TPTP Problem File: ITP134^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP134^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Number_Partition problem prob_94__5324562_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Number_Partition/prob_94__5324562_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 429 ( 135 unt; 79 typ; 0 def)
% Number of atoms : 1178 ( 279 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 3087 ( 93 ~; 21 |; 162 &;2318 @)
% ( 0 <=>; 493 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 820 ( 820 >; 0 *; 0 +; 0 <<)
% Number of symbols : 63 ( 62 usr; 3 con; 0-2 aty)
% Number of variables : 1092 ( 187 ^; 862 !; 43 ?;1092 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:42:27.163
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_na1064332610at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_na1201295426at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_se1657353at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_nat_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat_nat3: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat_nat4: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat_nat5: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_set_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (62)
thf(sy_c_Finite__Set_OFpow_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite252967095at_nat: set_nat_nat_nat_nat3 > set_se1657353at_nat ).
thf(sy_c_Finite__Set_OFpow_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
finite1633036872at_nat: set_nat_nat_nat2 > set_set_nat_nat_nat2 ).
thf(sy_c_Finite__Set_OFpow_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite90785608at_nat: set_nat_nat_nat > set_set_nat_nat_nat ).
thf(sy_c_Finite__Set_OFpow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite_Fpow_nat_nat: set_nat_nat > set_set_nat_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
finite1526199988at_nat: set_nat_nat_nat_nat5 > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
finite2081887156at_nat: set_nat_nat_nat_nat4 > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
finite1090277411at_nat: set_na1201295426at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
finite1397812515at_nat: set_na1064332610at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite1064868788at_nat: set_nat_nat_nat_nat3 > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
finite1440337093at_nat: set_nat_nat_nat2 > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
finite584233140at_nat: set_nat_nat_nat_nat2 > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
finite891768244at_nat: set_nat_nat_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite2045569477at_nat: set_nat_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite570312790at_nat: set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
finite547180394at_nat: set_se1657353at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
finite1977496699at_nat: set_set_nat_nat_nat2 > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
finite137548155at_nat: set_set_nat_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite604103692at_nat: set_set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite2012248349et_nat: set_set_nat > $o ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Number__Partition__Mirabelle__zerdlymyoj_Opartitions,type,
number1551313001itions: ( nat > nat ) > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le2059018749at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le2026740592_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
ord_le809907342at_nat: ( ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le1415139726at_nat: ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le2040082867at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le633272388at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le940807492at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le1415039317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Set_OCollect_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
collec2071977458at_nat: ( ( ( ( nat > nat ) > nat ) > nat ) > $o ) > set_nat_nat_nat_nat5 ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
collec480180978at_nat: ( ( ( nat > nat > nat ) > nat ) > $o ) > set_nat_nat_nat_nat4 ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
collec6783585at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > set_na1201295426at_nat ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
collec314318689at_nat: ( ( ( nat > nat ) > nat > nat > nat ) > $o ) > set_na1064332610at_nat ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec1610646258at_nat: ( ( ( nat > nat ) > nat > nat ) > $o ) > set_nat_nat_nat_nat3 ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
collect_nat_nat_nat: ( ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
collec1130010610at_nat: ( ( nat > ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat_nat2 ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
collec1437545714at_nat: ( ( nat > nat > nat > nat ) > $o ) > set_nat_nat_nat_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_nat_nat_nat2: ( ( nat > nat > nat ) > $o ) > set_nat_nat_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
collec1702444712at_nat: ( set_nat_nat_nat_nat3 > $o ) > set_se1657353at_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
collec309745849at_nat: ( set_nat_nat_nat2 > $o ) > set_set_nat_nat_nat2 ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
collec617280953at_nat: ( set_nat_nat_nat > $o ) > set_set_nat_nat_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_set_nat_nat: ( set_nat_nat > $o ) > set_set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member1128122036at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (349)
thf(fact_0__092_060open_062finite_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [F: nat > nat] :
( ! [I: nat] : ( ord_less_eq_nat @ ( F @ I ) @ n )
& ! [I: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ I )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% \<open>finite {f. (\<forall>i. f i \<le> n) \<and> (\<forall>i\<ge>n + 1. f i = 0)}\<close>
thf(fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
( ord_le1415039317at_nat
@ ( collect_nat_nat
@ ^ [P: nat > nat] : ( number1551313001itions @ P @ n ) )
@ ( collect_nat_nat
@ ^ [F: nat > nat] :
( ! [I: nat] : ( ord_less_eq_nat @ ( F @ I ) @ n )
& ! [I: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ I )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% \<open>{p. p partitions n} \<subseteq> {f. (\<forall>i. f i \<le> n) \<and> (\<forall>i\<ge>n + 1. f i = 0)}\<close>
thf(fact_2_finite__Collect__conjI,axiom,
! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ( ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ P2 ) )
| ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ Q ) ) )
=> ( finite2045569477at_nat
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3_finite__Collect__conjI,axiom,
! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
( ( ( finite1440337093at_nat @ ( collect_nat_nat_nat @ P2 ) )
| ( finite1440337093at_nat @ ( collect_nat_nat_nat @ Q ) ) )
=> ( finite1440337093at_nat
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_4_finite__Collect__conjI,axiom,
! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
( ( ( finite1064868788at_nat @ ( collec1610646258at_nat @ P2 ) )
| ( finite1064868788at_nat @ ( collec1610646258at_nat @ Q ) ) )
=> ( finite1064868788at_nat
@ ( collec1610646258at_nat
@ ^ [X: ( nat > nat ) > nat > nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_5_finite__Collect__conjI,axiom,
! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ( finite570312790at_nat @ ( collect_nat_nat @ P2 ) )
| ( finite570312790at_nat @ ( collect_nat_nat @ Q ) ) )
=> ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_6_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_7_finite__Collect__disjI,axiom,
! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ( finite2045569477at_nat
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ P2 ) )
& ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_8_finite__Collect__disjI,axiom,
! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
( ( finite1440337093at_nat
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite1440337093at_nat @ ( collect_nat_nat_nat @ P2 ) )
& ( finite1440337093at_nat @ ( collect_nat_nat_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_9_finite__Collect__disjI,axiom,
! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
( ( finite1064868788at_nat
@ ( collec1610646258at_nat
@ ^ [X: ( nat > nat ) > nat > nat] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite1064868788at_nat @ ( collec1610646258at_nat @ P2 ) )
& ( finite1064868788at_nat @ ( collec1610646258at_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_10_finite__Collect__disjI,axiom,
! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite570312790at_nat @ ( collect_nat_nat @ P2 ) )
& ( finite570312790at_nat @ ( collect_nat_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_11_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_12_partitions__zero,axiom,
! [P3: nat > nat] :
( ( number1551313001itions @ P3 @ zero_zero_nat )
= ( P3
= ( ^ [I: nat] : zero_zero_nat ) ) ) ).
