%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP077^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:22 AM UTC 2026
% Result : Theorem 5.55s 2.12s
% Output : Refutation 5.55s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP077^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n008.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:25:24 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.34/0.48 ---- Original DTF file ---
% 0.34/0.48 thf(spec,logic,$$dhol).
% 0.34/0.48 %------------------------------------------------------------------------------
% 0.34/0.48 % File : ITP077^2 : TPTP v9.2.1. Released v7.5.0.
% 0.34/0.48 % Domain : Interactive Theorem Proving
% 0.34/0.48 % Problem : Sledgehammer Hilbert_Function problem prob_146__11622158_1
% 0.34/0.48 % Version : Especial.
% 0.34/0.48 % English :
% 0.34/0.48
% 0.34/0.48 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.34/0.48 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.34/0.48 % Source : [Des21]
% 0.34/0.48 % Names : Hilbert_Function/prob_146__11622158_1 [Des21]
% 0.34/0.48
% 0.34/0.48 % Status : Theorem
% 0.34/0.48 % Rating : 0.33 v8.1.0, 0.75 v7.5.0
% 0.34/0.48 % Syntax : Number of formulae : 336 ( 75 unt; 48 typ; 0 def)
% 0.34/0.48 % Number of atoms : 922 ( 240 equ; 0 cnn)
% 0.34/0.48 % Maximal formula atoms : 13 ( 3 avg)
% 0.34/0.48 % Number of connectives : 4025 ( 72 ~; 13 |; 46 &;3378 @)
% 0.34/0.48 % ( 0 <=>; 516 =>; 0 <=; 0 <~>)
% 0.34/0.48 % Maximal formula depth : 20 ( 9 avg)
% 0.34/0.48 % Number of types : 3 ( 2 usr)
% 0.34/0.48 % Number of type conns : 186 ( 186 >; 0 *; 0 +; 0 <<)
% 0.34/0.48 % Number of symbols : 49 ( 46 usr; 8 con; 0-5 aty)
% 0.34/0.48 % Number of variables : 1125 ( 57 ^; 982 !; 47 ?;1125 :)
% 0.34/0.48 % ( 39 !>; 0 ?*; 0 @-; 0 @+)
% 0.34/0.48 % SPC : TH1_THM_EQU_NAR
% 0.34/0.48
% 0.34/0.48 % Comments : This file was generated by Sledgehammer 2021-02-23 16:22:15.198
% 0.34/0.48 %------------------------------------------------------------------------------
% 0.34/0.48 % Could-be-implicit typings (5)
% 0.34/0.48 thf(ty_t_Multiset_Omultiset,type,
% 0.34/0.48 multiset: $tType > $tType ).
% 0.34/0.48
% 0.34/0.48 thf(ty_t_List_Olist,type,
% 0.34/0.48 list: $tType > $tType ).
% 0.34/0.48
% 0.34/0.48 thf(ty_t_Set_Oset,type,
% 0.34/0.48 set: $tType > $tType ).
% 0.34/0.48
% 0.34/0.48 thf(ty_t_Nat_Onat,type,
% 0.34/0.48 nat: $tType ).
% 0.34/0.48
% 0.34/0.48 thf(ty_tf_a,type,
% 0.34/0.48 a: $tType ).
% 0.34/0.48
% 0.34/0.48 % Explicit typings (43)
% 0.34/0.48 thf(sy_cl_Rings_Odioid,type,
% 0.34/0.48 dioid:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_cl_HOL_Otype,type,
% 0.34/0.48 type:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_cl_Nat_Osize,type,
% 0.34/0.48 size:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_cl_Orderings_Oord,type,
% 0.34/0.48 ord:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_cl_Orderings_Oorder,type,
% 0.34/0.48 order:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_cl_Orderings_Ono__bot,type,
% 0.34/0.48 no_bot:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_cl_Orderings_Ono__top,type,
% 0.34/0.48 no_top:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_cl_Orderings_Olinorder,type,
% 0.34/0.48 linorder:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_cl_Orderings_Opreorder,type,
% 0.34/0.48 preorder:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_cl_Orderings_Owellorder,type,
% 0.34/0.48 wellorder:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_cl_Orderings_Odense__order,type,
% 0.34/0.48 dense_order:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.34/0.48 comm_monoid_add:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.34/0.48 dense_linorder:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.34/0.48 canoni770627133id_add:
% 0.34/0.48 !>[A: $tType] : $o ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Fun_Obij__betw,type,
% 0.34/0.48 bij_betw:
% 0.34/0.48 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Fun_Oinj__on,type,
% 0.34/0.48 inj_on:
% 0.34/0.48 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
% 0.34/0.48 groups501227526m_list:
% 0.34/0.48 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Hilbert__Choice_Oinv__into,type,
% 0.34/0.48 hilbert_inv_into:
% 0.34/0.48 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Hilbert__Function__Mirabelle__phhewjaonq_Odirect__decomp,type,
% 0.34/0.48 hilber580762119decomp:
% 0.34/0.48 !>[A: $tType] : ( ( set @ A ) > ( list @ ( set @ A ) ) > $o ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_List_Olist__ex,type,
% 0.34/0.48 list_ex:
% 0.34/0.48 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_List_Olist__update,type,
% 0.34/0.48 list_update:
% 0.34/0.48 !>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_List_Olistset,type,
% 0.34/0.48 listset:
% 0.34/0.48 !>[A: $tType] : ( ( list @ ( set @ A ) ) > ( set @ ( list @ A ) ) ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_List_Onth,type,
% 0.34/0.48 nth:
% 0.34/0.48 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_List_Orev,type,
% 0.34/0.48 rev:
% 0.34/0.48 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset,type,
% 0.34/0.48 comm_m1079429914m_mset:
% 0.34/0.48 !>[A: $tType] : ( ( multiset @ A ) > A ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Multiset_Omset,type,
% 0.34/0.48 mset:
% 0.34/0.48 !>[A: $tType] : ( ( list @ A ) > ( multiset @ A ) ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Multiset_Oset__mset,type,
% 0.34/0.48 set_mset:
% 0.34/0.48 !>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.34/0.48 size_size:
% 0.34/0.48 !>[A: $tType] : ( A > nat ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.34/0.48 ord_less:
% 0.34/0.48 !>[A: $tType] : ( A > A > $o ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.34/0.48 ord_less_eq:
% 0.34/0.48 !>[A: $tType] : ( A > A > $o ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Permutation_Operm,type,
% 0.34/0.48 perm:
% 0.34/0.48 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Set_OCollect,type,
% 0.34/0.48 collect:
% 0.34/0.48 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Set_Oimage,type,
% 0.34/0.48 image:
% 0.34/0.48 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
% 0.34/0.48 set_ord_lessThan:
% 0.34/0.48 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_c_member,type,
% 0.34/0.48 member:
% 0.34/0.48 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_v_A,type,
% 0.34/0.48 a2: set @ a ).
% 0.34/0.48
% 0.34/0.48 thf(sy_v_a____,type,
% 0.34/0.48 a3: a ).
% 0.34/0.48
% 0.34/0.48 thf(sy_v_f____,type,
% 0.34/0.48 f: nat > nat ).
% 0.34/0.48
% 0.34/0.48 thf(sy_v_g____,type,
% 0.34/0.48 g: nat > nat ).
% 0.34/0.48
% 0.34/0.48 thf(sy_v_qs2____,type,
% 0.34/0.48 qs2: list @ a ).
% 0.34/0.48
% 0.34/0.48 thf(sy_v_qs____,type,
% 0.34/0.48 qs: list @ a ).
% 0.34/0.48
% 0.34/0.48 thf(sy_v_ss1,type,
% 0.34/0.48 ss1: list @ ( set @ a ) ).
% 0.34/0.48
% 0.34/0.48 thf(sy_v_ss2,type,
% 0.34/0.48 ss2: list @ ( set @ a ) ).
% 0.34/0.48
% 0.34/0.48 % Relevant facts (256)
% 0.34/0.48 thf(fact_0__092_060open_062mset_Aqs2_A_061_Amset_Aqs_092_060close_062,axiom,
% 0.34/0.48 ( ( mset @ a @ qs2 )
% 0.34/0.48 = ( mset @ a @ qs ) ) ).
% 0.34/0.48
% 0.34/0.48 % \<open>mset qs2 = mset qs\<close>
% 0.34/0.48 thf(fact_1__092_060open_062a_A_092_060in_062_AA_092_060close_062,axiom,
% 0.34/0.48 member @ a @ a3 @ a2 ).
% 0.34/0.48
% 0.34/0.48 % \<open>a \<in> A\<close>
% 0.34/0.48 thf(fact_2_a,axiom,
% 0.34/0.48 ( a3
% 0.34/0.48 = ( groups501227526m_list @ a @ qs ) ) ).
% 0.34/0.48
% 0.34/0.48 % a
% 0.34/0.48 thf(fact_3_len__qs2,axiom,
% 0.34/0.48 ( ( size_size @ ( list @ a ) @ qs2 )
% 0.34/0.48 = ( size_size @ ( list @ a ) @ qs ) ) ).
% 0.34/0.48
% 0.34/0.48 % len_qs2
% 0.34/0.48 thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs_O_A_092_060lbrakk_062a_A_061_Asum__list_Aqs_059_Aqs_A_092_060in_062_Alistset_Ass1_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.34/0.48 ~ ! [Qs: list @ a] :
% 0.34/0.48 ( ( a3
% 0.34/0.48 = ( groups501227526m_list @ a @ Qs ) )
% 0.34/0.48 => ~ ( member @ ( list @ a ) @ Qs @ ( listset @ a @ ss1 ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % \<open>\<And>thesis. (\<And>qs. \<lbrakk>a = sum_list qs; qs \<in> listset ss1\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.34/0.48 thf(fact_5_qs2__in,axiom,
% 0.34/0.48 member @ ( list @ a ) @ qs2 @ ( listset @ a @ ss2 ) ).
% 0.34/0.48
% 0.34/0.48 % qs2_in
% 0.34/0.48 thf(fact_6_direct__decompD_I1_J,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( comm_monoid_add @ A )
% 0.34/0.48 => ! [A2: set @ A,Ss: list @ ( set @ A ),Qs2: list @ A] :
% 0.34/0.48 ( ( hilber580762119decomp @ A @ A2 @ Ss )
% 0.34/0.48 => ( ( member @ ( list @ A ) @ Qs2 @ ( listset @ A @ Ss ) )
% 0.34/0.48 => ( member @ A @ ( groups501227526m_list @ A @ Qs2 ) @ A2 ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % direct_decompD(1)
% 0.34/0.48 thf(fact_7_direct__decompE,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( comm_monoid_add @ A )
% 0.34/0.48 => ! [A2: set @ A,Ss: list @ ( set @ A ),A3: A] :
% 0.34/0.48 ( ( hilber580762119decomp @ A @ A2 @ Ss )
% 0.34/0.48 => ( ( member @ A @ A3 @ A2 )
% 0.34/0.48 => ~ ! [Qs: list @ A] :
% 0.34/0.48 ( ( member @ ( list @ A ) @ Qs @ ( listset @ A @ Ss ) )
% 0.34/0.48 => ( A3
% 0.34/0.48 != ( groups501227526m_list @ A @ Qs ) ) ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % direct_decompE
% 0.34/0.48 thf(fact_8_direct__decompI__alt,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( comm_monoid_add @ A )
% 0.34/0.48 => ! [Ss: list @ ( set @ A ),A2: set @ A] :
% 0.34/0.48 ( ! [Qs: list @ A] :
% 0.34/0.48 ( ( member @ ( list @ A ) @ Qs @ ( listset @ A @ Ss ) )
% 0.34/0.48 => ( member @ A @ ( groups501227526m_list @ A @ Qs ) @ A2 ) )
% 0.34/0.48 => ( ! [A4: A] :
% 0.34/0.48 ( ( member @ A @ A4 @ A2 )
% 0.34/0.48 => ? [X: list @ A] :
% 0.34/0.48 ( ( member @ ( list @ A ) @ X @ ( listset @ A @ Ss ) )
% 0.34/0.48 & ( A4
% 0.34/0.48 = ( groups501227526m_list @ A @ X ) )
% 0.34/0.48 & ! [Y: list @ A] :
% 0.34/0.48 ( ( ( member @ ( list @ A ) @ Y @ ( listset @ A @ Ss ) )
% 0.34/0.48 & ( A4
% 0.34/0.48 = ( groups501227526m_list @ A @ Y ) ) )
% 0.34/0.48 => ( Y = X ) ) ) )
% 0.34/0.48 => ( hilber580762119decomp @ A @ A2 @ Ss ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % direct_decompI_alt
% 0.34/0.48 thf(fact_9_direct__decomp__unique,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( comm_monoid_add @ A )
% 0.34/0.48 => ! [A2: set @ A,Ss: list @ ( set @ A ),Qs2: list @ A,Qs3: list @ A] :
% 0.34/0.48 ( ( hilber580762119decomp @ A @ A2 @ Ss )
% 0.34/0.48 => ( ( member @ ( list @ A ) @ Qs2 @ ( listset @ A @ Ss ) )
% 0.34/0.48 => ( ( member @ ( list @ A ) @ Qs3 @ ( listset @ A @ Ss ) )
% 0.34/0.48 => ( ( ( groups501227526m_list @ A @ Qs2 )
% 0.34/0.48 = ( groups501227526m_list @ A @ Qs3 ) )
% 0.34/0.48 => ( Qs2 = Qs3 ) ) ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % direct_decomp_unique
% 0.34/0.48 thf(fact_10_sum__list_Orev,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( comm_monoid_add @ A )
% 0.34/0.48 => ! [Xs: list @ A] :
% 0.34/0.48 ( ( groups501227526m_list @ A @ ( rev @ A @ Xs ) )
% 0.34/0.48 = ( groups501227526m_list @ A @ Xs ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % sum_list.rev
% 0.34/0.48 thf(fact_11__C1_C,axiom,
% 0.34/0.48 ! [I: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ a ) @ qs ) )
% 0.34/0.48 => ( ( nth @ a @ qs2 @ I )
% 0.34/0.48 = ( nth @ a @ qs @ ( g @ I ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % "1"
% 0.34/0.48 thf(fact_12_g__bij2,axiom,
% 0.34/0.48 bij_betw @ nat @ nat @ g @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ a ) @ qs2 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ a ) @ qs ) ) ).
% 0.34/0.48
% 0.34/0.48 % g_bij2
% 0.34/0.48 thf(fact_13_sum__mset__sum__list,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( comm_monoid_add @ A )
% 0.34/0.48 => ! [Xs: list @ A] :
% 0.34/0.48 ( ( comm_m1079429914m_mset @ A @ ( mset @ A @ Xs ) )
% 0.34/0.48 = ( groups501227526m_list @ A @ Xs ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % sum_mset_sum_list
% 0.34/0.48 thf(fact_14_direct__decompD_I3_J,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( comm_monoid_add @ A )
% 0.34/0.48 => ! [A2: set @ A,Ss: list @ ( set @ A )] :
% 0.34/0.48 ( ( hilber580762119decomp @ A @ A2 @ Ss )
% 0.34/0.48 => ( ( image @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss ) )
% 0.34/0.48 = A2 ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % direct_decompD(3)
% 0.34/0.48 thf(fact_15_assms_I1_J,axiom,
% 0.34/0.48 hilber580762119decomp @ a @ a2 @ ss1 ).
% 0.34/0.48
% 0.34/0.48 % assms(1)
% 0.34/0.48 thf(fact_16_qs__in,axiom,
% 0.34/0.48 member @ ( list @ a ) @ qs @ ( listset @ a @ ss1 ) ).
% 0.34/0.48
% 0.34/0.48 % qs_in
% 0.34/0.48 thf(fact_17_mset__rev,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A] :
% 0.34/0.48 ( ( mset @ A @ ( rev @ A @ Xs ) )
% 0.34/0.48 = ( mset @ A @ Xs ) ) ).
% 0.34/0.48
% 0.34/0.48 % mset_rev
% 0.34/0.48 thf(fact_18__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs2_O_A_092_060lbrakk_062qs2_A_092_060in_062_Alistset_Ass2_059_Alength_Aqs2_A_061_Alength_Aqs_059_A_092_060And_062i_O_Ai_A_060_Alength_Aqs_A_092_060Longrightarrow_062_Aqs2_A_B_Ai_A_061_Aqs_A_B_Ag_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.34/0.48 ~ ! [Qs22: list @ a] :
% 0.34/0.48 ( ( member @ ( list @ a ) @ Qs22 @ ( listset @ a @ ss2 ) )
% 0.34/0.48 => ( ( ( size_size @ ( list @ a ) @ Qs22 )
% 0.34/0.48 = ( size_size @ ( list @ a ) @ qs ) )
% 0.34/0.48 => ~ ! [I2: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ a ) @ qs ) )
% 0.34/0.48 => ( ( nth @ a @ Qs22 @ I2 )
% 0.34/0.48 = ( nth @ a @ qs @ ( g @ I2 ) ) ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % \<open>\<And>thesis. (\<And>qs2. \<lbrakk>qs2 \<in> listset ss2; length qs2 = length qs; \<And>i. i < length qs \<Longrightarrow> qs2 ! i = qs ! g i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.34/0.48 thf(fact_19_assms_I2_J,axiom,
% 0.34/0.48 perm @ ( set @ a ) @ ss1 @ ss2 ).
% 0.34/0.48
% 0.34/0.48 % assms(2)
% 0.34/0.48 thf(fact_20_ex__mset,axiom,
% 0.34/0.48 ! [A: $tType,X2: multiset @ A] :
% 0.34/0.48 ? [Xs2: list @ A] :
% 0.34/0.48 ( ( mset @ A @ Xs2 )
% 0.34/0.48 = X2 ) ).
% 0.34/0.48
% 0.34/0.48 % ex_mset
% 0.34/0.48 thf(fact_21_mset__eq__length,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.34/0.48 ( ( ( mset @ A @ Xs )
% 0.34/0.48 = ( mset @ A @ Ys ) )
% 0.34/0.48 => ( ( size_size @ ( list @ A ) @ Xs )
% 0.34/0.48 = ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % mset_eq_length
% 0.34/0.48 thf(fact_22_mset__bij,axiom,
% 0.34/0.48 ! [A: $tType,F: nat > nat,Xs: list @ A,Ys: list @ A] :
% 0.34/0.48 ( ( bij_betw @ nat @ nat @ F @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Ys ) ) )
% 0.34/0.48 => ( ! [I3: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.34/0.48 => ( ( nth @ A @ Xs @ I3 )
% 0.34/0.48 = ( nth @ A @ Ys @ ( F @ I3 ) ) ) )
% 0.34/0.48 => ( ( mset @ A @ Xs )
% 0.34/0.48 = ( mset @ A @ Ys ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % mset_bij
% 0.34/0.48 thf(fact_23_listset__permE,axiom,
% 0.34/0.48 ! [A: $tType,Ys: list @ A,Xs: list @ ( set @ A ),F: nat > nat,Xs3: list @ ( set @ A )] :
% 0.34/0.48 ( ( member @ ( list @ A ) @ Ys @ ( listset @ A @ Xs ) )
% 0.34/0.48 => ( ( bij_betw @ nat @ nat @ F @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ A ) ) @ Xs ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ A ) ) @ Xs3 ) ) )
% 0.34/0.48 => ( ! [I3: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( set @ A ) ) @ Xs ) )
% 0.34/0.48 => ( ( nth @ ( set @ A ) @ Xs3 @ I3 )
% 0.34/0.48 = ( nth @ ( set @ A ) @ Xs @ ( F @ I3 ) ) ) )
% 0.34/0.48 => ~ ! [Ys2: list @ A] :
% 0.34/0.48 ( ( member @ ( list @ A ) @ Ys2 @ ( listset @ A @ Xs3 ) )
% 0.34/0.48 => ( ( ( size_size @ ( list @ A ) @ Ys2 )
% 0.34/0.48 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.34/0.48 => ~ ! [I2: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys ) )
% 0.34/0.48 => ( ( nth @ A @ Ys2 @ I2 )
% 0.34/0.48 = ( nth @ A @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % listset_permE
% 0.34/0.48 thf(fact_24_g__bij,axiom,
% 0.34/0.48 bij_betw @ nat @ nat @ g @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) ).
% 0.34/0.48
% 0.34/0.48 % g_bij
% 0.34/0.48 thf(fact_25_length__rev,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A] :
% 0.34/0.48 ( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
% 0.34/0.48 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.34/0.48
% 0.34/0.48 % length_rev
% 0.34/0.48 thf(fact_26_lessThan__iff,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( ord @ A )
% 0.34/0.48 => ! [I: A,K: A] :
% 0.34/0.48 ( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
% 0.34/0.48 = ( ord_less @ A @ I @ K ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % lessThan_iff
% 0.34/0.48 thf(fact_27_listsetI,axiom,
% 0.34/0.48 ! [A: $tType,Ys: list @ A,Xs: list @ ( set @ A )] :
% 0.34/0.48 ( ( ( size_size @ ( list @ A ) @ Ys )
% 0.34/0.48 = ( size_size @ ( list @ ( set @ A ) ) @ Xs ) )
% 0.34/0.48 => ( ! [I3: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( set @ A ) ) @ Xs ) )
% 0.34/0.48 => ( member @ A @ ( nth @ A @ Ys @ I3 ) @ ( nth @ ( set @ A ) @ Xs @ I3 ) ) )
% 0.34/0.48 => ( member @ ( list @ A ) @ Ys @ ( listset @ A @ Xs ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % listsetI
% 0.34/0.48 thf(fact_28_len__qs,axiom,
% 0.34/0.48 ( ( size_size @ ( list @ a ) @ qs )
% 0.34/0.48 = ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) ).
% 0.34/0.48
% 0.34/0.48 % len_qs
% 0.34/0.48 thf(fact_29_f__bij,axiom,
% 0.34/0.48 bij_betw @ nat @ nat @ f @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) ).
% 0.34/0.48
% 0.34/0.48 % f_bij
% 0.34/0.48 thf(fact_30_nth__equalityI,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.34/0.48 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.34/0.48 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.34/0.48 => ( ! [I3: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.34/0.48 => ( ( nth @ A @ Xs @ I3 )
% 0.34/0.48 = ( nth @ A @ Ys @ I3 ) ) )
% 0.34/0.48 => ( Xs = Ys ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % nth_equalityI
% 0.34/0.48 thf(fact_31_Skolem__list__nth,axiom,
% 0.34/0.48 ! [A: $tType,K: nat,P: nat > A > $o] :
% 0.34/0.48 ( ( ! [I4: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I4 @ K )
% 0.34/0.48 => ? [X3: A] : ( P @ I4 @ X3 ) ) )
% 0.34/0.48 = ( ? [Xs4: list @ A] :
% 0.34/0.48 ( ( ( size_size @ ( list @ A ) @ Xs4 )
% 0.34/0.48 = K )
% 0.34/0.48 & ! [I4: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I4 @ K )
% 0.34/0.48 => ( P @ I4 @ ( nth @ A @ Xs4 @ I4 ) ) ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % Skolem_list_nth
% 0.34/0.48 thf(fact_32_list__eq__iff__nth__eq,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( ^ [Y2: list @ A,Z: list @ A] : ( Y2 = Z ) )
% 0.34/0.48 = ( ^ [Xs4: list @ A,Ys3: list @ A] :
% 0.34/0.48 ( ( ( size_size @ ( list @ A ) @ Xs4 )
% 0.34/0.48 = ( size_size @ ( list @ A ) @ Ys3 ) )
% 0.34/0.48 & ! [I4: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs4 ) )
% 0.34/0.48 => ( ( nth @ A @ Xs4 @ I4 )
% 0.34/0.48 = ( nth @ A @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % list_eq_iff_nth_eq
% 0.34/0.48 thf(fact_33_lessThan__eq__iff,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( linorder @ A )
% 0.34/0.48 => ! [X4: A,Y3: A] :
% 0.34/0.48 ( ( ( set_ord_lessThan @ A @ X4 )
% 0.34/0.48 = ( set_ord_lessThan @ A @ Y3 ) )
% 0.34/0.48 = ( X4 = Y3 ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % lessThan_eq_iff
% 0.34/0.48 thf(fact_34_rev__is__rev__conv,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.34/0.48 ( ( ( rev @ A @ Xs )
% 0.34/0.48 = ( rev @ A @ Ys ) )
% 0.34/0.48 = ( Xs = Ys ) ) ).
% 0.34/0.48
% 0.34/0.48 % rev_is_rev_conv
% 0.34/0.48 thf(fact_35_rev__rev__ident,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A] :
% 0.34/0.48 ( ( rev @ A @ ( rev @ A @ Xs ) )
% 0.34/0.48 = Xs ) ).
% 0.34/0.48
% 0.34/0.48 % rev_rev_ident
% 0.34/0.48 thf(fact_36_len__ss1,axiom,
% 0.34/0.48 ( ( size_size @ ( list @ ( set @ a ) ) @ ss1 )
% 0.34/0.48 = ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) ).
% 0.34/0.48
% 0.34/0.48 % len_ss1
% 0.34/0.48 thf(fact_37_g,axiom,
% 0.34/0.48 ! [I: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) )
% 0.34/0.48 => ( ( nth @ ( set @ a ) @ ss2 @ I )
% 0.34/0.48 = ( nth @ ( set @ a ) @ ss1 @ ( g @ I ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % g
% 0.34/0.48 thf(fact_38_f,axiom,
% 0.34/0.48 ! [I: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) )
% 0.34/0.48 => ( ( nth @ ( set @ a ) @ ss1 @ I )
% 0.34/0.48 = ( nth @ ( set @ a ) @ ss2 @ ( f @ I ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % f
% 0.34/0.48 thf(fact_39__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062f_O_A_092_060lbrakk_062bij__betw_Af_A_123_O_O_060length_Ass2_125_A_123_O_O_060length_Ass1_125_059_A_092_060And_062i_O_Ai_A_060_Alength_Ass2_A_092_060Longrightarrow_062_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.34/0.48 ~ ! [F2: nat > nat] :
% 0.34/0.48 ( ( bij_betw @ nat @ nat @ F2 @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) )
% 0.34/0.48 => ~ ! [I2: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) )
% 0.34/0.48 => ( ( nth @ ( set @ a ) @ ss1 @ I2 )
% 0.34/0.48 = ( nth @ ( set @ a ) @ ss2 @ ( F2 @ I2 ) ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % \<open>\<And>thesis. (\<And>f. \<lbrakk>bij_betw f {..<length ss2} {..<length ss1}; \<And>i. i < length ss2 \<Longrightarrow> ss1 ! i = ss2 ! f i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.34/0.48 thf(fact_40__092_060open_062_092_060exists_062f_O_Abij__betw_Af_A_123_O_O_060length_Ass1_125_A_123_O_O_060length_Ass2_125_A_092_060and_062_A_I_092_060forall_062i_060length_Ass1_O_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_J_092_060close_062,axiom,
% 0.34/0.48 ? [F2: nat > nat] :
% 0.34/0.48 ( ( bij_betw @ nat @ nat @ F2 @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) )
% 0.34/0.48 & ! [I2: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) )
% 0.34/0.48 => ( ( nth @ ( set @ a ) @ ss1 @ I2 )
% 0.34/0.48 = ( nth @ ( set @ a ) @ ss2 @ ( F2 @ I2 ) ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % \<open>\<exists>f. bij_betw f {..<length ss1} {..<length ss2} \<and> (\<forall>i<length ss1. ss1 ! i = ss2 ! f i)\<close>
% 0.34/0.48 thf(fact_41_f__g,axiom,
% 0.34/0.48 ! [I: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) )
% 0.34/0.48 => ( ( f @ ( g @ I ) )
% 0.34/0.48 = I ) ) ).
% 0.34/0.48
% 0.34/0.48 % f_g
% 0.34/0.48 thf(fact_42_g__f,axiom,
% 0.34/0.48 ! [I: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) )
% 0.34/0.48 => ( ( g @ ( f @ I ) )
% 0.34/0.48 = I ) ) ).
% 0.34/0.48
% 0.34/0.48 % g_f
% 0.34/0.48 thf(fact_43_listsetD_I2_J,axiom,
% 0.34/0.48 ! [A: $tType,Ys: list @ A,Xs: list @ ( set @ A ),I: nat] :
% 0.34/0.48 ( ( member @ ( list @ A ) @ Ys @ ( listset @ A @ Xs ) )
% 0.34/0.48 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ A ) ) @ Xs ) )
% 0.34/0.48 => ( member @ A @ ( nth @ A @ Ys @ I ) @ ( nth @ ( set @ A ) @ Xs @ I ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % listsetD(2)
% 0.34/0.48 thf(fact_44_neq__if__length__neq,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.34/0.48 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.34/0.48 != ( size_size @ ( list @ A ) @ Ys ) )
% 0.34/0.48 => ( Xs != Ys ) ) ).
% 0.34/0.48
% 0.34/0.48 % neq_if_length_neq
% 0.34/0.48 thf(fact_45_mem__Collect__eq,axiom,
% 0.34/0.48 ! [A: $tType,A3: A,P: A > $o] :
% 0.34/0.48 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.34/0.48 = ( P @ A3 ) ) ).
% 0.34/0.48
% 0.34/0.48 % mem_Collect_eq
% 0.34/0.48 thf(fact_46_Collect__mem__eq,axiom,
% 0.34/0.48 ! [A: $tType,A2: set @ A] :
% 0.34/0.48 ( ( collect @ A
% 0.34/0.48 @ ^ [X5: A] : ( member @ A @ X5 @ A2 ) )
% 0.34/0.48 = A2 ) ).
% 0.34/0.48
% 0.34/0.48 % Collect_mem_eq
% 0.34/0.48 thf(fact_47_Collect__cong,axiom,
% 0.34/0.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.34/0.48 ( ! [X6: A] :
% 0.34/0.48 ( ( P @ X6 )
% 0.34/0.48 = ( Q @ X6 ) )
% 0.34/0.48 => ( ( collect @ A @ P )
% 0.34/0.48 = ( collect @ A @ Q ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % Collect_cong
% 0.34/0.48 thf(fact_48_ext,axiom,
% 0.34/0.48 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.34/0.48 ( ! [X6: A] :
% 0.34/0.48 ( ( F @ X6 )
% 0.34/0.48 = ( G @ X6 ) )
% 0.34/0.48 => ( F = G ) ) ).
% 0.34/0.48
% 0.34/0.48 % ext
% 0.34/0.48 thf(fact_49_Ex__list__of__length,axiom,
% 0.34/0.48 ! [A: $tType,N: nat] :
% 0.34/0.48 ? [Xs2: list @ A] :
% 0.34/0.48 ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.34/0.48 = N ) ).
% 0.34/0.48
% 0.34/0.48 % Ex_list_of_length
% 0.34/0.48 thf(fact_50_rev__swap,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.34/0.48 ( ( ( rev @ A @ Xs )
% 0.34/0.48 = Ys )
% 0.34/0.48 = ( Xs
% 0.34/0.48 = ( rev @ A @ Ys ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % rev_swap
% 0.34/0.48 thf(fact_51_lessThan__strict__subset__iff,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( linorder @ A )
% 0.34/0.48 => ! [M: A,N: A] :
% 0.34/0.48 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
% 0.34/0.48 = ( ord_less @ A @ M @ N ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % lessThan_strict_subset_iff
% 0.34/0.48 thf(fact_52_length__induct,axiom,
% 0.34/0.48 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.34/0.48 ( ! [Xs2: list @ A] :
% 0.34/0.48 ( ! [Ys4: list @ A] :
% 0.34/0.48 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys4 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.34/0.48 => ( P @ Ys4 ) )
% 0.34/0.48 => ( P @ Xs2 ) )
% 0.34/0.48 => ( P @ Xs ) ) ).
% 0.34/0.48
% 0.34/0.48 % length_induct
% 0.34/0.48 thf(fact_53_listsetD_I1_J,axiom,
% 0.34/0.48 ! [A: $tType,Ys: list @ A,Xs: list @ ( set @ A )] :
% 0.34/0.48 ( ( member @ ( list @ A ) @ Ys @ ( listset @ A @ Xs ) )
% 0.34/0.48 => ( ( size_size @ ( list @ A ) @ Ys )
% 0.34/0.48 = ( size_size @ ( list @ ( set @ A ) ) @ Xs ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % listsetD(1)
% 0.34/0.48 thf(fact_54_permutation__Ex__bij,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.34/0.48 ( ( perm @ A @ Xs @ Ys )
% 0.34/0.48 => ? [F2: nat > nat] :
% 0.34/0.48 ( ( bij_betw @ nat @ nat @ F2 @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Ys ) ) )
% 0.34/0.48 & ! [I2: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.34/0.48 => ( ( nth @ A @ Xs @ I2 )
% 0.34/0.48 = ( nth @ A @ Ys @ ( F2 @ I2 ) ) ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % permutation_Ex_bij
% 0.34/0.48 thf(fact_55_g__def,axiom,
% 0.34/0.48 ( g
% 0.34/0.48 = ( hilbert_inv_into @ nat @ nat @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) @ f ) ) ).
% 0.34/0.48
% 0.34/0.48 % g_def
% 0.34/0.48 thf(fact_56_perm__refl,axiom,
% 0.34/0.48 ! [A: $tType,L: list @ A] : ( perm @ A @ L @ L ) ).
% 0.34/0.48
% 0.34/0.48 % perm_refl
% 0.34/0.48 thf(fact_57_image__eqI,axiom,
% 0.34/0.48 ! [A: $tType,B: $tType,B2: A,F: B > A,X4: B,A2: set @ B] :
% 0.34/0.48 ( ( B2
% 0.34/0.48 = ( F @ X4 ) )
% 0.34/0.48 => ( ( member @ B @ X4 @ A2 )
% 0.34/0.48 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % image_eqI
% 0.34/0.48 thf(fact_58_mset__eq__perm,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.34/0.48 ( ( ( mset @ A @ Xs )
% 0.34/0.48 = ( mset @ A @ Ys ) )
% 0.34/0.48 = ( perm @ A @ Xs @ Ys ) ) ).
% 0.34/0.48
% 0.34/0.48 % mset_eq_perm
% 0.34/0.48 thf(fact_59_perm__rev,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A] : ( perm @ A @ ( rev @ A @ Xs ) @ Xs ) ).
% 0.34/0.48
% 0.34/0.48 % perm_rev
% 0.34/0.48 thf(fact_60_perm__length,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.34/0.48 ( ( perm @ A @ Xs @ Ys )
% 0.34/0.48 => ( ( size_size @ ( list @ A ) @ Xs )
% 0.34/0.48 = ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % perm_length
% 0.34/0.48 thf(fact_61_bij__betw__imp__surj__on,axiom,
% 0.34/0.48 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B] :
% 0.34/0.48 ( ( bij_betw @ A @ B @ F @ A2 @ B3 )
% 0.34/0.48 => ( ( image @ A @ B @ F @ A2 )
% 0.34/0.48 = B3 ) ) ).
% 0.34/0.48
% 0.34/0.48 % bij_betw_imp_surj_on
% 0.34/0.48 thf(fact_62_mbs_Oless__not__eq,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( size @ A )
% 0.34/0.48 => ! [X4: A,A2: set @ A,Y3: A] :
% 0.34/0.48 ( ( member @ A @ X4 @ A2 )
% 0.34/0.48 => ( ( ord_less @ nat @ ( size_size @ A @ X4 ) @ ( size_size @ A @ Y3 ) )
% 0.34/0.48 => ( X4 != Y3 ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % mbs.less_not_eq
% 0.34/0.48 thf(fact_63_size__mset,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A] :
% 0.34/0.48 ( ( size_size @ ( multiset @ A ) @ ( mset @ A @ Xs ) )
% 0.34/0.48 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.34/0.48
% 0.34/0.48 % size_mset
% 0.34/0.48 thf(fact_64_psubsetD,axiom,
% 0.34/0.48 ! [A: $tType,A2: set @ A,B3: set @ A,C: A] :
% 0.34/0.48 ( ( ord_less @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.48 => ( ( member @ A @ C @ A2 )
% 0.34/0.48 => ( member @ A @ C @ B3 ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % psubsetD
% 0.34/0.48 thf(fact_65_psubset__trans,axiom,
% 0.34/0.48 ! [A: $tType,A2: set @ A,B3: set @ A,C2: set @ A] :
% 0.34/0.48 ( ( ord_less @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.48 => ( ( ord_less @ ( set @ A ) @ B3 @ C2 )
% 0.34/0.48 => ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % psubset_trans
% 0.34/0.48 thf(fact_66_imageI,axiom,
% 0.34/0.48 ! [B: $tType,A: $tType,X4: A,A2: set @ A,F: A > B] :
% 0.34/0.48 ( ( member @ A @ X4 @ A2 )
% 0.34/0.48 => ( member @ B @ ( F @ X4 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % imageI
% 0.34/0.48 thf(fact_67_image__iff,axiom,
% 0.34/0.48 ! [A: $tType,B: $tType,Z2: A,F: B > A,A2: set @ B] :
% 0.34/0.48 ( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A2 ) )
% 0.34/0.48 = ( ? [X5: B] :
% 0.34/0.48 ( ( member @ B @ X5 @ A2 )
% 0.34/0.48 & ( Z2
% 0.34/0.48 = ( F @ X5 ) ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % image_iff
% 0.34/0.48 thf(fact_68_bex__imageD,axiom,
% 0.34/0.48 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
% 0.34/0.48 ( ? [X: A] :
% 0.34/0.48 ( ( member @ A @ X @ ( image @ B @ A @ F @ A2 ) )
% 0.34/0.48 & ( P @ X ) )
% 0.34/0.48 => ? [X6: B] :
% 0.34/0.48 ( ( member @ B @ X6 @ A2 )
% 0.34/0.48 & ( P @ ( F @ X6 ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % bex_imageD
% 0.34/0.48 thf(fact_69_image__cong,axiom,
% 0.34/0.48 ! [B: $tType,A: $tType,M2: set @ A,N2: set @ A,F: A > B,G: A > B] :
% 0.34/0.48 ( ( M2 = N2 )
% 0.34/0.48 => ( ! [X6: A] :
% 0.34/0.48 ( ( member @ A @ X6 @ N2 )
% 0.34/0.48 => ( ( F @ X6 )
% 0.34/0.48 = ( G @ X6 ) ) )
% 0.34/0.48 => ( ( image @ A @ B @ F @ M2 )
% 0.34/0.48 = ( image @ A @ B @ G @ N2 ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % image_cong
% 0.34/0.48 thf(fact_70_ball__imageD,axiom,
% 0.34/0.48 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
% 0.34/0.48 ( ! [X6: A] :
% 0.34/0.48 ( ( member @ A @ X6 @ ( image @ B @ A @ F @ A2 ) )
% 0.34/0.48 => ( P @ X6 ) )
% 0.34/0.48 => ! [X: B] :
% 0.34/0.48 ( ( member @ B @ X @ A2 )
% 0.34/0.48 => ( P @ ( F @ X ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % ball_imageD
% 0.34/0.48 thf(fact_71_rev__image__eqI,axiom,
% 0.34/0.48 ! [B: $tType,A: $tType,X4: A,A2: set @ A,B2: B,F: A > B] :
% 0.34/0.48 ( ( member @ A @ X4 @ A2 )
% 0.34/0.48 => ( ( B2
% 0.34/0.48 = ( F @ X4 ) )
% 0.34/0.48 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % rev_image_eqI
% 0.34/0.48 thf(fact_72_direct__decomp__def,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( comm_monoid_add @ A )
% 0.34/0.48 => ( ( hilber580762119decomp @ A )
% 0.34/0.48 = ( ^ [A5: set @ A,Ss2: list @ ( set @ A )] : ( bij_betw @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss2 ) @ A5 ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % direct_decomp_def
% 0.34/0.48 thf(fact_73_bij__betwE,axiom,
% 0.34/0.48 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B] :
% 0.34/0.48 ( ( bij_betw @ A @ B @ F @ A2 @ B3 )
% 0.34/0.48 => ! [X: A] :
% 0.34/0.48 ( ( member @ A @ X @ A2 )
% 0.34/0.48 => ( member @ B @ ( F @ X ) @ B3 ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % bij_betwE
% 0.34/0.48 thf(fact_74_bij__betw__inv,axiom,
% 0.34/0.48 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B3: set @ B] :
% 0.34/0.48 ( ( bij_betw @ A @ B @ F @ A2 @ B3 )
% 0.34/0.48 => ? [G2: B > A] : ( bij_betw @ B @ A @ G2 @ B3 @ A2 ) ) ).
