%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP163^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n014.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 08:56:04 AM UTC 2026
% Result : Theorem 3.82s 1.90s
% Output : Refutation 3.82s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP163^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n014.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 08:28:27 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.47 ---- Original DTF file ---
% 0.30/0.47 thf(spec,logic,$$dhol).
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % File : ITP163^2 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.47 % Domain : Interactive Theorem Proving
% 0.30/0.47 % Problem : Sledgehammer Refine_Basic problem prob_374__3586996_1
% 0.30/0.47 % Version : Especial.
% 0.30/0.47 % English :
% 0.30/0.47
% 0.30/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.47 % Source : [Des21]
% 0.30/0.47 % Names : Refine_Basic/prob_374__3586996_1 [Des21]
% 0.30/0.47
% 0.30/0.47 % Status : Theorem
% 0.30/0.47 % Rating : 0.33 v8.2.0, 0.00 v8.1.0, 0.25 v7.5.0
% 0.30/0.47 % Syntax : Number of formulae : 306 ( 67 unt; 44 typ; 0 def)
% 0.30/0.47 % Number of atoms : 926 ( 241 equ; 0 cnn)
% 0.30/0.47 % Maximal formula atoms : 15 ( 3 avg)
% 0.30/0.47 % Number of connectives : 4080 ( 141 ~; 20 |; 41 &;3323 @)
% 0.30/0.47 % ( 0 <=>; 555 =>; 0 <=; 0 <~>)
% 0.30/0.47 % Maximal formula depth : 28 ( 10 avg)
% 0.30/0.47 % Number of types : 2 ( 1 usr)
% 0.30/0.47 % Number of type conns : 179 ( 179 >; 0 *; 0 +; 0 <<)
% 0.30/0.47 % Number of symbols : 46 ( 43 usr; 3 con; 0-5 aty)
% 0.30/0.47 % Number of variables : 1218 ( 64 ^;1061 !; 45 ?;1218 :)
% 0.30/0.47 % ( 48 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.47 % SPC : TH1_THM_EQU_NAR
% 0.30/0.47
% 0.30/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:19:02.950
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % Could-be-implicit typings (4)
% 0.30/0.47 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
% 0.30/0.47 refine1665802226e_nres: $tType > $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.30/0.47 product_prod: $tType > $tType > $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_t_Set_Oset,type,
% 0.30/0.47 set: $tType > $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_tf_a,type,
% 0.30/0.47 a: $tType ).
% 0.30/0.47
% 0.30/0.47 % Explicit typings (40)
% 0.30/0.47 thf(sy_cl_HOL_Otype,type,
% 0.30/0.47 type:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.30/0.47 ord:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.30/0.47 order:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Ono__bot,type,
% 0.30/0.47 no_bot:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Ono__top,type,
% 0.30/0.47 no_top:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/0.47 linorder:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/0.47 preorder:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Owellorder,type,
% 0.30/0.47 wellorder:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Odense__order,type,
% 0.30/0.47 dense_order:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.30/0.47 linordered_idom:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Fields_Olinordered__field,type,
% 0.30/0.47 linordered_field:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.30/0.47 dense_linorder:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.30/0.47 condit1656338222tinuum:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.30/0.47 condit1037483654norder:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.30/0.47 bNF_Ca1785829860lChain:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_If,type,
% 0.30/0.47 if:
% 0.30/0.47 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Misc_Ouncurry,type,
% 0.30/0.47 uncurry:
% 0.30/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.30/0.47 ord_less:
% 0.30/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.47 ord_less_eq:
% 0.30/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.30/0.47 order_Greatest:
% 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.30/0.47 product_Pair:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.30/0.47 produc2004651681e_prod:
% 0.30/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.30/0.47 product_rec_prod:
% 0.30/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
% 0.30/0.47 refine1687780735RETURN:
% 0.30/0.47 !>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres,type,
% 0.30/0.47 refine1315500908_inres:
% 0.30/0.47 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oless__eq__nres__rel,type,
% 0.30/0.47 refine1554218259es_rel:
% 0.30/0.47 !>[A: $tType] : ( ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) > ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oless__nres__rel,type,
% 0.30/0.47 refine1378444575es_rel:
% 0.30/0.47 !>[A: $tType] : ( ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) > ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres,type,
% 0.30/0.47 refine406925620_inres:
% 0.30/0.47 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail,type,
% 0.30/0.47 refine1102455758nofail:
% 0.30/0.47 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_OFAILi,type,
% 0.30/0.47 refine1767639642_FAILi:
% 0.30/0.47 !>[A: $tType] : ( refine1665802226e_nres @ A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES,type,
% 0.30/0.47 refine605929679le_RES:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( refine1665802226e_nres @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_Ocase__nres,type,
% 0.30/0.47 refine60818195e_nres:
% 0.30/0.47 !>[B: $tType,A: $tType] : ( B > ( ( set @ A ) > B ) > ( refine1665802226e_nres @ A ) > B ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_Orec__nres,type,
% 0.30/0.47 refine1442219249c_nres:
% 0.30/0.47 !>[C: $tType,A: $tType] : ( C > ( ( set @ A ) > C ) > ( refine1665802226e_nres @ A ) > C ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES,type,
% 0.30/0.47 refine1672542526he_RES:
% 0.30/0.47 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > ( set @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_OCollect,type,
% 0.30/0.47 collect:
% 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Wellfounded_Oaccp,type,
% 0.30/0.47 accp:
% 0.30/0.47 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Zorn_Ochain__subset,type,
% 0.30/0.47 chain_subset:
% 0.30/0.47 !>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_member,type,
% 0.30/0.47 member:
% 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v__092_060Phi_062,type,
% 0.30/0.47 phi: a > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_x,type,
% 0.30/0.47 x: a ).
% 0.30/0.47
% 0.30/0.47 % Relevant facts (245)
% 0.30/0.47 thf(fact_0_nres__more__simps_I4_J,axiom,
% 0.30/0.47 ! [A: $tType,X: set @ A,Y: set @ A] :
% 0.30/0.47 ( ( ( refine605929679le_RES @ A @ X )
% 0.30/0.47 = ( refine605929679le_RES @ A @ Y ) )
% 0.30/0.47 = ( X = Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % nres_more_simps(4)
% 0.30/0.47 thf(fact_1_nres_Oinject,axiom,
% 0.30/0.47 ! [A: $tType,X2: set @ A,Y2: set @ A] :
% 0.30/0.47 ( ( ( refine605929679le_RES @ A @ X2 )
% 0.30/0.47 = ( refine605929679le_RES @ A @ Y2 ) )
% 0.30/0.47 = ( X2 = Y2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % nres.inject
% 0.30/0.47 thf(fact_2_nf__inres__RES,axiom,
% 0.30/0.47 ! [A: $tType,X: set @ A,X3: A] :
% 0.30/0.47 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ X ) @ X3 )
% 0.30/0.47 = ( member @ A @ X3 @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % nf_inres_RES
% 0.30/0.47 thf(fact_3_the__RES_Osimps,axiom,
% 0.30/0.47 ! [A: $tType,X: set @ A] :
% 0.30/0.47 ( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X ) )
% 0.30/0.47 = X ) ).
% 0.30/0.47
% 0.30/0.47 % the_RES.simps
% 0.30/0.47 thf(fact_4_nres_Osimps_I7_J,axiom,
% 0.30/0.47 ! [C: $tType,A: $tType,F1: C,F2: ( set @ A ) > C,X2: set @ A] :
% 0.30/0.47 ( ( refine1442219249c_nres @ C @ A @ F1 @ F2 @ ( refine605929679le_RES @ A @ X2 ) )
% 0.30/0.47 = ( F2 @ X2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % nres.simps(7)
% 0.30/0.47 thf(fact_5_nres_Osimps_I5_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F1: B,F2: ( set @ A ) > B,X2: set @ A] :
% 0.30/0.47 ( ( refine60818195e_nres @ B @ A @ F1 @ F2 @ ( refine605929679le_RES @ A @ X2 ) )
% 0.30/0.47 = ( F2 @ X2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % nres.simps(5)
% 0.30/0.47 thf(fact_6_inres__simps_I2_J,axiom,
% 0.30/0.47 ! [B: $tType,X: set @ B] :
% 0.30/0.47 ( ( refine1315500908_inres @ B @ ( refine605929679le_RES @ B @ X ) )
% 0.30/0.47 = ( ^ [X4: B] : ( member @ B @ X4 @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inres_simps(2)
% 0.30/0.47 thf(fact_7_nofail__simps_I2_J,axiom,
% 0.30/0.47 ! [B: $tType,X: set @ B] : ( refine1102455758nofail @ B @ ( refine605929679le_RES @ B @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % nofail_simps(2)
% 0.30/0.47 thf(fact_8_nofail__RES__conv,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( refine1102455758nofail @ A )
% 0.30/0.47 = ( ^ [M: refine1665802226e_nres @ A] :
% 0.30/0.47 ? [M2: set @ A] :
% 0.30/0.47 ( M
% 0.30/0.47 = ( refine605929679le_RES @ A @ M2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nofail_RES_conv
% 0.30/0.47 thf(fact_9_nres_Odistinct_I1_J,axiom,
% 0.30/0.47 ! [A: $tType,X2: set @ A] :
% 0.30/0.47 ( ( refine1767639642_FAILi @ A )
% 0.30/0.47 != ( refine605929679le_RES @ A @ X2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % nres.distinct(1)
% 0.30/0.47 thf(fact_10_nres_Oinduct,axiom,
% 0.30/0.47 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > $o,Nres: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( P @ ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ( ! [X5: set @ A] : ( P @ ( refine605929679le_RES @ A @ X5 ) )
% 0.30/0.47 => ( P @ Nres ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nres.induct
% 0.30/0.47 thf(fact_11_nres_Oexhaust,axiom,
% 0.30/0.47 ! [A: $tType,Y3: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( Y3
% 0.30/0.47 != ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ~ ! [X22: set @ A] :
% 0.30/0.47 ( Y3
% 0.30/0.47 != ( refine605929679le_RES @ A @ X22 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nres.exhaust
% 0.30/0.47 thf(fact_12_sup__nres_Oinduct,axiom,
% 0.30/0.47 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ! [Uu: refine1665802226e_nres @ A] : ( P @ Uu @ ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ( ! [V: set @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V ) )
% 0.30/0.47 => ( ! [A2: set @ A,B2: set @ A] : ( P @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_nres.induct
% 0.30/0.47 thf(fact_13_less__nres_Oinduct,axiom,
% 0.30/0.47 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ! [X_1: refine1665802226e_nres @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ X_1 )
% 0.30/0.47 => ( ! [Uv: set @ A] : ( P @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ( ! [A2: set @ A,B2: set @ A] : ( P @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_nres.induct
% 0.30/0.47 thf(fact_14_less__eq__nres_Oinduct,axiom,
% 0.30/0.47 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ! [Uu: refine1665802226e_nres @ A] : ( P @ Uu @ ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ( ! [A2: set @ A,B2: set @ A] : ( P @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ( ! [Uv: set @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) )
% 0.30/0.47 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_nres.induct
% 0.30/0.47 thf(fact_15_the__RES__inv,axiom,
% 0.30/0.47 ! [A: $tType,M3: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( refine1102455758nofail @ A @ M3 )
% 0.30/0.47 => ( ( refine605929679le_RES @ A @ ( refine1672542526he_RES @ A @ M3 ) )
% 0.30/0.47 = M3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % the_RES_inv
% 0.30/0.47 thf(fact_16_nres_Osimps_I6_J,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,F1: C,F2: ( set @ A ) > C] :
% 0.30/0.47 ( ( refine1442219249c_nres @ C @ A @ F1 @ F2 @ ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 = F1 ) ).
% 0.30/0.47
% 0.30/0.47 % nres.simps(6)
% 0.30/0.47 thf(fact_17_nres_Osimps_I4_J,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F1: B,F2: ( set @ A ) > B] :
% 0.30/0.47 ( ( refine60818195e_nres @ B @ A @ F1 @ F2 @ ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 = F1 ) ).