% partitions_zero
thf(fact_13_partitions__bounds,axiom,
! [P3: nat > nat,N: nat,I2: nat] :
( ( number1551313001itions @ P3 @ N )
=> ( ord_less_eq_nat @ ( P3 @ I2 ) @ N ) ) ).
% partitions_bounds
thf(fact_14_finite__set__of__finite__funs,axiom,
! [A: set_nat,B: set_nat,D: nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [F: nat > nat] :
! [X: nat] :
( ( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( F @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_15_finite__set__of__finite__funs,axiom,
! [A: set_nat_nat,B: set_nat,D: nat] :
( ( finite570312790at_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite1440337093at_nat
@ ( collect_nat_nat_nat
@ ^ [F: ( nat > nat ) > nat] :
! [X: nat > nat] :
( ( ( member_nat_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
& ( ~ ( member_nat_nat @ X @ A )
=> ( ( F @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_16_finite__set__of__finite__funs,axiom,
! [A: set_nat,B: set_nat_nat,D: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( finite570312790at_nat @ B )
=> ( finite2045569477at_nat
@ ( collect_nat_nat_nat2
@ ^ [F: nat > nat > nat] :
! [X: nat] :
( ( ( member_nat @ X @ A )
=> ( member_nat_nat @ ( F @ X ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( F @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_17_finite__set__of__finite__funs,axiom,
! [A: set_nat_nat,B: set_nat_nat,D: nat > nat] :
( ( finite570312790at_nat @ A )
=> ( ( finite570312790at_nat @ B )
=> ( finite1064868788at_nat
@ ( collec1610646258at_nat
@ ^ [F: ( nat > nat ) > nat > nat] :
! [X: nat > nat] :
( ( ( member_nat_nat @ X @ A )
=> ( member_nat_nat @ ( F @ X ) @ B ) )
& ( ~ ( member_nat_nat @ X @ A )
=> ( ( F @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_18_finite__set__of__finite__funs,axiom,
! [A: set_nat,B: set_nat_nat_nat,D: nat > nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( finite2045569477at_nat @ B )
=> ( finite891768244at_nat
@ ( collec1437545714at_nat
@ ^ [F: nat > nat > nat > nat] :
! [X: nat] :
( ( ( member_nat @ X @ A )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( F @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_19_finite__set__of__finite__funs,axiom,
! [A: set_nat,B: set_nat_nat_nat2,D: ( nat > nat ) > nat] :
( ( finite_finite_nat @ A )
=> ( ( finite1440337093at_nat @ B )
=> ( finite584233140at_nat
@ ( collec1130010610at_nat
@ ^ [F: nat > ( nat > nat ) > nat] :
! [X: nat] :
( ( ( member_nat @ X @ A )
=> ( member_nat_nat_nat @ ( F @ X ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( F @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_20_finite__set__of__finite__funs,axiom,
! [A: set_nat_nat_nat,B: set_nat,D: nat] :
( ( finite2045569477at_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite2081887156at_nat
@ ( collec480180978at_nat
@ ^ [F: ( nat > nat > nat ) > nat] :
! [X: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
& ( ~ ( member_nat_nat_nat2 @ X @ A )
=> ( ( F @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_21_finite__set__of__finite__funs,axiom,
! [A: set_nat_nat_nat2,B: set_nat,D: nat] :
( ( finite1440337093at_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite1526199988at_nat
@ ( collec2071977458at_nat
@ ^ [F: ( ( nat > nat ) > nat ) > nat] :
! [X: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
& ( ~ ( member_nat_nat_nat @ X @ A )
=> ( ( F @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_22_finite__set__of__finite__funs,axiom,
! [A: set_nat_nat,B: set_nat_nat_nat,D: nat > nat > nat] :
( ( finite570312790at_nat @ A )
=> ( ( finite2045569477at_nat @ B )
=> ( finite1397812515at_nat
@ ( collec314318689at_nat
@ ^ [F: ( nat > nat ) > nat > nat > nat] :
! [X: nat > nat] :
( ( ( member_nat_nat @ X @ A )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ B ) )
& ( ~ ( member_nat_nat @ X @ A )
=> ( ( F @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_23_finite__set__of__finite__funs,axiom,
! [A: set_nat_nat,B: set_nat_nat_nat2,D: ( nat > nat ) > nat] :
( ( finite570312790at_nat @ A )
=> ( ( finite1440337093at_nat @ B )
=> ( finite1090277411at_nat
@ ( collec6783585at_nat
@ ^ [F: ( nat > nat ) > ( nat > nat ) > nat] :
! [X: nat > nat] :
( ( ( member_nat_nat @ X @ A )
=> ( member_nat_nat_nat @ ( F @ X ) @ B ) )
& ( ~ ( member_nat_nat @ X @ A )
=> ( ( F @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_24_not__finite__existsD,axiom,
! [P2: ( nat > nat > nat ) > $o] :
( ~ ( finite2045569477at_nat @ ( collect_nat_nat_nat2 @ P2 ) )
=> ? [X_1: nat > nat > nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_25_not__finite__existsD,axiom,
! [P2: ( ( nat > nat ) > nat ) > $o] :
( ~ ( finite1440337093at_nat @ ( collect_nat_nat_nat @ P2 ) )
=> ? [X_1: ( nat > nat ) > nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_26_not__finite__existsD,axiom,
! [P2: ( ( nat > nat ) > nat > nat ) > $o] :
( ~ ( finite1064868788at_nat @ ( collec1610646258at_nat @ P2 ) )
=> ? [X_1: ( nat > nat ) > nat > nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_27_not__finite__existsD,axiom,
! [P2: ( nat > nat ) > $o] :
( ~ ( finite570312790at_nat @ ( collect_nat_nat @ P2 ) )
=> ? [X_1: nat > nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_28_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_29_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A )
& ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_30_pigeonhole__infinite__rel,axiom,
! [A: set_nat_nat,B: set_nat,R: ( nat > nat ) > nat > $o] :
( ~ ( finite570312790at_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [A2: nat > nat] :
( ( member_nat_nat @ A2 @ A )
& ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_31_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat_nat,R: nat > ( nat > nat ) > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite570312790at_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A )
& ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_32_pigeonhole__infinite__rel,axiom,
! [A: set_nat_nat,B: set_nat_nat,R: ( nat > nat ) > ( nat > nat ) > $o] :
( ~ ( finite570312790at_nat @ A )
=> ( ( finite570312790at_nat @ B )
=> ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ? [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B )
& ~ ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [A2: nat > nat] :
( ( member_nat_nat @ A2 @ A )
& ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_33_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat_nat_nat,R: nat > ( nat > nat > nat ) > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite2045569477at_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat > nat > nat] :
( ( member_nat_nat_nat2 @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A )
& ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_34_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat_nat_nat2,R: nat > ( ( nat > nat ) > nat ) > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite1440337093at_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A )
& ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_35_pigeonhole__infinite__rel,axiom,
! [A: set_nat_nat_nat,B: set_nat,R: ( nat > nat > nat ) > nat > $o] :
( ~ ( finite2045569477at_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite2045569477at_nat
@ ( collect_nat_nat_nat2
@ ^ [A2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ A2 @ A )
& ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_36_pigeonhole__infinite__rel,axiom,
! [A: set_nat_nat_nat2,B: set_nat,R: ( ( nat > nat ) > nat ) > nat > $o] :
( ~ ( finite1440337093at_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite1440337093at_nat
@ ( collect_nat_nat_nat
@ ^ [A2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ A2 @ A )
& ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_37_pigeonhole__infinite__rel,axiom,
