%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP155^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n027.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 08:56:00 AM UTC 2026
% Result : Theorem 21.23s 4.34s
% Output : Refutation 21.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP155^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n027.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:28:22 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.45 ---- Original DTF file ---
% 0.30/0.45 thf(spec,logic,$$dhol).
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 % File : ITP155^2 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.45 % Domain : Interactive Theorem Proving
% 0.30/0.45 % Problem : Sledgehammer Preferences problem prob_260__6251520_1
% 0.30/0.45 % Version : Especial.
% 0.30/0.45 % English :
% 0.30/0.45
% 0.30/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.45 % Source : [Des21]
% 0.30/0.45 % Names : Preferences/prob_260__6251520_1 [Des21]
% 0.30/0.45
% 0.30/0.45 % Status : Theorem
% 0.30/0.45 % Rating : 0.00 v7.5.0
% 0.30/0.45 % Syntax : Number of formulae : 336 ( 87 unt; 43 typ; 0 def)
% 0.30/0.45 % Number of atoms : 932 ( 162 equ; 0 cnn)
% 0.30/0.45 % Maximal formula atoms : 13 ( 3 avg)
% 0.30/0.45 % Number of connectives : 4853 ( 92 ~; 21 |; 97 &;4183 @)
% 0.30/0.45 % ( 0 <=>; 460 =>; 0 <=; 0 <~>)
% 0.30/0.45 % Maximal formula depth : 28 ( 9 avg)
% 0.30/0.45 % Number of types : 3 ( 2 usr)
% 0.30/0.45 % Number of type conns : 118 ( 118 >; 0 *; 0 +; 0 <<)
% 0.30/0.45 % Number of symbols : 43 ( 41 usr; 5 con; 0-5 aty)
% 0.30/0.45 % Number of variables : 1060 ( 50 ^; 949 !; 20 ?;1060 :)
% 0.30/0.45 % ( 41 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.45 % SPC : TH1_THM_EQU_NAR
% 0.30/0.45
% 0.30/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 16:29:29.951
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 % Could-be-implicit typings (4)
% 0.30/0.45 thf(ty_t_Product__Type_Oprod,type,
% 0.30/0.45 product_prod: $tType > $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_Set_Oset,type,
% 0.30/0.45 set: $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_Nat_Onat,type,
% 0.30/0.45 nat: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_tf_a,type,
% 0.30/0.45 a: $tType ).
% 0.30/0.45
% 0.30/0.45 % Explicit typings (39)
% 0.30/0.45 thf(sy_cl_HOL_Otype,type,
% 0.30/0.45 type:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Groups_Ozero,type,
% 0.30/0.45 zero:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Obot,type,
% 0.30/0.45 bot:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Oord,type,
% 0.30/0.45 ord:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.30/0.45 order:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.30/0.45 finite_finite:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/0.45 linorder:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/0.45 preorder:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.30/0.45 order_bot:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.30/0.45 dense_linorder:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.30/0.45 linord1659791738miring:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.30/0.45 canoni770627133id_add:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.30/0.45 bNF_Ca1785829860lChain:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Finite__Set_Ocard,type,
% 0.30/0.45 finite_card:
% 0.30/0.45 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Finite__Set_Ofinite,type,
% 0.30/0.45 finite_finite2:
% 0.30/0.45 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.30/0.45 zero_zero:
% 0.30/0.45 !>[A: $tType] : A ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Order__Relation_Opreorder__on,type,
% 0.30/0.45 order_preorder_on:
% 0.30/0.45 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.30/0.45 bot_bot:
% 0.30/0.45 !>[A: $tType] : A ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.30/0.45 ord_less:
% 0.30/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.45 ord_less_eq:
% 0.30/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oas__good__as,type,
% 0.30/0.45 prefer951318096ood_as:
% 0.30/0.45 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good,type,
% 0.30/0.45 prefer310429814s_good:
% 0.30/0.45 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Ono__better__than,type,
% 0.30/0.45 prefer1532642881r_than:
% 0.30/0.45 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference,type,
% 0.30/0.45 prefer199794634erence:
% 0.30/0.45 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference,type,
% 0.30/0.45 prefer1997167224erence:
% 0.30/0.45 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference__axioms,type,
% 0.30/0.45 prefer1801827867axioms:
% 0.30/0.45 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_OPair,type,
% 0.30/0.45 product_Pair:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.30/0.45 produc2004651681e_prod:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.30/0.45 product_rec_prod:
% 0.30/0.45 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Relation_Orefl__on,type,
% 0.30/0.45 refl_on:
% 0.30/0.45 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Relation_Ototal__on,type,
% 0.30/0.45 total_on:
% 0.30/0.45 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Relation_Otrans,type,
% 0.30/0.45 trans:
% 0.30/0.45 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Set_OCollect,type,
% 0.30/0.45 collect:
% 0.30/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Set_Ois__empty,type,
% 0.30/0.45 is_empty:
% 0.30/0.45 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member,type,
% 0.30/0.45 member:
% 0.30/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_carrier,type,
% 0.30/0.45 carrier: set @ a ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_relation,type,
% 0.30/0.45 relation: set @ ( product_prod @ a @ a ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_x,type,
% 0.30/0.45 x: a ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_y,type,
% 0.30/0.45 y: a ).
% 0.30/0.45
% 0.30/0.45 % Relevant facts (256)
% 0.30/0.45 thf(fact_0_assms_I2_J,axiom,
% 0.30/0.45 member @ a @ y @ carrier ).
% 0.30/0.45
% 0.30/0.45 % assms(2)
% 0.30/0.45 thf(fact_1_assms_I1_J,axiom,
% 0.30/0.45 member @ a @ x @ carrier ).
% 0.30/0.45
% 0.30/0.45 % assms(1)
% 0.30/0.45 thf(fact_2_xy__in__eachothers__nbt,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( member @ a @ X @ carrier )
% 0.30/0.45 => ( ( member @ a @ Y @ carrier )
% 0.30/0.45 => ( ( member @ a @ X @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) )
% 0.30/0.45 | ( member @ a @ Y @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % xy_in_eachothers_nbt
% 0.30/0.45 thf(fact_3__092_060open_062card_A_Ino__better__than_Ax_Acarrier_Arelation_J_A_092_060le_062_Acard_A_Ino__better__than_Ay_Acarrier_Arelation_J_092_060close_062,axiom,
% 0.30/0.45 ord_less_eq @ nat @ ( finite_card @ a @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) ) @ ( finite_card @ a @ ( prefer1532642881r_than @ a @ y @ carrier @ relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % \<open>card (no_better_than x carrier relation) \<le> card (no_better_than y carrier relation)\<close>
% 0.30/0.45 thf(fact_4_nbt__nest,axiom,
% 0.30/0.45 ! [Y: a,X: a] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) )
% 0.30/0.45 | ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % nbt_nest
% 0.30/0.45 thf(fact_5_nbt__subset__carrier,axiom,
% 0.30/0.45 ! [X: a] :
% 0.30/0.45 ( ( member @ a @ X @ carrier )
% 0.30/0.45 => ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ carrier ) ) ).
% 0.30/0.45
% 0.30/0.45 % nbt_subset_carrier
% 0.30/0.45 thf(fact_6_rational__preference__axioms,axiom,
% 0.30/0.45 prefer1997167224erence @ a @ carrier @ relation ).
% 0.30/0.45
% 0.30/0.45 % rational_preference_axioms
% 0.30/0.45 thf(fact_7__092_060open_062finite_Acarrier_092_060close_062,axiom,
% 0.30/0.45 finite_finite2 @ a @ carrier ).
% 0.30/0.45
% 0.30/0.45 % \<open>finite carrier\<close>
% 0.30/0.45 thf(fact_8_preference__axioms,axiom,
% 0.30/0.45 prefer199794634erence @ a @ carrier @ relation ).
% 0.30/0.45
% 0.30/0.45 % preference_axioms
% 0.30/0.45 thf(fact_9_trans__refl,axiom,
% 0.30/0.45 order_preorder_on @ a @ carrier @ relation ).
% 0.30/0.45
% 0.30/0.45 % trans_refl
% 0.30/0.45 thf(fact_10_nbt__subset,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( finite_finite2 @ a @ carrier )
% 0.30/0.45 => ( ( member @ a @ X @ carrier )
% 0.30/0.45 => ( ( member @ a @ Y @ carrier )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) )
% 0.30/0.45 | ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % nbt_subset
% 0.30/0.45 thf(fact_11_no__better__subset__pref,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 => ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % no_better_subset_pref
% 0.30/0.45 thf(fact_12_no__better__thansubset__rel,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( member @ a @ X @ carrier )
% 0.30/0.45 => ( ( member @ a @ Y @ carrier )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) )
% 0.30/0.45 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % no_better_thansubset_rel
% 0.30/0.45 thf(fact_13_no__better__than__nonepty,axiom,
% 0.30/0.45 ! [X: a] :
% 0.30/0.45 ( ( carrier
% 0.30/0.45 != ( bot_bot @ ( set @ a ) ) )
% 0.30/0.45 => ( ( member @ a @ X @ carrier )
% 0.30/0.45 => ( ( prefer1532642881r_than @ a @ X @ carrier @ relation )
% 0.30/0.45 != ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % no_better_than_nonepty
% 0.30/0.45 thf(fact_14_total,axiom,
% 0.30/0.45 total_on @ a @ carrier @ relation ).
% 0.30/0.45
% 0.30/0.45 % total
% 0.30/0.45 thf(fact_15_fnt__carrier__fnt__nbt,axiom,
% 0.30/0.45 ! [X2: a] :
% 0.30/0.45 ( ( member @ a @ X2 @ carrier )
% 0.30/0.45 => ( finite_finite2 @ a @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % fnt_carrier_fnt_nbt
% 0.30/0.45 thf(fact_16_same__nbt__same__pref,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( member @ a @ X @ carrier )
% 0.30/0.45 => ( ( member @ a @ Y @ carrier )
% 0.30/0.45 => ( ( ( member @ a @ X @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) )
% 0.30/0.45 & ( member @ a @ Y @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) )
% 0.30/0.45 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % same_nbt_same_pref
% 0.30/0.45 thf(fact_17_worse__in__no__better,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 => ( member @ a @ Y @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % worse_in_no_better
% 0.30/0.45 thf(fact_18_reflexivity,axiom,
% 0.30/0.45 refl_on @ a @ carrier @ relation ).
% 0.30/0.45
% 0.30/0.45 % reflexivity
% 0.30/0.45 thf(fact_19_compl,axiom,
% 0.30/0.45 ! [X2: a] :
% 0.30/0.45 ( ( member @ a @ X2 @ carrier )
% 0.30/0.45 => ! [Xa: a] :
% 0.30/0.45 ( ( member @ a @ Xa @ carrier )
% 0.30/0.45 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Xa ) @ relation )
% 0.30/0.45 | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Xa @ X2 ) @ relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % compl
% 0.30/0.45 thf(fact_20_completeD,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( member @ a @ X @ carrier )
% 0.30/0.45 => ( ( member @ a @ Y @ carrier )
% 0.30/0.45 => ( ( X != Y )
% 0.30/0.45 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % completeD
% 0.30/0.45 thf(fact_21_strict__trans,axiom,
% 0.30/0.45 ! [X: a,Y: a,Z: a] :
% 0.30/0.45 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.30/0.45 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
% 0.30/0.45 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % strict_trans
% 0.30/0.45 thf(fact_22_indiff__trans,axiom,
% 0.30/0.45 ! [X: a,Y: a,Z: a] :
% 0.30/0.45 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.30/0.45 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 0.30/0.45 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
% 0.30/0.45 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
% 0.30/0.45 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % indiff_trans
% 0.30/0.45 thf(fact_23_indifferent__imp__weak__pref_I1_J,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.30/0.45 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % indifferent_imp_weak_pref(1)
% 0.30/0.45 thf(fact_24_indifferent__imp__weak__pref_I2_J,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.30/0.45 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % indifferent_imp_weak_pref(2)
% 0.30/0.45 thf(fact_25_weak__is__transitive,axiom,
% 0.30/0.45 ! [X: a,Y: a,Z: a] :
% 0.30/0.45 ( ( ( member @ a @ X @ carrier )
% 0.30/0.45 & ( member @ a @ Y @ carrier )
% 0.30/0.45 & ( member @ a @ Z @ carrier ) )
% 0.30/0.45 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 0.30/0.45 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % weak_is_transitive
% 0.30/0.45 thf(fact_26_strict__is__neg__transitive,axiom,
% 0.30/0.45 ! [X: a,Y: a,Z: a] :
% 0.30/0.45 ( ( ( member @ a @ X @ carrier )
% 0.30/0.45 & ( member @ a @ Y @ carrier )
% 0.30/0.45 & ( member @ a @ Z @ carrier ) )
% 0.30/0.45 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.30/0.45 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) )
% 0.30/0.45 | ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % strict_is_neg_transitive
% 0.30/0.45 thf(fact_27_not__outside,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 => ( member @ a @ X @ carrier ) ) ).
% 0.30/0.45
% 0.30/0.45 % not_outside
% 0.30/0.45 thf(fact_28_strict__not__refl__weak,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( ( member @ a @ X @ carrier )
% 0.30/0.45 & ( member @ a @ Y @ carrier ) )
% 0.30/0.45 => ( ( ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.30/0.45 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % strict_not_refl_weak
% 0.30/0.45 thf(fact_29_rational__preference_Oaxioms_I1_J,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.axioms(1)
% 0.30/0.45 thf(fact_30_rational__preference_Oindifferent__imp__weak__pref_I2_J,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.30/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.indifferent_imp_weak_pref(2)
% 0.30/0.45 thf(fact_31_rational__preference_Oindifferent__imp__weak__pref_I1_J,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.30/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.indifferent_imp_weak_pref(1)
% 0.30/0.45 thf(fact_32_preference__def,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( prefer199794634erence @ A )
% 0.30/0.45 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ! [X3: A,Y2: A] :
% 0.30/0.45 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
% 0.30/0.45 => ( member @ A @ X3 @ Carrier2 ) )
% 0.30/0.45 & ! [X3: A,Y2: A] :
% 0.30/0.45 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
% 0.30/0.45 => ( member @ A @ Y2 @ Carrier2 ) )
% 0.30/0.45 & ( order_preorder_on @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % preference_def
% 0.30/0.45 thf(fact_33_preference_Ointro,axiom,
% 0.30/0.45 ! [A: $tType,Relation: set @ ( product_prod @ A @ A ),Carrier: set @ A] :
% 0.30/0.45 ( ! [X4: A,Y3: A] :
% 0.30/0.45 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
% 0.30/0.45 => ( member @ A @ X4 @ Carrier ) )
% 0.30/0.45 => ( ! [X4: A,Y3: A] :
% 0.30/0.45 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
% 0.30/0.45 => ( member @ A @ Y3 @ Carrier ) )
% 0.30/0.45 => ( ( order_preorder_on @ A @ Carrier @ Relation )
% 0.30/0.45 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % preference.intro
% 0.30/0.45 thf(fact_34_preference_Otrans__refl,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( order_preorder_on @ A @ Carrier @ Relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % preference.trans_refl
% 0.30/0.45 thf(fact_35_preference_Onot__outside,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 => ( member @ A @ X @ Carrier ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % preference.not_outside
% 0.30/0.45 thf(fact_36_preference_Oreflexivity,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( refl_on @ A @ Carrier @ Relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % preference.reflexivity
% 0.30/0.45 thf(fact_37_preference_Oindiff__trans,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.30/0.45 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 0.30/0.45 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 0.30/0.45 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % preference.indiff_trans
% 0.30/0.45 thf(fact_38_rational__preference_Ocompl,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ! [X2: A] :
% 0.30/0.45 ( ( member @ A @ X2 @ Carrier )
% 0.30/0.45 => ! [Xa: A] :
% 0.30/0.45 ( ( member @ A @ Xa @ Carrier )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa ) @ Relation )
% 0.30/0.45 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.compl
% 0.30/0.45 thf(fact_39_rational__preference_Ototal,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( total_on @ A @ Carrier @ Relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.total
% 0.30/0.45 thf(fact_40_rational__preference_OcompleteD,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( member @ A @ X @ Carrier )
% 0.30/0.45 => ( ( member @ A @ Y @ Carrier )
% 0.30/0.45 => ( ( X != Y )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.completeD
% 0.30/0.45 thf(fact_41_rational__preference_Ostrict__trans,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.30/0.45 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.strict_trans
% 0.30/0.45 thf(fact_42_rational__preference_Oweak__is__transitive,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( ( member @ A @ X @ Carrier )
% 0.30/0.45 & ( member @ A @ Y @ Carrier )
% 0.30/0.45 & ( member @ A @ Z @ Carrier ) )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 0.30/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.weak_is_transitive
% 0.30/0.45 thf(fact_43_rational__preference_Ofnt__carrier__fnt__rel,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( finite_finite2 @ A @ Carrier )
% 0.30/0.45 => ( finite_finite2 @ ( product_prod @ A @ A ) @ Relation ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.fnt_carrier_fnt_rel
% 0.30/0.45 thf(fact_44_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( ( member @ A @ X @ Carrier )
% 0.30/0.45 & ( member @ A @ Y @ Carrier ) )
% 0.30/0.45 => ( ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.30/0.45 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.strict_not_refl_weak
% 0.30/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.45 ! [A: $tType,A2: A,P: A > $o] :
% 0.30/0.45 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.30/0.45 = ( P @ A2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % mem_Collect_eq
% 0.30/0.45 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A] :
% 0.30/0.45 ( ( collect @ A
% 0.30/0.45 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.30/0.45 = A3 ) ).
% 0.30/0.45
% 0.30/0.45 % Collect_mem_eq
% 0.30/0.45 thf(fact_47_Collect__cong,axiom,
% 0.30/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.45 ( ! [X4: A] :
% 0.30/0.45 ( ( P @ X4 )
% 0.30/0.45 = ( Q @ X4 ) )
% 0.30/0.45 => ( ( collect @ A @ P )
% 0.30/0.45 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Collect_cong
% 0.30/0.45 thf(fact_48_ext,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.45 ( ! [X4: A] :
% 0.30/0.45 ( ( F @ X4 )
% 0.30/0.45 = ( G @ X4 ) )
% 0.30/0.45 => ( F = G ) ) ).
% 0.30/0.45
% 0.30/0.45 % ext
% 0.30/0.45 thf(fact_49_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( ( member @ A @ X @ Carrier )
% 0.30/0.45 & ( member @ A @ Y @ Carrier )
% 0.30/0.45 & ( member @ A @ Z @ Carrier ) )
% 0.30/0.45 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.30/0.45 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) )
% 0.30/0.45 | ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 0.30/0.45 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.strict_is_neg_transitive
% 0.30/0.45 thf(fact_50_rational__preference_Oworse__in__no__better,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 => ( member @ A @ Y @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.worse_in_no_better
% 0.30/0.45 thf(fact_51_rational__preference_Osame__nbt__same__pref,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( member @ A @ X @ Carrier )
% 0.30/0.45 => ( ( member @ A @ Y @ Carrier )
% 0.30/0.45 => ( ( ( member @ A @ X @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) )
% 0.30/0.45 & ( member @ A @ Y @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) )
% 0.30/0.45 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.same_nbt_same_pref
% 0.30/0.45 thf(fact_52_rational__preference_Ofnt__carrier__fnt__nbt,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( finite_finite2 @ A @ Carrier )
% 0.30/0.45 => ! [X2: A] :
% 0.30/0.45 ( ( member @ A @ X2 @ Carrier )
% 0.30/0.45 => ( finite_finite2 @ A @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.fnt_carrier_fnt_nbt
% 0.30/0.45 thf(fact_53_rational__preference_Ono__better__than__nonepty,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( Carrier
% 0.30/0.45 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.45 => ( ( member @ A @ X @ Carrier )
% 0.30/0.45 => ( ( prefer1532642881r_than @ A @ X @ Carrier @ Relation )
% 0.30/0.45 != ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.no_better_than_nonepty
% 0.30/0.45 thf(fact_54_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( member @ A @ X @ Carrier )
% 0.30/0.45 => ( ( member @ A @ Y @ Carrier )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) )
% 0.30/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.no_better_thansubset_rel
% 0.30/0.45 thf(fact_55_rational__preference_Ono__better__subset__pref,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.no_better_subset_pref
% 0.30/0.45 thf(fact_56_rational__preference_Onbt__subset,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( finite_finite2 @ A @ Carrier )
% 0.30/0.45 => ( ( member @ A @ X @ Carrier )
% 0.30/0.45 => ( ( member @ A @ Y @ Carrier )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) )
% 0.30/0.45 | ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.nbt_subset
% 0.30/0.45 thf(fact_57_rational__preference_Oxy__in__eachothers__nbt,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( member @ A @ X @ Carrier )
% 0.30/0.45 => ( ( member @ A @ Y @ Carrier )
% 0.30/0.45 => ( ( member @ A @ X @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) )
% 0.30/0.45 | ( member @ A @ Y @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.xy_in_eachothers_nbt
% 0.30/0.45 thf(fact_58_rational__preference_Onbt__subset__carrier,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( member @ A @ X @ Carrier )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ Carrier ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.nbt_subset_carrier
% 0.30/0.45 thf(fact_59_rational__preference_Onbt__nest,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Y: A,X: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) )
% 0.30/0.45 | ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.nbt_nest
% 0.30/0.45 thf(fact_60_at__lst__asgd__not__ge,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( carrier
% 0.30/0.45 != ( bot_bot @ ( set @ a ) ) )
% 0.30/0.45 => ( ( member @ a @ X @ carrier )
% 0.30/0.45 => ( ( member @ a @ Y @ carrier )
% 0.30/0.45 => ( ~ ( member @ a @ X @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) )
% 0.30/0.45 => ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % at_lst_asgd_not_ge
% 0.30/0.45 thf(fact_61_same__at__least__as__equal,axiom,
% 0.30/0.45 ! [Z: a,Y: a] :
% 0.30/0.45 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 0.30/0.45 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) )
% 0.30/0.45 => ( ( prefer310429814s_good @ a @ Z @ carrier @ relation )
% 0.30/0.45 = ( prefer310429814s_good @ a @ Y @ carrier @ relation ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % same_at_least_as_equal
% 0.30/0.45 thf(fact_62_pref__in__at__least__as,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 => ( member @ a @ X @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % pref_in_at_least_as
% 0.30/0.45 thf(fact_63_empty__subsetI,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 0.30/0.45
% 0.30/0.45 % empty_subsetI
% 0.30/0.45 thf(fact_64_subset__empty,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.45 = ( A3
% 0.30/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subset_empty
% 0.30/0.45 thf(fact_65_Pair__le,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( ord @ B )
% 0.30/0.45 & ( ord @ A ) )
% 0.30/0.45 => ! [A2: A,B2: B,C2: A,D: B] :
% 0.30/0.45 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
% 0.30/0.45 = ( ( ord_less_eq @ A @ A2 @ C2 )
% 0.30/0.45 & ( ord_less_eq @ B @ B2 @ D ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Pair_le
% 0.30/0.45 thf(fact_66_as__good__as__sameIff,axiom,
% 0.30/0.45 ! [Z: a,Y: a] :
% 0.30/0.45 ( ( member @ a @ Z @ ( prefer951318096ood_as @ a @ Y @ carrier @ relation ) )
% 0.30/0.45 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 0.30/0.45 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % as_good_as_sameIff
% 0.30/0.45 thf(fact_67_as__good__asIff,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( member @ a @ X @ ( prefer951318096ood_as @ a @ Y @ carrier @ relation ) )
% 0.30/0.45 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/0.45 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % as_good_asIff
% 0.30/0.45 thf(fact_68_fnt__carrier__fnt__rel,axiom,
% 0.30/0.45 ( ( finite_finite2 @ a @ carrier )
% 0.30/0.45 => ( finite_finite2 @ ( product_prod @ a @ a ) @ relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % fnt_carrier_fnt_rel
% 0.30/0.45 thf(fact_69_card__mono,axiom,
% 0.30/0.45 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.30/0.45 ( ( finite_finite2 @ A @ B3 )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/0.45 => ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % card_mono
% 0.30/0.45 thf(fact_70_empty__Collect__eq,axiom,
% 0.30/0.45 ! [A: $tType,P: A > $o] :
% 0.30/0.45 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.45 = ( collect @ A @ P ) )
% 0.30/0.45 = ( ! [X3: A] :
% 0.30/0.45 ~ ( P @ X3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % empty_Collect_eq
% 0.30/0.45 thf(fact_71_Collect__empty__eq,axiom,
% 0.30/0.45 ! [A: $tType,P: A > $o] :
% 0.30/0.45 ( ( ( collect @ A @ P )
% 0.30/0.45 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.45 = ( ! [X3: A] :
% 0.30/0.45 ~ ( P @ X3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Collect_empty_eq
% 0.30/0.45 thf(fact_72_all__not__in__conv,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A] :
% 0.30/0.45 ( ( ! [X3: A] :
% 0.30/0.45 ~ ( member @ A @ X3 @ A3 ) )
% 0.30/0.45 = ( A3
% 0.30/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % all_not_in_conv
% 0.30/0.45 thf(fact_73_empty__iff,axiom,
% 0.30/0.45 ! [A: $tType,C2: A] :
% 0.30/0.45 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % empty_iff
% 0.30/0.45 thf(fact_74_subsetI,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/0.45 ( ! [X4: A] :
% 0.30/0.45 ( ( member @ A @ X4 @ A3 )
% 0.30/0.45 => ( member @ A @ X4 @ B3 ) )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % subsetI
% 0.30/0.45 thf(fact_75_subset__antisym,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 0.30/0.45 => ( A3 = B3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subset_antisym
% 0.30/0.45 thf(fact_76_finite__code,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( finite_finite @ A )
% 0.30/0.45 => ( ( finite_finite2 @ A )
% 0.30/0.45 = ( ^ [A4: set @ A] : $true ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % finite_code
% 0.30/0.45 thf(fact_77_at__least__as__goodD,axiom,
% 0.30/0.45 ! [A: $tType,Z: A,Y: A,B3: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( member @ A @ Z @ ( prefer310429814s_good @ A @ Y @ B3 @ Pr ) )
% 0.30/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Pr ) ) ).
% 0.30/0.45
% 0.30/0.45 % at_least_as_goodD
% 0.30/0.45 thf(fact_78_at__lst__asgd__ge,axiom,
% 0.30/0.45 ! [A: $tType,X: A,Y: A,B3: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ B3 @ Pr ) )
% 0.30/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Pr ) ) ).
% 0.30/0.45
% 0.30/0.45 % at_lst_asgd_ge
% 0.30/0.45 thf(fact_79_ex__in__conv,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A] :
% 0.30/0.45 ( ( ? [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.30/0.45 = ( A3
% 0.30/0.45 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ex_in_conv
% 0.30/0.45 thf(fact_80_equals0I,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A] :
% 0.30/0.45 ( ! [Y3: A] :
% 0.30/0.45 ~ ( member @ A @ Y3 @ A3 )
% 0.30/0.45 => ( A3
% 0.30/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % equals0I
% 0.30/0.45 thf(fact_81_equals0D,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,A2: A] :
% 0.30/0.45 ( ( A3
% 0.30/0.45 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.45 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % equals0D
% 0.30/0.45 thf(fact_82_emptyE,axiom,
% 0.30/0.45 ! [A: $tType,A2: A] :
% 0.30/0.45 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % emptyE
% 0.30/0.45 thf(fact_83_in__mono,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,B3: set @ A,X: A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/0.45 => ( ( member @ A @ X @ A3 )
% 0.30/0.45 => ( member @ A @ X @ B3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % in_mono
% 0.30/0.45 thf(fact_84_subsetD,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/0.45 => ( ( member @ A @ C2 @ A3 )
% 0.30/0.45 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subsetD
% 0.30/0.45 thf(fact_85_equalityE,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/0.45 ( ( A3 = B3 )
% 0.30/0.45 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/0.45 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % equalityE
% 0.30/0.45 thf(fact_86_subset__eq,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.45 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.30/0.45 ! [X3: A] :
% 0.30/0.45 ( ( member @ A @ X3 @ A4 )
% 0.30/0.45 => ( member @ A @ X3 @ B4 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subset_eq
% 0.30/0.45 thf(fact_87_equalityD1,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/0.45 ( ( A3 = B3 )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % equalityD1
% 0.30/0.45 thf(fact_88_equalityD2,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/0.45 ( ( A3 = B3 )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % equalityD2
% 0.30/0.45 thf(fact_89_subset__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.45 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.30/0.45 ! [T2: A] :
% 0.30/0.45 ( ( member @ A @ T2 @ A4 )
% 0.30/0.45 => ( member @ A @ T2 @ B4 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subset_iff
% 0.30/0.45 thf(fact_90_subset__refl,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.30/0.45
% 0.30/0.45 % subset_refl
% 0.30/0.45 thf(fact_91_Collect__mono,axiom,
% 0.30/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.45 ( ! [X4: A] :
% 0.30/0.45 ( ( P @ X4 )
% 0.30/0.45 => ( Q @ X4 ) )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Collect_mono
% 0.30/0.45 thf(fact_92_subset__trans,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subset_trans
% 0.30/0.45 thf(fact_93_set__eq__subset,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ^ [Y4: set @ A,Z2: set @ A] : ( Y4 = Z2 ) )
% 0.30/0.45 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.45 & ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % set_eq_subset
% 0.30/0.45 thf(fact_94_Collect__mono__iff,axiom,
% 0.30/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.30/0.45 = ( ! [X3: A] :
% 0.30/0.45 ( ( P @ X3 )
% 0.30/0.45 => ( Q @ X3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Collect_mono_iff
% 0.30/0.45 thf(fact_95_finite,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( finite_finite @ A )
% 0.30/0.45 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % finite
% 0.30/0.45 thf(fact_96_finite__set__choice,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 0.30/0.45 ( ( finite_finite2 @ A @ A3 )
% 0.30/0.45 => ( ! [X4: A] :
% 0.30/0.45 ( ( member @ A @ X4 @ A3 )
% 0.30/0.45 => ? [X_1: B] : ( P @ X4 @ X_1 ) )
% 0.30/0.45 => ? [F2: A > B] :
% 0.30/0.45 ! [X2: A] :
% 0.30/0.45 ( ( member @ A @ X2 @ A3 )
% 0.30/0.45 => ( P @ X2 @ ( F2 @ X2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % finite_set_choice
% 0.30/0.45 thf(fact_97_rational__preference_Osame__at__least__as__equal,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Z: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 0.30/0.45 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) )
% 0.30/0.45 => ( ( prefer310429814s_good @ A @ Z @ Carrier @ Relation )
% 0.30/0.45 = ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.same_at_least_as_equal
% 0.30/0.45 thf(fact_98_rational__preference_Opref__in__at__least__as,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 => ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.pref_in_at_least_as
% 0.30/0.45 thf(fact_99_rational__preference_Oas__good__as__sameIff,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Z: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( member @ A @ Z @ ( prefer951318096ood_as @ A @ Y @ Carrier @ Relation ) )
% 0.30/0.45 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 0.30/0.45 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.as_good_as_sameIff
% 0.30/0.45 thf(fact_100_rational__preference_Oas__good__asIff,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( member @ A @ X @ ( prefer951318096ood_as @ A @ Y @ Carrier @ Relation ) )
% 0.30/0.45 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.45 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.as_good_asIff
% 0.30/0.45 thf(fact_101_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( Carrier
% 0.30/0.45 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.45 => ( ( member @ A @ X @ Carrier )
% 0.30/0.45 => ( ( member @ A @ Y @ Carrier )
% 0.30/0.45 => ( ~ ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) )
% 0.30/0.45 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.at_lst_asgd_not_ge
% 0.30/0.45 thf(fact_102_Pair__mono,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( ord @ B )
% 0.30/0.45 & ( ord @ A ) )
% 0.30/0.45 => ! [X: A,X5: A,Y: B,Y5: B] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ X5 )
% 0.30/0.45 => ( ( ord_less_eq @ B @ Y @ Y5 )
% 0.30/0.45 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Pair_mono
% 0.30/0.45 thf(fact_103_finite__has__maximal2,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [A3: set @ A,A2: A] :
% 0.30/0.45 ( ( finite_finite2 @ A @ A3 )
% 0.30/0.45 => ( ( member @ A @ A2 @ A3 )
% 0.30/0.45 => ? [X4: A] :
% 0.30/0.45 ( ( member @ A @ X4 @ A3 )
% 0.30/0.45 & ( ord_less_eq @ A @ A2 @ X4 )
% 0.30/0.45 & ! [Xa: A] :
% 0.30/0.45 ( ( member @ A @ Xa @ A3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ X4 @ Xa )
% 0.30/0.45 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % finite_has_maximal2
% 0.30/0.45 thf(fact_104_finite__has__minimal2,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [A3: set @ A,A2: A] :
% 0.30/0.45 ( ( finite_finite2 @ A @ A3 )
% 0.30/0.45 => ( ( member @ A @ A2 @ A3 )
% 0.30/0.45 => ? [X4: A] :
% 0.30/0.45 ( ( member @ A @ X4 @ A3 )
% 0.30/0.45 & ( ord_less_eq @ A @ X4 @ A2 )
% 0.30/0.45 & ! [Xa: A] :
% 0.30/0.45 ( ( member @ A @ Xa @ A3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ Xa @ X4 )
% 0.30/0.45 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % finite_has_minimal2
% 0.30/0.45 thf(fact_105_bot__prod__def,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( bot @ B )
% 0.30/0.45 & ( bot @ A ) )
% 0.30/0.45 => ( ( bot_bot @ ( product_prod @ A @ B ) )
% 0.30/0.45 = ( product_Pair @ A @ B @ ( bot_bot @ A ) @ ( bot_bot @ B ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % bot_prod_def
% 0.30/0.45 thf(fact_106_infinite__imp__nonempty,axiom,
% 0.30/0.45 ! [A: $tType,S: set @ A] :
% 0.30/0.45 ( ~ ( finite_finite2 @ A @ S )
% 0.30/0.45 => ( S
% 0.30/0.45 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % infinite_imp_nonempty
% 0.30/0.45 thf(fact_107_finite_OemptyI,axiom,
% 0.30/0.45 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % finite.emptyI
% 0.30/0.45 thf(fact_108_finite__subset,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/0.45 => ( ( finite_finite2 @ A @ B3 )
% 0.30/0.45 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % finite_subset
% 0.30/0.45 thf(fact_109_infinite__super,axiom,
% 0.30/0.45 ! [A: $tType,S: set @ A,T3: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ S @ T3 )
% 0.30/0.45 => ( ~ ( finite_finite2 @ A @ S )
% 0.30/0.45 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % infinite_super
% 0.30/0.45 thf(fact_110_rev__finite__subset,axiom,
% 0.30/0.45 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.30/0.45 ( ( finite_finite2 @ A @ B3 )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/0.45 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rev_finite_subset
% 0.30/0.45 thf(fact_111_finite__has__minimal,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [A3: set @ A] :
% 0.30/0.45 ( ( finite_finite2 @ A @ A3 )
% 0.30/0.45 => ( ( A3
% 0.30/0.45 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.45 => ? [X4: A] :
% 0.30/0.45 ( ( member @ A @ X4 @ A3 )
% 0.30/0.45 & ! [Xa: A] :
% 0.30/0.45 ( ( member @ A @ Xa @ A3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ Xa @ X4 )
% 0.30/0.45 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % finite_has_minimal
% 0.30/0.45 thf(fact_112_finite__has__maximal,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [A3: set @ A] :
% 0.30/0.45 ( ( finite_finite2 @ A @ A3 )
% 0.30/0.45 => ( ( A3
% 0.30/0.45 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.45 => ? [X4: A] :
% 0.30/0.45 ( ( member @ A @ X4 @ A3 )
% 0.30/0.45 & ! [Xa: A] :
% 0.30/0.45 ( ( member @ A @ Xa @ A3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ X4 @ Xa )
% 0.30/0.45 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % finite_has_maximal
% 0.30/0.45 thf(fact_113_card__subset__eq,axiom,
% 0.30/0.45 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.30/0.45 ( ( finite_finite2 @ A @ B3 )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/0.45 => ( ( ( finite_card @ A @ A3 )
% 0.30/0.45 = ( finite_card @ A @ B3 ) )
% 0.30/0.45 => ( A3 = B3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % card_subset_eq
% 0.30/0.45 thf(fact_114_infinite__arbitrarily__large,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,N: nat] :
% 0.30/0.45 ( ~ ( finite_finite2 @ A @ A3 )
% 0.30/0.45 => ? [B5: set @ A] :
% 0.30/0.45 ( ( finite_finite2 @ A @ B5 )
% 0.30/0.45 & ( ( finite_card @ A @ B5 )
% 0.30/0.45 = N )
% 0.30/0.45 & ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % infinite_arbitrarily_large
% 0.30/0.45 thf(fact_115_finite__if__finite__subsets__card__bdd,axiom,
% 0.30/0.45 ! [A: $tType,F3: set @ A,C3: nat] :
% 0.30/0.45 ( ! [G2: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ G2 @ F3 )
% 0.30/0.45 => ( ( finite_finite2 @ A @ G2 )
% 0.30/0.45 => ( ord_less_eq @ nat @ ( finite_card @ A @ G2 ) @ C3 ) ) )
% 0.30/0.45 => ( ( finite_finite2 @ A @ F3 )
% 0.30/0.45 & ( ord_less_eq @ nat @ ( finite_card @ A @ F3 ) @ C3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % finite_if_finite_subsets_card_bdd
% 0.30/0.45 thf(fact_116_card__seteq,axiom,
% 0.30/0.45 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.30/0.45 ( ( finite_finite2 @ A @ B3 )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/0.45 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B3 ) @ ( finite_card @ A @ A3 ) )
% 0.30/0.45 => ( A3 = B3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % card_seteq
% 0.30/0.45 thf(fact_117_obtain__subset__with__card__n,axiom,
% 0.30/0.45 ! [A: $tType,N: nat,S: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S ) )
% 0.30/0.45 => ~ ! [T4: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ T4 @ S )
% 0.30/0.45 => ( ( ( finite_card @ A @ T4 )
% 0.30/0.45 = N )
% 0.30/0.45 => ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % obtain_subset_with_card_n
% 0.30/0.45 thf(fact_118_card__le__if__inj__on__rel,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,B3: set @ A,A3: set @ B,R: B > A > $o] :
% 0.30/0.45 ( ( finite_finite2 @ A @ B3 )
% 0.30/0.45 => ( ! [A5: B] :
% 0.30/0.45 ( ( member @ B @ A5 @ A3 )
% 0.30/0.45 => ? [B6: A] :
% 0.30/0.45 ( ( member @ A @ B6 @ B3 )
% 0.30/0.45 & ( R @ A5 @ B6 ) ) )
% 0.30/0.45 => ( ! [A1: B,A22: B,B7: A] :
% 0.30/0.45 ( ( member @ B @ A1 @ A3 )
% 0.30/0.45 => ( ( member @ B @ A22 @ A3 )
% 0.30/0.45 => ( ( member @ A @ B7 @ B3 )
% 0.30/0.45 => ( ( R @ A1 @ B7 )
% 0.30/0.45 => ( ( R @ A22 @ B7 )
% 0.30/0.45 => ( A1 = A22 ) ) ) ) ) )
% 0.30/0.45 => ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % card_le_if_inj_on_rel
% 0.30/0.45 thf(fact_119_ex__card,axiom,
% 0.30/0.45 ! [A: $tType,N: nat,A3: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ A3 ) )
% 0.30/0.45 => ? [S2: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ S2 @ A3 )
% 0.30/0.45 & ( ( finite_card @ A @ S2 )
% 0.30/0.45 = N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ex_card
% 0.30/0.45 thf(fact_120_old_Oprod_Oinject,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B8: B] :
% 0.30/0.45 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.30/0.45 = ( product_Pair @ A @ B @ A6 @ B8 ) )
% 0.30/0.45 = ( ( A2 = A6 )
% 0.30/0.45 & ( B2 = B8 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % old.prod.inject
% 0.30/0.45 thf(fact_121_prod_Oinject,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.30/0.45 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.30/0.45 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.30/0.45 = ( ( X1 = Y1 )
% 0.30/0.45 & ( X22 = Y22 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod.inject
% 0.30/0.45 thf(fact_122_bot__apply,axiom,
% 0.30/0.45 ! [C: $tType,D2: $tType] :
% 0.30/0.45 ( ( bot @ C )
% 0.30/0.45 => ( ( bot_bot @ ( D2 > C ) )
% 0.30/0.45 = ( ^ [X3: D2] : ( bot_bot @ C ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % bot_apply
% 0.30/0.45 thf(fact_123_order__refl,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( preorder @ A )
% 0.30/0.45 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_refl
% 0.30/0.45 thf(fact_124_bot__set__def,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( bot_bot @ ( set @ A ) )
% 0.30/0.45 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % bot_set_def
% 0.30/0.45 thf(fact_125_dual__order_Oantisym,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [B2: A,A2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( A2 = B2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % dual_order.antisym
% 0.30/0.45 thf(fact_126_dual__order_Oeq__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.30/0.45 = ( ^ [A7: A,B9: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ B9 @ A7 )
% 0.30/0.45 & ( ord_less_eq @ A @ A7 @ B9 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % dual_order.eq_iff
% 0.30/0.45 thf(fact_127_dual__order_Otrans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [B2: A,A2: A,C2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.30/0.45 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % dual_order.trans
% 0.30/0.45 thf(fact_128_linorder__wlog,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.30/0.45 ( ! [A5: A,B7: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A5 @ B7 )
% 0.30/0.45 => ( P @ A5 @ B7 ) )
% 0.30/0.45 => ( ! [A5: A,B7: A] :
% 0.30/0.45 ( ( P @ B7 @ A5 )
% 0.30/0.45 => ( P @ A5 @ B7 ) )
% 0.30/0.45 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % linorder_wlog
% 0.30/0.45 thf(fact_129_dual__order_Orefl,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % dual_order.refl
% 0.30/0.45 thf(fact_130_order__trans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( preorder @ A )
% 0.30/0.45 => ! [X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/0.45 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_trans
% 0.30/0.45 thf(fact_131_order__class_Oorder_Oantisym,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [A2: A,B2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.45 => ( A2 = B2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_class.order.antisym
% 0.30/0.45 thf(fact_132_ord__le__eq__trans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ord @ A )
% 0.30/0.45 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( B2 = C2 )
% 0.30/0.45 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ord_le_eq_trans
% 0.30/0.45 thf(fact_133_ord__eq__le__trans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ord @ A )
% 0.30/0.45 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.45 ( ( A2 = B2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.45 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ord_eq_le_trans
% 0.30/0.45 thf(fact_134_order__class_Oorder_Oeq__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.30/0.45 = ( ^ [A7: A,B9: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A7 @ B9 )
% 0.30/0.45 & ( ord_less_eq @ A @ B9 @ A7 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_class.order.eq_iff
% 0.30/0.45 thf(fact_135_antisym__conv,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [Y: A,X: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.45 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 = ( X = Y ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antisym_conv
% 0.30/0.45 thf(fact_136_le__cases3,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.30/0.45 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.30/0.45 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.30/0.45 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.30/0.45 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.30/0.45 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_cases3
% 0.30/0.45 thf(fact_137_order_Otrans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.45 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order.trans
% 0.30/0.45 thf(fact_138_le__cases,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_cases
% 0.30/0.45 thf(fact_139_eq__refl,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( preorder @ A )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( X = Y )
% 0.30/0.45 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % eq_refl
% 0.30/0.45 thf(fact_140_linear,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % linear
% 0.30/0.45 thf(fact_141_antisym,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.45 => ( X = Y ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antisym
% 0.30/0.45 thf(fact_142_eq__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.30/0.45 = ( ^ [X3: A,Y2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.30/0.45 & ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % eq_iff
% 0.30/0.45 thf(fact_143_ord__le__eq__subst,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( ord @ B )
% 0.30/0.45 & ( ord @ A ) )
% 0.30/0.45 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( ( F @ B2 )
% 0.30/0.45 = C2 )
% 0.30/0.45 => ( ! [X4: A,Y3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ord_le_eq_subst
% 0.30/0.45 thf(fact_144_ord__eq__le__subst,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( ord @ B )
% 0.30/0.45 & ( ord @ A ) )
% 0.30/0.45 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.30/0.45 ( ( A2
% 0.30/0.45 = ( F @ B2 ) )
% 0.30/0.45 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.30/0.45 => ( ! [X4: B,Y3: B] :
% 0.30/0.45 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.30/0.45 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/0.45 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ord_eq_le_subst
% 0.30/0.45 thf(fact_145_order__subst2,axiom,
% 0.30/0.45 ! [A: $tType,C: $tType] :
% 0.30/0.45 ( ( ( order @ C )
% 0.30/0.45 & ( order @ A ) )
% 0.30/0.45 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.30/0.45 => ( ! [X4: A,Y3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.30/0.45 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/0.45 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_subst2
% 0.30/0.45 thf(fact_146_order__subst1,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( order @ B )
% 0.30/0.45 & ( order @ A ) )
% 0.30/0.45 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.30/0.45 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.30/0.45 => ( ! [X4: B,Y3: B] :
% 0.30/0.45 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.30/0.45 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/0.45 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_subst1
% 0.30/0.45 thf(fact_147_le__fun__def,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ord @ B )
% 0.30/0.45 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/0.45 = ( ^ [F4: A > B,G3: A > B] :
% 0.30/0.45 ! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G3 @ X3 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_fun_def
% 0.30/0.45 thf(fact_148_le__funI,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ord @ B )
% 0.30/0.45 => ! [F: A > B,G: A > B] :
% 0.30/0.45 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.30/0.45 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_funI
% 0.30/0.45 thf(fact_149_le__funE,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ord @ B )
% 0.30/0.45 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.45 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_funE
% 0.30/0.45 thf(fact_150_le__funD,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ord @ B )
% 0.30/0.45 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.45 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_funD
% 0.30/0.45 thf(fact_151_bot__fun__def,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( bot @ B )
% 0.30/0.45 => ( ( bot_bot @ ( A > B ) )
% 0.30/0.45 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % bot_fun_def
% 0.30/0.45 thf(fact_152_surj__pair,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.30/0.45 ? [X4: A,Y3: B] :
% 0.30/0.45 ( P2
% 0.30/0.45 = ( product_Pair @ A @ B @ X4 @ Y3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % surj_pair
% 0.30/0.45 thf(fact_153_prod__cases,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.30/0.45 ( ! [A5: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B7 ) )
% 0.30/0.45 => ( P @ P2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_cases
% 0.30/0.45 thf(fact_154_Pair__inject,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B8: B] :
% 0.30/0.45 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.30/0.45 = ( product_Pair @ A @ B @ A6 @ B8 ) )
% 0.30/0.45 => ~ ( ( A2 = A6 )
% 0.30/0.45 => ( B2 != B8 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Pair_inject
% 0.30/0.45 thf(fact_155_prod__cases3,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.45 ~ ! [A5: A,B7: B,C4: C] :
% 0.30/0.45 ( Y
% 0.30/0.45 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_cases3
% 0.30/0.45 thf(fact_156_prod__cases4,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.30/0.45 ~ ! [A5: A,B7: B,C4: C,D3: D2] :
% 0.30/0.45 ( Y
% 0.30/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_cases4
% 0.30/0.45 thf(fact_157_prod__cases5,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.30/0.45 ~ ! [A5: A,B7: B,C4: C,D3: D2,E2: E] :
% 0.30/0.45 ( Y
% 0.30/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_cases5
% 0.30/0.45 thf(fact_158_prod__cases6,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
% 0.30/0.45 ~ ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F2: F5] :
% 0.30/0.45 ( Y
% 0.30/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E2 @ F2 ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_cases6
% 0.30/0.45 thf(fact_159_prod__cases7,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,G4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) )] :
% 0.30/0.45 ~ ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F2: F5,G5: G4] :
% 0.30/0.45 ( Y
% 0.30/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G4 ) @ E2 @ ( product_Pair @ F5 @ G4 @ F2 @ G5 ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_cases7
% 0.30/0.45 thf(fact_160_prod__induct3,axiom,
% 0.30/0.45 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.45 ( ! [A5: A,B7: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) )
% 0.30/0.45 => ( P @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_induct3
% 0.30/0.45 thf(fact_161_prod__induct4,axiom,
% 0.30/0.45 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.30/0.45 ( ! [A5: A,B7: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
% 0.30/0.45 => ( P @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_induct4
% 0.30/0.45 thf(fact_162_prod__induct5,axiom,
% 0.30/0.45 ! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.30/0.45 ( ! [A5: A,B7: B,C4: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
% 0.30/0.45 => ( P @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_induct5
% 0.30/0.45 thf(fact_163_prod__induct6,axiom,
% 0.30/0.45 ! [F5: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
% 0.30/0.45 ( ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F2: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E2 @ F2 ) ) ) ) ) )
% 0.30/0.45 => ( P @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_induct6
% 0.30/0.45 thf(fact_164_prod__induct7,axiom,
% 0.30/0.45 ! [G4: $tType,F5: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) )] :
% 0.30/0.45 ( ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F2: F5,G5: G4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G4 ) @ E2 @ ( product_Pair @ F5 @ G4 @ F2 @ G5 ) ) ) ) ) ) )
% 0.30/0.45 => ( P @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_induct7
% 0.30/0.45 thf(fact_165_old_Oprod_Oexhaust,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.30/0.45 ~ ! [A5: A,B7: B] :
% 0.30/0.45 ( Y
% 0.30/0.45 != ( product_Pair @ A @ B @ A5 @ B7 ) ) ).