% 0.30/0.47
% 0.30/0.47 % nres.simps(4)
% 0.30/0.47 thf(fact_18_pw__eqI,axiom,
% 0.30/0.47 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( ( refine1102455758nofail @ A @ S )
% 0.30/0.47 = ( refine1102455758nofail @ A @ S2 ) )
% 0.30/0.47 => ( ! [X5: A] :
% 0.30/0.47 ( ( refine1315500908_inres @ A @ S @ X5 )
% 0.30/0.47 = ( refine1315500908_inres @ A @ S2 @ X5 ) )
% 0.30/0.47 => ( S = S2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pw_eqI
% 0.30/0.47 thf(fact_19_pw__eq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ^ [Y4: refine1665802226e_nres @ A,Z: refine1665802226e_nres @ A] : ( Y4 = Z ) )
% 0.30/0.47 = ( ^ [S3: refine1665802226e_nres @ A,S4: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( ( refine1102455758nofail @ A @ S3 )
% 0.30/0.47 = ( refine1102455758nofail @ A @ S4 ) )
% 0.30/0.47 & ! [X4: A] :
% 0.30/0.47 ( ( refine1315500908_inres @ A @ S3 @ X4 )
% 0.30/0.47 = ( refine1315500908_inres @ A @ S4 @ X4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pw_eq_iff
% 0.30/0.47 thf(fact_20_not__nofail__inres,axiom,
% 0.30/0.47 ! [A: $tType,S: refine1665802226e_nres @ A,X3: A] :
% 0.30/0.47 ( ~ ( refine1102455758nofail @ A @ S )
% 0.30/0.47 => ( refine1315500908_inres @ A @ S @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_nofail_inres
% 0.30/0.47 thf(fact_21_nf__inres__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( refine406925620_inres @ A )
% 0.30/0.47 = ( ^ [M: refine1665802226e_nres @ A,X4: A] :
% 0.30/0.47 ( ( refine1102455758nofail @ A @ M )
% 0.30/0.47 & ( refine1315500908_inres @ A @ M @ X4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nf_inres_def
% 0.30/0.47 thf(fact_22_inres__simps_I3_J,axiom,
% 0.30/0.47 ! [C: $tType,X3: C] :
% 0.30/0.47 ( ( refine1315500908_inres @ C @ ( refine1687780735RETURN @ C @ X3 ) )
% 0.30/0.47 = ( ^ [Y4: C,Z: C] : ( Y4 = Z )
% 0.30/0.47 @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % inres_simps(3)
% 0.30/0.47 thf(fact_23_nofail__simps_I3_J,axiom,
% 0.30/0.47 ! [C: $tType,X3: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % nofail_simps(3)
% 0.30/0.47 thf(fact_24_pw__leI,axiom,
% 0.30/0.47 ! [A: $tType,S2: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( ( refine1102455758nofail @ A @ S2 )
% 0.30/0.47 => ( ( refine1102455758nofail @ A @ S )
% 0.30/0.47 & ! [X5: A] :
% 0.30/0.47 ( ( refine1315500908_inres @ A @ S @ X5 )
% 0.30/0.47 => ( refine1315500908_inres @ A @ S2 @ X5 ) ) ) )
% 0.30/0.47 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % pw_leI
% 0.30/0.47 thf(fact_25_pw__leI_H,axiom,
% 0.30/0.47 ! [A: $tType,S2: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( ( refine1102455758nofail @ A @ S2 )
% 0.30/0.47 => ( refine1102455758nofail @ A @ S ) )
% 0.30/0.47 => ( ! [X5: A] :
% 0.30/0.47 ( ( refine1102455758nofail @ A @ S2 )
% 0.30/0.47 => ( ( refine1315500908_inres @ A @ S @ X5 )
% 0.30/0.47 => ( refine1315500908_inres @ A @ S2 @ X5 ) ) )
% 0.30/0.47 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pw_leI'
% 0.30/0.47 thf(fact_26_pw__le__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) )
% 0.30/0.47 = ( ^ [S3: refine1665802226e_nres @ A,S4: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( refine1102455758nofail @ A @ S4 )
% 0.30/0.47 => ( ( refine1102455758nofail @ A @ S3 )
% 0.30/0.47 & ! [X4: A] :
% 0.30/0.47 ( ( refine1315500908_inres @ A @ S3 @ X4 )
% 0.30/0.47 => ( refine1315500908_inres @ A @ S4 @ X4 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pw_le_iff
% 0.30/0.47 thf(fact_27_less__eq__nres_Osimps_I3_J,axiom,
% 0.30/0.47 ! [A: $tType,Uv2: set @ A] :
% 0.30/0.47 ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_nres.simps(3)
% 0.30/0.47 thf(fact_28_less__nres_Osimps_I2_J,axiom,
% 0.30/0.47 ! [A: $tType,Uv2: set @ A] : ( ord_less @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv2 ) @ ( refine1767639642_FAILi @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_nres.simps(2)
% 0.30/0.47 thf(fact_29_less__eq__nres_Ocases,axiom,
% 0.30/0.47 ! [A: $tType,X3: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
% 0.30/0.47 ( ! [Uu: refine1665802226e_nres @ A] :
% 0.30/0.47 ( X3
% 0.30/0.47 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ Uu @ ( refine1767639642_FAILi @ A ) ) )
% 0.30/0.47 => ( ! [A2: set @ A,B2: set @ A] :
% 0.30/0.47 ( X3
% 0.30/0.47 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
% 0.30/0.47 => ~ ! [Uv: set @ A] :
% 0.30/0.47 ( X3
% 0.30/0.47 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_nres.cases
% 0.30/0.47 thf(fact_30_less__nres_Ocases,axiom,
% 0.30/0.47 ! [A: $tType,X3: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
% 0.30/0.47 ( ! [Uu: refine1665802226e_nres @ A] :
% 0.30/0.47 ( X3
% 0.30/0.47 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Uu ) )
% 0.30/0.47 => ( ! [Uv: set @ A] :
% 0.30/0.47 ( X3
% 0.30/0.47 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) ) )
% 0.30/0.47 => ~ ! [A2: set @ A,B2: set @ A] :
% 0.30/0.47 ( X3
% 0.30/0.47 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_nres.cases
% 0.30/0.47 thf(fact_31_sup__nres_Ocases,axiom,
% 0.30/0.47 ! [A: $tType,X3: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
% 0.30/0.47 ( ! [Uu: refine1665802226e_nres @ A] :
% 0.30/0.47 ( X3
% 0.30/0.47 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ Uu @ ( refine1767639642_FAILi @ A ) ) )
% 0.30/0.47 => ( ! [V: set @ A] :
% 0.30/0.47 ( X3
% 0.30/0.47 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V ) ) )
% 0.30/0.47 => ~ ! [A2: set @ A,B2: set @ A] :
% 0.30/0.47 ( X3
% 0.30/0.47 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sup_nres.cases
% 0.30/0.47 thf(fact_32_nres__more__simps_I6_J,axiom,
% 0.30/0.47 ! [A: $tType,X3: A,Y3: A] :
% 0.30/0.47 ( ( ( refine1687780735RETURN @ A @ X3 )
% 0.30/0.47 = ( refine1687780735RETURN @ A @ Y3 ) )
% 0.30/0.47 = ( X3 = Y3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % nres_more_simps(6)
% 0.30/0.47 thf(fact_33_nres__order__simps_I20_J,axiom,
% 0.30/0.47 ! [W: $tType,X3: W,Y3: W] :
% 0.30/0.47 ( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X3 ) @ ( refine1687780735RETURN @ W @ Y3 ) )
% 0.30/0.47 = ( X3 = Y3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % nres_order_simps(20)
% 0.30/0.47 thf(fact_34_nres__order__simps_I21_J,axiom,
% 0.30/0.47 ! [X6: $tType,X3: X6,Y: set @ X6] :
% 0.30/0.47 ( ( ord_less_eq @ ( refine1665802226e_nres @ X6 ) @ ( refine1687780735RETURN @ X6 @ X3 ) @ ( refine605929679le_RES @ X6 @ Y ) )
% 0.30/0.47 = ( member @ X6 @ X3 @ Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % nres_order_simps(21)
% 0.30/0.47 thf(fact_35_inres__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( refine1315500908_inres @ A )
% 0.30/0.47 = ( ^ [S3: refine1665802226e_nres @ A,X4: A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X4 ) @ S3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inres_def
% 0.30/0.47 thf(fact_36_less__nres_Osimps_I1_J,axiom,
% 0.30/0.47 ! [A: $tType,Uu2: refine1665802226e_nres @ A] :
% 0.30/0.47 ~ ( ord_less @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Uu2 ) ).
% 0.30/0.47
% 0.30/0.47 % less_nres.simps(1)
% 0.30/0.47 thf(fact_37_less__eq__nres_Osimps_I1_J,axiom,
% 0.30/0.47 ! [A: $tType,Uu2: refine1665802226e_nres @ A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Uu2 @ ( refine1767639642_FAILi @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_nres.simps(1)
% 0.30/0.47 thf(fact_38_le__nofailI,axiom,
% 0.30/0.47 ! [A: $tType,M4: refine1665802226e_nres @ A,M5: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( ( refine1102455758nofail @ A @ M4 )
% 0.30/0.47 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M5 @ M4 ) )
% 0.30/0.47 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M5 @ M4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_nofailI
% 0.30/0.47 thf(fact_39_pwD1,axiom,
% 0.30/0.47 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 )
% 0.30/0.47 => ( ( refine1102455758nofail @ A @ S2 )
% 0.30/0.47 => ( refine1102455758nofail @ A @ S ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pwD1
% 0.30/0.47 thf(fact_40_pwD2,axiom,
% 0.30/0.47 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A,X3: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 )
% 0.30/0.47 => ( ( refine1315500908_inres @ A @ S @ X3 )
% 0.30/0.47 => ( refine1315500908_inres @ A @ S2 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pwD2
% 0.30/0.47 thf(fact_41_old_Oprod_Oinject,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.30/0.47 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.30/0.47 = ( ( A3 = A4 )
% 0.30/0.47 & ( B3 = B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.inject
% 0.30/0.47 thf(fact_42_prod_Oinject,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 0.30/0.47 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.30/0.47 = ( ( X1 = Y1 )
% 0.30/0.47 & ( X2 = Y2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod.inject
% 0.30/0.47 thf(fact_43_order__mono__setup_Orefl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_mono_setup.refl
% 0.30/0.47 thf(fact_44_complete__interval,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( condit1037483654norder @ A )
% 0.30/0.47 => ! [A3: A,B3: A,P: A > $o] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( P @ A3 )
% 0.30/0.47 => ( ~ ( P @ B3 )
% 0.30/0.47 => ? [C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.30/0.47 & ( ord_less_eq @ A @ C2 @ B3 )
% 0.30/0.47 & ! [X7: A] :
% 0.30/0.47 ( ( ( ord_less_eq @ A @ A3 @ X7 )
% 0.30/0.47 & ( ord_less @ A @ X7 @ C2 ) )
% 0.30/0.47 => ( P @ X7 ) )
% 0.30/0.47 & ! [D: A] :
% 0.30/0.47 ( ! [X5: A] :
% 0.30/0.47 ( ( ( ord_less_eq @ A @ A3 @ X5 )
% 0.30/0.47 & ( ord_less @ A @ X5 @ D ) )
% 0.30/0.47 => ( P @ X5 ) )
% 0.30/0.47 => ( ord_less_eq @ A @ D @ C2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % complete_interval
% 0.30/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.47 ! [A: $tType,A3: A,P: A > $o] :
% 0.30/0.47 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.30/0.47 = ( P @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_Collect_eq
% 0.30/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A] :
% 0.30/0.47 ( ( collect @ A
% 0.30/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A5 ) )
% 0.30/0.47 = A5 ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_mem_eq
% 0.30/0.47 thf(fact_47_Collect__cong,axiom,
% 0.30/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.47 ( ! [X5: A] :
% 0.30/0.47 ( ( P @ X5 )
% 0.30/0.47 = ( Q @ X5 ) )
% 0.30/0.47 => ( ( collect @ A @ P )
% 0.30/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_cong
% 0.30/0.47 thf(fact_48_ext,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.47 ( ! [X5: A] :
% 0.30/0.47 ( ( F @ X5 )
% 0.30/0.47 = ( G @ X5 ) )
% 0.30/0.47 => ( F = G ) ) ).
% 0.30/0.47
% 0.30/0.47 % ext
% 0.30/0.47 thf(fact_49_order_Onot__eq__order__implies__strict,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A3: A,B3: A] :
% 0.30/0.47 ( ( A3 != B3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.47 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.not_eq_order_implies_strict
% 0.30/0.47 thf(fact_50_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B3: A,A3: A] :
% 0.30/0.47 ( ( ord_less @ A @ B3 @ A3 )
% 0.30/0.47 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_implies_order
% 0.30/0.47 thf(fact_51_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ord_less @ A )
% 0.30/0.47 = ( ^ [B5: A,A6: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B5 @ A6 )
% 0.30/0.47 & ( A6 != B5 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_iff_order
% 0.30/0.47 thf(fact_52_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ord_less_eq @ A )
% 0.30/0.47 = ( ^ [B5: A,A6: A] :
% 0.30/0.47 ( ( ord_less @ A @ B5 @ A6 )
% 0.30/0.47 | ( A6 = B5 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.order_iff_strict
% 0.30/0.47 thf(fact_53_order_Ostrict__implies__order,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A3: A,B3: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/0.47 => ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_implies_order
% 0.30/0.47 thf(fact_54_less__eq__nres_Osimps_I2_J,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A3 ) @ ( refine605929679le_RES @ A @ B3 ) )
% 0.30/0.47 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_nres.simps(2)
% 0.30/0.47 thf(fact_55_nres__order__simps_I5_J,axiom,
% 0.30/0.47 ! [E: $tType,X: set @ E,Y: set @ E] :
% 0.30/0.47 ( ( ord_less_eq @ ( refine1665802226e_nres @ E ) @ ( refine605929679le_RES @ E @ X ) @ ( refine605929679le_RES @ E @ Y ) )
% 0.30/0.47 = ( ord_less_eq @ ( set @ E ) @ X @ Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % nres_order_simps(5)
% 0.30/0.47 thf(fact_56_less__nres_Osimps_I3_J,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/0.47 ( ( ord_less @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A3 ) @ ( refine605929679le_RES @ A @ B3 ) )
% 0.30/0.47 = ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_nres.simps(3)
% 0.30/0.47 thf(fact_57_le__funD,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ! [F: A > B,G: A > B,X3: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_funD
% 0.30/0.47 thf(fact_58_le__funE,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ! [F: A > B,G: A > B,X3: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_funE
% 0.30/0.47 thf(fact_59_le__funI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ! [F: A > B,G: A > B] :
% 0.30/0.47 ( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
% 0.30/0.47 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_funI
% 0.30/0.47 thf(fact_60_le__fun__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/0.47 = ( ^ [F3: A > B,G2: A > B] :
% 0.30/0.47 ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_fun_def
% 0.30/0.47 thf(fact_61_order__subst1,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( order @ B )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.30/0.47 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 0.30/0.47 => ( ! [X5: B,Y5: B] :
% 0.30/0.47 ( ( ord_less_eq @ B @ X5 @ Y5 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst1
% 0.30/0.47 thf(fact_62_order__subst2,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType] :
% 0.30/0.47 ( ( ( order @ C )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 0.30/0.47 => ( ! [X5: A,Y5: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X5 @ Y5 )
% 0.30/0.47 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 0.30/0.47 => ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst2
% 0.30/0.47 thf(fact_63_ord__eq__le__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B )
% 0.30/0.47 & ( ord @ A ) )
% 0.30/0.47 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.30/0.47 ( ( A3
% 0.30/0.47 = ( F @ B3 ) )
% 0.30/0.47 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 0.30/0.47 => ( ! [X5: B,Y5: B] :
% 0.30/0.47 ( ( ord_less_eq @ B @ X5 @ Y5 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_le_subst
% 0.30/0.47 thf(fact_64_ord__le__eq__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B )
% 0.30/0.47 & ( ord @ A ) )
% 0.30/0.47 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( ( F @ B3 )
% 0.30/0.47 = C3 )
% 0.30/0.47 => ( ! [X5: A,Y5: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X5 @ Y5 )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_le_eq_subst
% 0.30/0.47 thf(fact_65_eq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.30/0.47 = ( ^ [X4: A,Y6: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X4 @ Y6 )
% 0.30/0.47 & ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_iff
% 0.30/0.47 thf(fact_66_antisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.30/0.47 => ( X3 = Y3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym
% 0.30/0.47 thf(fact_67_linear,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 | ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linear
% 0.30/0.47 thf(fact_68_eq__refl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( X3 = Y3 )
% 0.30/0.47 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_refl
% 0.30/0.47 thf(fact_69_le__cases,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ~ ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_cases
% 0.30/0.47 thf(fact_70_order_Otrans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A3: A,B3: A,C3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.30/0.47 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.trans
% 0.30/0.47 thf(fact_71_le__cases3,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A,Z2: A] :
% 0.30/0.47 ( ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 0.30/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_cases3
% 0.30/0.47 thf(fact_72_antisym__conv,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [Y3: A,X3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 = ( X3 = Y3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym_conv
% 0.30/0.47 thf(fact_73_order__class_Oorder_Oeq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.30/0.47 = ( ^ [A6: A,B5: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A6 @ B5 )
% 0.30/0.47 & ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_class.order.eq_iff
% 0.30/0.47 thf(fact_74_ord__eq__le__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A )
% 0.30/0.47 => ! [A3: A,B3: A,C3: A] :
% 0.30/0.47 ( ( A3 = B3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.30/0.47 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_le_trans
% 0.30/0.47 thf(fact_75_ord__le__eq__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A )
% 0.30/0.47 => ! [A3: A,B3: A,C3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( B3 = C3 )
% 0.30/0.47 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_le_eq_trans
% 0.30/0.47 thf(fact_76_order__class_Oorder_Oantisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A3: A,B3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.47 => ( A3 = B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_class.order.antisym
% 0.30/0.47 thf(fact_77_order__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A,Z2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.30/0.47 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_trans