! [A: set_nat_nat,B: set_nat_nat_nat,R: ( nat > nat ) > ( nat > nat > nat ) > $o] :
( ~ ( finite570312790at_nat @ A )
=> ( ( finite2045569477at_nat @ B )
=> ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ? [Xa: nat > nat > nat] :
( ( member_nat_nat_nat2 @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ B )
& ~ ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [A2: nat > nat] :
( ( member_nat_nat @ A2 @ A )
& ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_38_pigeonhole__infinite__rel,axiom,
! [A: set_nat_nat,B: set_nat_nat_nat2,R: ( nat > nat ) > ( ( nat > nat ) > nat ) > $o] :
( ~ ( finite570312790at_nat @ A )
=> ( ( finite1440337093at_nat @ B )
=> ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ? [Xa: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ B )
& ~ ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [A2: nat > nat] :
( ( member_nat_nat @ A2 @ A )
& ( R @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_39_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite2012248349et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_40_finite__Collect__subsets,axiom,
! [A: set_nat_nat_nat] :
( ( finite2045569477at_nat @ A )
=> ( finite137548155at_nat
@ ( collec617280953at_nat
@ ^ [B2: set_nat_nat_nat] : ( ord_le940807492at_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_41_finite__Collect__subsets,axiom,
! [A: set_nat_nat_nat2] :
( ( finite1440337093at_nat @ A )
=> ( finite1977496699at_nat
@ ( collec309745849at_nat
@ ^ [B2: set_nat_nat_nat2] : ( ord_le633272388at_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_42_finite__Collect__subsets,axiom,
! [A: set_nat_nat_nat_nat3] :
( ( finite1064868788at_nat @ A )
=> ( finite547180394at_nat
@ ( collec1702444712at_nat
@ ^ [B2: set_nat_nat_nat_nat3] : ( ord_le2040082867at_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_43_finite__Collect__subsets,axiom,
! [A: set_nat_nat] :
( ( finite570312790at_nat @ A )
=> ( finite604103692at_nat
@ ( collect_set_nat_nat
@ ^ [B2: set_nat_nat] : ( ord_le1415039317at_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_44_bound__domain__and__range__impl__finitely__many__functions,axiom,
! [N: nat,M: nat] :
( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [F: nat > nat] :
( ! [I: nat] : ( ord_less_eq_nat @ ( F @ I ) @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ M @ I )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% bound_domain_and_range_impl_finitely_many_functions
thf(fact_45_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_46_finite__has__minimal2,axiom,
! [A: set_nat_nat,A3: nat > nat] :
( ( finite570312790at_nat @ A )
=> ( ( member_nat_nat @ A3 @ A )
=> ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
& ( ord_less_eq_nat_nat @ X2 @ A3 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_47_finite__has__minimal2,axiom,
! [A: set_nat_nat_nat,A3: nat > nat > nat] :
( ( finite2045569477at_nat @ A )
=> ( ( member_nat_nat_nat2 @ A3 @ A )
=> ? [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A )
& ( ord_le1415139726at_nat @ X2 @ A3 )
& ! [Xa: nat > nat > nat] :
( ( member_nat_nat_nat2 @ Xa @ A )
=> ( ( ord_le1415139726at_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_48_finite__has__minimal2,axiom,
! [A: set_nat_nat_nat2,A3: ( nat > nat ) > nat] :
( ( finite1440337093at_nat @ A )
=> ( ( member_nat_nat_nat @ A3 @ A )
=> ? [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A )
& ( ord_le809907342at_nat @ X2 @ A3 )
& ! [Xa: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa @ A )
=> ( ( ord_le809907342at_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_49_finite__has__minimal2,axiom,
! [A: set_nat_nat_nat_nat3,A3: ( nat > nat ) > nat > nat] :
( ( finite1064868788at_nat @ A )
=> ( ( member1128122036at_nat @ A3 @ A )
=> ? [X2: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ X2 @ A )
& ( ord_le2059018749at_nat @ X2 @ A3 )
& ! [Xa: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ Xa @ A )
=> ( ( ord_le2059018749at_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_50_finite__has__minimal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_51_finite__has__minimal2,axiom,
! [A: set_set_nat_nat,A3: set_nat_nat] :
( ( finite604103692at_nat @ A )
=> ( ( member_set_nat_nat @ A3 @ A )
=> ? [X2: set_nat_nat] :
( ( member_set_nat_nat @ X2 @ A )
& ( ord_le1415039317at_nat @ X2 @ A3 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le1415039317at_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_52_finite__has__maximal2,axiom,
! [A: set_nat_nat,A3: nat > nat] :
( ( finite570312790at_nat @ A )
=> ( ( member_nat_nat @ A3 @ A )
=> ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
& ( ord_less_eq_nat_nat @ A3 @ X2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_53_finite__has__maximal2,axiom,
! [A: set_nat_nat_nat,A3: nat > nat > nat] :
( ( finite2045569477at_nat @ A )
=> ( ( member_nat_nat_nat2 @ A3 @ A )
=> ? [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A )
& ( ord_le1415139726at_nat @ A3 @ X2 )
& ! [Xa: nat > nat > nat] :
( ( member_nat_nat_nat2 @ Xa @ A )
=> ( ( ord_le1415139726at_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_54_finite__has__maximal2,axiom,
! [A: set_nat_nat_nat2,A3: ( nat > nat ) > nat] :
( ( finite1440337093at_nat @ A )
=> ( ( member_nat_nat_nat @ A3 @ A )
=> ? [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A )
& ( ord_le809907342at_nat @ A3 @ X2 )
& ! [Xa: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa @ A )
=> ( ( ord_le809907342at_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_55_finite__has__maximal2,axiom,
! [A: set_nat_nat_nat_nat3,A3: ( nat > nat ) > nat > nat] :
( ( finite1064868788at_nat @ A )
=> ( ( member1128122036at_nat @ A3 @ A )
=> ? [X2: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ X2 @ A )
& ( ord_le2059018749at_nat @ A3 @ X2 )
& ! [Xa: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ Xa @ A )
=> ( ( ord_le2059018749at_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_56_finite__has__maximal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A3 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_57_finite__has__maximal2,axiom,
! [A: set_set_nat_nat,A3: set_nat_nat] :
( ( finite604103692at_nat @ A )
=> ( ( member_set_nat_nat @ A3 @ A )
=> ? [X2: set_nat_nat] :
( ( member_set_nat_nat @ X2 @ A )
& ( ord_le1415039317at_nat @ A3 @ X2 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le1415039317at_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_58_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_59_rev__finite__subset,axiom,
! [B: set_nat_nat_nat,A: set_nat_nat_nat] :
( ( finite2045569477at_nat @ B )
=> ( ( ord_le940807492at_nat @ A @ B )
=> ( finite2045569477at_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_60_rev__finite__subset,axiom,
! [B: set_nat_nat_nat2,A: set_nat_nat_nat2] :
( ( finite1440337093at_nat @ B )
=> ( ( ord_le633272388at_nat @ A @ B )
=> ( finite1440337093at_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_61_rev__finite__subset,axiom,
! [B: set_nat_nat_nat_nat3,A: set_nat_nat_nat_nat3] :
( ( finite1064868788at_nat @ B )
=> ( ( ord_le2040082867at_nat @ A @ B )
=> ( finite1064868788at_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_62_rev__finite__subset,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( finite570312790at_nat @ B )
=> ( ( ord_le1415039317at_nat @ A @ B )
=> ( finite570312790at_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_63_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_64_infinite__super,axiom,
! [S: set_nat_nat_nat,T: set_nat_nat_nat] :
( ( ord_le940807492at_nat @ S @ T )
=> ( ~ ( finite2045569477at_nat @ S )
=> ~ ( finite2045569477at_nat @ T ) ) ) ).