% 0.30/0.45
% 0.30/0.45 % old.prod.exhaust
% 0.30/0.45 thf(fact_166_old_Oprod_Oinducts,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.30/0.45 ( ! [A5: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B7 ) )
% 0.30/0.45 => ( P @ Prod ) ) ).
% 0.30/0.45
% 0.30/0.45 % old.prod.inducts
% 0.30/0.45 thf(fact_167_bounded__Max__nat,axiom,
% 0.30/0.45 ! [P: nat > $o,X: nat,M: nat] :
% 0.30/0.45 ( ( P @ X )
% 0.30/0.45 => ( ! [X4: nat] :
% 0.30/0.45 ( ( P @ X4 )
% 0.30/0.45 => ( ord_less_eq @ nat @ X4 @ M ) )
% 0.30/0.45 => ~ ! [M2: nat] :
% 0.30/0.45 ( ( P @ M2 )
% 0.30/0.45 => ~ ! [X2: nat] :
% 0.30/0.45 ( ( P @ X2 )
% 0.30/0.45 => ( ord_less_eq @ nat @ X2 @ M2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % bounded_Max_nat
% 0.30/0.45 thf(fact_168_finite__nat__set__iff__bounded__le,axiom,
% 0.30/0.45 ( ( finite_finite2 @ nat )
% 0.30/0.45 = ( ^ [N2: set @ nat] :
% 0.30/0.45 ? [M3: nat] :
% 0.30/0.45 ! [X3: nat] :
% 0.30/0.45 ( ( member @ nat @ X3 @ N2 )
% 0.30/0.45 => ( ord_less_eq @ nat @ X3 @ M3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % finite_nat_set_iff_bounded_le
% 0.30/0.45 thf(fact_169_bot_Oextremum__uniqueI,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order_bot @ A )
% 0.30/0.45 => ! [A2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.30/0.45 => ( A2
% 0.30/0.45 = ( bot_bot @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % bot.extremum_uniqueI
% 0.30/0.45 thf(fact_170_bot_Oextremum__unique,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order_bot @ A )
% 0.30/0.45 => ! [A2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.30/0.45 = ( A2
% 0.30/0.45 = ( bot_bot @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % bot.extremum_unique
% 0.30/0.45 thf(fact_171_bot_Oextremum,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order_bot @ A )
% 0.30/0.45 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % bot.extremum
% 0.30/0.45 thf(fact_172_old_Oprod_Orec,axiom,
% 0.30/0.45 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.30/0.45 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.30/0.45 = ( F1 @ A2 @ B2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % old.prod.rec
% 0.30/0.45 thf(fact_173_preorder__on__empty,axiom,
% 0.30/0.45 ! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % preorder_on_empty
% 0.30/0.45 thf(fact_174_total__onI,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ! [X4: A,Y3: A] :
% 0.30/0.45 ( ( member @ A @ X4 @ A3 )
% 0.30/0.45 => ( ( member @ A @ Y3 @ A3 )
% 0.30/0.45 => ( ( X4 != Y3 )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R )
% 0.30/0.45 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R ) ) ) ) )
% 0.30/0.45 => ( total_on @ A @ A3 @ R ) ) ).
% 0.30/0.45
% 0.30/0.45 % total_onI
% 0.30/0.45 thf(fact_175_total__on__def,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( total_on @ A )
% 0.30/0.45 = ( ^ [A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ! [X3: A] :
% 0.30/0.45 ( ( member @ A @ X3 @ A4 )
% 0.30/0.45 => ! [Y2: A] :
% 0.30/0.45 ( ( member @ A @ Y2 @ A4 )
% 0.30/0.45 => ( ( X3 != Y2 )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
% 0.30/0.45 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % total_on_def
% 0.30/0.45 thf(fact_176_bot__empty__eq,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( bot_bot @ ( A > $o ) )
% 0.30/0.45 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % bot_empty_eq
% 0.30/0.45 thf(fact_177_subrelI,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.30/0.45 ( ! [X4: A,Y3: B] :
% 0.30/0.45 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
% 0.30/0.45 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S3 ) )
% 0.30/0.45 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % subrelI
% 0.30/0.45 thf(fact_178_refl__on__empty,axiom,
% 0.30/0.45 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % refl_on_empty
% 0.30/0.45 thf(fact_179_refl__on__domain,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 0.30/0.45 ( ( refl_on @ A @ A3 @ R )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
% 0.30/0.45 => ( ( member @ A @ A2 @ A3 )
% 0.30/0.45 & ( member @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % refl_on_domain
% 0.30/0.45 thf(fact_180_refl__onD2,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( refl_on @ A @ A3 @ R )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.30/0.45 => ( member @ A @ Y @ A3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % refl_onD2
% 0.30/0.45 thf(fact_181_refl__onD1,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.45 ( ( refl_on @ A @ A3 @ R )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.30/0.45 => ( member @ A @ X @ A3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % refl_onD1
% 0.30/0.45 thf(fact_182_refl__onD,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),A2: A] :
% 0.30/0.45 ( ( refl_on @ A @ A3 @ R )
% 0.30/0.45 => ( ( member @ A @ A2 @ A3 )
% 0.30/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % refl_onD
% 0.30/0.45 thf(fact_183_total__on__empty,axiom,
% 0.30/0.45 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R ) ).
% 0.30/0.45
% 0.30/0.45 % total_on_empty
% 0.30/0.45 thf(fact_184_Collect__empty__eq__bot,axiom,
% 0.30/0.45 ! [A: $tType,P: A > $o] :
% 0.30/0.45 ( ( ( collect @ A @ P )
% 0.30/0.45 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.45 = ( P
% 0.30/0.45 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Collect_empty_eq_bot
% 0.30/0.45 thf(fact_185_infinite__nat__iff__unbounded__le,axiom,
% 0.30/0.45 ! [S: set @ nat] :
% 0.30/0.45 ( ( ~ ( finite_finite2 @ nat @ S ) )
% 0.30/0.45 = ( ! [M3: nat] :
% 0.30/0.45 ? [N3: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 0.30/0.45 & ( member @ nat @ N3 @ S ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % infinite_nat_iff_unbounded_le
% 0.30/0.45 thf(fact_186_finite__transitivity__chain,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A,R3: A > A > $o] :
% 0.30/0.45 ( ( finite_finite2 @ A @ A3 )
% 0.30/0.45 => ( ! [X4: A] :
% 0.30/0.45 ~ ( R3 @ X4 @ X4 )
% 0.30/0.45 => ( ! [X4: A,Y3: A,Z3: A] :
% 0.30/0.45 ( ( R3 @ X4 @ Y3 )
% 0.30/0.45 => ( ( R3 @ Y3 @ Z3 )
% 0.30/0.45 => ( R3 @ X4 @ Z3 ) ) )
% 0.30/0.45 => ( ! [X4: A] :
% 0.30/0.45 ( ( member @ A @ X4 @ A3 )
% 0.30/0.45 => ? [Y6: A] :
% 0.30/0.45 ( ( member @ A @ Y6 @ A3 )
% 0.30/0.45 & ( R3 @ X4 @ Y6 ) ) )
% 0.30/0.45 => ( A3
% 0.30/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % finite_transitivity_chain
% 0.30/0.45 thf(fact_187_less__by__empty,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( A3
% 0.30/0.45 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.30/0.45 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % less_by_empty
% 0.30/0.45 thf(fact_188_subset__emptyI,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A] :
% 0.30/0.45 ( ! [X4: A] :
% 0.30/0.45 ~ ( member @ A @ X4 @ A3 )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subset_emptyI
% 0.30/0.45 thf(fact_189_rational__preference__def,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( prefer1997167224erence @ A )
% 0.30/0.45 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( prefer199794634erence @ A @ Carrier2 @ Relation2 )
% 0.30/0.45 & ( prefer1801827867axioms @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference_def
% 0.30/0.45 thf(fact_190_ssubst__Pair__rhs,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,R: A,S3: B,R3: set @ ( product_prod @ A @ B ),S4: B] :
% 0.30/0.45 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R3 )
% 0.30/0.45 => ( ( S4 = S3 )
% 0.30/0.45 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S4 ) @ R3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ssubst_Pair_rhs
% 0.30/0.45 thf(fact_191_rational__preference__axioms_Ointro,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( total_on @ A @ Carrier @ Relation )
% 0.30/0.45 => ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference_axioms.intro
% 0.30/0.45 thf(fact_192_rational__preference__axioms__def,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( prefer1801827867axioms @ A )
% 0.30/0.45 = ( total_on @ A ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference_axioms_def
% 0.30/0.45 thf(fact_193_rational__preference_Oaxioms_I2_J,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.axioms(2)
% 0.30/0.45 thf(fact_194_rational__preference_Ointro,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( ( prefer1801827867axioms @ A @ Carrier @ Relation )
% 0.30/0.45 => ( prefer1997167224erence @ A @ Carrier @ Relation ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rational_preference.intro
% 0.30/0.45 thf(fact_195_finite__indexed__bound,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [A3: set @ B,P: B > A > $o] :
% 0.30/0.45 ( ( finite_finite2 @ B @ A3 )
% 0.30/0.45 => ( ! [X4: B] :
% 0.30/0.45 ( ( member @ B @ X4 @ A3 )
% 0.30/0.45 => ? [X_1: A] : ( P @ X4 @ X_1 ) )
% 0.30/0.45 => ? [M2: A] :
% 0.30/0.45 ! [X2: B] :
% 0.30/0.45 ( ( member @ B @ X2 @ A3 )
% 0.30/0.45 => ? [K: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ K @ M2 )
% 0.30/0.45 & ( P @ X2 @ K ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % finite_indexed_bound
% 0.30/0.45 thf(fact_196_transitivity,axiom,
% 0.30/0.45 trans @ a @ relation ).
% 0.30/0.45
% 0.30/0.45 % transitivity
% 0.30/0.45 thf(fact_197_trans__empty,axiom,
% 0.30/0.45 ! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % trans_empty
% 0.30/0.45 thf(fact_198_transD,axiom,
% 0.30/0.45 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( trans @ A @ R )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R )
% 0.30/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % transD
% 0.30/0.45 thf(fact_199_transE,axiom,
% 0.30/0.45 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( trans @ A @ R )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R )
% 0.30/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % transE
% 0.30/0.45 thf(fact_200_transI,axiom,
% 0.30/0.45 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ! [X4: A,Y3: A,Z3: A] :
% 0.30/0.45 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R )
% 0.30/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z3 ) @ R ) ) )
% 0.30/0.45 => ( trans @ A @ R ) ) ).
% 0.30/0.45
% 0.30/0.45 % transI
% 0.30/0.45 thf(fact_201_trans__def,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( trans @ A )
% 0.30/0.45 = ( ^ [R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ! [X3: A,Y2: A,Z4: A] :
% 0.30/0.45 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
% 0.30/0.45 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R2 )
% 0.30/0.45 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z4 ) @ R2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % trans_def
% 0.30/0.45 thf(fact_202_preference_Otransitivity,axiom,
% 0.30/0.45 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/0.45 => ( trans @ A @ Relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % preference.transitivity
% 0.30/0.45 thf(fact_203_preorder__on__def,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order_preorder_on @ A )
% 0.30/0.45 = ( ^ [A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.45 ( ( refl_on @ A @ A4 @ R2 )
% 0.30/0.45 & ( trans @ A @ R2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % preorder_on_def
% 0.30/0.45 thf(fact_204_Set_Ois__empty__def,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( is_empty @ A )
% 0.30/0.45 = ( ^ [A4: set @ A] :
% 0.30/0.45 ( A4
% 0.30/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Set.is_empty_def
% 0.30/0.45 thf(fact_205_internal__case__prod__conv,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
% 0.30/0.45 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.30/0.45 = ( C2 @ A2 @ B2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % internal_case_prod_conv
% 0.30/0.45 thf(fact_206_card__0__eq,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A] :
% 0.30/0.45 ( ( finite_finite2 @ A @ A3 )
% 0.30/0.45 => ( ( ( finite_card @ A @ A3 )
% 0.30/0.45 = ( zero_zero @ nat ) )
% 0.30/0.45 = ( A3
% 0.30/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % card_0_eq
% 0.30/0.45 thf(fact_207_relChain__def,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ord @ B )
% 0.30/0.45 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.30/0.45 = ( ^ [R2: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.30/0.45 ! [I: A,J: A] :
% 0.30/0.45 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R2 )
% 0.30/0.45 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % relChain_def
% 0.30/0.45 thf(fact_208_card_Oempty,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.45 = ( zero_zero @ nat ) ) ).
% 0.30/0.45
% 0.30/0.45 % card.empty
% 0.30/0.45 thf(fact_209_card_Oinfinite,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A] :
% 0.30/0.45 ( ~ ( finite_finite2 @ A @ A3 )
% 0.30/0.45 => ( ( finite_card @ A @ A3 )
% 0.30/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % card.infinite
% 0.30/0.45 thf(fact_210_card__eq__0__iff,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A] :
% 0.30/0.45 ( ( ( finite_card @ A @ A3 )
% 0.30/0.45 = ( zero_zero @ nat ) )
% 0.30/0.45 = ( ( A3
% 0.30/0.45 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.45 | ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % card_eq_0_iff
% 0.30/0.45 thf(fact_211_bot__nat__0_Oextremum,axiom,
% 0.30/0.45 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.30/0.45
% 0.30/0.45 % bot_nat_0.extremum
% 0.30/0.45 thf(fact_212_le0,axiom,
% 0.30/0.45 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.30/0.45
% 0.30/0.45 % le0
% 0.30/0.45 thf(fact_213_Nat_Oex__has__greatest__nat,axiom,
% 0.30/0.45 ! [P: nat > $o,K2: nat,B2: nat] :
% 0.30/0.45 ( ( P @ K2 )
% 0.30/0.45 => ( ! [Y3: nat] :
% 0.30/0.45 ( ( P @ Y3 )
% 0.30/0.45 => ( ord_less_eq @ nat @ Y3 @ B2 ) )
% 0.30/0.45 => ? [X4: nat] :
% 0.30/0.45 ( ( P @ X4 )
% 0.30/0.45 & ! [Y6: nat] :
% 0.30/0.45 ( ( P @ Y6 )
% 0.30/0.45 => ( ord_less_eq @ nat @ Y6 @ X4 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Nat.ex_has_greatest_nat
% 0.30/0.45 thf(fact_214_nat__le__linear,axiom,
% 0.30/0.45 ! [M4: nat,N: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ M4 @ N )
% 0.30/0.45 | ( ord_less_eq @ nat @ N @ M4 ) ) ).
% 0.30/0.45
% 0.30/0.45 % nat_le_linear
% 0.30/0.45 thf(fact_215_le__antisym,axiom,
% 0.30/0.45 ! [M4: nat,N: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ M4 @ N )
% 0.30/0.45 => ( ( ord_less_eq @ nat @ N @ M4 )
% 0.30/0.45 => ( M4 = N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_antisym
% 0.30/0.45 thf(fact_216_eq__imp__le,axiom,
% 0.30/0.45 ! [M4: nat,N: nat] :
% 0.30/0.45 ( ( M4 = N )
% 0.30/0.45 => ( ord_less_eq @ nat @ M4 @ N ) ) ).
% 0.30/0.45
% 0.30/0.45 % eq_imp_le
% 0.30/0.45 thf(fact_217_le__trans,axiom,
% 0.30/0.45 ! [I2: nat,J2: nat,K2: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ I2 @ J2 )
% 0.30/0.45 => ( ( ord_less_eq @ nat @ J2 @ K2 )
% 0.30/0.45 => ( ord_less_eq @ nat @ I2 @ K2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_trans
% 0.30/0.45 thf(fact_218_le__refl,axiom,
% 0.30/0.45 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.30/0.45
% 0.30/0.45 % le_refl
% 0.30/0.45 thf(fact_219_less__eq__nat_Osimps_I1_J,axiom,
% 0.30/0.45 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.30/0.45
% 0.30/0.45 % less_eq_nat.simps(1)
% 0.30/0.45 thf(fact_220_le__0__eq,axiom,
% 0.30/0.45 ! [N: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.30/0.45 = ( N
% 0.30/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_0_eq
% 0.30/0.45 thf(fact_221_bot__nat__0_Oextremum__unique,axiom,
% 0.30/0.45 ! [A2: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.30/0.45 = ( A2
% 0.30/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % bot_nat_0.extremum_unique
% 0.30/0.45 thf(fact_222_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.30/0.45 ! [A2: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.30/0.45 => ( A2
% 0.30/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % bot_nat_0.extremum_uniqueI
% 0.30/0.45 thf(fact_223_le__zero__eq,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( canoni770627133id_add @ A )
% 0.30/0.45 => ! [N: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.30/0.45 = ( N
% 0.30/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_zero_eq
% 0.30/0.45 thf(fact_224_zero__prod__def,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( zero @ B )
% 0.30/0.45 & ( zero @ A ) )
% 0.30/0.45 => ( ( zero_zero @ ( product_prod @ A @ B ) )
% 0.30/0.45 = ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % zero_prod_def
% 0.30/0.45 thf(fact_225_zero__le,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( canoni770627133id_add @ A )
% 0.30/0.45 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % zero_le
% 0.30/0.45 thf(fact_226_le__numeral__extra_I3_J,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linord1659791738miring @ A )
% 0.30/0.45 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_numeral_extra(3)
% 0.30/0.45 thf(fact_227_card__gt__0__iff,axiom,
% 0.30/0.45 ! [A: $tType,A3: set @ A] :
% 0.30/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
% 0.30/0.45 = ( ( A3
% 0.30/0.45 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.45 & ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % card_gt_0_iff
% 0.30/0.45 thf(fact_228_ex__least__nat__le,axiom,
% 0.30/0.45 ! [P: nat > $o,N: nat] :
% 0.30/0.45 ( ( P @ N )
% 0.30/0.45 => ( ~ ( P @ ( zero_zero @ nat ) )
% 0.30/0.45 => ? [K: nat] :
% 0.30/0.45 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.45 & ! [I3: nat] :
% 0.30/0.45 ( ( ord_less @ nat @ I3 @ K )
% 0.30/0.45 => ~ ( P @ I3 ) )
% 0.30/0.45 & ( P @ K ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ex_least_nat_le
% 0.30/0.45 thf(fact_229_leD,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [Y: A,X: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.45 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % leD
% 0.30/0.45 thf(fact_230_leI,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.45 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % leI
% 0.30/0.45 thf(fact_231_le__less,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ( ( ord_less_eq @ A )
% 0.30/0.45 = ( ^ [X3: A,Y2: A] :
% 0.30/0.45 ( ( ord_less @ A @ X3 @ Y2 )
% 0.30/0.45 | ( X3 = Y2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_less
% 0.30/0.45 thf(fact_232_less__le,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ( ( ord_less @ A )
% 0.30/0.45 = ( ^ [X3: A,Y2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.30/0.45 & ( X3 != Y2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % less_le
% 0.30/0.45 thf(fact_233_order__le__less__subst1,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( order @ B )
% 0.30/0.45 & ( order @ A ) )
% 0.30/0.45 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.30/0.45 => ( ( ord_less @ B @ B2 @ C2 )
% 0.30/0.45 => ( ! [X4: B,Y3: B] :
% 0.30/0.45 ( ( ord_less @ B @ X4 @ Y3 )
% 0.30/0.45 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/0.45 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_le_less_subst1
% 0.30/0.45 thf(fact_234_order__le__less__subst2,axiom,
% 0.30/0.45 ! [A: $tType,C: $tType] :
% 0.30/0.45 ( ( ( order @ C )
% 0.30/0.45 & ( order @ A ) )
% 0.30/0.45 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 0.30/0.45 => ( ! [X4: A,Y3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.30/0.45 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/0.45 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_le_less_subst2
% 0.30/0.45 thf(fact_235_order__less__le__subst1,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( order @ B )
% 0.30/0.45 & ( order @ A ) )
% 0.30/0.45 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.30/0.45 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.30/0.45 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.30/0.45 => ( ! [X4: B,Y3: B] :
% 0.30/0.45 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.30/0.45 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/0.45 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_less_le_subst1
% 0.30/0.45 thf(fact_236_order__less__le__subst2,axiom,
% 0.30/0.45 ! [A: $tType,C: $tType] :
% 0.30/0.45 ( ( ( order @ C )
% 0.30/0.45 & ( order @ A ) )
% 0.30/0.45 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.30/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.30/0.45 => ( ! [X4: A,Y3: A] :
% 0.30/0.45 ( ( ord_less @ A @ X4 @ Y3 )
% 0.30/0.45 => ( ord_less @ C @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/0.45 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_less_le_subst2
% 0.30/0.45 thf(fact_237_not__le,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 0.30/0.45 = ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % not_le
% 0.30/0.45 thf(fact_238_not__less,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.45 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % not_less
% 0.30/0.45 thf(fact_239_le__neq__trans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [A2: A,B2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( A2 != B2 )
% 0.30/0.45 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_neq_trans
% 0.30/0.45 thf(fact_240_antisym__conv1,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.45 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 = ( X = Y ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antisym_conv1
% 0.30/0.45 thf(fact_241_antisym__conv2,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.45 = ( X = Y ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antisym_conv2
% 0.30/0.45 thf(fact_242_less__imp__le,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( preorder @ A )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( ord_less @ A @ X @ Y )
% 0.30/0.45 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % less_imp_le
% 0.30/0.45 thf(fact_243_le__less__trans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( preorder @ A )
% 0.30/0.45 => ! [X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ( ( ord_less @ A @ Y @ Z )
% 0.30/0.45 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_less_trans
% 0.30/0.45 thf(fact_244_less__le__trans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( preorder @ A )
% 0.30/0.45 => ! [X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( ord_less @ A @ X @ Y )
% 0.30/0.45 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/0.45 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % less_le_trans
% 0.30/0.45 thf(fact_245_dense__ge,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( dense_linorder @ A )
% 0.30/0.45 => ! [Z: A,Y: A] :
% 0.30/0.45 ( ! [X4: A] :
% 0.30/0.45 ( ( ord_less @ A @ Z @ X4 )
% 0.30/0.45 => ( ord_less_eq @ A @ Y @ X4 ) )
% 0.30/0.45 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % dense_ge
% 0.30/0.45 thf(fact_246_dense__le,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( dense_linorder @ A )
% 0.30/0.45 => ! [Y: A,Z: A] :
% 0.30/0.45 ( ! [X4: A] :
% 0.30/0.45 ( ( ord_less @ A @ X4 @ Y )
% 0.30/0.45 => ( ord_less_eq @ A @ X4 @ Z ) )
% 0.30/0.45 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % dense_le
% 0.30/0.45 thf(fact_247_le__less__linear,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_less_linear
% 0.30/0.45 thf(fact_248_le__imp__less__or__eq,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ( ( ord_less @ A @ X @ Y )
% 0.30/0.45 | ( X = Y ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_imp_less_or_eq
% 0.30/0.45 thf(fact_249_less__le__not__le,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( preorder @ A )
% 0.30/0.45 => ( ( ord_less @ A )
% 0.30/0.45 = ( ^ [X3: A,Y2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.30/0.45 & ~ ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % less_le_not_le
% 0.30/0.45 thf(fact_250_not__le__imp__less,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A )
% 0.30/0.45 => ! [Y: A,X: A] :
% 0.30/0.45 ( ~ ( ord_less_eq @ A @ Y @ X )
% 0.30/0.45 => ( ord_less @ A @ X @ Y ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % not_le_imp_less
% 0.30/0.45 thf(fact_251_order_Ostrict__trans1,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( ord_less @ A @ B2 @ C2 )
% 0.30/0.45 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order.strict_trans1
% 0.30/0.45 thf(fact_252_order_Ostrict__trans2,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.45 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.45 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order.strict_trans2
% 0.30/0.45 thf(fact_253_order_Oorder__iff__strict,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ( ( ord_less_eq @ A )
% 0.30/0.45 = ( ^ [A7: A,B9: A] :
% 0.30/0.45 ( ( ord_less @ A @ A7 @ B9 )
% 0.30/0.45 | ( A7 = B9 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order.order_iff_strict
% 0.30/0.45 thf(fact_254_order_Ostrict__iff__order,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ( ( ord_less @ A )
% 0.30/0.45 = ( ^ [A7: A,B9: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A7 @ B9 )
% 0.30/0.45 & ( A7 != B9 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order.strict_iff_order
% 0.30/0.45 thf(fact_255_dual__order_Ostrict__trans1,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A )
% 0.30/0.45 => ! [B2: A,A2: A,C2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.45 => ( ( ord_less @ A @ C2 @ B2 )
% 0.30/0.45 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % dual_order.strict_trans1
% 0.30/0.45
% 0.30/0.45 % Type constructors (36)
% 0.30/0.45 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.30/0.45 ! [A8: $tType,A9: $tType] :
% 0.30/0.45 ( ( order_bot @ A9 )
% 0.30/0.45 => ( order_bot @ ( A8 > A9 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.45 ! [A8: $tType,A9: $tType] :
% 0.30/0.45 ( ( preorder @ A9 )
% 0.30/0.45 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.30/0.45 ! [A8: $tType,A9: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ A8 )
% 0.30/0.45 & ( finite_finite @ A9 ) )
% 0.30/0.45 => ( finite_finite @ ( A8 > A9 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.45 ! [A8: $tType,A9: $tType] :
% 0.30/0.45 ( ( order @ A9 )
% 0.30/0.45 => ( order @ ( A8 > A9 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.45 ! [A8: $tType,A9: $tType] :
% 0.30/0.45 ( ( ord @ A9 )
% 0.30/0.45 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_fun___Orderings_Obot,axiom,
% 0.30/0.45 ! [A8: $tType,A9: $tType] :
% 0.30/0.45 ( ( bot @ A9 )
% 0.30/0.45 => ( bot @ ( A8 > A9 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.30/0.45 canoni770627133id_add @ nat ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.30/0.45 linord1659791738miring @ nat ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
% 0.30/0.45 order_bot @ nat ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 0.30/0.45 preorder @ nat ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.30/0.45 linorder @ nat ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 0.30/0.45 order @ nat ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 0.30/0.45 ord @ nat ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
% 0.30/0.45 bot @ nat ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.30/0.45 zero @ nat ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
% 0.30/0.45 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 0.30/0.45 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
% 0.30/0.45 ! [A8: $tType] :
% 0.30/0.45 ( ( finite_finite @ A8 )
% 0.30/0.45 => ( finite_finite @ ( set @ A8 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 0.30/0.45 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
% 0.30/0.45 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
% 0.30/0.45 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Set_Oset___Groups_Ozero_12,axiom,
% 0.30/0.45 ! [A8: $tType] :
% 0.30/0.45 ( ( zero @ A8 )
% 0.30/0.45 => ( zero @ ( set @ A8 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_HOL_Obool___Orderings_Oorder__bot_13,axiom,
% 0.30/0.45 order_bot @ $o ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
% 0.30/0.45 preorder @ $o ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_HOL_Obool___Orderings_Olinorder_15,axiom,
% 0.30/0.45 linorder @ $o ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_HOL_Obool___Finite__Set_Ofinite_16,axiom,
% 0.30/0.45 finite_finite @ $o ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_HOL_Obool___Orderings_Oorder_17,axiom,
% 0.30/0.45 order @ $o ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
% 0.30/0.45 ord @ $o ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_HOL_Obool___Orderings_Obot_19,axiom,
% 0.30/0.45 bot @ $o ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Product__Type_Oprod___Orderings_Oorder__bot_20,axiom,
% 0.30/0.45 ! [A8: $tType,A9: $tType] :
% 0.30/0.45 ( ( ( order_bot @ A8 )
% 0.30/0.45 & ( order_bot @ A9 ) )
% 0.30/0.45 => ( order_bot @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_21,axiom,
% 0.30/0.45 ! [A8: $tType,A9: $tType] :
% 0.30/0.45 ( ( ( preorder @ A8 )
% 0.30/0.45 & ( preorder @ A9 ) )
% 0.30/0.45 => ( preorder @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_22,axiom,
% 0.30/0.45 ! [A8: $tType,A9: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ A8 )
% 0.30/0.45 & ( finite_finite @ A9 ) )
% 0.30/0.45 => ( finite_finite @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Product__Type_Oprod___Orderings_Oorder_23,axiom,
% 0.30/0.45 ! [A8: $tType,A9: $tType] :
% 0.30/0.45 ( ( ( order @ A8 )
% 0.30/0.45 & ( order @ A9 ) )
% 0.30/0.45 => ( order @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Product__Type_Oprod___Orderings_Oord_24,axiom,
% 0.30/0.45 ! [A8: $tType,A9: $tType] :
% 0.30/0.45 ( ( ( ord @ A8 )
% 0.30/0.45 & ( ord @ A9 ) )
% 0.30/0.45 => ( ord @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Product__Type_Oprod___Orderings_Obot_25,axiom,
% 0.30/0.45 ! [A8: $tType,A9: $tType] :
% 0.30/0.45 ( ( ( bot @ A8 )
% 0.30/0.45 & ( bot @ A9 ) )
% 0.30/0.45 => ( bot @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(tcon_Product__Type_Oprod___Groups_Ozero_26,axiom,
% 0.30/0.45 ! [A8: $tType,A9: $tType] :
% 0.30/0.45 ( ( ( zero @ A8 )
% 0.30/0.45 & ( zero @ A9 ) )
% 0.30/0.45 => ( zero @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Conjectures (1)
% 0.30/0.45 thf(conj_0,conjecture,
% 0.30/0.45 ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ y @ carrier @ relation ) ).
% 0.30/0.45
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 ------------------------
% 0.30/1.51 ---- Embedded in THF ---
% 0.30/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.30/1.51 %%% Generated on Tue Feb 24 08:28:23 EST 2026
% 0.30/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 0.30/1.51 %%% Logic specification used:
% 0.30/1.51 %%% thf(spec, logic, $$dhol).
% 0.30/1.51
% 0.30/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.vFZO1l9sao/DTF2DTF28194.p
% 0.30/1.51 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 0.30/1.51 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.30/1.51 thf(ty_t_Nat_Onat, type, nat: $tType).
% 0.30/1.51 thf(ty_tf_a, type, a: $tType).
% 0.30/1.51 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 0.30/1.51 thf(sy_c_Finite__Set_Ocard, type, finite_card: (!> [B:$tType]: (((set @ B) > nat)))).
% 0.30/1.51 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 0.30/1.51 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 0.30/1.51 thf(sy_c_Order__Relation_Opreorder__on, type, order_preorder_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 0.30/1.51 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 0.30/1.51 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 0.30/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 0.30/1.51 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oas__good__as, type, prefer951318096ood_as: (!> [A:$tType]: ((A > ((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ A))))))).
% 0.30/1.51 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good, type, prefer310429814s_good: (!> [A:$tType]: ((A > ((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ A))))))).
% 0.30/1.51 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Ono__better__than, type, prefer1532642881r_than: (!> [A:$tType]: ((A > ((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ A))))))).
% 0.30/1.51 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference, type, prefer199794634erence: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 0.30/1.51 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference, type, prefer1997167224erence: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 0.30/1.51 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference__axioms, type, prefer1801827867axioms: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 0.30/1.51 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 0.30/1.51 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 0.30/1.51 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 0.30/1.51 thf(sy_c_Relation_Orefl__on, type, refl_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 0.30/1.51 thf(sy_c_Relation_Ototal__on, type, total_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 0.30/1.51 thf(sy_c_Relation_Otrans, type, trans: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 0.30/1.51 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.30/1.51 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 0.30/1.51 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.30/1.51 thf(sy_v_carrier, type, carrier: (set @ a)).