% 0.34/0.48
% 0.34/0.48 % bij_betw_inv
% 0.34/0.48 thf(fact_75_bij__betw__cong,axiom,
% 0.34/0.48 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,G: A > B,A6: set @ B] :
% 0.34/0.48 ( ! [A4: A] :
% 0.34/0.48 ( ( member @ A @ A4 @ A2 )
% 0.34/0.48 => ( ( F @ A4 )
% 0.34/0.48 = ( G @ A4 ) ) )
% 0.34/0.48 => ( ( bij_betw @ A @ B @ F @ A2 @ A6 )
% 0.34/0.48 = ( bij_betw @ A @ B @ G @ A2 @ A6 ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % bij_betw_cong
% 0.34/0.48 thf(fact_76_bij__betw__apply,axiom,
% 0.34/0.48 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B,A3: A] :
% 0.34/0.48 ( ( bij_betw @ A @ B @ F @ A2 @ B3 )
% 0.34/0.48 => ( ( member @ A @ A3 @ A2 )
% 0.34/0.48 => ( member @ B @ ( F @ A3 ) @ B3 ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % bij_betw_apply
% 0.34/0.48 thf(fact_77_bij__betw__iff__bijections,axiom,
% 0.34/0.48 ! [B: $tType,A: $tType] :
% 0.34/0.48 ( ( bij_betw @ A @ B )
% 0.34/0.48 = ( ^ [F3: A > B,A5: set @ A,B4: set @ B] :
% 0.34/0.48 ? [G3: B > A] :
% 0.34/0.48 ( ! [X5: A] :
% 0.34/0.48 ( ( member @ A @ X5 @ A5 )
% 0.34/0.48 => ( ( member @ B @ ( F3 @ X5 ) @ B4 )
% 0.34/0.48 & ( ( G3 @ ( F3 @ X5 ) )
% 0.34/0.48 = X5 ) ) )
% 0.34/0.48 & ! [X5: B] :
% 0.34/0.48 ( ( member @ B @ X5 @ B4 )
% 0.34/0.48 => ( ( member @ A @ ( G3 @ X5 ) @ A5 )
% 0.34/0.48 & ( ( F3 @ ( G3 @ X5 ) )
% 0.34/0.48 = X5 ) ) ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % bij_betw_iff_bijections
% 0.34/0.48 thf(fact_78_perm__sym,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.34/0.48 ( ( perm @ A @ Xs @ Ys )
% 0.34/0.48 => ( perm @ A @ Ys @ Xs ) ) ).
% 0.34/0.48
% 0.34/0.48 % perm_sym
% 0.34/0.48 thf(fact_79_perm_Otrans,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
% 0.34/0.48 ( ( perm @ A @ Xs @ Ys )
% 0.34/0.48 => ( ( perm @ A @ Ys @ Zs )
% 0.34/0.48 => ( perm @ A @ Xs @ Zs ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % perm.trans
% 0.34/0.48 thf(fact_80_bij__betw__inv__into__right,axiom,
% 0.34/0.48 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,A6: set @ B,A7: B] :
% 0.34/0.48 ( ( bij_betw @ A @ B @ F @ A2 @ A6 )
% 0.34/0.48 => ( ( member @ B @ A7 @ A6 )
% 0.34/0.48 => ( ( F @ ( hilbert_inv_into @ A @ B @ A2 @ F @ A7 ) )
% 0.34/0.48 = A7 ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % bij_betw_inv_into_right
% 0.34/0.48 thf(fact_81_bij__betw__inv__into__left,axiom,
% 0.34/0.48 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,A6: set @ B,A3: A] :
% 0.34/0.48 ( ( bij_betw @ A @ B @ F @ A2 @ A6 )
% 0.34/0.48 => ( ( member @ A @ A3 @ A2 )
% 0.34/0.48 => ( ( hilbert_inv_into @ A @ B @ A2 @ F @ ( F @ A3 ) )
% 0.34/0.48 = A3 ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % bij_betw_inv_into_left
% 0.34/0.48 thf(fact_82_inv__into__inv__into__eq,axiom,
% 0.34/0.48 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,A6: set @ B,A3: A] :
% 0.34/0.48 ( ( bij_betw @ A @ B @ F @ A2 @ A6 )
% 0.34/0.48 => ( ( member @ A @ A3 @ A2 )
% 0.34/0.48 => ( ( hilbert_inv_into @ B @ A @ A6 @ ( hilbert_inv_into @ A @ B @ A2 @ F ) @ A3 )
% 0.34/0.48 = ( F @ A3 ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % inv_into_inv_into_eq
% 0.34/0.48 thf(fact_83_bij__betw__inv__into,axiom,
% 0.34/0.48 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B3: set @ B] :
% 0.34/0.48 ( ( bij_betw @ A @ B @ F @ A2 @ B3 )
% 0.34/0.48 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A2 @ F ) @ B3 @ A2 ) ) ).
% 0.34/0.48
% 0.34/0.48 % bij_betw_inv_into
% 0.34/0.48 thf(fact_84_inv__into__injective,axiom,
% 0.34/0.48 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,X4: B,Y3: B] :
% 0.34/0.48 ( ( ( hilbert_inv_into @ A @ B @ A2 @ F @ X4 )
% 0.34/0.48 = ( hilbert_inv_into @ A @ B @ A2 @ F @ Y3 ) )
% 0.34/0.48 => ( ( member @ B @ X4 @ ( image @ A @ B @ F @ A2 ) )
% 0.34/0.48 => ( ( member @ B @ Y3 @ ( image @ A @ B @ F @ A2 ) )
% 0.34/0.48 => ( X4 = Y3 ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % inv_into_injective
% 0.34/0.48 thf(fact_85_inv__into__into,axiom,
% 0.34/0.48 ! [A: $tType,B: $tType,X4: A,F: B > A,A2: set @ B] :
% 0.34/0.48 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A2 ) )
% 0.34/0.48 => ( member @ B @ ( hilbert_inv_into @ B @ A @ A2 @ F @ X4 ) @ A2 ) ) ).
% 0.34/0.48
% 0.34/0.48 % inv_into_into
% 0.34/0.48 thf(fact_86_f__inv__into__f,axiom,
% 0.34/0.48 ! [B: $tType,A: $tType,Y3: A,F: B > A,A2: set @ B] :
% 0.34/0.48 ( ( member @ A @ Y3 @ ( image @ B @ A @ F @ A2 ) )
% 0.34/0.48 => ( ( F @ ( hilbert_inv_into @ B @ A @ A2 @ F @ Y3 ) )
% 0.34/0.48 = Y3 ) ) ).
% 0.34/0.48
% 0.34/0.48 % f_inv_into_f
% 0.34/0.48 thf(fact_87_list__ex__length,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( list_ex @ A )
% 0.34/0.48 = ( ^ [P2: A > $o,Xs4: list @ A] :
% 0.34/0.48 ? [N3: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs4 ) )
% 0.34/0.48 & ( P2 @ ( nth @ A @ Xs4 @ N3 ) ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % list_ex_length
% 0.34/0.48 thf(fact_88_direct__decompI,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( comm_monoid_add @ A )
% 0.34/0.48 => ! [Ss: list @ ( set @ A ),A2: set @ A] :
% 0.34/0.48 ( ( inj_on @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss ) )
% 0.34/0.48 => ( ( ( image @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss ) )
% 0.34/0.48 = A2 )
% 0.34/0.48 => ( hilber580762119decomp @ A @ A2 @ Ss ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % direct_decompI
% 0.34/0.48 thf(fact_89_elem__le__sum__list,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( canoni770627133id_add @ A )
% 0.34/0.48 => ! [K: nat,Ns: list @ A] :
% 0.34/0.48 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Ns ) )
% 0.34/0.48 => ( ord_less_eq @ A @ ( nth @ A @ Ns @ K ) @ ( groups501227526m_list @ A @ Ns ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % elem_le_sum_list
% 0.34/0.48 thf(fact_90_perm__swap,axiom,
% 0.34/0.48 ! [A: $tType,I: nat,Xs: list @ A,J: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.34/0.48 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.34/0.48 => ( perm @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) @ Xs ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % perm_swap
% 0.34/0.48 thf(fact_91_mset__swap,axiom,
% 0.34/0.48 ! [A: $tType,I: nat,Ls: list @ A,J: nat] :
% 0.34/0.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ls ) )
% 0.34/0.48 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Ls ) )
% 0.34/0.48 => ( ( mset @ A @ ( list_update @ A @ ( list_update @ A @ Ls @ J @ ( nth @ A @ Ls @ I ) ) @ I @ ( nth @ A @ Ls @ J ) ) )
% 0.34/0.48 = ( mset @ A @ Ls ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % mset_swap
% 0.34/0.48 thf(fact_92_nth__mem__mset,axiom,
% 0.34/0.48 ! [A: $tType,I: nat,Ls: list @ A] :
% 0.34/0.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ls ) )
% 0.34/0.48 => ( member @ A @ ( nth @ A @ Ls @ I ) @ ( set_mset @ A @ ( mset @ A @ Ls ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % nth_mem_mset
% 0.34/0.48 thf(fact_93_in__Union__mset__iff,axiom,
% 0.34/0.48 ! [A: $tType,X4: A,MM: multiset @ ( multiset @ A )] :
% 0.34/0.48 ( ( member @ A @ X4 @ ( set_mset @ A @ ( comm_m1079429914m_mset @ ( multiset @ A ) @ MM ) ) )
% 0.34/0.48 = ( ? [M3: multiset @ A] :
% 0.34/0.48 ( ( member @ ( multiset @ A ) @ M3 @ ( set_mset @ ( multiset @ A ) @ MM ) )
% 0.34/0.48 & ( member @ A @ X4 @ ( set_mset @ A @ M3 ) ) ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % in_Union_mset_iff
% 0.34/0.48 thf(fact_94_psubsetI,axiom,
% 0.34/0.48 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.34/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.48 => ( ( A2 != B3 )
% 0.34/0.48 => ( ord_less @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % psubsetI
% 0.34/0.48 thf(fact_95_inj__on__rev,axiom,
% 0.34/0.48 ! [A: $tType,A2: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ A2 ) ).
% 0.34/0.48
% 0.34/0.48 % inj_on_rev
% 0.34/0.48 thf(fact_96_list__update__overwrite,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A,I: nat,X4: A,Y3: A] :
% 0.34/0.48 ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ I @ Y3 )
% 0.34/0.48 = ( list_update @ A @ Xs @ I @ Y3 ) ) ).
% 0.34/0.48
% 0.34/0.48 % list_update_overwrite
% 0.34/0.48 thf(fact_97_lessThan__subset__iff,axiom,
% 0.34/0.48 ! [A: $tType] :
% 0.34/0.48 ( ( linorder @ A )
% 0.34/0.48 => ! [X4: A,Y3: A] :
% 0.34/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X4 ) @ ( set_ord_lessThan @ A @ Y3 ) )
% 0.34/0.48 = ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% 0.34/0.48
% 0.34/0.48 % lessThan_subset_iff
% 0.34/0.48 thf(fact_98_list__update__beyond,axiom,
% 0.34/0.48 ! [A: $tType,Xs: list @ A,I: nat,X4: A] :
% 0.34/0.49 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
% 0.34/0.49 => ( ( list_update @ A @ Xs @ I @ X4 )
% 0.34/0.49 = Xs ) ) ).
% 0.34/0.49
% 0.34/0.49 % list_update_beyond
% 0.34/0.49 thf(fact_99_length__list__update,axiom,
% 0.34/0.49 ! [A: $tType,Xs: list @ A,I: nat,X4: A] :
% 0.34/0.49 ( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I @ X4 ) )
% 0.34/0.49 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.34/0.49
% 0.34/0.49 % length_list_update
% 0.34/0.49 thf(fact_100_inv__into__f__f,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ A2 )
% 0.34/0.49 => ( ( member @ A @ X4 @ A2 )
% 0.34/0.49 => ( ( hilbert_inv_into @ A @ B @ A2 @ F @ ( F @ X4 ) )
% 0.34/0.49 = X4 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inv_into_f_f
% 0.34/0.49 thf(fact_101_list__update__id,axiom,
% 0.34/0.49 ! [A: $tType,Xs: list @ A,I: nat] :
% 0.34/0.49 ( ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ I ) )
% 0.34/0.49 = Xs ) ).
% 0.34/0.49
% 0.34/0.49 % list_update_id
% 0.34/0.49 thf(fact_102_nth__list__update__neq,axiom,
% 0.34/0.49 ! [A: $tType,I: nat,J: nat,Xs: list @ A,X4: A] :
% 0.34/0.49 ( ( I != J )
% 0.34/0.49 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ J )
% 0.34/0.49 = ( nth @ A @ Xs @ J ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % nth_list_update_neq
% 0.34/0.49 thf(fact_103_list__ex__rev,axiom,
% 0.34/0.49 ! [A: $tType,P: A > $o,Xs: list @ A] :
% 0.34/0.49 ( ( list_ex @ A @ P @ ( rev @ A @ Xs ) )
% 0.34/0.49 = ( list_ex @ A @ P @ Xs ) ) ).
% 0.34/0.49
% 0.34/0.49 % list_ex_rev
% 0.34/0.49 thf(fact_104_inv__into__image__cancel,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,S: set @ A] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ A2 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ S @ A2 )
% 0.34/0.49 => ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ S ) )
% 0.34/0.49 = S ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inv_into_image_cancel
% 0.34/0.49 thf(fact_105_nth__list__update__eq,axiom,
% 0.34/0.49 ! [A: $tType,I: nat,Xs: list @ A,X4: A] :
% 0.34/0.49 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.34/0.49 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ I )
% 0.34/0.49 = X4 ) ) ).
% 0.34/0.49
% 0.34/0.49 % nth_list_update_eq
% 0.34/0.49 thf(fact_106_subset__image__inj,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType,S: set @ A,F: B > A,T: set @ B] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ S @ ( image @ B @ A @ F @ T ) )
% 0.34/0.49 = ( ? [U: set @ B] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ B ) @ U @ T )
% 0.34/0.49 & ( inj_on @ B @ A @ F @ U )
% 0.34/0.49 & ( S
% 0.34/0.49 = ( image @ B @ A @ F @ U ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % subset_image_inj
% 0.34/0.49 thf(fact_107_inj__on__inv__into,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,B3: set @ A,F: B > A,A2: set @ B] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F @ A2 ) )
% 0.34/0.49 => ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ A2 @ F ) @ B3 ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_on_inv_into
% 0.34/0.49 thf(fact_108_list__update__swap,axiom,
% 0.34/0.49 ! [A: $tType,I: nat,I5: nat,Xs: list @ A,X4: A,X7: A] :
% 0.34/0.49 ( ( I != I5 )
% 0.34/0.49 => ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ I5 @ X7 )
% 0.34/0.49 = ( list_update @ A @ ( list_update @ A @ Xs @ I5 @ X7 ) @ I @ X4 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % list_update_swap
% 0.34/0.49 thf(fact_109_inj__on__image__eq__iff,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,C2: set @ A,A2: set @ A,B3: set @ A] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ C2 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C2 )
% 0.34/0.49 => ( ( ( image @ A @ B @ F @ A2 )
% 0.34/0.49 = ( image @ A @ B @ F @ B3 ) )
% 0.34/0.49 = ( A2 = B3 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_on_image_eq_iff
% 0.34/0.49 thf(fact_110_inj__on__image__mem__iff,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,B3: set @ A,A3: A,A2: set @ A] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ B3 )
% 0.34/0.49 => ( ( member @ A @ A3 @ B3 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ( ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) )
% 0.34/0.49 = ( member @ A @ A3 @ A2 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_on_image_mem_iff
% 0.34/0.49 thf(fact_111_inj__on__inverseI,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,A2: set @ A,G: B > A,F: A > B] :
% 0.34/0.49 ( ! [X6: A] :
% 0.34/0.49 ( ( member @ A @ X6 @ A2 )
% 0.34/0.49 => ( ( G @ ( F @ X6 ) )
% 0.34/0.49 = X6 ) )
% 0.34/0.49 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_on_inverseI
% 0.34/0.49 thf(fact_112_inj__on__contraD,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A,Y3: A] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ A2 )
% 0.34/0.49 => ( ( X4 != Y3 )
% 0.34/0.49 => ( ( member @ A @ X4 @ A2 )
% 0.34/0.49 => ( ( member @ A @ Y3 @ A2 )
% 0.34/0.49 => ( ( F @ X4 )
% 0.34/0.49 != ( F @ Y3 ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_on_contraD
% 0.34/0.49 thf(fact_113_subset__inj__on,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,B3: set @ A,A2: set @ A] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ B3 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % subset_inj_on
% 0.34/0.49 thf(fact_114_inj__on__subset,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B3: set @ A] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ A2 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 0.34/0.49 => ( inj_on @ A @ B @ F @ B3 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_on_subset
% 0.34/0.49 thf(fact_115_inj__on__eq__iff,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A,Y3: A] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ A2 )
% 0.34/0.49 => ( ( member @ A @ X4 @ A2 )
% 0.34/0.49 => ( ( member @ A @ Y3 @ A2 )
% 0.34/0.49 => ( ( ( F @ X4 )
% 0.34/0.49 = ( F @ Y3 ) )
% 0.34/0.49 = ( X4 = Y3 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_on_eq_iff
% 0.34/0.49 thf(fact_116_inj__on__cong,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,G: A > B] :
% 0.34/0.49 ( ! [A4: A] :
% 0.34/0.49 ( ( member @ A @ A4 @ A2 )
% 0.34/0.49 => ( ( F @ A4 )
% 0.34/0.49 = ( G @ A4 ) ) )
% 0.34/0.49 => ( ( inj_on @ A @ B @ F @ A2 )
% 0.34/0.49 = ( inj_on @ A @ B @ G @ A2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_on_cong
% 0.34/0.49 thf(fact_117_inj__on__def,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType] :
% 0.34/0.49 ( ( inj_on @ A @ B )
% 0.34/0.49 = ( ^ [F3: A > B,A5: set @ A] :
% 0.34/0.49 ! [X5: A] :
% 0.34/0.49 ( ( member @ A @ X5 @ A5 )
% 0.34/0.49 => ! [Y4: A] :
% 0.34/0.49 ( ( member @ A @ Y4 @ A5 )
% 0.34/0.49 => ( ( ( F3 @ X5 )
% 0.34/0.49 = ( F3 @ Y4 ) )
% 0.34/0.49 => ( X5 = Y4 ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_on_def
% 0.34/0.49 thf(fact_118_inj__onI,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,A2: set @ A,F: A > B] :
% 0.34/0.49 ( ! [X6: A,Y: A] :
% 0.34/0.49 ( ( member @ A @ X6 @ A2 )
% 0.34/0.49 => ( ( member @ A @ Y @ A2 )
% 0.34/0.49 => ( ( ( F @ X6 )
% 0.34/0.49 = ( F @ Y ) )
% 0.34/0.49 => ( X6 = Y ) ) ) )
% 0.34/0.49 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_onI
% 0.34/0.49 thf(fact_119_inj__onD,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A,Y3: A] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ A2 )
% 0.34/0.49 => ( ( ( F @ X4 )
% 0.34/0.49 = ( F @ Y3 ) )
% 0.34/0.49 => ( ( member @ A @ X4 @ A2 )
% 0.34/0.49 => ( ( member @ A @ Y3 @ A2 )
% 0.34/0.49 => ( X4 = Y3 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_onD
% 0.34/0.49 thf(fact_120_linorder__inj__onI,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [A2: set @ A,F: A > B] :
% 0.34/0.49 ( ! [X6: A,Y: A] :
% 0.34/0.49 ( ( ord_less @ A @ X6 @ Y )
% 0.34/0.49 => ( ( member @ A @ X6 @ A2 )
% 0.34/0.49 => ( ( member @ A @ Y @ A2 )
% 0.34/0.49 => ( ( F @ X6 )
% 0.34/0.49 != ( F @ Y ) ) ) ) )
% 0.34/0.49 => ( ! [X6: A,Y: A] :
% 0.34/0.49 ( ( member @ A @ X6 @ A2 )
% 0.34/0.49 => ( ( member @ A @ Y @ A2 )
% 0.34/0.49 => ( ( ord_less_eq @ A @ X6 @ Y )
% 0.34/0.49 | ( ord_less_eq @ A @ Y @ X6 ) ) ) )
% 0.34/0.49 => ( inj_on @ A @ B @ F @ A2 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % linorder_inj_onI
% 0.34/0.49 thf(fact_121_inj__on__image__iff,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,A2: set @ A,G: A > B,F: A > A] :
% 0.34/0.49 ( ! [X6: A] :
% 0.34/0.49 ( ( member @ A @ X6 @ A2 )
% 0.34/0.49 => ! [Xa: A] :
% 0.34/0.49 ( ( member @ A @ Xa @ A2 )
% 0.34/0.49 => ( ( ( G @ ( F @ X6 ) )
% 0.34/0.49 = ( G @ ( F @ Xa ) ) )
% 0.34/0.49 = ( ( G @ X6 )
% 0.34/0.49 = ( G @ Xa ) ) ) ) )
% 0.34/0.49 => ( ( inj_on @ A @ A @ F @ A2 )
% 0.34/0.49 => ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A2 ) )
% 0.34/0.49 = ( inj_on @ A @ B @ G @ A2 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_on_image_iff
% 0.34/0.49 thf(fact_122_bij__betw__imp__inj__on,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B3: set @ B] :
% 0.34/0.49 ( ( bij_betw @ A @ B @ F @ A2 @ B3 )
% 0.34/0.49 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 0.34/0.49
% 0.34/0.49 % bij_betw_imp_inj_on
% 0.34/0.49 thf(fact_123_inv__into__f__eq,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A,Y3: B] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ A2 )
% 0.34/0.49 => ( ( member @ A @ X4 @ A2 )
% 0.34/0.49 => ( ( ( F @ X4 )
% 0.34/0.49 = Y3 )
% 0.34/0.49 => ( ( hilbert_inv_into @ A @ B @ A2 @ F @ Y3 )
% 0.34/0.49 = X4 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inv_into_f_eq
% 0.34/0.49 thf(fact_124_subset__image__iff,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType,B3: set @ A,F: B > A,A2: set @ B] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F @ A2 ) )
% 0.34/0.49 = ( ? [AA: set @ B] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
% 0.34/0.49 & ( B3
% 0.34/0.49 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % subset_image_iff
% 0.34/0.49 thf(fact_125_image__subset__iff,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B3: set @ A] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B3 )
% 0.34/0.49 = ( ! [X5: B] :
% 0.34/0.49 ( ( member @ B @ X5 @ A2 )
% 0.34/0.49 => ( member @ A @ ( F @ X5 ) @ B3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % image_subset_iff
% 0.34/0.49 thf(fact_126_subset__imageE,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType,B3: set @ A,F: B > A,A2: set @ B] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F @ A2 ) )
% 0.34/0.49 => ~ ! [C3: set @ B] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ B ) @ C3 @ A2 )
% 0.34/0.49 => ( B3
% 0.34/0.49 != ( image @ B @ A @ F @ C3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % subset_imageE
% 0.34/0.49 thf(fact_127_image__subsetI,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,B3: set @ B] :
% 0.34/0.49 ( ! [X6: A] :
% 0.34/0.49 ( ( member @ A @ X6 @ A2 )
% 0.34/0.49 => ( member @ B @ ( F @ X6 ) @ B3 ) )
% 0.34/0.49 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B3 ) ) ).
% 0.34/0.49
% 0.34/0.49 % image_subsetI
% 0.34/0.49 thf(fact_128_image__mono,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,A2: set @ A,B3: set @ A,F: A > B] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B3 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % image_mono
% 0.34/0.49 thf(fact_129_psubsetE,axiom,
% 0.34/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.34/0.49 ( ( ord_less @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % psubsetE
% 0.34/0.49 thf(fact_130_psubset__eq,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( ord_less @ ( set @ A ) )
% 0.34/0.49 = ( ^ [A5: set @ A,B4: set @ A] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
% 0.34/0.49 & ( A5 != B4 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % psubset_eq
% 0.34/0.49 thf(fact_131_psubset__imp__subset,axiom,
% 0.34/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.34/0.49 ( ( ord_less @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 0.34/0.49
% 0.34/0.49 % psubset_imp_subset
% 0.34/0.49 thf(fact_132_psubset__subset__trans,axiom,
% 0.34/0.49 ! [A: $tType,A2: set @ A,B3: set @ A,C2: set @ A] :
% 0.34/0.49 ( ( ord_less @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C2 )
% 0.34/0.49 => ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % psubset_subset_trans
% 0.34/0.49 thf(fact_133_subset__not__subset__eq,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( ord_less @ ( set @ A ) )
% 0.34/0.49 = ( ^ [A5: set @ A,B4: set @ A] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
% 0.34/0.49 & ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % subset_not_subset_eq
% 0.34/0.49 thf(fact_134_subset__psubset__trans,axiom,
% 0.34/0.49 ! [A: $tType,A2: set @ A,B3: set @ A,C2: set @ A] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ( ( ord_less @ ( set @ A ) @ B3 @ C2 )
% 0.34/0.49 => ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % subset_psubset_trans
% 0.34/0.49 thf(fact_135_subset__iff__psubset__eq,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) )
% 0.34/0.49 = ( ^ [A5: set @ A,B4: set @ A] :
% 0.34/0.49 ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
% 0.34/0.49 | ( A5 = B4 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % subset_iff_psubset_eq
% 0.34/0.49 thf(fact_136_inj__on__strict__subset,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,B3: set @ A,A2: set @ A] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ B3 )
% 0.34/0.49 => ( ( ord_less @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_on_strict_subset
% 0.34/0.49 thf(fact_137_bij__betw__def,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType] :
% 0.34/0.49 ( ( bij_betw @ A @ B )
% 0.34/0.49 = ( ^ [F3: A > B,A5: set @ A,B4: set @ B] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F3 @ A5 )
% 0.34/0.49 & ( ( image @ A @ B @ F3 @ A5 )
% 0.34/0.49 = B4 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % bij_betw_def
% 0.34/0.49 thf(fact_138_bij__betw__imageI,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ A2 )
% 0.34/0.49 => ( ( ( image @ A @ B @ F @ A2 )
% 0.34/0.49 = B3 )
% 0.34/0.49 => ( bij_betw @ A @ B @ F @ A2 @ B3 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % bij_betw_imageI
% 0.34/0.49 thf(fact_139_inj__on__imp__bij__betw,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ A2 )
% 0.34/0.49 => ( bij_betw @ A @ B @ F @ A2 @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % inj_on_imp_bij_betw
% 0.34/0.49 thf(fact_140_mbs_Ominimal,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( size @ A )
% 0.34/0.49 => ! [X4: A,A2: set @ A,P: A > $o] :
% 0.34/0.49 ( ( member @ A @ X4 @ A2 )
% 0.34/0.49 => ( ( P @ X4 )
% 0.34/0.49 => ? [X6: A] :
% 0.34/0.49 ( ( member @ A @ X6 @ A2 )
% 0.34/0.49 & ( ord_less_eq @ nat @ ( size_size @ A @ X6 ) @ ( size_size @ A @ X4 ) )
% 0.34/0.49 & ( P @ X6 )
% 0.34/0.49 & ! [Xa2: A] :
% 0.34/0.49 ( ( member @ A @ Xa2 @ A2 )
% 0.34/0.49 => ( ( ord_less @ nat @ ( size_size @ A @ Xa2 ) @ ( size_size @ A @ X6 ) )
% 0.34/0.49 => ~ ( P @ Xa2 ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % mbs.minimal
% 0.34/0.49 thf(fact_141_bij__betw__subset,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,A6: set @ B,B3: set @ A,B5: set @ B] :
% 0.34/0.49 ( ( bij_betw @ A @ B @ F @ A2 @ A6 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 0.34/0.49 => ( ( ( image @ A @ B @ F @ B3 )
% 0.34/0.49 = B5 )
% 0.34/0.49 => ( bij_betw @ A @ B @ F @ B3 @ B5 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % bij_betw_subset
% 0.34/0.49 thf(fact_142_bij__betw__byWitness,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType,A2: set @ A,F4: B > A,F: A > B,A6: set @ B] :
% 0.34/0.49 ( ! [X6: A] :
% 0.34/0.49 ( ( member @ A @ X6 @ A2 )
% 0.34/0.49 => ( ( F4 @ ( F @ X6 ) )
% 0.34/0.49 = X6 ) )
% 0.34/0.49 => ( ! [X6: B] :
% 0.34/0.49 ( ( member @ B @ X6 @ A6 )
% 0.34/0.49 => ( ( F @ ( F4 @ X6 ) )
% 0.34/0.49 = X6 ) )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ A6 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ A2 )
% 0.34/0.49 => ( bij_betw @ A @ B @ F @ A2 @ A6 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % bij_betw_byWitness
% 0.34/0.49 thf(fact_143_image__inv__into__cancel,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,A6: set @ A,B5: set @ A] :
% 0.34/0.49 ( ( ( image @ B @ A @ F @ A2 )
% 0.34/0.49 = A6 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
% 0.34/0.49 => ( ( image @ B @ A @ F @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ A2 @ F ) @ B5 ) )
% 0.34/0.49 = B5 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % image_inv_into_cancel
% 0.34/0.49 thf(fact_144_listset__mono,axiom,
% 0.34/0.49 ! [A: $tType,Xs: list @ ( set @ A ),Ys: list @ ( set @ A )] :
% 0.34/0.49 ( ( ( size_size @ ( list @ ( set @ A ) ) @ Xs )
% 0.34/0.49 = ( size_size @ ( list @ ( set @ A ) ) @ Ys ) )
% 0.34/0.49 => ( ! [I3: nat] :
% 0.34/0.49 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( set @ A ) ) @ Ys ) )
% 0.34/0.49 => ( ord_less_eq @ ( set @ A ) @ ( nth @ ( set @ A ) @ Xs @ I3 ) @ ( nth @ ( set @ A ) @ Ys @ I3 ) ) )
% 0.34/0.49 => ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( listset @ A @ Xs ) @ ( listset @ A @ Ys ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % listset_mono
% 0.34/0.49 thf(fact_145_nth__list__update,axiom,
% 0.34/0.49 ! [A: $tType,I: nat,Xs: list @ A,J: nat,X4: A] :
% 0.34/0.49 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.34/0.49 => ( ( ( I = J )
% 0.34/0.49 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ J )
% 0.34/0.49 = X4 ) )
% 0.34/0.49 & ( ( I != J )
% 0.34/0.49 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ J )
% 0.34/0.49 = ( nth @ A @ Xs @ J ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % nth_list_update
% 0.34/0.49 thf(fact_146_list__update__same__conv,axiom,
% 0.34/0.49 ! [A: $tType,I: nat,Xs: list @ A,X4: A] :
% 0.34/0.49 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.34/0.49 => ( ( ( list_update @ A @ Xs @ I @ X4 )
% 0.34/0.49 = Xs )
% 0.34/0.49 = ( ( nth @ A @ Xs @ I )
% 0.34/0.49 = X4 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % list_update_same_conv
% 0.34/0.49 thf(fact_147_bij__betw__inv__into__subset,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,A6: set @ B,B3: set @ A,B5: set @ B] :
% 0.34/0.49 ( ( bij_betw @ A @ B @ F @ A2 @ A6 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 0.34/0.49 => ( ( ( image @ A @ B @ F @ B3 )
% 0.34/0.49 = B5 )
% 0.34/0.49 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A2 @ F ) @ B5 @ B3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % bij_betw_inv_into_subset
% 0.34/0.49 thf(fact_148_direct__decompD_I2_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( comm_monoid_add @ A )
% 0.34/0.49 => ! [A2: set @ A,Ss: list @ ( set @ A )] :
% 0.34/0.49 ( ( hilber580762119decomp @ A @ A2 @ Ss )
% 0.34/0.49 => ( inj_on @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % direct_decompD(2)
% 0.34/0.49 thf(fact_149_Schroeder__Bernstein,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B,G: B > A] :
% 0.34/0.49 ( ( inj_on @ A @ B @ F @ A2 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B3 )
% 0.34/0.49 => ( ( inj_on @ B @ A @ G @ B3 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B3 ) @ A2 )
% 0.34/0.49 => ? [H: A > B] : ( bij_betw @ A @ B @ H @ A2 @ B3 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % Schroeder_Bernstein
% 0.34/0.49 thf(fact_150_order__refl,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( preorder @ A )
% 0.34/0.49 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 0.34/0.49
% 0.34/0.49 % order_refl
% 0.34/0.49 thf(fact_151_subset__imageE__inj,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType,B3: set @ A,F: B > A,A2: set @ B] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F @ A2 ) )
% 0.34/0.49 => ~ ! [C3: set @ B] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ B ) @ C3 @ A2 )
% 0.34/0.49 => ( ( B3
% 0.34/0.49 = ( image @ B @ A @ F @ C3 ) )
% 0.34/0.49 => ~ ( inj_on @ B @ A @ F @ C3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % subset_imageE_inj
% 0.34/0.49 thf(fact_152_minf_I8_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [T2: A] :
% 0.34/0.49 ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ X @ Z3 )
% 0.34/0.49 => ~ ( ord_less_eq @ A @ T2 @ X ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % minf(8)
% 0.34/0.49 thf(fact_153_minf_I6_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [T2: A] :
% 0.34/0.49 ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ X @ Z3 )
% 0.34/0.49 => ( ord_less_eq @ A @ X @ T2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % minf(6)
% 0.34/0.49 thf(fact_154_subset__antisym,axiom,
% 0.34/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 0.34/0.49 => ( A2 = B3 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % subset_antisym
% 0.34/0.49 thf(fact_155_subsetI,axiom,
% 0.34/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.34/0.49 ( ! [X6: A] :
% 0.34/0.49 ( ( member @ A @ X6 @ A2 )
% 0.34/0.49 => ( member @ A @ X6 @ B3 ) )
% 0.34/0.49 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 0.34/0.49
% 0.34/0.49 % subsetI
% 0.34/0.49 thf(fact_156_bounded__Max__nat,axiom,
% 0.34/0.49 ! [P: nat > $o,X4: nat,M2: nat] :
% 0.34/0.49 ( ( P @ X4 )
% 0.34/0.49 => ( ! [X6: nat] :
% 0.34/0.49 ( ( P @ X6 )
% 0.34/0.49 => ( ord_less_eq @ nat @ X6 @ M2 ) )
% 0.34/0.49 => ~ ! [M4: nat] :
% 0.34/0.49 ( ( P @ M4 )
% 0.34/0.49 => ~ ! [X: nat] :
% 0.34/0.49 ( ( P @ X )
% 0.34/0.49 => ( ord_less_eq @ nat @ X @ M4 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % bounded_Max_nat
% 0.34/0.49 thf(fact_157_Collect__mono__iff,axiom,
% 0.34/0.49 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.34/0.49 = ( ! [X5: A] :
% 0.34/0.49 ( ( P @ X5 )
% 0.34/0.49 => ( Q @ X5 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % Collect_mono_iff
% 0.34/0.49 thf(fact_158_set__eq__subset,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( ^ [Y2: set @ A,Z: set @ A] : ( Y2 = Z ) )
% 0.34/0.49 = ( ^ [A5: set @ A,B4: set @ A] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
% 0.34/0.49 & ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % set_eq_subset
% 0.34/0.49 thf(fact_159_subset__trans,axiom,
% 0.34/0.49 ! [A: $tType,A2: set @ A,B3: set @ A,C2: set @ A] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C2 )
% 0.34/0.49 => ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % subset_trans
% 0.34/0.49 thf(fact_160_Collect__mono,axiom,
% 0.34/0.49 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.34/0.49 ( ! [X6: A] :
% 0.34/0.49 ( ( P @ X6 )
% 0.34/0.49 => ( Q @ X6 ) )
% 0.34/0.49 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % Collect_mono
% 0.34/0.49 thf(fact_161_subset__refl,axiom,
% 0.34/0.49 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.34/0.49
% 0.34/0.49 % subset_refl
% 0.34/0.49 thf(fact_162_subset__iff,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) )
% 0.34/0.49 = ( ^ [A5: set @ A,B4: set @ A] :
% 0.34/0.49 ! [T3: A] :
% 0.34/0.49 ( ( member @ A @ T3 @ A5 )
% 0.34/0.49 => ( member @ A @ T3 @ B4 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % subset_iff
% 0.34/0.49 thf(fact_163_equalityD2,axiom,
% 0.34/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.34/0.49 ( ( A2 = B3 )
% 0.34/0.49 => ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ).