% infinite_super
thf(fact_65_infinite__super,axiom,
! [S: set_nat_nat_nat2,T: set_nat_nat_nat2] :
( ( ord_le633272388at_nat @ S @ T )
=> ( ~ ( finite1440337093at_nat @ S )
=> ~ ( finite1440337093at_nat @ T ) ) ) ).
% infinite_super
thf(fact_66_infinite__super,axiom,
! [S: set_nat_nat_nat_nat3,T: set_nat_nat_nat_nat3] :
( ( ord_le2040082867at_nat @ S @ T )
=> ( ~ ( finite1064868788at_nat @ S )
=> ~ ( finite1064868788at_nat @ T ) ) ) ).
% infinite_super
thf(fact_67_infinite__super,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( ord_le1415039317at_nat @ S @ T )
=> ( ~ ( finite570312790at_nat @ S )
=> ~ ( finite570312790at_nat @ T ) ) ) ).
% infinite_super
thf(fact_68_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_69_finite__subset,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( ord_le940807492at_nat @ A @ B )
=> ( ( finite2045569477at_nat @ B )
=> ( finite2045569477at_nat @ A ) ) ) ).
% finite_subset
thf(fact_70_finite__subset,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( ord_le633272388at_nat @ A @ B )
=> ( ( finite1440337093at_nat @ B )
=> ( finite1440337093at_nat @ A ) ) ) ).
% finite_subset
thf(fact_71_finite__subset,axiom,
! [A: set_nat_nat_nat_nat3,B: set_nat_nat_nat_nat3] :
( ( ord_le2040082867at_nat @ A @ B )
=> ( ( finite1064868788at_nat @ B )
=> ( finite1064868788at_nat @ A ) ) ) ).
% finite_subset
thf(fact_72_finite__subset,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le1415039317at_nat @ A @ B )
=> ( ( finite570312790at_nat @ B )
=> ( finite570312790at_nat @ A ) ) ) ).
% finite_subset
thf(fact_73_add__le__same__cancel1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_74_add__le__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_75_le__add__same__cancel1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_76_le__add__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_77_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_78_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_79_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_80_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_81_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_82_add__right__cancel,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
= ( B3 = C ) ) ).
% add_right_cancel
thf(fact_83_add__left__cancel,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= ( plus_plus_nat @ A3 @ C ) )
= ( B3 = C ) ) ).
% add_left_cancel
thf(fact_84_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_85_add__le__cancel__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_86_add__le__cancel__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_87_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y ) )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_88_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_89_add__cancel__right__right,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( plus_plus_nat @ A3 @ B3 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_90_add__cancel__right__left,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ A3 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_91_add__cancel__left__right,axiom,
! [A3: nat,B3: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_92_add__cancel__left__left,axiom,
! [B3: nat,A3: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_93_mem__Collect__eq,axiom,
! [A3: nat > nat,P2: ( nat > nat ) > $o] :
( ( member_nat_nat @ A3 @ ( collect_nat_nat @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
! [A3: nat,P2: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
! [A3: nat > nat > nat,P2: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ A3 @ ( collect_nat_nat_nat2 @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_96_mem__Collect__eq,axiom,
! [A3: ( nat > nat ) > nat,P2: ( ( nat > nat ) > nat ) > $o] :
( ( member_nat_nat_nat @ A3 @ ( collect_nat_nat_nat @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_97_mem__Collect__eq,axiom,
! [A3: ( nat > nat ) > nat > nat,P2: ( ( nat > nat ) > nat > nat ) > $o] :
( ( member1128122036at_nat @ A3 @ ( collec1610646258at_nat @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_98_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_99_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_100_Collect__mem__eq,axiom,
! [A: set_nat_nat_nat] :
( ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] : ( member_nat_nat_nat2 @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
! [A: set_nat_nat_nat2] :
( ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_102_Collect__mem__eq,axiom,
! [A: set_nat_nat_nat_nat3] :
( ( collec1610646258at_nat
@ ^ [X: ( nat > nat ) > nat > nat] : ( member1128122036at_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_103_Collect__cong,axiom,
! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X2: nat > nat] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat_nat @ P2 )
= ( collect_nat_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_104_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_105_Collect__cong,axiom,
! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ! [X2: nat > nat > nat] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat_nat_nat2 @ P2 )
= ( collect_nat_nat_nat2 @ Q ) ) ) ).
% Collect_cong
thf(fact_106_Collect__cong,axiom,
! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
( ! [X2: ( nat > nat ) > nat] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat_nat_nat @ P2 )
= ( collect_nat_nat_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_107_Collect__cong,axiom,
! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
( ! [X2: ( nat > nat ) > nat > nat] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collec1610646258at_nat @ P2 )
= ( collec1610646258at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_108_add_Oright__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.right_neutral
thf(fact_109_add_Oleft__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% add.left_neutral
thf(fact_110_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_111_add__right__imp__eq,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
=> ( B3 = C ) ) ).
% add_right_imp_eq
thf(fact_112_add__left__imp__eq,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= ( plus_plus_nat @ A3 @ C ) )
=> ( B3 = C ) ) ).
% add_left_imp_eq
thf(fact_113_add_Oleft__commute,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add.left_commute
thf(fact_114_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B4: nat] : ( plus_plus_nat @ B4 @ A2 ) ) ) ).