% 0.30/1.51 thf(sy_v_relation, type, relation: (set @ ((product_prod @ a) @ a))).
% 0.30/1.51 thf(sy_v_x, type, x: a).
% 0.30/1.51 thf(sy_v_y, type, y: a).
% 0.30/1.51 thf(fact_0_assms_I2_J, axiom, (((member @ a) @ y) @ carrier)).
% 0.30/1.51 thf(fact_1_assms_I1_J, axiom, (((member @ a) @ x) @ carrier)).
% 0.30/1.51 thf(fact_2_xy__in__eachothers__nbt, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ carrier) => ((((member @ a) @ Y) @ carrier) => ((((member @ a) @ X) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)) | (((member @ a) @ Y) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation)))))))).
% 0.30/1.51 thf(fact_3__092_060open_062card_A_Ino__better__than_Ax_Acarrier_Arelation_J_A_092_060le_062_Acard_A_Ino__better__than_Ay_Acarrier_Arelation_J_092_060close_062, axiom, (((ord_less_eq @ nat) @ ((finite_card @ a) @ ((((prefer1532642881r_than @ a) @ x) @ carrier) @ relation))) @ ((finite_card @ a) @ ((((prefer1532642881r_than @ a) @ y) @ carrier) @ relation)))).
% 0.30/1.51 thf(fact_4_nbt__nest, axiom, (! [Y:a,X:a]: (((((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation)) | (((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)))))).
% 0.30/1.51 thf(fact_5_nbt__subset__carrier, axiom, (! [X:a]: (((((member @ a) @ X) @ carrier) => (((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation)) @ carrier))))).
% 0.30/1.51 thf(fact_6_rational__preference__axioms, axiom, (((prefer1997167224erence @ a) @ carrier) @ relation)).
% 0.30/1.51 thf(fact_7__092_060open_062finite_Acarrier_092_060close_062, axiom, ((finite_finite2 @ a) @ carrier)).
% 0.30/1.51 thf(fact_8_preference__axioms, axiom, (((prefer199794634erence @ a) @ carrier) @ relation)).
% 0.30/1.51 thf(fact_9_trans__refl, axiom, (((order_preorder_on @ a) @ carrier) @ relation)).
% 0.30/1.51 thf(fact_10_nbt__subset, axiom, (! [X:a,Y:a]: ((((finite_finite2 @ a) @ carrier) => ((((member @ a) @ X) @ carrier) => ((((member @ a) @ Y) @ carrier) => ((((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation)) | (((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation))))))))).
% 0.30/1.51 thf(fact_11_no__better__subset__pref, axiom, (! [X:a,Y:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) => (((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation)))))).
% 0.30/1.51 thf(fact_12_no__better__thansubset__rel, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ carrier) => ((((member @ a) @ Y) @ carrier) => ((((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation)) => (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation))))))).
% 0.30/1.51 thf(fact_13_no__better__than__nonepty, axiom, (! [X:a]: (((carrier != (bot_bot @ (set @ a))) => ((((member @ a) @ X) @ carrier) => (((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation) != (bot_bot @ (set @ a)))))))).
% 0.30/1.51 thf(fact_14_total, axiom, (((total_on @ a) @ carrier) @ relation)).
% 0.30/1.51 thf(fact_15_fnt__carrier__fnt__nbt, axiom, (! [X2:a]: (((((member @ a) @ X2) @ carrier) => ((finite_finite2 @ a) @ ((((prefer1532642881r_than @ a) @ X2) @ carrier) @ relation)))))).
% 0.30/1.51 thf(fact_16_same__nbt__same__pref, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ carrier) => ((((member @ a) @ Y) @ carrier) => (((((member @ a) @ X) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)) & (((member @ a) @ Y) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation))) = ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ relation)))))))).
% 0.30/1.51 thf(fact_17_worse__in__no__better, axiom, (! [X:a,Y:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) => (((member @ a) @ Y) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)))))).
% 0.30/1.51 thf(fact_18_reflexivity, axiom, (((refl_on @ a) @ carrier) @ relation)).
% 0.30/1.51 thf(fact_19_compl, axiom, (! [X2:a]: (((((member @ a) @ X2) @ carrier) => (! [Xa:a]: (((((member @ a) @ Xa) @ carrier) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Xa)) @ relation) | (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Xa) @ X2)) @ relation))))))))).
% 0.30/1.51 thf(fact_20_completeD, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ carrier) => ((((member @ a) @ Y) @ carrier) => ((X != Y) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) | (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ relation)))))))).
% 0.30/1.51 thf(fact_21_strict__trans, axiom, (! [X:a,Y:a,Z:a]: ((((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ relation)))) => (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ Z)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ Y)) @ relation)))) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Z)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ X)) @ relation))))))))).
% 0.30/1.51 thf(fact_22_indiff__trans, axiom, (! [X:a,Y:a,Z:a]: ((((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ relation)) => (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ Z)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ Y)) @ relation)) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Z)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ X)) @ relation))))))).
% 0.30/1.51 thf(fact_23_indifferent__imp__weak__pref_I1_J, axiom, (! [X:a,Y:a]: ((((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ relation)) => (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation))))).
% 0.30/1.51 thf(fact_24_indifferent__imp__weak__pref_I2_J, axiom, (! [X:a,Y:a]: ((((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ relation)) => (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ relation))))).
% 0.30/1.51 thf(fact_25_weak__is__transitive, axiom, (! [X:a,Y:a,Z:a]: ((((((member @ a) @ X) @ carrier) & ((((member @ a) @ Y) @ carrier) & (((member @ a) @ Z) @ carrier))) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ Z)) @ relation) => (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Z)) @ relation))))))).
% 0.30/1.51 thf(fact_26_strict__is__neg__transitive, axiom, (! [X:a,Y:a,Z:a]: ((((((member @ a) @ X) @ carrier) & ((((member @ a) @ Y) @ carrier) & (((member @ a) @ Z) @ carrier))) => (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ relation)))) => (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Z)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ X)) @ relation)))) | ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ Y)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ Z)) @ relation)))))))))).
% 0.30/1.51 thf(fact_27_not__outside, axiom, (! [X:a,Y:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) => (((member @ a) @ X) @ carrier))))).
% 0.30/1.51 thf(fact_28_strict__not__refl__weak, axiom, (! [X:a,Y:a]: ((((((member @ a) @ X) @ carrier) & (((member @ a) @ Y) @ carrier)) => ((~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ relation))) = ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ relation))))))))).
% 0.30/1.51 thf(fact_29_rational__preference_Oaxioms_I1_J, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((prefer199794634erence @ A) @ Carrier) @ Relation))))).
% 0.30/1.51 thf(fact_30_rational__preference_Oindifferent__imp__weak__pref_I2_J, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation)))))).
% 0.30/1.51 thf(fact_31_rational__preference_Oindifferent__imp__weak__pref_I1_J, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation)))))).
% 0.30/1.51 thf(fact_32_preference__def, axiom, (! [A:$tType]: (((prefer199794634erence @ A) = (^ [Carrier2:(set @ A),Relation2:(set @ ((product_prod @ A) @ A))]: (((! [X3:A,Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y2)) @ Relation2) => (((member @ A) @ X3) @ Carrier2)))) & ((! [X3:A,Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y2)) @ Relation2) => (((member @ A) @ Y2) @ Carrier2)))) & (((order_preorder_on @ A) @ Carrier2) @ Relation2))))))))).
% 0.30/1.51 thf(fact_33_preference_Ointro, axiom, (! [A:$tType,Relation:(set @ ((product_prod @ A) @ A)),Carrier:(set @ A)]: (((! [X4:A,Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y3)) @ Relation) => (((member @ A) @ X4) @ Carrier)))) => ((! [X4:A,Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y3)) @ Relation) => (((member @ A) @ Y3) @ Carrier)))) => ((((order_preorder_on @ A) @ Carrier) @ Relation) => (((prefer199794634erence @ A) @ Carrier) @ Relation))))))).
% 0.30/1.51 thf(fact_34_preference_Otrans__refl, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => (((order_preorder_on @ A) @ Carrier) @ Relation))))).
% 0.30/1.51 thf(fact_35_preference_Onot__outside, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) => (((member @ A) @ X) @ Carrier)))))).
% 0.30/1.51 thf(fact_36_preference_Oreflexivity, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => (((refl_on @ A) @ Carrier) @ Relation))))).
% 0.30/1.51 thf(fact_37_preference_Oindiff__trans, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z:A]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation)) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ Y)) @ Relation)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ X)) @ Relation)))))))).
% 0.30/1.51 thf(fact_38_rational__preference_Ocompl, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (! [X2:A]: (((((member @ A) @ X2) @ Carrier) => (! [Xa:A]: (((((member @ A) @ Xa) @ Carrier) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Xa)) @ Relation) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Xa) @ X2)) @ Relation)))))))))))).
% 0.30/1.51 thf(fact_39_rational__preference_Ototal, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((total_on @ A) @ Carrier) @ Relation))))).
% 0.30/1.51 thf(fact_40_rational__preference_OcompleteD, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ A) @ X) @ Carrier) => ((((member @ A) @ Y) @ Carrier) => ((X != Y) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation))))))))).
% 0.30/1.51 thf(fact_41_rational__preference_Ostrict__trans, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation)))) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ Y)) @ Relation)))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ X)) @ Relation)))))))))).
% 0.30/1.51 thf(fact_42_rational__preference_Oweak__is__transitive, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ A) @ X) @ Carrier) & ((((member @ A) @ Y) @ Carrier) & (((member @ A) @ Z) @ Carrier))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ Relation) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z)) @ Relation)))))))).
% 0.30/1.51 thf(fact_43_rational__preference_Ofnt__carrier__fnt__rel, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((finite_finite2 @ A) @ Carrier) => ((finite_finite2 @ ((product_prod @ A) @ A)) @ Relation)))))).
% 0.30/1.51 thf(fact_44_rational__preference_Ostrict__not__refl__weak, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ A) @ X) @ Carrier) & (((member @ A) @ Y) @ Carrier)) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation))) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation)))))))))).
% 0.30/1.51 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 0.30/1.51 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A3)))) = A3)))).
% 0.30/1.51 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 0.30/1.51 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 0.30/1.51 thf(fact_49_rational__preference_Ostrict__is__neg__transitive, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ A) @ X) @ Carrier) & ((((member @ A) @ Y) @ Carrier) & (((member @ A) @ Z) @ Carrier))) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation)))) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ X)) @ Relation)))) | ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ Y)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ Relation))))))))))).
% 0.30/1.51 thf(fact_50_rational__preference_Oworse__in__no__better, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) => (((member @ A) @ Y) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation))))))).
% 0.30/1.51 thf(fact_51_rational__preference_Osame__nbt__same__pref, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ A) @ X) @ Carrier) => ((((member @ A) @ Y) @ Carrier) => (((((member @ A) @ X) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation)) & (((member @ A) @ Y) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation))) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation))))))))).
% 0.30/1.51 thf(fact_52_rational__preference_Ofnt__carrier__fnt__nbt, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((finite_finite2 @ A) @ Carrier) => (! [X2:A]: (((((member @ A) @ X2) @ Carrier) => ((finite_finite2 @ A) @ ((((prefer1532642881r_than @ A) @ X2) @ Carrier) @ Relation)))))))))).
% 0.30/1.51 thf(fact_53_rational__preference_Ono__better__than__nonepty, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((Carrier != (bot_bot @ (set @ A))) => ((((member @ A) @ X) @ Carrier) => (((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation) != (bot_bot @ (set @ A))))))))).
% 0.30/1.51 thf(fact_54_rational__preference_Ono__better__thansubset__rel, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ A) @ X) @ Carrier) => ((((member @ A) @ Y) @ Carrier) => ((((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation)))))))).
% 0.30/1.51 thf(fact_55_rational__preference_Ono__better__subset__pref, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) => (((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation))))))).
% 0.30/1.51 thf(fact_56_rational__preference_Onbt__subset, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((finite_finite2 @ A) @ Carrier) => ((((member @ A) @ X) @ Carrier) => ((((member @ A) @ Y) @ Carrier) => ((((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation)) | (((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation)))))))))).
% 0.30/1.51 thf(fact_57_rational__preference_Oxy__in__eachothers__nbt, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ A) @ X) @ Carrier) => ((((member @ A) @ Y) @ Carrier) => ((((member @ A) @ X) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation)) | (((member @ A) @ Y) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation))))))))).
% 0.30/1.51 thf(fact_58_rational__preference_Onbt__subset__carrier, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ A) @ X) @ Carrier) => (((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation)) @ Carrier)))))).
% 0.30/1.51 thf(fact_59_rational__preference_Onbt__nest, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),Y:A,X:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation)) | (((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation))))))).
% 0.30/1.51 thf(fact_60_at__lst__asgd__not__ge, axiom, (! [X:a,Y:a]: (((carrier != (bot_bot @ (set @ a))) => ((((member @ a) @ X) @ carrier) => ((((member @ a) @ Y) @ carrier) => ((~ ((((member @ a) @ X) @ ((((prefer310429814s_good @ a) @ Y) @ carrier) @ relation)))) => (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation)))))))))).
% 0.30/1.51 thf(fact_61_same__at__least__as__equal, axiom, (! [Z:a,Y:a]: ((((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ Y)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ Z)) @ relation)) => (((((prefer310429814s_good @ a) @ Z) @ carrier) @ relation) = ((((prefer310429814s_good @ a) @ Y) @ carrier) @ relation)))))).
% 0.30/1.51 thf(fact_62_pref__in__at__least__as, axiom, (! [X:a,Y:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) => (((member @ a) @ X) @ ((((prefer310429814s_good @ a) @ Y) @ carrier) @ relation)))))).
% 0.30/1.51 thf(fact_63_empty__subsetI, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3)))).
% 0.30/1.51 thf(fact_64_subset__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = (A3 = (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_65_Pair__le, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:B,C2:A,D:B]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Pair @ A) @ B) @ C2) @ D)) = ((((ord_less_eq @ A) @ A2) @ C2) & (((ord_less_eq @ B) @ B2) @ D))))))))).
% 0.30/1.51 thf(fact_66_as__good__as__sameIff, axiom, (! [Z:a,Y:a]: (((((member @ a) @ Z) @ ((((prefer951318096ood_as @ a) @ Y) @ carrier) @ relation)) = ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ Y)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ Z)) @ relation)))))).
% 0.30/1.51 thf(fact_67_as__good__asIff, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ ((((prefer951318096ood_as @ a) @ Y) @ carrier) @ relation)) = ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ relation)))))).
% 0.30/1.51 thf(fact_68_fnt__carrier__fnt__rel, axiom, (((finite_finite2 @ a) @ carrier) => ((finite_finite2 @ ((product_prod @ a) @ a)) @ relation))).
% 0.30/1.51 thf(fact_69_card__mono, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3))))))).
% 0.30/1.51 thf(fact_70_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 0.30/1.51 thf(fact_71_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 0.30/1.51 thf(fact_72_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_73_empty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((member @ A) @ C2) @ (bot_bot @ (set @ A)))))))).
% 0.30/1.51 thf(fact_74_subsetI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A3) => (((member @ A) @ X4) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 0.30/1.51 thf(fact_75_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => (A3 = B3)))))).
% 0.30/1.51 thf(fact_76_finite__code, axiom, (! [A:$tType]: (((finite_finite @ A) => ((finite_finite2 @ A) = (^ [A4:(set @ A)]: ($true))))))).
% 0.30/1.51 thf(fact_77_at__least__as__goodD, axiom, (! [A:$tType,Z:A,Y:A,B3:(set @ A),Pr:(set @ ((product_prod @ A) @ A))]: (((((member @ A) @ Z) @ ((((prefer310429814s_good @ A) @ Y) @ B3) @ Pr)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ Y)) @ Pr))))).
% 0.30/1.51 thf(fact_78_at__lst__asgd__ge, axiom, (! [A:$tType,X:A,Y:A,B3:(set @ A),Pr:(set @ ((product_prod @ A) @ A))]: (((((member @ A) @ X) @ ((((prefer310429814s_good @ A) @ Y) @ B3) @ Pr)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Pr))))).
% 0.30/1.51 thf(fact_79_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X3:A]: ((((member @ A) @ X3) @ A3))) = (A3 != (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_80_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_81_equals0D, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A3))))))).
% 0.30/1.51 thf(fact_82_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 0.30/1.51 thf(fact_83_in__mono, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ X) @ A3) => (((member @ A) @ X) @ B3)))))).
% 0.30/1.51 thf(fact_84_subsetD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))).
% 0.30/1.51 thf(fact_85_equalityE, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (~ ((((ord_less_eq @ (set @ A)) @ B3) @ A3)))))))))).
% 0.30/1.51 thf(fact_86_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B4:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A4) => (((member @ A) @ X3) @ B4)))))))))).
% 0.30/1.51 thf(fact_87_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 0.30/1.51 thf(fact_88_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (((ord_less_eq @ (set @ A)) @ B3) @ A3))))).
% 0.30/1.51 thf(fact_89_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B4:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A4) => (((member @ A) @ T2) @ B4)))))))))).
% 0.30/1.51 thf(fact_90_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 0.30/1.51 thf(fact_91_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 0.30/1.51 thf(fact_92_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C3) => (((ord_less_eq @ (set @ A)) @ A3) @ C3)))))).
% 0.30/1.51 thf(fact_93_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z2:(set @ A)]: ((Y4 = Z2))) = (^ [A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) & (((ord_less_eq @ (set @ A)) @ B4) @ A4)))))))).
% 0.30/1.51 thf(fact_94_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 0.30/1.51 thf(fact_95_finite, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A3:(set @ A)]: (((finite_finite2 @ A) @ A3))))))).
% 0.30/1.51 thf(fact_96_finite__set__choice, axiom, (! [B:$tType,A:$tType,A3:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X4:A]: (((((member @ A) @ X4) @ A3) => (? [X_1:B]: (((P @ X4) @ X_1)))))) => (? [F2:(A > B)]: ((! [X2:A]: (((((member @ A) @ X2) @ A3) => ((P @ X2) @ (F2 @ X2)))))))))))).
% 0.30/1.51 thf(fact_97_rational__preference_Osame__at__least__as__equal, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),Z:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ Y)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ Relation)) => (((((prefer310429814s_good @ A) @ Z) @ Carrier) @ Relation) = ((((prefer310429814s_good @ A) @ Y) @ Carrier) @ Relation))))))).
% 0.30/1.51 thf(fact_98_rational__preference_Opref__in__at__least__as, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) => (((member @ A) @ X) @ ((((prefer310429814s_good @ A) @ Y) @ Carrier) @ Relation))))))).
% 0.30/1.51 thf(fact_99_rational__preference_Oas__good__as__sameIff, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),Z:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ A) @ Z) @ ((((prefer951318096ood_as @ A) @ Y) @ Carrier) @ Relation)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ Y)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ Relation))))))).
% 0.30/1.51 thf(fact_100_rational__preference_Oas__good__asIff, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ A) @ X) @ ((((prefer951318096ood_as @ A) @ Y) @ Carrier) @ Relation)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation))))))).
% 0.30/1.51 thf(fact_101_rational__preference_Oat__lst__asgd__not__ge, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((Carrier != (bot_bot @ (set @ A))) => ((((member @ A) @ X) @ Carrier) => ((((member @ A) @ Y) @ Carrier) => ((~ ((((member @ A) @ X) @ ((((prefer310429814s_good @ A) @ Y) @ Carrier) @ Relation)))) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation))))))))))).
% 0.30/1.51 thf(fact_102_Pair__mono, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [X:A,X5:A,Y:B,Y5:B]: (((((ord_less_eq @ A) @ X) @ X5) => ((((ord_less_eq @ B) @ Y) @ Y5) => (((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ ((((product_Pair @ A) @ B) @ X5) @ Y5)))))))))).
% 0.30/1.51 thf(fact_103_finite__has__maximal2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A2) @ A3) => (? [X4:A]: (((((member @ A) @ X4) @ A3) & ((((ord_less_eq @ A) @ A2) @ X4) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ X4) @ Xa) => (X4 = Xa))))))))))))))))).
% 0.30/1.51 thf(fact_104_finite__has__minimal2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A2) @ A3) => (? [X4:A]: (((((member @ A) @ X4) @ A3) & ((((ord_less_eq @ A) @ X4) @ A2) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ Xa) @ X4) => (X4 = Xa))))))))))))))))).
% 0.30/1.51 thf(fact_105_bot__prod__def, axiom, (! [A:$tType,B:$tType]: ((((bot @ B) & (bot @ A)) => ((bot_bot @ ((product_prod @ A) @ B)) = ((((product_Pair @ A) @ B) @ (bot_bot @ A)) @ (bot_bot @ B))))))).
% 0.30/1.51 thf(fact_106_infinite__imp__nonempty, axiom, (! [A:$tType,S:(set @ A)]: (((~ (((finite_finite2 @ A) @ S))) => (S != (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_107_finite_OemptyI, axiom, (! [A:$tType]: (((finite_finite2 @ A) @ (bot_bot @ (set @ A)))))).
% 0.30/1.51 thf(fact_108_finite__subset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((finite_finite2 @ A) @ B3) => ((finite_finite2 @ A) @ A3)))))).
% 0.30/1.51 thf(fact_109_infinite__super, axiom, (! [A:$tType,S:(set @ A),T3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S) @ T3) => ((~ (((finite_finite2 @ A) @ S))) => (~ (((finite_finite2 @ A) @ T3)))))))).
% 0.30/1.51 thf(fact_110_rev__finite__subset, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((finite_finite2 @ A) @ A3)))))).
% 0.30/1.51 thf(fact_111_finite__has__minimal, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (? [X4:A]: (((((member @ A) @ X4) @ A3) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ Xa) @ X4) => (X4 = Xa)))))))))))))))).
% 0.30/1.51 thf(fact_112_finite__has__maximal, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (? [X4:A]: (((((member @ A) @ X4) @ A3) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ X4) @ Xa) => (X4 = Xa)))))))))))))))).
% 0.30/1.51 thf(fact_113_card__subset__eq, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((finite_card @ A) @ A3) = ((finite_card @ A) @ B3)) => (A3 = B3))))))).
% 0.30/1.51 thf(fact_114_infinite__arbitrarily__large, axiom, (! [A:$tType,A3:(set @ A),N:nat]: (((~ (((finite_finite2 @ A) @ A3))) => (? [B5:(set @ A)]: ((((finite_finite2 @ A) @ B5) & ((((finite_card @ A) @ B5) = N) & (((ord_less_eq @ (set @ A)) @ B5) @ A3))))))))).
% 0.30/1.51 thf(fact_115_finite__if__finite__subsets__card__bdd, axiom, (! [A:$tType,F3:(set @ A),C3:nat]: (((! [G2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ G2) @ F3) => (((finite_finite2 @ A) @ G2) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ G2)) @ C3))))) => (((finite_finite2 @ A) @ F3) & (((ord_less_eq @ nat) @ ((finite_card @ A) @ F3)) @ C3)))))).
% 0.30/1.51 thf(fact_116_card__seteq, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ B3)) @ ((finite_card @ A) @ A3)) => (A3 = B3))))))).
% 0.30/1.51 thf(fact_117_obtain__subset__with__card__n, axiom, (! [A:$tType,N:nat,S:(set @ A)]: (((((ord_less_eq @ nat) @ N) @ ((finite_card @ A) @ S)) => (~ ((! [T4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ T4) @ S) => ((((finite_card @ A) @ T4) = N) => (~ (((finite_finite2 @ A) @ T4))))))))))))).
% 0.30/1.51 thf(fact_118_card__le__if__inj__on__rel, axiom, (! [B:$tType,A:$tType,B3:(set @ A),A3:(set @ B),R:(B > (A > $o))]: ((((finite_finite2 @ A) @ B3) => ((! [A5:B]: (((((member @ B) @ A5) @ A3) => (? [B6:A]: (((((member @ A) @ B6) @ B3) & ((R @ A5) @ B6))))))) => ((! [A1:B,A22:B,B7:A]: (((((member @ B) @ A1) @ A3) => ((((member @ B) @ A22) @ A3) => ((((member @ A) @ B7) @ B3) => (((R @ A1) @ B7) => (((R @ A22) @ B7) => (A1 = A22)))))))) => (((ord_less_eq @ nat) @ ((finite_card @ B) @ A3)) @ ((finite_card @ A) @ B3)))))))).
% 0.30/1.51 thf(fact_119_ex__card, axiom, (! [A:$tType,N:nat,A3:(set @ A)]: (((((ord_less_eq @ nat) @ N) @ ((finite_card @ A) @ A3)) => (? [S2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S2) @ A3) & (((finite_card @ A) @ S2) = N)))))))).
% 0.30/1.51 thf(fact_120_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A6:A,B8:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A6) @ B8)) = ((A2 = A6) & (B2 = B8)))))).
% 0.30/1.51 thf(fact_121_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 0.30/1.51 thf(fact_122_bot__apply, axiom, (! [C:$tType,D2:$tType]: (((bot @ C) => ((bot_bot @ (D2 > C)) = (^ [X3:D2]: ((bot_bot @ C)))))))).
% 0.30/1.51 thf(fact_123_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 0.30/1.51 thf(fact_124_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 0.30/1.51 thf(fact_125_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 0.30/1.51 thf(fact_126_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A7:A,B9:A]: (((((ord_less_eq @ A) @ B9) @ A7) & (((ord_less_eq @ A) @ A7) @ B9))))))))).
% 0.30/1.51 thf(fact_127_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 0.30/1.51 thf(fact_128_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A5:A,B7:A]: (((((ord_less_eq @ A) @ A5) @ B7) => ((P @ A5) @ B7)))) => ((! [A5:A,B7:A]: ((((P @ B7) @ A5) => ((P @ A5) @ B7)))) => ((P @ A2) @ B2))))))))).
% 0.30/1.51 thf(fact_129_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 0.30/1.51 thf(fact_130_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 0.30/1.51 thf(fact_131_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 0.30/1.51 thf(fact_132_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 0.30/1.51 thf(fact_133_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 0.30/1.51 thf(fact_134_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A7:A,B9:A]: (((((ord_less_eq @ A) @ A7) @ B9) & (((ord_less_eq @ A) @ B9) @ A7))))))))).
% 0.30/1.51 thf(fact_135_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 0.30/1.51 thf(fact_136_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z)))) => (((((ord_less_eq @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X)))) => (~ (((((ord_less_eq @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 0.30/1.51 thf(fact_137_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 0.30/1.51 thf(fact_138_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 0.30/1.51 thf(fact_139_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 0.30/1.51 thf(fact_140_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 0.30/1.51 thf(fact_141_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 0.30/1.51 thf(fact_142_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) & (((ord_less_eq @ A) @ Y2) @ X3))))))))).
% 0.30/1.51 thf(fact_143_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C2) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ B) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C2)))))))))).
% 0.30/1.51 thf(fact_144_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X4:B,Y3:B]: (((((ord_less_eq @ B) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 0.30/1.51 thf(fact_145_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ C) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ C) @ (F @ A2)) @ C2)))))))))).
% 0.30/1.51 thf(fact_146_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X4:B,Y3:B]: (((((ord_less_eq @ B) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 0.30/1.51 thf(fact_147_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F4:(A > B),G3:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F4 @ X3)) @ (G3 @ X3))))))))))).
% 0.30/1.51 thf(fact_148_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X4:A]: ((((ord_less_eq @ B) @ (F @ X4)) @ (G @ X4)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 0.30/1.51 thf(fact_149_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 0.30/1.51 thf(fact_150_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 0.30/1.51 thf(fact_151_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X3:A]: ((bot_bot @ B)))))))).
% 0.30/1.51 thf(fact_152_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X4:A,Y3:B]: ((P2 = ((((product_Pair @ A) @ B) @ X4) @ Y3))))))).
% 0.30/1.51 thf(fact_153_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B7:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B7)))) => (P @ P2))))).
% 0.30/1.51 thf(fact_154_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A6:A,B8:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A6) @ B8)) => (~ (((A2 = A6) => (B2 != B8)))))))).
% 0.30/1.51 thf(fact_155_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B7:B,C4:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B7) @ C4)))))))))).
% 0.30/1.51 thf(fact_156_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: ((~ ((! [A5:A,B7:B,C4:C,D3:D2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B7) @ ((((product_Pair @ C) @ D2) @ C4) @ D3))))))))))).
% 0.30/1.51 thf(fact_157_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))))]: ((~ ((! [A5:A,B7:B,C4:C,D3:D2,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E)) @ C4) @ ((((product_Pair @ D2) @ E) @ D3) @ E2)))))))))))).
% 0.30/1.51 thf(fact_158_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,F5:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5)))))]: ((~ ((! [A5:A,B7:B,C4:C,D3:D2,E2:E,F2:F5]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5)))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ F5)) @ D3) @ ((((product_Pair @ E) @ F5) @ E2) @ F2))))))))))))).
% 0.30/1.51 thf(fact_159_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,F5:$tType,G4:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G4))))))]: ((~ ((! [A5:A,B7:B,C4:C,D3:D2,E2:E,F2:F5,G5:G4]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G4)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G4))))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G4)))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G4))) @ D3) @ ((((product_Pair @ E) @ ((product_prod @ F5) @ G4)) @ E2) @ ((((product_Pair @ F5) @ G4) @ F2) @ G5)))))))))))))).
% 0.30/1.51 thf(fact_160_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B7:B,C4:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B7) @ C4))))) => (P @ X))))).
% 0.30/1.51 thf(fact_161_prod__induct4, axiom, (! [D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: (((! [A5:A,B7:B,C4:C,D3:D2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B7) @ ((((product_Pair @ C) @ D2) @ C4) @ D3)))))) => (P @ X))))).
% 0.30/1.51 thf(fact_162_prod__induct5, axiom, (! [E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))))]: (((! [A5:A,B7:B,C4:C,D3:D2,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E)) @ C4) @ ((((product_Pair @ D2) @ E) @ D3) @ E2))))))) => (P @ X))))).
% 0.30/1.51 thf(fact_163_prod__induct6, axiom, (! [F5:$tType,E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5)))))]: (((! [A5:A,B7:B,C4:C,D3:D2,E2:E,F2:F5]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5)))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ F5)) @ D3) @ ((((product_Pair @ E) @ F5) @ E2) @ F2)))))))) => (P @ X))))).
% 0.30/1.51 thf(fact_164_prod__induct7, axiom, (! [G4:$tType,F5:$tType,E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G4)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G4))))))]: (((! [A5:A,B7:B,C4:C,D3:D2,E2:E,F2:F5,G5:G4]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G4)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G4))))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G4)))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G4))) @ D3) @ ((((product_Pair @ E) @ ((product_prod @ F5) @ G4)) @ E2) @ ((((product_Pair @ F5) @ G4) @ F2) @ G5))))))))) => (P @ X))))).
% 0.30/1.51 thf(fact_165_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B7:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B7))))))))).
% 0.30/1.51 thf(fact_166_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B7:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B7)))) => (P @ Prod))))).
% 0.30/1.51 thf(fact_167_bounded__Max__nat, axiom, (! [P:(nat > $o),X:nat,M:nat]: (((P @ X) => ((! [X4:nat]: (((P @ X4) => (((ord_less_eq @ nat) @ X4) @ M)))) => (~ ((! [M2:nat]: (((P @ M2) => (~ ((! [X2:nat]: (((P @ X2) => (((ord_less_eq @ nat) @ X2) @ M2)))))))))))))))).
% 0.30/1.51 thf(fact_168_finite__nat__set__iff__bounded__le, axiom, ((finite_finite2 @ nat) = (^ [N2:(set @ nat)]: ((? [M3:nat]: ((! [X3:nat]: (((((member @ nat) @ X3) @ N2) => (((ord_less_eq @ nat) @ X3) @ M3)))))))))).
% 0.30/1.51 thf(fact_169_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) => (A2 = (bot_bot @ A))))))))).
% 0.30/1.51 thf(fact_170_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) = (A2 = (bot_bot @ A))))))))).
% 0.30/1.51 thf(fact_171_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A2))))))).
% 0.30/1.51 thf(fact_172_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 0.30/1.51 thf(fact_173_preorder__on__empty, axiom, (! [A:$tType]: ((((order_preorder_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 0.30/1.51 thf(fact_174_total__onI, axiom, (! [A:$tType,A3:(set @ A),R:(set @ ((product_prod @ A) @ A))]: (((! [X4:A,Y3:A]: (((((member @ A) @ X4) @ A3) => ((((member @ A) @ Y3) @ A3) => ((X4 != Y3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y3)) @ R) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X4)) @ R))))))) => (((total_on @ A) @ A3) @ R))))).
% 0.30/1.51 thf(fact_175_total__on__def, axiom, (! [A:$tType]: (((total_on @ A) = (^ [A4:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: ((! [X3:A]: (((((member @ A) @ X3) @ A4) => (! [Y2:A]: (((((member @ A) @ Y2) @ A4) => ((X3 != Y2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y2)) @ R2) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ X3)) @ R2))))))))))))))).
% 0.30/1.51 thf(fact_176_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X3:A]: ((((member @ A) @ X3) @ (bot_bot @ (set @ A))))))))).
% 0.30/1.51 thf(fact_177_subrelI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: (((! [X4:A,Y3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y3)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y3)) @ S3)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S3))))).
% 0.30/1.51 thf(fact_178_refl__on__empty, axiom, (! [A:$tType]: ((((refl_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 0.30/1.51 thf(fact_179_refl__on__domain, axiom, (! [A:$tType,A3:(set @ A),R:(set @ ((product_prod @ A) @ A)),A2:A,B2:A]: (((((refl_on @ A) @ A3) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R) => ((((member @ A) @ A2) @ A3) & (((member @ A) @ B2) @ A3))))))).
% 0.30/1.51 thf(fact_180_refl__onD2, axiom, (! [A:$tType,A3:(set @ A),R:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((refl_on @ A) @ A3) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => (((member @ A) @ Y) @ A3)))))).
% 0.30/1.51 thf(fact_181_refl__onD1, axiom, (! [A:$tType,A3:(set @ A),R:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((refl_on @ A) @ A3) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => (((member @ A) @ X) @ A3)))))).
% 0.30/1.51 thf(fact_182_refl__onD, axiom, (! [A:$tType,A3:(set @ A),R:(set @ ((product_prod @ A) @ A)),A2:A]: (((((refl_on @ A) @ A3) @ R) => ((((member @ A) @ A2) @ A3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R)))))).
% 0.30/1.51 thf(fact_183_total__on__empty, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((total_on @ A) @ (bot_bot @ (set @ A))) @ R)))).
% 0.30/1.51 thf(fact_184_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 0.30/1.51 thf(fact_185_infinite__nat__iff__unbounded__le, axiom, (! [S:(set @ nat)]: (((~ (((finite_finite2 @ nat) @ S))) = (! [M3:nat]: ((? [N3:nat]: (((((ord_less_eq @ nat) @ M3) @ N3) & (((member @ nat) @ N3) @ S)))))))))).
% 0.30/1.51 thf(fact_186_finite__transitivity__chain, axiom, (! [A:$tType,A3:(set @ A),R3:(A > (A > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X4:A]: ((~ (((R3 @ X4) @ X4))))) => ((! [X4:A,Y3:A,Z3:A]: ((((R3 @ X4) @ Y3) => (((R3 @ Y3) @ Z3) => ((R3 @ X4) @ Z3))))) => ((! [X4:A]: (((((member @ A) @ X4) @ A3) => (? [Y6:A]: (((((member @ A) @ Y6) @ A3) & ((R3 @ X4) @ Y6))))))) => (A3 = (bot_bot @ (set @ A)))))))))).
% 0.30/1.51 thf(fact_187_less__by__empty, axiom, (! [A:$tType,A3:(set @ ((product_prod @ A) @ A)),B3:(set @ ((product_prod @ A) @ A))]: (((A3 = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ A3) @ B3))))).
% 0.30/1.51 thf(fact_188_subset__emptyI, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A3))))) => (((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_189_rational__preference__def, axiom, (! [A:$tType]: (((prefer1997167224erence @ A) = (^ [Carrier2:(set @ A),Relation2:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier2) @ Relation2) & (((prefer1801827867axioms @ A) @ Carrier2) @ Relation2)))))))).
% 0.30/1.51 thf(fact_190_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R:A,S3:B,R3:(set @ ((product_prod @ A) @ B)),S4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S3)) @ R3) => ((S4 = S3) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S4)) @ R3)))))).
% 0.30/1.51 thf(fact_191_rational__preference__axioms_Ointro, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((total_on @ A) @ Carrier) @ Relation) => (((prefer1801827867axioms @ A) @ Carrier) @ Relation))))).
% 0.30/1.51 thf(fact_192_rational__preference__axioms__def, axiom, (! [A:$tType]: (((prefer1801827867axioms @ A) = (total_on @ A))))).
% 0.30/1.51 thf(fact_193_rational__preference_Oaxioms_I2_J, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((prefer1801827867axioms @ A) @ Carrier) @ Relation))))).
% 0.30/1.51 thf(fact_194_rational__preference_Ointro, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => ((((prefer1801827867axioms @ A) @ Carrier) @ Relation) => (((prefer1997167224erence @ A) @ Carrier) @ Relation)))))).
% 0.30/1.51 thf(fact_195_finite__indexed__bound, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [A3:(set @ B),P:(B > (A > $o))]: ((((finite_finite2 @ B) @ A3) => ((! [X4:B]: (((((member @ B) @ X4) @ A3) => (? [X_1:A]: (((P @ X4) @ X_1)))))) => (? [M2:A]: ((! [X2:B]: (((((member @ B) @ X2) @ A3) => (? [K:A]: (((((ord_less_eq @ A) @ K) @ M2) & ((P @ X2) @ K))))))))))))))))).
% 0.30/1.51 thf(fact_196_transitivity, axiom, ((trans @ a) @ relation)).
% 0.30/1.51 thf(fact_197_trans__empty, axiom, (! [A:$tType]: (((trans @ A) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 0.30/1.51 thf(fact_198_transD, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z:A]: ((((trans @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z)) @ R))))))).
% 0.30/1.51 thf(fact_199_transE, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z:A]: ((((trans @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z)) @ R))))))).
% 0.30/1.51 thf(fact_200_transI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((! [X4:A,Y3:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y3)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z3)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Z3)) @ R))))) => ((trans @ A) @ R))))).
% 0.30/1.51 thf(fact_201_trans__def, axiom, (! [A:$tType]: (((trans @ A) = (^ [R2:(set @ ((product_prod @ A) @ A))]: ((! [X3:A,Y2:A,Z4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y2)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ Z4)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z4)) @ R2))))))))))).
% 0.30/1.51 thf(fact_202_preference_Otransitivity, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => ((trans @ A) @ Relation))))).
% 0.30/1.51 thf(fact_203_preorder__on__def, axiom, (! [A:$tType]: (((order_preorder_on @ A) = (^ [A4:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((refl_on @ A) @ A4) @ R2) & ((trans @ A) @ R2)))))))).
% 0.30/1.51 thf(fact_204_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A4:(set @ A)]: ((A4 = (bot_bot @ (set @ A))))))))).
% 0.30/1.51 thf(fact_205_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C2 @ A2) @ B2))))).
% 0.30/1.51 thf(fact_206_card__0__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((((finite_card @ A) @ A3) = (zero_zero @ nat)) = (A3 = (bot_bot @ (set @ A)))))))).
% 0.30/1.51 thf(fact_207_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R2:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R2) => (((ord_less_eq @ B) @ (As @ I)) @ (As @ J)))))))))))).
% 0.30/1.51 thf(fact_208_card_Oempty, axiom, (! [A:$tType]: ((((finite_card @ A) @ (bot_bot @ (set @ A))) = (zero_zero @ nat))))).
% 0.30/1.51 thf(fact_209_card_Oinfinite, axiom, (! [A:$tType,A3:(set @ A)]: (((~ (((finite_finite2 @ A) @ A3))) => (((finite_card @ A) @ A3) = (zero_zero @ nat)))))).
% 0.30/1.51 thf(fact_210_card__eq__0__iff, axiom, (! [A:$tType,A3:(set @ A)]: (((((finite_card @ A) @ A3) = (zero_zero @ nat)) = ((A3 = (bot_bot @ (set @ A))) | (~ (((finite_finite2 @ A) @ A3)))))))).