% 0.34/0.49
% 0.34/0.49 % equalityD2
% 0.34/0.49 thf(fact_164_equalityD1,axiom,
% 0.34/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.34/0.49 ( ( A2 = B3 )
% 0.34/0.49 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 0.34/0.49
% 0.34/0.49 % equalityD1
% 0.34/0.49 thf(fact_165_subset__eq,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) )
% 0.34/0.49 = ( ^ [A5: set @ A,B4: set @ A] :
% 0.34/0.49 ! [X5: A] :
% 0.34/0.49 ( ( member @ A @ X5 @ A5 )
% 0.34/0.49 => ( member @ A @ X5 @ B4 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % subset_eq
% 0.34/0.49 thf(fact_166_equalityE,axiom,
% 0.34/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.34/0.49 ( ( A2 = B3 )
% 0.34/0.49 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % equalityE
% 0.34/0.49 thf(fact_167_subsetD,axiom,
% 0.34/0.49 ! [A: $tType,A2: set @ A,B3: set @ A,C: A] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ( ( member @ A @ C @ A2 )
% 0.34/0.49 => ( member @ A @ C @ B3 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % subsetD
% 0.34/0.49 thf(fact_168_in__mono,axiom,
% 0.34/0.49 ! [A: $tType,A2: set @ A,B3: set @ A,X4: A] :
% 0.34/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.34/0.49 => ( ( member @ A @ X4 @ A2 )
% 0.34/0.49 => ( member @ A @ X4 @ B3 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % in_mono
% 0.34/0.49 thf(fact_169_dual__order_Oantisym,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [B2: A,A3: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.34/0.49 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.34/0.49 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dual_order.antisym
% 0.34/0.49 thf(fact_170_dual__order_Oeq__iff,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.34/0.49 = ( ^ [A8: A,B6: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ B6 @ A8 )
% 0.34/0.49 & ( ord_less_eq @ A @ A8 @ B6 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dual_order.eq_iff
% 0.34/0.49 thf(fact_171_dual__order_Otrans,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [B2: A,A3: A,C: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.34/0.49 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.34/0.49 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dual_order.trans
% 0.34/0.49 thf(fact_172_linorder__wlog,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [P: A > A > $o,A3: A,B2: A] :
% 0.34/0.49 ( ! [A4: A,B7: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ A4 @ B7 )
% 0.34/0.49 => ( P @ A4 @ B7 ) )
% 0.34/0.49 => ( ! [A4: A,B7: A] :
% 0.34/0.49 ( ( P @ B7 @ A4 )
% 0.34/0.49 => ( P @ A4 @ B7 ) )
% 0.34/0.49 => ( P @ A3 @ B2 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % linorder_wlog
% 0.34/0.49 thf(fact_173_dual__order_Orefl,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.34/0.49
% 0.34/0.49 % dual_order.refl
% 0.34/0.49 thf(fact_174_order__trans,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( preorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A,Z2: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.34/0.49 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.34/0.49 => ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order_trans
% 0.34/0.49 thf(fact_175_order__class_Oorder_Oantisym,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [A3: A,B2: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.34/0.49 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.34/0.49 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order_class.order.antisym
% 0.34/0.49 thf(fact_176_ord__le__eq__trans,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( ord @ A )
% 0.34/0.49 => ! [A3: A,B2: A,C: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.34/0.49 => ( ( B2 = C )
% 0.34/0.49 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % ord_le_eq_trans
% 0.34/0.49 thf(fact_177_ord__eq__le__trans,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( ord @ A )
% 0.34/0.49 => ! [A3: A,B2: A,C: A] :
% 0.34/0.49 ( ( A3 = B2 )
% 0.34/0.49 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.34/0.49 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % ord_eq_le_trans
% 0.34/0.49 thf(fact_178_order__class_Oorder_Oeq__iff,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.34/0.49 = ( ^ [A8: A,B6: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ A8 @ B6 )
% 0.34/0.49 & ( ord_less_eq @ A @ B6 @ A8 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order_class.order.eq_iff
% 0.34/0.49 thf(fact_179_antisym__conv,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [Y3: A,X4: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ Y3 @ X4 )
% 0.34/0.49 => ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.34/0.49 = ( X4 = Y3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % antisym_conv
% 0.34/0.49 thf(fact_180_le__cases3,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A,Z2: A] :
% 0.34/0.49 ( ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.34/0.49 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 0.34/0.49 => ( ( ( ord_less_eq @ A @ Y3 @ X4 )
% 0.34/0.49 => ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
% 0.34/0.49 => ( ( ( ord_less_eq @ A @ X4 @ Z2 )
% 0.34/0.49 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 0.34/0.49 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 0.34/0.49 => ~ ( ord_less_eq @ A @ Y3 @ X4 ) )
% 0.34/0.49 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.34/0.49 => ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
% 0.34/0.49 => ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
% 0.34/0.49 => ~ ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % le_cases3
% 0.34/0.49 thf(fact_181_order_Otrans,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [A3: A,B2: A,C: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.34/0.49 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.34/0.49 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order.trans
% 0.34/0.49 thf(fact_182_le__cases,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ~ ( ord_less_eq @ A @ X4 @ Y3 )
% 0.34/0.49 => ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % le_cases
% 0.34/0.49 thf(fact_183_eq__refl,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( preorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( X4 = Y3 )
% 0.34/0.49 => ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % eq_refl
% 0.34/0.49 thf(fact_184_linear,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.34/0.49 | ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % linear
% 0.34/0.49 thf(fact_185_antisym,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.34/0.49 => ( ( ord_less_eq @ A @ Y3 @ X4 )
% 0.34/0.49 => ( X4 = Y3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % antisym
% 0.34/0.49 thf(fact_186_le__rel__bool__arg__iff,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( ord @ A )
% 0.34/0.49 => ( ( ord_less_eq @ ( $o > A ) )
% 0.34/0.49 = ( ^ [X8: $o > A,Y5: $o > A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ ( X8 @ $false ) @ ( Y5 @ $false ) )
% 0.34/0.49 & ( ord_less_eq @ A @ ( X8 @ $true ) @ ( Y5 @ $true ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % le_rel_bool_arg_iff
% 0.34/0.49 thf(fact_187_eq__iff,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.34/0.49 = ( ^ [X5: A,Y4: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ X5 @ Y4 )
% 0.34/0.49 & ( ord_less_eq @ A @ Y4 @ X5 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % eq_iff
% 0.34/0.49 thf(fact_188_ord__le__eq__subst,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType] :
% 0.34/0.49 ( ( ( ord @ B )
% 0.34/0.49 & ( ord @ A ) )
% 0.34/0.49 => ! [A3: A,B2: A,F: A > B,C: B] :
% 0.34/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.34/0.49 => ( ( ( F @ B2 )
% 0.34/0.49 = C )
% 0.34/0.49 => ( ! [X6: A,Y: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ X6 @ Y )
% 0.34/0.49 => ( ord_less_eq @ B @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.34/0.49 => ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % ord_le_eq_subst
% 0.34/0.49 thf(fact_189_ord__eq__le__subst,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType] :
% 0.34/0.49 ( ( ( ord @ B )
% 0.34/0.49 & ( ord @ A ) )
% 0.34/0.49 => ! [A3: A,F: B > A,B2: B,C: B] :
% 0.34/0.49 ( ( A3
% 0.34/0.49 = ( F @ B2 ) )
% 0.34/0.49 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.34/0.49 => ( ! [X6: B,Y: B] :
% 0.34/0.49 ( ( ord_less_eq @ B @ X6 @ Y )
% 0.34/0.49 => ( ord_less_eq @ A @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.34/0.49 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % ord_eq_le_subst
% 0.34/0.49 thf(fact_190_order__subst2,axiom,
% 0.34/0.49 ! [A: $tType,C4: $tType] :
% 0.34/0.49 ( ( ( order @ C4 )
% 0.34/0.49 & ( order @ A ) )
% 0.34/0.49 => ! [A3: A,B2: A,F: A > C4,C: C4] :
% 0.34/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.34/0.49 => ( ( ord_less_eq @ C4 @ ( F @ B2 ) @ C )
% 0.34/0.49 => ( ! [X6: A,Y: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ X6 @ Y )
% 0.34/0.49 => ( ord_less_eq @ C4 @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.34/0.49 => ( ord_less_eq @ C4 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order_subst2
% 0.34/0.49 thf(fact_191_order__subst1,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType] :
% 0.34/0.49 ( ( ( order @ B )
% 0.34/0.49 & ( order @ A ) )
% 0.34/0.49 => ! [A3: A,F: B > A,B2: B,C: B] :
% 0.34/0.49 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 0.34/0.49 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.34/0.49 => ( ! [X6: B,Y: B] :
% 0.34/0.49 ( ( ord_less_eq @ B @ X6 @ Y )
% 0.34/0.49 => ( ord_less_eq @ A @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.34/0.49 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order_subst1
% 0.34/0.49 thf(fact_192_le__fun__def,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType] :
% 0.34/0.49 ( ( ord @ B )
% 0.34/0.49 => ( ( ord_less_eq @ ( A > B ) )
% 0.34/0.49 = ( ^ [F3: A > B,G3: A > B] :
% 0.34/0.49 ! [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % le_fun_def
% 0.34/0.49 thf(fact_193_le__funI,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType] :
% 0.34/0.49 ( ( ord @ B )
% 0.34/0.49 => ! [F: A > B,G: A > B] :
% 0.34/0.49 ( ! [X6: A] : ( ord_less_eq @ B @ ( F @ X6 ) @ ( G @ X6 ) )
% 0.34/0.49 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % le_funI
% 0.34/0.49 thf(fact_194_le__funE,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType] :
% 0.34/0.49 ( ( ord @ B )
% 0.34/0.49 => ! [F: A > B,G: A > B,X4: A] :
% 0.34/0.49 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.34/0.49 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % le_funE
% 0.34/0.49 thf(fact_195_le__funD,axiom,
% 0.34/0.49 ! [B: $tType,A: $tType] :
% 0.34/0.49 ( ( ord @ B )
% 0.34/0.49 => ! [F: A > B,G: A > B,X4: A] :
% 0.34/0.49 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.34/0.49 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % le_funD
% 0.34/0.49 thf(fact_196_dual__order_Ostrict__implies__not__eq,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [B2: A,A3: A] :
% 0.34/0.49 ( ( ord_less @ A @ B2 @ A3 )
% 0.34/0.49 => ( A3 != B2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dual_order.strict_implies_not_eq
% 0.34/0.49 thf(fact_197_order_Ostrict__implies__not__eq,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [A3: A,B2: A] :
% 0.34/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.34/0.49 => ( A3 != B2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order.strict_implies_not_eq
% 0.34/0.49 thf(fact_198_not__less__iff__gr__or__eq,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( ~ ( ord_less @ A @ X4 @ Y3 ) )
% 0.34/0.49 = ( ( ord_less @ A @ Y3 @ X4 )
% 0.34/0.49 | ( X4 = Y3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % not_less_iff_gr_or_eq
% 0.34/0.49 thf(fact_199_dual__order_Ostrict__trans,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [B2: A,A3: A,C: A] :
% 0.34/0.49 ( ( ord_less @ A @ B2 @ A3 )
% 0.34/0.49 => ( ( ord_less @ A @ C @ B2 )
% 0.34/0.49 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dual_order.strict_trans
% 0.34/0.49 thf(fact_200_linorder__less__wlog,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [P: A > A > $o,A3: A,B2: A] :
% 0.34/0.49 ( ! [A4: A,B7: A] :
% 0.34/0.49 ( ( ord_less @ A @ A4 @ B7 )
% 0.34/0.49 => ( P @ A4 @ B7 ) )
% 0.34/0.49 => ( ! [A4: A] : ( P @ A4 @ A4 )
% 0.34/0.49 => ( ! [A4: A,B7: A] :
% 0.34/0.49 ( ( P @ B7 @ A4 )
% 0.34/0.49 => ( P @ A4 @ B7 ) )
% 0.34/0.49 => ( P @ A3 @ B2 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % linorder_less_wlog
% 0.34/0.49 thf(fact_201_exists__least__iff,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( wellorder @ A )
% 0.34/0.49 => ( ( ^ [P3: A > $o] :
% 0.34/0.49 ? [X3: A] : ( P3 @ X3 ) )
% 0.34/0.49 = ( ^ [P2: A > $o] :
% 0.34/0.49 ? [N3: A] :
% 0.34/0.49 ( ( P2 @ N3 )
% 0.34/0.49 & ! [M5: A] :
% 0.34/0.49 ( ( ord_less @ A @ M5 @ N3 )
% 0.34/0.49 => ~ ( P2 @ M5 ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % exists_least_iff
% 0.34/0.49 thf(fact_202_less__imp__not__less,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( preorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 => ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % less_imp_not_less
% 0.34/0.49 thf(fact_203_order_Ostrict__trans,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [A3: A,B2: A,C: A] :
% 0.34/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.34/0.49 => ( ( ord_less @ A @ B2 @ C )
% 0.34/0.49 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order.strict_trans
% 0.34/0.49 thf(fact_204_dual__order_Oirrefl,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [A3: A] :
% 0.34/0.49 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 0.34/0.49
% 0.34/0.49 % dual_order.irrefl
% 0.34/0.49 thf(fact_205_linorder__cases,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ~ ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 => ( ( X4 != Y3 )
% 0.34/0.49 => ( ord_less @ A @ Y3 @ X4 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % linorder_cases
% 0.34/0.49 thf(fact_206_less__imp__triv,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( preorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A,P: $o] :
% 0.34/0.49 ( ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 => ( ( ord_less @ A @ Y3 @ X4 )
% 0.34/0.49 => P ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % less_imp_triv
% 0.34/0.49 thf(fact_207_less__imp__not__eq2,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 => ( Y3 != X4 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % less_imp_not_eq2
% 0.34/0.49 thf(fact_208_antisym__conv3,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [Y3: A,X4: A] :
% 0.34/0.49 ( ~ ( ord_less @ A @ Y3 @ X4 )
% 0.34/0.49 => ( ( ~ ( ord_less @ A @ X4 @ Y3 ) )
% 0.34/0.49 = ( X4 = Y3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % antisym_conv3
% 0.34/0.49 thf(fact_209_less__induct,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( wellorder @ A )
% 0.34/0.49 => ! [P: A > $o,A3: A] :
% 0.34/0.49 ( ! [X6: A] :
% 0.34/0.49 ( ! [Y6: A] :
% 0.34/0.49 ( ( ord_less @ A @ Y6 @ X6 )
% 0.34/0.49 => ( P @ Y6 ) )
% 0.34/0.49 => ( P @ X6 ) )
% 0.34/0.49 => ( P @ A3 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % less_induct
% 0.34/0.49 thf(fact_210_less__not__sym,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( preorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 => ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % less_not_sym
% 0.34/0.49 thf(fact_211_less__imp__not__eq,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 => ( X4 != Y3 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % less_imp_not_eq
% 0.34/0.49 thf(fact_212_dual__order_Oasym,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [B2: A,A3: A] :
% 0.34/0.49 ( ( ord_less @ A @ B2 @ A3 )
% 0.34/0.49 => ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dual_order.asym
% 0.34/0.49 thf(fact_213_ord__less__eq__trans,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( ord @ A )
% 0.34/0.49 => ! [A3: A,B2: A,C: A] :
% 0.34/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.34/0.49 => ( ( B2 = C )
% 0.34/0.49 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % ord_less_eq_trans
% 0.34/0.49 thf(fact_214_ord__eq__less__trans,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( ord @ A )
% 0.34/0.49 => ! [A3: A,B2: A,C: A] :
% 0.34/0.49 ( ( A3 = B2 )
% 0.34/0.49 => ( ( ord_less @ A @ B2 @ C )
% 0.34/0.49 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % ord_eq_less_trans
% 0.34/0.49 thf(fact_215_less__irrefl,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( preorder @ A )
% 0.34/0.49 => ! [X4: A] :
% 0.34/0.49 ~ ( ord_less @ A @ X4 @ X4 ) ) ).
% 0.34/0.49
% 0.34/0.49 % less_irrefl
% 0.34/0.49 thf(fact_216_less__linear,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 | ( X4 = Y3 )
% 0.34/0.49 | ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % less_linear
% 0.34/0.49 thf(fact_217_less__trans,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( preorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A,Z2: A] :
% 0.34/0.49 ( ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 => ( ( ord_less @ A @ Y3 @ Z2 )
% 0.34/0.49 => ( ord_less @ A @ X4 @ Z2 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % less_trans
% 0.34/0.49 thf(fact_218_less__asym_H,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( preorder @ A )
% 0.34/0.49 => ! [A3: A,B2: A] :
% 0.34/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.34/0.49 => ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % less_asym'
% 0.34/0.49 thf(fact_219_less__asym,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( preorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 => ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % less_asym
% 0.34/0.49 thf(fact_220_less__imp__neq,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 => ( X4 != Y3 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % less_imp_neq
% 0.34/0.49 thf(fact_221_dense,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( dense_order @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 => ? [Z3: A] :
% 0.34/0.49 ( ( ord_less @ A @ X4 @ Z3 )
% 0.34/0.49 & ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dense
% 0.34/0.49 thf(fact_222_order_Oasym,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [A3: A,B2: A] :
% 0.34/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.34/0.49 => ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order.asym
% 0.34/0.49 thf(fact_223_neq__iff,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( X4 != Y3 )
% 0.34/0.49 = ( ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 | ( ord_less @ A @ Y3 @ X4 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % neq_iff
% 0.34/0.49 thf(fact_224_neqE,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A] :
% 0.34/0.49 ( ( X4 != Y3 )
% 0.34/0.49 => ( ~ ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 => ( ord_less @ A @ Y3 @ X4 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % neqE
% 0.34/0.49 thf(fact_225_gt__ex,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( no_top @ A )
% 0.34/0.49 => ! [X4: A] :
% 0.34/0.49 ? [X_1: A] : ( ord_less @ A @ X4 @ X_1 ) ) ).
% 0.34/0.49
% 0.34/0.49 % gt_ex
% 0.34/0.49 thf(fact_226_lt__ex,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( no_bot @ A )
% 0.34/0.49 => ! [X4: A] :
% 0.34/0.49 ? [Y: A] : ( ord_less @ A @ Y @ X4 ) ) ).
% 0.34/0.49
% 0.34/0.49 % lt_ex
% 0.34/0.49 thf(fact_227_order__less__subst2,axiom,
% 0.34/0.49 ! [A: $tType,C4: $tType] :
% 0.34/0.49 ( ( ( order @ C4 )
% 0.34/0.49 & ( order @ A ) )
% 0.34/0.49 => ! [A3: A,B2: A,F: A > C4,C: C4] :
% 0.34/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.34/0.49 => ( ( ord_less @ C4 @ ( F @ B2 ) @ C )
% 0.34/0.49 => ( ! [X6: A,Y: A] :
% 0.34/0.49 ( ( ord_less @ A @ X6 @ Y )
% 0.34/0.49 => ( ord_less @ C4 @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.34/0.49 => ( ord_less @ C4 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order_less_subst2
% 0.34/0.49 thf(fact_228_order__less__subst1,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType] :
% 0.34/0.49 ( ( ( order @ B )
% 0.34/0.49 & ( order @ A ) )
% 0.34/0.49 => ! [A3: A,F: B > A,B2: B,C: B] :
% 0.34/0.49 ( ( ord_less @ A @ A3 @ ( F @ B2 ) )
% 0.34/0.49 => ( ( ord_less @ B @ B2 @ C )
% 0.34/0.49 => ( ! [X6: B,Y: B] :
% 0.34/0.49 ( ( ord_less @ B @ X6 @ Y )
% 0.34/0.49 => ( ord_less @ A @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.34/0.49 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order_less_subst1
% 0.34/0.49 thf(fact_229_ord__less__eq__subst,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType] :
% 0.34/0.49 ( ( ( ord @ B )
% 0.34/0.49 & ( ord @ A ) )
% 0.34/0.49 => ! [A3: A,B2: A,F: A > B,C: B] :
% 0.34/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.34/0.49 => ( ( ( F @ B2 )
% 0.34/0.49 = C )
% 0.34/0.49 => ( ! [X6: A,Y: A] :
% 0.34/0.49 ( ( ord_less @ A @ X6 @ Y )
% 0.34/0.49 => ( ord_less @ B @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.34/0.49 => ( ord_less @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % ord_less_eq_subst
% 0.34/0.49 thf(fact_230_ord__eq__less__subst,axiom,
% 0.34/0.49 ! [A: $tType,B: $tType] :
% 0.34/0.49 ( ( ( ord @ B )
% 0.34/0.49 & ( ord @ A ) )
% 0.34/0.49 => ! [A3: A,F: B > A,B2: B,C: B] :
% 0.34/0.49 ( ( A3
% 0.34/0.49 = ( F @ B2 ) )
% 0.34/0.49 => ( ( ord_less @ B @ B2 @ C )
% 0.34/0.49 => ( ! [X6: B,Y: B] :
% 0.34/0.49 ( ( ord_less @ B @ X6 @ Y )
% 0.34/0.49 => ( ord_less @ A @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.34/0.49 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % ord_eq_less_subst
% 0.34/0.49 thf(fact_231_pinf_I1_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.34/0.49 ( ? [Z4: A] :
% 0.34/0.49 ! [X6: A] :
% 0.34/0.49 ( ( ord_less @ A @ Z4 @ X6 )
% 0.34/0.49 => ( ( P @ X6 )
% 0.34/0.49 = ( P4 @ X6 ) ) )
% 0.34/0.49 => ( ? [Z4: A] :
% 0.34/0.49 ! [X6: A] :
% 0.34/0.49 ( ( ord_less @ A @ Z4 @ X6 )
% 0.34/0.49 => ( ( Q @ X6 )
% 0.34/0.49 = ( Q2 @ X6 ) ) )
% 0.34/0.49 => ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ Z3 @ X )
% 0.34/0.49 => ( ( ( P @ X )
% 0.34/0.49 & ( Q @ X ) )
% 0.34/0.49 = ( ( P4 @ X )
% 0.34/0.49 & ( Q2 @ X ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % pinf(1)
% 0.34/0.49 thf(fact_232_pinf_I2_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.34/0.49 ( ? [Z4: A] :
% 0.34/0.49 ! [X6: A] :
% 0.34/0.49 ( ( ord_less @ A @ Z4 @ X6 )
% 0.34/0.49 => ( ( P @ X6 )
% 0.34/0.49 = ( P4 @ X6 ) ) )
% 0.34/0.49 => ( ? [Z4: A] :
% 0.34/0.49 ! [X6: A] :
% 0.34/0.49 ( ( ord_less @ A @ Z4 @ X6 )
% 0.34/0.49 => ( ( Q @ X6 )
% 0.34/0.49 = ( Q2 @ X6 ) ) )
% 0.34/0.49 => ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ Z3 @ X )
% 0.34/0.49 => ( ( ( P @ X )
% 0.34/0.49 | ( Q @ X ) )
% 0.34/0.49 = ( ( P4 @ X )
% 0.34/0.49 | ( Q2 @ X ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % pinf(2)
% 0.34/0.49 thf(fact_233_pinf_I3_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [T2: A] :
% 0.34/0.49 ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ Z3 @ X )
% 0.34/0.49 => ( X != T2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % pinf(3)
% 0.34/0.49 thf(fact_234_pinf_I4_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [T2: A] :
% 0.34/0.49 ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ Z3 @ X )
% 0.34/0.49 => ( X != T2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % pinf(4)
% 0.34/0.49 thf(fact_235_pinf_I5_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [T2: A] :
% 0.34/0.49 ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ Z3 @ X )
% 0.34/0.49 => ~ ( ord_less @ A @ X @ T2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % pinf(5)
% 0.34/0.49 thf(fact_236_pinf_I7_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [T2: A] :
% 0.34/0.49 ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ Z3 @ X )
% 0.34/0.49 => ( ord_less @ A @ T2 @ X ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % pinf(7)
% 0.34/0.49 thf(fact_237_pinf_I11_J,axiom,
% 0.34/0.49 ! [C4: $tType,D: $tType] :
% 0.34/0.49 ( ( ord @ C4 )
% 0.34/0.49 => ! [F5: D] :
% 0.34/0.49 ? [Z3: C4] :
% 0.34/0.49 ! [X: C4] :
% 0.34/0.49 ( ( ord_less @ C4 @ Z3 @ X )
% 0.34/0.49 => ( F5 = F5 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % pinf(11)
% 0.34/0.49 thf(fact_238_minf_I1_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.34/0.49 ( ? [Z4: A] :
% 0.34/0.49 ! [X6: A] :
% 0.34/0.49 ( ( ord_less @ A @ X6 @ Z4 )
% 0.34/0.49 => ( ( P @ X6 )
% 0.34/0.49 = ( P4 @ X6 ) ) )
% 0.34/0.49 => ( ? [Z4: A] :
% 0.34/0.49 ! [X6: A] :
% 0.34/0.49 ( ( ord_less @ A @ X6 @ Z4 )
% 0.34/0.49 => ( ( Q @ X6 )
% 0.34/0.49 = ( Q2 @ X6 ) ) )
% 0.34/0.49 => ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ X @ Z3 )
% 0.34/0.49 => ( ( ( P @ X )
% 0.34/0.49 & ( Q @ X ) )
% 0.34/0.49 = ( ( P4 @ X )
% 0.34/0.49 & ( Q2 @ X ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % minf(1)
% 0.34/0.49 thf(fact_239_minf_I2_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.34/0.49 ( ? [Z4: A] :
% 0.34/0.49 ! [X6: A] :
% 0.34/0.49 ( ( ord_less @ A @ X6 @ Z4 )
% 0.34/0.49 => ( ( P @ X6 )
% 0.34/0.49 = ( P4 @ X6 ) ) )
% 0.34/0.49 => ( ? [Z4: A] :
% 0.34/0.49 ! [X6: A] :
% 0.34/0.49 ( ( ord_less @ A @ X6 @ Z4 )
% 0.34/0.49 => ( ( Q @ X6 )
% 0.34/0.49 = ( Q2 @ X6 ) ) )
% 0.34/0.49 => ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ X @ Z3 )
% 0.34/0.49 => ( ( ( P @ X )
% 0.34/0.49 | ( Q @ X ) )
% 0.34/0.49 = ( ( P4 @ X )
% 0.34/0.49 | ( Q2 @ X ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % minf(2)
% 0.34/0.49 thf(fact_240_minf_I3_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [T2: A] :
% 0.34/0.49 ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ X @ Z3 )
% 0.34/0.49 => ( X != T2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % minf(3)
% 0.34/0.49 thf(fact_241_minf_I4_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [T2: A] :
% 0.34/0.49 ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ X @ Z3 )
% 0.34/0.49 => ( X != T2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % minf(4)
% 0.34/0.49 thf(fact_242_minf_I5_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [T2: A] :
% 0.34/0.49 ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ X @ Z3 )
% 0.34/0.49 => ( ord_less @ A @ X @ T2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % minf(5)
% 0.34/0.49 thf(fact_243_minf_I7_J,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( linorder @ A )
% 0.34/0.49 => ! [T2: A] :
% 0.34/0.49 ? [Z3: A] :
% 0.34/0.49 ! [X: A] :
% 0.34/0.49 ( ( ord_less @ A @ X @ Z3 )
% 0.34/0.49 => ~ ( ord_less @ A @ T2 @ X ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % minf(7)
% 0.34/0.49 thf(fact_244_minf_I11_J,axiom,
% 0.34/0.49 ! [C4: $tType,D: $tType] :
% 0.34/0.49 ( ( ord @ C4 )
% 0.34/0.49 => ! [F5: D] :
% 0.34/0.49 ? [Z3: C4] :
% 0.34/0.49 ! [X: C4] :
% 0.34/0.49 ( ( ord_less @ C4 @ X @ Z3 )
% 0.34/0.49 => ( F5 = F5 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % minf(11)
% 0.34/0.49 thf(fact_245_order_Onot__eq__order__implies__strict,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [A3: A,B2: A] :
% 0.34/0.49 ( ( A3 != B2 )
% 0.34/0.49 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.34/0.49 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order.not_eq_order_implies_strict
% 0.34/0.49 thf(fact_246_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [B2: A,A3: A] :
% 0.34/0.49 ( ( ord_less @ A @ B2 @ A3 )
% 0.34/0.49 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dual_order.strict_implies_order
% 0.34/0.49 thf(fact_247_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ( ( ord_less @ A )
% 0.34/0.49 = ( ^ [B6: A,A8: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ B6 @ A8 )
% 0.34/0.49 & ( A8 != B6 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dual_order.strict_iff_order
% 0.34/0.49 thf(fact_248_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ( ( ord_less_eq @ A )
% 0.34/0.49 = ( ^ [B6: A,A8: A] :
% 0.34/0.49 ( ( ord_less @ A @ B6 @ A8 )
% 0.34/0.49 | ( A8 = B6 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dual_order.order_iff_strict
% 0.34/0.49 thf(fact_249_order_Ostrict__implies__order,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [A3: A,B2: A] :
% 0.34/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.34/0.49 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order.strict_implies_order
% 0.34/0.49 thf(fact_250_dense__le__bounded,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( dense_linorder @ A )
% 0.34/0.49 => ! [X4: A,Y3: A,Z2: A] :
% 0.34/0.49 ( ( ord_less @ A @ X4 @ Y3 )
% 0.34/0.49 => ( ! [W: A] :
% 0.34/0.49 ( ( ord_less @ A @ X4 @ W )
% 0.34/0.49 => ( ( ord_less @ A @ W @ Y3 )
% 0.34/0.49 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.34/0.49 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dense_le_bounded
% 0.34/0.49 thf(fact_251_dense__ge__bounded,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( dense_linorder @ A )
% 0.34/0.49 => ! [Z2: A,X4: A,Y3: A] :
% 0.34/0.49 ( ( ord_less @ A @ Z2 @ X4 )
% 0.34/0.49 => ( ! [W: A] :
% 0.34/0.49 ( ( ord_less @ A @ Z2 @ W )
% 0.34/0.49 => ( ( ord_less @ A @ W @ X4 )
% 0.34/0.49 => ( ord_less_eq @ A @ Y3 @ W ) ) )
% 0.34/0.49 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dense_ge_bounded
% 0.34/0.49 thf(fact_252_dual__order_Ostrict__trans2,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [B2: A,A3: A,C: A] :
% 0.34/0.49 ( ( ord_less @ A @ B2 @ A3 )
% 0.34/0.49 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.34/0.49 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dual_order.strict_trans2
% 0.34/0.49 thf(fact_253_dual__order_Ostrict__trans1,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ! [B2: A,A3: A,C: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.34/0.49 => ( ( ord_less @ A @ C @ B2 )
% 0.34/0.49 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % dual_order.strict_trans1
% 0.34/0.49 thf(fact_254_order_Ostrict__iff__order,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ( ( ord_less @ A )
% 0.34/0.49 = ( ^ [A8: A,B6: A] :
% 0.34/0.49 ( ( ord_less_eq @ A @ A8 @ B6 )
% 0.34/0.49 & ( A8 != B6 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order.strict_iff_order
% 0.34/0.49 thf(fact_255_order_Oorder__iff__strict,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( order @ A )
% 0.34/0.49 => ( ( ord_less_eq @ A )
% 0.34/0.49 = ( ^ [A8: A,B6: A] :
% 0.34/0.49 ( ( ord_less @ A @ A8 @ B6 )
% 0.34/0.49 | ( A8 = B6 ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 % order.order_iff_strict
% 0.34/0.49
% 0.34/0.49 % Subclasses (1)
% 0.34/0.49 thf(subcl_Groups_Ocomm__monoid__add___HOL_Otype,axiom,
% 0.34/0.49 ! [A: $tType] :
% 0.34/0.49 ( ( comm_monoid_add @ A )
% 0.34/0.49 => ( type @ A ) ) ).
% 0.34/0.49
% 0.34/0.49 % Type constructors (29)
% 0.34/0.49 thf(tcon_Nat_Onat___Rings_Odioid,axiom,
% 0.34/0.49 dioid @ nat ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_fun___Rings_Odioid_1,axiom,
% 0.34/0.49 ! [A9: $tType,A10: $tType] :
% 0.34/0.49 ( ( dioid @ A10 )
% 0.34/0.49 => ( dioid @ ( A9 > A10 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_fun___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.34/0.49 ! [A9: $tType,A10: $tType] :
% 0.34/0.49 ( ( dioid @ A10 )
% 0.34/0.49 => ( canoni770627133id_add @ ( A9 > A10 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_fun___Groups_Ocomm__monoid__add,axiom,
% 0.34/0.49 ! [A9: $tType,A10: $tType] :
% 0.34/0.49 ( ( comm_monoid_add @ A10 )
% 0.34/0.49 => ( comm_monoid_add @ ( A9 > A10 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.34/0.49 ! [A9: $tType,A10: $tType] :
% 0.34/0.49 ( ( preorder @ A10 )
% 0.34/0.49 => ( preorder @ ( A9 > A10 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.34/0.49 ! [A9: $tType,A10: $tType] :
% 0.34/0.49 ( ( order @ A10 )
% 0.34/0.49 => ( order @ ( A9 > A10 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_fun___Orderings_Oord,axiom,
% 0.34/0.49 ! [A9: $tType,A10: $tType] :
% 0.34/0.49 ( ( ord @ A10 )
% 0.34/0.49 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add_2,axiom,
% 0.34/0.49 canoni770627133id_add @ nat ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_3,axiom,
% 0.34/0.49 comm_monoid_add @ nat ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.34/0.49 wellorder @ nat ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
% 0.34/0.49 preorder @ nat ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.34/0.49 linorder @ nat ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.34/0.49 no_top @ nat ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Nat_Onat___Orderings_Oorder_5,axiom,
% 0.34/0.49 order @ nat ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Nat_Onat___Orderings_Oord_6,axiom,
% 0.34/0.49 ord @ nat ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 0.34/0.49 size @ nat ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 0.34/0.49 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
% 0.34/0.49 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
% 0.34/0.49 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 0.34/0.49 preorder @ $o ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_HOL_Obool___Orderings_Olinorder_11,axiom,
% 0.34/0.49 linorder @ $o ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
% 0.34/0.49 order @ $o ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
% 0.34/0.49 ord @ $o ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_List_Olist___Nat_Osize_14,axiom,
% 0.34/0.49 ! [A9: $tType] : ( size @ ( list @ A9 ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_15,axiom,
% 0.34/0.49 ! [A9: $tType] : ( comm_monoid_add @ ( multiset @ A9 ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Multiset_Omultiset___Orderings_Opreorder_16,axiom,
% 0.34/0.49 ! [A9: $tType] :
% 0.34/0.49 ( ( preorder @ A9 )
% 0.34/0.49 => ( preorder @ ( multiset @ A9 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Multiset_Omultiset___Orderings_Oorder_17,axiom,
% 0.34/0.49 ! [A9: $tType] :
% 0.34/0.49 ( ( preorder @ A9 )
% 0.34/0.49 => ( order @ ( multiset @ A9 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Multiset_Omultiset___Orderings_Oord_18,axiom,
% 0.34/0.49 ! [A9: $tType] :
% 0.34/0.49 ( ( preorder @ A9 )
% 0.34/0.49 => ( ord @ ( multiset @ A9 ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(tcon_Multiset_Omultiset___Nat_Osize_19,axiom,
% 0.34/0.49 ! [A9: $tType] : ( size @ ( multiset @ A9 ) ) ).
% 0.34/0.49
% 0.34/0.49 % Free types (1)
% 0.34/0.49 thf(tfree_0,hypothesis,
% 0.34/0.49 comm_monoid_add @ a ).
% 0.34/0.49
% 0.34/0.49 % Conjectures (1)
% 0.34/0.49 thf(conj_0,conjecture,
% 0.34/0.49 ( a3
% 0.34/0.49 = ( groups501227526m_list @ a @ qs2 ) ) ).
% 0.34/0.49
% 0.34/0.49 %------------------------------------------------------------------------------
% 0.34/0.49 ------------------------
% 3.17/1.71 ---- Embedded in THF ---
% 3.17/1.71 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.17/1.71 %%% Generated on Tue Feb 24 08:25:26 EST 2026
% 3.17/1.71 %%% using '$$dhol' embedding, version 1.3.0.
% 3.17/1.71 %%% Logic specification used:
% 3.17/1.71 %%% thf(spec, logic, $$dhol).
% 3.17/1.71
% 3.17/1.71 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.DwfRHP4Hjf/DTF2DTF28982.p
% 3.17/1.71 thf(ty_t_Multiset_Omultiset, type, multiset: ($tType > $tType)).
% 3.17/1.71 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 3.17/1.71 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.17/1.71 thf(ty_t_Nat_Onat, type, nat: $tType).
% 3.17/1.71 thf(ty_tf_a, type, a: $tType).
% 3.17/1.71 thf(sy_cl_Rings_Odioid, type, dioid: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_cl_Nat_Osize, type, size: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 3.17/1.71 thf(sy_c_Fun_Obij__betw, type, bij_betw: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > ((set @ B) > $o)))))).
% 3.17/1.71 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 3.17/1.71 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list, type, groups501227526m_list: (!> [A:$tType]: (((list @ A) > A)))).
% 3.17/1.71 thf(sy_c_Hilbert__Choice_Oinv__into, type, hilbert_inv_into: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (B > A)))))).
% 3.17/1.71 thf(sy_c_Hilbert__Function__Mirabelle__phhewjaonq_Odirect__decomp, type, hilber580762119decomp: (!> [A:$tType]: (((set @ A) > ((list @ (set @ A)) > $o))))).
% 3.17/1.71 thf(sy_c_List_Olist__ex, type, list_ex: (!> [A:$tType]: (((A > $o) > ((list @ A) > $o))))).
% 3.17/1.71 thf(sy_c_List_Olist__update, type, list_update: (!> [A:$tType]: (((list @ A) > (nat > (A > (list @ A))))))).
% 3.17/1.71 thf(sy_c_List_Olistset, type, listset: (!> [A:$tType]: (((list @ (set @ A)) > (set @ (list @ A)))))).
% 3.17/1.71 thf(sy_c_List_Onth, type, nth: (!> [A:$tType]: (((list @ A) > (nat > A))))).
% 3.17/1.71 thf(sy_c_List_Orev, type, rev: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 3.17/1.71 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset, type, comm_m1079429914m_mset: (!> [A:$tType]: (((multiset @ A) > A)))).
% 3.17/1.71 thf(sy_c_Multiset_Omset, type, mset: (!> [A:$tType]: (((list @ A) > (multiset @ A))))).
% 3.17/1.71 thf(sy_c_Multiset_Oset__mset, type, set_mset: (!> [A:$tType]: (((multiset @ A) > (set @ A))))).
% 3.17/1.71 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 3.17/1.71 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 3.17/1.71 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.17/1.71 thf(sy_c_Permutation_Operm, type, perm: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))).
% 3.17/1.71 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.17/1.71 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 3.17/1.71 thf(sy_c_Set__Interval_Oord__class_OlessThan, type, set_ord_lessThan: (!> [A:$tType]: ((A > (set @ A))))).
% 3.17/1.71 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.17/1.71 thf(sy_v_A, type, a2: (set @ a)).
% 3.17/1.71 thf(sy_v_a____, type, a3: a).
% 3.17/1.71 thf(sy_v_f____, type, f: (nat > nat)).
% 3.17/1.71 thf(sy_v_g____, type, g: (nat > nat)).
% 3.17/1.71 thf(sy_v_qs2____, type, qs2: (list @ a)).
% 3.17/1.71 thf(sy_v_qs____, type, qs: (list @ a)).
% 3.17/1.71 thf(sy_v_ss1, type, ss1: (list @ (set @ a))).
% 3.17/1.71 thf(sy_v_ss2, type, ss2: (list @ (set @ a))).
% 3.17/1.71 thf(fact_0__092_060open_062mset_Aqs2_A_061_Amset_Aqs_092_060close_062, axiom, (((mset @ a) @ qs2) = ((mset @ a) @ qs))).
% 3.17/1.71 thf(fact_1__092_060open_062a_A_092_060in_062_AA_092_060close_062, axiom, (((member @ a) @ a3) @ a2)).
% 3.17/1.71 thf(fact_2_a, axiom, (a3 = ((groups501227526m_list @ a) @ qs))).
% 3.17/1.71 thf(fact_3_len__qs2, axiom, (((size_size @ (list @ a)) @ qs2) = ((size_size @ (list @ a)) @ qs))).
% 3.17/1.71 thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs_O_A_092_060lbrakk_062a_A_061_Asum__list_Aqs_059_Aqs_A_092_060in_062_Alistset_Ass1_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [Qs:(list @ a)]: (((a3 = ((groups501227526m_list @ a) @ Qs)) => (~ ((((member @ (list @ a)) @ Qs) @ ((listset @ a) @ ss1)))))))))).
% 3.17/1.71 thf(fact_5_qs2__in, axiom, (((member @ (list @ a)) @ qs2) @ ((listset @ a) @ ss2))).
% 3.17/1.71 thf(fact_6_direct__decompD_I1_J, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:(set @ A),Ss:(list @ (set @ A)),Qs2:(list @ A)]: (((((hilber580762119decomp @ A) @ A2) @ Ss) => ((((member @ (list @ A)) @ Qs2) @ ((listset @ A) @ Ss)) => (((member @ A) @ ((groups501227526m_list @ A) @ Qs2)) @ A2))))))))).
% 3.17/1.71 thf(fact_7_direct__decompE, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:(set @ A),Ss:(list @ (set @ A)),A3:A]: (((((hilber580762119decomp @ A) @ A2) @ Ss) => ((((member @ A) @ A3) @ A2) => (~ ((! [Qs:(list @ A)]: (((((member @ (list @ A)) @ Qs) @ ((listset @ A) @ Ss)) => (A3 != ((groups501227526m_list @ A) @ Qs))))))))))))))).
% 3.17/1.71 thf(fact_8_direct__decompI__alt, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [Ss:(list @ (set @ A)),A2:(set @ A)]: (((! [Qs:(list @ A)]: (((((member @ (list @ A)) @ Qs) @ ((listset @ A) @ Ss)) => (((member @ A) @ ((groups501227526m_list @ A) @ Qs)) @ A2)))) => ((! [A4:A]: (((((member @ A) @ A4) @ A2) => (? [X:(list @ A)]: (((((member @ (list @ A)) @ X) @ ((listset @ A) @ Ss)) & ((A4 = ((groups501227526m_list @ A) @ X)) & (! [Y:(list @ A)]: ((((((member @ (list @ A)) @ Y) @ ((listset @ A) @ Ss)) & (A4 = ((groups501227526m_list @ A) @ Y))) => (Y = X))))))))))) => (((hilber580762119decomp @ A) @ A2) @ Ss))))))))).
% 3.17/1.71 thf(fact_9_direct__decomp__unique, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:(set @ A),Ss:(list @ (set @ A)),Qs2:(list @ A),Qs3:(list @ A)]: (((((hilber580762119decomp @ A) @ A2) @ Ss) => ((((member @ (list @ A)) @ Qs2) @ ((listset @ A) @ Ss)) => ((((member @ (list @ A)) @ Qs3) @ ((listset @ A) @ Ss)) => ((((groups501227526m_list @ A) @ Qs2) = ((groups501227526m_list @ A) @ Qs3)) => (Qs2 = Qs3))))))))))).
% 3.17/1.71 thf(fact_10_sum__list_Orev, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [Xs:(list @ A)]: ((((groups501227526m_list @ A) @ ((rev @ A) @ Xs)) = ((groups501227526m_list @ A) @ Xs)))))))).
% 3.17/1.71 thf(fact_11__C1_C, axiom, (! [I:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ a)) @ qs)) => ((((nth @ a) @ qs2) @ I) = (((nth @ a) @ qs) @ (g @ I))))))).
% 3.17/1.71 thf(fact_12_g__bij2, axiom, (((((bij_betw @ nat) @ nat) @ g) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ a)) @ qs2))) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ a)) @ qs)))).
% 3.17/1.71 thf(fact_13_sum__mset__sum__list, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [Xs:(list @ A)]: ((((comm_m1079429914m_mset @ A) @ ((mset @ A) @ Xs)) = ((groups501227526m_list @ A) @ Xs)))))))).
% 3.17/1.71 thf(fact_14_direct__decompD_I3_J, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:(set @ A),Ss:(list @ (set @ A))]: (((((hilber580762119decomp @ A) @ A2) @ Ss) => (((((image @ (list @ A)) @ A) @ (groups501227526m_list @ A)) @ ((listset @ A) @ Ss)) = A2)))))))).
% 3.17/1.71 thf(fact_15_assms_I1_J, axiom, (((hilber580762119decomp @ a) @ a2) @ ss1)).
% 3.17/1.71 thf(fact_16_qs__in, axiom, (((member @ (list @ a)) @ qs) @ ((listset @ a) @ ss1))).
% 3.17/1.71 thf(fact_17_mset__rev, axiom, (! [A:$tType,Xs:(list @ A)]: ((((mset @ A) @ ((rev @ A) @ Xs)) = ((mset @ A) @ Xs))))).
% 3.17/1.71 thf(fact_18__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs2_O_A_092_060lbrakk_062qs2_A_092_060in_062_Alistset_Ass2_059_Alength_Aqs2_A_061_Alength_Aqs_059_A_092_060And_062i_O_Ai_A_060_Alength_Aqs_A_092_060Longrightarrow_062_Aqs2_A_B_Ai_A_061_Aqs_A_B_Ag_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [Qs22:(list @ a)]: (((((member @ (list @ a)) @ Qs22) @ ((listset @ a) @ ss2)) => ((((size_size @ (list @ a)) @ Qs22) = ((size_size @ (list @ a)) @ qs)) => (~ ((! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ a)) @ qs)) => ((((nth @ a) @ Qs22) @ I2) = (((nth @ a) @ qs) @ (g @ I2))))))))))))))).
% 3.17/1.71 thf(fact_19_assms_I2_J, axiom, (((perm @ (set @ a)) @ ss1) @ ss2)).