% 0.30/0.47 thf(fact_78_dual__order_Orefl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.refl
% 0.30/0.47 thf(fact_79_linorder__wlog,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.30/0.47 ( ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( P @ A2 @ B2 ) )
% 0.30/0.47 => ( ! [A2: A,B2: A] :
% 0.30/0.47 ( ( P @ B2 @ A2 )
% 0.30/0.47 => ( P @ A2 @ B2 ) )
% 0.30/0.47 => ( P @ A3 @ B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_wlog
% 0.30/0.47 thf(fact_80_dual__order_Otrans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B3: A,A3: A,C3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 0.30/0.47 => ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.trans
% 0.30/0.47 thf(fact_81_dual__order_Oeq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.30/0.47 = ( ^ [A6: A,B5: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B5 @ A6 )
% 0.30/0.47 & ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.eq_iff
% 0.30/0.47 thf(fact_82_dual__order_Oantisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B3: A,A3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.47 => ( A3 = B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.antisym
% 0.30/0.47 thf(fact_83_order__mono__setup_Omono__if,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [T2: A,T3: A,E2: A,E3: A,B3: $o] :
% 0.30/0.47 ( ( ord_less_eq @ A @ T2 @ T3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ E2 @ E3 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( if @ A @ B3 @ T2 @ E2 ) @ ( if @ A @ B3 @ T3 @ E3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_mono_setup.mono_if
% 0.30/0.47 thf(fact_84_order__mono__setup_Omono__let,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [F: B > A,F4: B > A,X3: B] :
% 0.30/0.47 ( ! [X5: B] : ( ord_less_eq @ A @ ( F @ X5 ) @ ( F4 @ X5 ) )
% 0.30/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F4 @ X3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_mono_setup.mono_let
% 0.30/0.47 thf(fact_85_surj__pair,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.30/0.47 ? [X5: A,Y5: B] :
% 0.30/0.47 ( P2
% 0.30/0.47 = ( product_Pair @ A @ B @ X5 @ Y5 ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_pair
% 0.30/0.47 thf(fact_86_prod__cases,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.30/0.47 ( ! [A2: A,B2: B] : ( P @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.30/0.47 => ( P @ P2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases
% 0.30/0.47 thf(fact_87_Pair__inject,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.30/0.47 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.30/0.47 => ~ ( ( A3 = A4 )
% 0.30/0.47 => ( B3 != B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pair_inject
% 0.30/0.47 thf(fact_88_prod__cases3,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.47 ~ ! [A2: A,B2: B,C2: C] :
% 0.30/0.47 ( Y3
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases3
% 0.30/0.47 thf(fact_89_prod__cases4,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.30/0.47 ~ ! [A2: A,B2: B,C2: C,D3: D2] :
% 0.30/0.47 ( Y3
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B2 @ ( product_Pair @ C @ D2 @ C2 @ D3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases4
% 0.30/0.47 thf(fact_90_prod__cases5,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.30/0.47 ~ ! [A2: A,B2: B,C2: C,D3: D2,E4: E] :
% 0.30/0.47 ( Y3
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C2 @ ( product_Pair @ D2 @ E @ D3 @ E4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases5
% 0.30/0.47 thf(fact_91_prod__cases6,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
% 0.30/0.47 ~ ! [A2: A,B2: B,C2: C,D3: D2,E4: E,F6: F5] :
% 0.30/0.47 ( Y3
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E4 @ F6 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases6
% 0.30/0.47 thf(fact_92_prod__cases7,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
% 0.30/0.47 ~ ! [A2: A,B2: B,C2: C,D3: D2,E4: E,F6: F5,G4: G3] :
% 0.30/0.47 ( Y3
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E4 @ ( product_Pair @ F5 @ G3 @ F6 @ G4 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases7
% 0.30/0.47 thf(fact_93_prod__induct3,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.47 ( ! [A2: A,B2: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C2 ) ) )
% 0.30/0.47 => ( P @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct3
% 0.30/0.47 thf(fact_94_prod__induct4,axiom,
% 0.30/0.47 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.30/0.47 ( ! [A2: A,B2: B,C2: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B2 @ ( product_Pair @ C @ D2 @ C2 @ D3 ) ) ) )
% 0.30/0.47 => ( P @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct4
% 0.30/0.47 thf(fact_95_prod__induct5,axiom,
% 0.30/0.47 ! [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,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.30/0.47 ( ! [A2: A,B2: B,C2: C,D3: D2,E4: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C2 @ ( product_Pair @ D2 @ E @ D3 @ E4 ) ) ) ) )
% 0.30/0.47 => ( P @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct5
% 0.30/0.47 thf(fact_96_prod__induct6,axiom,
% 0.30/0.47 ! [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,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
% 0.30/0.47 ( ! [A2: A,B2: B,C2: C,D3: D2,E4: E,F6: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E4 @ F6 ) ) ) ) ) )
% 0.30/0.47 => ( P @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct6
% 0.30/0.47 thf(fact_97_prod__induct7,axiom,
% 0.30/0.47 ! [G3: $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 @ G3 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
% 0.30/0.47 ( ! [A2: A,B2: B,C2: C,D3: D2,E4: E,F6: F5,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E4 @ ( product_Pair @ F5 @ G3 @ F6 @ G4 ) ) ) ) ) ) )
% 0.30/0.47 => ( P @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct7
% 0.30/0.47 thf(fact_98_old_Oprod_Oexhaust,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 0.30/0.47 ~ ! [A2: A,B2: B] :
% 0.30/0.47 ( Y3
% 0.30/0.47 != ( product_Pair @ A @ B @ A2 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.exhaust
% 0.30/0.47 thf(fact_99_old_Oprod_Oinducts,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.30/0.47 ( ! [A2: A,B2: B] : ( P @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.30/0.47 => ( P @ Prod ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.inducts
% 0.30/0.47 thf(fact_100_ord__eq__less__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B )
% 0.30/0.47 & ( ord @ A ) )
% 0.30/0.47 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.30/0.47 ( ( A3
% 0.30/0.47 = ( F @ B3 ) )
% 0.30/0.47 => ( ( ord_less @ B @ B3 @ C3 )
% 0.30/0.47 => ( ! [X5: B,Y5: B] :
% 0.30/0.47 ( ( ord_less @ B @ X5 @ Y5 )
% 0.30/0.47 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 0.30/0.47 => ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_less_subst
% 0.30/0.47 thf(fact_101_ord__less__eq__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B )
% 0.30/0.47 & ( ord @ A ) )
% 0.30/0.47 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( ( F @ B3 )
% 0.30/0.47 = C3 )
% 0.30/0.47 => ( ! [X5: A,Y5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Y5 )
% 0.30/0.47 => ( ord_less @ B @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 0.30/0.47 => ( ord_less @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_less_eq_subst
% 0.30/0.47 thf(fact_102_order__less__subst1,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( order @ B )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 0.30/0.47 => ( ( ord_less @ B @ B3 @ C3 )
% 0.30/0.47 => ( ! [X5: B,Y5: B] :
% 0.30/0.47 ( ( ord_less @ B @ X5 @ Y5 )
% 0.30/0.47 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 0.30/0.47 => ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_less_subst1
% 0.30/0.47 thf(fact_103_order__less__subst2,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType] :
% 0.30/0.47 ( ( ( order @ C )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( ord_less @ C @ ( F @ B3 ) @ C3 )
% 0.30/0.47 => ( ! [X5: A,Y5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Y5 )
% 0.30/0.47 => ( ord_less @ C @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 0.30/0.47 => ( ord_less @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_less_subst2
% 0.30/0.47 thf(fact_104_lt__ex,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( no_bot @ A )
% 0.30/0.47 => ! [X3: A] :
% 0.30/0.47 ? [Y5: A] : ( ord_less @ A @ Y5 @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % lt_ex
% 0.30/0.47 thf(fact_105_gt__ex,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( no_top @ A )
% 0.30/0.47 => ! [X3: A] :
% 0.30/0.47 ? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% 0.30/0.47
% 0.30/0.47 % gt_ex
% 0.30/0.47 thf(fact_106_neqE,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( X3 != Y3 )
% 0.30/0.47 => ( ~ ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neqE
% 0.30/0.47 thf(fact_107_neq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( X3 != Y3 )
% 0.30/0.47 = ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 | ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neq_iff
% 0.30/0.47 thf(fact_108_order_Oasym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A3: A,B3: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/0.47 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.asym
% 0.30/0.47 thf(fact_109_dense,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( dense_order @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ? [Z3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Z3 )
% 0.30/0.47 & ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dense
% 0.30/0.47 thf(fact_110_less__imp__neq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ( X3 != Y3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_neq
% 0.30/0.47 thf(fact_111_less__asym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_asym
% 0.30/0.47 thf(fact_112_less__asym_H,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [A3: A,B3: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/0.47 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_asym'
% 0.30/0.47 thf(fact_113_less__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A,Z2: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ( ord_less @ A @ Y3 @ Z2 )
% 0.30/0.47 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_trans
% 0.30/0.47 thf(fact_114_less__linear,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 | ( X3 = Y3 )
% 0.30/0.47 | ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_linear
% 0.30/0.47 thf(fact_115_less__irrefl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X3: A] :
% 0.30/0.47 ~ ( ord_less @ A @ X3 @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_irrefl
% 0.30/0.47 thf(fact_116_ord__eq__less__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A )
% 0.30/0.47 => ! [A3: A,B3: A,C3: A] :
% 0.30/0.47 ( ( A3 = B3 )
% 0.30/0.47 => ( ( ord_less @ A @ B3 @ C3 )
% 0.30/0.47 => ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_less_trans
% 0.30/0.47 thf(fact_117_ord__less__eq__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A )
% 0.30/0.47 => ! [A3: A,B3: A,C3: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( B3 = C3 )
% 0.30/0.47 => ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_less_eq_trans
% 0.30/0.47 thf(fact_118_dual__order_Oasym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B3: A,A3: A] :
% 0.30/0.47 ( ( ord_less @ A @ B3 @ A3 )
% 0.30/0.47 => ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.asym
% 0.30/0.47 thf(fact_119_less__imp__not__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ( X3 != Y3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_eq
% 0.30/0.47 thf(fact_120_less__not__sym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_not_sym
% 0.30/0.47 thf(fact_121_less__induct,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( wellorder @ A )
% 0.30/0.47 => ! [P: A > $o,A3: A] :
% 0.30/0.47 ( ! [X5: A] :
% 0.30/0.47 ( ! [Y7: A] :
% 0.30/0.47 ( ( ord_less @ A @ Y7 @ X5 )
% 0.30/0.47 => ( P @ Y7 ) )
% 0.30/0.47 => ( P @ X5 ) )
% 0.30/0.47 => ( P @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_induct
% 0.30/0.47 thf(fact_122_antisym__conv3,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [Y3: A,X3: A] :
% 0.30/0.47 ( ~ ( ord_less @ A @ Y3 @ X3 )
% 0.30/0.47 => ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 0.30/0.47 = ( X3 = Y3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym_conv3
% 0.30/0.47 thf(fact_123_less__imp__not__eq2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ( Y3 != X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_eq2
% 0.30/0.47 thf(fact_124_less__imp__triv,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A,P: $o] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ( ord_less @ A @ Y3 @ X3 )
% 0.30/0.47 => P ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_triv
% 0.30/0.47 thf(fact_125_linorder__cases,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ~ ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ( X3 != Y3 )
% 0.30/0.47 => ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_cases
% 0.30/0.47 thf(fact_126_dual__order_Oirrefl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A3: A] :
% 0.30/0.47 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.irrefl
% 0.30/0.47 thf(fact_127_order_Ostrict__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A3: A,B3: A,C3: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( ord_less @ A @ B3 @ C3 )
% 0.30/0.47 => ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_trans
% 0.30/0.47 thf(fact_128_less__imp__not__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_less
% 0.30/0.47 thf(fact_129_exists__least__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( wellorder @ A )
% 0.30/0.47 => ( ( ^ [P3: A > $o] :
% 0.30/0.47 ? [X8: A] : ( P3 @ X8 ) )
% 0.30/0.47 = ( ^ [P4: A > $o] :
% 0.30/0.47 ? [N: A] :
% 0.30/0.47 ( ( P4 @ N )
% 0.30/0.47 & ! [M: A] :
% 0.30/0.47 ( ( ord_less @ A @ M @ N )
% 0.30/0.47 => ~ ( P4 @ M ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % exists_least_iff
% 0.30/0.47 thf(fact_130_linorder__less__wlog,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.30/0.47 ( ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.47 => ( P @ A2 @ B2 ) )
% 0.30/0.47 => ( ! [A2: A] : ( P @ A2 @ A2 )
% 0.30/0.47 => ( ! [A2: A,B2: A] :
% 0.30/0.47 ( ( P @ B2 @ A2 )
% 0.30/0.47 => ( P @ A2 @ B2 ) )
% 0.30/0.47 => ( P @ A3 @ B3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_less_wlog
% 0.30/0.47 thf(fact_131_dual__order_Ostrict__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B3: A,A3: A,C3: A] :
% 0.30/0.47 ( ( ord_less @ A @ B3 @ A3 )
% 0.30/0.47 => ( ( ord_less @ A @ C3 @ B3 )
% 0.30/0.47 => ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_trans
% 0.30/0.47 thf(fact_132_not__less__iff__gr__or__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 0.30/0.47 = ( ( ord_less @ A @ Y3 @ X3 )
% 0.30/0.47 | ( X3 = Y3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_less_iff_gr_or_eq
% 0.30/0.47 thf(fact_133_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A3: A,B3: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/0.47 => ( A3 != B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_implies_not_eq
% 0.30/0.47 thf(fact_134_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B3: A,A3: A] :
% 0.30/0.47 ( ( ord_less @ A @ B3 @ A3 )
% 0.30/0.47 => ( A3 != B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_implies_not_eq
% 0.30/0.47 thf(fact_135_ex__gt__or__lt,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( condit1656338222tinuum @ A )
% 0.30/0.47 => ! [A3: A] :
% 0.30/0.47 ? [B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.47 | ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ex_gt_or_lt
% 0.30/0.47 thf(fact_136_less__eq__nres_Oelims_I3_J,axiom,
% 0.30/0.47 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 0.30/0.47 => ( ! [A2: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ A2 ) )
% 0.30/0.47 => ! [B2: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) )
% 0.30/0.47 => ~ ( ( X3
% 0.30/0.47 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ! [Uv: set @ A] :
% 0.30/0.47 ( Xa
% 0.30/0.47 != ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_nres.elims(3)
% 0.30/0.47 thf(fact_137_less__eq__nres_Oelims_I2_J,axiom,
% 0.30/0.47 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 0.30/0.47 => ( ( Xa
% 0.30/0.47 != ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ~ ! [A2: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ A2 ) )
% 0.30/0.47 => ! [B2: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_nres.elims(2)
% 0.30/0.47 thf(fact_138_less__eq__nres_Oelims_I1_J,axiom,
% 0.30/0.47 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
% 0.30/0.47 ( ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 0.30/0.47 = Y3 )
% 0.30/0.47 => ( ( ( Xa
% 0.30/0.47 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ~ Y3 )
% 0.30/0.47 => ( ! [A2: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ A2 ) )
% 0.30/0.47 => ! [B2: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ( Y3
% 0.30/0.47 = ( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) )
% 0.30/0.47 => ~ ( ( X3
% 0.30/0.47 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ( ? [Uv: set @ A] :
% 0.30/0.47 ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ Uv ) )
% 0.30/0.47 => Y3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_nres.elims(1)
% 0.30/0.47 thf(fact_139_less__nres_Oelims_I3_J,axiom,
% 0.30/0.47 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ~ ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 0.30/0.47 => ( ( X3
% 0.30/0.47 != ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ~ ! [A2: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ A2 ) )
% 0.30/0.47 => ! [B2: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_nres.elims(3)
% 0.30/0.47 thf(fact_140_less__nres_Oelims_I2_J,axiom,
% 0.30/0.47 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 0.30/0.47 => ( ( ? [Uv: set @ A] :
% 0.30/0.47 ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ Uv ) )
% 0.30/0.47 => ( Xa