% 0.30/1.51 thf(fact_211_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))).
% 0.30/1.51 thf(fact_212_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 0.30/1.51 thf(fact_213_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K2:nat,B2:nat]: (((P @ K2) => ((! [Y3:nat]: (((P @ Y3) => (((ord_less_eq @ nat) @ Y3) @ B2)))) => (? [X4:nat]: (((P @ X4) & (! [Y6:nat]: (((P @ Y6) => (((ord_less_eq @ nat) @ Y6) @ X4)))))))))))).
% 0.30/1.51 thf(fact_214_nat__le__linear, axiom, (! [M4:nat,N:nat]: (((((ord_less_eq @ nat) @ M4) @ N) | (((ord_less_eq @ nat) @ N) @ M4))))).
% 0.30/1.51 thf(fact_215_le__antisym, axiom, (! [M4:nat,N:nat]: (((((ord_less_eq @ nat) @ M4) @ N) => ((((ord_less_eq @ nat) @ N) @ M4) => (M4 = N)))))).
% 0.30/1.51 thf(fact_216_eq__imp__le, axiom, (! [M4:nat,N:nat]: (((M4 = N) => (((ord_less_eq @ nat) @ M4) @ N))))).
% 0.30/1.51 thf(fact_217_le__trans, axiom, (! [I2:nat,J2:nat,K2:nat]: (((((ord_less_eq @ nat) @ I2) @ J2) => ((((ord_less_eq @ nat) @ J2) @ K2) => (((ord_less_eq @ nat) @ I2) @ K2)))))).
% 0.30/1.51 thf(fact_218_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 0.30/1.51 thf(fact_219_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 0.30/1.51 thf(fact_220_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 0.30/1.51 thf(fact_221_bot__nat__0_Oextremum__unique, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) = (A2 = (zero_zero @ nat)))))).
% 0.30/1.51 thf(fact_222_bot__nat__0_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) => (A2 = (zero_zero @ nat)))))).
% 0.30/1.51 thf(fact_223_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 0.30/1.51 thf(fact_224_zero__prod__def, axiom, (! [A:$tType,B:$tType]: ((((zero @ B) & (zero @ A)) => ((zero_zero @ ((product_prod @ A) @ B)) = ((((product_Pair @ A) @ B) @ (zero_zero @ A)) @ (zero_zero @ B))))))).
% 0.30/1.51 thf(fact_225_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 0.30/1.51 thf(fact_226_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 0.30/1.51 thf(fact_227_card__gt__0__iff, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ A3)) = ((A3 != (bot_bot @ (set @ A))) & ((finite_finite2 @ A) @ A3)))))).
% 0.30/1.51 thf(fact_228_ex__least__nat__le, axiom, (! [P:(nat > $o),N:nat]: (((P @ N) => ((~ ((P @ (zero_zero @ nat)))) => (? [K:nat]: (((((ord_less_eq @ nat) @ K) @ N) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ K) => (~ ((P @ I3)))))) & (P @ K)))))))))).
% 0.30/1.51 thf(fact_229_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less @ A) @ X) @ Y)))))))))).
% 0.30/1.51 thf(fact_230_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 0.30/1.51 thf(fact_231_le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) | (X3 = Y2))))))))).
% 0.30/1.51 thf(fact_232_less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) & (X3 != Y2))))))))).
% 0.30/1.51 thf(fact_233_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C2) => ((! [X4:B,Y3:B]: (((((ord_less @ B) @ X4) @ Y3) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 0.30/1.51 thf(fact_234_order__le__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ C) @ (F @ B2)) @ C2) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ C) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ C) @ (F @ A2)) @ C2)))))))))).
% 0.30/1.51 thf(fact_235_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X4:B,Y3:B]: (((((ord_less_eq @ B) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 0.30/1.51 thf(fact_236_order__less__le__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) => (((ord_less @ C) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less @ C) @ (F @ A2)) @ C2)))))))))).
% 0.30/1.51 thf(fact_237_not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) = (((ord_less @ A) @ Y) @ X)))))))).
% 0.30/1.51 thf(fact_238_not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 0.30/1.51 thf(fact_239_le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((A2 != B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 0.30/1.51 thf(fact_240_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 0.30/1.51 thf(fact_241_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 0.30/1.51 thf(fact_242_less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 0.30/1.51 thf(fact_243_le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 0.30/1.51 thf(fact_244_less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 0.30/1.51 thf(fact_245_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z:A,Y:A]: (((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (((ord_less_eq @ A) @ Y) @ X4)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 0.30/1.51 thf(fact_246_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y:A,Z:A]: (((! [X4:A]: (((((ord_less @ A) @ X4) @ Y) => (((ord_less_eq @ A) @ X4) @ Z)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 0.30/1.51 thf(fact_247_le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X)))))))).
% 0.30/1.51 thf(fact_248_le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ X) @ Y) | (X = Y))))))))).
% 0.30/1.51 thf(fact_249_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) & (~ ((((ord_less_eq @ A) @ Y2) @ X3))))))))))).
% 0.30/1.51 thf(fact_250_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X))) => (((ord_less @ A) @ X) @ Y)))))))).
% 0.30/1.51 thf(fact_251_order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 0.30/1.51 thf(fact_252_order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 0.30/1.51 thf(fact_253_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A7:A,B9:A]: (((((ord_less @ A) @ A7) @ B9) | (A7 = B9))))))))).
% 0.30/1.51 thf(fact_254_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A7:A,B9:A]: (((((ord_less_eq @ A) @ A7) @ B9) & (A7 != B9))))))))).
% 0.30/1.51 thf(fact_255_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ A2))))))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A8:$tType,A9:$tType]: (((order_bot @ A9) => (order_bot @ (A8 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A8:$tType,A9:$tType]: ((((finite_finite @ A8) & (finite_finite @ A9)) => (finite_finite @ (A8 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Obot, axiom, (! [A8:$tType,A9:$tType]: (((bot @ A9) => (bot @ (A8 > A9)))))).
% 0.30/1.51 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 0.30/1.51 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ nat)).
% 0.30/1.51 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1, axiom, (order_bot @ nat)).
% 0.30/1.51 thf(tcon_Nat_Onat___Orderings_Opreorder_2, axiom, (preorder @ nat)).
% 0.30/1.51 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 0.30/1.51 thf(tcon_Nat_Onat___Orderings_Oorder_3, axiom, (order @ nat)).
% 0.30/1.51 thf(tcon_Nat_Onat___Orderings_Oord_4, axiom, (ord @ nat)).
% 0.30/1.51 thf(tcon_Nat_Onat___Orderings_Obot_5, axiom, (bot @ nat)).
% 0.30/1.51 thf(tcon_Nat_Onat___Groups_Ozero, axiom, (zero @ nat)).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Oorder__bot_6, axiom, (! [A8:$tType]: ((order_bot @ (set @ A8))))).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Opreorder_7, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 0.30/1.51 thf(tcon_Set_Oset___Finite__Set_Ofinite_8, axiom, (! [A8:$tType]: (((finite_finite @ A8) => (finite_finite @ (set @ A8)))))).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Oorder_9, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Oord_10, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Obot_11, axiom, (! [A8:$tType]: ((bot @ (set @ A8))))).
% 0.30/1.51 thf(tcon_Set_Oset___Groups_Ozero_12, axiom, (! [A8:$tType]: (((zero @ A8) => (zero @ (set @ A8)))))).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Oorder__bot_13, axiom, (order_bot @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Opreorder_14, axiom, (preorder @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Olinorder_15, axiom, (linorder @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Finite__Set_Ofinite_16, axiom, (finite_finite @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Oorder_17, axiom, (order @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Oord_18, axiom, (ord @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Obot_19, axiom, (bot @ $o)).
% 0.30/1.51 thf(tcon_Product__Type_Oprod___Orderings_Oorder__bot_20, axiom, (! [A8:$tType,A9:$tType]: ((((order_bot @ A8) & (order_bot @ A9)) => (order_bot @ ((product_prod @ A8) @ A9)))))).
% 0.30/1.51 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_21, axiom, (! [A8:$tType,A9:$tType]: ((((preorder @ A8) & (preorder @ A9)) => (preorder @ ((product_prod @ A8) @ A9)))))).
% 0.30/1.51 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_22, axiom, (! [A8:$tType,A9:$tType]: ((((finite_finite @ A8) & (finite_finite @ A9)) => (finite_finite @ ((product_prod @ A8) @ A9)))))).
% 0.30/1.51 thf(tcon_Product__Type_Oprod___Orderings_Oorder_23, axiom, (! [A8:$tType,A9:$tType]: ((((order @ A8) & (order @ A9)) => (order @ ((product_prod @ A8) @ A9)))))).
% 0.30/1.51 thf(tcon_Product__Type_Oprod___Orderings_Oord_24, axiom, (! [A8:$tType,A9:$tType]: ((((ord @ A8) & (ord @ A9)) => (ord @ ((product_prod @ A8) @ A9)))))).
% 0.30/1.51 thf(tcon_Product__Type_Oprod___Orderings_Obot_25, axiom, (! [A8:$tType,A9:$tType]: ((((bot @ A8) & (bot @ A9)) => (bot @ ((product_prod @ A8) @ A9)))))).
% 0.30/1.51 thf(tcon_Product__Type_Oprod___Groups_Ozero_26, axiom, (! [A8:$tType,A9:$tType]: ((((zero @ A8) & (zero @ A9)) => (zero @ ((product_prod @ A8) @ A9)))))).
% 0.30/1.51 thf(conj_0, conjecture, (((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ x) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ y) @ carrier) @ relation))).
% 0.30/1.51 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.vFZO1l9sao/DTF2DTF28194.p
% 0.30/1.51 ------------------------
% 0.30/1.54 ---- Cleaned THF ---
% 0.30/1.54 thf(ty_t_Product__Type_Oprod,type,
% 0.30/1.54 product_prod: $tType > $tType > $tType ).
% 0.30/1.54
% 0.30/1.54 thf(ty_t_Set_Oset,type,
% 0.30/1.54 set: $tType > $tType ).
% 0.30/1.54
% 0.30/1.54 thf(ty_t_Nat_Onat,type,
% 0.30/1.54 nat: $tType ).
% 0.30/1.54
% 0.30/1.54 thf(ty_tf_a,type,
% 0.30/1.54 a: $tType ).
% 0.30/1.54
% 0.30/1.54 thf(sy_cl_HOL_Otype,type,
% 0.30/1.54 type:
% 0.30/1.54 !>[A: $tType] : $o ).
% 0.30/1.54
% 0.30/1.54 thf(sy_cl_Groups_Ozero,type,
% 0.30/1.54 zero:
% 0.30/1.54 !>[A: $tType] : $o ).
% 0.30/1.54
% 0.30/1.54 thf(sy_cl_Orderings_Obot,type,
% 0.30/1.54 bot:
% 0.30/1.54 !>[A: $tType] : $o ).
% 0.30/1.54
% 0.30/1.54 thf(sy_cl_Orderings_Oord,type,
% 0.30/1.54 ord:
% 0.30/1.54 !>[A: $tType] : $o ).
% 0.30/1.54
% 0.30/1.54 thf(sy_cl_Orderings_Oorder,type,
% 0.30/1.54 order:
% 0.30/1.54 !>[A: $tType] : $o ).
% 0.30/1.54
% 0.30/1.54 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.30/1.54 finite_finite:
% 0.30/1.54 !>[A: $tType] : $o ).
% 0.30/1.54
% 0.30/1.54 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/1.54 linorder:
% 0.30/1.54 !>[A: $tType] : $o ).
% 0.30/1.54
% 0.30/1.54 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/1.54 preorder:
% 0.30/1.54 !>[A: $tType] : $o ).
% 0.30/1.54
% 0.30/1.54 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.30/1.54 order_bot:
% 0.30/1.54 !>[A: $tType] : $o ).
% 0.30/1.54
% 0.30/1.54 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.30/1.54 dense_linorder:
% 0.30/1.54 !>[A: $tType] : $o ).
% 0.30/1.54
% 0.30/1.54 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.30/1.54 linord1659791738miring:
% 0.30/1.54 !>[A: $tType] : $o ).
% 0.30/1.54
% 0.30/1.54 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.30/1.54 canoni770627133id_add:
% 0.30/1.54 !>[A: $tType] : $o ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.30/1.54 bNF_Ca1785829860lChain:
% 0.30/1.54 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Finite__Set_Ocard,type,
% 0.30/1.54 finite_card:
% 0.30/1.54 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Finite__Set_Ofinite,type,
% 0.30/1.54 finite_finite2:
% 0.30/1.54 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.30/1.54 zero_zero:
% 0.30/1.54 !>[A: $tType] : A ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Order__Relation_Opreorder__on,type,
% 0.30/1.54 order_preorder_on:
% 0.30/1.54 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.30/1.54 bot_bot:
% 0.30/1.54 !>[A: $tType] : A ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.30/1.54 ord_less:
% 0.30/1.54 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/1.54 ord_less_eq:
% 0.30/1.54 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oas__good__as,type,
% 0.30/1.54 prefer951318096ood_as:
% 0.30/1.54 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good,type,
% 0.30/1.54 prefer310429814s_good:
% 0.30/1.54 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Ono__better__than,type,
% 0.30/1.54 prefer1532642881r_than:
% 0.30/1.54 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference,type,
% 0.30/1.54 prefer199794634erence:
% 0.30/1.54 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference,type,
% 0.30/1.54 prefer1997167224erence:
% 0.30/1.54 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference__axioms,type,
% 0.30/1.54 prefer1801827867axioms:
% 0.30/1.54 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Product__Type_OPair,type,
% 0.30/1.54 product_Pair:
% 0.30/1.54 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.30/1.54 produc2004651681e_prod:
% 0.30/1.54 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.30/1.54 product_rec_prod:
% 0.30/1.54 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Relation_Orefl__on,type,
% 0.30/1.54 refl_on:
% 0.30/1.54 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Relation_Ototal__on,type,
% 0.30/1.54 total_on:
% 0.30/1.54 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Relation_Otrans,type,
% 0.30/1.54 trans:
% 0.30/1.54 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Set_OCollect,type,
% 0.30/1.54 collect:
% 0.30/1.54 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_Set_Ois__empty,type,
% 0.30/1.54 is_empty:
% 0.30/1.54 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_c_member,type,
% 0.30/1.54 member:
% 0.30/1.54 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_v_carrier,type,
% 0.30/1.54 carrier: set @ a ).
% 0.30/1.54
% 0.30/1.54 thf(sy_v_relation,type,
% 0.30/1.54 relation: set @ ( product_prod @ a @ a ) ).
% 0.30/1.54
% 0.30/1.54 thf(sy_v_x,type,
% 0.30/1.54 x: a ).
% 0.30/1.54
% 0.30/1.54 thf(sy_v_y,type,
% 0.30/1.54 y: a ).
% 0.30/1.54
% 0.30/1.54 thf(fact_0_assms_I2_J,axiom,
% 0.30/1.54 member @ a @ y @ carrier ).
% 0.30/1.54
% 0.30/1.54 thf(fact_1_assms_I1_J,axiom,
% 0.30/1.54 member @ a @ x @ carrier ).
% 0.30/1.54
% 0.30/1.54 thf(fact_2_xy__in__eachothers__nbt,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( member @ a @ X @ carrier )
% 0.30/1.54 => ( ( member @ a @ Y @ carrier )
% 0.30/1.54 => ( ( member @ a @ X @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) )
% 0.30/1.54 | ( member @ a @ Y @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_3__092_060open_062card_A_Ino__better__than_Ax_Acarrier_Arelation_J_A_092_060le_062_Acard_A_Ino__better__than_Ay_Acarrier_Arelation_J_092_060close_062,axiom,
% 0.30/1.54 ord_less_eq @ nat @ ( finite_card @ a @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) ) @ ( finite_card @ a @ ( prefer1532642881r_than @ a @ y @ carrier @ relation ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_4_nbt__nest,axiom,
% 0.30/1.54 ! [Y: a,X: a] :
% 0.30/1.54 ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) )
% 0.30/1.54 | ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_5_nbt__subset__carrier,axiom,
% 0.30/1.54 ! [X: a] :
% 0.30/1.54 ( ( member @ a @ X @ carrier )
% 0.30/1.54 => ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ carrier ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_6_rational__preference__axioms,axiom,
% 0.30/1.54 prefer1997167224erence @ a @ carrier @ relation ).
% 0.30/1.54
% 0.30/1.54 thf(fact_7__092_060open_062finite_Acarrier_092_060close_062,axiom,
% 0.30/1.54 finite_finite2 @ a @ carrier ).
% 0.30/1.54
% 0.30/1.54 thf(fact_8_preference__axioms,axiom,
% 0.30/1.54 prefer199794634erence @ a @ carrier @ relation ).
% 0.30/1.54
% 0.30/1.54 thf(fact_9_trans__refl,axiom,
% 0.30/1.54 order_preorder_on @ a @ carrier @ relation ).
% 0.30/1.54
% 0.30/1.54 thf(fact_10_nbt__subset,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( finite_finite2 @ a @ carrier )
% 0.30/1.54 => ( ( member @ a @ X @ carrier )
% 0.30/1.54 => ( ( member @ a @ Y @ carrier )
% 0.30/1.54 => ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) )
% 0.30/1.54 | ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_11_no__better__subset__pref,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 => ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_12_no__better__thansubset__rel,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( member @ a @ X @ carrier )
% 0.30/1.54 => ( ( member @ a @ Y @ carrier )
% 0.30/1.54 => ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) )
% 0.30/1.54 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_13_no__better__than__nonepty,axiom,
% 0.30/1.54 ! [X: a] :
% 0.30/1.54 ( ( carrier
% 0.30/1.54 != ( bot_bot @ ( set @ a ) ) )
% 0.30/1.54 => ( ( member @ a @ X @ carrier )
% 0.30/1.54 => ( ( prefer1532642881r_than @ a @ X @ carrier @ relation )
% 0.30/1.54 != ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_14_total,axiom,
% 0.30/1.54 total_on @ a @ carrier @ relation ).
% 0.30/1.54
% 0.30/1.54 thf(fact_15_fnt__carrier__fnt__nbt,axiom,
% 0.30/1.54 ! [X2: a] :
% 0.30/1.54 ( ( member @ a @ X2 @ carrier )
% 0.30/1.54 => ( finite_finite2 @ a @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_16_same__nbt__same__pref,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( member @ a @ X @ carrier )
% 0.30/1.54 => ( ( member @ a @ Y @ carrier )
% 0.30/1.54 => ( ( ( member @ a @ X @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) )
% 0.30/1.54 & ( member @ a @ Y @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) )
% 0.30/1.54 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_17_worse__in__no__better,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 => ( member @ a @ Y @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_18_reflexivity,axiom,
% 0.30/1.54 refl_on @ a @ carrier @ relation ).
% 0.30/1.54
% 0.30/1.54 thf(fact_19_compl,axiom,
% 0.30/1.54 ! [X2: a] :
% 0.30/1.54 ( ( member @ a @ X2 @ carrier )
% 0.30/1.54 => ! [Xa: a] :
% 0.30/1.54 ( ( member @ a @ Xa @ carrier )
% 0.30/1.54 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Xa ) @ relation )
% 0.30/1.54 | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Xa @ X2 ) @ relation ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_20_completeD,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( member @ a @ X @ carrier )
% 0.30/1.54 => ( ( member @ a @ Y @ carrier )
% 0.30/1.54 => ( ( X != Y )
% 0.30/1.54 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_21_strict__trans,axiom,
% 0.30/1.54 ! [X: a,Y: a,Z: a] :
% 0.30/1.54 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.30/1.54 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
% 0.30/1.54 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_22_indiff__trans,axiom,
% 0.30/1.54 ! [X: a,Y: a,Z: a] :
% 0.30/1.54 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.30/1.54 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 0.30/1.54 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
% 0.30/1.54 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
% 0.30/1.54 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_23_indifferent__imp__weak__pref_I1_J,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.30/1.54 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_24_indifferent__imp__weak__pref_I2_J,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.30/1.54 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_25_weak__is__transitive,axiom,
% 0.30/1.54 ! [X: a,Y: a,Z: a] :
% 0.30/1.54 ( ( ( member @ a @ X @ carrier )
% 0.30/1.54 & ( member @ a @ Y @ carrier )
% 0.30/1.54 & ( member @ a @ Z @ carrier ) )
% 0.30/1.54 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 0.30/1.54 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_26_strict__is__neg__transitive,axiom,
% 0.30/1.54 ! [X: a,Y: a,Z: a] :
% 0.30/1.54 ( ( ( member @ a @ X @ carrier )
% 0.30/1.54 & ( member @ a @ Y @ carrier )
% 0.30/1.54 & ( member @ a @ Z @ carrier ) )
% 0.30/1.54 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.30/1.54 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) )
% 0.30/1.54 | ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_27_not__outside,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 => ( member @ a @ X @ carrier ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_28_strict__not__refl__weak,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( ( member @ a @ X @ carrier )
% 0.30/1.54 & ( member @ a @ Y @ carrier ) )
% 0.30/1.54 => ( ( ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.30/1.54 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_29_rational__preference_Oaxioms_I1_J,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_30_rational__preference_Oindifferent__imp__weak__pref_I2_J,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.54 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.30/1.54 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_31_rational__preference_Oindifferent__imp__weak__pref_I1_J,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.54 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.30/1.54 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_32_preference__def,axiom,
% 0.30/1.54 ! [A: $tType] :
% 0.30/1.54 ( ( prefer199794634erence @ A )
% 0.30/1.54 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 0.30/1.54 ( ! [X3: A,Y2: A] :
% 0.30/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
% 0.30/1.54 => ( member @ A @ X3 @ Carrier2 ) )
% 0.30/1.54 & ! [X3: A,Y2: A] :
% 0.30/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
% 0.30/1.54 => ( member @ A @ Y2 @ Carrier2 ) )
% 0.30/1.54 & ( order_preorder_on @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_33_preference_Ointro,axiom,
% 0.30/1.54 ! [A: $tType,Relation: set @ ( product_prod @ A @ A ),Carrier: set @ A] :
% 0.30/1.54 ( ! [X4: A,Y3: A] :
% 0.30/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
% 0.30/1.54 => ( member @ A @ X4 @ Carrier ) )
% 0.30/1.54 => ( ! [X4: A,Y3: A] :
% 0.30/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
% 0.30/1.54 => ( member @ A @ Y3 @ Carrier ) )
% 0.30/1.54 => ( ( order_preorder_on @ A @ Carrier @ Relation )
% 0.30/1.54 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_34_preference_Otrans__refl,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/1.54 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( order_preorder_on @ A @ Carrier @ Relation ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_35_preference_Onot__outside,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.54 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.54 => ( member @ A @ X @ Carrier ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_36_preference_Oreflexivity,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/1.54 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( refl_on @ A @ Carrier @ Relation ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_37_preference_Oindiff__trans,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.30/1.54 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.54 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.30/1.54 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 0.30/1.54 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 0.30/1.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 0.30/1.54 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_38_rational__preference_Ocompl,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ! [X2: A] :
% 0.30/1.54 ( ( member @ A @ X2 @ Carrier )
% 0.30/1.54 => ! [Xa: A] :
% 0.30/1.54 ( ( member @ A @ Xa @ Carrier )
% 0.30/1.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa ) @ Relation )
% 0.30/1.54 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_39_rational__preference_Ototal,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( total_on @ A @ Carrier @ Relation ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_40_rational__preference_OcompleteD,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( member @ A @ X @ Carrier )
% 0.30/1.54 => ( ( member @ A @ Y @ Carrier )
% 0.30/1.54 => ( ( X != Y )
% 0.30/1.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.54 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_41_rational__preference_Ostrict__trans,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.30/1.54 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 0.30/1.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_42_rational__preference_Oweak__is__transitive,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( ( member @ A @ X @ Carrier )
% 0.30/1.54 & ( member @ A @ Y @ Carrier )
% 0.30/1.54 & ( member @ A @ Z @ Carrier ) )
% 0.30/1.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 0.30/1.54 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_43_rational__preference_Ofnt__carrier__fnt__rel,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( finite_finite2 @ A @ Carrier )
% 0.30/1.54 => ( finite_finite2 @ ( product_prod @ A @ A ) @ Relation ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_44_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( ( member @ A @ X @ Carrier )
% 0.30/1.54 & ( member @ A @ Y @ Carrier ) )
% 0.30/1.54 => ( ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.30/1.54 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/1.54 ! [A: $tType,A2: A,P: A > $o] :
% 0.30/1.54 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.30/1.54 = ( P @ A2 ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/1.54 ! [A: $tType,A3: set @ A] :
% 0.30/1.54 ( ( collect @ A
% 0.30/1.54 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.30/1.54 = A3 ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_47_Collect__cong,axiom,
% 0.30/1.54 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/1.54 ( ! [X4: A] :
% 0.30/1.54 ( ( P @ X4 )
% 0.30/1.54 = ( Q @ X4 ) )
% 0.30/1.54 => ( ( collect @ A @ P )
% 0.30/1.54 = ( collect @ A @ Q ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_48_ext,axiom,
% 0.30/1.54 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/1.54 ( ! [X4: A] :
% 0.30/1.54 ( ( F @ X4 )
% 0.30/1.54 = ( G @ X4 ) )
% 0.30/1.54 => ( F = G ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_49_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( ( member @ A @ X @ Carrier )
% 0.30/1.54 & ( member @ A @ Y @ Carrier )
% 0.30/1.54 & ( member @ A @ Z @ Carrier ) )
% 0.30/1.54 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.30/1.54 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) )
% 0.30/1.54 | ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 0.30/1.54 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_50_rational__preference_Oworse__in__no__better,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.54 => ( member @ A @ Y @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_51_rational__preference_Osame__nbt__same__pref,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( member @ A @ X @ Carrier )
% 0.30/1.54 => ( ( member @ A @ Y @ Carrier )
% 0.30/1.54 => ( ( ( member @ A @ X @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) )
% 0.30/1.54 & ( member @ A @ Y @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) )
% 0.30/1.54 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.54 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_52_rational__preference_Ofnt__carrier__fnt__nbt,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( finite_finite2 @ A @ Carrier )
% 0.30/1.54 => ! [X2: A] :
% 0.30/1.54 ( ( member @ A @ X2 @ Carrier )
% 0.30/1.54 => ( finite_finite2 @ A @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_53_rational__preference_Ono__better__than__nonepty,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( Carrier
% 0.30/1.54 != ( bot_bot @ ( set @ A ) ) )
% 0.30/1.54 => ( ( member @ A @ X @ Carrier )
% 0.30/1.54 => ( ( prefer1532642881r_than @ A @ X @ Carrier @ Relation )
% 0.30/1.54 != ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_54_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( member @ A @ X @ Carrier )
% 0.30/1.54 => ( ( member @ A @ Y @ Carrier )
% 0.30/1.54 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) )
% 0.30/1.54 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_55_rational__preference_Ono__better__subset__pref,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.54 => ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_56_rational__preference_Onbt__subset,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( finite_finite2 @ A @ Carrier )
% 0.30/1.54 => ( ( member @ A @ X @ Carrier )
% 0.30/1.54 => ( ( member @ A @ Y @ Carrier )
% 0.30/1.54 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) )
% 0.30/1.54 | ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_57_rational__preference_Oxy__in__eachothers__nbt,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( member @ A @ X @ Carrier )
% 0.30/1.54 => ( ( member @ A @ Y @ Carrier )
% 0.30/1.54 => ( ( member @ A @ X @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) )
% 0.30/1.54 | ( member @ A @ Y @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_58_rational__preference_Onbt__subset__carrier,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( member @ A @ X @ Carrier )
% 0.30/1.54 => ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ Carrier ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_59_rational__preference_Onbt__nest,axiom,
% 0.30/1.54 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Y: A,X: A] :
% 0.30/1.54 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.54 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) )
% 0.30/1.54 | ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_60_at__lst__asgd__not__ge,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( carrier
% 0.30/1.54 != ( bot_bot @ ( set @ a ) ) )
% 0.30/1.54 => ( ( member @ a @ X @ carrier )
% 0.30/1.54 => ( ( member @ a @ Y @ carrier )
% 0.30/1.54 => ( ~ ( member @ a @ X @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) )
% 0.30/1.54 => ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_61_same__at__least__as__equal,axiom,
% 0.30/1.54 ! [Z: a,Y: a] :
% 0.30/1.54 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 0.30/1.54 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) )
% 0.30/1.54 => ( ( prefer310429814s_good @ a @ Z @ carrier @ relation )
% 0.30/1.54 = ( prefer310429814s_good @ a @ Y @ carrier @ relation ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_62_pref__in__at__least__as,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 => ( member @ a @ X @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_63_empty__subsetI,axiom,
% 0.30/1.54 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_64_subset__empty,axiom,
% 0.30/1.54 ! [A: $tType,A3: set @ A] :
% 0.30/1.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/1.54 = ( A3
% 0.30/1.54 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_65_Pair__le,axiom,
% 0.30/1.54 ! [A: $tType,B: $tType] :
% 0.30/1.54 ( ( ( ord @ B )
% 0.30/1.54 & ( ord @ A ) )
% 0.30/1.54 => ! [A2: A,B2: B,C2: A,D: B] :
% 0.30/1.54 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
% 0.30/1.54 = ( ( ord_less_eq @ A @ A2 @ C2 )
% 0.30/1.54 & ( ord_less_eq @ B @ B2 @ D ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_66_as__good__as__sameIff,axiom,
% 0.30/1.54 ! [Z: a,Y: a] :
% 0.30/1.54 ( ( member @ a @ Z @ ( prefer951318096ood_as @ a @ Y @ carrier @ relation ) )
% 0.30/1.54 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 0.30/1.54 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_67_as__good__asIff,axiom,
% 0.30/1.54 ! [X: a,Y: a] :
% 0.30/1.54 ( ( member @ a @ X @ ( prefer951318096ood_as @ a @ Y @ carrier @ relation ) )
% 0.30/1.54 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.30/1.54 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_68_fnt__carrier__fnt__rel,axiom,
% 0.30/1.54 ( ( finite_finite2 @ a @ carrier )
% 0.30/1.54 => ( finite_finite2 @ ( product_prod @ a @ a ) @ relation ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(fact_69_card__mono,axiom,
% 0.30/1.54 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.30/1.54 ( ( finite_finite2 @ A @ B3 )
% 0.30/1.54 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/1.54 => ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_70_empty__Collect__eq,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o] :
% 0.30/1.55 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/1.55 = ( collect @ A @ P ) )
% 0.30/1.55 = ( ! [X3: A] :
% 0.30/1.55 ~ ( P @ X3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_71_Collect__empty__eq,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o] :
% 0.30/1.55 ( ( ( collect @ A @ P )
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.55 = ( ! [X3: A] :
% 0.30/1.55 ~ ( P @ X3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_72_all__not__in__conv,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A] :
% 0.30/1.55 ( ( ! [X3: A] :
% 0.30/1.55 ~ ( member @ A @ X3 @ A3 ) )
% 0.30/1.55 = ( A3
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_73_empty__iff,axiom,
% 0.30/1.55 ! [A: $tType,C2: A] :
% 0.30/1.55 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_74_subsetI,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/1.55 ( ! [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ A3 )
% 0.30/1.55 => ( member @ A @ X4 @ B3 ) )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_75_subset__antisym,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 0.30/1.55 => ( A3 = B3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_76_finite__code,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( finite_finite @ A )
% 0.30/1.55 => ( ( finite_finite2 @ A )
% 0.30/1.55 = ( ^ [A4: set @ A] : $true ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_77_at__least__as__goodD,axiom,
% 0.30/1.55 ! [A: $tType,Z: A,Y: A,B3: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 0.30/1.55 ( ( member @ A @ Z @ ( prefer310429814s_good @ A @ Y @ B3 @ Pr ) )
% 0.30/1.55 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Pr ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_78_at__lst__asgd__ge,axiom,
% 0.30/1.55 ! [A: $tType,X: A,Y: A,B3: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 0.30/1.55 ( ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ B3 @ Pr ) )
% 0.30/1.55 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Pr ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_79_ex__in__conv,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A] :
% 0.30/1.55 ( ( ? [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.30/1.55 = ( A3
% 0.30/1.55 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_80_equals0I,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A] :
% 0.30/1.55 ( ! [Y3: A] :
% 0.30/1.55 ~ ( member @ A @ Y3 @ A3 )
% 0.30/1.55 => ( A3
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_81_equals0D,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,A2: A] :
% 0.30/1.55 ( ( A3
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.55 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_82_emptyE,axiom,
% 0.30/1.55 ! [A: $tType,A2: A] :
% 0.30/1.55 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_83_in__mono,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,B3: set @ A,X: A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/1.55 => ( ( member @ A @ X @ A3 )
% 0.30/1.55 => ( member @ A @ X @ B3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_84_subsetD,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/1.55 => ( ( member @ A @ C2 @ A3 )
% 0.30/1.55 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_85_equalityE,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/1.55 ( ( A3 = B3 )
% 0.30/1.55 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/1.55 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_86_subset__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/1.55 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.30/1.55 ! [X3: A] :
% 0.30/1.55 ( ( member @ A @ X3 @ A4 )
% 0.30/1.55 => ( member @ A @ X3 @ B4 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_87_equalityD1,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/1.55 ( ( A3 = B3 )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_88_equalityD2,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/1.55 ( ( A3 = B3 )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_89_subset__iff,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/1.55 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.30/1.55 ! [T2: A] :
% 0.30/1.55 ( ( member @ A @ T2 @ A4 )
% 0.30/1.55 => ( member @ A @ T2 @ B4 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_90_subset__refl,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_91_Collect__mono,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/1.55 ( ! [X4: A] :
% 0.30/1.55 ( ( P @ X4 )
% 0.30/1.55 => ( Q @ X4 ) )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_92_subset__trans,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_93_set__eq__subset,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ^ [Y4: set @ A,Z2: set @ A] : ( Y4 = Z2 ) )
% 0.30/1.55 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/1.55 & ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_94_Collect__mono__iff,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.30/1.55 = ( ! [X3: A] :
% 0.30/1.55 ( ( P @ X3 )
% 0.30/1.55 => ( Q @ X3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_95_finite,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( finite_finite @ A )
% 0.30/1.55 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_96_finite__set__choice,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 0.30/1.55 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.55 => ( ! [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ A3 )
% 0.30/1.55 => ? [X_1: B] : ( P @ X4 @ X_1 ) )
% 0.30/1.55 => ? [F2: A > B] :
% 0.30/1.55 ! [X2: A] :
% 0.30/1.55 ( ( member @ A @ X2 @ A3 )
% 0.30/1.55 => ( P @ X2 @ ( F2 @ X2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_97_rational__preference_Osame__at__least__as__equal,axiom,
% 0.30/1.55 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Z: A,Y: A] :
% 0.30/1.55 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.55 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 0.30/1.55 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) )
% 0.30/1.55 => ( ( prefer310429814s_good @ A @ Z @ Carrier @ Relation )
% 0.30/1.55 = ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_98_rational__preference_Opref__in__at__least__as,axiom,
% 0.30/1.55 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.55 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.55 => ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_99_rational__preference_Oas__good__as__sameIff,axiom,
% 0.30/1.55 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Z: A,Y: A] :
% 0.30/1.55 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.55 => ( ( member @ A @ Z @ ( prefer951318096ood_as @ A @ Y @ Carrier @ Relation ) )
% 0.30/1.55 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 0.30/1.55 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_100_rational__preference_Oas__good__asIff,axiom,
% 0.30/1.55 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.55 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.55 => ( ( member @ A @ X @ ( prefer951318096ood_as @ A @ Y @ Carrier @ Relation ) )
% 0.30/1.55 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/1.55 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_101_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.30/1.55 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.55 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.55 => ( ( Carrier
% 0.30/1.55 != ( bot_bot @ ( set @ A ) ) )
% 0.30/1.55 => ( ( member @ A @ X @ Carrier )
% 0.30/1.55 => ( ( member @ A @ Y @ Carrier )
% 0.30/1.55 => ( ~ ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) )
% 0.30/1.55 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_102_Pair__mono,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( ord @ B )
% 0.30/1.55 & ( ord @ A ) )
% 0.30/1.55 => ! [X: A,X5: A,Y: B,Y5: B] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ X5 )
% 0.30/1.55 => ( ( ord_less_eq @ B @ Y @ Y5 )
% 0.30/1.55 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_103_finite__has__maximal2,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: set @ A,A2: A] :
% 0.30/1.55 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.55 => ( ( member @ A @ A2 @ A3 )
% 0.30/1.55 => ? [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ A3 )
% 0.30/1.55 & ( ord_less_eq @ A @ A2 @ X4 )
% 0.30/1.55 & ! [Xa: A] :
% 0.30/1.55 ( ( member @ A @ Xa @ A3 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ X4 @ Xa )
% 0.30/1.55 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_104_finite__has__minimal2,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: set @ A,A2: A] :
% 0.30/1.55 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.55 => ( ( member @ A @ A2 @ A3 )
% 0.30/1.55 => ? [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ A3 )
% 0.30/1.55 & ( ord_less_eq @ A @ X4 @ A2 )
% 0.30/1.55 & ! [Xa: A] :
% 0.30/1.55 ( ( member @ A @ Xa @ A3 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ Xa @ X4 )
% 0.30/1.55 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_105_bot__prod__def,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( bot @ B )
% 0.30/1.55 & ( bot @ A ) )
% 0.30/1.55 => ( ( bot_bot @ ( product_prod @ A @ B ) )
% 0.30/1.55 = ( product_Pair @ A @ B @ ( bot_bot @ A ) @ ( bot_bot @ B ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_106_infinite__imp__nonempty,axiom,
% 0.30/1.55 ! [A: $tType,S: set @ A] :
% 0.30/1.55 ( ~ ( finite_finite2 @ A @ S )
% 0.30/1.55 => ( S
% 0.30/1.55 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_107_finite_OemptyI,axiom,
% 0.30/1.55 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_108_finite__subset,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/1.55 => ( ( finite_finite2 @ A @ B3 )
% 0.30/1.55 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_109_infinite__super,axiom,
% 0.30/1.55 ! [A: $tType,S: set @ A,T3: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ S @ T3 )
% 0.30/1.55 => ( ~ ( finite_finite2 @ A @ S )
% 0.30/1.55 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_110_rev__finite__subset,axiom,
% 0.30/1.55 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.30/1.55 ( ( finite_finite2 @ A @ B3 )
% 0.30/1.55 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/1.55 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_111_finite__has__minimal,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: set @ A] :
% 0.30/1.55 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.55 => ( ( A3
% 0.30/1.55 != ( bot_bot @ ( set @ A ) ) )
% 0.30/1.55 => ? [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ A3 )
% 0.30/1.55 & ! [Xa: A] :
% 0.30/1.55 ( ( member @ A @ Xa @ A3 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ Xa @ X4 )
% 0.30/1.55 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_112_finite__has__maximal,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A3: set @ A] :
% 0.30/1.55 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.55 => ( ( A3
% 0.30/1.55 != ( bot_bot @ ( set @ A ) ) )
% 0.30/1.55 => ? [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ A3 )
% 0.30/1.55 & ! [Xa: A] :
% 0.30/1.55 ( ( member @ A @ Xa @ A3 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ X4 @ Xa )
% 0.30/1.55 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_113_card__subset__eq,axiom,
% 0.30/1.55 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.30/1.55 ( ( finite_finite2 @ A @ B3 )
% 0.30/1.55 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/1.55 => ( ( ( finite_card @ A @ A3 )
% 0.30/1.55 = ( finite_card @ A @ B3 ) )
% 0.30/1.55 => ( A3 = B3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_114_infinite__arbitrarily__large,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,N: nat] :
% 0.30/1.55 ( ~ ( finite_finite2 @ A @ A3 )
% 0.30/1.55 => ? [B5: set @ A] :
% 0.30/1.55 ( ( finite_finite2 @ A @ B5 )
% 0.30/1.55 & ( ( finite_card @ A @ B5 )
% 0.30/1.55 = N )
% 0.30/1.55 & ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_115_finite__if__finite__subsets__card__bdd,axiom,
% 0.30/1.55 ! [A: $tType,F3: set @ A,C3: nat] :
% 0.30/1.55 ( ! [G2: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ G2 @ F3 )
% 0.30/1.55 => ( ( finite_finite2 @ A @ G2 )
% 0.30/1.55 => ( ord_less_eq @ nat @ ( finite_card @ A @ G2 ) @ C3 ) ) )
% 0.30/1.55 => ( ( finite_finite2 @ A @ F3 )
% 0.30/1.55 & ( ord_less_eq @ nat @ ( finite_card @ A @ F3 ) @ C3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_116_card__seteq,axiom,
% 0.30/1.55 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.30/1.55 ( ( finite_finite2 @ A @ B3 )
% 0.30/1.55 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/1.55 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B3 ) @ ( finite_card @ A @ A3 ) )