% 3.17/1.71 thf(fact_20_ex__mset, axiom, (! [A:$tType,X2:(multiset @ A)]: ((? [Xs2:(list @ A)]: ((((mset @ A) @ Xs2) = X2)))))).
% 3.17/1.71 thf(fact_21_mset__eq__length, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((mset @ A) @ Xs) = ((mset @ A) @ Ys)) => (((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)))))).
% 3.17/1.71 thf(fact_22_mset__bij, axiom, (! [A:$tType,F:(nat > nat),Xs:(list @ A),Ys:(list @ A)]: (((((((bij_betw @ nat) @ nat) @ F) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ A)) @ Xs))) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ A)) @ Ys))) => ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I3) = (((nth @ A) @ Ys) @ (F @ I3)))))) => (((mset @ A) @ Xs) = ((mset @ A) @ Ys))))))).
% 3.17/1.71 thf(fact_23_listset__permE, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ (set @ A)),F:(nat > nat),Xs3:(list @ (set @ A))]: (((((member @ (list @ A)) @ Ys) @ ((listset @ A) @ Xs)) => ((((((bij_betw @ nat) @ nat) @ F) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ (set @ A))) @ Xs))) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ (set @ A))) @ Xs3))) => ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ (set @ A))) @ Xs)) => ((((nth @ (set @ A)) @ Xs3) @ I3) = (((nth @ (set @ A)) @ Xs) @ (F @ I3)))))) => (~ ((! [Ys2:(list @ A)]: (((((member @ (list @ A)) @ Ys2) @ ((listset @ A) @ Xs3)) => ((((size_size @ (list @ A)) @ Ys2) = ((size_size @ (list @ A)) @ Ys)) => (~ ((! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ A)) @ Ys)) => ((((nth @ A) @ Ys2) @ I2) = (((nth @ A) @ Ys) @ (F @ I2)))))))))))))))))))).
% 3.17/1.71 thf(fact_24_g__bij, axiom, (((((bij_betw @ nat) @ nat) @ g) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ (set @ a))) @ ss1))) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ (set @ a))) @ ss2)))).
% 3.17/1.71 thf(fact_25_length__rev, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ ((rev @ A) @ Xs)) = ((size_size @ (list @ A)) @ Xs))))).
% 3.17/1.71 thf(fact_26_lessThan__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_lessThan @ A) @ K)) = (((ord_less @ A) @ I) @ K)))))))).
% 3.17/1.71 thf(fact_27_listsetI, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ (set @ A))]: (((((size_size @ (list @ A)) @ Ys) = ((size_size @ (list @ (set @ A))) @ Xs)) => ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ (set @ A))) @ Xs)) => (((member @ A) @ (((nth @ A) @ Ys) @ I3)) @ (((nth @ (set @ A)) @ Xs) @ I3))))) => (((member @ (list @ A)) @ Ys) @ ((listset @ A) @ Xs))))))).
% 3.17/1.71 thf(fact_28_len__qs, axiom, (((size_size @ (list @ a)) @ qs) = ((size_size @ (list @ (set @ a))) @ ss1))).
% 3.17/1.71 thf(fact_29_f__bij, axiom, (((((bij_betw @ nat) @ nat) @ f) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ (set @ a))) @ ss2))) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ (set @ a))) @ ss1)))).
% 3.17/1.71 thf(fact_30_nth__equalityI, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)) => ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I3) = (((nth @ A) @ Ys) @ I3))))) => (Xs = Ys)))))).
% 3.17/1.71 thf(fact_31_Skolem__list__nth, axiom, (! [A:$tType,K:nat,P:(nat > (A > $o))]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ K) => (? [X3:A]: (((P @ I4) @ X3)))))) = (? [Xs4:(list @ A)]: (((((size_size @ (list @ A)) @ Xs4) = K) & (! [I4:nat]: (((((ord_less @ nat) @ I4) @ K) => ((P @ I4) @ (((nth @ A) @ Xs4) @ I4)))))))))))).
% 3.17/1.71 thf(fact_32_list__eq__iff__nth__eq, axiom, (! [A:$tType]: (((^ [Y2:(list @ A),Z:(list @ A)]: ((Y2 = Z))) = (^ [Xs4:(list @ A),Ys3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs4) = ((size_size @ (list @ A)) @ Ys3)) & (! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs4)) => ((((nth @ A) @ Xs4) @ I4) = (((nth @ A) @ Ys3) @ I4)))))))))))).
% 3.17/1.71 thf(fact_33_lessThan__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((((set_ord_lessThan @ A) @ X4) = ((set_ord_lessThan @ A) @ Y3)) = (X4 = Y3)))))))).
% 3.17/1.71 thf(fact_34_rev__is__rev__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((rev @ A) @ Xs) = ((rev @ A) @ Ys)) = (Xs = Ys))))).
% 3.17/1.71 thf(fact_35_rev__rev__ident, axiom, (! [A:$tType,Xs:(list @ A)]: ((((rev @ A) @ ((rev @ A) @ Xs)) = Xs)))).
% 3.17/1.71 thf(fact_36_len__ss1, axiom, (((size_size @ (list @ (set @ a))) @ ss1) = ((size_size @ (list @ (set @ a))) @ ss2))).
% 3.17/1.71 thf(fact_37_g, axiom, (! [I:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ (set @ a))) @ ss1)) => ((((nth @ (set @ a)) @ ss2) @ I) = (((nth @ (set @ a)) @ ss1) @ (g @ I))))))).
% 3.17/1.71 thf(fact_38_f, axiom, (! [I:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ (set @ a))) @ ss2)) => ((((nth @ (set @ a)) @ ss1) @ I) = (((nth @ (set @ a)) @ ss2) @ (f @ I))))))).
% 3.17/1.71 thf(fact_39__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062f_O_A_092_060lbrakk_062bij__betw_Af_A_123_O_O_060length_Ass2_125_A_123_O_O_060length_Ass1_125_059_A_092_060And_062i_O_Ai_A_060_Alength_Ass2_A_092_060Longrightarrow_062_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [F2:(nat > nat)]: (((((((bij_betw @ nat) @ nat) @ F2) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ (set @ a))) @ ss2))) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ (set @ a))) @ ss1))) => (~ ((! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ (set @ a))) @ ss2)) => ((((nth @ (set @ a)) @ ss1) @ I2) = (((nth @ (set @ a)) @ ss2) @ (F2 @ I2)))))))))))))).
% 3.17/1.71 thf(fact_40__092_060open_062_092_060exists_062f_O_Abij__betw_Af_A_123_O_O_060length_Ass1_125_A_123_O_O_060length_Ass2_125_A_092_060and_062_A_I_092_060forall_062i_060length_Ass1_O_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_J_092_060close_062, axiom, (? [F2:(nat > nat)]: (((((((bij_betw @ nat) @ nat) @ F2) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ (set @ a))) @ ss1))) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ (set @ a))) @ ss2))) & (! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ (set @ a))) @ ss1)) => ((((nth @ (set @ a)) @ ss1) @ I2) = (((nth @ (set @ a)) @ ss2) @ (F2 @ I2)))))))))).
% 3.17/1.71 thf(fact_41_f__g, axiom, (! [I:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ (set @ a))) @ ss1)) => ((f @ (g @ I)) = I))))).
% 3.17/1.71 thf(fact_42_g__f, axiom, (! [I:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ (set @ a))) @ ss2)) => ((g @ (f @ I)) = I))))).
% 3.17/1.71 thf(fact_43_listsetD_I2_J, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ (set @ A)),I:nat]: (((((member @ (list @ A)) @ Ys) @ ((listset @ A) @ Xs)) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ (set @ A))) @ Xs)) => (((member @ A) @ (((nth @ A) @ Ys) @ I)) @ (((nth @ (set @ A)) @ Xs) @ I))))))).
% 3.17/1.71 thf(fact_44_neq__if__length__neq, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) != ((size_size @ (list @ A)) @ Ys)) => (Xs != Ys))))).
% 3.17/1.71 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 3.17/1.71 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X5:A]: ((((member @ A) @ X5) @ A2)))) = A2)))).
% 3.17/1.71 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X6:A]: (((P @ X6) = (Q @ X6)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.17/1.71 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X6:A]: (((F @ X6) = (G @ X6)))) => (F = G))))).
% 3.17/1.71 thf(fact_49_Ex__list__of__length, axiom, (! [A:$tType,N:nat]: ((? [Xs2:(list @ A)]: ((((size_size @ (list @ A)) @ Xs2) = N)))))).
% 3.17/1.71 thf(fact_50_rev__swap, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((rev @ A) @ Xs) = Ys) = (Xs = ((rev @ A) @ Ys)))))).
% 3.17/1.71 thf(fact_51_lessThan__strict__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [M:A,N:A]: (((((ord_less @ (set @ A)) @ ((set_ord_lessThan @ A) @ M)) @ ((set_ord_lessThan @ A) @ N)) = (((ord_less @ A) @ M) @ N)))))))).
% 3.17/1.71 thf(fact_52_length__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((! [Xs2:(list @ A)]: (((! [Ys4:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys4)) @ ((size_size @ (list @ A)) @ Xs2)) => (P @ Ys4)))) => (P @ Xs2)))) => (P @ Xs))))).
% 3.17/1.71 thf(fact_53_listsetD_I1_J, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ (set @ A))]: (((((member @ (list @ A)) @ Ys) @ ((listset @ A) @ Xs)) => (((size_size @ (list @ A)) @ Ys) = ((size_size @ (list @ (set @ A))) @ Xs)))))).
% 3.17/1.71 thf(fact_54_permutation__Ex__bij, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((perm @ A) @ Xs) @ Ys) => (? [F2:(nat > nat)]: (((((((bij_betw @ nat) @ nat) @ F2) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ A)) @ Xs))) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ A)) @ Ys))) & (! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I2) = (((nth @ A) @ Ys) @ (F2 @ I2))))))))))))).
% 3.17/1.71 thf(fact_55_g__def, axiom, (g = ((((hilbert_inv_into @ nat) @ nat) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ (set @ a))) @ ss2))) @ f))).
% 3.17/1.71 thf(fact_56_perm__refl, axiom, (! [A:$tType,L:(list @ A)]: ((((perm @ A) @ L) @ L)))).
% 3.17/1.71 thf(fact_57_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),X4:B,A2:(set @ B)]: (((B2 = (F @ X4)) => ((((member @ B) @ X4) @ A2) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ A2))))))).
% 3.17/1.71 thf(fact_58_mset__eq__perm, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((mset @ A) @ Xs) = ((mset @ A) @ Ys)) = (((perm @ A) @ Xs) @ Ys))))).
% 3.17/1.71 thf(fact_59_perm__rev, axiom, (! [A:$tType,Xs:(list @ A)]: ((((perm @ A) @ ((rev @ A) @ Xs)) @ Xs)))).
% 3.17/1.71 thf(fact_60_perm__length, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((perm @ A) @ Xs) @ Ys) => (((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)))))).
% 3.17/1.71 thf(fact_61_bij__betw__imp__surj__on, axiom, (! [A:$tType,B:$tType,F:(A > B),A2:(set @ A),B3:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A2) @ B3) => (((((image @ A) @ B) @ F) @ A2) = B3))))).
% 3.17/1.71 thf(fact_62_mbs_Oless__not__eq, axiom, (! [A:$tType]: (((size @ A) => (! [X4:A,A2:(set @ A),Y3:A]: (((((member @ A) @ X4) @ A2) => ((((ord_less @ nat) @ ((size_size @ A) @ X4)) @ ((size_size @ A) @ Y3)) => (X4 != Y3))))))))).
% 3.17/1.71 thf(fact_63_size__mset, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (multiset @ A)) @ ((mset @ A) @ Xs)) = ((size_size @ (list @ A)) @ Xs))))).
% 3.17/1.71 thf(fact_64_psubsetD, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),C:A]: (((((ord_less @ (set @ A)) @ A2) @ B3) => ((((member @ A) @ C) @ A2) => (((member @ A) @ C) @ B3)))))).
% 3.17/1.71 thf(fact_65_psubset__trans, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),C2:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B3) => ((((ord_less @ (set @ A)) @ B3) @ C2) => (((ord_less @ (set @ A)) @ A2) @ C2)))))).
% 3.17/1.71 thf(fact_66_imageI, axiom, (! [B:$tType,A:$tType,X4:A,A2:(set @ A),F:(A > B)]: (((((member @ A) @ X4) @ A2) => (((member @ B) @ (F @ X4)) @ ((((image @ A) @ B) @ F) @ A2)))))).
% 3.17/1.71 thf(fact_67_image__iff, axiom, (! [A:$tType,B:$tType,Z2:A,F:(B > A),A2:(set @ B)]: (((((member @ A) @ Z2) @ ((((image @ B) @ A) @ F) @ A2)) = (? [X5:B]: (((((member @ B) @ X5) @ A2) & (Z2 = (F @ X5))))))))).
% 3.17/1.71 thf(fact_68_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P:(A > $o)]: (((? [X:A]: (((((member @ A) @ X) @ ((((image @ B) @ A) @ F) @ A2)) & (P @ X)))) => (? [X6:B]: (((((member @ B) @ X6) @ A2) & (P @ (F @ X6))))))))).
% 3.17/1.71 thf(fact_69_image__cong, axiom, (! [B:$tType,A:$tType,M2:(set @ A),N2:(set @ A),F:(A > B),G:(A > B)]: (((M2 = N2) => ((! [X6:A]: (((((member @ A) @ X6) @ N2) => ((F @ X6) = (G @ X6))))) => (((((image @ A) @ B) @ F) @ M2) = ((((image @ A) @ B) @ G) @ N2))))))).
% 3.17/1.71 thf(fact_70_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P:(A > $o)]: (((! [X6:A]: (((((member @ A) @ X6) @ ((((image @ B) @ A) @ F) @ A2)) => (P @ X6)))) => (! [X:B]: (((((member @ B) @ X) @ A2) => (P @ (F @ X))))))))).
% 3.17/1.71 thf(fact_71_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X4:A,A2:(set @ A),B2:B,F:(A > B)]: (((((member @ A) @ X4) @ A2) => ((B2 = (F @ X4)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F) @ A2))))))).
% 3.17/1.71 thf(fact_72_direct__decomp__def, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => ((hilber580762119decomp @ A) = (^ [A5:(set @ A),Ss2:(list @ (set @ A))]: ((((((bij_betw @ (list @ A)) @ A) @ (groups501227526m_list @ A)) @ ((listset @ A) @ Ss2)) @ A5)))))))).
% 3.17/1.71 thf(fact_73_bij__betwE, axiom, (! [A:$tType,B:$tType,F:(A > B),A2:(set @ A),B3:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A2) @ B3) => (! [X:A]: (((((member @ A) @ X) @ A2) => (((member @ B) @ (F @ X)) @ B3)))))))).
% 3.17/1.71 thf(fact_74_bij__betw__inv, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B3:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A2) @ B3) => (? [G2:(B > A)]: ((((((bij_betw @ B) @ A) @ G2) @ B3) @ A2))))))).
% 3.17/1.71 thf(fact_75_bij__betw__cong, axiom, (! [A:$tType,B:$tType,A2:(set @ A),F:(A > B),G:(A > B),A6:(set @ B)]: (((! [A4:A]: (((((member @ A) @ A4) @ A2) => ((F @ A4) = (G @ A4))))) => ((((((bij_betw @ A) @ B) @ F) @ A2) @ A6) = (((((bij_betw @ A) @ B) @ G) @ A2) @ A6)))))).
% 3.17/1.71 thf(fact_76_bij__betw__apply, axiom, (! [A:$tType,B:$tType,F:(A > B),A2:(set @ A),B3:(set @ B),A3:A]: (((((((bij_betw @ A) @ B) @ F) @ A2) @ B3) => ((((member @ A) @ A3) @ A2) => (((member @ B) @ (F @ A3)) @ B3)))))).
% 3.17/1.71 thf(fact_77_bij__betw__iff__bijections, axiom, (! [B:$tType,A:$tType]: ((((bij_betw @ A) @ B) = (^ [F3:(A > B),A5:(set @ A),B4:(set @ B)]: ((? [G3:(B > A)]: (((! [X5:A]: (((((member @ A) @ X5) @ A5) => ((((member @ B) @ (F3 @ X5)) @ B4) & ((G3 @ (F3 @ X5)) = X5))))) & (! [X5:B]: (((((member @ B) @ X5) @ B4) => ((((member @ A) @ (G3 @ X5)) @ A5) & ((F3 @ (G3 @ X5)) = X5)))))))))))))).
% 3.17/1.71 thf(fact_78_perm__sym, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((perm @ A) @ Xs) @ Ys) => (((perm @ A) @ Ys) @ Xs))))).
% 3.17/1.71 thf(fact_79_perm_Otrans, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs:(list @ A)]: (((((perm @ A) @ Xs) @ Ys) => ((((perm @ A) @ Ys) @ Zs) => (((perm @ A) @ Xs) @ Zs)))))).
% 3.17/1.71 thf(fact_80_bij__betw__inv__into__right, axiom, (! [A:$tType,B:$tType,F:(A > B),A2:(set @ A),A6:(set @ B),A7:B]: (((((((bij_betw @ A) @ B) @ F) @ A2) @ A6) => ((((member @ B) @ A7) @ A6) => ((F @ (((((hilbert_inv_into @ A) @ B) @ A2) @ F) @ A7)) = A7)))))).
% 3.17/1.71 thf(fact_81_bij__betw__inv__into__left, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),A6:(set @ B),A3:A]: (((((((bij_betw @ A) @ B) @ F) @ A2) @ A6) => ((((member @ A) @ A3) @ A2) => ((((((hilbert_inv_into @ A) @ B) @ A2) @ F) @ (F @ A3)) = A3)))))).
% 3.17/1.71 thf(fact_82_inv__into__inv__into__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),A6:(set @ B),A3:A]: (((((((bij_betw @ A) @ B) @ F) @ A2) @ A6) => ((((member @ A) @ A3) @ A2) => ((((((hilbert_inv_into @ B) @ A) @ A6) @ ((((hilbert_inv_into @ A) @ B) @ A2) @ F)) @ A3) = (F @ A3))))))).
% 3.17/1.71 thf(fact_83_bij__betw__inv__into, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B3:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A2) @ B3) => (((((bij_betw @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ A2) @ F)) @ B3) @ A2))))).
% 3.17/1.71 thf(fact_84_inv__into__injective, axiom, (! [A:$tType,B:$tType,A2:(set @ A),F:(A > B),X4:B,Y3:B]: ((((((((hilbert_inv_into @ A) @ B) @ A2) @ F) @ X4) = (((((hilbert_inv_into @ A) @ B) @ A2) @ F) @ Y3)) => ((((member @ B) @ X4) @ ((((image @ A) @ B) @ F) @ A2)) => ((((member @ B) @ Y3) @ ((((image @ A) @ B) @ F) @ A2)) => (X4 = Y3))))))).
% 3.17/1.71 thf(fact_85_inv__into__into, axiom, (! [A:$tType,B:$tType,X4:A,F:(B > A),A2:(set @ B)]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F) @ A2)) => (((member @ B) @ (((((hilbert_inv_into @ B) @ A) @ A2) @ F) @ X4)) @ A2))))).
% 3.17/1.71 thf(fact_86_f__inv__into__f, axiom, (! [B:$tType,A:$tType,Y3:A,F:(B > A),A2:(set @ B)]: (((((member @ A) @ Y3) @ ((((image @ B) @ A) @ F) @ A2)) => ((F @ (((((hilbert_inv_into @ B) @ A) @ A2) @ F) @ Y3)) = Y3))))).
% 3.17/1.71 thf(fact_87_list__ex__length, axiom, (! [A:$tType]: (((list_ex @ A) = (^ [P2:(A > $o),Xs4:(list @ A)]: ((? [N3:nat]: (((((ord_less @ nat) @ N3) @ ((size_size @ (list @ A)) @ Xs4)) & (P2 @ (((nth @ A) @ Xs4) @ N3))))))))))).
% 3.17/1.71 thf(fact_88_direct__decompI, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [Ss:(list @ (set @ A)),A2:(set @ A)]: ((((((inj_on @ (list @ A)) @ A) @ (groups501227526m_list @ A)) @ ((listset @ A) @ Ss)) => ((((((image @ (list @ A)) @ A) @ (groups501227526m_list @ A)) @ ((listset @ A) @ Ss)) = A2) => (((hilber580762119decomp @ A) @ A2) @ Ss))))))))).
% 3.17/1.71 thf(fact_89_elem__le__sum__list, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [K:nat,Ns:(list @ A)]: (((((ord_less @ nat) @ K) @ ((size_size @ (list @ A)) @ Ns)) => (((ord_less_eq @ A) @ (((nth @ A) @ Ns) @ K)) @ ((groups501227526m_list @ A) @ Ns))))))))).
% 3.17/1.71 thf(fact_90_perm__swap, axiom, (! [A:$tType,I:nat,Xs:(list @ A),J:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((perm @ A) @ ((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I) @ (((nth @ A) @ Xs) @ J))) @ J) @ (((nth @ A) @ Xs) @ I))) @ Xs)))))).
% 3.17/1.71 thf(fact_91_mset__swap, axiom, (! [A:$tType,I:nat,Ls:(list @ A),J:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Ls)) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Ls)) => (((mset @ A) @ ((((list_update @ A) @ ((((list_update @ A) @ Ls) @ J) @ (((nth @ A) @ Ls) @ I))) @ I) @ (((nth @ A) @ Ls) @ J))) = ((mset @ A) @ Ls))))))).
% 3.17/1.71 thf(fact_92_nth__mem__mset, axiom, (! [A:$tType,I:nat,Ls:(list @ A)]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Ls)) => (((member @ A) @ (((nth @ A) @ Ls) @ I)) @ ((set_mset @ A) @ ((mset @ A) @ Ls))))))).
% 3.17/1.71 thf(fact_93_in__Union__mset__iff, axiom, (! [A:$tType,X4:A,MM:(multiset @ (multiset @ A))]: (((((member @ A) @ X4) @ ((set_mset @ A) @ ((comm_m1079429914m_mset @ (multiset @ A)) @ MM))) = (? [M3:(multiset @ A)]: (((((member @ (multiset @ A)) @ M3) @ ((set_mset @ (multiset @ A)) @ MM)) & (((member @ A) @ X4) @ ((set_mset @ A) @ M3))))))))).
% 3.17/1.71 thf(fact_94_psubsetI, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((A2 != B3) => (((ord_less @ (set @ A)) @ A2) @ B3)))))).
% 3.17/1.71 thf(fact_95_inj__on__rev, axiom, (! [A:$tType,A2:(set @ (list @ A))]: (((((inj_on @ (list @ A)) @ (list @ A)) @ (rev @ A)) @ A2)))).
% 3.17/1.71 thf(fact_96_list__update__overwrite, axiom, (! [A:$tType,Xs:(list @ A),I:nat,X4:A,Y3:A]: ((((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I) @ X4)) @ I) @ Y3) = ((((list_update @ A) @ Xs) @ I) @ Y3))))).
% 3.17/1.71 thf(fact_97_lessThan__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_lessThan @ A) @ X4)) @ ((set_ord_lessThan @ A) @ Y3)) = (((ord_less_eq @ A) @ X4) @ Y3)))))))).
% 3.17/1.71 thf(fact_98_list__update__beyond, axiom, (! [A:$tType,Xs:(list @ A),I:nat,X4:A]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ I) => (((((list_update @ A) @ Xs) @ I) @ X4) = Xs))))).
% 3.17/1.71 thf(fact_99_length__list__update, axiom, (! [A:$tType,Xs:(list @ A),I:nat,X4:A]: ((((size_size @ (list @ A)) @ ((((list_update @ A) @ Xs) @ I) @ X4)) = ((size_size @ (list @ A)) @ Xs))))).
% 3.17/1.71 thf(fact_100_inv__into__f__f, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X4:A]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((member @ A) @ X4) @ A2) => ((((((hilbert_inv_into @ A) @ B) @ A2) @ F) @ (F @ X4)) = X4)))))).
% 3.17/1.71 thf(fact_101_list__update__id, axiom, (! [A:$tType,Xs:(list @ A),I:nat]: ((((((list_update @ A) @ Xs) @ I) @ (((nth @ A) @ Xs) @ I)) = Xs)))).
% 3.17/1.71 thf(fact_102_nth__list__update__neq, axiom, (! [A:$tType,I:nat,J:nat,Xs:(list @ A),X4:A]: (((I != J) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I) @ X4)) @ J) = (((nth @ A) @ Xs) @ J)))))).
% 3.17/1.71 thf(fact_103_list__ex__rev, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: (((((list_ex @ A) @ P) @ ((rev @ A) @ Xs)) = (((list_ex @ A) @ P) @ Xs))))).
% 3.17/1.71 thf(fact_104_inv__into__image__cancel, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),S:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((ord_less_eq @ (set @ A)) @ S) @ A2) => (((((image @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ A2) @ F)) @ ((((image @ A) @ B) @ F) @ S)) = S)))))).
% 3.17/1.71 thf(fact_105_nth__list__update__eq, axiom, (! [A:$tType,I:nat,Xs:(list @ A),X4:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I) @ X4)) @ I) = X4))))).
% 3.17/1.71 thf(fact_106_subset__image__inj, axiom, (! [A:$tType,B:$tType,S:(set @ A),F:(B > A),T:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ S) @ ((((image @ B) @ A) @ F) @ T)) = (? [U:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ U) @ T) & (((((inj_on @ B) @ A) @ F) @ U) & (S = ((((image @ B) @ A) @ F) @ U)))))))))).
% 3.17/1.71 thf(fact_107_inj__on__inv__into, axiom, (! [B:$tType,A:$tType,B3:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B3) @ ((((image @ B) @ A) @ F) @ A2)) => ((((inj_on @ A) @ B) @ ((((hilbert_inv_into @ B) @ A) @ A2) @ F)) @ B3))))).
% 3.17/1.71 thf(fact_108_list__update__swap, axiom, (! [A:$tType,I:nat,I5:nat,Xs:(list @ A),X4:A,X7:A]: (((I != I5) => (((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I) @ X4)) @ I5) @ X7) = ((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I5) @ X7)) @ I) @ X4)))))).
% 3.17/1.71 thf(fact_109_inj__on__image__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),C2:(set @ A),A2:(set @ A),B3:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ C2) => ((((ord_less_eq @ (set @ A)) @ A2) @ C2) => ((((ord_less_eq @ (set @ A)) @ B3) @ C2) => ((((((image @ A) @ B) @ F) @ A2) = ((((image @ A) @ B) @ F) @ B3)) = (A2 = B3)))))))).
% 3.17/1.71 thf(fact_110_inj__on__image__mem__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),B3:(set @ A),A3:A,A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B3) => ((((member @ A) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((member @ B) @ (F @ A3)) @ ((((image @ A) @ B) @ F) @ A2)) = (((member @ A) @ A3) @ A2)))))))).
% 3.17/1.71 thf(fact_111_inj__on__inverseI, axiom, (! [B:$tType,A:$tType,A2:(set @ A),G:(B > A),F:(A > B)]: (((! [X6:A]: (((((member @ A) @ X6) @ A2) => ((G @ (F @ X6)) = X6)))) => ((((inj_on @ A) @ B) @ F) @ A2))))).
% 3.17/1.71 thf(fact_112_inj__on__contraD, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X4:A,Y3:A]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((X4 != Y3) => ((((member @ A) @ X4) @ A2) => ((((member @ A) @ Y3) @ A2) => ((F @ X4) != (F @ Y3))))))))).
% 3.17/1.71 thf(fact_113_subset__inj__on, axiom, (! [B:$tType,A:$tType,F:(A > B),B3:(set @ A),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B3) => ((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((inj_on @ A) @ B) @ F) @ A2)))))).
% 3.17/1.71 thf(fact_114_inj__on__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B3:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((ord_less_eq @ (set @ A)) @ B3) @ A2) => ((((inj_on @ A) @ B) @ F) @ B3)))))).
% 3.17/1.71 thf(fact_115_inj__on__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X4:A,Y3:A]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((member @ A) @ X4) @ A2) => ((((member @ A) @ Y3) @ A2) => (((F @ X4) = (F @ Y3)) = (X4 = Y3)))))))).
% 3.17/1.71 thf(fact_116_inj__on__cong, axiom, (! [B:$tType,A:$tType,A2:(set @ A),F:(A > B),G:(A > B)]: (((! [A4:A]: (((((member @ A) @ A4) @ A2) => ((F @ A4) = (G @ A4))))) => (((((inj_on @ A) @ B) @ F) @ A2) = ((((inj_on @ A) @ B) @ G) @ A2)))))).
% 3.17/1.71 thf(fact_117_inj__on__def, axiom, (! [B:$tType,A:$tType]: ((((inj_on @ A) @ B) = (^ [F3:(A > B),A5:(set @ A)]: ((! [X5:A]: (((((member @ A) @ X5) @ A5) => (! [Y4:A]: (((((member @ A) @ Y4) @ A5) => (((F3 @ X5) = (F3 @ Y4)) => (X5 = Y4)))))))))))))).
% 3.17/1.71 thf(fact_118_inj__onI, axiom, (! [B:$tType,A:$tType,A2:(set @ A),F:(A > B)]: (((! [X6:A,Y:A]: (((((member @ A) @ X6) @ A2) => ((((member @ A) @ Y) @ A2) => (((F @ X6) = (F @ Y)) => (X6 = Y)))))) => ((((inj_on @ A) @ B) @ F) @ A2))))).
% 3.17/1.71 thf(fact_119_inj__onD, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X4:A,Y3:A]: ((((((inj_on @ A) @ B) @ F) @ A2) => (((F @ X4) = (F @ Y3)) => ((((member @ A) @ X4) @ A2) => ((((member @ A) @ Y3) @ A2) => (X4 = Y3)))))))).
% 3.17/1.71 thf(fact_120_linorder__inj__onI, axiom, (! [B:$tType,A:$tType]: (((order @ A) => (! [A2:(set @ A),F:(A > B)]: (((! [X6:A,Y:A]: (((((ord_less @ A) @ X6) @ Y) => ((((member @ A) @ X6) @ A2) => ((((member @ A) @ Y) @ A2) => ((F @ X6) != (F @ Y))))))) => ((! [X6:A,Y:A]: (((((member @ A) @ X6) @ A2) => ((((member @ A) @ Y) @ A2) => ((((ord_less_eq @ A) @ X6) @ Y) | (((ord_less_eq @ A) @ Y) @ X6)))))) => ((((inj_on @ A) @ B) @ F) @ A2))))))))).
% 3.17/1.71 thf(fact_121_inj__on__image__iff, axiom, (! [B:$tType,A:$tType,A2:(set @ A),G:(A > B),F:(A > A)]: (((! [X6:A]: (((((member @ A) @ X6) @ A2) => (! [Xa:A]: (((((member @ A) @ Xa) @ A2) => (((G @ (F @ X6)) = (G @ (F @ Xa))) = ((G @ X6) = (G @ Xa))))))))) => (((((inj_on @ A) @ A) @ F) @ A2) => (((((inj_on @ A) @ B) @ G) @ ((((image @ A) @ A) @ F) @ A2)) = ((((inj_on @ A) @ B) @ G) @ A2))))))).
% 3.17/1.71 thf(fact_122_bij__betw__imp__inj__on, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B3:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A2) @ B3) => ((((inj_on @ A) @ B) @ F) @ A2))))).
% 3.17/1.71 thf(fact_123_inv__into__f__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X4:A,Y3:B]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((member @ A) @ X4) @ A2) => (((F @ X4) = Y3) => ((((((hilbert_inv_into @ A) @ B) @ A2) @ F) @ Y3) = X4))))))).
% 3.17/1.71 thf(fact_124_subset__image__iff, axiom, (! [A:$tType,B:$tType,B3:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B3) @ ((((image @ B) @ A) @ F) @ A2)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A2) & (B3 = ((((image @ B) @ A) @ F) @ AA))))))))).
% 3.17/1.71 thf(fact_125_image__subset__iff, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A2)) @ B3) = (! [X5:B]: (((((member @ B) @ X5) @ A2) => (((member @ A) @ (F @ X5)) @ B3)))))))).
% 3.17/1.71 thf(fact_126_subset__imageE, axiom, (! [A:$tType,B:$tType,B3:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B3) @ ((((image @ B) @ A) @ F) @ A2)) => (~ ((! [C3:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C3) @ A2) => (B3 != ((((image @ B) @ A) @ F) @ C3))))))))))).
% 3.17/1.71 thf(fact_127_image__subsetI, axiom, (! [A:$tType,B:$tType,A2:(set @ A),F:(A > B),B3:(set @ B)]: (((! [X6:A]: (((((member @ A) @ X6) @ A2) => (((member @ B) @ (F @ X6)) @ B3)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ B3))))).
% 3.17/1.71 thf(fact_128_image__mono, axiom, (! [B:$tType,A:$tType,A2:(set @ A),B3:(set @ A),F:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ ((((image @ A) @ B) @ F) @ B3)))))).
% 3.17/1.71 thf(fact_129_psubsetE, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B3) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => (((ord_less_eq @ (set @ A)) @ B3) @ A2)))))))).
% 3.17/1.71 thf(fact_130_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A5:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B4) & (A5 != B4)))))))).
% 3.17/1.71 thf(fact_131_psubset__imp__subset, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B3) => (((ord_less_eq @ (set @ A)) @ A2) @ B3))))).
% 3.17/1.71 thf(fact_132_psubset__subset__trans, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),C2:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C2) => (((ord_less @ (set @ A)) @ A2) @ C2)))))).
% 3.17/1.71 thf(fact_133_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A5:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B4) & (~ ((((ord_less_eq @ (set @ A)) @ B4) @ A5)))))))))).
% 3.17/1.71 thf(fact_134_subset__psubset__trans, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),C2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((ord_less @ (set @ A)) @ B3) @ C2) => (((ord_less @ (set @ A)) @ A2) @ C2)))))).
% 3.17/1.71 thf(fact_135_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B4:(set @ A)]: (((((ord_less @ (set @ A)) @ A5) @ B4) | (A5 = B4)))))))).
% 3.17/1.71 thf(fact_136_inj__on__strict__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),B3:(set @ A),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B3) => ((((ord_less @ (set @ A)) @ A2) @ B3) => (((ord_less @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ ((((image @ A) @ B) @ F) @ B3))))))).
% 3.17/1.71 thf(fact_137_bij__betw__def, axiom, (! [B:$tType,A:$tType]: ((((bij_betw @ A) @ B) = (^ [F3:(A > B),A5:(set @ A),B4:(set @ B)]: ((((((inj_on @ A) @ B) @ F3) @ A5) & (((((image @ A) @ B) @ F3) @ A5) = B4)))))))).
% 3.17/1.71 thf(fact_138_bij__betw__imageI, axiom, (! [A:$tType,B:$tType,F:(A > B),A2:(set @ A),B3:(set @ B)]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((((image @ A) @ B) @ F) @ A2) = B3) => (((((bij_betw @ A) @ B) @ F) @ A2) @ B3)))))).
% 3.17/1.71 thf(fact_139_inj__on__imp__bij__betw, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A2) => (((((bij_betw @ A) @ B) @ F) @ A2) @ ((((image @ A) @ B) @ F) @ A2)))))).
% 3.17/1.71 thf(fact_140_mbs_Ominimal, axiom, (! [A:$tType]: (((size @ A) => (! [X4:A,A2:(set @ A),P:(A > $o)]: (((((member @ A) @ X4) @ A2) => ((P @ X4) => (? [X6:A]: (((((member @ A) @ X6) @ A2) & ((((ord_less_eq @ nat) @ ((size_size @ A) @ X6)) @ ((size_size @ A) @ X4)) & ((P @ X6) & (! [Xa2:A]: (((((member @ A) @ Xa2) @ A2) => ((((ord_less @ nat) @ ((size_size @ A) @ Xa2)) @ ((size_size @ A) @ X6)) => (~ ((P @ Xa2)))))))))))))))))))).
% 3.17/1.71 thf(fact_141_bij__betw__subset, axiom, (! [A:$tType,B:$tType,F:(A > B),A2:(set @ A),A6:(set @ B),B3:(set @ A),B5:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A2) @ A6) => ((((ord_less_eq @ (set @ A)) @ B3) @ A2) => ((((((image @ A) @ B) @ F) @ B3) = B5) => (((((bij_betw @ A) @ B) @ F) @ B3) @ B5))))))).
% 3.17/1.71 thf(fact_142_bij__betw__byWitness, axiom, (! [A:$tType,B:$tType,A2:(set @ A),F4:(B > A),F:(A > B),A6:(set @ B)]: (((! [X6:A]: (((((member @ A) @ X6) @ A2) => ((F4 @ (F @ X6)) = X6)))) => ((! [X6:B]: (((((member @ B) @ X6) @ A6) => ((F @ (F4 @ X6)) = X6)))) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ A6) => ((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F4) @ A6)) @ A2) => (((((bij_betw @ A) @ B) @ F) @ A2) @ A6)))))))).
% 3.17/1.71 thf(fact_143_image__inv__into__cancel, axiom, (! [B:$tType,A:$tType,F:(B > A),A2:(set @ B),A6:(set @ A),B5:(set @ A)]: (((((((image @ B) @ A) @ F) @ A2) = A6) => ((((ord_less_eq @ (set @ A)) @ B5) @ A6) => (((((image @ B) @ A) @ F) @ ((((image @ A) @ B) @ ((((hilbert_inv_into @ B) @ A) @ A2) @ F)) @ B5)) = B5)))))).
% 3.17/1.71 thf(fact_144_listset__mono, axiom, (! [A:$tType,Xs:(list @ (set @ A)),Ys:(list @ (set @ A))]: (((((size_size @ (list @ (set @ A))) @ Xs) = ((size_size @ (list @ (set @ A))) @ Ys)) => ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ (set @ A))) @ Ys)) => (((ord_less_eq @ (set @ A)) @ (((nth @ (set @ A)) @ Xs) @ I3)) @ (((nth @ (set @ A)) @ Ys) @ I3))))) => (((ord_less_eq @ (set @ (list @ A))) @ ((listset @ A) @ Xs)) @ ((listset @ A) @ Ys))))))).
% 3.17/1.71 thf(fact_145_nth__list__update, axiom, (! [A:$tType,I:nat,Xs:(list @ A),J:nat,X4:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => (((I = J) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I) @ X4)) @ J) = X4)) & ((I != J) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I) @ X4)) @ J) = (((nth @ A) @ Xs) @ J)))))))).
% 3.17/1.71 thf(fact_146_list__update__same__conv, axiom, (! [A:$tType,I:nat,Xs:(list @ A),X4:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((((list_update @ A) @ Xs) @ I) @ X4) = Xs) = ((((nth @ A) @ Xs) @ I) = X4)))))).
% 3.17/1.71 thf(fact_147_bij__betw__inv__into__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),A6:(set @ B),B3:(set @ A),B5:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A2) @ A6) => ((((ord_less_eq @ (set @ A)) @ B3) @ A2) => ((((((image @ A) @ B) @ F) @ B3) = B5) => (((((bij_betw @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ A2) @ F)) @ B5) @ B3))))))).
% 3.17/1.71 thf(fact_148_direct__decompD_I2_J, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:(set @ A),Ss:(list @ (set @ A))]: (((((hilber580762119decomp @ A) @ A2) @ Ss) => ((((inj_on @ (list @ A)) @ A) @ (groups501227526m_list @ A)) @ ((listset @ A) @ Ss))))))))).
% 3.17/1.71 thf(fact_149_Schroeder__Bernstein, axiom, (! [A:$tType,B:$tType,F:(A > B),A2:(set @ A),B3:(set @ B),G:(B > A)]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ B3) => (((((inj_on @ B) @ A) @ G) @ B3) => ((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ G) @ B3)) @ A2) => (? [H:(A > B)]: ((((((bij_betw @ A) @ B) @ H) @ A2) @ B3)))))))))).
% 3.17/1.71 thf(fact_150_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A]: ((((ord_less_eq @ A) @ X4) @ X4))))))).
% 3.17/1.71 thf(fact_151_subset__imageE__inj, axiom, (! [A:$tType,B:$tType,B3:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B3) @ ((((image @ B) @ A) @ F) @ A2)) => (~ ((! [C3:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C3) @ A2) => ((B3 = ((((image @ B) @ A) @ F) @ C3)) => (~ (((((inj_on @ B) @ A) @ F) @ C3))))))))))))).
% 3.17/1.71 thf(fact_152_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z3) => (~ ((((ord_less_eq @ A) @ T2) @ X)))))))))))))).
% 3.17/1.71 thf(fact_153_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z3) => (((ord_less_eq @ A) @ X) @ T2)))))))))))).
% 3.17/1.71 thf(fact_154_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A2) => (A2 = B3)))))).
% 3.17/1.71 thf(fact_155_subsetI, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((! [X6:A]: (((((member @ A) @ X6) @ A2) => (((member @ A) @ X6) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B3))))).