% 0.30/0.47 != ( refine1767639642_FAILi @ A ) ) )
% 0.30/0.47 => ~ ! [A2: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ A2 ) )
% 0.30/0.47 => ! [B2: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ~ ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_nres.elims(2)
% 0.30/0.47 thf(fact_141_less__nres_Oelims_I1_J,axiom,
% 0.30/0.47 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
% 0.30/0.47 ( ( ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 0.30/0.47 = Y3 )
% 0.30/0.47 => ( ( ( X3
% 0.30/0.47 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => Y3 )
% 0.30/0.47 => ( ( ? [Uv: set @ A] :
% 0.30/0.47 ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ Uv ) )
% 0.30/0.47 => ( ( Xa
% 0.30/0.47 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ~ Y3 ) )
% 0.30/0.47 => ~ ! [A2: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ A2 ) )
% 0.30/0.47 => ! [B2: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ( Y3
% 0.30/0.47 = ( ~ ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_nres.elims(1)
% 0.30/0.47 thf(fact_142_leD,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [Y3: A,X3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.30/0.47 => ~ ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % leD
% 0.30/0.47 thf(fact_143_leI,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ~ ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % leI
% 0.30/0.47 thf(fact_144_le__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ord_less_eq @ A )
% 0.30/0.47 = ( ^ [X4: A,Y6: A] :
% 0.30/0.47 ( ( ord_less @ A @ X4 @ Y6 )
% 0.30/0.47 | ( X4 = Y6 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_less
% 0.30/0.47 thf(fact_145_less__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ord_less @ A )
% 0.30/0.47 = ( ^ [X4: A,Y6: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X4 @ Y6 )
% 0.30/0.47 & ( X4 != Y6 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_le
% 0.30/0.47 thf(fact_146_order__le__less__subst1,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( order @ B )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.30/0.47 => ( ( ord_less @ B @ B3 @ C3 )
% 0.30/0.47 => ( ! [X5: B,Y5: B] :
% 0.30/0.47 ( ( ord_less @ B @ X5 @ Y5 )
% 0.30/0.47 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 0.30/0.47 => ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_le_less_subst1
% 0.30/0.47 thf(fact_147_order__le__less__subst2,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType] :
% 0.30/0.47 ( ( ( order @ C )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( ord_less @ C @ ( F @ B3 ) @ C3 )
% 0.30/0.47 => ( ! [X5: A,Y5: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X5 @ Y5 )
% 0.30/0.47 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 0.30/0.47 => ( ord_less @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_le_less_subst2
% 0.30/0.47 thf(fact_148_order__less__le__subst1,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( order @ B )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 0.30/0.47 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 0.30/0.47 => ( ! [X5: B,Y5: B] :
% 0.30/0.47 ( ( ord_less_eq @ B @ X5 @ Y5 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 0.30/0.47 => ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_less_le_subst1
% 0.30/0.47 thf(fact_149_order__less__le__subst2,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType] :
% 0.30/0.47 ( ( ( order @ C )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 0.30/0.47 => ( ! [X5: A,Y5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Y5 )
% 0.30/0.47 => ( ord_less @ C @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 0.30/0.47 => ( ord_less @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_less_le_subst2
% 0.30/0.47 thf(fact_150_not__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ~ ( ord_less_eq @ A @ X3 @ Y3 ) )
% 0.30/0.47 = ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_le
% 0.30/0.47 thf(fact_151_not__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_less
% 0.30/0.47 thf(fact_152_le__neq__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A3: A,B3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( A3 != B3 )
% 0.30/0.47 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_neq_trans
% 0.30/0.47 thf(fact_153_antisym__conv1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ~ ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 = ( X3 = Y3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym_conv1
% 0.30/0.47 thf(fact_154_antisym__conv2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 0.30/0.47 = ( X3 = Y3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym_conv2
% 0.30/0.47 thf(fact_155_less__imp__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_le
% 0.30/0.47 thf(fact_156_le__less__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A,Z2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ( ord_less @ A @ Y3 @ Z2 )
% 0.30/0.47 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_less_trans
% 0.30/0.47 thf(fact_157_less__le__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A,Z2: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.30/0.47 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_le_trans
% 0.30/0.47 thf(fact_158_dense__ge,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( dense_linorder @ A )
% 0.30/0.47 => ! [Z2: A,Y3: A] :
% 0.30/0.47 ( ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z2 @ X5 )
% 0.30/0.47 => ( ord_less_eq @ A @ Y3 @ X5 ) )
% 0.30/0.47 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dense_ge
% 0.30/0.47 thf(fact_159_dense__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( dense_linorder @ A )
% 0.30/0.47 => ! [Y3: A,Z2: A] :
% 0.30/0.47 ( ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Y3 )
% 0.30/0.47 => ( ord_less_eq @ A @ X5 @ Z2 ) )
% 0.30/0.47 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dense_le
% 0.30/0.47 thf(fact_160_le__less__linear,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 | ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_less_linear
% 0.30/0.47 thf(fact_161_le__imp__less__or__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 | ( X3 = Y3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_imp_less_or_eq
% 0.30/0.47 thf(fact_162_less__le__not__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ( ( ord_less @ A )
% 0.30/0.47 = ( ^ [X4: A,Y6: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X4 @ Y6 )
% 0.30/0.47 & ~ ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_le_not_le
% 0.30/0.47 thf(fact_163_not__le__imp__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [Y3: A,X3: A] :
% 0.30/0.47 ( ~ ( ord_less_eq @ A @ Y3 @ X3 )
% 0.30/0.47 => ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_le_imp_less
% 0.30/0.47 thf(fact_164_order_Ostrict__trans1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A3: A,B3: A,C3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( ord_less @ A @ B3 @ C3 )
% 0.30/0.47 => ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_trans1
% 0.30/0.47 thf(fact_165_order_Ostrict__trans2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A3: A,B3: A,C3: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.30/0.47 => ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_trans2
% 0.30/0.47 thf(fact_166_order_Oorder__iff__strict,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ord_less_eq @ A )
% 0.30/0.47 = ( ^ [A6: A,B5: A] :
% 0.30/0.47 ( ( ord_less @ A @ A6 @ B5 )
% 0.30/0.47 | ( A6 = B5 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.order_iff_strict
% 0.30/0.47 thf(fact_167_order_Ostrict__iff__order,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ord_less @ A )
% 0.30/0.47 = ( ^ [A6: A,B5: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A6 @ B5 )
% 0.30/0.47 & ( A6 != B5 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_iff_order
% 0.30/0.47 thf(fact_168_dual__order_Ostrict__trans1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B3: A,A3: A,C3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.47 => ( ( ord_less @ A @ C3 @ B3 )
% 0.30/0.47 => ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_trans1
% 0.30/0.47 thf(fact_169_dual__order_Ostrict__trans2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B3: A,A3: A,C3: A] :
% 0.30/0.47 ( ( ord_less @ A @ B3 @ A3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 0.30/0.47 => ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_trans2
% 0.30/0.47 thf(fact_170_dense__ge__bounded,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( dense_linorder @ A )
% 0.30/0.47 => ! [Z2: A,X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z2 @ X3 )
% 0.30/0.47 => ( ! [W2: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z2 @ W2 )
% 0.30/0.47 => ( ( ord_less @ A @ W2 @ X3 )
% 0.30/0.47 => ( ord_less_eq @ A @ Y3 @ W2 ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dense_ge_bounded
% 0.30/0.47 thf(fact_171_dense__le__bounded,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( dense_linorder @ A )
% 0.30/0.47 => ! [X3: A,Y3: A,Z2: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ! [W2: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ W2 )
% 0.30/0.47 => ( ( ord_less @ A @ W2 @ Y3 )
% 0.30/0.47 => ( ord_less_eq @ A @ W2 @ Z2 ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dense_le_bounded
% 0.30/0.47 thf(fact_172_minf_I8_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ T2 @ X7 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(8)
% 0.30/0.47 thf(fact_173_minf_I6_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/0.47 => ( ord_less_eq @ A @ X7 @ T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(6)
% 0.30/0.47 thf(fact_174_pinf_I8_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/0.47 => ( ord_less_eq @ A @ T2 @ X7 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(8)
% 0.30/0.47 thf(fact_175_pinf_I6_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ X7 @ T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(6)
% 0.30/0.47 thf(fact_176_verit__comp__simplify1_I3_J,axiom,
% 0.30/0.47 ! [B: $tType] :
% 0.30/0.47 ( ( linorder @ B )
% 0.30/0.47 => ! [B4: B,A4: B] :
% 0.30/0.47 ( ( ~ ( ord_less_eq @ B @ B4 @ A4 ) )
% 0.30/0.47 = ( ord_less @ B @ A4 @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % verit_comp_simplify1(3)
% 0.30/0.47 thf(fact_177_internal__case__prod__conv,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B3: C] :
% 0.30/0.47 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 0.30/0.47 = ( C3 @ A3 @ B3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % internal_case_prod_conv
% 0.30/0.47 thf(fact_178_old_Oprod_Orec,axiom,
% 0.30/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 0.30/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.30/0.47 = ( F1 @ A3 @ B3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.rec
% 0.30/0.47 thf(fact_179_relChain__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.30/0.47 = ( ^ [R: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.30/0.47 ! [I: A,J: A] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
% 0.30/0.47 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % relChain_def
% 0.30/0.47 thf(fact_180_less__fun__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ( ( ord_less @ ( A > B ) )
% 0.30/0.47 = ( ^ [F3: A > B,G2: A > B] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 0.30/0.47 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_fun_def
% 0.30/0.47 thf(fact_181_verit__la__disequality,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [A3: A,B3: A] :
% 0.30/0.47 ( ( A3 = B3 )
% 0.30/0.47 | ~ ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.47 | ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % verit_la_disequality
% 0.30/0.47 thf(fact_182_verit__comp__simplify1_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A3: A] :
% 0.30/0.47 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % verit_comp_simplify1(1)
% 0.30/0.47 thf(fact_183_pinf_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.47 ( ? [Z4: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z4 @ X5 )
% 0.30/0.47 => ( ( P @ X5 )
% 0.30/0.47 = ( P5 @ X5 ) ) )
% 0.30/0.47 => ( ? [Z4: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z4 @ X5 )
% 0.30/0.47 => ( ( Q @ X5 )
% 0.30/0.47 = ( Q2 @ X5 ) ) )
% 0.30/0.47 => ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/0.47 => ( ( ( P @ X7 )
% 0.30/0.47 & ( Q @ X7 ) )
% 0.30/0.47 = ( ( P5 @ X7 )
% 0.30/0.47 & ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(1)
% 0.30/0.47 thf(fact_184_pinf_I2_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.47 ( ? [Z4: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z4 @ X5 )
% 0.30/0.47 => ( ( P @ X5 )
% 0.30/0.47 = ( P5 @ X5 ) ) )
% 0.30/0.47 => ( ? [Z4: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z4 @ X5 )
% 0.30/0.47 => ( ( Q @ X5 )
% 0.30/0.47 = ( Q2 @ X5 ) ) )
% 0.30/0.47 => ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/0.47 => ( ( ( P @ X7 )
% 0.30/0.47 | ( Q @ X7 ) )
% 0.30/0.47 = ( ( P5 @ X7 )
% 0.30/0.47 | ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(2)
% 0.30/0.47 thf(fact_185_pinf_I3_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/0.47 => ( X7 != T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(3)
% 0.30/0.47 thf(fact_186_pinf_I4_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/0.47 => ( X7 != T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(4)
% 0.30/0.47 thf(fact_187_pinf_I5_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/0.47 => ~ ( ord_less @ A @ X7 @ T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(5)
% 0.30/0.47 thf(fact_188_pinf_I7_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ Z3 @ X7 )
% 0.30/0.47 => ( ord_less @ A @ T2 @ X7 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(7)
% 0.30/0.47 thf(fact_189_pinf_I11_J,axiom,
% 0.30/0.47 ! [C: $tType,D2: $tType] :
% 0.30/0.47 ( ( ord @ C )
% 0.30/0.47 => ! [F7: D2] :
% 0.30/0.47 ? [Z3: C] :
% 0.30/0.47 ! [X7: C] :
% 0.30/0.47 ( ( ord_less @ C @ Z3 @ X7 )
% 0.30/0.47 => ( F7 = F7 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pinf(11)
% 0.30/0.47 thf(fact_190_minf_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.47 ( ? [Z4: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Z4 )
% 0.30/0.47 => ( ( P @ X5 )
% 0.30/0.47 = ( P5 @ X5 ) ) )
% 0.30/0.47 => ( ? [Z4: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Z4 )
% 0.30/0.47 => ( ( Q @ X5 )
% 0.30/0.47 = ( Q2 @ X5 ) ) )
% 0.30/0.47 => ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/0.47 => ( ( ( P @ X7 )
% 0.30/0.47 & ( Q @ X7 ) )
% 0.30/0.47 = ( ( P5 @ X7 )
% 0.30/0.47 & ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(1)
% 0.30/0.47 thf(fact_191_minf_I2_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.47 ( ? [Z4: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Z4 )
% 0.30/0.47 => ( ( P @ X5 )
% 0.30/0.47 = ( P5 @ X5 ) ) )
% 0.30/0.47 => ( ? [Z4: A] :
% 0.30/0.47 ! [X5: A] :
% 0.30/0.47 ( ( ord_less @ A @ X5 @ Z4 )
% 0.30/0.47 => ( ( Q @ X5 )
% 0.30/0.47 = ( Q2 @ X5 ) ) )
% 0.30/0.47 => ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/0.47 => ( ( ( P @ X7 )
% 0.30/0.47 | ( Q @ X7 ) )
% 0.30/0.47 = ( ( P5 @ X7 )
% 0.30/0.47 | ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(2)
% 0.30/0.47 thf(fact_192_minf_I3_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/0.47 => ( X7 != T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(3)
% 0.30/0.47 thf(fact_193_minf_I4_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/0.47 => ( X7 != T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(4)
% 0.30/0.47 thf(fact_194_minf_I5_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/0.47 => ( ord_less @ A @ X7 @ T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(5)
% 0.30/0.47 thf(fact_195_minf_I7_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [T2: A] :
% 0.30/0.47 ? [Z3: A] :
% 0.30/0.47 ! [X7: A] :
% 0.30/0.47 ( ( ord_less @ A @ X7 @ Z3 )
% 0.30/0.47 => ~ ( ord_less @ A @ T2 @ X7 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(7)
% 0.30/0.47 thf(fact_196_minf_I11_J,axiom,
% 0.30/0.47 ! [C: $tType,D2: $tType] :
% 0.30/0.47 ( ( ord @ C )
% 0.30/0.47 => ! [F7: D2] :
% 0.30/0.47 ? [Z3: C] :
% 0.30/0.47 ! [X7: C] :
% 0.30/0.47 ( ( ord_less @ C @ X7 @ Z3 )
% 0.30/0.47 => ( F7 = F7 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minf(11)
% 0.30/0.47 thf(fact_197_psubsetI,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ( ( A5 != B6 )
% 0.30/0.47 => ( ord_less @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubsetI
% 0.30/0.47 thf(fact_198_subset__antisym,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
% 0.30/0.47 => ( A5 = B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_antisym
% 0.30/0.47 thf(fact_199_subsetI,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ! [X5: A] :
% 0.30/0.47 ( ( member @ A @ X5 @ A5 )
% 0.30/0.47 => ( member @ A @ X5 @ B6 ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % subsetI
% 0.30/0.47 thf(fact_200_in__mono,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A,X3: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ( ( member @ A @ X3 @ A5 )
% 0.30/0.47 => ( member @ A @ X3 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_mono
% 0.30/0.47 thf(fact_201_subsetD,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A,C3: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ( ( member @ A @ C3 @ A5 )
% 0.30/0.47 => ( member @ A @ C3 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subsetD
% 0.30/0.47 thf(fact_202_equalityE,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( A5 = B6 )
% 0.30/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % equalityE
% 0.30/0.47 thf(fact_203_subset__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ A7 )
% 0.30/0.47 => ( member @ A @ X4 @ B7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_eq
% 0.30/0.47 thf(fact_204_equalityD1,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( A5 = B6 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % equalityD1
% 0.30/0.47 thf(fact_205_equalityD2,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( A5 = B6 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ).