% 0.30/1.55 => ( A3 = B3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_117_obtain__subset__with__card__n,axiom,
% 0.30/1.55 ! [A: $tType,N: nat,S: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S ) )
% 0.30/1.55 => ~ ! [T4: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ T4 @ S )
% 0.30/1.55 => ( ( ( finite_card @ A @ T4 )
% 0.30/1.55 = N )
% 0.30/1.55 => ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_118_card__le__if__inj__on__rel,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType,B3: set @ A,A3: set @ B,R: B > A > $o] :
% 0.30/1.55 ( ( finite_finite2 @ A @ B3 )
% 0.30/1.55 => ( ! [A5: B] :
% 0.30/1.55 ( ( member @ B @ A5 @ A3 )
% 0.30/1.55 => ? [B6: A] :
% 0.30/1.55 ( ( member @ A @ B6 @ B3 )
% 0.30/1.55 & ( R @ A5 @ B6 ) ) )
% 0.30/1.55 => ( ! [A1: B,A22: B,B7: A] :
% 0.30/1.55 ( ( member @ B @ A1 @ A3 )
% 0.30/1.55 => ( ( member @ B @ A22 @ A3 )
% 0.30/1.55 => ( ( member @ A @ B7 @ B3 )
% 0.30/1.55 => ( ( R @ A1 @ B7 )
% 0.30/1.55 => ( ( R @ A22 @ B7 )
% 0.30/1.55 => ( A1 = A22 ) ) ) ) ) )
% 0.30/1.55 => ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_119_ex__card,axiom,
% 0.30/1.55 ! [A: $tType,N: nat,A3: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ A3 ) )
% 0.30/1.55 => ? [S2: set @ A] :
% 0.30/1.55 ( ( ord_less_eq @ ( set @ A ) @ S2 @ A3 )
% 0.30/1.55 & ( ( finite_card @ A @ S2 )
% 0.30/1.55 = N ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_120_old_Oprod_Oinject,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B8: B] :
% 0.30/1.55 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.30/1.55 = ( product_Pair @ A @ B @ A6 @ B8 ) )
% 0.30/1.55 = ( ( A2 = A6 )
% 0.30/1.55 & ( B2 = B8 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_121_prod_Oinject,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.30/1.55 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.30/1.55 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.30/1.55 = ( ( X1 = Y1 )
% 0.30/1.55 & ( X22 = Y22 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_122_bot__apply,axiom,
% 0.30/1.55 ! [C: $tType,D2: $tType] :
% 0.30/1.55 ( ( bot @ C )
% 0.30/1.55 => ( ( bot_bot @ ( D2 > C ) )
% 0.30/1.55 = ( ^ [X3: D2] : ( bot_bot @ C ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_123_order__refl,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_124_bot__set__def,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( bot_bot @ ( set @ A ) )
% 0.30/1.55 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_125_dual__order_Oantisym,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [B2: A,A2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/1.55 => ( A2 = B2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_126_dual__order_Oeq__iff,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.30/1.55 = ( ^ [A7: A,B9: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ B9 @ A7 )
% 0.30/1.55 & ( ord_less_eq @ A @ A7 @ B9 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_127_dual__order_Otrans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [B2: A,A2: A,C2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.30/1.55 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_128_linorder__wlog,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.30/1.55 ( ! [A5: A,B7: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A5 @ B7 )
% 0.30/1.55 => ( P @ A5 @ B7 ) )
% 0.30/1.55 => ( ! [A5: A,B7: A] :
% 0.30/1.55 ( ( P @ B7 @ A5 )
% 0.30/1.55 => ( P @ A5 @ B7 ) )
% 0.30/1.55 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_129_dual__order_Orefl,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_130_order__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A,Z: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/1.55 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_131_order__class_Oorder_Oantisym,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A2: A,B2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/1.55 => ( A2 = B2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_132_ord__le__eq__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord @ A )
% 0.30/1.55 => ! [A2: A,B2: A,C2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/1.55 => ( ( B2 = C2 )
% 0.30/1.55 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_133_ord__eq__le__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( ord @ A )
% 0.30/1.55 => ! [A2: A,B2: A,C2: A] :
% 0.30/1.55 ( ( A2 = B2 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/1.55 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_134_order__class_Oorder_Oeq__iff,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.30/1.55 = ( ^ [A7: A,B9: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A7 @ B9 )
% 0.30/1.55 & ( ord_less_eq @ A @ B9 @ A7 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_135_antisym__conv,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [Y: A,X: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/1.55 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 = ( X = Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_136_le__cases3,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A,Z: A] :
% 0.30/1.55 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.30/1.55 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.30/1.55 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.30/1.55 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.30/1.55 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.30/1.55 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.30/1.55 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_137_order_Otrans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A2: A,B2: A,C2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/1.55 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_138_le__cases,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_139_eq__refl,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( X = Y )
% 0.30/1.55 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_140_linear,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_141_antisym,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less_eq @ A @ Y @ X )
% 0.30/1.55 => ( X = Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_142_eq__iff,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.30/1.55 = ( ^ [X3: A,Y2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.30/1.55 & ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_143_ord__le__eq__subst,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( ord @ B )
% 0.30/1.55 & ( ord @ A ) )
% 0.30/1.55 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/1.55 => ( ( ( F @ B2 )
% 0.30/1.55 = C2 )
% 0.30/1.55 => ( ! [X4: A,Y3: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.30/1.55 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/1.55 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_144_ord__eq__le__subst,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( ord @ B )
% 0.30/1.55 & ( ord @ A ) )
% 0.30/1.55 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.30/1.55 ( ( A2
% 0.30/1.55 = ( F @ B2 ) )
% 0.30/1.55 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.30/1.55 => ( ! [X4: B,Y3: B] :
% 0.30/1.55 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.30/1.55 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/1.55 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_145_order__subst2,axiom,
% 0.30/1.55 ! [A: $tType,C: $tType] :
% 0.30/1.55 ( ( ( order @ C )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/1.55 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.30/1.55 => ( ! [X4: A,Y3: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.30/1.55 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/1.55 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_146_order__subst1,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( order @ B )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.30/1.55 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.30/1.55 => ( ! [X4: B,Y3: B] :
% 0.30/1.55 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.30/1.55 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/1.55 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_147_le__fun__def,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( ord @ B )
% 0.30/1.55 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/1.55 = ( ^ [F4: A > B,G3: A > B] :
% 0.30/1.55 ! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G3 @ X3 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_148_le__funI,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( ord @ B )
% 0.30/1.55 => ! [F: A > B,G: A > B] :
% 0.30/1.55 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.30/1.55 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_149_le__funE,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( ord @ B )
% 0.30/1.55 => ! [F: A > B,G: A > B,X: A] :
% 0.30/1.55 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/1.55 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_150_le__funD,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( ord @ B )
% 0.30/1.55 => ! [F: A > B,G: A > B,X: A] :
% 0.30/1.55 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/1.55 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_151_bot__fun__def,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( bot @ B )
% 0.30/1.55 => ( ( bot_bot @ ( A > B ) )
% 0.30/1.55 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_152_surj__pair,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.30/1.55 ? [X4: A,Y3: B] :
% 0.30/1.55 ( P2
% 0.30/1.55 = ( product_Pair @ A @ B @ X4 @ Y3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_153_prod__cases,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.30/1.55 ( ! [A5: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B7 ) )
% 0.30/1.55 => ( P @ P2 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_154_Pair__inject,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B8: B] :
% 0.30/1.55 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.30/1.55 = ( product_Pair @ A @ B @ A6 @ B8 ) )
% 0.30/1.55 => ~ ( ( A2 = A6 )
% 0.30/1.55 => ( B2 != B8 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_155_prod__cases3,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/1.55 ~ ! [A5: A,B7: B,C4: C] :
% 0.30/1.55 ( Y
% 0.30/1.55 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_156_prod__cases4,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.30/1.55 ~ ! [A5: A,B7: B,C4: C,D3: D2] :
% 0.30/1.55 ( Y
% 0.30/1.55 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_157_prod__cases5,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.30/1.55 ~ ! [A5: A,B7: B,C4: C,D3: D2,E2: E] :
% 0.30/1.55 ( Y
% 0.30/1.55 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_158_prod__cases6,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
% 0.30/1.55 ~ ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F2: F5] :
% 0.30/1.55 ( Y
% 0.30/1.55 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E2 @ F2 ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_159_prod__cases7,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,G4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) )] :
% 0.30/1.55 ~ ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F2: F5,G5: G4] :
% 0.30/1.55 ( Y
% 0.30/1.55 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G4 ) @ E2 @ ( product_Pair @ F5 @ G4 @ F2 @ G5 ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_160_prod__induct3,axiom,
% 0.30/1.55 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/1.55 ( ! [A5: A,B7: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) )
% 0.30/1.55 => ( P @ X ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_161_prod__induct4,axiom,
% 0.30/1.55 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.30/1.55 ( ! [A5: A,B7: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
% 0.30/1.55 => ( P @ X ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_162_prod__induct5,axiom,
% 0.30/1.55 ! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.30/1.55 ( ! [A5: A,B7: B,C4: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
% 0.30/1.55 => ( P @ X ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_163_prod__induct6,axiom,
% 0.30/1.55 ! [F5: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
% 0.30/1.55 ( ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F2: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E2 @ F2 ) ) ) ) ) )
% 0.30/1.55 => ( P @ X ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_164_prod__induct7,axiom,
% 0.30/1.55 ! [G4: $tType,F5: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) )] :
% 0.30/1.55 ( ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F2: F5,G5: G4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G4 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G4 ) @ E2 @ ( product_Pair @ F5 @ G4 @ F2 @ G5 ) ) ) ) ) ) )
% 0.30/1.55 => ( P @ X ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_165_old_Oprod_Oexhaust,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.30/1.55 ~ ! [A5: A,B7: B] :
% 0.30/1.55 ( Y
% 0.30/1.55 != ( product_Pair @ A @ B @ A5 @ B7 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_166_old_Oprod_Oinducts,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.30/1.55 ( ! [A5: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B7 ) )
% 0.30/1.55 => ( P @ Prod ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_167_bounded__Max__nat,axiom,
% 0.30/1.55 ! [P: nat > $o,X: nat,M: nat] :
% 0.30/1.55 ( ( P @ X )
% 0.30/1.55 => ( ! [X4: nat] :
% 0.30/1.55 ( ( P @ X4 )
% 0.30/1.55 => ( ord_less_eq @ nat @ X4 @ M ) )
% 0.30/1.55 => ~ ! [M2: nat] :
% 0.30/1.55 ( ( P @ M2 )
% 0.30/1.55 => ~ ! [X2: nat] :
% 0.30/1.55 ( ( P @ X2 )
% 0.30/1.55 => ( ord_less_eq @ nat @ X2 @ M2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_168_finite__nat__set__iff__bounded__le,axiom,
% 0.30/1.55 ( ( finite_finite2 @ nat )
% 0.30/1.55 = ( ^ [N2: set @ nat] :
% 0.30/1.55 ? [M3: nat] :
% 0.30/1.55 ! [X3: nat] :
% 0.30/1.55 ( ( member @ nat @ X3 @ N2 )
% 0.30/1.55 => ( ord_less_eq @ nat @ X3 @ M3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_169_bot_Oextremum__uniqueI,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order_bot @ A )
% 0.30/1.55 => ! [A2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.30/1.55 => ( A2
% 0.30/1.55 = ( bot_bot @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_170_bot_Oextremum__unique,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order_bot @ A )
% 0.30/1.55 => ! [A2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.30/1.55 = ( A2
% 0.30/1.55 = ( bot_bot @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_171_bot_Oextremum,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order_bot @ A )
% 0.30/1.55 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_172_old_Oprod_Orec,axiom,
% 0.30/1.55 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.30/1.55 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.30/1.55 = ( F1 @ A2 @ B2 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_173_preorder__on__empty,axiom,
% 0.30/1.55 ! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_174_total__onI,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.30/1.55 ( ! [X4: A,Y3: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ A3 )
% 0.30/1.55 => ( ( member @ A @ Y3 @ A3 )
% 0.30/1.55 => ( ( X4 != Y3 )
% 0.30/1.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R )
% 0.30/1.55 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R ) ) ) ) )
% 0.30/1.55 => ( total_on @ A @ A3 @ R ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_175_total__on__def,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( total_on @ A )
% 0.30/1.55 = ( ^ [A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.30/1.55 ! [X3: A] :
% 0.30/1.55 ( ( member @ A @ X3 @ A4 )
% 0.30/1.55 => ! [Y2: A] :
% 0.30/1.55 ( ( member @ A @ Y2 @ A4 )
% 0.30/1.55 => ( ( X3 != Y2 )
% 0.30/1.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
% 0.30/1.55 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_176_bot__empty__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( bot_bot @ ( A > $o ) )
% 0.30/1.55 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_177_subrelI,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.30/1.55 ( ! [X4: A,Y3: B] :
% 0.30/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
% 0.30/1.55 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S3 ) )
% 0.30/1.55 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_178_refl__on__empty,axiom,
% 0.30/1.55 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_179_refl__on__domain,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 0.30/1.55 ( ( refl_on @ A @ A3 @ R )
% 0.30/1.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
% 0.30/1.55 => ( ( member @ A @ A2 @ A3 )
% 0.30/1.55 & ( member @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_180_refl__onD2,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.55 ( ( refl_on @ A @ A3 @ R )
% 0.30/1.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.30/1.55 => ( member @ A @ Y @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_181_refl__onD1,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/1.55 ( ( refl_on @ A @ A3 @ R )
% 0.30/1.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.30/1.55 => ( member @ A @ X @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_182_refl__onD,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),A2: A] :
% 0.30/1.55 ( ( refl_on @ A @ A3 @ R )
% 0.30/1.55 => ( ( member @ A @ A2 @ A3 )
% 0.30/1.55 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_183_total__on__empty,axiom,
% 0.30/1.55 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_184_Collect__empty__eq__bot,axiom,
% 0.30/1.55 ! [A: $tType,P: A > $o] :
% 0.30/1.55 ( ( ( collect @ A @ P )
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.55 = ( P
% 0.30/1.55 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_185_infinite__nat__iff__unbounded__le,axiom,
% 0.30/1.55 ! [S: set @ nat] :
% 0.30/1.55 ( ( ~ ( finite_finite2 @ nat @ S ) )
% 0.30/1.55 = ( ! [M3: nat] :
% 0.30/1.55 ? [N3: nat] :
% 0.30/1.55 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 0.30/1.55 & ( member @ nat @ N3 @ S ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_186_finite__transitivity__chain,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A,R3: A > A > $o] :
% 0.30/1.55 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.55 => ( ! [X4: A] :
% 0.30/1.55 ~ ( R3 @ X4 @ X4 )
% 0.30/1.55 => ( ! [X4: A,Y3: A,Z3: A] :
% 0.30/1.55 ( ( R3 @ X4 @ Y3 )
% 0.30/1.55 => ( ( R3 @ Y3 @ Z3 )
% 0.30/1.55 => ( R3 @ X4 @ Z3 ) ) )
% 0.30/1.55 => ( ! [X4: A] :
% 0.30/1.55 ( ( member @ A @ X4 @ A3 )
% 0.30/1.55 => ? [Y6: A] :
% 0.30/1.55 ( ( member @ A @ Y6 @ A3 )
% 0.30/1.55 & ( R3 @ X4 @ Y6 ) ) )
% 0.30/1.55 => ( A3
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_187_less__by__empty,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
% 0.30/1.55 ( ( A3
% 0.30/1.55 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.30/1.55 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_188_subset__emptyI,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A] :
% 0.30/1.55 ( ! [X4: A] :
% 0.30/1.55 ~ ( member @ A @ X4 @ A3 )
% 0.30/1.55 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_189_rational__preference__def,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( prefer1997167224erence @ A )
% 0.30/1.55 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 0.30/1.55 ( ( prefer199794634erence @ A @ Carrier2 @ Relation2 )
% 0.30/1.55 & ( prefer1801827867axioms @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_190_ssubst__Pair__rhs,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType,R: A,S3: B,R3: set @ ( product_prod @ A @ B ),S4: B] :
% 0.30/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R3 )
% 0.30/1.55 => ( ( S4 = S3 )
% 0.30/1.55 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S4 ) @ R3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_191_rational__preference__axioms_Ointro,axiom,
% 0.30/1.55 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/1.55 ( ( total_on @ A @ Carrier @ Relation )
% 0.30/1.55 => ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_192_rational__preference__axioms__def,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( prefer1801827867axioms @ A )
% 0.30/1.55 = ( total_on @ A ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_193_rational__preference_Oaxioms_I2_J,axiom,
% 0.30/1.55 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/1.55 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.30/1.55 => ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_194_rational__preference_Ointro,axiom,
% 0.30/1.55 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/1.55 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/1.55 => ( ( prefer1801827867axioms @ A @ Carrier @ Relation )
% 0.30/1.55 => ( prefer1997167224erence @ A @ Carrier @ Relation ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_195_finite__indexed__bound,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [A3: set @ B,P: B > A > $o] :
% 0.30/1.55 ( ( finite_finite2 @ B @ A3 )
% 0.30/1.55 => ( ! [X4: B] :
% 0.30/1.55 ( ( member @ B @ X4 @ A3 )
% 0.30/1.55 => ? [X_1: A] : ( P @ X4 @ X_1 ) )
% 0.30/1.55 => ? [M2: A] :
% 0.30/1.55 ! [X2: B] :
% 0.30/1.55 ( ( member @ B @ X2 @ A3 )
% 0.30/1.55 => ? [K: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ K @ M2 )
% 0.30/1.55 & ( P @ X2 @ K ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_196_transitivity,axiom,
% 0.30/1.55 trans @ a @ relation ).
% 0.30/1.55
% 0.30/1.55 thf(fact_197_trans__empty,axiom,
% 0.30/1.55 ! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_198_transD,axiom,
% 0.30/1.55 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.30/1.55 ( ( trans @ A @ R )
% 0.30/1.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.30/1.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R )
% 0.30/1.55 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_199_transE,axiom,
% 0.30/1.55 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.30/1.55 ( ( trans @ A @ R )
% 0.30/1.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.30/1.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R )
% 0.30/1.55 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_200_transI,axiom,
% 0.30/1.55 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.30/1.55 ( ! [X4: A,Y3: A,Z3: A] :
% 0.30/1.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R )
% 0.30/1.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R )
% 0.30/1.55 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z3 ) @ R ) ) )
% 0.30/1.55 => ( trans @ A @ R ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_201_trans__def,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( trans @ A )
% 0.30/1.55 = ( ^ [R2: set @ ( product_prod @ A @ A )] :
% 0.30/1.55 ! [X3: A,Y2: A,Z4: A] :
% 0.30/1.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
% 0.30/1.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R2 )
% 0.30/1.55 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z4 ) @ R2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_202_preference_Otransitivity,axiom,
% 0.30/1.55 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/1.55 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/1.55 => ( trans @ A @ Relation ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_203_preorder__on__def,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order_preorder_on @ A )
% 0.30/1.55 = ( ^ [A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.30/1.55 ( ( refl_on @ A @ A4 @ R2 )
% 0.30/1.55 & ( trans @ A @ R2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_204_Set_Ois__empty__def,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( is_empty @ A )
% 0.30/1.55 = ( ^ [A4: set @ A] :
% 0.30/1.55 ( A4
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_205_internal__case__prod__conv,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
% 0.30/1.55 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.30/1.55 = ( C2 @ A2 @ B2 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_206_card__0__eq,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A] :
% 0.30/1.55 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.55 => ( ( ( finite_card @ A @ A3 )
% 0.30/1.55 = ( zero_zero @ nat ) )
% 0.30/1.55 = ( A3
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_207_relChain__def,axiom,
% 0.30/1.55 ! [B: $tType,A: $tType] :
% 0.30/1.55 ( ( ord @ B )
% 0.30/1.55 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.30/1.55 = ( ^ [R2: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.30/1.55 ! [I: A,J: A] :
% 0.30/1.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R2 )
% 0.30/1.55 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_208_card_Oempty,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.30/1.55 = ( zero_zero @ nat ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_209_card_Oinfinite,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A] :
% 0.30/1.55 ( ~ ( finite_finite2 @ A @ A3 )
% 0.30/1.55 => ( ( finite_card @ A @ A3 )
% 0.30/1.55 = ( zero_zero @ nat ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_210_card__eq__0__iff,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A] :
% 0.30/1.55 ( ( ( finite_card @ A @ A3 )
% 0.30/1.55 = ( zero_zero @ nat ) )
% 0.30/1.55 = ( ( A3
% 0.30/1.55 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.55 | ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_211_bot__nat__0_Oextremum,axiom,
% 0.30/1.55 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_212_le0,axiom,
% 0.30/1.55 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_213_Nat_Oex__has__greatest__nat,axiom,
% 0.30/1.55 ! [P: nat > $o,K2: nat,B2: nat] :
% 0.30/1.55 ( ( P @ K2 )
% 0.30/1.55 => ( ! [Y3: nat] :
% 0.30/1.55 ( ( P @ Y3 )
% 0.30/1.55 => ( ord_less_eq @ nat @ Y3 @ B2 ) )
% 0.30/1.55 => ? [X4: nat] :
% 0.30/1.55 ( ( P @ X4 )
% 0.30/1.55 & ! [Y6: nat] :
% 0.30/1.55 ( ( P @ Y6 )
% 0.30/1.55 => ( ord_less_eq @ nat @ Y6 @ X4 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_214_nat__le__linear,axiom,
% 0.30/1.55 ! [M4: nat,N: nat] :
% 0.30/1.55 ( ( ord_less_eq @ nat @ M4 @ N )
% 0.30/1.55 | ( ord_less_eq @ nat @ N @ M4 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_215_le__antisym,axiom,
% 0.30/1.55 ! [M4: nat,N: nat] :
% 0.30/1.55 ( ( ord_less_eq @ nat @ M4 @ N )
% 0.30/1.55 => ( ( ord_less_eq @ nat @ N @ M4 )
% 0.30/1.55 => ( M4 = N ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_216_eq__imp__le,axiom,
% 0.30/1.55 ! [M4: nat,N: nat] :
% 0.30/1.55 ( ( M4 = N )
% 0.30/1.55 => ( ord_less_eq @ nat @ M4 @ N ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_217_le__trans,axiom,
% 0.30/1.55 ! [I2: nat,J2: nat,K2: nat] :
% 0.30/1.55 ( ( ord_less_eq @ nat @ I2 @ J2 )
% 0.30/1.55 => ( ( ord_less_eq @ nat @ J2 @ K2 )
% 0.30/1.55 => ( ord_less_eq @ nat @ I2 @ K2 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_218_le__refl,axiom,
% 0.30/1.55 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_219_less__eq__nat_Osimps_I1_J,axiom,
% 0.30/1.55 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_220_le__0__eq,axiom,
% 0.30/1.55 ! [N: nat] :
% 0.30/1.55 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.30/1.55 = ( N
% 0.30/1.55 = ( zero_zero @ nat ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_221_bot__nat__0_Oextremum__unique,axiom,
% 0.30/1.55 ! [A2: nat] :
% 0.30/1.55 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.30/1.55 = ( A2
% 0.30/1.55 = ( zero_zero @ nat ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_222_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.30/1.55 ! [A2: nat] :
% 0.30/1.55 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.30/1.55 => ( A2
% 0.30/1.55 = ( zero_zero @ nat ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_223_le__zero__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( canoni770627133id_add @ A )
% 0.30/1.55 => ! [N: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.30/1.55 = ( N
% 0.30/1.55 = ( zero_zero @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_224_zero__prod__def,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( zero @ B )
% 0.30/1.55 & ( zero @ A ) )
% 0.30/1.55 => ( ( zero_zero @ ( product_prod @ A @ B ) )
% 0.30/1.55 = ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_225_zero__le,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( canoni770627133id_add @ A )
% 0.30/1.55 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_226_le__numeral__extra_I3_J,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linord1659791738miring @ A )
% 0.30/1.55 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_227_card__gt__0__iff,axiom,
% 0.30/1.55 ! [A: $tType,A3: set @ A] :
% 0.30/1.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
% 0.30/1.55 = ( ( A3
% 0.30/1.55 != ( bot_bot @ ( set @ A ) ) )
% 0.30/1.55 & ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_228_ex__least__nat__le,axiom,
% 0.30/1.55 ! [P: nat > $o,N: nat] :
% 0.30/1.55 ( ( P @ N )
% 0.30/1.55 => ( ~ ( P @ ( zero_zero @ nat ) )
% 0.30/1.55 => ? [K: nat] :
% 0.30/1.55 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/1.55 & ! [I3: nat] :
% 0.30/1.55 ( ( ord_less @ nat @ I3 @ K )
% 0.30/1.55 => ~ ( P @ I3 ) )
% 0.30/1.55 & ( P @ K ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_229_leD,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [Y: A,X: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/1.55 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_230_leI,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_231_le__less,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ord_less_eq @ A )
% 0.30/1.55 = ( ^ [X3: A,Y2: A] :
% 0.30/1.55 ( ( ord_less @ A @ X3 @ Y2 )
% 0.30/1.55 | ( X3 = Y2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_232_less__le,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ord_less @ A )
% 0.30/1.55 = ( ^ [X3: A,Y2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.30/1.55 & ( X3 != Y2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_233_order__le__less__subst1,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( order @ B )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.30/1.55 => ( ( ord_less @ B @ B2 @ C2 )
% 0.30/1.55 => ( ! [X4: B,Y3: B] :
% 0.30/1.55 ( ( ord_less @ B @ X4 @ Y3 )
% 0.30/1.55 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/1.55 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_234_order__le__less__subst2,axiom,
% 0.30/1.55 ! [A: $tType,C: $tType] :
% 0.30/1.55 ( ( ( order @ C )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/1.55 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 0.30/1.55 => ( ! [X4: A,Y3: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.30/1.55 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/1.55 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_235_order__less__le__subst1,axiom,
% 0.30/1.55 ! [A: $tType,B: $tType] :
% 0.30/1.55 ( ( ( order @ B )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.30/1.55 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.30/1.55 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.30/1.55 => ( ! [X4: B,Y3: B] :
% 0.30/1.55 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.30/1.55 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/1.55 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_236_order__less__le__subst2,axiom,
% 0.30/1.55 ! [A: $tType,C: $tType] :
% 0.30/1.55 ( ( ( order @ C )
% 0.30/1.55 & ( order @ A ) )
% 0.30/1.55 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.30/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/1.55 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.30/1.55 => ( ! [X4: A,Y3: A] :
% 0.30/1.55 ( ( ord_less @ A @ X4 @ Y3 )
% 0.30/1.55 => ( ord_less @ C @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.30/1.55 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_237_not__le,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 0.30/1.55 = ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_238_not__less,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/1.55 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_239_le__neq__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A2: A,B2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/1.55 => ( ( A2 != B2 )
% 0.30/1.55 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_240_antisym__conv1,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 = ( X = Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_241_antisym__conv2,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/1.55 = ( X = Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_242_less__imp__le,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_243_le__less__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A,Z: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less @ A @ Y @ Z )
% 0.30/1.55 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_244_less__le__trans,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ! [X: A,Y: A,Z: A] :
% 0.30/1.55 ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/1.55 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_245_dense__ge,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( dense_linorder @ A )
% 0.30/1.55 => ! [Z: A,Y: A] :
% 0.30/1.55 ( ! [X4: A] :
% 0.30/1.55 ( ( ord_less @ A @ Z @ X4 )
% 0.30/1.55 => ( ord_less_eq @ A @ Y @ X4 ) )
% 0.30/1.55 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_246_dense__le,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( dense_linorder @ A )
% 0.30/1.55 => ! [Y: A,Z: A] :
% 0.30/1.55 ( ! [X4: A] :
% 0.30/1.55 ( ( ord_less @ A @ X4 @ Y )
% 0.30/1.55 => ( ord_less_eq @ A @ X4 @ Z ) )
% 0.30/1.55 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_247_le__less__linear,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_248_le__imp__less__or__eq,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [X: A,Y: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/1.55 => ( ( ord_less @ A @ X @ Y )
% 0.30/1.55 | ( X = Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_249_less__le__not__le,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( preorder @ A )
% 0.30/1.55 => ( ( ord_less @ A )
% 0.30/1.55 = ( ^ [X3: A,Y2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.30/1.55 & ~ ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_250_not__le__imp__less,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( linorder @ A )
% 0.30/1.55 => ! [Y: A,X: A] :
% 0.30/1.55 ( ~ ( ord_less_eq @ A @ Y @ X )
% 0.30/1.55 => ( ord_less @ A @ X @ Y ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_251_order_Ostrict__trans1,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A2: A,B2: A,C2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/1.55 => ( ( ord_less @ A @ B2 @ C2 )
% 0.30/1.55 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_252_order_Ostrict__trans2,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [A2: A,B2: A,C2: A] :
% 0.30/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/1.55 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/1.55 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_253_order_Oorder__iff__strict,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ord_less_eq @ A )
% 0.30/1.55 = ( ^ [A7: A,B9: A] :
% 0.30/1.55 ( ( ord_less @ A @ A7 @ B9 )
% 0.30/1.55 | ( A7 = B9 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_254_order_Ostrict__iff__order,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ( ( ord_less @ A )
% 0.30/1.55 = ( ^ [A7: A,B9: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ A7 @ B9 )
% 0.30/1.55 & ( A7 != B9 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_255_dual__order_Ostrict__trans1,axiom,
% 0.30/1.55 ! [A: $tType] :
% 0.30/1.55 ( ( order @ A )
% 0.30/1.55 => ! [B2: A,A2: A,C2: A] :
% 0.30/1.55 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/1.55 => ( ( ord_less @ A @ C2 @ B2 )
% 0.30/1.55 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.30/1.55 ! [A8: $tType,A9: $tType] :
% 0.30/1.55 ( ( order_bot @ A9 )
% 0.30/1.55 => ( order_bot @ ( A8 > A9 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/1.55 ! [A8: $tType,A9: $tType] :
% 0.30/1.55 ( ( preorder @ A9 )
% 0.30/1.55 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.30/1.55 ! [A8: $tType,A9: $tType] :
% 0.30/1.55 ( ( ( finite_finite @ A8 )
% 0.30/1.55 & ( finite_finite @ A9 ) )
% 0.30/1.55 => ( finite_finite @ ( A8 > A9 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/1.55 ! [A8: $tType,A9: $tType] :
% 0.30/1.55 ( ( order @ A9 )
% 0.30/1.55 => ( order @ ( A8 > A9 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/1.55 ! [A8: $tType,A9: $tType] :
% 0.30/1.55 ( ( ord @ A9 )
% 0.30/1.55 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_fun___Orderings_Obot,axiom,
% 0.30/1.55 ! [A8: $tType,A9: $tType] :
% 0.30/1.55 ( ( bot @ A9 )
% 0.30/1.55 => ( bot @ ( A8 > A9 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.30/1.55 canoni770627133id_add @ nat ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.30/1.55 linord1659791738miring @ nat ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
% 0.30/1.55 order_bot @ nat ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 0.30/1.55 preorder @ nat ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.30/1.55 linorder @ nat ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 0.30/1.55 order @ nat ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 0.30/1.55 ord @ nat ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
% 0.30/1.55 bot @ nat ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.30/1.55 zero @ nat ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
% 0.30/1.55 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 0.30/1.55 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
% 0.30/1.55 ! [A8: $tType] :
% 0.30/1.55 ( ( finite_finite @ A8 )
% 0.30/1.55 => ( finite_finite @ ( set @ A8 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 0.30/1.55 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
% 0.30/1.55 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
% 0.30/1.55 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Set_Oset___Groups_Ozero_12,axiom,
% 0.30/1.55 ! [A8: $tType] :
% 0.30/1.55 ( ( zero @ A8 )
% 0.30/1.55 => ( zero @ ( set @ A8 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_HOL_Obool___Orderings_Oorder__bot_13,axiom,
% 0.30/1.55 order_bot @ $o ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
% 0.30/1.55 preorder @ $o ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_HOL_Obool___Orderings_Olinorder_15,axiom,
% 0.30/1.55 linorder @ $o ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_HOL_Obool___Finite__Set_Ofinite_16,axiom,
% 0.30/1.55 finite_finite @ $o ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_HOL_Obool___Orderings_Oorder_17,axiom,
% 0.30/1.55 order @ $o ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
% 0.30/1.55 ord @ $o ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_HOL_Obool___Orderings_Obot_19,axiom,
% 0.30/1.55 bot @ $o ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Product__Type_Oprod___Orderings_Oorder__bot_20,axiom,
% 0.30/1.55 ! [A8: $tType,A9: $tType] :
% 0.30/1.55 ( ( ( order_bot @ A8 )
% 0.30/1.55 & ( order_bot @ A9 ) )
% 0.30/1.55 => ( order_bot @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_21,axiom,
% 0.30/1.55 ! [A8: $tType,A9: $tType] :
% 0.30/1.55 ( ( ( preorder @ A8 )
% 0.30/1.55 & ( preorder @ A9 ) )
% 0.30/1.55 => ( preorder @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_22,axiom,
% 0.30/1.55 ! [A8: $tType,A9: $tType] :
% 0.30/1.55 ( ( ( finite_finite @ A8 )
% 0.30/1.55 & ( finite_finite @ A9 ) )
% 0.30/1.55 => ( finite_finite @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Product__Type_Oprod___Orderings_Oorder_23,axiom,
% 0.30/1.55 ! [A8: $tType,A9: $tType] :
% 0.30/1.55 ( ( ( order @ A8 )
% 0.30/1.55 & ( order @ A9 ) )
% 0.30/1.55 => ( order @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Product__Type_Oprod___Orderings_Oord_24,axiom,
% 0.30/1.55 ! [A8: $tType,A9: $tType] :
% 0.30/1.55 ( ( ( ord @ A8 )
% 0.30/1.55 & ( ord @ A9 ) )
% 0.30/1.55 => ( ord @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Product__Type_Oprod___Orderings_Obot_25,axiom,
% 0.30/1.55 ! [A8: $tType,A9: $tType] :
% 0.30/1.55 ( ( ( bot @ A8 )
% 0.30/1.55 & ( bot @ A9 ) )
% 0.30/1.55 => ( bot @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(tcon_Product__Type_Oprod___Groups_Ozero_26,axiom,
% 0.30/1.55 ! [A8: $tType,A9: $tType] :
% 0.30/1.55 ( ( ( zero @ A8 )
% 0.30/1.55 & ( zero @ A9 ) )
% 0.30/1.55 => ( zero @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(conj_0,conjecture,
% 0.30/1.55 ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ y @ carrier @ relation ) ).
% 0.30/1.55 ------------------------
% 0.30/1.58 % (28414)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_28194 for (2999ds/4Mi)
% 0.30/1.58 % (28417)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_28194 for (2999ds/2Mi)
% 0.30/1.58 % (28417)Instruction limit reached!
% 0.30/1.58 % (28417)------------------------------
% 0.30/1.58 % (28417)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.58 % (28417)Termination reason: Unknown
% 0.30/1.58 % (28417)Termination phase: shuffling
% 0.30/1.58
% 0.30/1.58 % (28417)Memory used [KB]: 1407
% 0.30/1.58 % (28417)Time elapsed: 0.002 s
% 0.30/1.58 % (28417)Instructions burned: 3 (million)
% 0.30/1.58 % (28417)------------------------------
% 0.30/1.58 % (28417)------------------------------
% 0.30/1.58 % (28414)Instruction limit reached!
% 0.30/1.58 % (28414)------------------------------
% 0.30/1.58 % (28414)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.58 % (28414)Termination reason: Unknown
% 0.30/1.58 % (28414)Termination phase: shuffling
% 0.30/1.58
% 0.30/1.58 % (28414)Memory used [KB]: 1407
% 0.30/1.58 % (28414)Time elapsed: 0.003 s
% 0.30/1.58 % (28414)Instructions burned: 6 (million)
% 0.30/1.58 % (28414)------------------------------
% 0.30/1.58 % (28414)------------------------------
% 0.30/1.58 % (28418)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_28194 for (2999ds/275Mi)
% 0.30/1.59 % (28413)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_28194 for (2999ds/183Mi)
% 0.30/1.59 % (28415)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_28194 for (2999ds/27Mi)
% 0.30/1.59 % (28416)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_28194 for (2999ds/2Mi)
% 0.30/1.59 % (28419)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_28194 for (2999ds/18Mi)
% 0.30/1.59 % (28416)Instruction limit reached!
% 0.30/1.59 % (28416)------------------------------
% 0.30/1.59 % (28416)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.59 % (28416)Termination reason: Unknown
% 0.30/1.59 % (28416)Termination phase: shuffling
% 0.30/1.59
% 0.30/1.59 % (28416)Memory used [KB]: 1407
% 0.30/1.59 % (28416)Time elapsed: 0.004 s
% 0.30/1.59 % (28416)Instructions burned: 3 (million)
% 0.30/1.59 % (28416)------------------------------
% 0.30/1.59 % (28416)------------------------------
% 0.30/1.59 % (28420)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_28194 for (2999ds/3Mi)
% 0.30/1.59 % (28420)Instruction limit reached!
% 0.30/1.59 % (28420)------------------------------
% 0.30/1.59 % (28420)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.59 % (28420)Termination reason: Unknown
% 0.30/1.59 % (28420)Termination phase: shuffling
% 0.30/1.59
% 0.30/1.59 % (28420)Memory used [KB]: 1407
% 0.30/1.59 % (28420)Time elapsed: 0.002 s
% 0.30/1.59 % (28420)Instructions burned: 3 (million)
% 0.30/1.59 % (28420)------------------------------
% 0.30/1.59 % (28420)------------------------------
% 0.30/1.59 % (28421)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_28194 for (2999ds/37Mi)
% 2.73/1.60 % (28419)Instruction limit reached!
% 2.73/1.60 % (28419)------------------------------
% 2.73/1.60 % (28419)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (28419)Termination reason: Unknown
% 2.73/1.60 % (28419)Termination phase: shuffling
% 2.73/1.60
% 2.73/1.60 % (28419)Memory used [KB]: 1663
% 2.73/1.60 % (28419)Time elapsed: 0.014 s
% 2.73/1.60 % (28419)Instructions burned: 19 (million)
% 2.73/1.60 % (28419)------------------------------
% 2.73/1.60 % (28419)------------------------------
% 2.73/1.60 % (28423)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_28194 for (2999ds/3Mi)
% 2.73/1.60 % (28423)Instruction limit reached!
% 2.73/1.60 % (28423)------------------------------
% 2.73/1.60 % (28423)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (28423)Termination reason: Unknown
% 2.73/1.60 % (28423)Termination phase: shuffling
% 2.73/1.60
% 2.73/1.60 % (28423)Memory used [KB]: 1407
% 2.73/1.60 % (28423)Time elapsed: 0.003 s
% 2.73/1.60 % (28423)Instructions burned: 5 (million)
% 2.73/1.60 % (28423)------------------------------
% 2.73/1.60 % (28423)------------------------------
% 2.73/1.60 % (28421)Instruction limit reached!
% 2.73/1.60 % (28421)------------------------------
% 2.73/1.60 % (28421)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (28421)Termination reason: Unknown
% 2.73/1.60 % (28421)Termination phase: shuffling
% 2.73/1.60
% 2.73/1.60 % (28421)Memory used [KB]: 2174
% 2.73/1.60 % (28421)Time elapsed: 0.014 s
% 2.73/1.60 % (28421)Instructions burned: 40 (million)
% 2.73/1.60 % (28421)------------------------------
% 2.73/1.60 % (28421)------------------------------
% 2.73/1.60 % (28415)Instruction limit reached!