% add.commute
thf(fact_115_add_Oassoc,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add.assoc
thf(fact_116_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B3: nat,A3: nat] :
( ( B
= ( plus_plus_nat @ K @ B3 ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_117_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A3: nat,B3: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_118_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_119_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_120_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_121_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B3: nat] :
( ( P2 @ K )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ? [X2: nat] :
( ( P2 @ X2 )
& ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_122_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_123_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_124_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_125_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_126_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_127_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_128_add__le__imp__le__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_129_add__le__imp__le__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_130_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B4: nat] :
? [C2: nat] :
( B4
= ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_131_add__right__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add_right_mono
thf(fact_132_less__eqE,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ~ ! [C3: nat] :
( B3
!= ( plus_plus_nat @ A3 @ C3 ) ) ) ).
% less_eqE
thf(fact_133_add__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% add_left_mono
thf(fact_134_add__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% add_mono
thf(fact_135_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_136_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_137_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_138_add_Ocomm__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.comm_neutral
thf(fact_139_comm__monoid__add__class_Oadd__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_140_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_141_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_142_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_143_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_144_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_145_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_146_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_147_trans__le__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_148_trans__le__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_149_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_150_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_151_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_152_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_153_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_154_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_155_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_156_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_157_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_158_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_159_add__nonpos__nonpos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_160_add__nonneg__nonneg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_161_add__increasing2,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_162_add__decreasing2,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_163_add__increasing,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_164_add__decreasing,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_165_sum_Ofinite__Collect__op,axiom,
! [I3: set_nat_nat,X3: ( nat > nat ) > nat,Y: ( nat > nat ) > nat] :
( ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [I: nat > nat] :
( ( member_nat_nat @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [I: nat > nat] :
( ( member_nat_nat @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_nat ) ) ) )
=> ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [I: nat > nat] :
( ( member_nat_nat @ I @ I3 )
& ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_166_sum_Ofinite__Collect__op,axiom,
! [I3: set_nat,X3: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_167_sum_Ofinite__Collect__op,axiom,
! [I3: set_nat_nat_nat,X3: ( nat > nat > nat ) > nat,Y: ( nat > nat > nat ) > nat] :
( ( finite2045569477at_nat
@ ( collect_nat_nat_nat2
@ ^ [I: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite2045569477at_nat
@ ( collect_nat_nat_nat2
@ ^ [I: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_nat ) ) ) )
=> ( finite2045569477at_nat
@ ( collect_nat_nat_nat2
@ ^ [I: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ I3 )
& ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_168_sum_Ofinite__Collect__op,axiom,
! [I3: set_nat_nat_nat2,X3: ( ( nat > nat ) > nat ) > nat,Y: ( ( nat > nat ) > nat ) > nat] :
( ( finite1440337093at_nat
@ ( collect_nat_nat_nat
@ ^ [I: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite1440337093at_nat
@ ( collect_nat_nat_nat
@ ^ [I: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_nat ) ) ) )
=> ( finite1440337093at_nat
@ ( collect_nat_nat_nat
@ ^ [I: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ I3 )
& ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_169_sum_Ofinite__Collect__op,axiom,
! [I3: set_nat_nat_nat_nat3,X3: ( ( nat > nat ) > nat > nat ) > nat,Y: ( ( nat > nat ) > nat > nat ) > nat] :
( ( finite1064868788at_nat
@ ( collec1610646258at_nat
@ ^ [I: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite1064868788at_nat
@ ( collec1610646258at_nat
@ ^ [I: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_nat ) ) ) )
=> ( finite1064868788at_nat
@ ( collec1610646258at_nat
@ ^ [I: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ I @ I3 )
& ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_170_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_171_zero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_le_one
thf(fact_172_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F2 @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_173_subset__antisym,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le1415039317at_nat @ A @ B )
=> ( ( ord_le1415039317at_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_174_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_175_subsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( member_nat_nat @ X2 @ B ) )
=> ( ord_le1415039317at_nat @ A @ B ) ) ).
% subsetI
thf(fact_176_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_177_order__refl,axiom,
! [X3: set_nat_nat] : ( ord_le1415039317at_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_178_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ord_less_eq_nat @ X @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_179_Euclid__induct,axiom,
! [P2: nat > nat > $o,A3: nat,B3: nat] :
( ! [A4: nat,B5: nat] :
( ( P2 @ A4 @ B5 )
= ( P2 @ B5 @ A4 ) )
=> ( ! [A4: nat] : ( P2 @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B5: nat] :
( ( P2 @ A4 @ B5 )
=> ( P2 @ A4 @ ( plus_plus_nat @ A4 @ B5 ) ) )
=> ( P2 @ A3 @ B3 ) ) ) ) ).
% Euclid_induct
thf(fact_180_order__subst1,axiom,
! [A3: nat,F2: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_181_order__subst1,axiom,
! [A3: nat,F2: set_nat_nat > nat,B3: set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_le1415039317at_nat @ B3 @ C )
=> ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le1415039317at_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_182_order__subst1,axiom,
! [A3: set_nat_nat,F2: nat > set_nat_nat,B3: nat,C: nat] :
( ( ord_le1415039317at_nat @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1415039317at_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_183_order__subst1,axiom,
! [A3: set_nat_nat,F2: set_nat_nat > set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
( ( ord_le1415039317at_nat @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_le1415039317at_nat @ B3 @ C )
=> ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le1415039317at_nat @ X2 @ Y2 )
=> ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1415039317at_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_184_order__subst2,axiom,
! [A3: nat,B3: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F2 @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_185_order__subst2,axiom,
! [A3: nat,B3: nat,F2: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_le1415039317at_nat @ ( F2 @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1415039317at_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_186_order__subst2,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,F2: set_nat_nat > nat,C: nat] :
( ( ord_le1415039317at_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F2 @ B3 ) @ C )
=> ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le1415039317at_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_187_order__subst2,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,F2: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le1415039317at_nat @ A3 @ B3 )
=> ( ( ord_le1415039317at_nat @ ( F2 @ B3 ) @ C )
=> ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le1415039317at_nat @ X2 @ Y2 )
=> ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1415039317at_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_188_ord__eq__le__subst,axiom,
! [A3: nat,F2: nat > nat,B3: nat,C: nat] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_189_ord__eq__le__subst,axiom,
! [A3: set_nat_nat,F2: nat > set_nat_nat,B3: nat,C: nat] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1415039317at_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_190_ord__eq__le__subst,axiom,
! [A3: nat,F2: set_nat_nat > nat,B3: set_nat_nat,C: set_nat_nat] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_le1415039317at_nat @ B3 @ C )
=> ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le1415039317at_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_191_ord__eq__le__subst,axiom,
! [A3: set_nat_nat,F2: set_nat_nat > set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_le1415039317at_nat @ B3 @ C )
=> ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le1415039317at_nat @ X2 @ Y2 )
=> ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1415039317at_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_192_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_193_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F2: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1415039317at_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_194_ord__le__eq__subst,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,F2: set_nat_nat > nat,C: nat] :
( ( ord_le1415039317at_nat @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le1415039317at_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_195_ord__le__eq__subst,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,F2: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le1415039317at_nat @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le1415039317at_nat @ X2 @ Y2 )
=> ( ord_le1415039317at_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1415039317at_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_196_eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).
