! [K: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M2 )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1250_nat__dvd__not__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N )
=> ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).
% nat_dvd_not_less
thf(fact_1251_dvd__pos__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M2 @ N )
=> ( ord_less_nat @ zero_zero_nat @ M2 ) ) ) ).
% dvd_pos_nat
thf(fact_1252_dvd__minus__self,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) )
= ( ( ord_less_nat @ N @ M2 )
| ( dvd_dvd_nat @ M2 @ N ) ) ) ).
% dvd_minus_self
thf(fact_1253_even__even__mod__4__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_1254_mod__nat__eqI,axiom,
! [R2: nat,N: nat,M2: nat] :
( ( ord_less_nat @ R2 @ N )
=> ( ( ord_less_eq_nat @ R2 @ M2 )
=> ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M2 @ R2 ) )
=> ( ( modulo_modulo_nat @ M2 @ N )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_1255_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_1256_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_1257_real__arch__pow,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ? [N4: nat] : ( ord_less_real @ Y2 @ ( power_power_real @ X2 @ N4 ) ) ) ).
% real_arch_pow
thf(fact_1258_real__arch__pow__inv,axiom,
! [Y2: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N4 ) @ Y2 ) ) ) ).
% real_arch_pow_inv
thf(fact_1259_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ ( suc @ N ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1260_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_1261_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1262_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ( power_power_real @ X4 @ N )
= A )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N )
= A ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y2: list_a] :
( ( if_list_a @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y2: list_a] :
( ( if_list_a @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [C5: list_a] :
( ( mset_a @ ya )
= ( mset_a @ ( cons_a @ y @ ( cons_a @ z @ C5 ) ) ) ) ).
%------------------------------------------------------------------------------