% 2.73/1.60 % (28415)------------------------------
% 2.73/1.60 % (28415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (28415)Termination reason: Unknown
% 2.73/1.60 % (28415)Termination phase: Property scanning
% 2.73/1.60
% 2.73/1.60 % (28415)Memory used [KB]: 1918
% 2.73/1.60 % (28415)Time elapsed: 0.020 s
% 2.73/1.60 % (28415)Instructions burned: 28 (million)
% 2.73/1.60 % (28415)------------------------------
% 2.73/1.60 % (28415)------------------------------
% 2.73/1.61 % (28422)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_28194 for (2999ds/15Mi)
% 2.73/1.61 % (28425)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_28194 for (2999ds/7Mi)
% 2.73/1.61 % (28426)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_28194 for (2999ds/16Mi)
% 2.73/1.61 % (28425)Instruction limit reached!
% 2.73/1.61 % (28425)------------------------------
% 2.73/1.61 % (28425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (28425)Termination reason: Unknown
% 2.73/1.61 % (28425)Termination phase: shuffling
% 2.73/1.61
% 2.73/1.61 % (28425)Memory used [KB]: 1535
% 2.73/1.61 % (28425)Time elapsed: 0.004 s
% 2.73/1.61 % (28425)Instructions burned: 9 (million)
% 2.73/1.61 % (28425)------------------------------
% 2.73/1.61 % (28425)------------------------------
% 2.73/1.62 % (28422)Instruction limit reached!
% 2.73/1.62 % (28422)------------------------------
% 2.73/1.62 % (28422)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.62 % (28422)Termination reason: Unknown
% 2.73/1.62 % (28422)Termination phase: shuffling
% 2.73/1.62
% 2.73/1.62 % (28422)Memory used [KB]: 1535
% 2.73/1.62 % (28422)Time elapsed: 0.011 s
% 2.73/1.62 % (28422)Instructions burned: 15 (million)
% 2.73/1.62 % (28422)------------------------------
% 2.73/1.62 % (28422)------------------------------
% 2.73/1.62 % (28424)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_28194 for (2999ds/1041Mi)
% 2.73/1.62 % (28426)Instruction limit reached!
% 2.73/1.62 % (28426)------------------------------
% 2.73/1.62 % (28426)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.62 % (28426)Termination reason: Unknown
% 2.73/1.62 % (28426)Termination phase: shuffling
% 2.73/1.62
% 2.73/1.62 % (28426)Memory used [KB]: 1663
% 2.73/1.62 % (28426)Time elapsed: 0.008 s
% 2.73/1.62 % (28426)Instructions burned: 17 (million)
% 2.73/1.62 % (28426)------------------------------
% 2.73/1.62 % (28426)------------------------------
% 2.73/1.62 % (28427)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_28194 for (2999ds/3Mi)
% 2.73/1.62 % (28427)Instruction limit reached!
% 2.73/1.62 % (28427)------------------------------
% 2.73/1.62 % (28427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.62 % (28427)Termination reason: Unknown
% 2.73/1.62 % (28427)Termination phase: shuffling
% 2.73/1.62
% 2.73/1.62 % (28427)Memory used [KB]: 1407
% 2.73/1.62 % (28427)Time elapsed: 0.004 s
% 2.73/1.62 % (28427)Instructions burned: 3 (million)
% 2.73/1.62 % (28427)------------------------------
% 2.73/1.62 % (28427)------------------------------
% 2.73/1.62 % (28428)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_28194 for (2999ds/3Mi)
% 2.73/1.63 % (28428)Instruction limit reached!
% 2.73/1.63 % (28428)------------------------------
% 2.73/1.63 % (28428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.63 % (28428)Termination reason: Unknown
% 2.73/1.63 % (28428)Termination phase: shuffling
% 2.73/1.63
% 2.73/1.63 % (28428)Memory used [KB]: 1407
% 2.73/1.63 % (28428)Time elapsed: 0.002 s
% 2.73/1.63 % (28428)Instructions burned: 3 (million)
% 2.73/1.63 % (28428)------------------------------
% 2.73/1.63 % (28428)------------------------------
% 2.73/1.63 % (28430)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_28194 for (2999ds/3Mi)
% 2.73/1.63 % (28429)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_28194 for (2999ds/7Mi)
% 2.73/1.63 % (28430)Instruction limit reached!
% 2.73/1.63 % (28430)------------------------------
% 2.73/1.63 % (28430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.63 % (28430)Termination reason: Unknown
% 2.73/1.63 % (28430)Termination phase: shuffling
% 2.73/1.63
% 2.73/1.63 % (28430)Memory used [KB]: 1407
% 2.73/1.63 % (28430)Time elapsed: 0.003 s
% 2.73/1.63 % (28430)Instructions burned: 4 (million)
% 2.73/1.63 % (28430)------------------------------
% 2.73/1.63 % (28430)------------------------------
% 2.73/1.63 % (28429)Instruction limit reached!
% 2.73/1.63 % (28429)------------------------------
% 2.73/1.63 % (28429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.63 % (28429)Termination reason: Unknown
% 2.73/1.63 % (28429)Termination phase: shuffling
% 2.73/1.63
% 2.73/1.63 % (28429)Memory used [KB]: 1535
% 2.73/1.63 % (28429)Time elapsed: 0.005 s
% 2.73/1.63 % (28429)Instructions burned: 9 (million)
% 2.73/1.63 % (28429)------------------------------
% 2.73/1.63 % (28429)------------------------------
% 2.73/1.63 % (28432)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_28194 for (2999ds/18Mi)
% 2.73/1.64 % (28432)Instruction limit reached!
% 2.73/1.64 % (28432)------------------------------
% 2.73/1.64 % (28432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.64 % (28432)Termination reason: Unknown
% 2.73/1.64 % (28432)Termination phase: shuffling
% 2.73/1.64
% 2.73/1.64 % (28432)Memory used [KB]: 1663
% 2.73/1.64 % (28432)Time elapsed: 0.007 s
% 2.73/1.64 % (28432)Instructions burned: 18 (million)
% 2.73/1.64 % (28432)------------------------------
% 2.73/1.64 % (28432)------------------------------
% 2.73/1.64 % (28431)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_28194 for (2999ds/4Mi)
% 2.73/1.64 % (28433)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_28194 for (2999ds/710Mi)
% 2.73/1.64 % (28434)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_28194 for (2999ds/6Mi)
% 2.73/1.64 % (28431)Instruction limit reached!
% 2.73/1.64 % (28431)------------------------------
% 2.73/1.64 % (28431)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.64 % (28431)Termination reason: Unknown
% 2.73/1.64 % (28431)Termination phase: shuffling
% 2.73/1.64
% 2.73/1.64 % (28431)Memory used [KB]: 1407
% 2.73/1.64 % (28431)Time elapsed: 0.005 s
% 2.73/1.64 % (28431)Instructions burned: 5 (million)
% 2.73/1.64 % (28431)------------------------------
% 2.73/1.64 % (28431)------------------------------
% 2.73/1.65 % (28434)Instruction limit reached!
% 2.73/1.65 % (28434)------------------------------
% 2.73/1.65 % (28434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.65 % (28434)Termination reason: Unknown
% 2.73/1.65 % (28434)Termination phase: shuffling
% 2.73/1.65
% 2.73/1.65 % (28434)Memory used [KB]: 1407
% 2.73/1.65 % (28434)Time elapsed: 0.004 s
% 2.73/1.65 % (28434)Instructions burned: 7 (million)
% 2.73/1.65 % (28434)------------------------------
% 2.73/1.65 % (28434)------------------------------
% 2.73/1.65 % (28435)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_28194 for (2999ds/902Mi)
% 2.73/1.66 % (28437)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_28194 for (2999ds/5Mi)
% 2.73/1.66 % (28437)Instruction limit reached!
% 2.73/1.66 % (28437)------------------------------
% 2.73/1.66 % (28437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.66 % (28437)Termination reason: Unknown
% 2.73/1.66 % (28437)Termination phase: shuffling
% 2.73/1.66
% 2.73/1.66 % (28437)Memory used [KB]: 1535
% 2.73/1.66 % (28437)Time elapsed: 0.004 s
% 2.73/1.66 % (28437)Instructions burned: 7 (million)
% 2.73/1.66 % (28437)------------------------------
% 2.73/1.66 % (28437)------------------------------
% 2.73/1.66 % (28436)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_28194 for (2999ds/21Mi)
% 2.73/1.67 % (28438)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_28194 for (2999ds/6Mi)
% 2.73/1.67 % (28438)Instruction limit reached!
% 2.73/1.67 % (28438)------------------------------
% 2.73/1.67 % (28438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.67 % (28438)Termination reason: Unknown
% 2.73/1.67 % (28438)Termination phase: shuffling
% 2.73/1.67
% 2.73/1.67 % (28438)Memory used [KB]: 1407
% 2.73/1.67 % (28438)Time elapsed: 0.004 s
% 2.73/1.67 % (28438)Instructions burned: 7 (million)
% 2.73/1.67 % (28438)------------------------------
% 2.73/1.67 % (28438)------------------------------
% 2.73/1.68 % (28436)Instruction limit reached!
% 2.73/1.68 % (28436)------------------------------
% 2.73/1.68 % (28436)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.68 % (28436)Termination reason: Unknown
% 2.73/1.68 % (28436)Termination phase: shuffling
% 2.73/1.68
% 2.73/1.68 % (28436)Memory used [KB]: 1663
% 2.73/1.68 % (28436)Time elapsed: 0.015 s
% 2.73/1.68 % (28436)Instructions burned: 21 (million)
% 2.73/1.68 % (28436)------------------------------
% 2.73/1.68 % (28436)------------------------------
% 2.73/1.68 % (28439)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_28194 for (2998ds/377Mi)
% 2.73/1.68 % (28418)Instruction limit reached!
% 2.73/1.68 % (28418)------------------------------
% 2.73/1.68 % (28418)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.68 % (28418)Termination reason: Unknown
% 2.73/1.68 % (28418)Termination phase: Saturation
% 2.73/1.68
% 2.73/1.68 % (28418)Memory used [KB]: 7419
% 2.73/1.68 % (28418)Time elapsed: 0.100 s
% 2.73/1.68 % (28418)Instructions burned: 276 (million)
% 2.73/1.68 % (28418)------------------------------
% 2.73/1.68 % (28418)------------------------------
% 2.73/1.68 % (28413)Instruction limit reached!
% 2.73/1.68 % (28413)------------------------------
% 2.73/1.68 % (28413)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.68 % (28413)Termination reason: Unknown
% 2.73/1.68 % (28413)Termination phase: Saturation
% 2.73/1.68
% 2.73/1.68 % (28413)Memory used [KB]: 7164
% 2.73/1.68 % (28413)Time elapsed: 0.099 s
% 2.73/1.68 % (28413)Instructions burned: 184 (million)
% 2.73/1.68 % (28413)------------------------------
% 2.73/1.68 % (28413)------------------------------
% 2.73/1.69 % (28441)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_28194 for (2998ds/19Mi)
% 2.73/1.69 % (28440)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_28194 for (2998ds/779Mi)
% 2.73/1.70 % (28441)Instruction limit reached!
% 2.73/1.70 % (28441)------------------------------
% 2.73/1.70 % (28441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.70 % (28441)Termination reason: Unknown
% 2.73/1.70 % (28441)Termination phase: shuffling
% 2.73/1.70
% 2.73/1.70 % (28441)Memory used [KB]: 1791
% 2.73/1.70 % (28441)Time elapsed: 0.008 s
% 2.73/1.70 % (28441)Instructions burned: 20 (million)
% 2.73/1.70 % (28441)------------------------------
% 2.73/1.70 % (28441)------------------------------
% 2.73/1.70 % (28442)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_28194 for (2998ds/879Mi)
% 2.73/1.71 % (28443)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_28194 for (2998ds/17Mi)
% 2.73/1.71 % (28443)Instruction limit reached!
% 2.73/1.71 % (28443)------------------------------
% 2.73/1.71 % (28443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.71 % (28443)Termination reason: Unknown
% 2.73/1.71 % (28443)Termination phase: shuffling
% 2.73/1.71
% 2.73/1.71 % (28443)Memory used [KB]: 1663
% 2.73/1.71 % (28443)Time elapsed: 0.008 s
% 2.73/1.71 % (28443)Instructions burned: 17 (million)
% 2.73/1.71 % (28443)------------------------------
% 2.73/1.71 % (28443)------------------------------
% 2.73/1.72 % (28444)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_28194 for (2998ds/3Mi)
% 2.73/1.73 % (28444)Instruction limit reached!
% 2.73/1.73 % (28444)------------------------------
% 2.73/1.73 % (28444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.73 % (28444)Termination reason: Unknown
% 2.73/1.73 % (28444)Termination phase: shuffling
% 2.73/1.73
% 2.73/1.73 % (28444)Memory used [KB]: 1407
% 2.73/1.73 % (28444)Time elapsed: 0.003 s
% 2.73/1.73 % (28444)Instructions burned: 4 (million)
% 2.73/1.73 % (28444)------------------------------
% 2.73/1.73 % (28444)------------------------------
% 2.73/1.74 % (28445)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_28194 for (2998ds/30Mi)
% 2.73/1.75 % (28445)Instruction limit reached!
% 2.73/1.75 % (28445)------------------------------
% 2.73/1.75 % (28445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.75 % (28445)Termination reason: Unknown
% 2.73/1.75 % (28445)Termination phase: Property scanning
% 2.73/1.75
% 2.73/1.75 % (28445)Memory used [KB]: 1918
% 2.73/1.75 % (28445)Time elapsed: 0.013 s
% 2.73/1.75 % (28445)Instructions burned: 31 (million)
% 2.73/1.75 % (28445)------------------------------
% 2.73/1.75 % (28445)------------------------------
% 2.73/1.76 % (28446)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_28194 for (2998ds/127Mi)
% 2.73/1.79 % (28439)Instruction limit reached!
% 2.73/1.79 % (28439)------------------------------
% 2.73/1.79 % (28439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.79 % (28439)Termination reason: Unknown
% 2.73/1.79 % (28439)Termination phase: Saturation
% 2.73/1.79
% 2.73/1.79 % (28439)Memory used [KB]: 8059
% 2.73/1.79 % (28439)Time elapsed: 0.114 s
% 2.73/1.79 % (28439)Instructions burned: 378 (million)
% 2.73/1.79 % (28439)------------------------------
% 2.73/1.79 % (28439)------------------------------
% 2.73/1.80 % (28446)Instruction limit reached!
% 2.73/1.80 % (28446)------------------------------
% 2.73/1.80 % (28446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.80 % (28446)Termination reason: Unknown
% 2.73/1.80 % (28446)Termination phase: Property scanning
% 2.73/1.80
% 2.73/1.80 % (28446)Memory used [KB]: 2302
% 2.73/1.80 % (28446)Time elapsed: 0.041 s
% 2.73/1.80 % (28446)Instructions burned: 127 (million)
% 2.73/1.80 % (28446)------------------------------
% 2.73/1.80 % (28446)------------------------------
% 2.73/1.80 % (28447)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_28194 for (2997ds/100Mi)
% 2.73/1.81 % (28448)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_28194 for (2997ds/3Mi)
% 2.73/1.81 % (28448)Instruction limit reached!
% 2.73/1.81 % (28448)------------------------------
% 2.73/1.81 % (28448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.81 % (28448)Termination reason: Unknown
% 2.73/1.81 % (28448)Termination phase: shuffling
% 2.73/1.81
% 2.73/1.81 % (28448)Memory used [KB]: 1407
% 2.73/1.81 % (28448)Time elapsed: 0.003 s
% 2.73/1.81 % (28448)Instructions burned: 4 (million)
% 2.73/1.81 % (28448)------------------------------
% 2.73/1.81 % (28448)------------------------------
% 2.73/1.82 % (28449)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_28194 for (2997ds/20Mi)
% 2.73/1.83 % (28449)Instruction limit reached!
% 2.73/1.83 % (28449)------------------------------
% 2.73/1.83 % (28449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.83 % (28449)Termination reason: Unknown
% 2.73/1.83 % (28449)Termination phase: shuffling
% 2.73/1.83
% 2.73/1.83 % (28449)Memory used [KB]: 1663
% 2.73/1.83 % (28449)Time elapsed: 0.009 s
% 2.73/1.83 % (28449)Instructions burned: 20 (million)
% 2.73/1.83 % (28449)------------------------------
% 2.73/1.83 % (28449)------------------------------
% 2.73/1.83 % (28447)Instruction limit reached!
% 2.73/1.83 % (28447)------------------------------
% 2.73/1.83 % (28447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.83 % (28447)Termination reason: Unknown
% 2.73/1.83 % (28447)Termination phase: Property scanning
% 2.73/1.83
% 2.73/1.83 % (28447)Memory used [KB]: 2302
% 2.73/1.83 % (28447)Time elapsed: 0.031 s
% 2.73/1.83 % (28447)Instructions burned: 100 (million)
% 2.73/1.83 % (28447)------------------------------
% 2.73/1.83 % (28447)------------------------------
% 2.73/1.84 % (28450)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_28194 for (2997ds/86Mi)
% 2.73/1.84 % (28451)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_28194 for (2997ds/107Mi)
% 2.73/1.86 % (28450)Instruction limit reached!
% 2.73/1.86 % (28450)------------------------------
% 2.73/1.86 % (28450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.86 % (28450)Termination reason: Unknown
% 2.73/1.86 % (28450)Termination phase: Function definition elimination
% 2.73/1.86
% 2.73/1.86 % (28450)Memory used [KB]: 2430
% 2.73/1.86 % (28450)Time elapsed: 0.027 s
% 2.73/1.86 % (28450)Instructions burned: 87 (million)
% 2.73/1.86 % (28450)------------------------------
% 2.73/1.86 % (28450)------------------------------
% 2.73/1.87 % (28452)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_28194 for (2996ds/39Mi)
% 2.73/1.87 % (28433)Instruction limit reached!
% 2.73/1.87 % (28433)------------------------------
% 2.73/1.87 % (28433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.87 % (28433)Termination reason: Unknown
% 2.73/1.87 % (28433)Termination phase: Saturation
% 2.73/1.87
% 2.73/1.87 % (28433)Memory used [KB]: 8187
% 2.73/1.87 % (28433)Time elapsed: 0.234 s
% 2.73/1.87 % (28433)Instructions burned: 711 (million)
% 2.73/1.87 % (28433)------------------------------
% 2.73/1.87 % (28433)------------------------------
% 2.73/1.88 % (28451)Instruction limit reached!
% 2.73/1.88 % (28451)------------------------------
% 2.73/1.88 % (28451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.88 % (28451)Termination reason: Unknown
% 2.73/1.88 % (28451)Termination phase: Property scanning
% 2.73/1.88
% 2.73/1.88 % (28451)Memory used [KB]: 2302
% 2.73/1.88 % (28451)Time elapsed: 0.036 s
% 2.73/1.88 % (28451)Instructions burned: 110 (million)
% 2.73/1.88 % (28451)------------------------------
% 2.73/1.88 % (28451)------------------------------
% 2.73/1.88 % (28452)Instruction limit reached!
% 2.73/1.88 % (28452)------------------------------
% 2.73/1.88 % (28452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.88 % (28452)Termination reason: Unknown
% 2.73/1.88 % (28452)Termination phase: shuffling
% 2.73/1.88
% 2.73/1.88 % (28452)Memory used [KB]: 2174
% 2.73/1.88 % (28452)Time elapsed: 0.012 s
% 2.73/1.88 % (28452)Instructions burned: 39 (million)
% 2.73/1.88 % (28452)------------------------------
% 2.73/1.88 % (28452)------------------------------
% 2.73/1.88 % (28453)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_28194 for (2996ds/448Mi)
% 2.73/1.89 % (28454)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_28194 for (2996ds/46Mi)
% 2.73/1.89 % (28455)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_28194 for (2996ds/98Mi)
% 2.73/1.90 % (28454)Instruction limit reached!
% 2.73/1.90 % (28454)------------------------------
% 2.73/1.90 % (28454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.90 % (28454)Termination reason: Unknown
% 2.73/1.90 % (28454)Termination phase: SInE selection
% 2.73/1.90
% 2.73/1.90 % (28454)Memory used [KB]: 1791
% 2.73/1.90 % (28454)Time elapsed: 0.017 s
% 2.73/1.90 % (28454)Instructions burned: 46 (million)
% 2.73/1.90 % (28454)------------------------------
% 2.73/1.90 % (28454)------------------------------
% 2.73/1.91 % (28456)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_28194 for (2996ds/507Mi)
% 2.73/1.91 % (28435)Instruction limit reached!
% 2.73/1.91 % (28435)------------------------------
% 2.73/1.91 % (28435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.91 % (28435)Termination reason: Unknown
% 2.73/1.91 % (28435)Termination phase: Saturation
% 2.73/1.91
% 2.73/1.91 % (28435)Memory used [KB]: 8443
% 2.73/1.91 % (28435)Time elapsed: 0.264 s
% 2.73/1.91 % (28435)Instructions burned: 903 (million)
% 2.73/1.91 % (28435)------------------------------
% 2.73/1.91 % (28435)------------------------------
% 2.73/1.92 % (28455)Instruction limit reached!
% 2.73/1.92 % (28455)------------------------------
% 2.73/1.92 % (28455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.92 % (28455)Termination reason: Unknown
% 2.73/1.92 % (28455)Termination phase: Property scanning
% 2.73/1.92
% 2.73/1.92 % (28455)Memory used [KB]: 2302
% 2.73/1.92 % (28455)Time elapsed: 0.028 s
% 2.73/1.92 % (28455)Instructions burned: 101 (million)
% 2.73/1.92 % (28455)------------------------------
% 2.73/1.92 % (28455)------------------------------
% 2.73/1.92 % (28457)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_28194 for (2996ds/149Mi)
% 2.73/1.93 % (28458)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_28194 for (2996ds/39Mi)
% 2.73/1.94 % (28424)Instruction limit reached!
% 2.73/1.94 % (28424)------------------------------
% 2.73/1.94 % (28424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.94 % (28424)Termination reason: Unknown
% 2.73/1.94 % (28424)Termination phase: Saturation
% 2.73/1.94
% 2.73/1.94 % (28424)Memory used [KB]: 10106
% 2.73/1.94 % (28424)Time elapsed: 0.322 s
% 2.73/1.94 % (28424)Instructions burned: 1044 (million)
% 2.73/1.94 % (28424)------------------------------
% 2.73/1.94 % (28424)------------------------------
% 2.73/1.94 % (28458)Instruction limit reached!
% 2.73/1.94 % (28458)------------------------------
% 2.73/1.94 % (28458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.94 % (28458)Termination reason: Unknown
% 2.73/1.94 % (28458)Termination phase: Preprocessing 1
% 2.73/1.94
% 2.73/1.94 % (28458)Memory used [KB]: 1791
% 2.73/1.94 % (28458)Time elapsed: 0.014 s
% 2.73/1.94 % (28458)Instructions burned: 42 (million)
% 2.73/1.94 % (28458)------------------------------
% 2.73/1.94 % (28458)------------------------------
% 2.73/1.95 % (28459)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_28194 for (2996ds/6Mi)
% 2.73/1.95 % (28459)Instruction limit reached!
% 2.73/1.95 % (28459)------------------------------
% 2.73/1.95 % (28459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.95 % (28459)Termination reason: Unknown
% 2.73/1.95 % (28459)Termination phase: shuffling
% 2.73/1.95
% 2.73/1.95 % (28459)Memory used [KB]: 1407
% 2.73/1.95 % (28459)Time elapsed: 0.003 s
% 2.73/1.95 % (28459)Instructions burned: 6 (million)
% 2.73/1.95 % (28459)------------------------------
% 2.73/1.95 % (28459)------------------------------
% 2.73/1.95 % (28460)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_28194 for (2996ds/1087Mi)
% 2.73/1.96 % (28461)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_28194 for (2996ds/47Mi)
% 2.73/1.96 % (28440)Instruction limit reached!
% 2.73/1.96 % (28440)------------------------------
% 2.73/1.96 % (28440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.96 % (28440)Termination reason: Unknown
% 2.73/1.96 % (28440)Termination phase: Saturation
% 2.73/1.96
% 2.73/1.96 % (28440)Memory used [KB]: 8187
% 2.73/1.96 % (28440)Time elapsed: 0.292 s
% 2.73/1.96 % (28440)Instructions burned: 781 (million)
% 2.73/1.96 % (28440)------------------------------
% 2.73/1.96 % (28440)------------------------------
% 2.73/1.96 % (28457)Instruction limit reached!
% 2.73/1.96 % (28457)------------------------------
% 2.73/1.96 % (28457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.96 % (28457)Termination reason: Unknown
% 2.73/1.96 % (28457)Termination phase: Saturation
% 2.73/1.96
% 2.73/1.96 % (28457)Memory used [KB]: 6908
% 2.73/1.96 % (28457)Time elapsed: 0.043 s
% 2.73/1.96 % (28457)Instructions burned: 150 (million)
% 2.73/1.96 % (28457)------------------------------
% 2.73/1.96 % (28457)------------------------------
% 2.73/1.97 % (28462)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_28194 for (2995ds/44Mi)
% 2.73/1.97 % (28461)Instruction limit reached!
% 2.73/1.97 % (28461)------------------------------
% 2.73/1.97 % (28461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.97 % (28461)Termination reason: Unknown
% 2.73/1.97 % (28461)Termination phase: Preprocessing 3
% 2.73/1.97
% 2.73/1.97 % (28461)Memory used [KB]: 2174
% 2.73/1.97 % (28461)Time elapsed: 0.016 s
% 2.73/1.97 % (28461)Instructions burned: 48 (million)
% 2.73/1.97 % (28461)------------------------------
% 2.73/1.97 % (28461)------------------------------
% 2.73/1.97 % (28463)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_28194 for (2995ds/432Mi)
% 2.73/1.98 % (28464)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_28194 for (2995ds/62Mi)
% 2.73/1.98 % (28462)Instruction limit reached!
% 2.73/1.98 % (28462)------------------------------
% 2.73/1.98 % (28462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.98 % (28462)Termination reason: Unknown
% 2.73/1.98 % (28462)Termination phase: Property scanning
% 2.73/1.98
% 2.73/1.98 % (28462)Memory used [KB]: 2174
% 2.73/1.98 % (28462)Time elapsed: 0.015 s
% 2.73/1.98 % (28462)Instructions burned: 44 (million)
% 2.73/1.98 % (28462)------------------------------
% 2.73/1.98 % (28462)------------------------------
% 2.73/1.99 % (28442)Instruction limit reached!
% 2.73/1.99 % (28442)------------------------------
% 2.73/1.99 % (28442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.99 % (28442)Termination reason: Unknown
% 2.73/1.99 % (28442)Termination phase: Saturation
% 2.73/1.99
% 2.73/1.99 % (28442)Memory used [KB]: 7675
% 2.73/1.99 % (28442)Time elapsed: 0.311 s
% 2.73/1.99 % (28442)Instructions burned: 881 (million)
% 2.73/1.99 % (28442)------------------------------
% 2.73/1.99 % (28442)------------------------------
% 2.73/1.99 % (28465)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_28194 for (2995ds/31Mi)
% 2.73/2.00 % (28464)Instruction limit reached!
% 2.73/2.00 % (28464)------------------------------
% 2.73/2.00 % (28464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/2.00 % (28464)Termination reason: Unknown
% 2.73/2.00 % (28464)Termination phase: Property scanning
% 2.73/2.00
% 2.73/2.00 % (28464)Memory used [KB]: 2174
% 2.73/2.00 % (28464)Time elapsed: 0.018 s
% 2.73/2.00 % (28464)Instructions burned: 64 (million)
% 2.73/2.00 % (28464)------------------------------
% 2.73/2.00 % (28464)------------------------------
% 2.73/2.00 % (28466)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_28194 for (2995ds/96Mi)
% 2.73/2.01 % (28465)Instruction limit reached!
% 2.73/2.01 % (28465)------------------------------
% 2.73/2.01 % (28465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/2.01 % (28465)Termination reason: Unknown
% 2.73/2.01 % (28465)Termination phase: Property scanning
% 2.73/2.01
% 2.73/2.01 % (28465)Memory used [KB]: 1791
% 2.73/2.01 % (28465)Time elapsed: 0.013 s
% 2.73/2.01 % (28465)Instructions burned: 32 (million)
% 2.73/2.01 % (28465)------------------------------
% 2.73/2.01 % (28465)------------------------------
% 2.73/2.01 % (28467)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_28194 for (2995ds/453Mi)
% 5.76/2.02 % (28468)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_28194 for (2995ds/71Mi)
% 5.76/2.02 % (28453)Instruction limit reached!
% 5.76/2.02 % (28453)------------------------------
% 5.76/2.02 % (28453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.76/2.02 % (28453)Termination reason: Unknown
% 5.76/2.02 % (28453)Termination phase: Saturation
% 5.76/2.02
% 5.76/2.02 % (28453)Memory used [KB]: 8059
% 5.76/2.02 % (28453)Time elapsed: 0.138 s
% 5.76/2.02 % (28453)Instructions burned: 449 (million)
% 5.76/2.02 % (28453)------------------------------
% 5.76/2.02 % (28453)------------------------------
% 5.76/2.03 % (28469)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_28194 for (2995ds/83Mi)
% 5.76/2.04 % (28468)Instruction limit reached!
% 5.76/2.04 % (28468)------------------------------
% 5.76/2.04 % (28468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.76/2.04 % (28468)Termination reason: Unknown
% 5.76/2.04 % (28468)Termination phase: Property scanning
% 5.76/2.04
% 5.76/2.04 % (28468)Memory used [KB]: 2302
% 5.76/2.04 % (28468)Time elapsed: 0.026 s
% 5.76/2.04 % (28468)Instructions burned: 74 (million)
% 5.76/2.04 % (28468)------------------------------
% 5.76/2.04 % (28468)------------------------------
% 5.76/2.05 % (28470)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_28194 for (2995ds/11Mi)
% 5.76/2.05 % (28470)Instruction limit reached!
% 5.76/2.05 % (28470)------------------------------
% 5.76/2.05 % (28470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.76/2.05 % (28470)Termination reason: Unknown
% 5.76/2.05 % (28470)Termination phase: shuffling
% 5.76/2.05
% 5.76/2.05 % (28470)Memory used [KB]: 1535
% 5.76/2.05 % (28470)Time elapsed: 0.005 s
% 5.76/2.05 % (28470)Instructions burned: 11 (million)
% 5.76/2.05 % (28470)------------------------------
% 5.76/2.05 % (28470)------------------------------
% 5.76/2.06 % (28469)Instruction limit reached!
% 5.76/2.06 % (28469)------------------------------
% 5.76/2.06 % (28469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.76/2.06 % (28469)Termination reason: Unknown
% 5.76/2.06 % (28469)Termination phase: Property scanning
% 5.76/2.06
% 5.76/2.06 % (28469)Memory used [KB]: 2174
% 5.76/2.06 % (28469)Time elapsed: 0.026 s
% 5.76/2.06 % (28469)Instructions burned: 84 (million)
% 5.76/2.06 % (28469)------------------------------
% 5.76/2.06 % (28469)------------------------------
% 5.76/2.06 % (28466)Instruction limit reached!
% 5.76/2.06 % (28466)------------------------------
% 5.76/2.06 % (28466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.76/2.06 % (28466)Termination reason: Unknown
% 5.76/2.06 % (28466)Termination phase: Property scanning
% 5.76/2.06
% 5.76/2.06 % (28466)Memory used [KB]: 2302
% 5.76/2.06 % (28466)Time elapsed: 0.056 s
% 5.76/2.06 % (28466)Instructions burned: 97 (million)
% 5.76/2.06 % (28466)------------------------------
% 5.76/2.06 % (28466)------------------------------
% 5.76/2.06 % (28471)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_28194 for (2995ds/41Mi)
% 5.76/2.06 % (28472)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_28194 for (2995ds/6Mi)
% 5.76/2.07 % (28472)Instruction limit reached!
% 5.76/2.07 % (28472)------------------------------
% 5.76/2.07 % (28472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.76/2.07 % (28472)Termination reason: Unknown
% 5.76/2.07 % (28472)Termination phase: shuffling
% 5.76/2.07
% 5.76/2.07 % (28472)Memory used [KB]: 1407
% 5.76/2.07 % (28472)Time elapsed: 0.004 s
% 5.76/2.07 % (28472)Instructions burned: 8 (million)
% 5.76/2.07 % (28472)------------------------------
% 5.76/2.07 % (28472)------------------------------
% 5.76/2.07 % (28456)Instruction limit reached!
% 5.76/2.07 % (28456)------------------------------
% 5.76/2.07 % (28456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.76/2.07 % (28456)Termination reason: Unknown
% 5.76/2.07 % (28456)Termination phase: Saturation
% 5.76/2.07
% 5.76/2.07 % (28456)Memory used [KB]: 8059
% 5.76/2.07 % (28456)Time elapsed: 0.162 s
% 5.76/2.07 % (28456)Instructions burned: 508 (million)
% 5.76/2.07 % (28456)------------------------------
% 5.76/2.07 % (28456)------------------------------
% 5.76/2.08 % (28473)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_28194 for (2995ds/150Mi)
% 5.76/2.08 % (28471)Instruction limit reached!
% 5.76/2.08 % (28471)------------------------------
% 5.76/2.08 % (28471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.76/2.08 % (28471)Termination reason: Unknown
% 5.76/2.08 % (28471)Termination phase: shuffling
% 5.76/2.08
% 5.76/2.08 % (28471)Memory used [KB]: 2174
% 5.76/2.08 % (28471)Time elapsed: 0.014 s
% 5.76/2.08 % (28471)Instructions burned: 41 (million)
% 5.76/2.08 % (28471)------------------------------
% 5.76/2.08 % (28471)------------------------------
% 5.76/2.08 % (28474)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_28194 for (2994ds/343Mi)
% 5.76/2.08 % (28475)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_28194 for (2994ds/109Mi)
% 5.76/2.09 % (28476)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_28194 for (2994ds/253Mi)
% 5.76/2.11 % (28463)Instruction limit reached!
% 5.76/2.11 % (28463)------------------------------
% 5.76/2.11 % (28463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.76/2.11 % (28463)Termination reason: Unknown
% 5.76/2.11 % (28463)Termination phase: Saturation
% 5.76/2.11
% 5.76/2.11 % (28463)Memory used [KB]: 4605
% 5.76/2.11 % (28463)Time elapsed: 0.134 s
% 5.76/2.11 % (28463)Instructions burned: 434 (million)
% 5.76/2.11 % (28463)------------------------------
% 5.76/2.11 % (28463)------------------------------
% 6.25/2.12 % (28475)Instruction limit reached!
% 6.25/2.12 % (28475)------------------------------
% 6.25/2.12 % (28475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.25/2.12 % (28475)Termination reason: Unknown
% 6.25/2.12 % (28475)Termination phase: Property scanning
% 6.25/2.12
% 6.25/2.12 % (28475)Memory used [KB]: 2302
% 6.25/2.12 % (28475)Time elapsed: 0.034 s
% 6.25/2.12 % (28475)Instructions burned: 110 (million)
% 6.25/2.12 % (28475)------------------------------
% 6.25/2.12 % (28475)------------------------------
% 6.25/2.12 % (28477)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_28194 for (2994ds/140Mi)
% 6.25/2.12 % (28478)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_28194 for (2994ds/614Mi)
% 6.25/2.14 % (28467)Instruction limit reached!
% 6.25/2.14 % (28467)------------------------------
% 6.25/2.14 % (28467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.25/2.14 % (28467)Termination reason: Unknown
% 6.25/2.14 % (28467)Termination phase: Saturation
% 6.25/2.14
% 6.25/2.14 % (28467)Memory used [KB]: 7931
% 6.25/2.14 % (28467)Time elapsed: 0.133 s
% 6.25/2.14 % (28467)Instructions burned: 455 (million)
% 6.25/2.14 % (28467)------------------------------
% 6.25/2.14 % (28467)------------------------------
% 6.25/2.15 % (28473)Instruction limit reached!
% 6.25/2.15 % (28473)------------------------------
% 6.25/2.15 % (28473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.25/2.15 % (28473)Termination reason: Unknown
% 6.25/2.15 % (28473)Termination phase: Saturation
% 6.25/2.15
% 6.25/2.15 % (28473)Memory used [KB]: 6908
% 6.25/2.15 % (28473)Time elapsed: 0.075 s
% 6.25/2.15 % (28473)Instructions burned: 151 (million)
% 6.25/2.15 % (28473)------------------------------
% 6.25/2.15 % (28473)------------------------------
% 6.25/2.15 % (28479)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_28194 for (2994ds/202Mi)
% 6.25/2.16 % (28480)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_28194 for (2993ds/339Mi)
% 6.25/2.16 % (28477)Instruction limit reached!
% 6.25/2.16 % (28477)------------------------------
% 6.25/2.16 % (28477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.25/2.16 % (28477)Termination reason: Unknown
% 6.25/2.16 % (28477)Termination phase: Saturation
% 6.25/2.16
% 6.25/2.16 % (28477)Memory used [KB]: 6908
% 6.25/2.16 % (28477)Time elapsed: 0.044 s
% 6.25/2.16 % (28477)Instructions burned: 142 (million)
% 6.25/2.16 % (28477)------------------------------
% 6.25/2.16 % (28477)------------------------------
% 6.25/2.17 % (28476)Instruction limit reached!
% 6.25/2.17 % (28476)------------------------------
% 6.25/2.17 % (28476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.25/2.17 % (28476)Termination reason: Unknown
% 6.25/2.17 % (28476)Termination phase: Saturation
% 6.25/2.17
% 6.25/2.17 % (28476)Memory used [KB]: 7803
% 6.25/2.17 % (28476)Time elapsed: 0.081 s
% 6.25/2.17 % (28476)Instructions burned: 255 (million)
% 6.25/2.17 % (28476)------------------------------
% 6.25/2.17 % (28476)------------------------------
% 6.25/2.17 % (28481)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=283:si=on:rtra=on_0 on DTF2THF_28194 for (2993ds/283Mi)
% 6.25/2.17 % (28482)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_28194 for (2993ds/134Mi)
% 6.25/2.18 % (28474)Instruction limit reached!
% 6.25/2.18 % (28474)------------------------------
% 6.25/2.18 % (28474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.25/2.18 % (28474)Termination reason: Unknown
% 6.25/2.18 % (28474)Termination phase: Saturation
% 6.25/2.18
% 6.25/2.18 % (28474)Memory used [KB]: 7803
% 6.25/2.18 % (28474)Time elapsed: 0.108 s
% 6.25/2.18 % (28474)Instructions burned: 343 (million)
% 6.25/2.18 % (28474)------------------------------
% 6.25/2.18 % (28474)------------------------------
% 6.25/2.19 % (28483)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_28194 for (2993ds/298Mi)
% 6.87/2.21 % (28479)Instruction limit reached!
% 6.87/2.21 % (28479)------------------------------
% 6.87/2.21 % (28479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.21 % (28479)Termination reason: Unknown
% 6.87/2.21 % (28479)Termination phase: Saturation
% 6.87/2.21
% 6.87/2.21 % (28479)Memory used [KB]: 7547
% 6.87/2.21 % (28479)Time elapsed: 0.061 s
% 6.87/2.21 % (28479)Instructions burned: 204 (million)
% 6.87/2.21 % (28479)------------------------------
% 6.87/2.21 % (28479)------------------------------
% 6.87/2.21 % (28482)Instruction limit reached!
% 6.87/2.21 % (28482)------------------------------
% 6.87/2.21 % (28482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.21 % (28482)Termination reason: Unknown
% 6.87/2.21 % (28482)Termination phase: Saturation
% 6.87/2.21
% 6.87/2.21 % (28482)Memory used [KB]: 6908
% 6.87/2.21 % (28482)Time elapsed: 0.040 s
% 6.87/2.21 % (28482)Instructions burned: 136 (million)
% 6.87/2.21 % (28482)------------------------------
% 6.87/2.21 % (28482)------------------------------
% 6.87/2.22 % (28484)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_28194 for (2993ds/192Mi)
% 6.87/2.22 % (28485)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_28194 for (2993ds/258Mi)
% 6.87/2.26 % (28481)Instruction limit reached!
% 6.87/2.26 % (28481)------------------------------
% 6.87/2.26 % (28481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.26 % (28481)Termination reason: Unknown
% 6.87/2.26 % (28481)Termination phase: Saturation
% 6.87/2.26
% 6.87/2.26 % (28481)Memory used [KB]: 7803
% 6.87/2.26 % (28481)Time elapsed: 0.089 s
% 6.87/2.26 % (28481)Instructions burned: 283 (million)
% 6.87/2.26 % (28481)------------------------------
% 6.87/2.26 % (28481)------------------------------
% 6.87/2.26 % (28480)Instruction limit reached!