% eq_iff
thf(fact_197_eq__iff,axiom,
( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : Y4 = Z )
= ( ^ [X: set_nat_nat,Y5: set_nat_nat] :
( ( ord_le1415039317at_nat @ X @ Y5 )
& ( ord_le1415039317at_nat @ Y5 @ X ) ) ) ) ).
% eq_iff
thf(fact_198_antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% antisym
thf(fact_199_antisym,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] :
( ( ord_le1415039317at_nat @ X3 @ Y )
=> ( ( ord_le1415039317at_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% antisym
thf(fact_200_linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linear
thf(fact_201_eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% eq_refl
thf(fact_202_eq__refl,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] :
( ( X3 = Y )
=> ( ord_le1415039317at_nat @ X3 @ Y ) ) ).
% eq_refl
thf(fact_203_le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% le_cases
thf(fact_204_order_Otrans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_205_order_Otrans,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
( ( ord_le1415039317at_nat @ A3 @ B3 )
=> ( ( ord_le1415039317at_nat @ B3 @ C )
=> ( ord_le1415039317at_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_206_le__cases3,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_207_antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv
thf(fact_208_antisym__conv,axiom,
! [Y: set_nat_nat,X3: set_nat_nat] :
( ( ord_le1415039317at_nat @ Y @ X3 )
=> ( ( ord_le1415039317at_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv
thf(fact_209_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
& ( ord_less_eq_nat @ B4 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_210_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : Y4 = Z )
= ( ^ [A2: set_nat_nat,B4: set_nat_nat] :
( ( ord_le1415039317at_nat @ A2 @ B4 )
& ( ord_le1415039317at_nat @ B4 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_211_ord__eq__le__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_212_ord__eq__le__trans,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
( ( A3 = B3 )
=> ( ( ord_le1415039317at_nat @ B3 @ C )
=> ( ord_le1415039317at_nat @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_213_ord__le__eq__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_214_ord__le__eq__trans,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
( ( ord_le1415039317at_nat @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_le1415039317at_nat @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_215_order__class_Oorder_Oantisym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% order_class.order.antisym
thf(fact_216_order__class_Oorder_Oantisym,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le1415039317at_nat @ A3 @ B3 )
=> ( ( ord_le1415039317at_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% order_class.order.antisym
thf(fact_217_order__trans,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_218_order__trans,axiom,
! [X3: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le1415039317at_nat @ X3 @ Y )
=> ( ( ord_le1415039317at_nat @ Y @ Z2 )
=> ( ord_le1415039317at_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_219_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_220_dual__order_Orefl,axiom,
! [A3: set_nat_nat] : ( ord_le1415039317at_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_221_linorder__wlog,axiom,
! [P2: nat > nat > $o,A3: nat,B3: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
=> ( P2 @ A4 @ B5 ) )
=> ( ! [A4: nat,B5: nat] :
( ( P2 @ B5 @ A4 )
=> ( P2 @ A4 @ B5 ) )
=> ( P2 @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_222_dual__order_Otrans,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_223_dual__order_Otrans,axiom,
! [B3: set_nat_nat,A3: set_nat_nat,C: set_nat_nat] :
( ( ord_le1415039317at_nat @ B3 @ A3 )
=> ( ( ord_le1415039317at_nat @ C @ B3 )
=> ( ord_le1415039317at_nat @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_224_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
& ( ord_less_eq_nat @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_225_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : Y4 = Z )
= ( ^ [A2: set_nat_nat,B4: set_nat_nat] :
( ( ord_le1415039317at_nat @ B4 @ A2 )
& ( ord_le1415039317at_nat @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_226_dual__order_Oantisym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_227_dual__order_Oantisym,axiom,
! [B3: set_nat_nat,A3: set_nat_nat] :
( ( ord_le1415039317at_nat @ B3 @ A3 )
=> ( ( ord_le1415039317at_nat @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_228_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_229_in__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,X3: nat > nat] :
( ( ord_le1415039317at_nat @ A @ B )
=> ( ( member_nat_nat @ X3 @ A )
=> ( member_nat_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_230_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_231_subsetD,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: nat > nat] :
( ( ord_le1415039317at_nat @ A @ B )
=> ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_232_equalityE,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ~ ( ( ord_le1415039317at_nat @ A @ B )
=> ~ ( ord_le1415039317at_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_233_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_234_subset__eq,axiom,
( ord_le1415039317at_nat
= ( ^ [A5: set_nat_nat,B2: set_nat_nat] :
! [X: nat > nat] :
( ( member_nat_nat @ X @ A5 )
=> ( member_nat_nat @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_235_equalityD1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le1415039317at_nat @ A @ B ) ) ).
% equalityD1
thf(fact_236_equalityD2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le1415039317at_nat @ B @ A ) ) ).
% equalityD2
thf(fact_237_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_238_subset__iff,axiom,
( ord_le1415039317at_nat
= ( ^ [A5: set_nat_nat,B2: set_nat_nat] :
! [T2: nat > nat] :
( ( member_nat_nat @ T2 @ A5 )
=> ( member_nat_nat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_239_subset__refl,axiom,
! [A: set_nat_nat] : ( ord_le1415039317at_nat @ A @ A ) ).
% subset_refl
thf(fact_240_Collect__mono,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_241_Collect__mono,axiom,
! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ! [X2: nat > nat > nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le940807492at_nat @ ( collect_nat_nat_nat2 @ P2 ) @ ( collect_nat_nat_nat2 @ Q ) ) ) ).
% Collect_mono
thf(fact_242_Collect__mono,axiom,
! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
( ! [X2: ( nat > nat ) > nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le633272388at_nat @ ( collect_nat_nat_nat @ P2 ) @ ( collect_nat_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_243_Collect__mono,axiom,
! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
( ! [X2: ( nat > nat ) > nat > nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le2040082867at_nat @ ( collec1610646258at_nat @ P2 ) @ ( collec1610646258at_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_244_Collect__mono,axiom,
! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X2: nat > nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le1415039317at_nat @ ( collect_nat_nat @ P2 ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_245_subset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C4: set_nat_nat] :
( ( ord_le1415039317at_nat @ A @ B )
=> ( ( ord_le1415039317at_nat @ B @ C4 )
=> ( ord_le1415039317at_nat @ A @ C4 ) ) ) ).