% 0.30/0.47
% 0.30/0.47 % equalityD2
% 0.30/0.47 thf(fact_206_subset__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.30/0.47 ! [T4: A] :
% 0.30/0.47 ( ( member @ A @ T4 @ A7 )
% 0.30/0.47 => ( member @ A @ T4 @ B7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_iff
% 0.30/0.47 thf(fact_207_subset__refl,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 0.30/0.47
% 0.30/0.47 % subset_refl
% 0.30/0.47 thf(fact_208_Collect__mono,axiom,
% 0.30/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.47 ( ! [X5: A] :
% 0.30/0.47 ( ( P @ X5 )
% 0.30/0.47 => ( Q @ X5 ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_mono
% 0.30/0.47 thf(fact_209_subset__trans,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A,C4: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_trans
% 0.30/0.47 thf(fact_210_set__eq__subset,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ^ [Y4: set @ A,Z: set @ A] : ( Y4 = Z ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.47 & ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % set_eq_subset
% 0.30/0.47 thf(fact_211_Collect__mono__iff,axiom,
% 0.30/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.30/0.47 = ( ! [X4: A] :
% 0.30/0.47 ( ( P @ X4 )
% 0.30/0.47 => ( Q @ X4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_mono_iff
% 0.30/0.47 thf(fact_212_psubsetD,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A,C3: A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ( ( member @ A @ C3 @ A5 )
% 0.30/0.47 => ( member @ A @ C3 @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubsetD
% 0.30/0.47 thf(fact_213_psubset__trans,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A,C4: set @ A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
% 0.30/0.47 => ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubset_trans
% 0.30/0.47 thf(fact_214_psubsetE,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubsetE
% 0.30/0.47 thf(fact_215_psubset__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.47 & ( A7 != B7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubset_eq
% 0.30/0.47 thf(fact_216_psubset__imp__subset,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubset_imp_subset
% 0.30/0.47 thf(fact_217_psubset__subset__trans,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A,C4: set @ A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.30/0.47 => ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % psubset_subset_trans
% 0.30/0.47 thf(fact_218_subset__not__subset__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.47 & ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_not_subset_eq
% 0.30/0.47 thf(fact_219_subset__psubset__trans,axiom,
% 0.30/0.47 ! [A: $tType,A5: set @ A,B6: set @ A,C4: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.47 => ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
% 0.30/0.47 => ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_psubset_trans
% 0.30/0.47 thf(fact_220_subset__iff__psubset__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.30/0.47 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.47 | ( A7 = B7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_iff_psubset_eq
% 0.30/0.47 thf(fact_221_less__nres_Opelims_I1_J,axiom,
% 0.30/0.47 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
% 0.30/0.47 ( ( ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 0.30/0.47 = Y3 )
% 0.30/0.47 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
% 0.30/0.47 => ( ( ( X3
% 0.30/0.47 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ( ~ Y3
% 0.30/0.47 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Xa ) ) ) )
% 0.30/0.47 => ( ! [Uv: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ Uv ) )
% 0.30/0.47 => ( ( Xa
% 0.30/0.47 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ( Y3
% 0.30/0.47 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) ) ) ) ) )
% 0.30/0.47 => ~ ! [A2: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ A2 ) )
% 0.30/0.47 => ! [B2: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ( ( Y3
% 0.30/0.47 = ( ord_less @ ( set @ A ) @ A2 @ B2 ) )
% 0.30/0.47 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_nres.pelims(1)
% 0.30/0.47 thf(fact_222_less__nres_Opelims_I2_J,axiom,
% 0.30/0.47 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 0.30/0.47 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
% 0.30/0.47 => ( ! [Uv: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ Uv ) )
% 0.30/0.47 => ( ( Xa
% 0.30/0.47 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) ) ) ) )
% 0.30/0.47 => ~ ! [A2: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ A2 ) )
% 0.30/0.47 => ! [B2: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
% 0.30/0.47 => ~ ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_nres.pelims(2)
% 0.30/0.47 thf(fact_223_less__nres_Opelims_I3_J,axiom,
% 0.30/0.47 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ~ ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 0.30/0.47 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
% 0.30/0.47 => ( ( ( X3
% 0.30/0.47 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Xa ) ) )
% 0.30/0.47 => ~ ! [A2: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ A2 ) )
% 0.30/0.47 => ! [B2: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
% 0.30/0.47 => ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_nres.pelims(3)
% 0.30/0.47 thf(fact_224_less__eq__nres_Opelims_I3_J,axiom,
% 0.30/0.47 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 0.30/0.47 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
% 0.30/0.47 => ( ! [A2: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ A2 ) )
% 0.30/0.47 => ! [B2: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) )
% 0.30/0.47 => ~ ( ( X3
% 0.30/0.47 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ! [Uv: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ Uv ) )
% 0.30/0.47 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_nres.pelims(3)
% 0.30/0.47 thf(fact_225_less__eq__nres_Opelims_I2_J,axiom,
% 0.30/0.47 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 0.30/0.47 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
% 0.30/0.47 => ( ( ( Xa
% 0.30/0.47 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ ( refine1767639642_FAILi @ A ) ) ) )
% 0.30/0.47 => ~ ! [A2: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ A2 ) )
% 0.30/0.47 => ! [B2: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
% 0.30/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_nres.pelims(2)
% 0.30/0.47 thf(fact_226_less__eq__nres_Opelims_I1_J,axiom,
% 0.30/0.47 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
% 0.30/0.47 ( ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 0.30/0.47 = Y3 )
% 0.30/0.47 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
% 0.30/0.47 => ( ( ( Xa
% 0.30/0.47 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ( Y3
% 0.30/0.47 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ ( refine1767639642_FAILi @ A ) ) ) ) )
% 0.30/0.47 => ( ! [A2: set @ A] :
% 0.30/0.47 ( ( X3
% 0.30/0.47 = ( refine605929679le_RES @ A @ A2 ) )
% 0.30/0.47 => ! [B2: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.47 => ( ( Y3
% 0.30/0.47 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) )
% 0.30/0.47 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) ) ) ) )
% 0.30/0.47 => ~ ( ( X3
% 0.30/0.47 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.47 => ! [Uv: set @ A] :
% 0.30/0.47 ( ( Xa
% 0.30/0.47 = ( refine605929679le_RES @ A @ Uv ) )
% 0.30/0.47 => ( ~ Y3
% 0.30/0.47 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_nres.pelims(1)
% 0.30/0.47 thf(fact_227_GreatestI2__order,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [P: A > $o,X3: A,Q: A > $o] :
% 0.30/0.47 ( ( P @ X3 )
% 0.30/0.47 => ( ! [Y5: A] :
% 0.30/0.47 ( ( P @ Y5 )
% 0.30/0.47 => ( ord_less_eq @ A @ Y5 @ X3 ) )
% 0.30/0.47 => ( ! [X5: A] :
% 0.30/0.47 ( ( P @ X5 )
% 0.30/0.47 => ( ! [Y7: A] :
% 0.30/0.47 ( ( P @ Y7 )
% 0.30/0.47 => ( ord_less_eq @ A @ Y7 @ X5 ) )
% 0.30/0.47 => ( Q @ X5 ) ) )
% 0.30/0.47 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % GreatestI2_order
% 0.30/0.47 thf(fact_228_Greatest__equality,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [P: A > $o,X3: A] :
% 0.30/0.47 ( ( P @ X3 )
% 0.30/0.47 => ( ! [Y5: A] :
% 0.30/0.47 ( ( P @ Y5 )
% 0.30/0.47 => ( ord_less_eq @ A @ Y5 @ X3 ) )
% 0.30/0.47 => ( ( order_Greatest @ A @ P )
% 0.30/0.47 = X3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Greatest_equality
% 0.30/0.47 thf(fact_229_subset__Collect__conv,axiom,
% 0.30/0.47 ! [A: $tType,S: set @ A,P: A > $o] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
% 0.30/0.47 = ( ! [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ S )
% 0.30/0.47 => ( P @ X4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_Collect_conv
% 0.30/0.47 thf(fact_230_relprop__triggers_I6_J,axiom,
% 0.30/0.47 ! [I2: $tType,R2: set @ I2,R3: set @ I2] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ I2 ) @ R2 @ R3 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ I2 ) @ R2 @ R3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % relprop_triggers(6)
% 0.30/0.47 thf(fact_231_ord__eq__le__eq__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A )
% 0.30/0.47 => ! [A3: A,B3: A,C3: A,D4: A] :
% 0.30/0.47 ( ( A3 = B3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.30/0.47 => ( ( C3 = D4 )
% 0.30/0.47 => ( ord_less_eq @ A @ A3 @ D4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_le_eq_trans
% 0.30/0.47 thf(fact_232_bex2I,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A3: A,B3: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ S )
% 0.30/0.47 => ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ S )
% 0.30/0.47 => ( P @ A3 @ B3 ) )
% 0.30/0.47 => ? [A2: A,B2: B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S )
% 0.30/0.47 & ( P @ A2 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bex2I
% 0.30/0.47 thf(fact_233_pairself_Ocases,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X3: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
% 0.30/0.47 ~ ! [F6: A > B,A2: A,B2: A] :
% 0.30/0.47 ( X3
% 0.30/0.47 != ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F6 @ ( product_Pair @ A @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pairself.cases
% 0.30/0.47 thf(fact_234_pairself_Oinduct,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P: ( A > B ) > ( product_prod @ A @ A ) > $o,A0: A > B,A1: product_prod @ A @ A] :
% 0.30/0.47 ( ! [F6: A > B,A2: A,B2: A] : ( P @ F6 @ ( product_Pair @ A @ A @ A2 @ B2 ) )
% 0.30/0.47 => ( P @ A0 @ A1 ) ) ).
% 0.30/0.47
% 0.30/0.47 % pairself.induct
% 0.30/0.47 thf(fact_235_uncurry__apply,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B3: C] :
% 0.30/0.48 ( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 0.30/0.48 = ( F @ A3 @ B3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % uncurry_apply
% 0.30/0.48 thf(fact_236_measure__induct,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( wellorder @ B )
% 0.30/0.48 => ! [F: A > B,P: A > $o,A3: A] :
% 0.30/0.48 ( ! [X5: A] :
% 0.30/0.48 ( ! [Y7: A] :
% 0.30/0.48 ( ( ord_less @ B @ ( F @ Y7 ) @ ( F @ X5 ) )
% 0.30/0.48 => ( P @ Y7 ) )
% 0.30/0.48 => ( P @ X5 ) )
% 0.30/0.48 => ( P @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % measure_induct
% 0.30/0.48 thf(fact_237_measure__induct__rule,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( wellorder @ B )
% 0.30/0.48 => ! [F: A > B,P: A > $o,A3: A] :
% 0.30/0.48 ( ! [X5: A] :
% 0.30/0.48 ( ! [Y7: A] :
% 0.30/0.48 ( ( ord_less @ B @ ( F @ Y7 ) @ ( F @ X5 ) )
% 0.30/0.48 => ( P @ Y7 ) )
% 0.30/0.48 => ( P @ X5 ) )
% 0.30/0.48 => ( P @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % measure_induct_rule
% 0.30/0.48 thf(fact_238_linorder__neqE__linordered__idom,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [X3: A,Y3: A] :
% 0.30/0.48 ( ( X3 != Y3 )
% 0.30/0.48 => ( ~ ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.48 => ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_neqE_linordered_idom
% 0.30/0.48 thf(fact_239_linordered__field__no__ub,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_field @ A )
% 0.30/0.48 => ! [X7: A] :
% 0.30/0.48 ? [X_1: A] : ( ord_less @ A @ X7 @ X_1 ) ) ).
% 0.30/0.48
% 0.30/0.48 % linordered_field_no_ub
% 0.30/0.48 thf(fact_240_linordered__field__no__lb,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_field @ A )
% 0.30/0.48 => ! [X7: A] :
% 0.30/0.48 ? [Y5: A] : ( ord_less @ A @ Y5 @ X7 ) ) ).