% 6.87/2.26 % (28480)------------------------------
% 6.87/2.26 % (28480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.26 % (28480)Termination reason: Unknown
% 6.87/2.26 % (28480)Termination phase: Saturation
% 6.87/2.26
% 6.87/2.26 % (28480)Memory used [KB]: 7419
% 6.87/2.26 % (28480)Time elapsed: 0.107 s
% 6.87/2.26 % (28480)Instructions burned: 340 (million)
% 6.87/2.26 % (28480)------------------------------
% 6.87/2.26 % (28480)------------------------------
% 6.87/2.27 % (28486)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_28194 for (2992ds/90Mi)
% 6.87/2.27 % (28487)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_28194 for (2992ds/231Mi)
% 6.87/2.28 % (28484)Instruction limit reached!
% 6.87/2.28 % (28484)------------------------------
% 6.87/2.28 % (28484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.28 % (28484)Termination reason: Unknown
% 6.87/2.28 % (28484)Termination phase: Saturation
% 6.87/2.28
% 6.87/2.28 % (28484)Memory used [KB]: 3070
% 6.87/2.28 % (28484)Time elapsed: 0.058 s
% 6.87/2.28 % (28484)Instructions burned: 195 (million)
% 6.87/2.28 % (28484)------------------------------
% 6.87/2.28 % (28484)------------------------------
% 6.87/2.28 % (28483)Instruction limit reached!
% 6.87/2.28 % (28483)------------------------------
% 6.87/2.28 % (28483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.28 % (28483)Termination reason: Unknown
% 6.87/2.28 % (28483)Termination phase: Saturation
% 6.87/2.28
% 6.87/2.28 % (28483)Memory used [KB]: 8571
% 6.87/2.28 % (28483)Time elapsed: 0.088 s
% 6.87/2.28 % (28483)Instructions burned: 299 (million)
% 6.87/2.28 % (28483)------------------------------
% 6.87/2.28 % (28483)------------------------------
% 6.87/2.28 % (28460)Instruction limit reached!
% 6.87/2.28 % (28460)------------------------------
% 6.87/2.28 % (28460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.28 % (28460)Termination reason: Unknown
% 6.87/2.28 % (28460)Termination phase: Saturation
% 6.87/2.28
% 6.87/2.28 % (28460)Memory used [KB]: 7675
% 6.87/2.28 % (28460)Time elapsed: 0.333 s
% 6.87/2.28 % (28460)Instructions burned: 1088 (million)
% 6.87/2.28 % (28460)------------------------------
% 6.87/2.28 % (28460)------------------------------
% 6.87/2.29 % (28488)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_28194 for (2992ds/458Mi)
% 6.87/2.29 % (28478)Instruction limit reached!
% 6.87/2.29 % (28478)------------------------------
% 6.87/2.29 % (28478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.29 % (28478)Termination reason: Unknown
% 6.87/2.29 % (28478)Termination phase: Saturation
% 6.87/2.29
% 6.87/2.29 % (28478)Memory used [KB]: 8571
% 6.87/2.29 % (28478)Time elapsed: 0.166 s
% 6.87/2.29 % (28478)Instructions burned: 614 (million)
% 6.87/2.29 % (28478)------------------------------
% 6.87/2.29 % (28478)------------------------------
% 6.87/2.29 % (28489)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_28194 for (2992ds/31Mi)
% 6.87/2.29 % (28490)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_28194 for (2992ds/297Mi)
% 6.87/2.29 % (28486)Instruction limit reached!
% 6.87/2.29 % (28486)------------------------------
% 6.87/2.29 % (28486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.29 % (28486)Termination reason: Unknown
% 6.87/2.29 % (28486)Termination phase: Property scanning
% 6.87/2.29
% 6.87/2.29 % (28486)Memory used [KB]: 2302
% 6.87/2.29 % (28486)Time elapsed: 0.028 s
% 6.87/2.29 % (28486)Instructions burned: 90 (million)
% 6.87/2.29 % (28486)------------------------------
% 6.87/2.29 % (28486)------------------------------
% 7.67/2.30 % (28485)Instruction limit reached!
% 7.67/2.30 % (28485)------------------------------
% 7.67/2.30 % (28485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.30 % (28485)Termination reason: Unknown
% 7.67/2.30 % (28485)Termination phase: Saturation
% 7.67/2.30
% 7.67/2.30 % (28485)Memory used [KB]: 7675
% 7.67/2.30 % (28485)Time elapsed: 0.077 s
% 7.67/2.30 % (28485)Instructions burned: 258 (million)
% 7.67/2.30 % (28485)------------------------------
% 7.67/2.30 % (28485)------------------------------
% 7.67/2.30 % (28491)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_28194 for (2992ds/987Mi)
% 7.67/2.30 % (28489)Instruction limit reached!
% 7.67/2.30 % (28489)------------------------------
% 7.67/2.30 % (28489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.30 % (28489)Termination reason: Unknown
% 7.67/2.30 % (28489)Termination phase: Property scanning
% 7.67/2.30
% 7.67/2.30 % (28489)Memory used [KB]: 1791
% 7.67/2.30 % (28489)Time elapsed: 0.012 s
% 7.67/2.30 % (28489)Instructions burned: 34 (million)
% 7.67/2.30 % (28489)------------------------------
% 7.67/2.30 % (28489)------------------------------
% 7.67/2.30 % (28492)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_28194 for (2992ds/182Mi)
% 7.67/2.31 % (28493)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_28194 for (2992ds/182Mi)
% 7.67/2.31 % (28494)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_28194 for (2992ds/154Mi)
% 7.67/2.34 % (28487)Instruction limit reached!
% 7.67/2.34 % (28487)------------------------------
% 7.67/2.34 % (28487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.34 % (28487)Termination reason: Unknown
% 7.67/2.34 % (28487)Termination phase: Saturation
% 7.67/2.34
% 7.67/2.34 % (28487)Memory used [KB]: 7036
% 7.67/2.34 % (28487)Time elapsed: 0.069 s
% 7.67/2.34 % (28487)Instructions burned: 233 (million)
% 7.67/2.34 % (28487)------------------------------
% 7.67/2.34 % (28487)------------------------------
% 7.67/2.35 % (28495)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_28194 for (2992ds/79Mi)
% 7.67/2.35 % (28494)Instruction limit reached!
% 7.67/2.35 % (28494)------------------------------
% 7.67/2.35 % (28494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.35 % (28494)Termination reason: Unknown
% 7.67/2.35 % (28494)Termination phase: Saturation
% 7.67/2.35
% 7.67/2.35 % (28494)Memory used [KB]: 2942
% 7.67/2.35 % (28494)Time elapsed: 0.045 s
% 7.67/2.35 % (28494)Instructions burned: 155 (million)
% 7.67/2.35 % (28494)------------------------------
% 7.67/2.35 % (28494)------------------------------
% 7.67/2.36 % (28492)Instruction limit reached!
% 7.67/2.36 % (28492)------------------------------
% 7.67/2.36 % (28492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.36 % (28492)Termination reason: Unknown
% 7.67/2.36 % (28492)Termination phase: Saturation
% 7.67/2.36
% 7.67/2.36 % (28492)Memory used [KB]: 7164
% 7.67/2.36 % (28492)Time elapsed: 0.053 s
% 7.67/2.36 % (28492)Instructions burned: 184 (million)
% 7.67/2.36 % (28492)------------------------------
% 7.67/2.36 % (28492)------------------------------
% 7.67/2.36 % (28496)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_28194 for (2991ds/216Mi)
% 7.67/2.36 % (28497)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_28194 for (2991ds/114Mi)
% 7.67/2.37 % (28493)Instruction limit reached!
% 7.67/2.37 % (28493)------------------------------
% 7.67/2.37 % (28493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.37 % (28493)Termination reason: Unknown
% 7.67/2.37 % (28493)Termination phase: Saturation
% 7.67/2.37
% 7.67/2.37 % (28493)Memory used [KB]: 7291
% 7.67/2.37 % (28493)Time elapsed: 0.058 s
% 7.67/2.37 % (28493)Instructions burned: 182 (million)
% 7.67/2.37 % (28493)------------------------------
% 7.67/2.37 % (28493)------------------------------
% 7.67/2.38 % (28498)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_28194 for (2991ds/922Mi)
% 7.67/2.38 % (28495)Instruction limit reached!
% 7.67/2.38 % (28495)------------------------------
% 7.67/2.38 % (28495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.38 % (28495)Termination reason: Unknown
% 7.67/2.38 % (28495)Termination phase: Function definition elimination
% 7.67/2.38
% 7.67/2.38 % (28495)Memory used [KB]: 2302
% 7.67/2.38 % (28495)Time elapsed: 0.028 s
% 7.67/2.38 % (28495)Instructions burned: 80 (million)
% 7.67/2.38 % (28495)------------------------------
% 7.67/2.38 % (28495)------------------------------
% 7.67/2.38 % (28490)Instruction limit reached!
% 7.67/2.38 % (28490)------------------------------
% 7.67/2.38 % (28490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.38 % (28490)Termination reason: Unknown
% 7.67/2.38 % (28490)Termination phase: Saturation
% 7.67/2.38
% 7.67/2.38 % (28490)Memory used [KB]: 7675
% 7.67/2.38 % (28490)Time elapsed: 0.089 s
% 7.67/2.38 % (28490)Instructions burned: 298 (million)
% 7.67/2.38 % (28490)------------------------------
% 7.67/2.38 % (28490)------------------------------
% 7.67/2.39 % (28499)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_28194 for (2991ds/182Mi)
% 7.67/2.39 % (28500)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_28194 for (2991ds/164Mi)
% 8.17/2.40 % (28497)Instruction limit reached!
% 8.17/2.40 % (28497)------------------------------
% 8.17/2.40 % (28497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.17/2.40 % (28497)Termination reason: Unknown
% 8.17/2.40 % (28497)Termination phase: Property scanning
% 8.17/2.40
% 8.17/2.40 % (28497)Memory used [KB]: 2302
% 8.17/2.40 % (28497)Time elapsed: 0.034 s
% 8.17/2.40 % (28497)Instructions burned: 115 (million)
% 8.17/2.40 % (28497)------------------------------
% 8.17/2.40 % (28497)------------------------------
% 8.17/2.41 % (28501)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_28194 for (2991ds/50Mi)
% 8.17/2.41 % (28488)Instruction limit reached!
% 8.17/2.41 % (28488)------------------------------
% 8.17/2.41 % (28488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.17/2.41 % (28488)Termination reason: Unknown
% 8.17/2.41 % (28488)Termination phase: Saturation
% 8.17/2.41
% 8.17/2.41 % (28488)Memory used [KB]: 8315
% 8.17/2.41 % (28488)Time elapsed: 0.129 s
% 8.17/2.41 % (28488)Instructions burned: 461 (million)
% 8.17/2.41 % (28488)------------------------------
% 8.17/2.41 % (28488)------------------------------
% 8.17/2.42 % (28501)Instruction limit reached!
% 8.17/2.42 % (28501)------------------------------
% 8.17/2.42 % (28501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.17/2.42 % (28501)Termination reason: Unknown
% 8.17/2.42 % (28501)Termination phase: Property scanning
% 8.17/2.42
% 8.17/2.42 % (28501)Memory used [KB]: 2174
% 8.17/2.42 % (28501)Time elapsed: 0.015 s
% 8.17/2.42 % (28501)Instructions burned: 52 (million)
% 8.17/2.42 % (28501)------------------------------
% 8.17/2.42 % (28501)------------------------------
% 8.17/2.42 % (28502)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_28194 for (2991ds/368Mi)
% 8.17/2.42 % (28496)Instruction limit reached!
% 8.17/2.42 % (28496)------------------------------
% 8.17/2.42 % (28496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.17/2.42 % (28496)Termination reason: Unknown
% 8.17/2.42 % (28496)Termination phase: Saturation
% 8.17/2.42
% 8.17/2.42 % (28496)Memory used [KB]: 7291
% 8.17/2.42 % (28496)Time elapsed: 0.061 s
% 8.17/2.42 % (28496)Instructions burned: 216 (million)
% 8.17/2.42 % (28496)------------------------------
% 8.17/2.42 % (28496)------------------------------
% 8.17/2.43 % (28503)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_28194 for (2991ds/1021Mi)
% 8.17/2.43 % (28504)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_28194 for (2991ds/267Mi)
% 8.17/2.43 % (28500)Instruction limit reached!
% 8.17/2.43 % (28500)------------------------------
% 8.17/2.43 % (28500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.17/2.43 % (28500)Termination reason: Unknown
% 8.17/2.43 % (28500)Termination phase: Saturation
% 8.17/2.43
% 8.17/2.43 % (28500)Memory used [KB]: 2558
% 8.17/2.43 % (28500)Time elapsed: 0.047 s
% 8.17/2.43 % (28500)Instructions burned: 166 (million)
% 8.17/2.43 % (28500)------------------------------
% 8.17/2.43 % (28500)------------------------------
% 8.17/2.44 % (28499)Instruction limit reached!
% 8.17/2.44 % (28499)------------------------------
% 8.17/2.44 % (28499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.17/2.44 % (28499)Termination reason: Unknown
% 8.17/2.44 % (28499)Termination phase: Saturation
% 8.17/2.44
% 8.17/2.44 % (28499)Memory used [KB]: 7291
% 8.17/2.44 % (28499)Time elapsed: 0.057 s
% 8.17/2.44 % (28499)Instructions burned: 184 (million)
% 8.17/2.44 % (28499)------------------------------
% 8.17/2.44 % (28499)------------------------------
% 8.17/2.44 % (28505)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_28194 for (2991ds/204Mi)
% 8.17/2.45 % (28506)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_28194 for (2991ds/26Mi)
% 8.17/2.46 % (28506)Instruction limit reached!
% 8.17/2.46 % (28506)------------------------------
% 8.17/2.46 % (28506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.17/2.46 % (28506)Termination reason: Unknown
% 8.17/2.46 % (28506)Termination phase: shuffling
% 8.17/2.46
% 8.17/2.46 % (28506)Memory used [KB]: 1918
% 8.17/2.46 % (28506)Time elapsed: 0.010 s
% 8.17/2.46 % (28506)Instructions burned: 27 (million)
% 8.17/2.46 % (28506)------------------------------
% 8.17/2.46 % (28506)------------------------------
% 8.17/2.47 % (28507)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_28194 for (2990ds/53Mi)
% 8.17/2.49 % (28507)Instruction limit reached!
% 8.17/2.49 % (28507)------------------------------
% 8.17/2.49 % (28507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.17/2.49 % (28507)Termination reason: Unknown
% 8.17/2.49 % (28507)Termination phase: Property scanning
% 8.17/2.49
% 8.17/2.49 % (28507)Memory used [KB]: 2174
% 8.17/2.49 % (28507)Time elapsed: 0.018 s
% 8.17/2.49 % (28507)Instructions burned: 55 (million)
% 8.17/2.49 % (28507)------------------------------
% 8.17/2.49 % (28507)------------------------------
% 8.62/2.50 % (28508)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_28194 for (2990ds/4436Mi)
% 8.62/2.50 % (28505)Instruction limit reached!
% 8.62/2.50 % (28505)------------------------------
% 8.62/2.50 % (28505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.62/2.50 % (28505)Termination reason: Unknown
% 8.62/2.50 % (28505)Termination phase: Saturation
% 8.62/2.50
% 8.62/2.50 % (28505)Memory used [KB]: 7164
% 8.62/2.50 % (28505)Time elapsed: 0.059 s
% 8.62/2.50 % (28505)Instructions burned: 204 (million)
% 8.62/2.50 % (28505)------------------------------
% 8.62/2.50 % (28505)------------------------------
% 8.62/2.51 % (28509)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_28194 for (2990ds/1539Mi)
% 8.62/2.52 % (28504)Instruction limit reached!
% 8.62/2.52 % (28504)------------------------------
% 8.62/2.52 % (28504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.62/2.52 % (28504)Termination reason: Unknown
% 8.62/2.52 % (28504)Termination phase: Saturation
% 8.62/2.52
% 8.62/2.52 % (28504)Memory used [KB]: 7675
% 8.62/2.52 % (28504)Time elapsed: 0.083 s
% 8.62/2.52 % (28504)Instructions burned: 269 (million)
% 8.62/2.52 % (28504)------------------------------
% 8.62/2.52 % (28504)------------------------------
% 8.62/2.52 % (28510)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_28194 for (2990ds/408Mi)
% 8.62/2.53 % (28502)Instruction limit reached!
% 8.62/2.53 % (28502)------------------------------
% 8.62/2.53 % (28502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.62/2.53 % (28502)Termination reason: Unknown
% 8.62/2.53 % (28502)Termination phase: Saturation
% 8.62/2.53
% 8.62/2.53 % (28502)Memory used [KB]: 8059
% 8.62/2.53 % (28502)Time elapsed: 0.103 s
% 8.62/2.53 % (28502)Instructions burned: 370 (million)
% 8.62/2.53 % (28502)------------------------------
% 8.62/2.53 % (28502)------------------------------
% 8.62/2.54 % (28511)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_28194 for (2990ds/5205Mi)
% 8.88/2.61 % (28491)Instruction limit reached!
% 8.88/2.61 % (28491)------------------------------
% 8.88/2.61 % (28491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.88/2.61 % (28491)Termination reason: Unknown
% 8.88/2.61 % (28491)Termination phase: Saturation
% 8.88/2.61
% 8.88/2.61 % (28491)Memory used [KB]: 9978
% 8.88/2.61 % (28491)Time elapsed: 0.310 s
% 8.88/2.61 % (28491)Instructions burned: 987 (million)
% 8.88/2.61 % (28491)------------------------------
% 8.88/2.61 % (28491)------------------------------
% 8.88/2.62 % (28512)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_28194 for (2989ds/1913Mi)
% 8.88/2.65 % (28510)Instruction limit reached!
% 8.88/2.65 % (28510)------------------------------
% 8.88/2.65 % (28510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.88/2.65 % (28510)Termination reason: Unknown
% 8.88/2.65 % (28510)Termination phase: Saturation
% 8.88/2.65
% 8.88/2.65 % (28510)Memory used [KB]: 8315
% 8.88/2.65 % (28510)Time elapsed: 0.121 s
% 8.88/2.65 % (28510)Instructions burned: 408 (million)
% 8.88/2.65 % (28510)------------------------------
% 8.88/2.65 % (28510)------------------------------
% 8.88/2.65 % (28513)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_28194 for (2989ds/4350Mi)
% 8.88/2.67 % (28498)Instruction limit reached!
% 8.88/2.67 % (28498)------------------------------
% 8.88/2.67 % (28498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.88/2.67 % (28498)Termination reason: Unknown
% 8.88/2.67 % (28498)Termination phase: Saturation
% 8.88/2.67
% 8.88/2.67 % (28498)Memory used [KB]: 7931
% 8.88/2.67 % (28498)Time elapsed: 0.296 s
% 8.88/2.67 % (28498)Instructions burned: 922 (million)
% 8.88/2.67 % (28498)------------------------------
% 8.88/2.67 % (28498)------------------------------
% 8.88/2.68 % (28514)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=68:si=on:rtra=on_0 on DTF2THF_28194 for (2988ds/68Mi)
% 9.64/2.70 % (28514)Instruction limit reached!
% 9.64/2.70 % (28514)------------------------------
% 9.64/2.70 % (28514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.64/2.70 % (28514)Termination reason: Unknown
% 9.64/2.70 % (28514)Termination phase: Property scanning
% 9.64/2.70
% 9.64/2.70 % (28514)Memory used [KB]: 2302
% 9.64/2.70 % (28514)Time elapsed: 0.023 s
% 9.64/2.70 % (28514)Instructions burned: 71 (million)
% 9.64/2.70 % (28514)------------------------------
% 9.64/2.70 % (28514)------------------------------
% 9.64/2.71 % (28515)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_28194 for (2988ds/358Mi)
% 9.64/2.74 % (28503)Instruction limit reached!
% 9.64/2.74 % (28503)------------------------------
% 9.64/2.74 % (28503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.64/2.74 % (28503)Termination reason: Unknown
% 9.64/2.74 % (28503)Termination phase: Saturation
% 9.64/2.74
% 9.64/2.74 % (28503)Memory used [KB]: 10106
% 9.64/2.74 % (28503)Time elapsed: 0.309 s
% 9.64/2.74 % (28503)Instructions burned: 1023 (million)
% 9.64/2.74 % (28503)------------------------------
% 9.64/2.74 % (28503)------------------------------
% 9.64/2.75 % (28516)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_28194 for (2988ds/31Mi)
% 9.64/2.76 % (28516)Instruction limit reached!
% 9.64/2.76 % (28516)------------------------------
% 9.64/2.76 % (28516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.64/2.76 % (28516)Termination reason: Unknown
% 9.64/2.76 % (28516)Termination phase: Property scanning
% 9.64/2.76
% 9.64/2.76 % (28516)Memory used [KB]: 1791
% 9.64/2.76 % (28516)Time elapsed: 0.012 s
% 9.64/2.76 % (28516)Instructions burned: 34 (million)
% 9.64/2.76 % (28516)------------------------------
% 9.64/2.76 % (28516)------------------------------
% 9.64/2.77 % (28517)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_28194 for (2988ds/279Mi)
% 10.01/2.81 % (28517)Refutation not found, incomplete strategy
% 10.01/2.81 % (28517)------------------------------
% 10.01/2.81 % (28517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.01/2.81 % (28517)Termination reason: Refutation not found, incomplete strategy
% 10.01/2.81
% 10.01/2.81
% 10.01/2.81 % (28517)Memory used [KB]: 7164
% 10.01/2.81 % (28517)Time elapsed: 0.065 s
% 10.01/2.81 % (28517)Instructions burned: 153 (million)
% 10.01/2.81 % (28517)------------------------------
% 10.01/2.81 % (28517)------------------------------
% 10.01/2.81 % (28515)Instruction limit reached!
% 10.01/2.81 % (28515)------------------------------
% 10.01/2.81 % (28515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.01/2.81 % (28515)Termination reason: Unknown
% 10.01/2.81 % (28515)Termination phase: Saturation
% 10.01/2.81
% 10.01/2.81 % (28515)Memory used [KB]: 7931
% 10.01/2.81 % (28515)Time elapsed: 0.102 s
% 10.01/2.81 % (28515)Instructions burned: 361 (million)
% 10.01/2.81 % (28515)------------------------------
% 10.01/2.81 % (28515)------------------------------
% 10.01/2.82 % (28518)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_28194 for (2987ds/258Mi)
% 10.01/2.82 % (28519)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_28194 for (2987ds/26Mi)
% 10.01/2.83 % (28519)Instruction limit reached!
% 10.01/2.83 % (28519)------------------------------
% 10.01/2.83 % (28519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.01/2.83 % (28519)Termination reason: Unknown
% 10.01/2.83 % (28519)Termination phase: shuffling
% 10.01/2.83
% 10.01/2.83 % (28519)Memory used [KB]: 2046
% 10.01/2.83 % (28519)Time elapsed: 0.009 s
% 10.01/2.83 % (28519)Instructions burned: 29 (million)
% 10.01/2.83 % (28519)------------------------------
% 10.01/2.83 % (28519)------------------------------
% 10.01/2.84 % (28520)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=995:si=on:rtra=on_0 on DTF2THF_28194 for (2987ds/995Mi)
% 10.19/2.90 % (28518)Instruction limit reached!
% 10.19/2.90 % (28518)------------------------------
% 10.19/2.90 % (28518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.19/2.90 % (28518)Termination reason: Unknown
% 10.19/2.90 % (28518)Termination phase: Saturation
% 10.19/2.90
% 10.19/2.90 % (28518)Memory used [KB]: 7675
% 10.19/2.90 % (28518)Time elapsed: 0.082 s
% 10.19/2.90 % (28518)Instructions burned: 259 (million)
% 10.19/2.90 % (28518)------------------------------
% 10.19/2.90 % (28518)------------------------------
% 10.19/2.91 % (28521)dis+10_1:8_amm=sco:cbe=off:hflw=5:hfsq=on:hfsqc=14:hfsql=off:hfsqr=4,1:hud=5:ile=on:lma=on:nm=2:ntd=on:prag=on:sac=on:sp=frequency:i=373:si=on:rtra=on_0 on DTF2THF_28194 for (2986ds/373Mi)
% 10.19/2.98 % (28509)Instruction limit reached!
% 10.19/2.98 % (28509)------------------------------
% 10.19/2.98 % (28509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.19/2.98 % (28509)Termination reason: Unknown
% 10.19/2.98 % (28509)Termination phase: Saturation
% 10.19/2.98
% 10.19/2.98 % (28509)Memory used [KB]: 13304
% 10.19/2.98 % (28509)Time elapsed: 0.465 s
% 10.19/2.98 % (28509)Instructions burned: 1539 (million)
% 10.19/2.98 % (28509)------------------------------
% 10.19/2.98 % (28509)------------------------------
% 10.19/2.99 % (28522)lrs+1010_1:128_au=on:cnfonf=lazy_pi_sigma_gen:hflw=4:hfsq=on:hfsqc=20:hfsql=off:i=253:si=on:rtra=on_0 on DTF2THF_28194 for (2985ds/253Mi)
% 10.19/3.01 % (28521)Instruction limit reached!
% 10.19/3.01 % (28521)------------------------------
% 10.19/3.01 % (28521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.19/3.01 % (28521)Termination reason: Unknown
% 10.19/3.01 % (28521)Termination phase: Saturation
% 10.19/3.01
% 10.19/3.01 % (28521)Memory used [KB]: 8059
% 10.19/3.01 % (28521)Time elapsed: 0.104 s
% 10.19/3.01 % (28521)Instructions burned: 376 (million)
% 10.19/3.01 % (28521)------------------------------
% 10.19/3.01 % (28521)------------------------------
% 12.88/3.02 % (28523)lrs+1002_1:14_au=on:bd=off:e2e=on:fde=unused:sac=on:sos=on:i=832:si=on:rtra=on_0 on DTF2THF_28194 for (2985ds/832Mi)
% 12.88/3.06 % (28522)Instruction limit reached!
% 12.88/3.06 % (28522)------------------------------
% 12.88/3.06 % (28522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.88/3.06 % (28522)Termination reason: Unknown
% 12.88/3.06 % (28522)Termination phase: Saturation
% 12.88/3.06
% 12.88/3.06 % (28522)Memory used [KB]: 7675
% 12.88/3.06 % (28522)Time elapsed: 0.073 s
% 12.88/3.06 % (28522)Instructions burned: 253 (million)
% 12.88/3.06 % (28522)------------------------------
% 12.88/3.06 % (28522)------------------------------
% 12.88/3.07 % (28524)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=668:si=on:rtra=on_0 on DTF2THF_28194 for (2985ds/668Mi)
% 13.42/3.14 % (28520)Instruction limit reached!
% 13.42/3.14 % (28520)------------------------------
% 13.42/3.14 % (28520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.42/3.14 % (28520)Termination reason: Unknown
% 13.42/3.14 % (28520)Termination phase: Saturation
% 13.42/3.14
% 13.42/3.14 % (28520)Memory used [KB]: 9466
% 13.42/3.14 % (28520)Time elapsed: 0.300 s
% 13.42/3.14 % (28520)Instructions burned: 995 (million)
% 13.42/3.14 % (28520)------------------------------
% 13.42/3.14 % (28520)------------------------------
% 13.42/3.15 % (28525)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=40:si=on:rtra=on_0 on DTF2THF_28194 for (2984ds/40Mi)
% 13.42/3.16 % (28525)Instruction limit reached!
% 13.42/3.16 % (28525)------------------------------
% 13.42/3.16 % (28525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.42/3.16 % (28525)Termination reason: Unknown
% 13.42/3.16 % (28525)Termination phase: shuffling
% 13.42/3.16
% 13.42/3.16 % (28525)Memory used [KB]: 2174
% 13.42/3.16 % (28525)Time elapsed: 0.013 s
% 13.42/3.16 % (28525)Instructions burned: 41 (million)
% 13.42/3.16 % (28525)------------------------------
% 13.42/3.16 % (28525)------------------------------
% 13.42/3.17 % (28526)lrs+1010_1:1_cnfonf=lazy_pi_sigma_gen:fd=off:ntd=on:sd=1:sgt=32:sos=on:ss=axioms:i=294:si=on:rtra=on_0 on DTF2THF_28194 for (2983ds/294Mi)
% 13.42/3.20 % (28526)Refutation not found, incomplete strategy
% 13.42/3.20 % (28526)------------------------------
% 13.42/3.20 % (28526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.42/3.20 % (28526)Termination reason: Refutation not found, incomplete strategy
% 13.42/3.20
% 13.42/3.20
% 13.42/3.20 % (28526)Memory used [KB]: 6780
% 13.42/3.20 % (28526)Time elapsed: 0.027 s
% 13.42/3.20 % (28526)Instructions burned: 96 (million)
% 13.42/3.20 % (28526)------------------------------
% 13.42/3.20 % (28526)------------------------------
% 13.80/3.20 % (28512)Instruction limit reached!
% 13.80/3.20 % (28512)------------------------------
% 13.80/3.20 % (28512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.80/3.20 % (28512)Termination reason: Unknown
% 13.80/3.20 % (28512)Termination phase: Saturation
% 13.80/3.20
% 13.80/3.20 % (28512)Memory used [KB]: 8059
% 13.80/3.20 % (28512)Time elapsed: 0.601 s
% 13.80/3.20 % (28512)Instructions burned: 1913 (million)
% 13.80/3.20 % (28512)------------------------------
% 13.80/3.20 % (28512)------------------------------
% 13.80/3.21 % (28527)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:plsq=on:plsqr=32,1:prag=on:i=187:si=on:rtra=on_0 on DTF2THF_28194 for (2983ds/187Mi)
% 13.80/3.21 % (28528)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_28194 for (2983ds/371Mi)
% 13.80/3.26 % (28527)Instruction limit reached!
% 13.80/3.26 % (28527)------------------------------
% 13.80/3.26 % (28527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.80/3.26 % (28527)Termination reason: Unknown
% 13.80/3.26 % (28527)Termination phase: Saturation
% 13.80/3.26
% 13.80/3.26 % (28527)Memory used [KB]: 7164
% 13.80/3.26 % (28527)Time elapsed: 0.055 s
% 13.80/3.26 % (28527)Instructions burned: 188 (million)
% 13.80/3.26 % (28527)------------------------------
% 13.80/3.26 % (28527)------------------------------
% 13.80/3.27 % (28529)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_28194 for (2982ds/1827Mi)
% 13.80/3.27 % (28524)Instruction limit reached!
% 13.80/3.27 % (28524)------------------------------
% 13.80/3.27 % (28524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.80/3.27 % (28524)Termination reason: Unknown
% 13.80/3.27 % (28524)Termination phase: Saturation
% 13.80/3.27
% 13.80/3.27 % (28524)Memory used [KB]: 13816
% 13.80/3.27 % (28524)Time elapsed: 0.203 s
% 13.80/3.27 % (28524)Instructions burned: 670 (million)
% 13.80/3.27 % (28524)------------------------------
% 13.80/3.27 % (28524)------------------------------
% 13.80/3.28 % (28530)lrs+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:tnu=1:i=70:si=on:rtra=on_0 on DTF2THF_28194 for (2982ds/70Mi)
% 13.80/3.28 % (28523)Instruction limit reached!
% 13.80/3.28 % (28523)------------------------------
% 13.80/3.28 % (28523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.80/3.28 % (28523)Termination reason: Unknown
% 13.80/3.28 % (28523)Termination phase: Saturation
% 13.80/3.28
% 13.80/3.28 % (28523)Memory used [KB]: 8187
% 13.80/3.28 % (28523)Time elapsed: 0.281 s
% 13.80/3.28 % (28523)Instructions burned: 833 (million)
% 13.80/3.28 % (28523)------------------------------
% 13.80/3.28 % (28523)------------------------------
% 13.80/3.29 % (28531)lrs+10_1:20_bd=off:cnfonf=lazy_not_be_gen:fe=axiom:hud=10:ins=1:nm=20:prag=on:sos=on:i=316:si=on:rtra=on_0 on DTF2THF_28194 for (2982ds/316Mi)
% 14.66/3.30 % (28530)Instruction limit reached!
% 14.66/3.30 % (28530)------------------------------
% 14.66/3.30 % (28530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.66/3.30 % (28530)Termination reason: Unknown
% 14.66/3.30 % (28530)Termination phase: Preprocessing 1
% 14.66/3.30
% 14.66/3.30 % (28530)Memory used [KB]: 2174
% 14.66/3.30 % (28530)Time elapsed: 0.020 s
% 14.66/3.30 % (28530)Instructions burned: 71 (million)
% 14.66/3.30 % (28530)------------------------------
% 14.66/3.30 % (28530)------------------------------
% 14.66/3.31 % (28532)ott+1010_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=3548551,65536:s2a=on:i=932:si=on:rtra=on_0 on DTF2THF_28194 for (2982ds/932Mi)
% 14.66/3.32 % (28528)Instruction limit reached!
% 14.66/3.32 % (28528)------------------------------
% 14.66/3.32 % (28528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.66/3.32 % (28528)Termination reason: Unknown
% 14.66/3.32 % (28528)Termination phase: Saturation
% 14.66/3.32
% 14.66/3.32 % (28528)Memory used [KB]: 8187
% 14.66/3.32 % (28528)Time elapsed: 0.118 s
% 14.66/3.32 % (28528)Instructions burned: 372 (million)
% 14.66/3.32 % (28528)------------------------------
% 14.66/3.32 % (28528)------------------------------
% 14.66/3.33 % (28533)lrs+1010_1:128_au=on:bd=preordered:cnfonf=lazy_pi_sigma_gen:plsq=on:plsqc=1:plsqr=32,1:sos=all:sp=occurrence:i=4714:si=on:rtra=on_0 on DTF2THF_28194 for (2982ds/4714Mi)
% 14.66/3.35 % (28531)Refutation not found, incomplete strategy
% 14.66/3.35 % (28531)------------------------------
% 14.66/3.35 % (28531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.66/3.35 % (28531)Termination reason: Refutation not found, incomplete strategy
% 14.66/3.35
% 14.66/3.35
% 14.66/3.35 % (28531)Memory used [KB]: 7547
% 14.66/3.35 % (28531)Time elapsed: 0.058 s
% 14.66/3.35 % (28531)Instructions burned: 206 (million)
% 14.66/3.35 % (28531)------------------------------
% 14.66/3.35 % (28531)------------------------------
% 14.66/3.36 % (28534)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=8250:si=on:rtra=on_0 on DTF2THF_28194 for (2981ds/8250Mi)
% 14.66/3.39 % (28533)Refutation not found, incomplete strategy
% 14.66/3.39 % (28533)------------------------------
% 14.66/3.39 % (28533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.66/3.39 % (28533)Termination reason: Refutation not found, incomplete strategy
% 14.66/3.39
% 14.66/3.39
% 14.66/3.39 % (28533)Memory used [KB]: 7675
% 14.66/3.39 % (28533)Time elapsed: 0.058 s
% 14.66/3.39 % (28533)Instructions burned: 207 (million)
% 14.66/3.39 % (28533)------------------------------
% 14.66/3.39 % (28533)------------------------------
% 15.24/3.40 % (28536)lrs+1002_1:1_apa=on:cnfonf=off:fd=off:flr=on:hud=1:lcm=reverse:prag=on:sd=1:sos=on:ss=axioms:i=770:si=on:rtra=on_0 on DTF2THF_28194 for (2981ds/770Mi)
% 15.55/3.58 % (28532)Instruction limit reached!
% 15.55/3.58 % (28532)------------------------------
% 15.55/3.58 % (28532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.55/3.58 % (28532)Termination reason: Unknown
% 15.55/3.58 % (28532)Termination phase: Saturation
% 15.55/3.58
% 15.55/3.58 % (28532)Memory used [KB]: 9978
% 15.55/3.58 % (28532)Time elapsed: 0.271 s
% 15.55/3.58 % (28532)Instructions burned: 934 (million)
% 15.55/3.58 % (28532)------------------------------
% 15.55/3.58 % (28532)------------------------------
% 15.55/3.59 % (28537)ott+1010_5:1_av=off:fe=off:ins=1:kws=precedence:sp=reverse_arity:i=794:si=on:rtra=on_0 on DTF2THF_28194 for (2979ds/794Mi)
% 16.13/3.62 % (28536)Instruction limit reached!
% 16.13/3.62 % (28536)------------------------------
% 16.13/3.62 % (28536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.13/3.62 % (28536)Termination reason: Unknown
% 16.13/3.62 % (28536)Termination phase: Saturation
% 16.13/3.62
% 16.13/3.62 % (28536)Memory used [KB]: 7164
% 16.13/3.62 % (28536)Time elapsed: 0.220 s
% 16.13/3.62 % (28536)Instructions burned: 772 (million)
% 16.13/3.62 % (28536)------------------------------
% 16.13/3.62 % (28536)------------------------------
% 16.13/3.63 % (28538)dis+1002_1:1_au=on:avsq=on:avsqr=1,16:cond=on:sp=frequency:i=464:si=on:rtra=on_0 on DTF2THF_28194 for (2979ds/464Mi)
% 16.66/3.76 % (28538)Instruction limit reached!
% 16.66/3.76 % (28538)------------------------------
% 16.66/3.76 % (28538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.66/3.76 % (28538)Termination reason: Unknown
% 16.66/3.76 % (28538)Termination phase: Saturation
% 16.66/3.76
% 16.66/3.76 % (28538)Memory used [KB]: 7675
% 16.66/3.76 % (28538)Time elapsed: 0.135 s
% 16.66/3.76 % (28538)Instructions burned: 465 (million)
% 16.66/3.76 % (28538)------------------------------
% 16.66/3.76 % (28538)------------------------------
% 16.66/3.77 % (28539)ott+21_1:3_avsq=on:bd=off:cnfonf=lazy_gen:fe=off:hud=10:ixr=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=1376:si=on:rtra=on_0 on DTF2THF_28194 for (2977ds/1376Mi)
% 17.12/3.83 % (28537)Instruction limit reached!
% 17.12/3.83 % (28537)------------------------------
% 17.12/3.83 % (28537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.12/3.83 % (28537)Termination reason: Unknown
% 17.12/3.83 % (28537)Termination phase: Saturation
% 17.12/3.83
% 17.12/3.83 % (28537)Memory used [KB]: 5373
% 17.12/3.83 % (28537)Time elapsed: 0.242 s
% 17.12/3.83 % (28537)Instructions burned: 795 (million)
% 17.12/3.83 % (28537)------------------------------
% 17.12/3.83 % (28537)------------------------------
% 17.12/3.84 % (28540)dis+10_1:1_etr=on:plsq=on:plsqc=1:plsqr=32,1:i=906:si=on:rtra=on_0 on DTF2THF_28194 for (2977ds/906Mi)
% 17.12/3.85 % (28508)Instruction limit reached!
% 17.12/3.85 % (28508)------------------------------
% 17.12/3.85 % (28508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.12/3.85 % (28508)Termination reason: Unknown
% 17.12/3.85 % (28508)Termination phase: Saturation
% 17.12/3.85
% 17.12/3.85 % (28508)Memory used [KB]: 10490
% 17.12/3.85 % (28508)Time elapsed: 1.352 s
% 17.12/3.85 % (28508)Instructions burned: 4439 (million)
% 17.12/3.85 % (28508)------------------------------
% 17.12/3.85 % (28508)------------------------------
% 17.12/3.85 % (28529)Instruction limit reached!
% 17.12/3.85 % (28529)------------------------------
% 17.12/3.85 % (28529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.12/3.85 % (28529)Termination reason: Unknown
% 17.12/3.85 % (28529)Termination phase: Saturation
% 17.12/3.85
% 17.12/3.85 % (28529)Memory used [KB]: 11513
% 17.12/3.85 % (28529)Time elapsed: 0.584 s
% 17.12/3.85 % (28529)Instructions burned: 1827 (million)
% 17.12/3.85 % (28529)------------------------------
% 17.12/3.85 % (28529)------------------------------
% 17.12/3.85 % (28513)Instruction limit reached!