% subset_trans
thf(fact_246_set__eq__subset,axiom,
( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : Y4 = Z )
= ( ^ [A5: set_nat_nat,B2: set_nat_nat] :
( ( ord_le1415039317at_nat @ A5 @ B2 )
& ( ord_le1415039317at_nat @ B2 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_247_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_248_Collect__mono__iff,axiom,
! [P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ( ord_le940807492at_nat @ ( collect_nat_nat_nat2 @ P2 ) @ ( collect_nat_nat_nat2 @ Q ) )
= ( ! [X: nat > nat > nat] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_249_Collect__mono__iff,axiom,
! [P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
( ( ord_le633272388at_nat @ ( collect_nat_nat_nat @ P2 ) @ ( collect_nat_nat_nat @ Q ) )
= ( ! [X: ( nat > nat ) > nat] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_250_Collect__mono__iff,axiom,
! [P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
( ( ord_le2040082867at_nat @ ( collec1610646258at_nat @ P2 ) @ ( collec1610646258at_nat @ Q ) )
= ( ! [X: ( nat > nat ) > nat > nat] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_251_Collect__mono__iff,axiom,
! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le1415039317at_nat @ ( collect_nat_nat @ P2 ) @ ( collect_nat_nat @ Q ) )
= ( ! [X: nat > nat] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_252_bounded__Max__nat,axiom,
! [P2: nat > $o,X3: nat,M3: nat] :
( ( P2 @ X3 )
=> ( ! [X2: nat] :
( ( P2 @ X2 )
=> ( ord_less_eq_nat @ X2 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P2 @ M4 )
=> ~ ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_253_Collect__subset,axiom,
! [A: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_254_Collect__subset,axiom,
! [A: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o] :
( ord_le940807492at_nat
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_255_Collect__subset,axiom,
! [A: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o] :
( ord_le633272388at_nat
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_256_Collect__subset,axiom,
! [A: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o] :
( ord_le2040082867at_nat
@ ( collec1610646258at_nat
@ ^ [X: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_257_Collect__subset,axiom,
! [A: set_nat_nat,P2: ( nat > nat ) > $o] :
( ord_le1415039317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_258_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_259_less__eq__set__def,axiom,
( ord_le1415039317at_nat
= ( ^ [A5: set_nat_nat,B2: set_nat_nat] :
( ord_le2026740592_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A5 )
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_260_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_261_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_262_verit__sum__simplify,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% verit_sum_simplify
thf(fact_263_add__0__iff,axiom,
! [B3: nat,A3: nat] :
( ( B3
= ( plus_plus_nat @ B3 @ A3 ) )
= ( A3 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_264_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_265_verit__la__disequality,axiom,
! [A3: nat,B3: nat] :
( ( A3 = B3 )
| ~ ( ord_less_eq_nat @ A3 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% verit_la_disequality
thf(fact_266_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_267_pred__subset__eq,axiom,
! [R: set_nat_nat,S: set_nat_nat] :
( ( ord_le2026740592_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ R )
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ S ) )
= ( ord_le1415039317at_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_268_conj__subset__def,axiom,
! [A: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P2 ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_269_conj__subset__def,axiom,
! [A: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ( ord_le940807492at_nat @ A
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le940807492at_nat @ A @ ( collect_nat_nat_nat2 @ P2 ) )
& ( ord_le940807492at_nat @ A @ ( collect_nat_nat_nat2 @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_270_conj__subset__def,axiom,
! [A: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
( ( ord_le633272388at_nat @ A
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le633272388at_nat @ A @ ( collect_nat_nat_nat @ P2 ) )
& ( ord_le633272388at_nat @ A @ ( collect_nat_nat_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_271_conj__subset__def,axiom,
! [A: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
( ( ord_le2040082867at_nat @ A
@ ( collec1610646258at_nat
@ ^ [X: ( nat > nat ) > nat > nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le2040082867at_nat @ A @ ( collec1610646258at_nat @ P2 ) )
& ( ord_le2040082867at_nat @ A @ ( collec1610646258at_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_272_conj__subset__def,axiom,
! [A: set_nat_nat,P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le1415039317at_nat @ A
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le1415039317at_nat @ A @ ( collect_nat_nat @ P2 ) )
& ( ord_le1415039317at_nat @ A @ ( collect_nat_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_273_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_274_subset__Collect__iff,axiom,
! [B: set_nat_nat_nat,A: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o] :
( ( ord_le940807492at_nat @ B @ A )
=> ( ( ord_le940807492at_nat @ B
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ B )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_275_subset__Collect__iff,axiom,
! [B: set_nat_nat_nat2,A: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o] :
( ( ord_le633272388at_nat @ B @ A )
=> ( ( ord_le633272388at_nat @ B
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ B )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_276_subset__Collect__iff,axiom,
! [B: set_nat_nat_nat_nat3,A: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o] :
( ( ord_le2040082867at_nat @ B @ A )
=> ( ( ord_le2040082867at_nat @ B
@ ( collec1610646258at_nat
@ ^ [X: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ X @ B )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_277_subset__Collect__iff,axiom,
! [B: set_nat_nat,A: set_nat_nat,P2: ( nat > nat ) > $o] :
( ( ord_le1415039317at_nat @ B @ A )
=> ( ( ord_le1415039317at_nat @ B
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_278_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q: nat > $o,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P2 @ X2 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_279_subset__CollectI,axiom,
! [B: set_nat_nat_nat,A: set_nat_nat_nat,Q: ( nat > nat > nat ) > $o,P2: ( nat > nat > nat ) > $o] :
( ( ord_le940807492at_nat @ B @ A )
=> ( ! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P2 @ X2 ) ) )
=> ( ord_le940807492at_nat
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ B )
& ( Q @ X ) ) )
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_280_subset__CollectI,axiom,
! [B: set_nat_nat_nat2,A: set_nat_nat_nat2,Q: ( ( nat > nat ) > nat ) > $o,P2: ( ( nat > nat ) > nat ) > $o] :
( ( ord_le633272388at_nat @ B @ A )
=> ( ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P2 @ X2 ) ) )
=> ( ord_le633272388at_nat
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_281_subset__CollectI,axiom,
! [B: set_nat_nat_nat_nat3,A: set_nat_nat_nat_nat3,Q: ( ( nat > nat ) > nat > nat ) > $o,P2: ( ( nat > nat ) > nat > nat ) > $o] :
( ( ord_le2040082867at_nat @ B @ A )
=> ( ! [X2: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P2 @ X2 ) ) )
=> ( ord_le2040082867at_nat
@ ( collec1610646258at_nat
@ ^ [X: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collec1610646258at_nat
@ ^ [X: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_282_subset__CollectI,axiom,
! [B: set_nat_nat,A: set_nat_nat,Q: ( nat > nat ) > $o,P2: ( nat > nat ) > $o] :
( ( ord_le1415039317at_nat @ B @ A )
=> ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P2 @ X2 ) ) )
=> ( ord_le1415039317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_283_Collect__restrict,axiom,
! [X5: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_284_Collect__restrict,axiom,
! [X5: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o] :
( ord_le940807492at_nat
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_285_Collect__restrict,axiom,
! [X5: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o] :
( ord_le633272388at_nat
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_286_Collect__restrict,axiom,
! [X5: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o] :
( ord_le2040082867at_nat