% 0.30/0.48
% 0.30/0.48 % linordered_field_no_lb
% 0.30/0.48 thf(fact_241_all__nat__split__at,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [K: A,P: A > $o] :
% 0.30/0.48 ( ! [I3: A] :
% 0.30/0.48 ( ( ord_less @ A @ I3 @ K )
% 0.30/0.48 => ( P @ I3 ) )
% 0.30/0.48 => ( ( P @ K )
% 0.30/0.48 => ( ! [I3: A] :
% 0.30/0.48 ( ( ord_less @ A @ K @ I3 )
% 0.30/0.48 => ( P @ I3 ) )
% 0.30/0.48 => ! [X_12: A] : ( P @ X_12 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % all_nat_split_at
% 0.30/0.48 thf(fact_242_dependent__wellorder__choice,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( wellorder @ A )
% 0.30/0.48 => ! [P: ( A > B ) > A > B > $o] :
% 0.30/0.48 ( ! [R4: B,F6: A > B,G4: A > B,X5: A] :
% 0.30/0.48 ( ! [Y7: A] :
% 0.30/0.48 ( ( ord_less @ A @ Y7 @ X5 )
% 0.30/0.48 => ( ( F6 @ Y7 )
% 0.30/0.48 = ( G4 @ Y7 ) ) )
% 0.30/0.48 => ( ( P @ F6 @ X5 @ R4 )
% 0.30/0.48 = ( P @ G4 @ X5 @ R4 ) ) )
% 0.30/0.48 => ( ! [X5: A,F6: A > B] :
% 0.30/0.48 ( ! [Y7: A] :
% 0.30/0.48 ( ( ord_less @ A @ Y7 @ X5 )
% 0.30/0.48 => ( P @ F6 @ Y7 @ ( F6 @ Y7 ) ) )
% 0.30/0.48 => ? [X_12: B] : ( P @ F6 @ X5 @ X_12 ) )
% 0.30/0.48 => ? [F6: A > B] :
% 0.30/0.48 ! [X7: A] : ( P @ F6 @ X7 @ ( F6 @ X7 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dependent_wellorder_choice
% 0.30/0.48 thf(fact_243_Refine__Misc_Oif__mono,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A )
% 0.30/0.48 => ! [B3: $o,M1: A,M12: A,M22: A,M23: A] :
% 0.30/0.48 ( ( B3
% 0.30/0.48 => ( ord_less_eq @ A @ M1 @ M12 ) )
% 0.30/0.48 => ( ( ~ B3
% 0.30/0.48 => ( ord_less_eq @ A @ M22 @ M23 ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( if @ A @ B3 @ M1 @ M22 ) @ ( if @ A @ B3 @ M12 @ M23 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Refine_Misc.if_mono
% 0.30/0.48 thf(fact_244_chain__subset__def,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( chain_subset @ A )
% 0.30/0.48 = ( ^ [C5: set @ ( set @ A )] :
% 0.30/0.48 ! [X4: set @ A] :
% 0.30/0.48 ( ( member @ ( set @ A ) @ X4 @ C5 )
% 0.30/0.48 => ! [Y6: set @ A] :
% 0.30/0.48 ( ( member @ ( set @ A ) @ Y6 @ C5 )
% 0.30/0.48 => ( ( ord_less_eq @ ( set @ A ) @ X4 @ Y6 )
% 0.30/0.48 | ( ord_less_eq @ ( set @ A ) @ Y6 @ X4 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % chain_subset_def
% 0.30/0.48
% 0.30/0.48 % Type constructors (13)
% 0.30/0.48 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.48 ! [A8: $tType,A9: $tType] :
% 0.30/0.48 ( ( preorder @ A9 )
% 0.30/0.48 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.48 ! [A8: $tType,A9: $tType] :
% 0.30/0.48 ( ( order @ A9 )
% 0.30/0.48 => ( order @ ( A8 > A9 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.48 ! [A8: $tType,A9: $tType] :
% 0.30/0.48 ( ( ord @ A9 )
% 0.30/0.48 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
% 0.30/0.48 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 0.30/0.48 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
% 0.30/0.48 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
% 0.30/0.48 preorder @ $o ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.30/0.48 linorder @ $o ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
% 0.30/0.48 order @ $o ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
% 0.30/0.48 ord @ $o ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_7,axiom,
% 0.30/0.48 ! [A8: $tType] : ( preorder @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_8,axiom,
% 0.30/0.48 ! [A8: $tType] : ( order @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_9,axiom,
% 0.30/0.48 ! [A8: $tType] : ( ord @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.30/0.48
% 0.30/0.48 % Helper facts (3)
% 0.30/0.48 thf(help_If_3_1_T,axiom,
% 0.30/0.48 ! [P: $o] :
% 0.30/0.48 ( ( P = $true )
% 0.30/0.48 | ( P = $false ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(help_If_2_1_T,axiom,
% 0.30/0.48 ! [A: $tType,X3: A,Y3: A] :
% 0.30/0.48 ( ( if @ A @ $false @ X3 @ Y3 )
% 0.30/0.48 = Y3 ) ).
% 0.30/0.48
% 0.30/0.48 thf(help_If_1_1_T,axiom,
% 0.30/0.48 ! [A: $tType,X3: A,Y3: A] :
% 0.30/0.48 ( ( if @ A @ $true @ X3 @ Y3 )
% 0.30/0.48 = X3 ) ).
% 0.30/0.48
% 0.30/0.48 % Conjectures (1)
% 0.30/0.48 thf(conj_0,conjecture,
% 0.30/0.48 ( ( refine406925620_inres @ a @ ( refine605929679le_RES @ a @ ( collect @ a @ phi ) ) @ x )
% 0.30/0.48 = ( phi @ x ) ) ).
% 0.30/0.48
% 0.30/0.48 %------------------------------------------------------------------------------
% 0.30/0.48 ------------------------
% 3.60/1.76 ---- Embedded in THF ---
% 3.60/1.76 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.60/1.76 %%% Generated on Tue Feb 24 08:28:28 EST 2026
% 3.60/1.76 %%% using '$$dhol' embedding, version 1.3.0.
% 3.60/1.76 %%% Logic specification used:
% 3.60/1.76 %%% thf(spec, logic, $$dhol).
% 3.60/1.76
% 3.60/1.76 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.jdFRlkfssT/DTF2DTF14407.p
% 3.60/1.76 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres, type, refine1665802226e_nres: ($tType > $tType)).
% 3.60/1.76 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 3.60/1.76 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.60/1.76 thf(ty_tf_a, type, a: $tType).
% 3.60/1.76 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: ($o))).
% 3.60/1.76 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 3.60/1.76 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 3.60/1.76 thf(sy_c_Misc_Ouncurry, type, uncurry: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.60/1.76 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 3.60/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.60/1.76 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 3.60/1.76 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 3.60/1.76 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.60/1.76 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 3.60/1.76 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN, type, refine1687780735RETURN: (!> [A:$tType]: ((A > (refine1665802226e_nres @ A))))).
% 3.60/1.76 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres, type, refine1315500908_inres: (!> [A:$tType]: (((refine1665802226e_nres @ A) > (A > $o))))).
% 3.60/1.76 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oless__eq__nres__rel, type, refine1554218259es_rel: (!> [A:$tType]: ((((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) > (((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) > $o))))).
% 3.60/1.76 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oless__nres__rel, type, refine1378444575es_rel: (!> [A:$tType]: ((((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) > (((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) > $o))))).
% 3.60/1.76 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres, type, refine406925620_inres: (!> [A:$tType]: (((refine1665802226e_nres @ A) > (A > $o))))).
% 3.60/1.76 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail, type, refine1102455758nofail: (!> [A:$tType]: (((refine1665802226e_nres @ A) > $o)))).
% 3.60/1.76 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_OFAILi, type, refine1767639642_FAILi: (!> [A:$tType]: ((refine1665802226e_nres @ A)))).
% 3.60/1.76 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES, type, refine605929679le_RES: (!> [A:$tType]: (((set @ A) > (refine1665802226e_nres @ A))))).
% 3.60/1.76 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_Ocase__nres, type, refine60818195e_nres: (!> [B:$tType,A:$tType]: ((B > (((set @ A) > B) > ((refine1665802226e_nres @ A) > B)))))).
% 3.60/1.76 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_Orec__nres, type, refine1442219249c_nres: (!> [C:$tType,A:$tType]: ((C > (((set @ A) > C) > ((refine1665802226e_nres @ A) > C)))))).
% 3.60/1.76 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES, type, refine1672542526he_RES: (!> [A:$tType]: (((refine1665802226e_nres @ A) > (set @ A))))).
% 3.60/1.76 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.60/1.76 thf(sy_c_Wellfounded_Oaccp, type, accp: (!> [A:$tType]: (((A > (A > $o)) > (A > $o))))).
% 3.60/1.76 thf(sy_c_Zorn_Ochain__subset, type, chain_subset: (!> [A:$tType]: (((set @ (set @ A)) > $o)))).
% 3.60/1.76 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.60/1.76 thf(sy_v__092_060Phi_062, type, phi: (a > $o)).
% 3.60/1.76 thf(sy_v_x, type, x: a).
% 3.60/1.76 thf(fact_0_nres__more__simps_I4_J, axiom, (! [A:$tType,X:(set @ A),Y:(set @ A)]: (((((refine605929679le_RES @ A) @ X) = ((refine605929679le_RES @ A) @ Y)) = (X = Y))))).
% 3.60/1.76 thf(fact_1_nres_Oinject, axiom, (! [A:$tType,X2:(set @ A),Y2:(set @ A)]: (((((refine605929679le_RES @ A) @ X2) = ((refine605929679le_RES @ A) @ Y2)) = (X2 = Y2))))).
% 3.60/1.76 thf(fact_2_nf__inres__RES, axiom, (! [A:$tType,X:(set @ A),X3:A]: (((((refine406925620_inres @ A) @ ((refine605929679le_RES @ A) @ X)) @ X3) = (((member @ A) @ X3) @ X))))).
% 3.60/1.76 thf(fact_3_the__RES_Osimps, axiom, (! [A:$tType,X:(set @ A)]: ((((refine1672542526he_RES @ A) @ ((refine605929679le_RES @ A) @ X)) = X)))).
% 3.60/1.76 thf(fact_4_nres_Osimps_I7_J, axiom, (! [C:$tType,A:$tType,F1:C,F2:((set @ A) > C),X2:(set @ A)]: (((((((refine1442219249c_nres @ C) @ A) @ F1) @ F2) @ ((refine605929679le_RES @ A) @ X2)) = (F2 @ X2))))).
% 3.60/1.76 thf(fact_5_nres_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,F1:B,F2:((set @ A) > B),X2:(set @ A)]: (((((((refine60818195e_nres @ B) @ A) @ F1) @ F2) @ ((refine605929679le_RES @ A) @ X2)) = (F2 @ X2))))).
% 3.60/1.76 thf(fact_6_inres__simps_I2_J, axiom, (! [B:$tType,X:(set @ B)]: ((((refine1315500908_inres @ B) @ ((refine605929679le_RES @ B) @ X)) = (^ [X4:B]: ((((member @ B) @ X4) @ X))))))).
% 3.60/1.76 thf(fact_7_nofail__simps_I2_J, axiom, (! [B:$tType,X:(set @ B)]: (((refine1102455758nofail @ B) @ ((refine605929679le_RES @ B) @ X))))).
% 3.60/1.76 thf(fact_8_nofail__RES__conv, axiom, (! [A:$tType]: (((refine1102455758nofail @ A) = (^ [M:(refine1665802226e_nres @ A)]: ((? [M2:(set @ A)]: ((M = ((refine605929679le_RES @ A) @ M2)))))))))).
% 3.60/1.76 thf(fact_9_nres_Odistinct_I1_J, axiom, (! [A:$tType,X2:(set @ A)]: (((refine1767639642_FAILi @ A) != ((refine605929679le_RES @ A) @ X2))))).
% 3.60/1.76 thf(fact_10_nres_Oinduct, axiom, (! [A:$tType,P:((refine1665802226e_nres @ A) > $o),Nres:(refine1665802226e_nres @ A)]: (((P @ (refine1767639642_FAILi @ A)) => ((! [X5:(set @ A)]: ((P @ ((refine605929679le_RES @ A) @ X5)))) => (P @ Nres)))))).
% 3.60/1.76 thf(fact_11_nres_Oexhaust, axiom, (! [A:$tType,Y3:(refine1665802226e_nres @ A)]: (((Y3 != (refine1767639642_FAILi @ A)) => (~ ((! [X22:(set @ A)]: ((Y3 != ((refine605929679le_RES @ A) @ X22)))))))))).
% 3.60/1.76 thf(fact_12_sup__nres_Oinduct, axiom, (! [A:$tType,P:((refine1665802226e_nres @ A) > ((refine1665802226e_nres @ A) > $o)),A0:(refine1665802226e_nres @ A),A1:(refine1665802226e_nres @ A)]: (((! [Uu:(refine1665802226e_nres @ A)]: (((P @ Uu) @ (refine1767639642_FAILi @ A)))) => ((! [V:(set @ A)]: (((P @ (refine1767639642_FAILi @ A)) @ ((refine605929679le_RES @ A) @ V)))) => ((! [A2:(set @ A),B2:(set @ A)]: (((P @ ((refine605929679le_RES @ A) @ A2)) @ ((refine605929679le_RES @ A) @ B2)))) => ((P @ A0) @ A1))))))).
% 3.60/1.76 thf(fact_13_less__nres_Oinduct, axiom, (! [A:$tType,P:((refine1665802226e_nres @ A) > ((refine1665802226e_nres @ A) > $o)),A0:(refine1665802226e_nres @ A),A1:(refine1665802226e_nres @ A)]: (((! [X_1:(refine1665802226e_nres @ A)]: (((P @ (refine1767639642_FAILi @ A)) @ X_1))) => ((! [Uv:(set @ A)]: (((P @ ((refine605929679le_RES @ A) @ Uv)) @ (refine1767639642_FAILi @ A)))) => ((! [A2:(set @ A),B2:(set @ A)]: (((P @ ((refine605929679le_RES @ A) @ A2)) @ ((refine605929679le_RES @ A) @ B2)))) => ((P @ A0) @ A1))))))).
% 3.60/1.76 thf(fact_14_less__eq__nres_Oinduct, axiom, (! [A:$tType,P:((refine1665802226e_nres @ A) > ((refine1665802226e_nres @ A) > $o)),A0:(refine1665802226e_nres @ A),A1:(refine1665802226e_nres @ A)]: (((! [Uu:(refine1665802226e_nres @ A)]: (((P @ Uu) @ (refine1767639642_FAILi @ A)))) => ((! [A2:(set @ A),B2:(set @ A)]: (((P @ ((refine605929679le_RES @ A) @ A2)) @ ((refine605929679le_RES @ A) @ B2)))) => ((! [Uv:(set @ A)]: (((P @ (refine1767639642_FAILi @ A)) @ ((refine605929679le_RES @ A) @ Uv)))) => ((P @ A0) @ A1))))))).
% 3.60/1.76 thf(fact_15_the__RES__inv, axiom, (! [A:$tType,M3:(refine1665802226e_nres @ A)]: ((((refine1102455758nofail @ A) @ M3) => (((refine605929679le_RES @ A) @ ((refine1672542526he_RES @ A) @ M3)) = M3))))).
% 3.60/1.76 thf(fact_16_nres_Osimps_I6_J, axiom, (! [A:$tType,C:$tType,F1:C,F2:((set @ A) > C)]: (((((((refine1442219249c_nres @ C) @ A) @ F1) @ F2) @ (refine1767639642_FAILi @ A)) = F1)))).