% 3.17/1.71 thf(fact_156_bounded__Max__nat, axiom, (! [P:(nat > $o),X4:nat,M2:nat]: (((P @ X4) => ((! [X6:nat]: (((P @ X6) => (((ord_less_eq @ nat) @ X6) @ M2)))) => (~ ((! [M4:nat]: (((P @ M4) => (~ ((! [X:nat]: (((P @ X) => (((ord_less_eq @ nat) @ X) @ M4)))))))))))))))).
% 3.17/1.71 thf(fact_157_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X5:A]: (((P @ X5) => (Q @ X5)))))))).
% 3.17/1.71 thf(fact_158_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y2:(set @ A),Z:(set @ A)]: ((Y2 = Z))) = (^ [A5:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B4) & (((ord_less_eq @ (set @ A)) @ B4) @ A5)))))))).
% 3.17/1.71 thf(fact_159_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),C2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C2) => (((ord_less_eq @ (set @ A)) @ A2) @ C2)))))).
% 3.17/1.71 thf(fact_160_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X6:A]: (((P @ X6) => (Q @ X6)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 3.17/1.71 thf(fact_161_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 3.17/1.71 thf(fact_162_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B4:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A5) => (((member @ A) @ T3) @ B4)))))))))).
% 3.17/1.71 thf(fact_163_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((A2 = B3) => (((ord_less_eq @ (set @ A)) @ B3) @ A2))))).
% 3.17/1.71 thf(fact_164_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((A2 = B3) => (((ord_less_eq @ (set @ A)) @ A2) @ B3))))).
% 3.17/1.71 thf(fact_165_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B4:(set @ A)]: ((! [X5:A]: (((((member @ A) @ X5) @ A5) => (((member @ A) @ X5) @ B4)))))))))).
% 3.17/1.71 thf(fact_166_equalityE, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((A2 = B3) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => (~ ((((ord_less_eq @ (set @ A)) @ B3) @ A2)))))))))).
% 3.17/1.71 thf(fact_167_subsetD, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((member @ A) @ C) @ A2) => (((member @ A) @ C) @ B3)))))).
% 3.17/1.71 thf(fact_168_in__mono, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),X4:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((member @ A) @ X4) @ A2) => (((member @ A) @ X4) @ B3)))))).
% 3.17/1.71 thf(fact_169_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ A3) @ B2) => (A3 = B2))))))))).
% 3.17/1.71 thf(fact_170_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [A8:A,B6:A]: (((((ord_less_eq @ A) @ B6) @ A8) & (((ord_less_eq @ A) @ A8) @ B6))))))))).
% 3.17/1.71 thf(fact_171_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ A3))))))))).
% 3.17/1.71 thf(fact_172_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B2:A]: (((! [A4:A,B7:A]: (((((ord_less_eq @ A) @ A4) @ B7) => ((P @ A4) @ B7)))) => ((! [A4:A,B7:A]: ((((P @ B7) @ A4) => ((P @ A4) @ B7)))) => ((P @ A3) @ B2))))))))).
% 3.17/1.71 thf(fact_173_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 3.17/1.71 thf(fact_174_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z2) => (((ord_less_eq @ A) @ X4) @ Z2))))))))).
% 3.17/1.71 thf(fact_175_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ B2) @ A3) => (A3 = B2))))))))).
% 3.17/1.71 thf(fact_176_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((B2 = C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 3.17/1.71 thf(fact_177_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C:A]: (((A3 = B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 3.17/1.71 thf(fact_178_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [A8:A,B6:A]: (((((ord_less_eq @ A) @ A8) @ B6) & (((ord_less_eq @ A) @ B6) @ A8))))))))).
% 3.17/1.71 thf(fact_179_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X4:A]: (((((ord_less_eq @ A) @ Y3) @ X4) => ((((ord_less_eq @ A) @ X4) @ Y3) = (X4 = Y3))))))))).
% 3.17/1.71 thf(fact_180_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A,Z2:A]: ((((((ord_less_eq @ A) @ X4) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z2)))) => (((((ord_less_eq @ A) @ Y3) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Z2)))) => (((((ord_less_eq @ A) @ X4) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y3)))) => (((((ord_less_eq @ A) @ Z2) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X4)))) => (((((ord_less_eq @ A) @ Y3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X4)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Y3))))))))))))))))).
% 3.17/1.71 thf(fact_181_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 3.17/1.71 thf(fact_182_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X4) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X4)))))))).
% 3.17/1.71 thf(fact_183_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y3:A]: (((X4 = Y3) => (((ord_less_eq @ A) @ X4) @ Y3)))))))).
% 3.17/1.71 thf(fact_184_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X4)))))))).
% 3.17/1.71 thf(fact_185_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X4) => (X4 = Y3))))))))).
% 3.17/1.71 thf(fact_186_le__rel__bool__arg__iff, axiom, (! [A:$tType]: (((ord @ A) => ((ord_less_eq @ ($o > A)) = (^ [X8:($o > A),Y5:($o > A)]: (((((ord_less_eq @ A) @ (X8 @ $false)) @ (Y5 @ $false)) & (((ord_less_eq @ A) @ (X8 @ $true)) @ (Y5 @ $true)))))))))).
% 3.17/1.71 thf(fact_187_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [X5:A,Y4:A]: (((((ord_less_eq @ A) @ X5) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X5))))))))).
% 3.17/1.71 thf(fact_188_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B2:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A3) @ B2) => (((F @ B2) = C) => ((! [X6:A,Y:A]: (((((ord_less_eq @ A) @ X6) @ Y) => (((ord_less_eq @ B) @ (F @ X6)) @ (F @ Y))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C)))))))))).
% 3.17/1.71 thf(fact_189_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B2:B,C:B]: (((A3 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X6:B,Y:B]: (((((ord_less_eq @ B) @ X6) @ Y) => (((ord_less_eq @ A) @ (F @ X6)) @ (F @ Y))))) => (((ord_less_eq @ A) @ A3) @ (F @ C))))))))))).
% 3.17/1.71 thf(fact_190_order__subst2, axiom, (! [A:$tType,C4:$tType]: ((((order @ C4) & (order @ A)) => (! [A3:A,B2:A,F:(A > C4),C:C4]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ C4) @ (F @ B2)) @ C) => ((! [X6:A,Y:A]: (((((ord_less_eq @ A) @ X6) @ Y) => (((ord_less_eq @ C4) @ (F @ X6)) @ (F @ Y))))) => (((ord_less_eq @ C4) @ (F @ A3)) @ C)))))))))).
% 3.17/1.71 thf(fact_191_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X6:B,Y:B]: (((((ord_less_eq @ B) @ X6) @ Y) => (((ord_less_eq @ A) @ (F @ X6)) @ (F @ Y))))) => (((ord_less_eq @ A) @ A3) @ (F @ C))))))))))).
% 3.17/1.71 thf(fact_192_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G3:(A > B)]: ((! [X5:A]: ((((ord_less_eq @ B) @ (F3 @ X5)) @ (G3 @ X5))))))))))).
% 3.17/1.71 thf(fact_193_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X6:A]: ((((ord_less_eq @ B) @ (F @ X6)) @ (G @ X6)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 3.17/1.71 thf(fact_194_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X4)) @ (G @ X4))))))))).
% 3.17/1.71 thf(fact_195_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X4)) @ (G @ X4))))))))).
% 3.17/1.71 thf(fact_196_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => (A3 != B2)))))))).
% 3.17/1.71 thf(fact_197_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (A3 != B2)))))))).
% 3.17/1.71 thf(fact_198_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((~ ((((ord_less @ A) @ X4) @ Y3))) = ((((ord_less @ A) @ Y3) @ X4) | (X4 = Y3))))))))).
% 3.17/1.71 thf(fact_199_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A,C:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A3))))))))).
% 3.17/1.71 thf(fact_200_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B2:A]: (((! [A4:A,B7:A]: (((((ord_less @ A) @ A4) @ B7) => ((P @ A4) @ B7)))) => ((! [A4:A]: (((P @ A4) @ A4))) => ((! [A4:A,B7:A]: ((((P @ B7) @ A4) => ((P @ A4) @ B7)))) => ((P @ A3) @ B2)))))))))).
% 3.17/1.71 thf(fact_201_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P3:(A > $o)]: ((? [X3:A]: ((P3 @ X3))))) = (^ [P2:(A > $o)]: ((? [N3:A]: (((P2 @ N3) & (! [M5:A]: (((((ord_less @ A) @ M5) @ N3) => (~ ((P2 @ M5)))))))))))))))).
% 3.17/1.71 thf(fact_202_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) => (~ ((((ord_less @ A) @ Y3) @ X4)))))))))).
% 3.17/1.71 thf(fact_203_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A,C:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A3) @ C))))))))).
% 3.17/1.71 thf(fact_204_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ A3))))))))).
% 3.17/1.71 thf(fact_205_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((~ ((((ord_less @ A) @ X4) @ Y3))) => ((X4 != Y3) => (((ord_less @ A) @ Y3) @ X4))))))))).
% 3.17/1.71 thf(fact_206_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y3:A,P:$o]: (((((ord_less @ A) @ X4) @ Y3) => ((((ord_less @ A) @ Y3) @ X4) => P)))))))).
% 3.17/1.71 thf(fact_207_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) => (Y3 != X4)))))))).
% 3.17/1.71 thf(fact_208_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y3:A,X4:A]: (((~ ((((ord_less @ A) @ Y3) @ X4))) => ((~ ((((ord_less @ A) @ X4) @ Y3))) = (X4 = Y3))))))))).
% 3.17/1.71 thf(fact_209_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A3:A]: (((! [X6:A]: (((! [Y6:A]: (((((ord_less @ A) @ Y6) @ X6) => (P @ Y6)))) => (P @ X6)))) => (P @ A3)))))))).
% 3.17/1.71 thf(fact_210_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) => (~ ((((ord_less @ A) @ Y3) @ X4)))))))))).
% 3.17/1.71 thf(fact_211_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) => (X4 != Y3)))))))).
% 3.17/1.71 thf(fact_212_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => (~ ((((ord_less @ A) @ A3) @ B2)))))))))).
% 3.17/1.71 thf(fact_213_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C:A]: (((((ord_less @ A) @ A3) @ B2) => ((B2 = C) => (((ord_less @ A) @ A3) @ C))))))))).
% 3.17/1.71 thf(fact_214_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C:A]: (((A3 = B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A3) @ C))))))))).
% 3.17/1.71 thf(fact_215_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A]: ((~ ((((ord_less @ A) @ X4) @ X4))))))))).
% 3.17/1.71 thf(fact_216_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) | ((X4 = Y3) | (((ord_less @ A) @ Y3) @ X4))))))))).
% 3.17/1.71 thf(fact_217_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y3:A,Z2:A]: (((((ord_less @ A) @ X4) @ Y3) => ((((ord_less @ A) @ Y3) @ Z2) => (((ord_less @ A) @ X4) @ Z2))))))))).
% 3.17/1.71 thf(fact_218_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ ((((ord_less @ A) @ B2) @ A3)))))))))).
% 3.17/1.71 thf(fact_219_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) => (~ ((((ord_less @ A) @ Y3) @ X4)))))))))).
% 3.17/1.71 thf(fact_220_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) => (X4 != Y3)))))))).
% 3.17/1.71 thf(fact_221_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) => (? [Z3:A]: (((((ord_less @ A) @ X4) @ Z3) & (((ord_less @ A) @ Z3) @ Y3))))))))))).
% 3.17/1.71 thf(fact_222_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ ((((ord_less @ A) @ B2) @ A3)))))))))).
% 3.17/1.71 thf(fact_223_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((X4 != Y3) = ((((ord_less @ A) @ X4) @ Y3) | (((ord_less @ A) @ Y3) @ X4))))))))).
% 3.17/1.71 thf(fact_224_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((X4 != Y3) => ((~ ((((ord_less @ A) @ X4) @ Y3))) => (((ord_less @ A) @ Y3) @ X4))))))))).
% 3.17/1.71 thf(fact_225_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X4:A]: ((? [X_1:A]: ((((ord_less @ A) @ X4) @ X_1))))))))).
% 3.17/1.71 thf(fact_226_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X4:A]: ((? [Y:A]: ((((ord_less @ A) @ Y) @ X4))))))))).
% 3.17/1.71 thf(fact_227_order__less__subst2, axiom, (! [A:$tType,C4:$tType]: ((((order @ C4) & (order @ A)) => (! [A3:A,B2:A,F:(A > C4),C:C4]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ C4) @ (F @ B2)) @ C) => ((! [X6:A,Y:A]: (((((ord_less @ A) @ X6) @ Y) => (((ord_less @ C4) @ (F @ X6)) @ (F @ Y))))) => (((ord_less @ C4) @ (F @ A3)) @ C)))))))))).
% 3.17/1.71 thf(fact_228_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B2:B,C:B]: (((((ord_less @ A) @ A3) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X6:B,Y:B]: (((((ord_less @ B) @ X6) @ Y) => (((ord_less @ A) @ (F @ X6)) @ (F @ Y))))) => (((ord_less @ A) @ A3) @ (F @ C))))))))))).
% 3.17/1.71 thf(fact_229_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B2:A,F:(A > B),C:B]: (((((ord_less @ A) @ A3) @ B2) => (((F @ B2) = C) => ((! [X6:A,Y:A]: (((((ord_less @ A) @ X6) @ Y) => (((ord_less @ B) @ (F @ X6)) @ (F @ Y))))) => (((ord_less @ B) @ (F @ A3)) @ C)))))))))).
% 3.17/1.71 thf(fact_230_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B2:B,C:B]: (((A3 = (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X6:B,Y:B]: (((((ord_less @ B) @ X6) @ Y) => (((ord_less @ A) @ (F @ X6)) @ (F @ Y))))) => (((ord_less @ A) @ A3) @ (F @ C))))))))))).
% 3.17/1.71 thf(fact_231_pinf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ Z4) @ X6) => ((P @ X6) = (P4 @ X6))))))) => ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ Z4) @ X6) => ((Q @ X6) = (Q2 @ X6))))))) => (? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ Z3) @ X) => (((P @ X) & (Q @ X)) = ((P4 @ X) & (Q2 @ X)))))))))))))))).
% 3.17/1.71 thf(fact_232_pinf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ Z4) @ X6) => ((P @ X6) = (P4 @ X6))))))) => ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ Z4) @ X6) => ((Q @ X6) = (Q2 @ X6))))))) => (? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ Z3) @ X) => (((P @ X) | (Q @ X)) = ((P4 @ X) | (Q2 @ X)))))))))))))))).
% 3.17/1.71 thf(fact_233_pinf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ Z3) @ X) => (X != T2)))))))))))).
% 3.17/1.71 thf(fact_234_pinf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ Z3) @ X) => (X != T2)))))))))))).
% 3.17/1.71 thf(fact_235_pinf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ Z3) @ X) => (~ ((((ord_less @ A) @ X) @ T2)))))))))))))).
% 3.17/1.71 thf(fact_236_pinf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ Z3) @ X) => (((ord_less @ A) @ T2) @ X)))))))))))).
% 3.17/1.71 thf(fact_237_pinf_I11_J, axiom, (! [C4:$tType,D:$tType]: (((ord @ C4) => (! [F5:D]: ((? [Z3:C4]: ((! [X:C4]: (((((ord_less @ C4) @ Z3) @ X) => (F5 = F5)))))))))))).
% 3.17/1.71 thf(fact_238_minf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z4) => ((P @ X6) = (P4 @ X6))))))) => ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z4) => ((Q @ X6) = (Q2 @ X6))))))) => (? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z3) => (((P @ X) & (Q @ X)) = ((P4 @ X) & (Q2 @ X)))))))))))))))).
% 3.17/1.71 thf(fact_239_minf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z4) => ((P @ X6) = (P4 @ X6))))))) => ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z4) => ((Q @ X6) = (Q2 @ X6))))))) => (? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z3) => (((P @ X) | (Q @ X)) = ((P4 @ X) | (Q2 @ X)))))))))))))))).
% 3.17/1.71 thf(fact_240_minf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z3) => (X != T2)))))))))))).
% 3.17/1.71 thf(fact_241_minf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z3) => (X != T2)))))))))))).
% 3.17/1.71 thf(fact_242_minf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z3) => (((ord_less @ A) @ X) @ T2)))))))))))).
% 3.17/1.71 thf(fact_243_minf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z3) => (~ ((((ord_less @ A) @ T2) @ X)))))))))))))).
% 3.17/1.71 thf(fact_244_minf_I11_J, axiom, (! [C4:$tType,D:$tType]: (((ord @ C4) => (! [F5:D]: ((? [Z3:C4]: ((! [X:C4]: (((((ord_less @ C4) @ X) @ Z3) => (F5 = F5)))))))))))).
% 3.17/1.71 thf(fact_245_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((A3 != B2) => ((((ord_less_eq @ A) @ A3) @ B2) => (((ord_less @ A) @ A3) @ B2))))))))).
% 3.17/1.71 thf(fact_246_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => (((ord_less_eq @ A) @ B2) @ A3)))))))).
% 3.17/1.71 thf(fact_247_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B6:A,A8:A]: (((((ord_less_eq @ A) @ B6) @ A8) & (A8 != B6))))))))).
% 3.17/1.71 thf(fact_248_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B6:A,A8:A]: (((((ord_less @ A) @ B6) @ A8) | (A8 = B6))))))))).
% 3.17/1.71 thf(fact_249_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 3.17/1.71 thf(fact_250_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X4:A,Y3:A,Z2:A]: (((((ord_less @ A) @ X4) @ Y3) => ((! [W:A]: (((((ord_less @ A) @ X4) @ W) => ((((ord_less @ A) @ W) @ Y3) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y3) @ Z2))))))))).
% 3.17/1.71 thf(fact_251_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,X4:A,Y3:A]: (((((ord_less @ A) @ Z2) @ X4) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X4) => (((ord_less_eq @ A) @ Y3) @ W))))) => (((ord_less_eq @ A) @ Y3) @ Z2))))))))).
% 3.17/1.71 thf(fact_252_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A,C:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A3))))))))).
% 3.17/1.71 thf(fact_253_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A3))))))))).
% 3.17/1.71 thf(fact_254_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A8:A,B6:A]: (((((ord_less_eq @ A) @ A8) @ B6) & (A8 != B6))))))))).
% 3.17/1.71 thf(fact_255_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A8:A,B6:A]: (((((ord_less @ A) @ A8) @ B6) | (A8 = B6))))))))).
% 3.17/1.71 thf(subcl_Groups_Ocomm__monoid__add___HOL_Otype, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (type @ A))))).
% 3.17/1.71 thf(tcon_Nat_Onat___Rings_Odioid, axiom, (dioid @ nat)).
% 3.17/1.71 thf(tcon_fun___Rings_Odioid_1, axiom, (! [A9:$tType,A10:$tType]: (((dioid @ A10) => (dioid @ (A9 > A10)))))).
% 3.17/1.71 thf(tcon_fun___Groups_Ocanonically__ordered__monoid__add, axiom, (! [A9:$tType,A10:$tType]: (((dioid @ A10) => (canoni770627133id_add @ (A9 > A10)))))).
% 3.17/1.71 thf(tcon_fun___Groups_Ocomm__monoid__add, axiom, (! [A9:$tType,A10:$tType]: (((comm_monoid_add @ A10) => (comm_monoid_add @ (A9 > A10)))))).
% 3.17/1.71 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A10:$tType]: (((preorder @ A10) => (preorder @ (A9 > A10)))))).
% 3.17/1.71 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: (((order @ A10) => (order @ (A9 > A10)))))).
% 3.17/1.71 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 3.17/1.71 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add_2, axiom, (canoni770627133id_add @ nat)).
% 3.17/1.71 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_3, axiom, (comm_monoid_add @ nat)).
% 3.17/1.71 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 3.17/1.71 thf(tcon_Nat_Onat___Orderings_Opreorder_4, axiom, (preorder @ nat)).
% 3.17/1.71 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 3.17/1.71 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, (no_top @ nat)).
% 3.17/1.71 thf(tcon_Nat_Onat___Orderings_Oorder_5, axiom, (order @ nat)).
% 3.17/1.71 thf(tcon_Nat_Onat___Orderings_Oord_6, axiom, (ord @ nat)).
% 3.17/1.71 thf(tcon_Nat_Onat___Nat_Osize, axiom, (size @ nat)).
% 3.17/1.71 thf(tcon_Set_Oset___Orderings_Opreorder_7, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 3.17/1.71 thf(tcon_Set_Oset___Orderings_Oorder_8, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 3.17/1.71 thf(tcon_Set_Oset___Orderings_Oord_9, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 3.17/1.71 thf(tcon_HOL_Obool___Orderings_Opreorder_10, axiom, (preorder @ $o)).
% 3.17/1.71 thf(tcon_HOL_Obool___Orderings_Olinorder_11, axiom, (linorder @ $o)).
% 3.17/1.71 thf(tcon_HOL_Obool___Orderings_Oorder_12, axiom, (order @ $o)).
% 3.17/1.71 thf(tcon_HOL_Obool___Orderings_Oord_13, axiom, (ord @ $o)).
% 3.17/1.71 thf(tcon_List_Olist___Nat_Osize_14, axiom, (! [A9:$tType]: ((size @ (list @ A9))))).
% 3.17/1.71 thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_15, axiom, (! [A9:$tType]: ((comm_monoid_add @ (multiset @ A9))))).
% 3.17/1.71 thf(tcon_Multiset_Omultiset___Orderings_Opreorder_16, axiom, (! [A9:$tType]: (((preorder @ A9) => (preorder @ (multiset @ A9)))))).
% 3.17/1.71 thf(tcon_Multiset_Omultiset___Orderings_Oorder_17, axiom, (! [A9:$tType]: (((preorder @ A9) => (order @ (multiset @ A9)))))).
% 3.17/1.71 thf(tcon_Multiset_Omultiset___Orderings_Oord_18, axiom, (! [A9:$tType]: (((preorder @ A9) => (ord @ (multiset @ A9)))))).
% 3.17/1.71 thf(tcon_Multiset_Omultiset___Nat_Osize_19, axiom, (! [A9:$tType]: ((size @ (multiset @ A9))))).
% 3.17/1.71 thf(tfree_0, hypothesis, (comm_monoid_add @ a)).
% 3.17/1.71 thf(conj_0, conjecture, (a3 = ((groups501227526m_list @ a) @ qs2))).
% 3.17/1.71 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.DwfRHP4Hjf/DTF2DTF28982.p
% 3.17/1.71 ------------------------
% 3.17/1.74 ---- Cleaned THF ---
% 3.17/1.74 thf(ty_t_Multiset_Omultiset,type,
% 3.17/1.74 multiset: $tType > $tType ).
% 3.17/1.74
% 3.17/1.74 thf(ty_t_List_Olist,type,
% 3.17/1.74 list: $tType > $tType ).
% 3.17/1.74
% 3.17/1.74 thf(ty_t_Set_Oset,type,
% 3.17/1.74 set: $tType > $tType ).
% 3.17/1.74
% 3.17/1.74 thf(ty_t_Nat_Onat,type,
% 3.17/1.74 nat: $tType ).
% 3.17/1.74
% 3.17/1.74 thf(ty_tf_a,type,
% 3.17/1.74 a: $tType ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_Rings_Odioid,type,
% 3.17/1.74 dioid:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_HOL_Otype,type,
% 3.17/1.74 type:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_Nat_Osize,type,
% 3.17/1.74 size:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_Orderings_Oord,type,
% 3.17/1.74 ord:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_Orderings_Oorder,type,
% 3.17/1.74 order:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_Orderings_Ono__bot,type,
% 3.17/1.74 no_bot:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_Orderings_Ono__top,type,
% 3.17/1.74 no_top:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_Orderings_Olinorder,type,
% 3.17/1.74 linorder:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_Orderings_Opreorder,type,
% 3.17/1.74 preorder:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_Orderings_Owellorder,type,
% 3.17/1.74 wellorder:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_Orderings_Odense__order,type,
% 3.17/1.74 dense_order:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 3.17/1.74 comm_monoid_add:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_Orderings_Odense__linorder,type,
% 3.17/1.74 dense_linorder:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 3.17/1.74 canoni770627133id_add:
% 3.17/1.74 !>[A: $tType] : $o ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Fun_Obij__betw,type,
% 3.17/1.74 bij_betw:
% 3.17/1.74 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Fun_Oinj__on,type,
% 3.17/1.74 inj_on:
% 3.17/1.74 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
% 3.17/1.74 groups501227526m_list:
% 3.17/1.74 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Hilbert__Choice_Oinv__into,type,
% 3.17/1.74 hilbert_inv_into:
% 3.17/1.74 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Hilbert__Function__Mirabelle__phhewjaonq_Odirect__decomp,type,
% 3.17/1.74 hilber580762119decomp:
% 3.17/1.74 !>[A: $tType] : ( ( set @ A ) > ( list @ ( set @ A ) ) > $o ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_List_Olist__ex,type,
% 3.17/1.74 list_ex:
% 3.17/1.74 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_List_Olist__update,type,
% 3.17/1.74 list_update:
% 3.17/1.74 !>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_List_Olistset,type,
% 3.17/1.74 listset:
% 3.17/1.74 !>[A: $tType] : ( ( list @ ( set @ A ) ) > ( set @ ( list @ A ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_List_Onth,type,
% 3.17/1.74 nth:
% 3.17/1.74 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_List_Orev,type,
% 3.17/1.74 rev:
% 3.17/1.74 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset,type,
% 3.17/1.74 comm_m1079429914m_mset:
% 3.17/1.74 !>[A: $tType] : ( ( multiset @ A ) > A ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Multiset_Omset,type,
% 3.17/1.74 mset:
% 3.17/1.74 !>[A: $tType] : ( ( list @ A ) > ( multiset @ A ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Multiset_Oset__mset,type,
% 3.17/1.74 set_mset:
% 3.17/1.74 !>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Nat_Osize__class_Osize,type,
% 3.17/1.74 size_size:
% 3.17/1.74 !>[A: $tType] : ( A > nat ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Orderings_Oord__class_Oless,type,
% 3.17/1.74 ord_less:
% 3.17/1.74 !>[A: $tType] : ( A > A > $o ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.17/1.74 ord_less_eq:
% 3.17/1.74 !>[A: $tType] : ( A > A > $o ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Permutation_Operm,type,
% 3.17/1.74 perm:
% 3.17/1.74 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Set_OCollect,type,
% 3.17/1.74 collect:
% 3.17/1.74 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Set_Oimage,type,
% 3.17/1.74 image:
% 3.17/1.74 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
% 3.17/1.74 set_ord_lessThan:
% 3.17/1.74 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_c_member,type,
% 3.17/1.74 member:
% 3.17/1.74 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_v_A,type,
% 3.17/1.74 a2: set @ a ).
% 3.17/1.74
% 3.17/1.74 thf(sy_v_a____,type,
% 3.17/1.74 a3: a ).
% 3.17/1.74
% 3.17/1.74 thf(sy_v_f____,type,
% 3.17/1.74 f: nat > nat ).
% 3.17/1.74
% 3.17/1.74 thf(sy_v_g____,type,
% 3.17/1.74 g: nat > nat ).
% 3.17/1.74
% 3.17/1.74 thf(sy_v_qs2____,type,
% 3.17/1.74 qs2: list @ a ).
% 3.17/1.74
% 3.17/1.74 thf(sy_v_qs____,type,
% 3.17/1.74 qs: list @ a ).
% 3.17/1.74
% 3.17/1.74 thf(sy_v_ss1,type,
% 3.17/1.74 ss1: list @ ( set @ a ) ).
% 3.17/1.74
% 3.17/1.74 thf(sy_v_ss2,type,
% 3.17/1.74 ss2: list @ ( set @ a ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_0__092_060open_062mset_Aqs2_A_061_Amset_Aqs_092_060close_062,axiom,
% 3.17/1.74 ( ( mset @ a @ qs2 )
% 3.17/1.74 = ( mset @ a @ qs ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_1__092_060open_062a_A_092_060in_062_AA_092_060close_062,axiom,
% 3.17/1.74 member @ a @ a3 @ a2 ).
% 3.17/1.74
% 3.17/1.74 thf(fact_2_a,axiom,
% 3.17/1.74 ( a3
% 3.17/1.74 = ( groups501227526m_list @ a @ qs ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_3_len__qs2,axiom,
% 3.17/1.74 ( ( size_size @ ( list @ a ) @ qs2 )
% 3.17/1.74 = ( size_size @ ( list @ a ) @ qs ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs_O_A_092_060lbrakk_062a_A_061_Asum__list_Aqs_059_Aqs_A_092_060in_062_Alistset_Ass1_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 3.17/1.74 ~ ! [Qs: list @ a] :
% 3.17/1.74 ( ( a3
% 3.17/1.74 = ( groups501227526m_list @ a @ Qs ) )
% 3.17/1.74 => ~ ( member @ ( list @ a ) @ Qs @ ( listset @ a @ ss1 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_5_qs2__in,axiom,
% 3.17/1.74 member @ ( list @ a ) @ qs2 @ ( listset @ a @ ss2 ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_6_direct__decompD_I1_J,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( comm_monoid_add @ A )
% 3.17/1.74 => ! [A2: set @ A,Ss: list @ ( set @ A ),Qs2: list @ A] :
% 3.17/1.74 ( ( hilber580762119decomp @ A @ A2 @ Ss )
% 3.17/1.74 => ( ( member @ ( list @ A ) @ Qs2 @ ( listset @ A @ Ss ) )
% 3.17/1.74 => ( member @ A @ ( groups501227526m_list @ A @ Qs2 ) @ A2 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_7_direct__decompE,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( comm_monoid_add @ A )
% 3.17/1.74 => ! [A2: set @ A,Ss: list @ ( set @ A ),A3: A] :
% 3.17/1.74 ( ( hilber580762119decomp @ A @ A2 @ Ss )
% 3.17/1.74 => ( ( member @ A @ A3 @ A2 )
% 3.17/1.74 => ~ ! [Qs: list @ A] :
% 3.17/1.74 ( ( member @ ( list @ A ) @ Qs @ ( listset @ A @ Ss ) )
% 3.17/1.74 => ( A3
% 3.17/1.74 != ( groups501227526m_list @ A @ Qs ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_8_direct__decompI__alt,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( comm_monoid_add @ A )
% 3.17/1.74 => ! [Ss: list @ ( set @ A ),A2: set @ A] :
% 3.17/1.74 ( ! [Qs: list @ A] :
% 3.17/1.74 ( ( member @ ( list @ A ) @ Qs @ ( listset @ A @ Ss ) )
% 3.17/1.74 => ( member @ A @ ( groups501227526m_list @ A @ Qs ) @ A2 ) )
% 3.17/1.74 => ( ! [A4: A] :
% 3.17/1.74 ( ( member @ A @ A4 @ A2 )
% 3.17/1.74 => ? [X: list @ A] :
% 3.17/1.74 ( ( member @ ( list @ A ) @ X @ ( listset @ A @ Ss ) )
% 3.17/1.74 & ( A4
% 3.17/1.74 = ( groups501227526m_list @ A @ X ) )
% 3.17/1.74 & ! [Y: list @ A] :
% 3.17/1.74 ( ( ( member @ ( list @ A ) @ Y @ ( listset @ A @ Ss ) )
% 3.17/1.74 & ( A4
% 3.17/1.74 = ( groups501227526m_list @ A @ Y ) ) )
% 3.17/1.74 => ( Y = X ) ) ) )
% 3.17/1.74 => ( hilber580762119decomp @ A @ A2 @ Ss ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_9_direct__decomp__unique,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( comm_monoid_add @ A )
% 3.17/1.74 => ! [A2: set @ A,Ss: list @ ( set @ A ),Qs2: list @ A,Qs3: list @ A] :
% 3.17/1.74 ( ( hilber580762119decomp @ A @ A2 @ Ss )
% 3.17/1.74 => ( ( member @ ( list @ A ) @ Qs2 @ ( listset @ A @ Ss ) )
% 3.17/1.74 => ( ( member @ ( list @ A ) @ Qs3 @ ( listset @ A @ Ss ) )
% 3.17/1.74 => ( ( ( groups501227526m_list @ A @ Qs2 )
% 3.17/1.74 = ( groups501227526m_list @ A @ Qs3 ) )
% 3.17/1.74 => ( Qs2 = Qs3 ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_10_sum__list_Orev,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( comm_monoid_add @ A )
% 3.17/1.74 => ! [Xs: list @ A] :
% 3.17/1.74 ( ( groups501227526m_list @ A @ ( rev @ A @ Xs ) )
% 3.17/1.74 = ( groups501227526m_list @ A @ Xs ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_11__C1_C,axiom,
% 3.17/1.74 ! [I: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ a ) @ qs ) )
% 3.17/1.74 => ( ( nth @ a @ qs2 @ I )
% 3.17/1.74 = ( nth @ a @ qs @ ( g @ I ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_12_g__bij2,axiom,
% 3.17/1.74 bij_betw @ nat @ nat @ g @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ a ) @ qs2 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ a ) @ qs ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_13_sum__mset__sum__list,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( comm_monoid_add @ A )
% 3.17/1.74 => ! [Xs: list @ A] :
% 3.17/1.74 ( ( comm_m1079429914m_mset @ A @ ( mset @ A @ Xs ) )
% 3.17/1.74 = ( groups501227526m_list @ A @ Xs ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_14_direct__decompD_I3_J,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( comm_monoid_add @ A )
% 3.17/1.74 => ! [A2: set @ A,Ss: list @ ( set @ A )] :
% 3.17/1.74 ( ( hilber580762119decomp @ A @ A2 @ Ss )
% 3.17/1.74 => ( ( image @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss ) )
% 3.17/1.74 = A2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_15_assms_I1_J,axiom,
% 3.17/1.74 hilber580762119decomp @ a @ a2 @ ss1 ).
% 3.17/1.74
% 3.17/1.74 thf(fact_16_qs__in,axiom,
% 3.17/1.74 member @ ( list @ a ) @ qs @ ( listset @ a @ ss1 ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_17_mset__rev,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A] :
% 3.17/1.74 ( ( mset @ A @ ( rev @ A @ Xs ) )
% 3.17/1.74 = ( mset @ A @ Xs ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_18__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs2_O_A_092_060lbrakk_062qs2_A_092_060in_062_Alistset_Ass2_059_Alength_Aqs2_A_061_Alength_Aqs_059_A_092_060And_062i_O_Ai_A_060_Alength_Aqs_A_092_060Longrightarrow_062_Aqs2_A_B_Ai_A_061_Aqs_A_B_Ag_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 3.17/1.74 ~ ! [Qs22: list @ a] :
% 3.17/1.74 ( ( member @ ( list @ a ) @ Qs22 @ ( listset @ a @ ss2 ) )
% 3.17/1.74 => ( ( ( size_size @ ( list @ a ) @ Qs22 )
% 3.17/1.74 = ( size_size @ ( list @ a ) @ qs ) )
% 3.17/1.74 => ~ ! [I2: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ a ) @ qs ) )
% 3.17/1.74 => ( ( nth @ a @ Qs22 @ I2 )
% 3.17/1.74 = ( nth @ a @ qs @ ( g @ I2 ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_19_assms_I2_J,axiom,
% 3.17/1.74 perm @ ( set @ a ) @ ss1 @ ss2 ).