@ ( collec1610646258at_nat
@ ^ [X: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_287_Collect__restrict,axiom,
! [X5: set_nat_nat,P2: ( nat > nat ) > $o] :
( ord_le1415039317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_288_prop__restrict,axiom,
! [X3: nat,Z3: set_nat,X5: set_nat,P2: nat > $o] :
( ( member_nat @ X3 @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_289_prop__restrict,axiom,
! [X3: nat > nat > nat,Z3: set_nat_nat_nat,X5: set_nat_nat_nat,P2: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ X3 @ Z3 )
=> ( ( ord_le940807492at_nat @ Z3
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_290_prop__restrict,axiom,
! [X3: ( nat > nat ) > nat,Z3: set_nat_nat_nat2,X5: set_nat_nat_nat2,P2: ( ( nat > nat ) > nat ) > $o] :
( ( member_nat_nat_nat @ X3 @ Z3 )
=> ( ( ord_le633272388at_nat @ Z3
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_291_prop__restrict,axiom,
! [X3: ( nat > nat ) > nat > nat,Z3: set_nat_nat_nat_nat3,X5: set_nat_nat_nat_nat3,P2: ( ( nat > nat ) > nat > nat ) > $o] :
( ( member1128122036at_nat @ X3 @ Z3 )
=> ( ( ord_le2040082867at_nat @ Z3
@ ( collec1610646258at_nat
@ ^ [X: ( nat > nat ) > nat > nat] :
( ( member1128122036at_nat @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_292_prop__restrict,axiom,
! [X3: nat > nat,Z3: set_nat_nat,X5: set_nat_nat,P2: ( nat > nat ) > $o] :
( ( member_nat_nat @ X3 @ Z3 )
=> ( ( ord_le1415039317at_nat @ Z3
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_293_Fpow__def,axiom,
( finite_Fpow_nat
= ( ^ [A5: set_nat] :
( collect_set_nat
@ ^ [X6: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ A5 )
& ( finite_finite_nat @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_294_Fpow__def,axiom,
( finite90785608at_nat
= ( ^ [A5: set_nat_nat_nat] :
( collec617280953at_nat
@ ^ [X6: set_nat_nat_nat] :
( ( ord_le940807492at_nat @ X6 @ A5 )
& ( finite2045569477at_nat @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_295_Fpow__def,axiom,
( finite1633036872at_nat
= ( ^ [A5: set_nat_nat_nat2] :
( collec309745849at_nat
@ ^ [X6: set_nat_nat_nat2] :
( ( ord_le633272388at_nat @ X6 @ A5 )
& ( finite1440337093at_nat @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_296_Fpow__def,axiom,
( finite252967095at_nat
= ( ^ [A5: set_nat_nat_nat_nat3] :
( collec1702444712at_nat
@ ^ [X6: set_nat_nat_nat_nat3] :
( ( ord_le2040082867at_nat @ X6 @ A5 )
& ( finite1064868788at_nat @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_297_Fpow__def,axiom,
( finite_Fpow_nat_nat
= ( ^ [A5: set_nat_nat] :
( collect_set_nat_nat
@ ^ [X6: set_nat_nat] :
( ( ord_le1415039317at_nat @ X6 @ A5 )
& ( finite570312790at_nat @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_298_partitions__imp__finite__elements,axiom,
! [P3: nat > nat,N: nat] :
( ( number1551313001itions @ P3 @ N )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ zero_zero_nat @ ( P3 @ I ) ) ) ) ) ).
% partitions_imp_finite_elements
thf(fact_299_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_300_add__less__cancel__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_301_add__less__cancel__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_302_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_303_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_304_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_305_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_306_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_307_add__less__same__cancel1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_308_add__less__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_309_less__add__same__cancel1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_310_less__add__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_311_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_312_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_313_add__less__le__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_314_add__le__less__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_315_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_316_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_317_pos__add__strict,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_318_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ! [C3: nat] :
( ( B3
= ( plus_plus_nat @ A3 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_319_add__pos__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_320_add__neg__neg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_321_order_Onot__eq__order__implies__strict,axiom,
! [A3: nat,B3: nat] :
( ( A3 != B3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_322_order_Onot__eq__order__implies__strict,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( A3 != B3 )
=> ( ( ord_le1415039317at_nat @ A3 @ B3 )
=> ( ord_less_set_nat_nat @ A3 @ B3 ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_323_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_324_dual__order_Ostrict__implies__order,axiom,
! [B3: set_nat_nat,A3: set_nat_nat] :
( ( ord_less_set_nat_nat @ B3 @ A3 )
=> ( ord_le1415039317at_nat @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_325_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
& ( A2 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_326_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [B4: set_nat_nat,A2: set_nat_nat] :
( ( ord_le1415039317at_nat @ B4 @ A2 )
& ( A2 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_327_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A2: nat] :
( ( ord_less_nat @ B4 @ A2 )
| ( A2 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_328_dual__order_Oorder__iff__strict,axiom,
( ord_le1415039317at_nat
= ( ^ [B4: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B4 @ A2 )
| ( A2 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_329_order_Ostrict__implies__order,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_330_order_Ostrict__implies__order,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B3 )
=> ( ord_le1415039317at_nat @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_331_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_332_dual__order_Ostrict__trans2,axiom,
! [B3: set_nat_nat,A3: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ B3 @ A3 )
=> ( ( ord_le1415039317at_nat @ C @ B3 )
=> ( ord_less_set_nat_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_333_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_334_dual__order_Ostrict__trans1,axiom,
! [B3: set_nat_nat,A3: set_nat_nat,C: set_nat_nat] :
( ( ord_le1415039317at_nat @ B3 @ A3 )
=> ( ( ord_less_set_nat_nat @ C @ B3 )
=> ( ord_less_set_nat_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_335_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
& ( A2 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_336_order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [A2: set_nat_nat,B4: set_nat_nat] :
( ( ord_le1415039317at_nat @ A2 @ B4 )
& ( A2 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_337_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_nat @ A2 @ B4 )
| ( A2 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_338_order_Oorder__iff__strict,axiom,
( ord_le1415039317at_nat
= ( ^ [A2: set_nat_nat,B4: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B4 )
| ( A2 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_339_order_Ostrict__trans2,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B3 )
=> ( ( ord_le1415039317at_nat @ B3 @ C )
=> ( ord_less_set_nat_nat @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_340_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_341_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_342_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_343_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_344_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_345_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I2: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F2 @ I4 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_346_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_347_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ord_less_nat @ X @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_348_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P2 @ K2 )
& ( ord_less_nat @ K2 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
% Conjectures (1)
thf(conj_0,conjecture,
( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [P: nat > nat] : ( number1551313001itions @ P @ n ) ) ) ).
%------------------------------------------------------------------------------