% 17.12/3.85 % (28513)------------------------------
% 17.12/3.85 % (28513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.12/3.85 % (28513)Termination reason: Unknown
% 17.12/3.85 % (28513)Termination phase: Saturation
% 17.12/3.85
% 17.12/3.85 % (28513)Memory used [KB]: 9083
% 17.12/3.85 % (28513)Time elapsed: 1.220 s
% 17.12/3.85 % (28513)Instructions burned: 4353 (million)
% 17.12/3.85 % (28513)------------------------------
% 17.12/3.85 % (28513)------------------------------
% 17.12/3.86 % (28541)lrs+1002_1:16_bd=off:e2e=on:fde=unused:kws=precedence:sac=on:sos=on:ss=axioms:i=434:si=on:rtra=on_0 on DTF2THF_28194 for (2977ds/434Mi)
% 17.12/3.86 % (28542)lrs+10_5:4_cnfonf=conj_eager:fe=axiom:hflw=5:hfsq=on:hfsqc=10:hfsql=off:hud=10:plsq=on:sgt=5:spb=goal_then_units:ss=axioms:i=865:si=on:rtra=on_0 on DTF2THF_28194 for (2976ds/865Mi)
% 17.12/3.86 % (28543)dis+1002_1:1_cnfonf=lazy_simp:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=593:si=on:rtra=on_0 on DTF2THF_28194 for (2976ds/593Mi)
% 18.19/3.99 % (28541)Instruction limit reached!
% 18.19/3.99 % (28541)------------------------------
% 18.19/3.99 % (28541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.19/3.99 % (28541)Termination reason: Unknown
% 18.19/3.99 % (28541)Termination phase: Saturation
% 18.19/3.99
% 18.19/3.99 % (28541)Memory used [KB]: 7803
% 18.19/3.99 % (28541)Time elapsed: 0.133 s
% 18.19/3.99 % (28541)Instructions burned: 434 (million)
% 18.19/3.99 % (28541)------------------------------
% 18.19/3.99 % (28541)------------------------------
% 18.19/4.00 % (28544)dis+10_1:1_sd=2:sgt=20:slsq=on:ss=axioms:st=6.0:i=443:si=on:rtra=on_0 on DTF2THF_28194 for (2975ds/443Mi)
% 19.88/4.04 % (28543)Instruction limit reached!
% 19.88/4.04 % (28543)------------------------------
% 19.88/4.04 % (28543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.88/4.04 % (28543)Termination reason: Unknown
% 19.88/4.04 % (28543)Termination phase: Saturation
% 19.88/4.04
% 19.88/4.04 % (28543)Memory used [KB]: 9594
% 19.88/4.04 % (28543)Time elapsed: 0.182 s
% 19.88/4.04 % (28543)Instructions burned: 595 (million)
% 19.88/4.04 % (28543)------------------------------
% 19.88/4.04 % (28543)------------------------------
% 19.88/4.05 % (28545)dis+10_1:1_atotf=0.1:cbe=off:sd=3:ss=axioms:st=2.0:i=594:si=on:rtra=on_0 on DTF2THF_28194 for (2975ds/594Mi)
% 19.88/4.12 % (28511)Instruction limit reached!
% 19.88/4.12 % (28511)------------------------------
% 19.88/4.12 % (28511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.88/4.12 % (28511)Termination reason: Unknown
% 19.88/4.12 % (28511)Termination phase: Saturation
% 19.88/4.12
% 19.88/4.12 % (28511)Memory used [KB]: 18038
% 19.88/4.12 % (28511)Time elapsed: 1.607 s
% 19.88/4.12 % (28511)Instructions burned: 5207 (million)
% 19.88/4.12 % (28511)------------------------------
% 19.88/4.12 % (28511)------------------------------
% 19.88/4.12 % (28544)Instruction limit reached!
% 19.88/4.12 % (28544)------------------------------
% 19.88/4.12 % (28544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.88/4.12 % (28544)Termination reason: Unknown
% 19.88/4.12 % (28544)Termination phase: Saturation
% 19.88/4.12
% 19.88/4.12 % (28544)Memory used [KB]: 8187
% 19.88/4.12 % (28544)Time elapsed: 0.124 s
% 19.88/4.12 % (28544)Instructions burned: 445 (million)
% 19.88/4.12 % (28544)------------------------------
% 19.88/4.12 % (28544)------------------------------
% 20.43/4.13 % (28542)Instruction limit reached!
% 20.43/4.13 % (28542)------------------------------
% 20.43/4.13 % (28542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.43/4.13 % (28542)Termination reason: Unknown
% 20.43/4.13 % (28542)Termination phase: Saturation
% 20.43/4.13
% 20.43/4.13 % (28542)Memory used [KB]: 9083
% 20.43/4.13 % (28542)Time elapsed: 0.265 s
% 20.43/4.13 % (28542)Instructions burned: 868 (million)
% 20.43/4.13 % (28542)------------------------------
% 20.43/4.13 % (28542)------------------------------
% 20.43/4.13 % (28540)Instruction limit reached!
% 20.43/4.13 % (28540)------------------------------
% 20.43/4.13 % (28540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.43/4.13 % (28540)Termination reason: Unknown
% 20.43/4.13 % (28540)Termination phase: Saturation
% 20.43/4.13
% 20.43/4.13 % (28540)Memory used [KB]: 12920
% 20.43/4.13 % (28540)Time elapsed: 0.293 s
% 20.43/4.13 % (28540)Instructions burned: 906 (million)
% 20.43/4.13 % (28540)------------------------------
% 20.43/4.13 % (28540)------------------------------
% 20.43/4.13 % (28546)ott+10_1:1024_anc=none:bd=off:e2e=on:fde=none:fsr=off:hud=10:ixr=off:nm=4:nwc=5.0:spb=goal:i=327:si=on:rtra=on_0 on DTF2THF_28194 for (2974ds/327Mi)
% 20.43/4.13 % (28547)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=5411:si=on:rtra=on_0 on DTF2THF_28194 for (2974ds/5411Mi)
% 20.43/4.14 % (28548)lrs+10_1:20_au=on:c=on:cbe=off:cnfonf=lazy_gen:ins=1:plsq=on:plsqc=1:plsqr=46,31:sac=on:i=1762:si=on:rtra=on_0 on DTF2THF_28194 for (2974ds/1762Mi)
% 20.43/4.14 % (28549)dis+1003_1:64_cha=on:cnfonf=lazy_gen:hud=9:nwc=8.0:prag=on:i=96:si=on:rtra=on_0 on DTF2THF_28194 for (2974ds/96Mi)
% 20.43/4.17 % (28549)Instruction limit reached!
% 20.43/4.17 % (28549)------------------------------
% 20.43/4.17 % (28549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.43/4.17 % (28549)Termination reason: Unknown
% 20.43/4.17 % (28549)Termination phase: Property scanning
% 20.43/4.17
% 20.43/4.17 % (28549)Memory used [KB]: 2302
% 20.43/4.17 % (28549)Time elapsed: 0.028 s
% 20.43/4.17 % (28549)Instructions burned: 96 (million)
% 20.43/4.17 % (28549)------------------------------
% 20.43/4.17 % (28549)------------------------------
% 20.43/4.18 % (28550)lrs+21_1:1_hud=5:kws=inv_frequency:nm=16:nwc=2.0:prag=on:sos=on:ss=axioms:i=503:si=on:rtra=on_0 on DTF2THF_28194 for (2973ds/503Mi)
% 20.76/4.20 % (28539)Instruction limit reached!
% 20.76/4.20 % (28539)------------------------------
% 20.76/4.20 % (28539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.76/4.20 % (28539)Termination reason: Unknown
% 20.76/4.20 % (28539)Termination phase: Saturation
% 20.76/4.20
% 20.76/4.20 % (28539)Memory used [KB]: 15863
% 20.76/4.20 % (28539)Time elapsed: 0.428 s
% 20.76/4.20 % (28539)Instructions burned: 1378 (million)
% 20.76/4.20 % (28539)------------------------------
% 20.76/4.20 % (28539)------------------------------
% 20.76/4.21 % (28551)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_28194 for (2973ds/90Mi)
% 20.76/4.24 % (28546)Instruction limit reached!
% 20.76/4.24 % (28546)------------------------------
% 20.76/4.24 % (28546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.76/4.24 % (28546)Termination reason: Unknown
% 20.76/4.24 % (28546)Termination phase: Saturation
% 20.76/4.24
% 20.76/4.24 % (28546)Memory used [KB]: 7803
% 20.76/4.24 % (28546)Time elapsed: 0.106 s
% 20.76/4.24 % (28546)Instructions burned: 327 (million)
% 20.76/4.24 % (28546)------------------------------
% 20.76/4.24 % (28546)------------------------------
% 20.76/4.24 % (28551)Instruction limit reached!
% 20.76/4.24 % (28551)------------------------------
% 20.76/4.24 % (28551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.76/4.24 % (28551)Termination reason: Unknown
% 20.76/4.24 % (28551)Termination phase: Property scanning
% 20.76/4.24
% 20.76/4.24 % (28551)Memory used [KB]: 2302
% 20.76/4.24 % (28551)Time elapsed: 0.029 s
% 20.76/4.24 % (28551)Instructions burned: 91 (million)
% 20.76/4.24 % (28551)------------------------------
% 20.76/4.24 % (28551)------------------------------
% 20.76/4.24 % (28545)Instruction limit reached!
% 20.76/4.24 % (28545)------------------------------
% 20.76/4.24 % (28545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.76/4.24 % (28545)Termination reason: Unknown
% 20.76/4.24 % (28545)Termination phase: Saturation
% 20.76/4.24
% 20.76/4.24 % (28545)Memory used [KB]: 8443
% 20.76/4.24 % (28545)Time elapsed: 0.192 s
% 20.76/4.24 % (28545)Instructions burned: 597 (million)
% 20.76/4.24 % (28545)------------------------------
% 20.76/4.24 % (28545)------------------------------
% 20.76/4.25 % (28552)lrs+10_1:64_hud=5:plsq=on:plsqr=2011171,524288:i=593:si=on:rtra=on_0 on DTF2THF_28194 for (2973ds/593Mi)
% 20.76/4.25 % (28553)dis-1002_1:1_alpa=false:apa=on:au=on:bd=off:cnfonf=off:e2e=on:er=filter:fde=unused:plsq=on:plsqr=9699663,1048576:s2a=on:sos=on:sp=unary_first:i=1512:si=on:rtra=on_0 on DTF2THF_28194 for (2973ds/1512Mi)
% 20.76/4.25 % (28554)lrs+1002_1:128_cbe=off:cnfonf=lazy_not_gen_be_off:ins=1:i=569:si=on:rtra=on_0 on DTF2THF_28194 for (2973ds/569Mi)
% 21.23/4.33 % (28550)Instruction limit reached!
% 21.23/4.33 % (28550)------------------------------
% 21.23/4.33 % (28550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.23/4.33 % (28550)Termination reason: Unknown
% 21.23/4.33 % (28550)Termination phase: Saturation
% 21.23/4.33
% 21.23/4.33 % (28550)Memory used [KB]: 8443
% 21.23/4.33 % (28550)Time elapsed: 0.153 s
% 21.23/4.33 % (28550)Instructions burned: 503 (million)
% 21.23/4.33 % (28550)------------------------------
% 21.23/4.33 % (28550)------------------------------
% 21.23/4.34 % (28534)First to succeed.
% 21.23/4.34 % (28555)ott+1002_1:1_bd=off:cnfonf=off:e2e=on:fd=off:fe=axiom:hud=5:prag=on:sos=on:i=501:si=on:rtra=on_0 on DTF2THF_28194 for (2972ds/501Mi)
% 21.23/4.34 % (28534)Refutation found. Thanks to Tanya!
% 21.23/4.34 % SZS status Theorem for DTF2THF_28194
% 21.23/4.34 % SZS output start Proof for DTF2THF_28194
% 21.23/4.34 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 21.23/4.34 thf(type_def_6, type, set: ($tType) > $tType).
% 21.23/4.34 thf(type_def_8, type, nat: $tType).
% 21.23/4.34 thf(type_def_9, type, a: $tType).
% 21.23/4.34 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 21.23/4.34 thf(func_def_1, type, set: ($tType) > $tType).
% 21.23/4.34 thf(func_def_2, type, nat: $tType).
% 21.23/4.34 thf(func_def_3, type, a: $tType).
% 21.23/4.34 thf(func_def_4, type, type: !>[X0: $tType]:($o)).
% 21.23/4.34 thf(func_def_5, type, zero: !>[X0: $tType]:($o)).
% 21.23/4.34 thf(func_def_6, type, bot: !>[X0: $tType]:($o)).
% 21.23/4.34 thf(func_def_7, type, ord: !>[X0: $tType]:($o)).
% 21.23/4.34 thf(func_def_8, type, order: !>[X0: $tType]:($o)).
% 21.23/4.34 thf(func_def_9, type, finite_finite: !>[X0: $tType]:($o)).
% 21.23/4.34 thf(func_def_10, type, linorder: !>[X0: $tType]:($o)).
% 21.23/4.34 thf(func_def_11, type, preorder: !>[X0: $tType]:($o)).
% 21.23/4.34 thf(func_def_12, type, order_bot: !>[X0: $tType]:($o)).
% 21.23/4.34 thf(func_def_13, type, dense_linorder: !>[X0: $tType]:($o)).
% 21.23/4.34 thf(func_def_14, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 21.23/4.34 thf(func_def_15, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 21.23/4.34 thf(func_def_16, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 21.23/4.34 thf(func_def_17, type, finite_card: !>[X0: $tType]:(set @ X0 > nat)).
% 21.23/4.34 thf(func_def_18, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 21.23/4.34 thf(func_def_19, type, zero_zero: !>[X0: $tType]:(X0)).
% 21.23/4.34 thf(func_def_20, type, order_preorder_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 21.23/4.34 thf(func_def_21, type, bot_bot: !>[X0: $tType]:(X0)).
% 21.23/4.34 thf(func_def_22, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 21.23/4.34 thf(func_def_23, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 21.23/4.34 thf(func_def_24, type, prefer951318096ood_as: !>[X0: $tType]:(X0 > set @ X0 > set @ product_prod @ X0 @ X0 > set @ X0)).
% 21.23/4.34 thf(func_def_25, type, prefer310429814s_good: !>[X0: $tType]:(X0 > set @ X0 > set @ product_prod @ X0 @ X0 > set @ X0)).
% 21.23/4.34 thf(func_def_26, type, prefer1532642881r_than: !>[X0: $tType]:(X0 > set @ X0 > set @ product_prod @ X0 @ X0 > set @ X0)).
% 21.23/4.34 thf(func_def_27, type, prefer199794634erence: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 21.23/4.34 thf(func_def_28, type, prefer1997167224erence: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 21.23/4.34 thf(func_def_29, type, prefer1801827867axioms: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 21.23/4.34 thf(func_def_30, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 21.23/4.34 thf(func_def_31, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 21.23/4.34 thf(func_def_32, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 21.23/4.34 thf(func_def_33, type, refl_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 21.23/4.34 thf(func_def_34, type, total_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 21.23/4.34 thf(func_def_35, type, trans: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 21.23/4.34 thf(func_def_36, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 21.23/4.34 thf(func_def_37, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 21.23/4.34 thf(func_def_38, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 21.23/4.34 thf(func_def_39, type, carrier: set @ a).
% 21.23/4.34 thf(func_def_40, type, relation: set @ product_prod @ a @ a).
% 21.23/4.34 thf(func_def_41, type, x: a).
% 21.23/4.34 thf(func_def_42, type, y: a).
% 21.23/4.34 thf(func_def_58, type, sP0: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > (X1 > X0 > $o) > $o)).
% 21.23/4.34 thf(func_def_59, type, sP1: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 21.23/4.34 thf(func_def_60, type, sP2: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 21.23/4.34 thf(func_def_61, type, sP3: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 21.23/4.34 thf(func_def_62, type, sK4: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > (X1 > X0 > $o) > X1)).
% 21.23/4.34 thf(func_def_63, type, sK5: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > (X1 > X0 > $o) > X1)).
% 21.23/4.34 thf(func_def_64, type, sK6: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > (X1 > X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_65, type, sK7: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > set @ X0 > set @ X1 > X0)).
% 21.23/4.34 thf(func_def_66, type, sK8: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_67, type, sK9: !>[X0: $tType]:(set @ X0 > nat > set @ X0)).
% 21.23/4.34 thf(func_def_68, type, sK10: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > X0)).
% 21.23/4.34 thf(func_def_69, type, sK11: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > X0)).
% 21.23/4.34 thf(func_def_70, type, sK12: !>[X0: $tType]:(X0 > set @ X0 > X0)).
% 21.23/4.34 thf(func_def_71, type, sK13: !>[X0: $tType]:(set @ X0 > X0)).
% 21.23/4.34 thf(func_def_72, type, sK14: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 21.23/4.34 thf(func_def_73, type, sK15: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 21.23/4.34 thf(func_def_74, type, sK16: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 21.23/4.34 thf(func_def_75, type, sK17: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 21.23/4.34 thf(func_def_76, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 21.23/4.34 thf(func_def_77, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 21.23/4.34 thf(func_def_78, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 21.23/4.34 thf(func_def_79, type, sK21: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 21.23/4.34 thf(func_def_80, type, sK22: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_81, type, sK23: nat > (nat > $o) > nat).
% 21.23/4.34 thf(func_def_82, type, sK24: (nat > $o) > nat).
% 21.23/4.34 thf(func_def_83, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 21.23/4.34 thf(func_def_84, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 21.23/4.34 thf(func_def_85, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 21.23/4.34 thf(func_def_86, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 21.23/4.34 thf(func_def_87, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 21.23/4.34 thf(func_def_88, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 21.23/4.34 thf(func_def_89, type, sK31: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 21.23/4.34 thf(func_def_90, type, sK32: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 21.23/4.34 thf(func_def_91, type, sK33: nat > (nat > $o) > nat).
% 21.23/4.34 thf(func_def_92, type, sK34: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 21.23/4.34 thf(func_def_93, type, sK35: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 21.23/4.34 thf(func_def_94, type, sK36: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 21.23/4.34 thf(func_def_95, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 21.23/4.34 thf(func_def_96, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 21.23/4.34 thf(func_def_97, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 21.23/4.34 thf(func_def_98, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 21.23/4.34 thf(func_def_99, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 21.23/4.34 thf(func_def_100, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_101, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 21.23/4.34 thf(func_def_102, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 21.23/4.34 thf(func_def_103, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 21.23/4.34 thf(func_def_104, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 21.23/4.34 thf(func_def_105, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 21.23/4.34 thf(func_def_106, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 21.23/4.34 thf(func_def_107, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_108, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 21.23/4.34 thf(func_def_109, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 21.23/4.34 thf(func_def_110, type, sK52: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 21.23/4.34 thf(func_def_111, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 21.23/4.34 thf(func_def_112, type, sK54: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 21.23/4.34 thf(func_def_113, type, sK55: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 21.23/4.34 thf(func_def_114, type, sK56: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 21.23/4.34 thf(func_def_115, type, sK57: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 21.23/4.34 thf(func_def_116, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 21.23/4.34 thf(func_def_117, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 21.23/4.34 thf(func_def_118, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 21.23/4.34 thf(func_def_119, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 21.23/4.34 thf(func_def_120, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_121, type, sK63: !>[X0: $tType]:(set @ X0 > X0)).
% 21.23/4.34 thf(func_def_122, type, sK64: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 21.23/4.34 thf(func_def_123, type, sK65: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 21.23/4.34 thf(func_def_124, type, sK66: !>[X0: $tType]:(X0 > X0 > X0)).
% 21.23/4.34 thf(func_def_125, type, sK67: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 21.23/4.34 thf(func_def_126, type, sK68: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 21.23/4.34 thf(func_def_127, type, sK69: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 21.23/4.34 thf(func_def_128, type, sK70: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 21.23/4.34 thf(func_def_129, type, sK71: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 21.23/4.34 thf(func_def_130, type, sK72: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 21.23/4.34 thf(func_def_131, type, sK73: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 21.23/4.34 thf(func_def_132, type, sK74: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 21.23/4.34 thf(func_def_133, type, sK75: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_134, type, sK76: !>[X0: $tType]:(nat > set @ X0 > set @ X0)).
% 21.23/4.34 thf(func_def_135, type, sK77: set @ nat > nat).
% 21.23/4.34 thf(func_def_136, type, sK78: nat > set @ nat > nat).
% 21.23/4.34 thf(func_def_137, type, sK79: !>[X0: $tType]:(set @ X0 > X0)).
% 21.23/4.34 thf(func_def_138, type, sK80: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > set @ X0 > X0)).
% 21.23/4.34 thf(func_def_139, type, sK81: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > set @ X0 > X1)).
% 21.23/4.34 thf(func_def_140, type, sK82: !>[X0: $tType, X1: $tType]:(X0 > (X0 > X1 > $o) > set @ X0 > X1)).
% 21.23/4.34 thf(func_def_141, type, sK83: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_142, type, sK84: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_143, type, sK85: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_144, type, sK86: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > X0)).
% 21.23/4.34 thf(func_def_145, type, sK87: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_146, type, sK88: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0)).
% 21.23/4.34 thf(func_def_147, type, sK89: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0)).
% 21.23/4.34 thf(func_def_148, type, sK90: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0)).
% 21.23/4.34 thf(func_def_149, type, sK91: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 21.23/4.34 thf(func_def_150, type, sK92: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 21.23/4.34 thf(func_def_151, type, sK93: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 21.23/4.34 thf(func_def_152, type, sK94: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_153, type, sK95: !>[X0: $tType]:((X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_154, type, sK96: nat > (nat > $o) > nat).
% 21.23/4.34 thf(func_def_155, type, sK97: (nat > $o) > nat).
% 21.23/4.34 thf(func_def_156, type, sK98: !>[X0: $tType]:(set @ X0 > X0)).
% 21.23/4.34 thf(func_def_157, type, sK99: !>[X0: $tType]:(X0 > X0 > X0)).
% 21.23/4.34 thf(func_def_158, type, sK100: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_159, type, sK101: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1)).
% 21.23/4.34 thf(func_def_160, type, sK102: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1 > X0)).
% 21.23/4.34 thf(func_def_161, type, sK103: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 21.23/4.34 thf(func_def_162, type, sK104: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 21.23/4.34 thf(func_def_163, type, sK105: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 21.23/4.34 thf(func_def_164, type, sK106: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_165, type, sK107: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 21.23/4.34 thf(func_def_166, type, sK108: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 21.23/4.34 thf(func_def_167, type, sK109: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 21.23/4.34 thf(func_def_168, type, sK110: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 21.23/4.34 thf(func_def_169, type, sK111: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 21.23/4.34 thf(func_def_170, type, sK112: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 21.23/4.34 thf(func_def_171, type, sK113: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 21.23/4.34 thf(func_def_172, type, sK114: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > X0)).
% 21.23/4.34 thf(func_def_173, type, sK115: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > X0)).
% 21.23/4.34 thf(func_def_174, type, sK116: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > X0)).
% 21.23/4.34 thf(func_def_175, type, sK117: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > X0)).
% 21.23/4.34 thf(func_def_176, type, sK118: !>[X0: $tType]:(X0 > set @ X0 > X0)).
% 21.23/4.34 thf(func_def_177, type, sK119: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 21.23/4.34 thf(func_def_178, type, sK120: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 21.23/4.34 thf(func_def_179, type, sK121: !>[X0: $tType]:((X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_180, type, sK122: !>[X0: $tType]:(set @ X0 > X0)).
% 21.23/4.34 thf(func_def_181, type, sK123: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_182, type, sK124: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_183, type, sK125: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_184, type, sK126: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_185, type, sK127: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 21.23/4.34 thf(func_def_186, type, sK128: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 21.23/4.34 thf(func_def_187, type, sK129: !>[X0: $tType]:(nat > set @ X0 > set @ X0)).
% 21.23/4.34 thf(func_def_188, type, sK130: !>[X0: $tType]:(set @ X0 > X0)).
% 21.23/4.34 thf(func_def_189, type, sK131: !>[X0: $tType]:(set @ X0 > nat > set @ X0)).
% 21.23/4.34 thf(func_def_190, type, sK132: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 21.23/4.34 thf(func_def_191, type, sK133: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 21.23/4.34 thf(func_def_192, type, sK134: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 21.23/4.34 thf(func_def_193, type, sK135: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 21.23/4.34 thf(func_def_195, type, ph137: !>[X0: $tType]:(X0)).
% 21.23/4.34 thf(f6410,plain,(
% 21.23/4.34 $false),
% 21.23/4.34 inference(avatar_sat_refutation,[status(thm)],[f3344,f4211,f6400])).
% 21.23/4.34 thf(f6400,plain,(
% 21.23/4.34 ~spl136_78),
% 21.23/4.34 inference(avatar_contradiction_clause,[status(thm)],[f6399])).
% 21.23/4.34 thf(f6399,plain,(
% 21.23/4.34 $false | ~spl136_78),
% 21.23/4.34 inference(subsumption_resolution,[status(thm)],[f6373,f1735])).
% 21.23/4.34 thf(f1735,plain,(
% 21.23/4.34 ( ! [X0 : $tType,X1 : set @ X0] : (((ord_less_eq @ set @ X0 @ X1 @ X1) = $true)) )),
% 21.23/4.34 inference(cnf_transformation,[status(thm)],[f505])).
% 21.23/4.34 thf(f505,plain,(
% 21.23/4.34 ! [X0 : $tType,X1 : set @ X0] : ((ord_less_eq @ set @ X0 @ X1 @ X1) = $true)),
% 21.23/4.34 inference(fool_elimination,[status(thm)],[f504])).
% 21.23/4.34 thf(f504,plain,(
% 21.23/4.34 ! [X0 : $tType,X1 : set @ X0] : (ord_less_eq @ set @ X0 @ X1 @ X1)),
% 21.23/4.34 inference(rectify,[status(thm)],[f91])).
% 21.23/4.34 thf(f91,axiom,(
% 21.23/4.34 ! [X0 : $tType,X19 : set @ X0] : (ord_less_eq @ set @ X0 @ X19 @ X19)),
% 21.23/4.34 file('/export/starexec/sandbox2/tmp/tmp.vFZO1l9sao/DTF2THF_28194.p',fact_90_subset__refl)).
% 21.23/4.34 thf(f6373,plain,(
% 21.23/4.34 ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) != $true) | ~spl136_78),
% 21.23/4.34 inference(backward_demodulation,[status(thm)],[f1732,f3488])).
% 21.23/4.34 thf(f3488,plain,(
% 21.23/4.34 ((prefer1532642881r_than @ a @ x @ carrier @ relation) = (prefer1532642881r_than @ a @ y @ carrier @ relation)) | ~spl136_78),
% 21.23/4.34 inference(avatar_component_clause,[status(thm)],[f3486])).
% 21.23/4.34 thf(f3486,definition,(
% 21.23/4.34 spl136_78 <=> ((prefer1532642881r_than @ a @ x @ carrier @ relation) = (prefer1532642881r_than @ a @ y @ carrier @ relation))),
% 21.23/4.34 introduced(definition,[new_symbols(naming,[spl136_78])],[avatar_definition])).
% 21.23/4.34 thf(f1732,plain,(
% 21.23/4.34 ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ y @ carrier @ relation)) != $true)),
% 21.23/4.34 inference(cnf_transformation,[status(thm)],[f874])).
% 21.23/4.34 thf(f874,plain,(
% 21.23/4.34 ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ y @ carrier @ relation)) != $true)),
% 21.23/4.34 inference(flattening,[status(thm)],[f553])).
% 21.23/4.34 thf(f553,plain,(
% 21.23/4.34 ~((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ y @ carrier @ relation)) = $true)),
% 21.23/4.34 inference(fool_elimination,[status(thm)],[f552])).
% 21.23/4.34 thf(f552,plain,(
% 21.23/4.34 ~(ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ y @ carrier @ relation))),
% 21.23/4.34 inference(rectify,[status(thm)],[f294])).
% 21.23/4.34 thf(f294,negated_conjecture,(
% 21.23/4.34 ~(ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ y @ carrier @ relation))),
% 21.23/4.34 inference(negated_conjecture,[status(cth)],[f293])).
% 21.23/4.34 thf(f293,conjecture,(
% 21.23/4.34 (ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ y @ carrier @ relation))),
% 21.23/4.34 file('/export/starexec/sandbox2/tmp/tmp.vFZO1l9sao/DTF2THF_28194.p',conj_0)).
% 21.23/4.34 thf(f4211,plain,(
% 21.23/4.34 spl136_78 | ~spl136_13),
% 21.23/4.34 inference(avatar_split_clause,[status(thm)],[f4210,f2447,f3486])).
% 21.23/4.34 thf(f2447,definition,(
% 21.23/4.34 spl136_13 <=> ((finite_finite2 @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) = $true)),
% 21.23/4.34 introduced(definition,[new_symbols(naming,[spl136_13])],[avatar_definition])).
% 21.23/4.34 thf(f4210,plain,(
% 21.23/4.34 ((prefer1532642881r_than @ a @ x @ carrier @ relation) = (prefer1532642881r_than @ a @ y @ carrier @ relation)) | ((finite_finite2 @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) != $true)),
% 21.23/4.34 inference(subsumption_resolution,[status(thm)],[f4204,f2155])).
% 21.23/4.34 thf(f2155,plain,(
% 21.23/4.34 ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ y @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) = $true)),
% 21.23/4.34 inference(trivial_inequality_removal,[status(thm)],[f2147])).
% 21.23/4.34 thf(f2147,plain,(
% 21.23/4.34 ($true != $true) | ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ y @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) = $true)),
% 21.23/4.34 inference(superposition,[status(thm)],[f1732,f1530])).
% 21.23/4.34 thf(f1530,plain,(
% 21.23/4.34 ( ! [X0 : a,X1 : a] : (((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ X1 @ carrier @ relation) @ (prefer1532642881r_than @ a @ X0 @ carrier @ relation)) = $true) | ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ X0 @ carrier @ relation) @ (prefer1532642881r_than @ a @ X1 @ carrier @ relation)) = $true)) )),
% 21.23/4.34 inference(cnf_transformation,[status(thm)],[f467])).
% 21.23/4.34 thf(f467,plain,(
% 21.23/4.34 ! [X0 : a,X1 : a] : (((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ X0 @ carrier @ relation) @ (prefer1532642881r_than @ a @ X1 @ carrier @ relation)) = $true) | ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ X1 @ carrier @ relation) @ (prefer1532642881r_than @ a @ X0 @ carrier @ relation)) = $true))),
% 21.23/4.34 inference(fool_elimination,[status(thm)],[f466])).
% 21.23/4.34 thf(f466,plain,(
% 21.23/4.34 ! [X0 : a,X1 : a] : ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ X0 @ carrier @ relation) @ (prefer1532642881r_than @ a @ X1 @ carrier @ relation)) | (ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ X1 @ carrier @ relation) @ (prefer1532642881r_than @ a @ X0 @ carrier @ relation)))),
% 21.23/4.34 inference(rectify,[status(thm)],[f5])).
% 21.23/4.34 thf(f5,axiom,(
% 21.23/4.34 ! [X5 : a,X4 : a] : ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ X5 @ carrier @ relation) @ (prefer1532642881r_than @ a @ X4 @ carrier @ relation)) | (ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ X4 @ carrier @ relation) @ (prefer1532642881r_than @ a @ X5 @ carrier @ relation)))),
% 21.23/4.34 file('/export/starexec/sandbox2/tmp/tmp.vFZO1l9sao/DTF2THF_28194.p',fact_4_nbt__nest)).
% 21.23/4.34 thf(f4204,plain,(
% 21.23/4.34 ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ y @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) != $true) | ((finite_finite2 @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) != $true) | ((prefer1532642881r_than @ a @ x @ carrier @ relation) = (prefer1532642881r_than @ a @ y @ carrier @ relation))),
% 21.23/4.34 inference(trivial_inequality_removal,[status(thm)],[f4199])).
% 21.23/4.34 thf(f4199,plain,(
% 21.23/4.34 ((finite_finite2 @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) != $true) | ($true != $true) | ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ y @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) != $true) | ((prefer1532642881r_than @ a @ x @ carrier @ relation) = (prefer1532642881r_than @ a @ y @ carrier @ relation))),
% 21.23/4.34 inference(superposition,[status(thm)],[f1627,f1531])).
% 21.23/4.34 thf(f1531,plain,(
% 21.23/4.34 ((ord_less_eq @ nat @ (finite_card @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) @ (finite_card @ a @ (prefer1532642881r_than @ a @ y @ carrier @ relation))) = $true)),
% 21.23/4.34 inference(cnf_transformation,[status(thm)],[f451])).
% 21.23/4.34 thf(f451,plain,(
% 21.23/4.34 ((ord_less_eq @ nat @ (finite_card @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) @ (finite_card @ a @ (prefer1532642881r_than @ a @ y @ carrier @ relation))) = $true)),
% 21.23/4.34 inference(fool_elimination,[status(thm)],[f450])).
% 21.23/4.34 thf(f450,plain,(
% 21.23/4.34 (ord_less_eq @ nat @ (finite_card @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) @ (finite_card @ a @ (prefer1532642881r_than @ a @ y @ carrier @ relation)))),
% 21.23/4.34 inference(rectify,[status(thm)],[f4])).
% 21.23/4.34 thf(f4,axiom,(
% 21.23/4.34 (ord_less_eq @ nat @ (finite_card @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) @ (finite_card @ a @ (prefer1532642881r_than @ a @ y @ carrier @ relation)))),
% 21.23/4.34 file('/export/starexec/sandbox2/tmp/tmp.vFZO1l9sao/DTF2THF_28194.p',fact_3__092_060open_062card_A_Ino__better__than_Ax_Acarrier_Arelation_J_A_092_060le_062_Acard_A_Ino__better__than_Ay_Acarrier_Arelation_J_092_060close_062)).
% 21.23/4.34 thf(f1627,plain,(
% 21.23/4.34 ( ! [X0 : $tType,X2 : set @ X0,X1 : set @ X0] : (((ord_less_eq @ nat @ (finite_card @ X0 @ X1) @ (finite_card @ X0 @ X2)) != $true) | ((ord_less_eq @ set @ X0 @ X2 @ X1) != $true) | ((finite_finite2 @ X0 @ X1) != $true) | (X1 = X2)) )),
% 21.23/4.34 inference(cnf_transformation,[status(thm)],[f1049])).
% 21.23/4.34 thf(f1049,plain,(
% 21.23/4.34 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : ((X1 = X2) | ((finite_finite2 @ X0 @ X1) != $true) | ((ord_less_eq @ set @ X0 @ X2 @ X1) != $true) | ((ord_less_eq @ nat @ (finite_card @ X0 @ X1) @ (finite_card @ X0 @ X2)) != $true))),
% 21.23/4.34 inference(flattening,[status(thm)],[f1048])).
% 21.23/4.34 thf(f1048,plain,(
% 21.23/4.34 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : ((((X1 = X2) | ((ord_less_eq @ nat @ (finite_card @ X0 @ X1) @ (finite_card @ X0 @ X2)) != $true)) | ((ord_less_eq @ set @ X0 @ X2 @ X1) != $true)) | ((finite_finite2 @ X0 @ X1) != $true))),
% 21.23/4.34 inference(ennf_transformation,[status(thm)],[f585])).
% 21.23/4.34 thf(f585,plain,(
% 21.23/4.34 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : (((finite_finite2 @ X0 @ X1) = $true) => (((ord_less_eq @ set @ X0 @ X2 @ X1) = $true) => (((ord_less_eq @ nat @ (finite_card @ X0 @ X1) @ (finite_card @ X0 @ X2)) = $true) => (X1 = X2))))),
% 21.23/4.34 inference(fool_elimination,[status(thm)],[f584])).
% 21.23/4.34 thf(f584,plain,(
% 21.23/4.34 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : ((finite_finite2 @ X0 @ X1) => ((ord_less_eq @ set @ X0 @ X2 @ X1) => ((ord_less_eq @ nat @ (finite_card @ X0 @ X1) @ (finite_card @ X0 @ X2)) => (X1 = X2))))),
% 21.23/4.34 inference(rectify,[status(thm)],[f117])).
% 21.23/4.34 thf(f117,axiom,(
% 21.23/4.34 ! [X0 : $tType,X26 : set @ X0,X19 : set @ X0] : ((finite_finite2 @ X0 @ X26) => ((ord_less_eq @ set @ X0 @ X19 @ X26) => ((ord_less_eq @ nat @ (finite_card @ X0 @ X26) @ (finite_card @ X0 @ X19)) => (X19 = X26))))),
% 21.23/4.34 file('/export/starexec/sandbox2/tmp/tmp.vFZO1l9sao/DTF2THF_28194.p',fact_116_card__seteq)).
% 21.23/4.34 thf(f3344,plain,(
% 21.23/4.34 spl136_13),
% 21.23/4.34 inference(avatar_split_clause,[status(thm)],[f1998,f2447])).
% 21.23/4.34 thf(f1998,plain,(
% 21.23/4.34 ((finite_finite2 @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) = $true)),
% 21.23/4.34 inference(trivial_inequality_removal,[status(thm)],[f1994])).
% 21.23/4.34 thf(f1994,plain,(
% 21.23/4.34 ($true != $true) | ((finite_finite2 @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) = $true)),
% 21.23/4.34 inference(superposition,[status(thm)],[f1566,f1705])).
% 21.23/4.34 thf(f1705,plain,(
% 21.23/4.34 ((member @ a @ x @ carrier) = $true)),
% 21.23/4.34 inference(cnf_transformation,[status(thm)],[f359])).
% 21.23/4.34 thf(f359,plain,(
% 21.23/4.34 ((member @ a @ x @ carrier) = $true)),
% 21.23/4.34 inference(fool_elimination,[status(thm)],[f358])).
% 21.23/4.34 thf(f358,plain,(
% 21.23/4.34 (member @ a @ x @ carrier)),
% 21.23/4.34 inference(rectify,[status(thm)],[f2])).
% 21.23/4.34 thf(f2,axiom,(
% 21.23/4.34 (member @ a @ x @ carrier)),
% 21.23/4.34 file('/export/starexec/sandbox2/tmp/tmp.vFZO1l9sao/DTF2THF_28194.p',fact_1_assms_I1_J)).
% 21.23/4.34 thf(f1566,plain,(
% 21.23/4.34 ( ! [X0 : a] : (((member @ a @ X0 @ carrier) != $true) | ((finite_finite2 @ a @ (prefer1532642881r_than @ a @ X0 @ carrier @ relation)) = $true)) )),
% 21.23/4.34 inference(cnf_transformation,[status(thm)],[f1087])).
% 21.23/4.34 thf(f1087,plain,(
% 21.23/4.34 ! [X0 : a] : (((finite_finite2 @ a @ (prefer1532642881r_than @ a @ X0 @ carrier @ relation)) = $true) | ((member @ a @ X0 @ carrier) != $true))),
% 21.23/4.34 inference(ennf_transformation,[status(thm)],[f393])).
% 21.23/4.34 thf(f393,plain,(
% 21.23/4.34 ! [X0 : a] : (((member @ a @ X0 @ carrier) = $true) => ((finite_finite2 @ a @ (prefer1532642881r_than @ a @ X0 @ carrier @ relation)) = $true))),
% 21.23/4.34 inference(fool_elimination,[status(thm)],[f392])).
% 21.23/4.34 thf(f392,plain,(
% 21.23/4.34 ! [X0 : a] : ((member @ a @ X0 @ carrier) => (finite_finite2 @ a @ (prefer1532642881r_than @ a @ X0 @ carrier @ relation)))),
% 21.23/4.34 inference(rectify,[status(thm)],[f16])).
% 21.23/4.34 thf(f16,axiom,(
% 21.23/4.34 ! [X6 : a] : ((member @ a @ X6 @ carrier) => (finite_finite2 @ a @ (prefer1532642881r_than @ a @ X6 @ carrier @ relation)))),
% 21.23/4.34 file('/export/starexec/sandbox2/tmp/tmp.vFZO1l9sao/DTF2THF_28194.p',fact_15_fnt__carrier__fnt__nbt)).
% 21.23/4.34 % SZS output end Proof for DTF2THF_28194
% 21.23/4.34 % (28534)------------------------------
% 21.23/4.34 % (28534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.23/4.34 % (28534)Termination reason: Refutation
% 21.23/4.34
% 21.23/4.34 % (28534)Memory used [KB]: 11129
% 21.23/4.34 % (28534)Time elapsed: 0.980 s
% 21.23/4.34 % (28534)Instructions burned: 3204 (million)
% 21.23/4.34 % (28534)------------------------------
% 21.23/4.34 % (28534)------------------------------
% 21.23/4.34 % (28412)Success in time 2.791 s
%------------------------------------------------------------------------------