% 3.17/1.74
% 3.17/1.74 thf(fact_20_ex__mset,axiom,
% 3.17/1.74 ! [A: $tType,X2: multiset @ A] :
% 3.17/1.74 ? [Xs2: list @ A] :
% 3.17/1.74 ( ( mset @ A @ Xs2 )
% 3.17/1.74 = X2 ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_21_mset__eq__length,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 3.17/1.74 ( ( ( mset @ A @ Xs )
% 3.17/1.74 = ( mset @ A @ Ys ) )
% 3.17/1.74 => ( ( size_size @ ( list @ A ) @ Xs )
% 3.17/1.74 = ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_22_mset__bij,axiom,
% 3.17/1.74 ! [A: $tType,F: nat > nat,Xs: list @ A,Ys: list @ A] :
% 3.17/1.74 ( ( bij_betw @ nat @ nat @ F @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Ys ) ) )
% 3.17/1.74 => ( ! [I3: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 3.17/1.74 => ( ( nth @ A @ Xs @ I3 )
% 3.17/1.74 = ( nth @ A @ Ys @ ( F @ I3 ) ) ) )
% 3.17/1.74 => ( ( mset @ A @ Xs )
% 3.17/1.74 = ( mset @ A @ Ys ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_23_listset__permE,axiom,
% 3.17/1.74 ! [A: $tType,Ys: list @ A,Xs: list @ ( set @ A ),F: nat > nat,Xs3: list @ ( set @ A )] :
% 3.17/1.74 ( ( member @ ( list @ A ) @ Ys @ ( listset @ A @ Xs ) )
% 3.17/1.74 => ( ( bij_betw @ nat @ nat @ F @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ A ) ) @ Xs ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ A ) ) @ Xs3 ) ) )
% 3.17/1.74 => ( ! [I3: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( set @ A ) ) @ Xs ) )
% 3.17/1.74 => ( ( nth @ ( set @ A ) @ Xs3 @ I3 )
% 3.17/1.74 = ( nth @ ( set @ A ) @ Xs @ ( F @ I3 ) ) ) )
% 3.17/1.74 => ~ ! [Ys2: list @ A] :
% 3.17/1.74 ( ( member @ ( list @ A ) @ Ys2 @ ( listset @ A @ Xs3 ) )
% 3.17/1.74 => ( ( ( size_size @ ( list @ A ) @ Ys2 )
% 3.17/1.74 = ( size_size @ ( list @ A ) @ Ys ) )
% 3.17/1.74 => ~ ! [I2: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys ) )
% 3.17/1.74 => ( ( nth @ A @ Ys2 @ I2 )
% 3.17/1.74 = ( nth @ A @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_24_g__bij,axiom,
% 3.17/1.74 bij_betw @ nat @ nat @ g @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_25_length__rev,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A] :
% 3.17/1.74 ( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
% 3.17/1.74 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_26_lessThan__iff,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( ord @ A )
% 3.17/1.74 => ! [I: A,K: A] :
% 3.17/1.74 ( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
% 3.17/1.74 = ( ord_less @ A @ I @ K ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_27_listsetI,axiom,
% 3.17/1.74 ! [A: $tType,Ys: list @ A,Xs: list @ ( set @ A )] :
% 3.17/1.74 ( ( ( size_size @ ( list @ A ) @ Ys )
% 3.17/1.74 = ( size_size @ ( list @ ( set @ A ) ) @ Xs ) )
% 3.17/1.74 => ( ! [I3: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( set @ A ) ) @ Xs ) )
% 3.17/1.74 => ( member @ A @ ( nth @ A @ Ys @ I3 ) @ ( nth @ ( set @ A ) @ Xs @ I3 ) ) )
% 3.17/1.74 => ( member @ ( list @ A ) @ Ys @ ( listset @ A @ Xs ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_28_len__qs,axiom,
% 3.17/1.74 ( ( size_size @ ( list @ a ) @ qs )
% 3.17/1.74 = ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_29_f__bij,axiom,
% 3.17/1.74 bij_betw @ nat @ nat @ f @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_30_nth__equalityI,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 3.17/1.74 ( ( ( size_size @ ( list @ A ) @ Xs )
% 3.17/1.74 = ( size_size @ ( list @ A ) @ Ys ) )
% 3.17/1.74 => ( ! [I3: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 3.17/1.74 => ( ( nth @ A @ Xs @ I3 )
% 3.17/1.74 = ( nth @ A @ Ys @ I3 ) ) )
% 3.17/1.74 => ( Xs = Ys ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_31_Skolem__list__nth,axiom,
% 3.17/1.74 ! [A: $tType,K: nat,P: nat > A > $o] :
% 3.17/1.74 ( ( ! [I4: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I4 @ K )
% 3.17/1.74 => ? [X3: A] : ( P @ I4 @ X3 ) ) )
% 3.17/1.74 = ( ? [Xs4: list @ A] :
% 3.17/1.74 ( ( ( size_size @ ( list @ A ) @ Xs4 )
% 3.17/1.74 = K )
% 3.17/1.74 & ! [I4: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I4 @ K )
% 3.17/1.74 => ( P @ I4 @ ( nth @ A @ Xs4 @ I4 ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_32_list__eq__iff__nth__eq,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( ^ [Y2: list @ A,Z: list @ A] : ( Y2 = Z ) )
% 3.17/1.74 = ( ^ [Xs4: list @ A,Ys3: list @ A] :
% 3.17/1.74 ( ( ( size_size @ ( list @ A ) @ Xs4 )
% 3.17/1.74 = ( size_size @ ( list @ A ) @ Ys3 ) )
% 3.17/1.74 & ! [I4: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs4 ) )
% 3.17/1.74 => ( ( nth @ A @ Xs4 @ I4 )
% 3.17/1.74 = ( nth @ A @ Ys3 @ I4 ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_33_lessThan__eq__iff,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( linorder @ A )
% 3.17/1.74 => ! [X4: A,Y3: A] :
% 3.17/1.74 ( ( ( set_ord_lessThan @ A @ X4 )
% 3.17/1.74 = ( set_ord_lessThan @ A @ Y3 ) )
% 3.17/1.74 = ( X4 = Y3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_34_rev__is__rev__conv,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 3.17/1.74 ( ( ( rev @ A @ Xs )
% 3.17/1.74 = ( rev @ A @ Ys ) )
% 3.17/1.74 = ( Xs = Ys ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_35_rev__rev__ident,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A] :
% 3.17/1.74 ( ( rev @ A @ ( rev @ A @ Xs ) )
% 3.17/1.74 = Xs ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_36_len__ss1,axiom,
% 3.17/1.74 ( ( size_size @ ( list @ ( set @ a ) ) @ ss1 )
% 3.17/1.74 = ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_37_g,axiom,
% 3.17/1.74 ! [I: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) )
% 3.17/1.74 => ( ( nth @ ( set @ a ) @ ss2 @ I )
% 3.17/1.74 = ( nth @ ( set @ a ) @ ss1 @ ( g @ I ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_38_f,axiom,
% 3.17/1.74 ! [I: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) )
% 3.17/1.74 => ( ( nth @ ( set @ a ) @ ss1 @ I )
% 3.17/1.74 = ( nth @ ( set @ a ) @ ss2 @ ( f @ I ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_39__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062f_O_A_092_060lbrakk_062bij__betw_Af_A_123_O_O_060length_Ass2_125_A_123_O_O_060length_Ass1_125_059_A_092_060And_062i_O_Ai_A_060_Alength_Ass2_A_092_060Longrightarrow_062_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 3.17/1.74 ~ ! [F2: nat > nat] :
% 3.17/1.74 ( ( bij_betw @ nat @ nat @ F2 @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) )
% 3.17/1.74 => ~ ! [I2: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) )
% 3.17/1.74 => ( ( nth @ ( set @ a ) @ ss1 @ I2 )
% 3.17/1.74 = ( nth @ ( set @ a ) @ ss2 @ ( F2 @ I2 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_40__092_060open_062_092_060exists_062f_O_Abij__betw_Af_A_123_O_O_060length_Ass1_125_A_123_O_O_060length_Ass2_125_A_092_060and_062_A_I_092_060forall_062i_060length_Ass1_O_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_J_092_060close_062,axiom,
% 3.17/1.74 ? [F2: nat > nat] :
% 3.17/1.74 ( ( bij_betw @ nat @ nat @ F2 @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) )
% 3.17/1.74 & ! [I2: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) )
% 3.17/1.74 => ( ( nth @ ( set @ a ) @ ss1 @ I2 )
% 3.17/1.74 = ( nth @ ( set @ a ) @ ss2 @ ( F2 @ I2 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_41_f__g,axiom,
% 3.17/1.74 ! [I: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) )
% 3.17/1.74 => ( ( f @ ( g @ I ) )
% 3.17/1.74 = I ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_42_g__f,axiom,
% 3.17/1.74 ! [I: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) )
% 3.17/1.74 => ( ( g @ ( f @ I ) )
% 3.17/1.74 = I ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_43_listsetD_I2_J,axiom,
% 3.17/1.74 ! [A: $tType,Ys: list @ A,Xs: list @ ( set @ A ),I: nat] :
% 3.17/1.74 ( ( member @ ( list @ A ) @ Ys @ ( listset @ A @ Xs ) )
% 3.17/1.74 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ A ) ) @ Xs ) )
% 3.17/1.74 => ( member @ A @ ( nth @ A @ Ys @ I ) @ ( nth @ ( set @ A ) @ Xs @ I ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_44_neq__if__length__neq,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 3.17/1.74 ( ( ( size_size @ ( list @ A ) @ Xs )
% 3.17/1.74 != ( size_size @ ( list @ A ) @ Ys ) )
% 3.17/1.74 => ( Xs != Ys ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_45_mem__Collect__eq,axiom,
% 3.17/1.74 ! [A: $tType,A3: A,P: A > $o] :
% 3.17/1.74 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 3.17/1.74 = ( P @ A3 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_46_Collect__mem__eq,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A] :
% 3.17/1.74 ( ( collect @ A
% 3.17/1.74 @ ^ [X5: A] : ( member @ A @ X5 @ A2 ) )
% 3.17/1.74 = A2 ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_47_Collect__cong,axiom,
% 3.17/1.74 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.17/1.74 ( ! [X6: A] :
% 3.17/1.74 ( ( P @ X6 )
% 3.17/1.74 = ( Q @ X6 ) )
% 3.17/1.74 => ( ( collect @ A @ P )
% 3.17/1.74 = ( collect @ A @ Q ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_48_ext,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.17/1.74 ( ! [X6: A] :
% 3.17/1.74 ( ( F @ X6 )
% 3.17/1.74 = ( G @ X6 ) )
% 3.17/1.74 => ( F = G ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_49_Ex__list__of__length,axiom,
% 3.17/1.74 ! [A: $tType,N: nat] :
% 3.17/1.74 ? [Xs2: list @ A] :
% 3.17/1.74 ( ( size_size @ ( list @ A ) @ Xs2 )
% 3.17/1.74 = N ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_50_rev__swap,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 3.17/1.74 ( ( ( rev @ A @ Xs )
% 3.17/1.74 = Ys )
% 3.17/1.74 = ( Xs
% 3.17/1.74 = ( rev @ A @ Ys ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_51_lessThan__strict__subset__iff,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( linorder @ A )
% 3.17/1.74 => ! [M: A,N: A] :
% 3.17/1.74 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
% 3.17/1.74 = ( ord_less @ A @ M @ N ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_52_length__induct,axiom,
% 3.17/1.74 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 3.17/1.74 ( ! [Xs2: list @ A] :
% 3.17/1.74 ( ! [Ys4: list @ A] :
% 3.17/1.74 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys4 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 3.17/1.74 => ( P @ Ys4 ) )
% 3.17/1.74 => ( P @ Xs2 ) )
% 3.17/1.74 => ( P @ Xs ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_53_listsetD_I1_J,axiom,
% 3.17/1.74 ! [A: $tType,Ys: list @ A,Xs: list @ ( set @ A )] :
% 3.17/1.74 ( ( member @ ( list @ A ) @ Ys @ ( listset @ A @ Xs ) )
% 3.17/1.74 => ( ( size_size @ ( list @ A ) @ Ys )
% 3.17/1.74 = ( size_size @ ( list @ ( set @ A ) ) @ Xs ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_54_permutation__Ex__bij,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 3.17/1.74 ( ( perm @ A @ Xs @ Ys )
% 3.17/1.74 => ? [F2: nat > nat] :
% 3.17/1.74 ( ( bij_betw @ nat @ nat @ F2 @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Ys ) ) )
% 3.17/1.74 & ! [I2: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 3.17/1.74 => ( ( nth @ A @ Xs @ I2 )
% 3.17/1.74 = ( nth @ A @ Ys @ ( F2 @ I2 ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_55_g__def,axiom,
% 3.17/1.74 ( g
% 3.17/1.74 = ( hilbert_inv_into @ nat @ nat @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) @ f ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_56_perm__refl,axiom,
% 3.17/1.74 ! [A: $tType,L: list @ A] : ( perm @ A @ L @ L ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_57_image__eqI,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,B2: A,F: B > A,X4: B,A2: set @ B] :
% 3.17/1.74 ( ( B2
% 3.17/1.74 = ( F @ X4 ) )
% 3.17/1.74 => ( ( member @ B @ X4 @ A2 )
% 3.17/1.74 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_58_mset__eq__perm,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 3.17/1.74 ( ( ( mset @ A @ Xs )
% 3.17/1.74 = ( mset @ A @ Ys ) )
% 3.17/1.74 = ( perm @ A @ Xs @ Ys ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_59_perm__rev,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A] : ( perm @ A @ ( rev @ A @ Xs ) @ Xs ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_60_perm__length,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 3.17/1.74 ( ( perm @ A @ Xs @ Ys )
% 3.17/1.74 => ( ( size_size @ ( list @ A ) @ Xs )
% 3.17/1.74 = ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_61_bij__betw__imp__surj__on,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B] :
% 3.17/1.74 ( ( bij_betw @ A @ B @ F @ A2 @ B3 )
% 3.17/1.74 => ( ( image @ A @ B @ F @ A2 )
% 3.17/1.74 = B3 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_62_mbs_Oless__not__eq,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( size @ A )
% 3.17/1.74 => ! [X4: A,A2: set @ A,Y3: A] :
% 3.17/1.74 ( ( member @ A @ X4 @ A2 )
% 3.17/1.74 => ( ( ord_less @ nat @ ( size_size @ A @ X4 ) @ ( size_size @ A @ Y3 ) )
% 3.17/1.74 => ( X4 != Y3 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_63_size__mset,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A] :
% 3.17/1.74 ( ( size_size @ ( multiset @ A ) @ ( mset @ A @ Xs ) )
% 3.17/1.74 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_64_psubsetD,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A,C: A] :
% 3.17/1.74 ( ( ord_less @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ( member @ A @ C @ A2 )
% 3.17/1.74 => ( member @ A @ C @ B3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_65_psubset__trans,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A,C2: set @ A] :
% 3.17/1.74 ( ( ord_less @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ( ord_less @ ( set @ A ) @ B3 @ C2 )
% 3.17/1.74 => ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_66_imageI,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,X4: A,A2: set @ A,F: A > B] :
% 3.17/1.74 ( ( member @ A @ X4 @ A2 )
% 3.17/1.74 => ( member @ B @ ( F @ X4 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_67_image__iff,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,Z2: A,F: B > A,A2: set @ B] :
% 3.17/1.74 ( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A2 ) )
% 3.17/1.74 = ( ? [X5: B] :
% 3.17/1.74 ( ( member @ B @ X5 @ A2 )
% 3.17/1.74 & ( Z2
% 3.17/1.74 = ( F @ X5 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_68_bex__imageD,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
% 3.17/1.74 ( ? [X: A] :
% 3.17/1.74 ( ( member @ A @ X @ ( image @ B @ A @ F @ A2 ) )
% 3.17/1.74 & ( P @ X ) )
% 3.17/1.74 => ? [X6: B] :
% 3.17/1.74 ( ( member @ B @ X6 @ A2 )
% 3.17/1.74 & ( P @ ( F @ X6 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_69_image__cong,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,M2: set @ A,N2: set @ A,F: A > B,G: A > B] :
% 3.17/1.74 ( ( M2 = N2 )
% 3.17/1.74 => ( ! [X6: A] :
% 3.17/1.74 ( ( member @ A @ X6 @ N2 )
% 3.17/1.74 => ( ( F @ X6 )
% 3.17/1.74 = ( G @ X6 ) ) )
% 3.17/1.74 => ( ( image @ A @ B @ F @ M2 )
% 3.17/1.74 = ( image @ A @ B @ G @ N2 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_70_ball__imageD,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
% 3.17/1.74 ( ! [X6: A] :
% 3.17/1.74 ( ( member @ A @ X6 @ ( image @ B @ A @ F @ A2 ) )
% 3.17/1.74 => ( P @ X6 ) )
% 3.17/1.74 => ! [X: B] :
% 3.17/1.74 ( ( member @ B @ X @ A2 )
% 3.17/1.74 => ( P @ ( F @ X ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_71_rev__image__eqI,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,X4: A,A2: set @ A,B2: B,F: A > B] :
% 3.17/1.74 ( ( member @ A @ X4 @ A2 )
% 3.17/1.74 => ( ( B2
% 3.17/1.74 = ( F @ X4 ) )
% 3.17/1.74 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_72_direct__decomp__def,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( comm_monoid_add @ A )
% 3.17/1.74 => ( ( hilber580762119decomp @ A )
% 3.17/1.74 = ( ^ [A5: set @ A,Ss2: list @ ( set @ A )] : ( bij_betw @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss2 ) @ A5 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_73_bij__betwE,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B] :
% 3.17/1.74 ( ( bij_betw @ A @ B @ F @ A2 @ B3 )
% 3.17/1.74 => ! [X: A] :
% 3.17/1.74 ( ( member @ A @ X @ A2 )
% 3.17/1.74 => ( member @ B @ ( F @ X ) @ B3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_74_bij__betw__inv,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B3: set @ B] :
% 3.17/1.74 ( ( bij_betw @ A @ B @ F @ A2 @ B3 )
% 3.17/1.74 => ? [G2: B > A] : ( bij_betw @ B @ A @ G2 @ B3 @ A2 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_75_bij__betw__cong,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,G: A > B,A6: set @ B] :
% 3.17/1.74 ( ! [A4: A] :
% 3.17/1.74 ( ( member @ A @ A4 @ A2 )
% 3.17/1.74 => ( ( F @ A4 )
% 3.17/1.74 = ( G @ A4 ) ) )
% 3.17/1.74 => ( ( bij_betw @ A @ B @ F @ A2 @ A6 )
% 3.17/1.74 = ( bij_betw @ A @ B @ G @ A2 @ A6 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_76_bij__betw__apply,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B,A3: A] :
% 3.17/1.74 ( ( bij_betw @ A @ B @ F @ A2 @ B3 )
% 3.17/1.74 => ( ( member @ A @ A3 @ A2 )
% 3.17/1.74 => ( member @ B @ ( F @ A3 ) @ B3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_77_bij__betw__iff__bijections,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType] :
% 3.17/1.74 ( ( bij_betw @ A @ B )
% 3.17/1.74 = ( ^ [F3: A > B,A5: set @ A,B4: set @ B] :
% 3.17/1.74 ? [G3: B > A] :
% 3.17/1.74 ( ! [X5: A] :
% 3.17/1.74 ( ( member @ A @ X5 @ A5 )
% 3.17/1.74 => ( ( member @ B @ ( F3 @ X5 ) @ B4 )
% 3.17/1.74 & ( ( G3 @ ( F3 @ X5 ) )
% 3.17/1.74 = X5 ) ) )
% 3.17/1.74 & ! [X5: B] :
% 3.17/1.74 ( ( member @ B @ X5 @ B4 )
% 3.17/1.74 => ( ( member @ A @ ( G3 @ X5 ) @ A5 )
% 3.17/1.74 & ( ( F3 @ ( G3 @ X5 ) )
% 3.17/1.74 = X5 ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_78_perm__sym,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 3.17/1.74 ( ( perm @ A @ Xs @ Ys )
% 3.17/1.74 => ( perm @ A @ Ys @ Xs ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_79_perm_Otrans,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
% 3.17/1.74 ( ( perm @ A @ Xs @ Ys )
% 3.17/1.74 => ( ( perm @ A @ Ys @ Zs )
% 3.17/1.74 => ( perm @ A @ Xs @ Zs ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_80_bij__betw__inv__into__right,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,A6: set @ B,A7: B] :
% 3.17/1.74 ( ( bij_betw @ A @ B @ F @ A2 @ A6 )
% 3.17/1.74 => ( ( member @ B @ A7 @ A6 )
% 3.17/1.74 => ( ( F @ ( hilbert_inv_into @ A @ B @ A2 @ F @ A7 ) )
% 3.17/1.74 = A7 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_81_bij__betw__inv__into__left,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,A6: set @ B,A3: A] :
% 3.17/1.74 ( ( bij_betw @ A @ B @ F @ A2 @ A6 )
% 3.17/1.74 => ( ( member @ A @ A3 @ A2 )
% 3.17/1.74 => ( ( hilbert_inv_into @ A @ B @ A2 @ F @ ( F @ A3 ) )
% 3.17/1.74 = A3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_82_inv__into__inv__into__eq,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,A6: set @ B,A3: A] :
% 3.17/1.74 ( ( bij_betw @ A @ B @ F @ A2 @ A6 )
% 3.17/1.74 => ( ( member @ A @ A3 @ A2 )
% 3.17/1.74 => ( ( hilbert_inv_into @ B @ A @ A6 @ ( hilbert_inv_into @ A @ B @ A2 @ F ) @ A3 )
% 3.17/1.74 = ( F @ A3 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_83_bij__betw__inv__into,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B3: set @ B] :
% 3.17/1.74 ( ( bij_betw @ A @ B @ F @ A2 @ B3 )
% 3.17/1.74 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A2 @ F ) @ B3 @ A2 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_84_inv__into__injective,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,X4: B,Y3: B] :
% 3.17/1.74 ( ( ( hilbert_inv_into @ A @ B @ A2 @ F @ X4 )
% 3.17/1.74 = ( hilbert_inv_into @ A @ B @ A2 @ F @ Y3 ) )
% 3.17/1.74 => ( ( member @ B @ X4 @ ( image @ A @ B @ F @ A2 ) )
% 3.17/1.74 => ( ( member @ B @ Y3 @ ( image @ A @ B @ F @ A2 ) )
% 3.17/1.74 => ( X4 = Y3 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_85_inv__into__into,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,X4: A,F: B > A,A2: set @ B] :
% 3.17/1.74 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A2 ) )
% 3.17/1.74 => ( member @ B @ ( hilbert_inv_into @ B @ A @ A2 @ F @ X4 ) @ A2 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_86_f__inv__into__f,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,Y3: A,F: B > A,A2: set @ B] :
% 3.17/1.74 ( ( member @ A @ Y3 @ ( image @ B @ A @ F @ A2 ) )
% 3.17/1.74 => ( ( F @ ( hilbert_inv_into @ B @ A @ A2 @ F @ Y3 ) )
% 3.17/1.74 = Y3 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_87_list__ex__length,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( list_ex @ A )
% 3.17/1.74 = ( ^ [P2: A > $o,Xs4: list @ A] :
% 3.17/1.74 ? [N3: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs4 ) )
% 3.17/1.74 & ( P2 @ ( nth @ A @ Xs4 @ N3 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_88_direct__decompI,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( comm_monoid_add @ A )
% 3.17/1.74 => ! [Ss: list @ ( set @ A ),A2: set @ A] :
% 3.17/1.74 ( ( inj_on @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss ) )
% 3.17/1.74 => ( ( ( image @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss ) )
% 3.17/1.74 = A2 )
% 3.17/1.74 => ( hilber580762119decomp @ A @ A2 @ Ss ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_89_elem__le__sum__list,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( canoni770627133id_add @ A )
% 3.17/1.74 => ! [K: nat,Ns: list @ A] :
% 3.17/1.74 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Ns ) )
% 3.17/1.74 => ( ord_less_eq @ A @ ( nth @ A @ Ns @ K ) @ ( groups501227526m_list @ A @ Ns ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_90_perm__swap,axiom,
% 3.17/1.74 ! [A: $tType,I: nat,Xs: list @ A,J: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 3.17/1.74 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 3.17/1.74 => ( perm @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) @ Xs ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_91_mset__swap,axiom,
% 3.17/1.74 ! [A: $tType,I: nat,Ls: list @ A,J: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ls ) )
% 3.17/1.74 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Ls ) )
% 3.17/1.74 => ( ( mset @ A @ ( list_update @ A @ ( list_update @ A @ Ls @ J @ ( nth @ A @ Ls @ I ) ) @ I @ ( nth @ A @ Ls @ J ) ) )
% 3.17/1.74 = ( mset @ A @ Ls ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_92_nth__mem__mset,axiom,
% 3.17/1.74 ! [A: $tType,I: nat,Ls: list @ A] :
% 3.17/1.74 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ls ) )
% 3.17/1.74 => ( member @ A @ ( nth @ A @ Ls @ I ) @ ( set_mset @ A @ ( mset @ A @ Ls ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_93_in__Union__mset__iff,axiom,
% 3.17/1.74 ! [A: $tType,X4: A,MM: multiset @ ( multiset @ A )] :
% 3.17/1.74 ( ( member @ A @ X4 @ ( set_mset @ A @ ( comm_m1079429914m_mset @ ( multiset @ A ) @ MM ) ) )
% 3.17/1.74 = ( ? [M3: multiset @ A] :
% 3.17/1.74 ( ( member @ ( multiset @ A ) @ M3 @ ( set_mset @ ( multiset @ A ) @ MM ) )
% 3.17/1.74 & ( member @ A @ X4 @ ( set_mset @ A @ M3 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_94_psubsetI,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ( A2 != B3 )
% 3.17/1.74 => ( ord_less @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_95_inj__on__rev,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ A2 ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_96_list__update__overwrite,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,I: nat,X4: A,Y3: A] :
% 3.17/1.74 ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ I @ Y3 )
% 3.17/1.74 = ( list_update @ A @ Xs @ I @ Y3 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_97_lessThan__subset__iff,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( linorder @ A )
% 3.17/1.74 => ! [X4: A,Y3: A] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X4 ) @ ( set_ord_lessThan @ A @ Y3 ) )
% 3.17/1.74 = ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_98_list__update__beyond,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,I: nat,X4: A] :
% 3.17/1.74 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
% 3.17/1.74 => ( ( list_update @ A @ Xs @ I @ X4 )
% 3.17/1.74 = Xs ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_99_length__list__update,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,I: nat,X4: A] :
% 3.17/1.74 ( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I @ X4 ) )
% 3.17/1.74 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_100_inv__into__f__f,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ A2 )
% 3.17/1.74 => ( ( member @ A @ X4 @ A2 )
% 3.17/1.74 => ( ( hilbert_inv_into @ A @ B @ A2 @ F @ ( F @ X4 ) )
% 3.17/1.74 = X4 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_101_list__update__id,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ A,I: nat] :
% 3.17/1.74 ( ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ I ) )
% 3.17/1.74 = Xs ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_102_nth__list__update__neq,axiom,
% 3.17/1.74 ! [A: $tType,I: nat,J: nat,Xs: list @ A,X4: A] :
% 3.17/1.74 ( ( I != J )
% 3.17/1.74 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ J )
% 3.17/1.74 = ( nth @ A @ Xs @ J ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_103_list__ex__rev,axiom,
% 3.17/1.74 ! [A: $tType,P: A > $o,Xs: list @ A] :
% 3.17/1.74 ( ( list_ex @ A @ P @ ( rev @ A @ Xs ) )
% 3.17/1.74 = ( list_ex @ A @ P @ Xs ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_104_inv__into__image__cancel,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,S: set @ A] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ A2 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ S @ A2 )
% 3.17/1.74 => ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ S ) )
% 3.17/1.74 = S ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_105_nth__list__update__eq,axiom,
% 3.17/1.74 ! [A: $tType,I: nat,Xs: list @ A,X4: A] :
% 3.17/1.74 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 3.17/1.74 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ I )
% 3.17/1.74 = X4 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_106_subset__image__inj,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,S: set @ A,F: B > A,T: set @ B] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ S @ ( image @ B @ A @ F @ T ) )
% 3.17/1.74 = ( ? [U: set @ B] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ B ) @ U @ T )
% 3.17/1.74 & ( inj_on @ B @ A @ F @ U )
% 3.17/1.74 & ( S
% 3.17/1.74 = ( image @ B @ A @ F @ U ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_107_inj__on__inv__into,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,B3: set @ A,F: B > A,A2: set @ B] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F @ A2 ) )
% 3.17/1.74 => ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ A2 @ F ) @ B3 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_108_list__update__swap,axiom,
% 3.17/1.74 ! [A: $tType,I: nat,I5: nat,Xs: list @ A,X4: A,X7: A] :
% 3.17/1.74 ( ( I != I5 )
% 3.17/1.74 => ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ I5 @ X7 )
% 3.17/1.74 = ( list_update @ A @ ( list_update @ A @ Xs @ I5 @ X7 ) @ I @ X4 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_109_inj__on__image__eq__iff,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,C2: set @ A,A2: set @ A,B3: set @ A] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ C2 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C2 )
% 3.17/1.74 => ( ( ( image @ A @ B @ F @ A2 )
% 3.17/1.74 = ( image @ A @ B @ F @ B3 ) )
% 3.17/1.74 = ( A2 = B3 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_110_inj__on__image__mem__iff,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,B3: set @ A,A3: A,A2: set @ A] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ B3 )
% 3.17/1.74 => ( ( member @ A @ A3 @ B3 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) )
% 3.17/1.74 = ( member @ A @ A3 @ A2 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_111_inj__on__inverseI,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,A2: set @ A,G: B > A,F: A > B] :
% 3.17/1.74 ( ! [X6: A] :
% 3.17/1.74 ( ( member @ A @ X6 @ A2 )
% 3.17/1.74 => ( ( G @ ( F @ X6 ) )
% 3.17/1.74 = X6 ) )
% 3.17/1.74 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_112_inj__on__contraD,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A,Y3: A] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ A2 )
% 3.17/1.74 => ( ( X4 != Y3 )
% 3.17/1.74 => ( ( member @ A @ X4 @ A2 )
% 3.17/1.74 => ( ( member @ A @ Y3 @ A2 )
% 3.17/1.74 => ( ( F @ X4 )
% 3.17/1.74 != ( F @ Y3 ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_113_subset__inj__on,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,B3: set @ A,A2: set @ A] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ B3 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_114_inj__on__subset,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B3: set @ A] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ A2 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 3.17/1.74 => ( inj_on @ A @ B @ F @ B3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_115_inj__on__eq__iff,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A,Y3: A] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ A2 )
% 3.17/1.74 => ( ( member @ A @ X4 @ A2 )
% 3.17/1.74 => ( ( member @ A @ Y3 @ A2 )
% 3.17/1.74 => ( ( ( F @ X4 )
% 3.17/1.74 = ( F @ Y3 ) )
% 3.17/1.74 = ( X4 = Y3 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_116_inj__on__cong,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,G: A > B] :
% 3.17/1.74 ( ! [A4: A] :
% 3.17/1.74 ( ( member @ A @ A4 @ A2 )
% 3.17/1.74 => ( ( F @ A4 )
% 3.17/1.74 = ( G @ A4 ) ) )
% 3.17/1.74 => ( ( inj_on @ A @ B @ F @ A2 )
% 3.17/1.74 = ( inj_on @ A @ B @ G @ A2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_117_inj__on__def,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType] :
% 3.17/1.74 ( ( inj_on @ A @ B )
% 3.17/1.74 = ( ^ [F3: A > B,A5: set @ A] :
% 3.17/1.74 ! [X5: A] :
% 3.17/1.74 ( ( member @ A @ X5 @ A5 )
% 3.17/1.74 => ! [Y4: A] :
% 3.17/1.74 ( ( member @ A @ Y4 @ A5 )
% 3.17/1.74 => ( ( ( F3 @ X5 )
% 3.17/1.74 = ( F3 @ Y4 ) )
% 3.17/1.74 => ( X5 = Y4 ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_118_inj__onI,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,A2: set @ A,F: A > B] :
% 3.17/1.74 ( ! [X6: A,Y: A] :
% 3.17/1.74 ( ( member @ A @ X6 @ A2 )
% 3.17/1.74 => ( ( member @ A @ Y @ A2 )
% 3.17/1.74 => ( ( ( F @ X6 )
% 3.17/1.74 = ( F @ Y ) )
% 3.17/1.74 => ( X6 = Y ) ) ) )
% 3.17/1.74 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_119_inj__onD,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A,Y3: A] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ A2 )
% 3.17/1.74 => ( ( ( F @ X4 )
% 3.17/1.74 = ( F @ Y3 ) )
% 3.17/1.74 => ( ( member @ A @ X4 @ A2 )
% 3.17/1.74 => ( ( member @ A @ Y3 @ A2 )
% 3.17/1.74 => ( X4 = Y3 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_120_linorder__inj__onI,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ! [A2: set @ A,F: A > B] :
% 3.17/1.74 ( ! [X6: A,Y: A] :
% 3.17/1.74 ( ( ord_less @ A @ X6 @ Y )
% 3.17/1.74 => ( ( member @ A @ X6 @ A2 )
% 3.17/1.74 => ( ( member @ A @ Y @ A2 )
% 3.17/1.74 => ( ( F @ X6 )
% 3.17/1.74 != ( F @ Y ) ) ) ) )
% 3.17/1.74 => ( ! [X6: A,Y: A] :
% 3.17/1.74 ( ( member @ A @ X6 @ A2 )
% 3.17/1.74 => ( ( member @ A @ Y @ A2 )
% 3.17/1.74 => ( ( ord_less_eq @ A @ X6 @ Y )
% 3.17/1.74 | ( ord_less_eq @ A @ Y @ X6 ) ) ) )
% 3.17/1.74 => ( inj_on @ A @ B @ F @ A2 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_121_inj__on__image__iff,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,A2: set @ A,G: A > B,F: A > A] :
% 3.17/1.74 ( ! [X6: A] :
% 3.17/1.74 ( ( member @ A @ X6 @ A2 )
% 3.17/1.74 => ! [Xa: A] :
% 3.17/1.74 ( ( member @ A @ Xa @ A2 )
% 3.17/1.74 => ( ( ( G @ ( F @ X6 ) )
% 3.17/1.74 = ( G @ ( F @ Xa ) ) )
% 3.17/1.74 = ( ( G @ X6 )
% 3.17/1.74 = ( G @ Xa ) ) ) ) )
% 3.17/1.74 => ( ( inj_on @ A @ A @ F @ A2 )
% 3.17/1.74 => ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A2 ) )
% 3.17/1.74 = ( inj_on @ A @ B @ G @ A2 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_122_bij__betw__imp__inj__on,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B3: set @ B] :
% 3.17/1.74 ( ( bij_betw @ A @ B @ F @ A2 @ B3 )
% 3.17/1.74 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_123_inv__into__f__eq,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A,Y3: B] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ A2 )
% 3.17/1.74 => ( ( member @ A @ X4 @ A2 )
% 3.17/1.74 => ( ( ( F @ X4 )
% 3.17/1.74 = Y3 )
% 3.17/1.74 => ( ( hilbert_inv_into @ A @ B @ A2 @ F @ Y3 )
% 3.17/1.74 = X4 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_124_subset__image__iff,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,B3: set @ A,F: B > A,A2: set @ B] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F @ A2 ) )
% 3.17/1.74 = ( ? [AA: set @ B] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
% 3.17/1.74 & ( B3
% 3.17/1.74 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_125_image__subset__iff,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B3: set @ A] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B3 )
% 3.17/1.74 = ( ! [X5: B] :
% 3.17/1.74 ( ( member @ B @ X5 @ A2 )
% 3.17/1.74 => ( member @ A @ ( F @ X5 ) @ B3 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_126_subset__imageE,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,B3: set @ A,F: B > A,A2: set @ B] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F @ A2 ) )
% 3.17/1.74 => ~ ! [C3: set @ B] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ B ) @ C3 @ A2 )
% 3.17/1.74 => ( B3
% 3.17/1.74 != ( image @ B @ A @ F @ C3 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_127_image__subsetI,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,B3: set @ B] :
% 3.17/1.74 ( ! [X6: A] :
% 3.17/1.74 ( ( member @ A @ X6 @ A2 )
% 3.17/1.74 => ( member @ B @ ( F @ X6 ) @ B3 ) )
% 3.17/1.74 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B3 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_128_image__mono,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,A2: set @ A,B3: set @ A,F: A > B] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_129_psubsetE,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.17/1.74 ( ( ord_less @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_130_psubset__eq,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( ord_less @ ( set @ A ) )
% 3.17/1.74 = ( ^ [A5: set @ A,B4: set @ A] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
% 3.17/1.74 & ( A5 != B4 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_131_psubset__imp__subset,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.17/1.74 ( ( ord_less @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_132_psubset__subset__trans,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A,C2: set @ A] :
% 3.17/1.74 ( ( ord_less @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C2 )
% 3.17/1.74 => ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_133_subset__not__subset__eq,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( ord_less @ ( set @ A ) )
% 3.17/1.74 = ( ^ [A5: set @ A,B4: set @ A] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
% 3.17/1.74 & ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_134_subset__psubset__trans,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A,C2: set @ A] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ( ord_less @ ( set @ A ) @ B3 @ C2 )
% 3.17/1.74 => ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_135_subset__iff__psubset__eq,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) )
% 3.17/1.74 = ( ^ [A5: set @ A,B4: set @ A] :
% 3.17/1.74 ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
% 3.17/1.74 | ( A5 = B4 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_136_inj__on__strict__subset,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,B3: set @ A,A2: set @ A] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ B3 )
% 3.17/1.74 => ( ( ord_less @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B3 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_137_bij__betw__def,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType] :
% 3.17/1.74 ( ( bij_betw @ A @ B )
% 3.17/1.74 = ( ^ [F3: A > B,A5: set @ A,B4: set @ B] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F3 @ A5 )
% 3.17/1.74 & ( ( image @ A @ B @ F3 @ A5 )
% 3.17/1.74 = B4 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_138_bij__betw__imageI,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ A2 )
% 3.17/1.74 => ( ( ( image @ A @ B @ F @ A2 )
% 3.17/1.74 = B3 )
% 3.17/1.74 => ( bij_betw @ A @ B @ F @ A2 @ B3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_139_inj__on__imp__bij__betw,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ A2 )
% 3.17/1.74 => ( bij_betw @ A @ B @ F @ A2 @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_140_mbs_Ominimal,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( size @ A )
% 3.17/1.74 => ! [X4: A,A2: set @ A,P: A > $o] :
% 3.17/1.74 ( ( member @ A @ X4 @ A2 )
% 3.17/1.74 => ( ( P @ X4 )
% 3.17/1.74 => ? [X6: A] :
% 3.17/1.74 ( ( member @ A @ X6 @ A2 )
% 3.17/1.74 & ( ord_less_eq @ nat @ ( size_size @ A @ X6 ) @ ( size_size @ A @ X4 ) )
% 3.17/1.74 & ( P @ X6 )
% 3.17/1.74 & ! [Xa2: A] :
% 3.17/1.74 ( ( member @ A @ Xa2 @ A2 )
% 3.17/1.74 => ( ( ord_less @ nat @ ( size_size @ A @ Xa2 ) @ ( size_size @ A @ X6 ) )
% 3.17/1.74 => ~ ( P @ Xa2 ) ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_141_bij__betw__subset,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,A6: set @ B,B3: set @ A,B5: set @ B] :
% 3.17/1.74 ( ( bij_betw @ A @ B @ F @ A2 @ A6 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 3.17/1.74 => ( ( ( image @ A @ B @ F @ B3 )
% 3.17/1.74 = B5 )
% 3.17/1.74 => ( bij_betw @ A @ B @ F @ B3 @ B5 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_142_bij__betw__byWitness,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,A2: set @ A,F4: B > A,F: A > B,A6: set @ B] :
% 3.17/1.74 ( ! [X6: A] :
% 3.17/1.74 ( ( member @ A @ X6 @ A2 )
% 3.17/1.74 => ( ( F4 @ ( F @ X6 ) )
% 3.17/1.74 = X6 ) )
% 3.17/1.74 => ( ! [X6: B] :
% 3.17/1.74 ( ( member @ B @ X6 @ A6 )
% 3.17/1.74 => ( ( F @ ( F4 @ X6 ) )
% 3.17/1.74 = X6 ) )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ A6 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ A2 )
% 3.17/1.74 => ( bij_betw @ A @ B @ F @ A2 @ A6 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_143_image__inv__into__cancel,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,A6: set @ A,B5: set @ A] :
% 3.17/1.74 ( ( ( image @ B @ A @ F @ A2 )
% 3.17/1.74 = A6 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
% 3.17/1.74 => ( ( image @ B @ A @ F @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ A2 @ F ) @ B5 ) )
% 3.17/1.74 = B5 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_144_listset__mono,axiom,
% 3.17/1.74 ! [A: $tType,Xs: list @ ( set @ A ),Ys: list @ ( set @ A )] :
% 3.17/1.74 ( ( ( size_size @ ( list @ ( set @ A ) ) @ Xs )
% 3.17/1.74 = ( size_size @ ( list @ ( set @ A ) ) @ Ys ) )
% 3.17/1.74 => ( ! [I3: nat] :
% 3.17/1.74 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( set @ A ) ) @ Ys ) )
% 3.17/1.74 => ( ord_less_eq @ ( set @ A ) @ ( nth @ ( set @ A ) @ Xs @ I3 ) @ ( nth @ ( set @ A ) @ Ys @ I3 ) ) )
% 3.17/1.74 => ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( listset @ A @ Xs ) @ ( listset @ A @ Ys ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_145_nth__list__update,axiom,
% 3.17/1.74 ! [A: $tType,I: nat,Xs: list @ A,J: nat,X4: A] :
% 3.17/1.74 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 3.17/1.74 => ( ( ( I = J )
% 3.17/1.74 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ J )
% 3.17/1.74 = X4 ) )
% 3.17/1.74 & ( ( I != J )
% 3.17/1.74 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ J )
% 3.17/1.74 = ( nth @ A @ Xs @ J ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_146_list__update__same__conv,axiom,
% 3.17/1.74 ! [A: $tType,I: nat,Xs: list @ A,X4: A] :
% 3.17/1.74 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 3.17/1.74 => ( ( ( list_update @ A @ Xs @ I @ X4 )
% 3.17/1.74 = Xs )
% 3.17/1.74 = ( ( nth @ A @ Xs @ I )
% 3.17/1.74 = X4 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_147_bij__betw__inv__into__subset,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,A6: set @ B,B3: set @ A,B5: set @ B] :
% 3.17/1.74 ( ( bij_betw @ A @ B @ F @ A2 @ A6 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 3.17/1.74 => ( ( ( image @ A @ B @ F @ B3 )
% 3.17/1.74 = B5 )
% 3.17/1.74 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A2 @ F ) @ B5 @ B3 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_148_direct__decompD_I2_J,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( comm_monoid_add @ A )
% 3.17/1.74 => ! [A2: set @ A,Ss: list @ ( set @ A )] :
% 3.17/1.74 ( ( hilber580762119decomp @ A @ A2 @ Ss )
% 3.17/1.74 => ( inj_on @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_149_Schroeder__Bernstein,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B,G: B > A] :
% 3.17/1.74 ( ( inj_on @ A @ B @ F @ A2 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B3 )
% 3.17/1.74 => ( ( inj_on @ B @ A @ G @ B3 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B3 ) @ A2 )
% 3.17/1.74 => ? [H: A > B] : ( bij_betw @ A @ B @ H @ A2 @ B3 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_150_order__refl,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( preorder @ A )
% 3.17/1.74 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_151_subset__imageE__inj,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType,B3: set @ A,F: B > A,A2: set @ B] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F @ A2 ) )
% 3.17/1.74 => ~ ! [C3: set @ B] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ B ) @ C3 @ A2 )
% 3.17/1.74 => ( ( B3
% 3.17/1.74 = ( image @ B @ A @ F @ C3 ) )
% 3.17/1.74 => ~ ( inj_on @ B @ A @ F @ C3 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_152_minf_I8_J,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( linorder @ A )
% 3.17/1.74 => ! [T2: A] :
% 3.17/1.74 ? [Z3: A] :
% 3.17/1.74 ! [X: A] :
% 3.17/1.74 ( ( ord_less @ A @ X @ Z3 )
% 3.17/1.74 => ~ ( ord_less_eq @ A @ T2 @ X ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_153_minf_I6_J,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( linorder @ A )
% 3.17/1.74 => ! [T2: A] :
% 3.17/1.74 ? [Z3: A] :
% 3.17/1.74 ! [X: A] :
% 3.17/1.74 ( ( ord_less @ A @ X @ Z3 )
% 3.17/1.74 => ( ord_less_eq @ A @ X @ T2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_154_subset__antisym,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 3.17/1.74 => ( A2 = B3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_155_subsetI,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.17/1.74 ( ! [X6: A] :
% 3.17/1.74 ( ( member @ A @ X6 @ A2 )
% 3.17/1.74 => ( member @ A @ X6 @ B3 ) )
% 3.17/1.74 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_156_bounded__Max__nat,axiom,
% 3.17/1.74 ! [P: nat > $o,X4: nat,M2: nat] :
% 3.17/1.74 ( ( P @ X4 )
% 3.17/1.74 => ( ! [X6: nat] :
% 3.17/1.74 ( ( P @ X6 )
% 3.17/1.74 => ( ord_less_eq @ nat @ X6 @ M2 ) )
% 3.17/1.74 => ~ ! [M4: nat] :
% 3.17/1.74 ( ( P @ M4 )
% 3.17/1.74 => ~ ! [X: nat] :
% 3.17/1.74 ( ( P @ X )
% 3.17/1.74 => ( ord_less_eq @ nat @ X @ M4 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_157_Collect__mono__iff,axiom,
% 3.17/1.74 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 3.17/1.74 = ( ! [X5: A] :
% 3.17/1.74 ( ( P @ X5 )
% 3.17/1.74 => ( Q @ X5 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_158_set__eq__subset,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( ^ [Y2: set @ A,Z: set @ A] : ( Y2 = Z ) )
% 3.17/1.74 = ( ^ [A5: set @ A,B4: set @ A] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
% 3.17/1.74 & ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_159_subset__trans,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A,C2: set @ A] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C2 )
% 3.17/1.74 => ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_160_Collect__mono,axiom,
% 3.17/1.74 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.17/1.74 ( ! [X6: A] :
% 3.17/1.74 ( ( P @ X6 )
% 3.17/1.74 => ( Q @ X6 ) )
% 3.17/1.74 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_161_subset__refl,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_162_subset__iff,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) )
% 3.17/1.74 = ( ^ [A5: set @ A,B4: set @ A] :
% 3.17/1.74 ! [T3: A] :
% 3.17/1.74 ( ( member @ A @ T3 @ A5 )
% 3.17/1.74 => ( member @ A @ T3 @ B4 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_163_equalityD2,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.17/1.74 ( ( A2 = B3 )
% 3.17/1.74 => ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_164_equalityD1,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.17/1.74 ( ( A2 = B3 )
% 3.17/1.74 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_165_subset__eq,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) )
% 3.17/1.74 = ( ^ [A5: set @ A,B4: set @ A] :
% 3.17/1.74 ! [X5: A] :
% 3.17/1.74 ( ( member @ A @ X5 @ A5 )
% 3.17/1.74 => ( member @ A @ X5 @ B4 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_166_equalityE,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.17/1.74 ( ( A2 = B3 )
% 3.17/1.74 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_167_subsetD,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A,C: A] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ( member @ A @ C @ A2 )
% 3.17/1.74 => ( member @ A @ C @ B3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_168_in__mono,axiom,
% 3.17/1.74 ! [A: $tType,A2: set @ A,B3: set @ A,X4: A] :
% 3.17/1.74 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.17/1.74 => ( ( member @ A @ X4 @ A2 )
% 3.17/1.74 => ( member @ A @ X4 @ B3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_169_dual__order_Oantisym,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ! [B2: A,A3: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ B2 @ A3 )
% 3.17/1.74 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.17/1.74 => ( A3 = B2 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_170_dual__order_Oeq__iff,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 3.17/1.74 = ( ^ [A8: A,B6: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ B6 @ A8 )
% 3.17/1.74 & ( ord_less_eq @ A @ A8 @ B6 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_171_dual__order_Otrans,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ! [B2: A,A3: A,C: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ B2 @ A3 )
% 3.17/1.74 => ( ( ord_less_eq @ A @ C @ B2 )
% 3.17/1.74 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_172_linorder__wlog,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( linorder @ A )
% 3.17/1.74 => ! [P: A > A > $o,A3: A,B2: A] :
% 3.17/1.74 ( ! [A4: A,B7: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ A4 @ B7 )
% 3.17/1.74 => ( P @ A4 @ B7 ) )
% 3.17/1.74 => ( ! [A4: A,B7: A] :
% 3.17/1.74 ( ( P @ B7 @ A4 )
% 3.17/1.74 => ( P @ A4 @ B7 ) )
% 3.17/1.74 => ( P @ A3 @ B2 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_173_dual__order_Orefl,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_174_order__trans,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( preorder @ A )
% 3.17/1.74 => ! [X4: A,Y3: A,Z2: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 3.17/1.74 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 3.17/1.74 => ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_175_order__class_Oorder_Oantisym,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ! [A3: A,B2: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.17/1.74 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 3.17/1.74 => ( A3 = B2 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_176_ord__le__eq__trans,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( ord @ A )
% 3.17/1.74 => ! [A3: A,B2: A,C: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.17/1.74 => ( ( B2 = C )
% 3.17/1.74 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_177_ord__eq__le__trans,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( ord @ A )
% 3.17/1.74 => ! [A3: A,B2: A,C: A] :
% 3.17/1.74 ( ( A3 = B2 )
% 3.17/1.74 => ( ( ord_less_eq @ A @ B2 @ C )
% 3.17/1.74 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_178_order__class_Oorder_Oeq__iff,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 3.17/1.74 = ( ^ [A8: A,B6: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ A8 @ B6 )
% 3.17/1.74 & ( ord_less_eq @ A @ B6 @ A8 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_179_antisym__conv,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ! [Y3: A,X4: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ Y3 @ X4 )
% 3.17/1.74 => ( ( ord_less_eq @ A @ X4 @ Y3 )
% 3.17/1.74 = ( X4 = Y3 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_180_le__cases3,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( linorder @ A )
% 3.17/1.74 => ! [X4: A,Y3: A,Z2: A] :
% 3.17/1.74 ( ( ( ord_less_eq @ A @ X4 @ Y3 )
% 3.17/1.74 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 3.17/1.74 => ( ( ( ord_less_eq @ A @ Y3 @ X4 )
% 3.17/1.74 => ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
% 3.17/1.74 => ( ( ( ord_less_eq @ A @ X4 @ Z2 )
% 3.17/1.74 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 3.17/1.74 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 3.17/1.74 => ~ ( ord_less_eq @ A @ Y3 @ X4 ) )
% 3.17/1.74 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 3.17/1.74 => ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
% 3.17/1.74 => ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
% 3.17/1.74 => ~ ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_181_order_Otrans,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ! [A3: A,B2: A,C: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.17/1.74 => ( ( ord_less_eq @ A @ B2 @ C )
% 3.17/1.74 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_182_le__cases,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( linorder @ A )
% 3.17/1.74 => ! [X4: A,Y3: A] :
% 3.17/1.74 ( ~ ( ord_less_eq @ A @ X4 @ Y3 )
% 3.17/1.74 => ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_183_eq__refl,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( preorder @ A )
% 3.17/1.74 => ! [X4: A,Y3: A] :
% 3.17/1.74 ( ( X4 = Y3 )
% 3.17/1.74 => ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_184_linear,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( linorder @ A )
% 3.17/1.74 => ! [X4: A,Y3: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 3.17/1.74 | ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_185_antisym,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ! [X4: A,Y3: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 3.17/1.74 => ( ( ord_less_eq @ A @ Y3 @ X4 )
% 3.17/1.74 => ( X4 = Y3 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_186_le__rel__bool__arg__iff,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( ord @ A )
% 3.17/1.74 => ( ( ord_less_eq @ ( $o > A ) )
% 3.17/1.74 = ( ^ [X8: $o > A,Y5: $o > A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ ( X8 @ $false ) @ ( Y5 @ $false ) )
% 3.17/1.74 & ( ord_less_eq @ A @ ( X8 @ $true ) @ ( Y5 @ $true ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_187_eq__iff,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 3.17/1.74 = ( ^ [X5: A,Y4: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ X5 @ Y4 )
% 3.17/1.74 & ( ord_less_eq @ A @ Y4 @ X5 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_188_ord__le__eq__subst,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType] :
% 3.17/1.74 ( ( ( ord @ B )
% 3.17/1.74 & ( ord @ A ) )
% 3.17/1.74 => ! [A3: A,B2: A,F: A > B,C: B] :
% 3.17/1.74 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.17/1.74 => ( ( ( F @ B2 )
% 3.17/1.74 = C )
% 3.17/1.74 => ( ! [X6: A,Y: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ X6 @ Y )
% 3.17/1.74 => ( ord_less_eq @ B @ ( F @ X6 ) @ ( F @ Y ) ) )
% 3.17/1.74 => ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_189_ord__eq__le__subst,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType] :
% 3.17/1.74 ( ( ( ord @ B )
% 3.17/1.74 & ( ord @ A ) )
% 3.17/1.74 => ! [A3: A,F: B > A,B2: B,C: B] :
% 3.17/1.74 ( ( A3
% 3.17/1.74 = ( F @ B2 ) )
% 3.17/1.74 => ( ( ord_less_eq @ B @ B2 @ C )
% 3.17/1.74 => ( ! [X6: B,Y: B] :
% 3.17/1.74 ( ( ord_less_eq @ B @ X6 @ Y )
% 3.17/1.74 => ( ord_less_eq @ A @ ( F @ X6 ) @ ( F @ Y ) ) )
% 3.17/1.74 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_190_order__subst2,axiom,
% 3.17/1.74 ! [A: $tType,C4: $tType] :
% 3.17/1.74 ( ( ( order @ C4 )
% 3.17/1.74 & ( order @ A ) )
% 3.17/1.74 => ! [A3: A,B2: A,F: A > C4,C: C4] :
% 3.17/1.74 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.17/1.74 => ( ( ord_less_eq @ C4 @ ( F @ B2 ) @ C )
% 3.17/1.74 => ( ! [X6: A,Y: A] :
% 3.17/1.74 ( ( ord_less_eq @ A @ X6 @ Y )
% 3.17/1.74 => ( ord_less_eq @ C4 @ ( F @ X6 ) @ ( F @ Y ) ) )
% 3.17/1.74 => ( ord_less_eq @ C4 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_191_order__subst1,axiom,
% 3.17/1.74 ! [A: $tType,B: $tType] :
% 3.17/1.74 ( ( ( order @ B )
% 3.17/1.74 & ( order @ A ) )
% 3.17/1.74 => ! [A3: A,F: B > A,B2: B,C: B] :
% 3.17/1.74 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 3.17/1.74 => ( ( ord_less_eq @ B @ B2 @ C )
% 3.17/1.74 => ( ! [X6: B,Y: B] :
% 3.17/1.74 ( ( ord_less_eq @ B @ X6 @ Y )
% 3.17/1.74 => ( ord_less_eq @ A @ ( F @ X6 ) @ ( F @ Y ) ) )
% 3.17/1.74 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_192_le__fun__def,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType] :
% 3.17/1.74 ( ( ord @ B )
% 3.17/1.74 => ( ( ord_less_eq @ ( A > B ) )
% 3.17/1.74 = ( ^ [F3: A > B,G3: A > B] :
% 3.17/1.74 ! [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_193_le__funI,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType] :
% 3.17/1.74 ( ( ord @ B )
% 3.17/1.74 => ! [F: A > B,G: A > B] :
% 3.17/1.74 ( ! [X6: A] : ( ord_less_eq @ B @ ( F @ X6 ) @ ( G @ X6 ) )
% 3.17/1.74 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_194_le__funE,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType] :
% 3.17/1.74 ( ( ord @ B )
% 3.17/1.74 => ! [F: A > B,G: A > B,X4: A] :
% 3.17/1.74 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.17/1.74 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_195_le__funD,axiom,
% 3.17/1.74 ! [B: $tType,A: $tType] :
% 3.17/1.74 ( ( ord @ B )
% 3.17/1.74 => ! [F: A > B,G: A > B,X4: A] :
% 3.17/1.74 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.17/1.74 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_196_dual__order_Ostrict__implies__not__eq,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ! [B2: A,A3: A] :
% 3.17/1.74 ( ( ord_less @ A @ B2 @ A3 )
% 3.17/1.74 => ( A3 != B2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_197_order_Ostrict__implies__not__eq,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ! [A3: A,B2: A] :
% 3.17/1.74 ( ( ord_less @ A @ A3 @ B2 )
% 3.17/1.74 => ( A3 != B2 ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_198_not__less__iff__gr__or__eq,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( linorder @ A )
% 3.17/1.74 => ! [X4: A,Y3: A] :
% 3.17/1.74 ( ( ~ ( ord_less @ A @ X4 @ Y3 ) )
% 3.17/1.74 = ( ( ord_less @ A @ Y3 @ X4 )
% 3.17/1.74 | ( X4 = Y3 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_199_dual__order_Ostrict__trans,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( order @ A )
% 3.17/1.74 => ! [B2: A,A3: A,C: A] :
% 3.17/1.74 ( ( ord_less @ A @ B2 @ A3 )
% 3.17/1.74 => ( ( ord_less @ A @ C @ B2 )
% 3.17/1.74 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_200_linorder__less__wlog,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( linorder @ A )
% 3.17/1.74 => ! [P: A > A > $o,A3: A,B2: A] :
% 3.17/1.74 ( ! [A4: A,B7: A] :
% 3.17/1.74 ( ( ord_less @ A @ A4 @ B7 )
% 3.17/1.74 => ( P @ A4 @ B7 ) )
% 3.17/1.74 => ( ! [A4: A] : ( P @ A4 @ A4 )
% 3.17/1.74 => ( ! [A4: A,B7: A] :
% 3.17/1.74 ( ( P @ B7 @ A4 )
% 3.17/1.74 => ( P @ A4 @ B7 ) )
% 3.17/1.74 => ( P @ A3 @ B2 ) ) ) ) ) ).