% 3.60/1.76 thf(fact_17_nres_Osimps_I4_J, axiom, (! [A:$tType,B:$tType,F1:B,F2:((set @ A) > B)]: (((((((refine60818195e_nres @ B) @ A) @ F1) @ F2) @ (refine1767639642_FAILi @ A)) = F1)))).
% 3.60/1.76 thf(fact_18_pw__eqI, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S2:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S) = ((refine1102455758nofail @ A) @ S2)) => ((! [X5:A]: (((((refine1315500908_inres @ A) @ S) @ X5) = (((refine1315500908_inres @ A) @ S2) @ X5)))) => (S = S2)))))).
% 3.60/1.76 thf(fact_19_pw__eq__iff, axiom, (! [A:$tType]: (((^ [Y4:(refine1665802226e_nres @ A),Z:(refine1665802226e_nres @ A)]: ((Y4 = Z))) = (^ [S3:(refine1665802226e_nres @ A),S4:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S3) = ((refine1102455758nofail @ A) @ S4)) & (! [X4:A]: (((((refine1315500908_inres @ A) @ S3) @ X4) = (((refine1315500908_inres @ A) @ S4) @ X4))))))))))).
% 3.60/1.76 thf(fact_20_not__nofail__inres, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),X3:A]: (((~ (((refine1102455758nofail @ A) @ S))) => (((refine1315500908_inres @ A) @ S) @ X3))))).
% 3.60/1.76 thf(fact_21_nf__inres__def, axiom, (! [A:$tType]: (((refine406925620_inres @ A) = (^ [M:(refine1665802226e_nres @ A),X4:A]: ((((refine1102455758nofail @ A) @ M) & (((refine1315500908_inres @ A) @ M) @ X4)))))))).
% 3.60/1.76 thf(fact_22_inres__simps_I3_J, axiom, (! [C:$tType,X3:C]: ((((refine1315500908_inres @ C) @ ((refine1687780735RETURN @ C) @ X3)) = ((^ [Y4:C,Z:C]: ((Y4 = Z))) @ X3))))).
% 3.60/1.76 thf(fact_23_nofail__simps_I3_J, axiom, (! [C:$tType,X3:C]: (((refine1102455758nofail @ C) @ ((refine1687780735RETURN @ C) @ X3))))).
% 3.60/1.76 thf(fact_24_pw__leI, axiom, (! [A:$tType,S2:(refine1665802226e_nres @ A),S:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S2) => (((refine1102455758nofail @ A) @ S) & (! [X5:A]: (((((refine1315500908_inres @ A) @ S) @ X5) => (((refine1315500908_inres @ A) @ S2) @ X5)))))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S2))))).
% 3.60/1.76 thf(fact_25_pw__leI_H, axiom, (! [A:$tType,S2:(refine1665802226e_nres @ A),S:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S2) => ((refine1102455758nofail @ A) @ S)) => ((! [X5:A]: ((((refine1102455758nofail @ A) @ S2) => ((((refine1315500908_inres @ A) @ S) @ X5) => (((refine1315500908_inres @ A) @ S2) @ X5))))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S2)))))).
% 3.60/1.76 thf(fact_26_pw__le__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (refine1665802226e_nres @ A)) = (^ [S3:(refine1665802226e_nres @ A),S4:(refine1665802226e_nres @ A)]: ((((refine1102455758nofail @ A) @ S4) => (((refine1102455758nofail @ A) @ S3) & (! [X4:A]: (((((refine1315500908_inres @ A) @ S3) @ X4) => (((refine1315500908_inres @ A) @ S4) @ X4)))))))))))).
% 3.60/1.76 thf(fact_27_less__eq__nres_Osimps_I3_J, axiom, (! [A:$tType,Uv2:(set @ A)]: ((~ ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ (refine1767639642_FAILi @ A)) @ ((refine605929679le_RES @ A) @ Uv2))))))).
% 3.60/1.76 thf(fact_28_less__nres_Osimps_I2_J, axiom, (! [A:$tType,Uv2:(set @ A)]: ((((ord_less @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ Uv2)) @ (refine1767639642_FAILi @ A))))).
% 3.60/1.76 thf(fact_29_less__eq__nres_Ocases, axiom, (! [A:$tType,X3:((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))]: (((! [Uu:(refine1665802226e_nres @ A)]: ((X3 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ Uu) @ (refine1767639642_FAILi @ A))))) => ((! [A2:(set @ A),B2:(set @ A)]: ((X3 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A2)) @ ((refine605929679le_RES @ A) @ B2))))) => (~ ((! [Uv:(set @ A)]: ((X3 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ (refine1767639642_FAILi @ A)) @ ((refine605929679le_RES @ A) @ Uv)))))))))))).
% 3.60/1.76 thf(fact_30_less__nres_Ocases, axiom, (! [A:$tType,X3:((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))]: (((! [Uu:(refine1665802226e_nres @ A)]: ((X3 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ (refine1767639642_FAILi @ A)) @ Uu)))) => ((! [Uv:(set @ A)]: ((X3 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ Uv)) @ (refine1767639642_FAILi @ A))))) => (~ ((! [A2:(set @ A),B2:(set @ A)]: ((X3 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A2)) @ ((refine605929679le_RES @ A) @ B2)))))))))))).
% 3.60/1.76 thf(fact_31_sup__nres_Ocases, axiom, (! [A:$tType,X3:((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))]: (((! [Uu:(refine1665802226e_nres @ A)]: ((X3 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ Uu) @ (refine1767639642_FAILi @ A))))) => ((! [V:(set @ A)]: ((X3 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ (refine1767639642_FAILi @ A)) @ ((refine605929679le_RES @ A) @ V))))) => (~ ((! [A2:(set @ A),B2:(set @ A)]: ((X3 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A2)) @ ((refine605929679le_RES @ A) @ B2)))))))))))).
% 3.60/1.76 thf(fact_32_nres__more__simps_I6_J, axiom, (! [A:$tType,X3:A,Y3:A]: (((((refine1687780735RETURN @ A) @ X3) = ((refine1687780735RETURN @ A) @ Y3)) = (X3 = Y3))))).
% 3.60/1.76 thf(fact_33_nres__order__simps_I20_J, axiom, (! [W:$tType,X3:W,Y3:W]: (((((ord_less_eq @ (refine1665802226e_nres @ W)) @ ((refine1687780735RETURN @ W) @ X3)) @ ((refine1687780735RETURN @ W) @ Y3)) = (X3 = Y3))))).
% 3.60/1.76 thf(fact_34_nres__order__simps_I21_J, axiom, (! [X6:$tType,X3:X6,Y:(set @ X6)]: (((((ord_less_eq @ (refine1665802226e_nres @ X6)) @ ((refine1687780735RETURN @ X6) @ X3)) @ ((refine605929679le_RES @ X6) @ Y)) = (((member @ X6) @ X3) @ Y))))).
% 3.60/1.76 thf(fact_35_inres__def, axiom, (! [A:$tType]: (((refine1315500908_inres @ A) = (^ [S3:(refine1665802226e_nres @ A),X4:A]: ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X4)) @ S3))))))).
% 3.60/1.76 thf(fact_36_less__nres_Osimps_I1_J, axiom, (! [A:$tType,Uu2:(refine1665802226e_nres @ A)]: ((~ ((((ord_less @ (refine1665802226e_nres @ A)) @ (refine1767639642_FAILi @ A)) @ Uu2)))))).
% 3.60/1.76 thf(fact_37_less__eq__nres_Osimps_I1_J, axiom, (! [A:$tType,Uu2:(refine1665802226e_nres @ A)]: ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ Uu2) @ (refine1767639642_FAILi @ A))))).
% 3.60/1.76 thf(fact_38_le__nofailI, axiom, (! [A:$tType,M4:(refine1665802226e_nres @ A),M5:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ M4) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ M5) @ M4)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ M5) @ M4))))).
% 3.60/1.76 thf(fact_39_pwD1, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S2:(refine1665802226e_nres @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S2) => (((refine1102455758nofail @ A) @ S2) => ((refine1102455758nofail @ A) @ S)))))).
% 3.60/1.76 thf(fact_40_pwD2, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S2:(refine1665802226e_nres @ A),X3:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S2) => ((((refine1315500908_inres @ A) @ S) @ X3) => (((refine1315500908_inres @ A) @ S2) @ X3)))))).
% 3.60/1.76 thf(fact_41_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A4:A,B4:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A4) @ B4)) = ((A3 = A4) & (B3 = B4)))))).
% 3.60/1.76 thf(fact_42_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X2:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X2) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 3.60/1.76 thf(fact_43_order__mono__setup_Orefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ X3) @ X3))))))).
% 3.60/1.76 thf(fact_44_complete__interval, axiom, (! [A:$tType]: (((condit1037483654norder @ A) => (! [A3:A,B3:A,P:(A > $o)]: (((((ord_less @ A) @ A3) @ B3) => ((P @ A3) => ((~ ((P @ B3))) => (? [C2:A]: (((((ord_less_eq @ A) @ A3) @ C2) & ((((ord_less_eq @ A) @ C2) @ B3) & ((! [X7:A]: ((((((ord_less_eq @ A) @ A3) @ X7) & (((ord_less @ A) @ X7) @ C2)) => (P @ X7)))) & (! [D:A]: (((! [X5:A]: ((((((ord_less_eq @ A) @ A3) @ X5) & (((ord_less @ A) @ X5) @ D)) => (P @ X5)))) => (((ord_less_eq @ A) @ D) @ C2)))))))))))))))))).
% 3.60/1.76 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 3.60/1.76 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A5)))) = A5)))).
% 3.60/1.76 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) = (Q @ X5)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.60/1.76 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X5:A]: (((F @ X5) = (G @ X5)))) => (F = G))))).
% 3.60/1.76 thf(fact_49_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((A3 != B3) => ((((ord_less_eq @ A) @ A3) @ B3) => (((ord_less @ A) @ A3) @ B3))))))))).
% 3.60/1.76 thf(fact_50_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less @ A) @ B3) @ A3) => (((ord_less_eq @ A) @ B3) @ A3)))))))).
% 3.60/1.76 thf(fact_51_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B5:A,A6:A]: (((((ord_less_eq @ A) @ B5) @ A6) & (A6 != B5))))))))).
% 3.60/1.76 thf(fact_52_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B5:A,A6:A]: (((((ord_less @ A) @ B5) @ A6) | (A6 = B5))))))))).
% 3.60/1.76 thf(fact_53_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (((ord_less_eq @ A) @ A3) @ B3)))))))).
% 3.60/1.76 thf(fact_54_less__eq__nres_Osimps_I2_J, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A3)) @ ((refine605929679le_RES @ A) @ B3)) = (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 3.60/1.76 thf(fact_55_nres__order__simps_I5_J, axiom, (! [E:$tType,X:(set @ E),Y:(set @ E)]: (((((ord_less_eq @ (refine1665802226e_nres @ E)) @ ((refine605929679le_RES @ E) @ X)) @ ((refine605929679le_RES @ E) @ Y)) = (((ord_less_eq @ (set @ E)) @ X) @ Y))))).
% 3.60/1.76 thf(fact_56_less__nres_Osimps_I3_J, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A3)) @ ((refine605929679le_RES @ A) @ B3)) = (((ord_less @ (set @ A)) @ A3) @ B3))))).
% 3.60/1.76 thf(fact_57_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3))))))))).
% 3.60/1.76 thf(fact_58_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3))))))))).
% 3.60/1.76 thf(fact_59_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X5:A]: ((((ord_less_eq @ B) @ (F @ X5)) @ (G @ X5)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 3.60/1.76 thf(fact_60_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: ((! [X4:A]: ((((ord_less_eq @ B) @ (F3 @ X4)) @ (G2 @ X4))))))))))).
% 3.60/1.76 thf(fact_61_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C3) => ((! [X5:B,Y5:B]: (((((ord_less_eq @ B) @ X5) @ Y5) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y5))))) => (((ord_less_eq @ A) @ A3) @ (F @ C3))))))))))).
% 3.60/1.76 thf(fact_62_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B3:A,F:(A > C),C3:C]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C3) => ((! [X5:A,Y5:A]: (((((ord_less_eq @ A) @ X5) @ Y5) => (((ord_less_eq @ C) @ (F @ X5)) @ (F @ Y5))))) => (((ord_less_eq @ C) @ (F @ A3)) @ C3)))))))))).
% 3.60/1.76 thf(fact_63_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((A3 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C3) => ((! [X5:B,Y5:B]: (((((ord_less_eq @ B) @ X5) @ Y5) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y5))))) => (((ord_less_eq @ A) @ A3) @ (F @ C3))))))))))).
% 3.60/1.76 thf(fact_64_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B3:A,F:(A > B),C3:B]: (((((ord_less_eq @ A) @ A3) @ B3) => (((F @ B3) = C3) => ((! [X5:A,Y5:A]: (((((ord_less_eq @ A) @ X5) @ Y5) => (((ord_less_eq @ B) @ (F @ X5)) @ (F @ Y5))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C3)))))))))).
% 3.60/1.76 thf(fact_65_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [X4:A,Y6:A]: (((((ord_less_eq @ A) @ X4) @ Y6) & (((ord_less_eq @ A) @ Y6) @ X4))))))))).
% 3.60/1.76 thf(fact_66_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X3) => (X3 = Y3))))))))).
% 3.60/1.76 thf(fact_67_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 3.60/1.76 thf(fact_68_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A]: (((X3 = Y3) => (((ord_less_eq @ A) @ X3) @ Y3)))))))).
% 3.60/1.76 thf(fact_69_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 3.60/1.76 thf(fact_70_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 3.60/1.76 thf(fact_71_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A,Z2:A]: ((((((ord_less_eq @ A) @ X3) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z2)))) => (((((ord_less_eq @ A) @ Y3) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Z2)))) => (((((ord_less_eq @ A) @ X3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y3)))) => (((((ord_less_eq @ A) @ Z2) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X3)))) => (((((ord_less_eq @ A) @ Y3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X3)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Y3))))))))))))))))).
% 3.60/1.76 thf(fact_72_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X3:A]: (((((ord_less_eq @ A) @ Y3) @ X3) => ((((ord_less_eq @ A) @ X3) @ Y3) = (X3 = Y3))))))))).
% 3.60/1.76 thf(fact_73_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [A6:A,B5:A]: (((((ord_less_eq @ A) @ A6) @ B5) & (((ord_less_eq @ A) @ B5) @ A6))))))))).
% 3.60/1.76 thf(fact_74_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 3.60/1.76 thf(fact_75_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((B3 = C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 3.60/1.76 thf(fact_76_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ A3) => (A3 = B3))))))))).
% 3.60/1.76 thf(fact_77_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z2) => (((ord_less_eq @ A) @ X3) @ Z2))))))))).
% 3.60/1.76 thf(fact_78_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 3.60/1.76 thf(fact_79_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B3:A]: (((! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((P @ A2) @ B2)))) => ((! [A2:A,B2:A]: ((((P @ B2) @ A2) => ((P @ A2) @ B2)))) => ((P @ A3) @ B3))))))))).