% 3.17/1.74
% 3.17/1.74 thf(fact_201_exists__least__iff,axiom,
% 3.17/1.74 ! [A: $tType] :
% 3.17/1.74 ( ( wellorder @ A )
% 3.17/1.74 => ( ( ^ [P3: A > $o] :
% 3.17/1.74 ? [X3: A] : ( P3 @ X3 ) )
% 3.17/1.74 = ( ^ [P2: A > $o] :
% 3.17/1.74 ? [N3: A] :
% 3.17/1.75 ( ( P2 @ N3 )
% 3.17/1.75 & ! [M5: A] :
% 3.17/1.75 ( ( ord_less @ A @ M5 @ N3 )
% 3.17/1.75 => ~ ( P2 @ M5 ) ) ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_202_less__imp__not__less,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( preorder @ A )
% 3.17/1.75 => ! [X4: A,Y3: A] :
% 3.17/1.75 ( ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 => ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_203_order_Ostrict__trans,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ! [A3: A,B2: A,C: A] :
% 3.17/1.75 ( ( ord_less @ A @ A3 @ B2 )
% 3.17/1.75 => ( ( ord_less @ A @ B2 @ C )
% 3.17/1.75 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_204_dual__order_Oirrefl,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ! [A3: A] :
% 3.17/1.75 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_205_linorder__cases,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [X4: A,Y3: A] :
% 3.17/1.75 ( ~ ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 => ( ( X4 != Y3 )
% 3.17/1.75 => ( ord_less @ A @ Y3 @ X4 ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_206_less__imp__triv,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( preorder @ A )
% 3.17/1.75 => ! [X4: A,Y3: A,P: $o] :
% 3.17/1.75 ( ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 => ( ( ord_less @ A @ Y3 @ X4 )
% 3.17/1.75 => P ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_207_less__imp__not__eq2,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ! [X4: A,Y3: A] :
% 3.17/1.75 ( ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 => ( Y3 != X4 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_208_antisym__conv3,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [Y3: A,X4: A] :
% 3.17/1.75 ( ~ ( ord_less @ A @ Y3 @ X4 )
% 3.17/1.75 => ( ( ~ ( ord_less @ A @ X4 @ Y3 ) )
% 3.17/1.75 = ( X4 = Y3 ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_209_less__induct,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( wellorder @ A )
% 3.17/1.75 => ! [P: A > $o,A3: A] :
% 3.17/1.75 ( ! [X6: A] :
% 3.17/1.75 ( ! [Y6: A] :
% 3.17/1.75 ( ( ord_less @ A @ Y6 @ X6 )
% 3.17/1.75 => ( P @ Y6 ) )
% 3.17/1.75 => ( P @ X6 ) )
% 3.17/1.75 => ( P @ A3 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_210_less__not__sym,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( preorder @ A )
% 3.17/1.75 => ! [X4: A,Y3: A] :
% 3.17/1.75 ( ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 => ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_211_less__imp__not__eq,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ! [X4: A,Y3: A] :
% 3.17/1.75 ( ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 => ( X4 != Y3 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_212_dual__order_Oasym,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ! [B2: A,A3: A] :
% 3.17/1.75 ( ( ord_less @ A @ B2 @ A3 )
% 3.17/1.75 => ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_213_ord__less__eq__trans,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( ord @ A )
% 3.17/1.75 => ! [A3: A,B2: A,C: A] :
% 3.17/1.75 ( ( ord_less @ A @ A3 @ B2 )
% 3.17/1.75 => ( ( B2 = C )
% 3.17/1.75 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_214_ord__eq__less__trans,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( ord @ A )
% 3.17/1.75 => ! [A3: A,B2: A,C: A] :
% 3.17/1.75 ( ( A3 = B2 )
% 3.17/1.75 => ( ( ord_less @ A @ B2 @ C )
% 3.17/1.75 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_215_less__irrefl,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( preorder @ A )
% 3.17/1.75 => ! [X4: A] :
% 3.17/1.75 ~ ( ord_less @ A @ X4 @ X4 ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_216_less__linear,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [X4: A,Y3: A] :
% 3.17/1.75 ( ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 | ( X4 = Y3 )
% 3.17/1.75 | ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_217_less__trans,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( preorder @ A )
% 3.17/1.75 => ! [X4: A,Y3: A,Z2: A] :
% 3.17/1.75 ( ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 => ( ( ord_less @ A @ Y3 @ Z2 )
% 3.17/1.75 => ( ord_less @ A @ X4 @ Z2 ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_218_less__asym_H,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( preorder @ A )
% 3.17/1.75 => ! [A3: A,B2: A] :
% 3.17/1.75 ( ( ord_less @ A @ A3 @ B2 )
% 3.17/1.75 => ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_219_less__asym,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( preorder @ A )
% 3.17/1.75 => ! [X4: A,Y3: A] :
% 3.17/1.75 ( ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 => ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_220_less__imp__neq,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ! [X4: A,Y3: A] :
% 3.17/1.75 ( ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 => ( X4 != Y3 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_221_dense,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( dense_order @ A )
% 3.17/1.75 => ! [X4: A,Y3: A] :
% 3.17/1.75 ( ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 => ? [Z3: A] :
% 3.17/1.75 ( ( ord_less @ A @ X4 @ Z3 )
% 3.17/1.75 & ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_222_order_Oasym,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ! [A3: A,B2: A] :
% 3.17/1.75 ( ( ord_less @ A @ A3 @ B2 )
% 3.17/1.75 => ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_223_neq__iff,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [X4: A,Y3: A] :
% 3.17/1.75 ( ( X4 != Y3 )
% 3.17/1.75 = ( ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 | ( ord_less @ A @ Y3 @ X4 ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_224_neqE,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [X4: A,Y3: A] :
% 3.17/1.75 ( ( X4 != Y3 )
% 3.17/1.75 => ( ~ ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 => ( ord_less @ A @ Y3 @ X4 ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_225_gt__ex,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( no_top @ A )
% 3.17/1.75 => ! [X4: A] :
% 3.17/1.75 ? [X_1: A] : ( ord_less @ A @ X4 @ X_1 ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_226_lt__ex,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( no_bot @ A )
% 3.17/1.75 => ! [X4: A] :
% 3.17/1.75 ? [Y: A] : ( ord_less @ A @ Y @ X4 ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_227_order__less__subst2,axiom,
% 3.17/1.75 ! [A: $tType,C4: $tType] :
% 3.17/1.75 ( ( ( order @ C4 )
% 3.17/1.75 & ( order @ A ) )
% 3.17/1.75 => ! [A3: A,B2: A,F: A > C4,C: C4] :
% 3.17/1.75 ( ( ord_less @ A @ A3 @ B2 )
% 3.17/1.75 => ( ( ord_less @ C4 @ ( F @ B2 ) @ C )
% 3.17/1.75 => ( ! [X6: A,Y: A] :
% 3.17/1.75 ( ( ord_less @ A @ X6 @ Y )
% 3.17/1.75 => ( ord_less @ C4 @ ( F @ X6 ) @ ( F @ Y ) ) )
% 3.17/1.75 => ( ord_less @ C4 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_228_order__less__subst1,axiom,
% 3.17/1.75 ! [A: $tType,B: $tType] :
% 3.17/1.75 ( ( ( order @ B )
% 3.17/1.75 & ( order @ A ) )
% 3.17/1.75 => ! [A3: A,F: B > A,B2: B,C: B] :
% 3.17/1.75 ( ( ord_less @ A @ A3 @ ( F @ B2 ) )
% 3.17/1.75 => ( ( ord_less @ B @ B2 @ C )
% 3.17/1.75 => ( ! [X6: B,Y: B] :
% 3.17/1.75 ( ( ord_less @ B @ X6 @ Y )
% 3.17/1.75 => ( ord_less @ A @ ( F @ X6 ) @ ( F @ Y ) ) )
% 3.17/1.75 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_229_ord__less__eq__subst,axiom,
% 3.17/1.75 ! [A: $tType,B: $tType] :
% 3.17/1.75 ( ( ( ord @ B )
% 3.17/1.75 & ( ord @ A ) )
% 3.17/1.75 => ! [A3: A,B2: A,F: A > B,C: B] :
% 3.17/1.75 ( ( ord_less @ A @ A3 @ B2 )
% 3.17/1.75 => ( ( ( F @ B2 )
% 3.17/1.75 = C )
% 3.17/1.75 => ( ! [X6: A,Y: A] :
% 3.17/1.75 ( ( ord_less @ A @ X6 @ Y )
% 3.17/1.75 => ( ord_less @ B @ ( F @ X6 ) @ ( F @ Y ) ) )
% 3.17/1.75 => ( ord_less @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_230_ord__eq__less__subst,axiom,
% 3.17/1.75 ! [A: $tType,B: $tType] :
% 3.17/1.75 ( ( ( ord @ B )
% 3.17/1.75 & ( ord @ A ) )
% 3.17/1.75 => ! [A3: A,F: B > A,B2: B,C: B] :
% 3.17/1.75 ( ( A3
% 3.17/1.75 = ( F @ B2 ) )
% 3.17/1.75 => ( ( ord_less @ B @ B2 @ C )
% 3.17/1.75 => ( ! [X6: B,Y: B] :
% 3.17/1.75 ( ( ord_less @ B @ X6 @ Y )
% 3.17/1.75 => ( ord_less @ A @ ( F @ X6 ) @ ( F @ Y ) ) )
% 3.17/1.75 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_231_pinf_I1_J,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 3.17/1.75 ( ? [Z4: A] :
% 3.17/1.75 ! [X6: A] :
% 3.17/1.75 ( ( ord_less @ A @ Z4 @ X6 )
% 3.17/1.75 => ( ( P @ X6 )
% 3.17/1.75 = ( P4 @ X6 ) ) )
% 3.17/1.75 => ( ? [Z4: A] :
% 3.17/1.75 ! [X6: A] :
% 3.17/1.75 ( ( ord_less @ A @ Z4 @ X6 )
% 3.17/1.75 => ( ( Q @ X6 )
% 3.17/1.75 = ( Q2 @ X6 ) ) )
% 3.17/1.75 => ? [Z3: A] :
% 3.17/1.75 ! [X: A] :
% 3.17/1.75 ( ( ord_less @ A @ Z3 @ X )
% 3.17/1.75 => ( ( ( P @ X )
% 3.17/1.75 & ( Q @ X ) )
% 3.17/1.75 = ( ( P4 @ X )
% 3.17/1.75 & ( Q2 @ X ) ) ) ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_232_pinf_I2_J,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 3.17/1.75 ( ? [Z4: A] :
% 3.17/1.75 ! [X6: A] :
% 3.17/1.75 ( ( ord_less @ A @ Z4 @ X6 )
% 3.17/1.75 => ( ( P @ X6 )
% 3.17/1.75 = ( P4 @ X6 ) ) )
% 3.17/1.75 => ( ? [Z4: A] :
% 3.17/1.75 ! [X6: A] :
% 3.17/1.75 ( ( ord_less @ A @ Z4 @ X6 )
% 3.17/1.75 => ( ( Q @ X6 )
% 3.17/1.75 = ( Q2 @ X6 ) ) )
% 3.17/1.75 => ? [Z3: A] :
% 3.17/1.75 ! [X: A] :
% 3.17/1.75 ( ( ord_less @ A @ Z3 @ X )
% 3.17/1.75 => ( ( ( P @ X )
% 3.17/1.75 | ( Q @ X ) )
% 3.17/1.75 = ( ( P4 @ X )
% 3.17/1.75 | ( Q2 @ X ) ) ) ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_233_pinf_I3_J,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [T2: A] :
% 3.17/1.75 ? [Z3: A] :
% 3.17/1.75 ! [X: A] :
% 3.17/1.75 ( ( ord_less @ A @ Z3 @ X )
% 3.17/1.75 => ( X != T2 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_234_pinf_I4_J,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [T2: A] :
% 3.17/1.75 ? [Z3: A] :
% 3.17/1.75 ! [X: A] :
% 3.17/1.75 ( ( ord_less @ A @ Z3 @ X )
% 3.17/1.75 => ( X != T2 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_235_pinf_I5_J,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [T2: A] :
% 3.17/1.75 ? [Z3: A] :
% 3.17/1.75 ! [X: A] :
% 3.17/1.75 ( ( ord_less @ A @ Z3 @ X )
% 3.17/1.75 => ~ ( ord_less @ A @ X @ T2 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_236_pinf_I7_J,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [T2: A] :
% 3.17/1.75 ? [Z3: A] :
% 3.17/1.75 ! [X: A] :
% 3.17/1.75 ( ( ord_less @ A @ Z3 @ X )
% 3.17/1.75 => ( ord_less @ A @ T2 @ X ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_237_pinf_I11_J,axiom,
% 3.17/1.75 ! [C4: $tType,D: $tType] :
% 3.17/1.75 ( ( ord @ C4 )
% 3.17/1.75 => ! [F5: D] :
% 3.17/1.75 ? [Z3: C4] :
% 3.17/1.75 ! [X: C4] :
% 3.17/1.75 ( ( ord_less @ C4 @ Z3 @ X )
% 3.17/1.75 => ( F5 = F5 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_238_minf_I1_J,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 3.17/1.75 ( ? [Z4: A] :
% 3.17/1.75 ! [X6: A] :
% 3.17/1.75 ( ( ord_less @ A @ X6 @ Z4 )
% 3.17/1.75 => ( ( P @ X6 )
% 3.17/1.75 = ( P4 @ X6 ) ) )
% 3.17/1.75 => ( ? [Z4: A] :
% 3.17/1.75 ! [X6: A] :
% 3.17/1.75 ( ( ord_less @ A @ X6 @ Z4 )
% 3.17/1.75 => ( ( Q @ X6 )
% 3.17/1.75 = ( Q2 @ X6 ) ) )
% 3.17/1.75 => ? [Z3: A] :
% 3.17/1.75 ! [X: A] :
% 3.17/1.75 ( ( ord_less @ A @ X @ Z3 )
% 3.17/1.75 => ( ( ( P @ X )
% 3.17/1.75 & ( Q @ X ) )
% 3.17/1.75 = ( ( P4 @ X )
% 3.17/1.75 & ( Q2 @ X ) ) ) ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_239_minf_I2_J,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 3.17/1.75 ( ? [Z4: A] :
% 3.17/1.75 ! [X6: A] :
% 3.17/1.75 ( ( ord_less @ A @ X6 @ Z4 )
% 3.17/1.75 => ( ( P @ X6 )
% 3.17/1.75 = ( P4 @ X6 ) ) )
% 3.17/1.75 => ( ? [Z4: A] :
% 3.17/1.75 ! [X6: A] :
% 3.17/1.75 ( ( ord_less @ A @ X6 @ Z4 )
% 3.17/1.75 => ( ( Q @ X6 )
% 3.17/1.75 = ( Q2 @ X6 ) ) )
% 3.17/1.75 => ? [Z3: A] :
% 3.17/1.75 ! [X: A] :
% 3.17/1.75 ( ( ord_less @ A @ X @ Z3 )
% 3.17/1.75 => ( ( ( P @ X )
% 3.17/1.75 | ( Q @ X ) )
% 3.17/1.75 = ( ( P4 @ X )
% 3.17/1.75 | ( Q2 @ X ) ) ) ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_240_minf_I3_J,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [T2: A] :
% 3.17/1.75 ? [Z3: A] :
% 3.17/1.75 ! [X: A] :
% 3.17/1.75 ( ( ord_less @ A @ X @ Z3 )
% 3.17/1.75 => ( X != T2 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_241_minf_I4_J,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [T2: A] :
% 3.17/1.75 ? [Z3: A] :
% 3.17/1.75 ! [X: A] :
% 3.17/1.75 ( ( ord_less @ A @ X @ Z3 )
% 3.17/1.75 => ( X != T2 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_242_minf_I5_J,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [T2: A] :
% 3.17/1.75 ? [Z3: A] :
% 3.17/1.75 ! [X: A] :
% 3.17/1.75 ( ( ord_less @ A @ X @ Z3 )
% 3.17/1.75 => ( ord_less @ A @ X @ T2 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_243_minf_I7_J,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( linorder @ A )
% 3.17/1.75 => ! [T2: A] :
% 3.17/1.75 ? [Z3: A] :
% 3.17/1.75 ! [X: A] :
% 3.17/1.75 ( ( ord_less @ A @ X @ Z3 )
% 3.17/1.75 => ~ ( ord_less @ A @ T2 @ X ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_244_minf_I11_J,axiom,
% 3.17/1.75 ! [C4: $tType,D: $tType] :
% 3.17/1.75 ( ( ord @ C4 )
% 3.17/1.75 => ! [F5: D] :
% 3.17/1.75 ? [Z3: C4] :
% 3.17/1.75 ! [X: C4] :
% 3.17/1.75 ( ( ord_less @ C4 @ X @ Z3 )
% 3.17/1.75 => ( F5 = F5 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_245_order_Onot__eq__order__implies__strict,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ! [A3: A,B2: A] :
% 3.17/1.75 ( ( A3 != B2 )
% 3.17/1.75 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.17/1.75 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_246_dual__order_Ostrict__implies__order,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ! [B2: A,A3: A] :
% 3.17/1.75 ( ( ord_less @ A @ B2 @ A3 )
% 3.17/1.75 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_247_dual__order_Ostrict__iff__order,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ( ( ord_less @ A )
% 3.17/1.75 = ( ^ [B6: A,A8: A] :
% 3.17/1.75 ( ( ord_less_eq @ A @ B6 @ A8 )
% 3.17/1.75 & ( A8 != B6 ) ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_248_dual__order_Oorder__iff__strict,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ( ( ord_less_eq @ A )
% 3.17/1.75 = ( ^ [B6: A,A8: A] :
% 3.17/1.75 ( ( ord_less @ A @ B6 @ A8 )
% 3.17/1.75 | ( A8 = B6 ) ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_249_order_Ostrict__implies__order,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ! [A3: A,B2: A] :
% 3.17/1.75 ( ( ord_less @ A @ A3 @ B2 )
% 3.17/1.75 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_250_dense__le__bounded,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( dense_linorder @ A )
% 3.17/1.75 => ! [X4: A,Y3: A,Z2: A] :
% 3.17/1.75 ( ( ord_less @ A @ X4 @ Y3 )
% 3.17/1.75 => ( ! [W: A] :
% 3.17/1.75 ( ( ord_less @ A @ X4 @ W )
% 3.17/1.75 => ( ( ord_less @ A @ W @ Y3 )
% 3.17/1.75 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 3.17/1.75 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_251_dense__ge__bounded,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( dense_linorder @ A )
% 3.17/1.75 => ! [Z2: A,X4: A,Y3: A] :
% 3.17/1.75 ( ( ord_less @ A @ Z2 @ X4 )
% 3.17/1.75 => ( ! [W: A] :
% 3.17/1.75 ( ( ord_less @ A @ Z2 @ W )
% 3.17/1.75 => ( ( ord_less @ A @ W @ X4 )
% 3.17/1.75 => ( ord_less_eq @ A @ Y3 @ W ) ) )
% 3.17/1.75 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_252_dual__order_Ostrict__trans2,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ! [B2: A,A3: A,C: A] :
% 3.17/1.75 ( ( ord_less @ A @ B2 @ A3 )
% 3.17/1.75 => ( ( ord_less_eq @ A @ C @ B2 )
% 3.17/1.75 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_253_dual__order_Ostrict__trans1,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ! [B2: A,A3: A,C: A] :
% 3.17/1.75 ( ( ord_less_eq @ A @ B2 @ A3 )
% 3.17/1.75 => ( ( ord_less @ A @ C @ B2 )
% 3.17/1.75 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_254_order_Ostrict__iff__order,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ( ( ord_less @ A )
% 3.17/1.75 = ( ^ [A8: A,B6: A] :
% 3.17/1.75 ( ( ord_less_eq @ A @ A8 @ B6 )
% 3.17/1.75 & ( A8 != B6 ) ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(fact_255_order_Oorder__iff__strict,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( order @ A )
% 3.17/1.75 => ( ( ord_less_eq @ A )
% 3.17/1.75 = ( ^ [A8: A,B6: A] :
% 3.17/1.75 ( ( ord_less @ A @ A8 @ B6 )
% 3.17/1.75 | ( A8 = B6 ) ) ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(subcl_Groups_Ocomm__monoid__add___HOL_Otype,axiom,
% 3.17/1.75 ! [A: $tType] :
% 3.17/1.75 ( ( comm_monoid_add @ A )
% 3.17/1.75 => ( type @ A ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Nat_Onat___Rings_Odioid,axiom,
% 3.17/1.75 dioid @ nat ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_fun___Rings_Odioid_1,axiom,
% 3.17/1.75 ! [A9: $tType,A10: $tType] :
% 3.17/1.75 ( ( dioid @ A10 )
% 3.17/1.75 => ( dioid @ ( A9 > A10 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_fun___Groups_Ocanonically__ordered__monoid__add,axiom,
% 3.17/1.75 ! [A9: $tType,A10: $tType] :
% 3.17/1.75 ( ( dioid @ A10 )
% 3.17/1.75 => ( canoni770627133id_add @ ( A9 > A10 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_fun___Groups_Ocomm__monoid__add,axiom,
% 3.17/1.75 ! [A9: $tType,A10: $tType] :
% 3.17/1.75 ( ( comm_monoid_add @ A10 )
% 3.17/1.75 => ( comm_monoid_add @ ( A9 > A10 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.17/1.75 ! [A9: $tType,A10: $tType] :
% 3.17/1.75 ( ( preorder @ A10 )
% 3.17/1.75 => ( preorder @ ( A9 > A10 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.17/1.75 ! [A9: $tType,A10: $tType] :
% 3.17/1.75 ( ( order @ A10 )
% 3.17/1.75 => ( order @ ( A9 > A10 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_fun___Orderings_Oord,axiom,
% 3.17/1.75 ! [A9: $tType,A10: $tType] :
% 3.17/1.75 ( ( ord @ A10 )
% 3.17/1.75 => ( ord @ ( A9 > A10 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add_2,axiom,
% 3.17/1.75 canoni770627133id_add @ nat ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_3,axiom,
% 3.17/1.75 comm_monoid_add @ nat ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 3.17/1.75 wellorder @ nat ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
% 3.17/1.75 preorder @ nat ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 3.17/1.75 linorder @ nat ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 3.17/1.75 no_top @ nat ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Nat_Onat___Orderings_Oorder_5,axiom,
% 3.17/1.75 order @ nat ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Nat_Onat___Orderings_Oord_6,axiom,
% 3.17/1.75 ord @ nat ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 3.17/1.75 size @ nat ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 3.17/1.75 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
% 3.17/1.75 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
% 3.17/1.75 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 3.17/1.75 preorder @ $o ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_HOL_Obool___Orderings_Olinorder_11,axiom,
% 3.17/1.75 linorder @ $o ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
% 3.17/1.75 order @ $o ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
% 3.17/1.75 ord @ $o ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_List_Olist___Nat_Osize_14,axiom,
% 3.17/1.75 ! [A9: $tType] : ( size @ ( list @ A9 ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_15,axiom,
% 3.17/1.75 ! [A9: $tType] : ( comm_monoid_add @ ( multiset @ A9 ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Multiset_Omultiset___Orderings_Opreorder_16,axiom,
% 3.17/1.75 ! [A9: $tType] :
% 3.17/1.75 ( ( preorder @ A9 )
% 3.17/1.75 => ( preorder @ ( multiset @ A9 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Multiset_Omultiset___Orderings_Oorder_17,axiom,
% 3.17/1.75 ! [A9: $tType] :
% 3.17/1.75 ( ( preorder @ A9 )
% 3.17/1.75 => ( order @ ( multiset @ A9 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Multiset_Omultiset___Orderings_Oord_18,axiom,
% 3.17/1.75 ! [A9: $tType] :
% 3.17/1.75 ( ( preorder @ A9 )
% 3.17/1.75 => ( ord @ ( multiset @ A9 ) ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tcon_Multiset_Omultiset___Nat_Osize_19,axiom,
% 3.17/1.75 ! [A9: $tType] : ( size @ ( multiset @ A9 ) ) ).
% 3.17/1.75
% 3.17/1.75 thf(tfree_0,hypothesis,
% 3.17/1.75 comm_monoid_add @ a ).
% 3.17/1.75
% 3.17/1.75 thf(conj_0,conjecture,
% 3.17/1.75 ( a3
% 3.17/1.75 = ( groups501227526m_list @ a @ qs2 ) ) ).
% 3.17/1.75 ------------------------
% 3.17/1.76 % (29089)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/4Mi)
% 3.17/1.77 % (29089)Instruction limit reached!
% 3.17/1.77 % (29089)------------------------------
% 3.17/1.77 % (29089)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.77 % (29092)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/2Mi)
% 3.17/1.77 % (29089)Termination reason: Unknown
% 3.17/1.77 % (29089)Termination phase: shuffling
% 3.17/1.77
% 3.17/1.77 % (29089)Memory used [KB]: 1535
% 3.17/1.77 % (29089)Time elapsed: 0.004 s
% 3.17/1.77 % (29089)Instructions burned: 7 (million)
% 3.17/1.77 % (29089)------------------------------
% 3.17/1.77 % (29089)------------------------------
% 3.17/1.77 % (29092)Instruction limit reached!
% 3.17/1.77 % (29092)------------------------------
% 3.17/1.77 % (29092)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.77 % (29092)Termination reason: Unknown
% 3.17/1.77 % (29092)Termination phase: shuffling
% 3.17/1.77
% 3.17/1.77 % (29092)Memory used [KB]: 1407
% 3.17/1.77 % (29092)Time elapsed: 0.003 s
% 3.17/1.77 % (29092)Instructions burned: 4 (million)
% 3.17/1.77 % (29092)------------------------------
% 3.17/1.77 % (29092)------------------------------
% 3.17/1.77 % (29093)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/275Mi)
% 3.17/1.77 % (29094)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/18Mi)
% 3.17/1.77 % (29088)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/183Mi)
% 3.17/1.77 % (29091)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/2Mi)
% 3.17/1.77 % (29090)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/27Mi)
% 3.17/1.77 % (29091)Instruction limit reached!
% 3.17/1.77 % (29091)------------------------------
% 3.17/1.77 % (29091)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.77 % (29091)Termination reason: Unknown
% 3.17/1.77 % (29091)Termination phase: shuffling
% 3.17/1.77
% 3.17/1.77 % (29091)Memory used [KB]: 1407
% 3.17/1.77 % (29091)Time elapsed: 0.003 s
% 3.17/1.77 % (29091)Instructions burned: 2 (million)
% 3.17/1.77 % (29091)------------------------------
% 3.17/1.77 % (29091)------------------------------
% 3.17/1.78 % (29095)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/3Mi)
% 3.17/1.78 % (29096)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/37Mi)
% 3.17/1.78 % (29094)Instruction limit reached!
% 3.17/1.78 % (29094)------------------------------
% 3.17/1.78 % (29094)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.78 % (29094)Termination reason: Unknown
% 3.17/1.78 % (29094)Termination phase: shuffling
% 3.17/1.78
% 3.17/1.78 % (29094)Memory used [KB]: 1791
% 3.17/1.78 % (29094)Time elapsed: 0.010 s
% 3.17/1.78 % (29094)Instructions burned: 18 (million)
% 3.17/1.78 % (29094)------------------------------
% 3.17/1.78 % (29094)------------------------------
% 3.17/1.78 % (29095)Instruction limit reached!
% 3.17/1.78 % (29095)------------------------------
% 3.17/1.78 % (29095)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.78 % (29095)Termination reason: Unknown
% 3.17/1.78 % (29095)Termination phase: shuffling
% 3.17/1.78
% 3.17/1.78 % (29095)Memory used [KB]: 1407
% 3.17/1.78 % (29095)Time elapsed: 0.003 s
% 3.17/1.78 % (29095)Instructions burned: 5 (million)
% 3.17/1.78 % (29095)------------------------------
% 3.17/1.78 % (29095)------------------------------
% 3.17/1.78 % (29090)Instruction limit reached!
% 3.17/1.78 % (29090)------------------------------
% 3.17/1.78 % (29090)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.78 % (29090)Termination reason: Unknown
% 3.17/1.78 % (29090)Termination phase: Property scanning
% 3.17/1.78
% 3.17/1.78 % (29090)Memory used [KB]: 1918
% 3.17/1.78 % (29090)Time elapsed: 0.011 s
% 3.17/1.78 % (29090)Instructions burned: 30 (million)
% 3.17/1.78 % (29090)------------------------------
% 3.17/1.78 % (29090)------------------------------
% 3.17/1.78 % (29097)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/15Mi)
% 3.17/1.79 % (29099)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/1041Mi)
% 3.17/1.79 % (29096)Instruction limit reached!
% 3.17/1.79 % (29096)------------------------------
% 3.17/1.79 % (29096)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.79 % (29096)Termination reason: Unknown
% 3.17/1.79 % (29096)Termination phase: shuffling
% 3.17/1.79
% 3.17/1.79 % (29096)Memory used [KB]: 2174
% 3.17/1.79 % (29096)Time elapsed: 0.012 s
% 3.17/1.79 % (29096)Instructions burned: 39 (million)
% 3.17/1.79 % (29096)------------------------------
% 3.17/1.79 % (29096)------------------------------
% 3.17/1.79 % (29097)Instruction limit reached!
% 3.17/1.79 % (29097)------------------------------
% 3.17/1.79 % (29097)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.79 % (29097)Termination reason: Unknown
% 3.17/1.79 % (29097)Termination phase: shuffling
% 3.17/1.79
% 3.17/1.79 % (29097)Memory used [KB]: 1663
% 3.17/1.79 % (29097)Time elapsed: 0.007 s
% 3.17/1.79 % (29097)Instructions burned: 17 (million)
% 3.17/1.79 % (29097)------------------------------
% 3.17/1.79 % (29097)------------------------------
% 3.17/1.79 % (29093)Refutation not found, incomplete strategy
% 3.17/1.79 % (29093)------------------------------
% 3.17/1.79 % (29093)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.79 % (29100)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/7Mi)
% 3.17/1.79 % (29093)Termination reason: Refutation not found, incomplete strategy
% 3.17/1.79
% 3.17/1.79
% 3.17/1.79 % (29093)Memory used [KB]: 6652
% 3.17/1.79 % (29093)Time elapsed: 0.025 s
% 3.17/1.79 % (29093)Instructions burned: 63 (million)
% 3.17/1.79 % (29093)------------------------------
% 3.17/1.79 % (29093)------------------------------
% 3.17/1.79 % (29098)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/3Mi)
% 3.17/1.79 % (29100)Instruction limit reached!
% 3.17/1.79 % (29100)------------------------------
% 3.17/1.79 % (29100)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.79 % (29100)Termination reason: Unknown
% 3.17/1.79 % (29100)Termination phase: shuffling
% 3.17/1.79
% 3.17/1.79 % (29100)Memory used [KB]: 1535
% 3.17/1.79 % (29100)Time elapsed: 0.004 s
% 3.17/1.79 % (29100)Instructions burned: 9 (million)
% 3.17/1.79 % (29100)------------------------------
% 3.17/1.79 % (29100)------------------------------
% 3.17/1.80 % (29098)Instruction limit reached!
% 3.17/1.80 % (29098)------------------------------
% 3.17/1.80 % (29098)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.80 % (29098)Termination reason: Unknown
% 3.17/1.80 % (29098)Termination phase: shuffling
% 3.17/1.80
% 3.17/1.80 % (29098)Memory used [KB]: 1407
% 3.17/1.80 % (29098)Time elapsed: 0.002 s
% 3.17/1.80 % (29098)Instructions burned: 3 (million)
% 3.17/1.80 % (29098)------------------------------
% 3.17/1.80 % (29098)------------------------------
% 3.22/1.80 % (29101)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/16Mi)
% 3.22/1.80 % (29102)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/3Mi)
% 3.22/1.80 % (29102)Instruction limit reached!
% 3.22/1.80 % (29102)------------------------------
% 3.22/1.80 % (29102)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.80 % (29102)Termination reason: Unknown
% 3.22/1.80 % (29102)Termination phase: shuffling
% 3.22/1.80
% 3.22/1.80 % (29102)Memory used [KB]: 1407
% 3.22/1.80 % (29102)Time elapsed: 0.003 s
% 3.22/1.80 % (29102)Instructions burned: 5 (million)
% 3.22/1.80 % (29102)------------------------------
% 3.22/1.80 % (29102)------------------------------
% 3.22/1.80 % (29103)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/3Mi)
% 3.22/1.80 % (29103)Instruction limit reached!
% 3.22/1.80 % (29103)------------------------------
% 3.22/1.80 % (29103)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.80 % (29103)Termination reason: Unknown
% 3.22/1.80 % (29103)Termination phase: shuffling
% 3.22/1.80
% 3.22/1.80 % (29103)Memory used [KB]: 1407
% 3.22/1.80 % (29103)Time elapsed: 0.002 s
% 3.22/1.80 % (29103)Instructions burned: 3 (million)
% 3.22/1.80 % (29103)------------------------------
% 3.22/1.80 % (29103)------------------------------
% 3.22/1.80 % (29101)Instruction limit reached!
% 3.22/1.80 % (29101)------------------------------
% 3.22/1.80 % (29101)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.80 % (29101)Termination reason: Unknown
% 3.22/1.80 % (29101)Termination phase: shuffling
% 3.22/1.80
% 3.22/1.80 % (29101)Memory used [KB]: 1791
% 3.22/1.80 % (29101)Time elapsed: 0.007 s
% 3.22/1.80 % (29101)Instructions burned: 18 (million)
% 3.22/1.80 % (29101)------------------------------
% 3.22/1.80 % (29101)------------------------------
% 3.22/1.80 % (29104)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/7Mi)
% 3.22/1.80 % (29105)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/3Mi)
% 3.22/1.81 % (29105)Instruction limit reached!
% 3.22/1.81 % (29105)------------------------------
% 3.22/1.81 % (29105)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.81 % (29105)Termination reason: Unknown
% 3.22/1.81 % (29105)Termination phase: shuffling
% 3.22/1.81
% 3.22/1.81 % (29105)Memory used [KB]: 1407
% 3.22/1.81 % (29105)Time elapsed: 0.002 s
% 3.22/1.81 % (29105)Instructions burned: 4 (million)
% 3.22/1.81 % (29105)------------------------------
% 3.22/1.81 % (29105)------------------------------
% 3.22/1.81 % (29104)Instruction limit reached!
% 3.22/1.81 % (29104)------------------------------
% 3.22/1.81 % (29104)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.81 % (29104)Termination reason: Unknown
% 3.22/1.81 % (29104)Termination phase: shuffling
% 3.22/1.81
% 3.22/1.81 % (29104)Memory used [KB]: 1535
% 3.22/1.81 % (29104)Time elapsed: 0.004 s
% 3.22/1.81 % (29104)Instructions burned: 9 (million)
% 3.22/1.81 % (29104)------------------------------
% 3.22/1.81 % (29104)------------------------------
% 3.22/1.81 % (29106)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/4Mi)
% 3.22/1.81 % (29106)Instruction limit reached!
% 3.22/1.81 % (29106)------------------------------
% 3.22/1.81 % (29106)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.81 % (29106)Termination reason: Unknown
% 3.22/1.81 % (29106)Termination phase: shuffling
% 3.22/1.81
% 3.22/1.81 % (29106)Memory used [KB]: 1535
% 3.22/1.81 % (29106)Time elapsed: 0.003 s
% 3.22/1.81 % (29106)Instructions burned: 7 (million)
% 3.22/1.81 % (29106)------------------------------
% 3.22/1.81 % (29106)------------------------------
% 3.22/1.81 % (29107)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/18Mi)
% 3.22/1.81 % (29108)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/710Mi)
% 3.22/1.82 % (29109)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/6Mi)
% 3.22/1.82 % (29110)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/902Mi)
% 3.22/1.82 % (29109)Instruction limit reached!
% 3.22/1.82 % (29109)------------------------------
% 3.22/1.82 % (29109)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.82 % (29109)Termination reason: Unknown
% 3.22/1.82 % (29109)Termination phase: shuffling
% 3.22/1.82
% 3.22/1.82 % (29109)Memory used [KB]: 1535
% 3.22/1.82 % (29109)Time elapsed: 0.003 s
% 3.22/1.82 % (29109)Instructions burned: 6 (million)
% 3.22/1.82 % (29109)------------------------------
% 3.22/1.82 % (29109)------------------------------
% 3.22/1.82 % (29107)Instruction limit reached!
% 3.22/1.82 % (29107)------------------------------
% 3.22/1.82 % (29107)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.82 % (29107)Termination reason: Unknown
% 3.22/1.82 % (29107)Termination phase: shuffling
% 3.22/1.82
% 3.22/1.82 % (29107)Memory used [KB]: 1791
% 3.22/1.82 % (29107)Time elapsed: 0.007 s
% 3.22/1.82 % (29107)Instructions burned: 19 (million)
% 3.22/1.82 % (29107)------------------------------
% 3.22/1.82 % (29107)------------------------------
% 3.22/1.82 % (29111)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/21Mi)
% 3.22/1.83 % (29112)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/5Mi)
% 3.22/1.83 % (29113)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/6Mi)
% 3.22/1.83 % (29111)Instruction limit reached!
% 3.22/1.83 % (29111)------------------------------
% 3.22/1.83 % (29111)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.83 % (29111)Termination reason: Unknown
% 3.22/1.83 % (29111)Termination phase: shuffling
% 3.22/1.83
% 3.22/1.83 % (29111)Memory used [KB]: 1918
% 3.22/1.83 % (29111)Time elapsed: 0.008 s
% 3.22/1.83 % (29111)Instructions burned: 21 (million)
% 3.22/1.83 % (29111)------------------------------
% 3.22/1.83 % (29111)------------------------------
% 3.22/1.83 % (29112)Instruction limit reached!
% 3.22/1.83 % (29112)------------------------------
% 3.22/1.83 % (29112)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.83 % (29112)Termination reason: Unknown
% 3.22/1.83 % (29112)Termination phase: shuffling
% 3.22/1.83
% 3.22/1.83 % (29112)Memory used [KB]: 1535
% 3.22/1.83 % (29112)Time elapsed: 0.003 s
% 3.22/1.83 % (29112)Instructions burned: 7 (million)
% 3.22/1.83 % (29112)------------------------------
% 3.22/1.83 % (29112)------------------------------
% 3.22/1.83 % (29113)Instruction limit reached!
% 3.22/1.83 % (29113)------------------------------
% 3.22/1.83 % (29113)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.83 % (29113)Termination reason: Unknown
% 3.22/1.83 % (29113)Termination phase: shuffling
% 3.22/1.83
% 3.22/1.83 % (29113)Memory used [KB]: 1535
% 3.22/1.83 % (29113)Time elapsed: 0.003 s
% 3.22/1.83 % (29113)Instructions burned: 7 (million)
% 3.22/1.83 % (29113)------------------------------
% 3.22/1.83 % (29113)------------------------------
% 3.22/1.84 % (29114)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/377Mi)
% 3.22/1.84 % (29115)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/779Mi)
% 3.22/1.84 % (29116)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/19Mi)
% 3.22/1.85 % (29116)Instruction limit reached!
% 3.22/1.85 % (29116)------------------------------
% 3.22/1.85 % (29116)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.85 % (29116)Termination reason: Unknown
% 3.22/1.85 % (29116)Termination phase: shuffling
% 3.22/1.85
% 3.22/1.85 % (29116)Memory used [KB]: 1791
% 3.22/1.85 % (29116)Time elapsed: 0.007 s
% 3.22/1.85 % (29116)Instructions burned: 22 (million)
% 3.22/1.85 % (29116)------------------------------
% 3.22/1.85 % (29116)------------------------------
% 3.22/1.85 % (29117)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_28982 for (2999ds/879Mi)
% 3.22/1.86 % (29110)Refutation not found, incomplete strategy
% 3.22/1.86 % (29110)------------------------------
% 3.22/1.86 % (29110)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.86 % (29110)Termination reason: Refutation not found, incomplete strategy
% 3.22/1.86
% 3.22/1.86
% 3.22/1.86 % (29110)Memory used [KB]: 7419
% 3.22/1.86 % (29110)Time elapsed: 0.041 s
% 3.22/1.86 % (29110)Instructions burned: 132 (million)
% 3.22/1.86 % (29110)------------------------------
% 3.22/1.86 % (29110)------------------------------
% 3.22/1.86 % (29088)Instruction limit reached!
% 3.22/1.86 % (29088)------------------------------
% 3.22/1.86 % (29088)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.86 % (29088)Termination reason: Unknown
% 3.22/1.86 % (29088)Termination phase: Saturation
% 3.22/1.86
% 3.22/1.86 % (29088)Memory used [KB]: 7419
% 3.22/1.86 % (29088)Time elapsed: 0.093 s
% 3.22/1.86 % (29088)Instructions burned: 184 (million)
% 3.22/1.86 % (29088)------------------------------
% 3.22/1.86 % (29088)------------------------------
% 3.22/1.87 % (29118)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_28982 for (2998ds/17Mi)
% 3.22/1.87 % (29118)Instruction limit reached!
% 3.22/1.87 % (29118)------------------------------
% 3.22/1.87 % (29118)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.87 % (29118)Termination reason: Unknown
% 3.22/1.87 % (29118)Termination phase: shuffling
% 3.22/1.87
% 3.22/1.87 % (29118)Memory used [KB]: 1663
% 3.22/1.87 % (29118)Time elapsed: 0.007 s
% 3.22/1.87 % (29118)Instructions burned: 17 (million)
% 3.22/1.87 % (29118)------------------------------
% 3.22/1.87 % (29118)------------------------------
% 3.22/1.88 % (29119)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_28982 for (2998ds/3Mi)
% 3.22/1.88 % (29119)Instruction limit reached!
% 3.22/1.88 % (29119)------------------------------
% 3.22/1.88 % (29119)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.88 % (29119)Termination reason: Unknown
% 3.22/1.88 % (29119)Termination phase: shuffling
% 3.22/1.88
% 3.22/1.88 % (29119)Memory used [KB]: 1407
% 3.22/1.88 % (29119)Time elapsed: 0.003 s
% 3.22/1.88 % (29119)Instructions burned: 3 (million)
% 3.22/1.88 % (29119)------------------------------
% 3.22/1.88 % (29119)------------------------------
% 3.22/1.88 % (29120)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_28982 for (2998ds/30Mi)
% 3.22/1.89 % (29121)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_28982 for (2998ds/127Mi)
% 3.22/1.89 % (29120)Instruction limit reached!
% 3.22/1.89 % (29120)------------------------------
% 3.22/1.89 % (29120)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.89 % (29120)Termination reason: Unknown
% 3.22/1.89 % (29120)Termination phase: Property scanning
% 3.22/1.89
% 3.22/1.89 % (29120)Memory used [KB]: 1918
% 3.22/1.89 % (29120)Time elapsed: 0.012 s
% 3.22/1.89 % (29120)Instructions burned: 32 (million)
% 3.22/1.89 % (29120)------------------------------
% 3.22/1.89 % (29120)------------------------------
% 3.22/1.90 % (29122)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_28982 for (2998ds/100Mi)
% 3.22/1.93 % (29121)Instruction limit reached!
% 3.22/1.93 % (29121)------------------------------
% 3.22/1.93 % (29121)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.93 % (29121)Termination reason: Unknown
% 3.22/1.93 % (29121)Termination phase: Saturation
% 3.22/1.93
% 3.22/1.93 % (29121)Memory used [KB]: 6908
% 3.22/1.93 % (29121)Time elapsed: 0.039 s
% 3.22/1.93 % (29121)Instructions burned: 127 (million)
% 3.22/1.93 % (29121)------------------------------
% 3.22/1.93 % (29121)------------------------------
% 3.22/1.93 % (29122)Instruction limit reached!
% 3.22/1.93 % (29122)------------------------------
% 3.22/1.93 % (29122)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.93 % (29122)Termination reason: Unknown
% 3.22/1.93 % (29122)Termination phase: Property scanning
% 3.22/1.93
% 3.22/1.93 % (29122)Memory used [KB]: 2430
% 3.22/1.93 % (29122)Time elapsed: 0.030 s
% 3.22/1.93 % (29122)Instructions burned: 103 (million)
% 3.22/1.93 % (29122)------------------------------
% 3.22/1.93 % (29122)------------------------------
% 3.22/1.94 % (29123)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_28982 for (2998ds/3Mi)
% 3.22/1.94 % (29123)Instruction limit reached!
% 3.22/1.94 % (29123)------------------------------
% 3.22/1.94 % (29123)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.94 % (29123)Termination reason: Unknown
% 3.22/1.94 % (29123)Termination phase: shuffling
% 3.22/1.94
% 3.22/1.94 % (29123)Memory used [KB]: 1407
% 3.22/1.94 % (29123)Time elapsed: 0.003 s
% 3.22/1.94 % (29123)Instructions burned: 6 (million)
% 3.22/1.94 % (29123)------------------------------
% 3.22/1.94 % (29123)------------------------------
% 3.22/1.94 % (29124)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_28982 for (2998ds/20Mi)
% 3.22/1.95 % (29124)Instruction limit reached!
% 3.22/1.95 % (29124)------------------------------
% 3.22/1.95 % (29124)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.95 % (29124)Termination reason: Unknown
% 3.22/1.95 % (29124)Termination phase: shuffling
% 3.22/1.95
% 3.22/1.95 % (29124)Memory used [KB]: 1791
% 3.22/1.95 % (29124)Time elapsed: 0.008 s
% 3.22/1.95 % (29124)Instructions burned: 20 (million)
% 3.22/1.95 % (29124)------------------------------
% 3.22/1.95 % (29124)------------------------------
% 3.22/1.95 % (29125)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_28982 for (2998ds/86Mi)
% 3.22/1.96 % (29114)Instruction limit reached!
% 3.22/1.96 % (29114)------------------------------
% 3.22/1.96 % (29114)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.96 % (29114)Termination reason: Unknown
% 3.22/1.96 % (29114)Termination phase: Saturation
% 3.22/1.96
% 3.22/1.96 % (29114)Memory used [KB]: 8187
% 3.22/1.96 % (29114)Time elapsed: 0.121 s
% 3.22/1.96 % (29114)Instructions burned: 377 (million)
% 3.22/1.96 % (29114)------------------------------
% 3.22/1.96 % (29114)------------------------------
% 3.22/1.96 % (29126)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_28982 for (2998ds/107Mi)
% 3.22/1.97 % (29127)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_28982 for (2998ds/39Mi)
% 3.22/1.98 % (29125)Instruction limit reached!
% 3.22/1.98 % (29125)------------------------------
% 3.22/1.98 % (29125)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.98 % (29125)Termination reason: Unknown
% 3.22/1.98 % (29125)Termination phase: Property scanning
% 3.22/1.98
% 3.22/1.98 % (29125)Memory used [KB]: 2430
% 3.22/1.98 % (29125)Time elapsed: 0.028 s
% 3.22/1.98 % (29125)Instructions burned: 88 (million)
% 3.22/1.98 % (29125)------------------------------
% 3.22/1.98 % (29125)------------------------------
% 3.22/1.98 % (29127)Instruction limit reached!
% 3.22/1.98 % (29127)------------------------------
% 3.22/1.98 % (29127)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.98 % (29127)Termination reason: Unknown
% 3.22/1.98 % (29127)Termination phase: shuffling
% 3.22/1.98
% 3.22/1.98 % (29127)Memory used [KB]: 2174
% 3.22/1.98 % (29127)Time elapsed: 0.012 s
% 3.22/1.98 % (29127)Instructions burned: 39 (million)
% 3.22/1.98 % (29127)------------------------------
% 3.22/1.98 % (29127)------------------------------
% 3.22/1.99 % (29128)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_28982 for (2997ds/448Mi)
% 3.22/1.99 % (29129)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_28982 for (2997ds/46Mi)
% 3.22/1.99 % (29126)Instruction limit reached!
% 3.22/1.99 % (29126)------------------------------
% 3.22/1.99 % (29126)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/1.99 % (29126)Termination reason: Unknown
% 3.22/1.99 % (29126)Termination phase: Saturation
% 3.22/1.99
% 3.22/1.99 % (29126)Memory used [KB]: 6908
% 3.22/1.99 % (29126)Time elapsed: 0.034 s
% 3.22/1.99 % (29126)Instructions burned: 110 (million)
% 3.22/1.99 % (29126)------------------------------
% 3.22/1.99 % (29126)------------------------------
% 3.22/2.00 % (29132)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_28982 for (2997ds/98Mi)
% 3.22/2.00 % (29129)Instruction limit reached!
% 3.22/2.00 % (29129)------------------------------
% 3.22/2.00 % (29129)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.22/2.00 % (29129)Termination reason: Unknown
% 3.22/2.00 % (29129)Termination phase: shuffling
% 3.22/2.00
% 3.22/2.00 % (29129)Memory used [KB]: 2046
% 3.22/2.00 % (29129)Time elapsed: 0.016 s
% 3.22/2.00 % (29129)Instructions burned: 48 (million)
% 3.22/2.00 % (29129)------------------------------
% 3.22/2.00 % (29129)------------------------------
% 4.94/2.01 % (29138)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_28982 for (2997ds/507Mi)
% 4.94/2.03 % (29132)Instruction limit reached!
% 4.94/2.03 % (29132)------------------------------
% 4.94/2.03 % (29132)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.03 % (29132)Termination reason: Unknown
% 4.94/2.03 % (29132)Termination phase: Property scanning
% 4.94/2.03
% 4.94/2.03 % (29132)Memory used [KB]: 2430
% 4.94/2.03 % (29132)Time elapsed: 0.028 s
% 4.94/2.03 % (29132)Instructions burned: 100 (million)
% 4.94/2.03 % (29132)------------------------------
% 4.94/2.03 % (29132)------------------------------
% 4.94/2.03 % (29108)Instruction limit reached!
% 4.94/2.03 % (29108)------------------------------
% 4.94/2.03 % (29108)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.03 % (29108)Termination reason: Unknown
% 4.94/2.03 % (29108)Termination phase: Saturation
% 4.94/2.03
% 4.94/2.03 % (29108)Memory used [KB]: 8059
% 4.94/2.03 % (29108)Time elapsed: 0.221 s
% 4.94/2.03 % (29108)Instructions burned: 710 (million)
% 4.94/2.03 % (29108)------------------------------
% 4.94/2.03 % (29108)------------------------------
% 4.94/2.04 % (29150)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_28982 for (2997ds/149Mi)
% 4.94/2.04 % (29153)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_28982 for (2997ds/39Mi)
% 4.94/2.06 % (29153)Instruction limit reached!
% 4.94/2.06 % (29153)------------------------------
% 4.94/2.06 % (29153)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.06 % (29153)Termination reason: Unknown
% 4.94/2.06 % (29153)Termination phase: Preprocessing 1
% 4.94/2.06
% 4.94/2.06 % (29153)Memory used [KB]: 1918
% 4.94/2.06 % (29153)Time elapsed: 0.014 s
% 4.94/2.06 % (29153)Instructions burned: 39 (million)
% 4.94/2.06 % (29153)------------------------------
% 4.94/2.06 % (29153)------------------------------
% 4.94/2.06 % (29138)Refutation not found, incomplete strategy
% 4.94/2.06 % (29138)------------------------------
% 4.94/2.06 % (29138)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.06 % (29138)Termination reason: Refutation not found, incomplete strategy
% 4.94/2.06
% 4.94/2.06
% 4.94/2.06 % (29138)Memory used [KB]: 7675
% 4.94/2.06 % (29138)Time elapsed: 0.052 s
% 4.94/2.06 % (29138)Instructions burned: 170 (million)
% 4.94/2.06 % (29138)------------------------------
% 4.94/2.06 % (29138)------------------------------
% 4.94/2.07 % (29163)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_28982 for (2996ds/6Mi)
% 4.94/2.07 % (29163)Instruction limit reached!
% 4.94/2.07 % (29163)------------------------------
% 4.94/2.07 % (29163)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.07 % (29163)Termination reason: Unknown
% 4.94/2.07 % (29163)Termination phase: shuffling
% 4.94/2.07
% 4.94/2.07 % (29163)Memory used [KB]: 1535
% 4.94/2.07 % (29163)Time elapsed: 0.003 s
% 4.94/2.07 % (29163)Instructions burned: 7 (million)
% 4.94/2.07 % (29163)------------------------------
% 4.94/2.07 % (29163)------------------------------
% 4.94/2.07 % (29170)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_28982 for (2996ds/1087Mi)
% 4.94/2.08 % (29115)Instruction limit reached!
% 4.94/2.08 % (29115)------------------------------
% 4.94/2.08 % (29115)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.08 % (29115)Termination reason: Unknown
% 4.94/2.08 % (29115)Termination phase: Saturation
% 4.94/2.08
% 4.94/2.08 % (29115)Memory used [KB]: 9083
% 4.94/2.08 % (29115)Time elapsed: 0.239 s
% 4.94/2.08 % (29115)Instructions burned: 779 (million)
% 4.94/2.08 % (29115)------------------------------
% 4.94/2.08 % (29115)------------------------------
% 4.94/2.08 % (29172)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_28982 for (2996ds/47Mi)
% 4.94/2.08 % (29150)Instruction limit reached!
% 4.94/2.08 % (29150)------------------------------
% 4.94/2.08 % (29150)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.08 % (29150)Termination reason: Unknown
% 4.94/2.08 % (29150)Termination phase: Saturation
% 4.94/2.08
% 4.94/2.08 % (29150)Memory used [KB]: 7164
% 4.94/2.08 % (29150)Time elapsed: 0.044 s
% 4.94/2.08 % (29150)Instructions burned: 149 (million)
% 4.94/2.08 % (29150)------------------------------
% 4.94/2.08 % (29150)------------------------------
% 4.94/2.09 % (29177)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_28982 for (2996ds/44Mi)
% 4.94/2.09 % (29178)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_28982 for (2996ds/432Mi)
% 4.94/2.09 % (29172)Instruction limit reached!
% 4.94/2.09 % (29172)------------------------------
% 4.94/2.09 % (29172)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.09 % (29172)Termination reason: Unknown
% 4.94/2.09 % (29172)Termination phase: Preprocessing 3
% 4.94/2.09
% 4.94/2.09 % (29172)Memory used [KB]: 2302
% 4.94/2.09 % (29172)Time elapsed: 0.017 s
% 4.94/2.09 % (29172)Instructions burned: 49 (million)
% 4.94/2.09 % (29172)------------------------------
% 4.94/2.09 % (29172)------------------------------
% 4.94/2.09 % (29099)Instruction limit reached!
% 4.94/2.09 % (29099)------------------------------
% 4.94/2.09 % (29099)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.09 % (29099)Termination reason: Unknown
% 4.94/2.09 % (29099)Termination phase: Saturation
% 4.94/2.09
% 4.94/2.09 % (29099)Memory used [KB]: 9722
% 4.94/2.09 % (29099)Time elapsed: 0.308 s
% 4.94/2.09 % (29099)Instructions burned: 1043 (million)
% 4.94/2.09 % (29099)------------------------------
% 4.94/2.09 % (29099)------------------------------
% 4.94/2.10 % (29177)Instruction limit reached!
% 4.94/2.10 % (29177)------------------------------
% 4.94/2.10 % (29177)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.10 % (29177)Termination reason: Unknown
% 4.94/2.10 % (29177)Termination phase: Property scanning
% 4.94/2.10
% 4.94/2.10 % (29177)Memory used [KB]: 2174
% 4.94/2.10 % (29177)Time elapsed: 0.014 s
% 4.94/2.10 % (29177)Instructions burned: 44 (million)
% 4.94/2.10 % (29177)------------------------------
% 4.94/2.10 % (29177)------------------------------
% 4.94/2.10 % (29184)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_28982 for (2996ds/62Mi)
% 4.94/2.10 % (29186)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_28982 for (2996ds/31Mi)
% 4.94/2.11 % (29188)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_28982 for (2996ds/96Mi)
% 5.55/2.11 % (29186)Instruction limit reached!
% 5.55/2.11 % (29186)------------------------------
% 5.55/2.11 % (29186)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.11 % (29186)Termination reason: Unknown
% 5.55/2.11 % (29186)Termination phase: Property scanning
% 5.55/2.11
% 5.55/2.11 % (29186)Memory used [KB]: 1918
% 5.55/2.11 % (29186)Time elapsed: 0.011 s
% 5.55/2.11 % (29186)Instructions burned: 32 (million)
% 5.55/2.11 % (29186)------------------------------
% 5.55/2.11 % (29186)------------------------------
% 5.55/2.12 % (29128)Instruction limit reached!
% 5.55/2.12 % (29128)------------------------------
% 5.55/2.12 % (29128)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.12 % (29128)Termination reason: Unknown
% 5.55/2.12 % (29128)Termination phase: Saturation
% 5.55/2.12
% 5.55/2.12 % (29128)Memory used [KB]: 8827
% 5.55/2.12 % (29128)Time elapsed: 0.155 s
% 5.55/2.12 % (29128)Instructions burned: 450 (million)
% 5.55/2.12 % (29128)------------------------------
% 5.55/2.12 % (29128)------------------------------
% 5.55/2.12 % (29184)Instruction limit reached!
% 5.55/2.12 % (29184)------------------------------
% 5.55/2.12 % (29184)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.12 % (29184)Termination reason: Unknown
% 5.55/2.12 % (29184)Termination phase: Property scanning
% 5.55/2.12
% 5.55/2.12 % (29184)Memory used [KB]: 2174
% 5.55/2.12 % (29184)Time elapsed: 0.018 s
% 5.55/2.12 % (29184)Instructions burned: 63 (million)
% 5.55/2.12 % (29184)------------------------------
% 5.55/2.12 % (29184)------------------------------
% 5.55/2.12 % (29117)First to succeed.
% 5.55/2.12 % (29117)Refutation found. Thanks to Tanya!
% 5.55/2.12 % SZS status Theorem for DTF2THF_28982
% 5.55/2.12 % SZS output start Proof for DTF2THF_28982
% 5.55/2.12 thf(type_def_6, type, set: ($tType) > $tType).
% 5.55/2.12 thf(type_def_7, type, list: ($tType) > $tType).
% 5.55/2.12 thf(type_def_8, type, nat: $tType).
% 5.55/2.12 thf(type_def_9, type, multiset: ($tType) > $tType).
% 5.55/2.12 thf(type_def_10, type, a: $tType).
% 5.55/2.12 thf(func_def_0, type, multiset: ($tType) > $tType).
% 5.55/2.12 thf(func_def_1, type, list: ($tType) > $tType).
% 5.55/2.12 thf(func_def_2, type, set: ($tType) > $tType).
% 5.55/2.12 thf(func_def_3, type, nat: $tType).
% 5.55/2.12 thf(func_def_4, type, a: $tType).
% 5.55/2.12 thf(func_def_5, type, dioid: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_6, type, type: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_7, type, size: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_9, type, order: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_10, type, no_bot: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_11, type, no_top: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_12, type, linorder: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_13, type, preorder: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_14, type, wellorder: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_15, type, dense_order: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_16, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_17, type, dense_linorder: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_18, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 5.55/2.12 thf(func_def_19, type, bij_betw: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1 > $o)).
% 5.55/2.12 thf(func_def_20, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 5.55/2.12 thf(func_def_21, type, groups501227526m_list: !>[X0: $tType]:(list @ X0 > X0)).
% 5.55/2.12 thf(func_def_22, type, hilbert_inv_into: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X1 > X0)).
% 5.55/2.12 thf(func_def_23, type, hilber580762119decomp: !>[X0: $tType]:(set @ X0 > list @ set @ X0 > $o)).
% 5.55/2.12 thf(func_def_24, type, list_ex: !>[X0: $tType]:((X0 > $o) > list @ X0 > $o)).
% 5.55/2.12 thf(func_def_25, type, list_update: !>[X0: $tType]:(list @ X0 > nat > X0 > list @ X0)).
% 5.55/2.12 thf(func_def_26, type, listset: !>[X0: $tType]:(list @ set @ X0 > set @ list @ X0)).
% 5.55/2.12 thf(func_def_27, type, nth: !>[X0: $tType]:(list @ X0 > nat > X0)).
% 5.55/2.12 thf(func_def_28, type, rev: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 5.55/2.12 thf(func_def_29, type, comm_m1079429914m_mset: !>[X0: $tType]:(multiset @ X0 > X0)).
% 5.55/2.12 thf(func_def_30, type, mset: !>[X0: $tType]:(list @ X0 > multiset @ X0)).
% 5.55/2.12 thf(func_def_31, type, set_mset: !>[X0: $tType]:(multiset @ X0 > set @ X0)).
% 5.55/2.12 thf(func_def_32, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 5.55/2.12 thf(func_def_33, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 5.55/2.12 thf(func_def_34, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 5.55/2.12 thf(func_def_35, type, perm: !>[X0: $tType]:(list @ X0 > list @ X0 > $o)).
% 5.55/2.12 thf(func_def_36, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 5.55/2.12 thf(func_def_37, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 5.55/2.12 thf(func_def_38, type, set_ord_lessThan: !>[X0: $tType]:(X0 > set @ X0)).
% 5.55/2.12 thf(func_def_39, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 5.55/2.12 thf(func_def_40, type, a2: set @ a).
% 5.55/2.12 thf(func_def_41, type, a3: a).
% 5.55/2.12 thf(func_def_42, type, f: nat > nat).
% 5.55/2.12 thf(func_def_43, type, g: nat > nat).
% 5.55/2.12 thf(func_def_44, type, qs2: list @ a).
% 5.55/2.12 thf(func_def_45, type, qs: list @ a).
% 5.55/2.12 thf(func_def_46, type, ss1: list @ set @ a).
% 5.55/2.12 thf(func_def_47, type, ss2: list @ set @ a).
% 5.55/2.12 thf(func_def_64, type, sK0: list @ a).
% 5.55/2.12 thf(func_def_65, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 5.55/2.12 thf(func_def_66, type, sK2: nat > nat).
% 5.55/2.12 thf(func_def_67, type, sK3: list @ a).
% 5.55/2.12 thf(func_def_68, type, sK4: nat > nat).
% 5.55/2.12 thf(func_def_69, type, sK5: !>[X0: $tType]:(X0 > list @ set @ X0 > list @ X0)).
% 5.55/2.12 thf(func_def_70, type, sK6: !>[X0: $tType]:(set @ X0 > list @ set @ X0 > X0)).
% 5.55/2.12 thf(func_def_71, type, sK7: !>[X0: $tType]:(list @ X0 > set @ X0 > list @ set @ X0 > list @ X0)).
% 5.55/2.12 thf(func_def_72, type, sK8: !>[X0: $tType]:(set @ X0 > list @ set @ X0 > list @ X0)).
% 5.55/2.12 thf(func_def_74, type, ph10: !>[X0: $tType]:(X0)).
% 5.55/2.12 thf(f2051,plain,(
% 5.55/2.12 $false),
% 5.55/2.12 inference(subsumption_resolution,[status(thm)],[f2050,f931])).
% 5.55/2.12 thf(f931,plain,(
% 5.55/2.12 ((groups501227526m_list @ a @ qs) != (groups501227526m_list @ a @ qs2))),
% 5.55/2.12 inference(definition_unfolding,[status(thm)],[f921,f869])).
% 5.55/2.12 thf(f869,plain,(
% 5.55/2.12 (a3 = (groups501227526m_list @ a @ qs))),
% 5.55/2.12 inference(cnf_transformation,[status(thm)],[f3])).
% 5.55/2.12 thf(f3,axiom,(
% 5.55/2.12 (a3 = (groups501227526m_list @ a @ qs))),
% 5.55/2.12 file('/export/starexec/sandbox/tmp/tmp.DwfRHP4Hjf/DTF2THF_28982.p',fact_2_a)).
% 5.55/2.12 thf(f921,plain,(
% 5.55/2.12 (a3 != (groups501227526m_list @ a @ qs2))),
% 5.55/2.12 inference(cnf_transformation,[status(thm)],[f819])).
% 5.55/2.12 thf(f819,plain,(
% 5.55/2.12 (a3 != (groups501227526m_list @ a @ qs2))),
% 5.55/2.12 inference(flattening,[status(thm)],[f289])).
% 5.55/2.12 thf(f289,negated_conjecture,(
% 5.55/2.12 ~(a3 = (groups501227526m_list @ a @ qs2))),
% 5.55/2.12 inference(negated_conjecture,[status(cth)],[f288])).
% 5.55/2.12 thf(f288,conjecture,(
% 5.55/2.12 (a3 = (groups501227526m_list @ a @ qs2))),
% 5.55/2.12 file('/export/starexec/sandbox/tmp/tmp.DwfRHP4Hjf/DTF2THF_28982.p',conj_0)).
% 5.55/2.12 thf(f2050,plain,(
% 5.55/2.12 ((groups501227526m_list @ a @ qs) = (groups501227526m_list @ a @ qs2))),
% 5.55/2.12 inference(forward_demodulation,[status(thm)],[f2043,f2017])).
% 5.55/2.12 thf(f2017,plain,(
% 5.55/2.12 ( ! [X0 : list @ a] : (((groups501227526m_list @ a @ X0) = (comm_m1079429914m_mset @ a @ (mset @ a @ X0)))) )),
% 5.55/2.12 inference(trivial_inequality_removal,[status(thm)],[f1988])).
% 5.55/2.12 thf(f1988,plain,(
% 5.55/2.12 ( ! [X0 : list @ a] : (((groups501227526m_list @ a @ X0) = (comm_m1079429914m_mset @ a @ (mset @ a @ X0))) | ($true != $true)) )),
% 5.55/2.12 inference(superposition,[status(thm)],[f920,f900])).
% 5.55/2.12 thf(f900,plain,(
% 5.55/2.12 (comm_monoid_add @ a = $true)),
% 5.55/2.12 inference(cnf_transformation,[status(thm)],[f330])).
% 5.55/2.12 thf(f330,plain,(
% 5.55/2.12 (comm_monoid_add @ a = $true)),
% 5.55/2.12 inference(fool_elimination,[status(thm)],[f329])).
% 5.55/2.12 thf(f329,plain,(
% 5.55/2.12 comm_monoid_add @ a),
% 5.55/2.12 inference(rectify,[status(thm)],[f287])).
% 5.55/2.12 thf(f287,axiom,(
% 5.55/2.12 comm_monoid_add @ a),
% 5.55/2.12 file('/export/starexec/sandbox/tmp/tmp.DwfRHP4Hjf/DTF2THF_28982.p',tfree_0)).
% 5.55/2.12 thf(f920,plain,(
% 5.55/2.12 ( ! [X0 : $tType,X1 : list @ X0] : ((comm_monoid_add @ X0 != $true) | ((groups501227526m_list @ X0 @ X1) = (comm_m1079429914m_mset @ X0 @ (mset @ X0 @ X1)))) )),
% 5.55/2.12 inference(cnf_transformation,[status(thm)],[f836])).
% 5.55/2.12 thf(f836,plain,(
% 5.55/2.12 ! [X0 : $tType] : ((comm_monoid_add @ X0 != $true) | ! [X1 : list @ X0] : ((groups501227526m_list @ X0 @ X1) = (comm_m1079429914m_mset @ X0 @ (mset @ X0 @ X1))))),
% 5.55/2.12 inference(ennf_transformation,[status(thm)],[f618])).
% 5.55/2.12 thf(f618,plain,(
% 5.55/2.12 ! [X0 : $tType] : ((comm_monoid_add @ X0 = $true) => ! [X1 : list @ X0] : ((groups501227526m_list @ X0 @ X1) = (comm_m1079429914m_mset @ X0 @ (mset @ X0 @ X1))))),
% 5.55/2.12 inference(fool_elimination,[status(thm)],[f617])).
% 5.55/2.12 thf(f617,plain,(
% 5.55/2.12 ! [X0 : $tType] : (comm_monoid_add @ X0 => ! [X1 : list @ X0] : ((groups501227526m_list @ X0 @ X1) = (comm_m1079429914m_mset @ X0 @ (mset @ X0 @ X1))))),
% 5.55/2.12 inference(rectify,[status(thm)],[f14])).
% 5.55/2.12 thf(f14,axiom,(
% 5.55/2.12 ! [X0 : $tType] : (comm_monoid_add @ X0 => ! [X11 : list @ X0] : ((groups501227526m_list @ X0 @ X11) = (comm_m1079429914m_mset @ X0 @ (mset @ X0 @ X11))))),
% 5.55/2.12 file('/export/starexec/sandbox/tmp/tmp.DwfRHP4Hjf/DTF2THF_28982.p',fact_13_sum__mset__sum__list)).
% 5.55/2.12 thf(f2043,plain,(
% 5.55/2.12 ((groups501227526m_list @ a @ qs) = (comm_m1079429914m_mset @ a @ (mset @ a @ qs2)))),
% 5.55/2.12 inference(superposition,[status(thm)],[f2017,f883])).
% 5.55/2.12 thf(f883,plain,(
% 5.55/2.12 ((mset @ a @ qs2) = (mset @ a @ qs))),
% 5.55/2.12 inference(cnf_transformation,[status(thm)],[f1])).
% 5.55/2.12 thf(f1,axiom,(
% 5.55/2.12 ((mset @ a @ qs2) = (mset @ a @ qs))),
% 5.55/2.12 file('/export/starexec/sandbox/tmp/tmp.DwfRHP4Hjf/DTF2THF_28982.p',fact_0__092_060open_062mset_Aqs2_A_061_Amset_Aqs_092_060close_062)).
% 5.55/2.12 % SZS output end Proof for DTF2THF_28982
% 5.55/2.12 % (29117)------------------------------
% 5.55/2.12 % (29117)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.12 % (29117)Termination reason: Refutation
% 5.55/2.12
% 5.55/2.12 % (29117)Memory used [KB]: 8187
% 5.55/2.12 % (29117)Time elapsed: 0.266 s
% 5.55/2.12 % (29117)Instructions burned: 832 (million)
% 5.55/2.12 % (29117)------------------------------
% 5.55/2.12 % (29117)------------------------------
% 5.55/2.12 % (29087)Success in time 0.372 s
%------------------------------------------------------------------------------