% 3.60/1.76 thf(fact_80_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C3) @ B3) => (((ord_less_eq @ A) @ C3) @ A3))))))))).
% 3.60/1.76 thf(fact_81_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [A6:A,B5:A]: (((((ord_less_eq @ A) @ B5) @ A6) & (((ord_less_eq @ A) @ A6) @ B5))))))))).
% 3.60/1.76 thf(fact_82_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ A3) @ B3) => (A3 = B3))))))))).
% 3.60/1.76 thf(fact_83_order__mono__setup_Omono__if, axiom, (! [A:$tType]: (((preorder @ A) => (! [T2:A,T3:A,E2:A,E3:A,B3:$o]: (((((ord_less_eq @ A) @ T2) @ T3) => ((((ord_less_eq @ A) @ E2) @ E3) => (((ord_less_eq @ A) @ ((((if @ A) @ B3) @ T2) @ E2)) @ ((((if @ A) @ B3) @ T3) @ E3)))))))))).
% 3.60/1.76 thf(fact_84_order__mono__setup_Omono__let, axiom, (! [A:$tType,B:$tType]: (((preorder @ A) => (! [F:(B > A),F4:(B > A),X3:B]: (((! [X5:B]: ((((ord_less_eq @ A) @ (F @ X5)) @ (F4 @ X5)))) => (((ord_less_eq @ A) @ (F @ X3)) @ (F4 @ X3))))))))).
% 3.60/1.76 thf(fact_85_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X5:A,Y5:B]: ((P2 = ((((product_Pair @ A) @ B) @ X5) @ Y5))))))).
% 3.60/1.76 thf(fact_86_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A2:A,B2:B]: ((P @ ((((product_Pair @ A) @ B) @ A2) @ B2)))) => (P @ P2))))).
% 3.60/1.76 thf(fact_87_Pair__inject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A4:A,B4:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A4) @ B4)) => (~ (((A3 = A4) => (B3 != B4)))))))).
% 3.60/1.76 thf(fact_88_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A2:A,B2:B,C2:C]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A2) @ ((((product_Pair @ B) @ C) @ B2) @ C2)))))))))).
% 3.60/1.76 thf(fact_89_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: ((~ ((! [A2:A,B2:B,C2:C,D3:D2]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B2) @ ((((product_Pair @ C) @ D2) @ C2) @ D3))))))))))).
% 3.60/1.76 thf(fact_90_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))))]: ((~ ((! [A2:A,B2:B,C2:C,D3:D2,E4:E]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))) @ B2) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E)) @ C2) @ ((((product_Pair @ D2) @ E) @ D3) @ E4)))))))))))).
% 3.60/1.76 thf(fact_91_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,F5:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5)))))]: ((~ ((! [A2:A,B2:B,C2:C,D3:D2,E4:E,F6:F5]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5))))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5)))) @ B2) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5))) @ C2) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ F5)) @ D3) @ ((((product_Pair @ E) @ F5) @ E4) @ F6))))))))))))).
% 3.60/1.76 thf(fact_92_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,F5:$tType,G3:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3))))))]: ((~ ((! [A2:A,B2:B,C2:C,D3:D2,E4:E,F6:F5,G4:G3]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3)))))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3))))) @ B2) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3)))) @ C2) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3))) @ D3) @ ((((product_Pair @ E) @ ((product_prod @ F5) @ G3)) @ E4) @ ((((product_Pair @ F5) @ G3) @ F6) @ G4)))))))))))))).
% 3.60/1.76 thf(fact_93_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A2:A,B2:B,C2:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A2) @ ((((product_Pair @ B) @ C) @ B2) @ C2))))) => (P @ X3))))).
% 3.60/1.76 thf(fact_94_prod__induct4, axiom, (! [D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: (((! [A2:A,B2:B,C2:C,D3:D2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B2) @ ((((product_Pair @ C) @ D2) @ C2) @ D3)))))) => (P @ X3))))).
% 3.60/1.76 thf(fact_95_prod__induct5, axiom, (! [E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))))]: (((! [A2:A,B2:B,C2:C,D3:D2,E4:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))) @ B2) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E)) @ C2) @ ((((product_Pair @ D2) @ E) @ D3) @ E4))))))) => (P @ X3))))).
% 3.60/1.76 thf(fact_96_prod__induct6, axiom, (! [F5:$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))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5)))))]: (((! [A2:A,B2:B,C2:C,D3:D2,E4:E,F6:F5]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5))))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5)))) @ B2) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F5))) @ C2) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ F5)) @ D3) @ ((((product_Pair @ E) @ F5) @ E4) @ F6)))))))) => (P @ X3))))).
% 3.60/1.76 thf(fact_97_prod__induct7, axiom, (! [G3:$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) @ G3)))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3))))))]: (((! [A2:A,B2:B,C2:C,D3:D2,E4:E,F6:F5,G4:G3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3)))))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3))))) @ B2) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3)))) @ C2) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3))) @ D3) @ ((((product_Pair @ E) @ ((product_prod @ F5) @ G3)) @ E4) @ ((((product_Pair @ F5) @ G3) @ F6) @ G4))))))))) => (P @ X3))))).
% 3.60/1.76 thf(fact_98_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y3:((product_prod @ A) @ B)]: ((~ ((! [A2:A,B2:B]: ((Y3 != ((((product_Pair @ A) @ B) @ A2) @ B2))))))))).
% 3.60/1.76 thf(fact_99_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A2:A,B2:B]: ((P @ ((((product_Pair @ A) @ B) @ A2) @ B2)))) => (P @ Prod))))).
% 3.60/1.76 thf(fact_100_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((A3 = (F @ B3)) => ((((ord_less @ B) @ B3) @ C3) => ((! [X5:B,Y5:B]: (((((ord_less @ B) @ X5) @ Y5) => (((ord_less @ A) @ (F @ X5)) @ (F @ Y5))))) => (((ord_less @ A) @ A3) @ (F @ C3))))))))))).
% 3.60/1.76 thf(fact_101_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B3:A,F:(A > B),C3:B]: (((((ord_less @ A) @ A3) @ B3) => (((F @ B3) = C3) => ((! [X5:A,Y5:A]: (((((ord_less @ A) @ X5) @ Y5) => (((ord_less @ B) @ (F @ X5)) @ (F @ Y5))))) => (((ord_less @ B) @ (F @ A3)) @ C3)))))))))).
% 3.60/1.76 thf(fact_102_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((((ord_less @ A) @ A3) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C3) => ((! [X5:B,Y5:B]: (((((ord_less @ B) @ X5) @ Y5) => (((ord_less @ A) @ (F @ X5)) @ (F @ Y5))))) => (((ord_less @ A) @ A3) @ (F @ C3))))))))))).
% 3.60/1.76 thf(fact_103_order__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B3:A,F:(A > C),C3:C]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less @ C) @ (F @ B3)) @ C3) => ((! [X5:A,Y5:A]: (((((ord_less @ A) @ X5) @ Y5) => (((ord_less @ C) @ (F @ X5)) @ (F @ Y5))))) => (((ord_less @ C) @ (F @ A3)) @ C3)))))))))).
% 3.60/1.76 thf(fact_104_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X3:A]: ((? [Y5:A]: ((((ord_less @ A) @ Y5) @ X3))))))))).
% 3.60/1.76 thf(fact_105_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X3:A]: ((? [X_1:A]: ((((ord_less @ A) @ X3) @ X_1))))))))).
% 3.60/1.76 thf(fact_106_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((X3 != Y3) => ((~ ((((ord_less @ A) @ X3) @ Y3))) => (((ord_less @ A) @ Y3) @ X3))))))))).
% 3.60/1.76 thf(fact_107_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((X3 != Y3) = ((((ord_less @ A) @ X3) @ Y3) | (((ord_less @ A) @ Y3) @ X3))))))))).
% 3.60/1.76 thf(fact_108_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (~ ((((ord_less @ A) @ B3) @ A3)))))))))).
% 3.60/1.76 thf(fact_109_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (? [Z3:A]: (((((ord_less @ A) @ X3) @ Z3) & (((ord_less @ A) @ Z3) @ Y3))))))))))).
% 3.60/1.76 thf(fact_110_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (X3 != Y3)))))))).
% 3.60/1.76 thf(fact_111_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (~ ((((ord_less @ A) @ Y3) @ X3)))))))))).
% 3.60/1.76 thf(fact_112_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (~ ((((ord_less @ A) @ B3) @ A3)))))))))).
% 3.60/1.76 thf(fact_113_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A,Z2:A]: (((((ord_less @ A) @ X3) @ Y3) => ((((ord_less @ A) @ Y3) @ Z2) => (((ord_less @ A) @ X3) @ Z2))))))))).
% 3.60/1.76 thf(fact_114_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) | ((X3 = Y3) | (((ord_less @ A) @ Y3) @ X3))))))))).
% 3.60/1.76 thf(fact_115_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A]: ((~ ((((ord_less @ A) @ X3) @ X3))))))))).
% 3.60/1.76 thf(fact_116_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((A3 = B3) => ((((ord_less @ A) @ B3) @ C3) => (((ord_less @ A) @ A3) @ C3))))))))).
% 3.60/1.76 thf(fact_117_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((((ord_less @ A) @ A3) @ B3) => ((B3 = C3) => (((ord_less @ A) @ A3) @ C3))))))))).
% 3.60/1.76 thf(fact_118_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less @ A) @ B3) @ A3) => (~ ((((ord_less @ A) @ A3) @ B3)))))))))).
% 3.60/1.76 thf(fact_119_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (X3 != Y3)))))))).
% 3.60/1.76 thf(fact_120_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (~ ((((ord_less @ A) @ Y3) @ X3)))))))))).
% 3.60/1.76 thf(fact_121_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A3:A]: (((! [X5:A]: (((! [Y7:A]: (((((ord_less @ A) @ Y7) @ X5) => (P @ Y7)))) => (P @ X5)))) => (P @ A3)))))))).
% 3.60/1.76 thf(fact_122_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y3:A,X3:A]: (((~ ((((ord_less @ A) @ Y3) @ X3))) => ((~ ((((ord_less @ A) @ X3) @ Y3))) = (X3 = Y3))))))))).
% 3.60/1.76 thf(fact_123_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (Y3 != X3)))))))).
% 3.60/1.76 thf(fact_124_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A,P:$o]: (((((ord_less @ A) @ X3) @ Y3) => ((((ord_less @ A) @ Y3) @ X3) => P)))))))).
% 3.60/1.76 thf(fact_125_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((~ ((((ord_less @ A) @ X3) @ Y3))) => ((X3 != Y3) => (((ord_less @ A) @ Y3) @ X3))))))))).
% 3.60/1.76 thf(fact_126_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ A3))))))))).
% 3.60/1.76 thf(fact_127_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C3:A]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less @ A) @ B3) @ C3) => (((ord_less @ A) @ A3) @ C3))))))))).
% 3.60/1.76 thf(fact_128_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (~ ((((ord_less @ A) @ Y3) @ X3)))))))))).
% 3.60/1.76 thf(fact_129_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P3:(A > $o)]: ((? [X8:A]: ((P3 @ X8))))) = (^ [P4:(A > $o)]: ((? [N:A]: (((P4 @ N) & (! [M:A]: (((((ord_less @ A) @ M) @ N) => (~ ((P4 @ M)))))))))))))))).
% 3.60/1.76 thf(fact_130_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B3:A]: (((! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => ((P @ A2) @ B2)))) => ((! [A2:A]: (((P @ A2) @ A2))) => ((! [A2:A,B2:A]: ((((P @ B2) @ A2) => ((P @ A2) @ B2)))) => ((P @ A3) @ B3)))))))))).
% 3.60/1.76 thf(fact_131_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C3:A]: (((((ord_less @ A) @ B3) @ A3) => ((((ord_less @ A) @ C3) @ B3) => (((ord_less @ A) @ C3) @ A3))))))))).
% 3.60/1.76 thf(fact_132_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((~ ((((ord_less @ A) @ X3) @ Y3))) = ((((ord_less @ A) @ Y3) @ X3) | (X3 = Y3))))))))).
% 3.60/1.76 thf(fact_133_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less @ A) @ A3) @ B3) => (A3 != B3)))))))).
% 3.60/1.76 thf(fact_134_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less @ A) @ B3) @ A3) => (A3 != B3)))))))).
% 3.60/1.76 thf(fact_135_ex__gt__or__lt, axiom, (! [A:$tType]: (((condit1656338222tinuum @ A) => (! [A3:A]: ((? [B2:A]: (((((ord_less @ A) @ A3) @ B2) | (((ord_less @ A) @ B2) @ A3)))))))))).
% 3.60/1.76 thf(fact_136_less__eq__nres_Oelims_I3_J, axiom, (! [A:$tType,X3:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A)]: (((~ ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ X3) @ Xa))) => ((! [A2:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ A2)) => (! [B2:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B2)) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))))) => (~ (((X3 = (refine1767639642_FAILi @ A)) => (! [Uv:(set @ A)]: ((Xa != ((refine605929679le_RES @ A) @ Uv)))))))))))).
% 3.60/1.76 thf(fact_137_less__eq__nres_Oelims_I2_J, axiom, (! [A:$tType,X3:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ X3) @ Xa) => ((Xa != (refine1767639642_FAILi @ A)) => (~ ((! [A2:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ A2)) => (! [B2:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B2)) => (~ ((((ord_less_eq @ (set @ A)) @ A2) @ B2)))))))))))))))).
% 3.60/1.76 thf(fact_138_less__eq__nres_Oelims_I1_J, axiom, (! [A:$tType,X3:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A),Y3:$o]: ((((((ord_less_eq @ (refine1665802226e_nres @ A)) @ X3) @ Xa) = Y3) => (((Xa = (refine1767639642_FAILi @ A)) => (~ (Y3))) => ((! [A2:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ A2)) => (! [B2:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B2)) => (Y3 = (~ ((((ord_less_eq @ (set @ A)) @ A2) @ B2)))))))))) => (~ (((X3 = (refine1767639642_FAILi @ A)) => ((? [Uv:(set @ A)]: ((Xa = ((refine605929679le_RES @ A) @ Uv)))) => Y3)))))))))).
% 3.60/1.76 thf(fact_139_less__nres_Oelims_I3_J, axiom, (! [A:$tType,X3:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A)]: (((~ ((((ord_less @ (refine1665802226e_nres @ A)) @ X3) @ Xa))) => ((X3 != (refine1767639642_FAILi @ A)) => (~ ((! [A2:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ A2)) => (! [B2:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B2)) => (((ord_less @ (set @ A)) @ A2) @ B2)))))))))))))).
% 3.60/1.76 thf(fact_140_less__nres_Oelims_I2_J, axiom, (! [A:$tType,X3:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A)]: (((((ord_less @ (refine1665802226e_nres @ A)) @ X3) @ Xa) => (((? [Uv:(set @ A)]: ((X3 = ((refine605929679le_RES @ A) @ Uv)))) => (Xa != (refine1767639642_FAILi @ A))) => (~ ((! [A2:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ A2)) => (! [B2:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B2)) => (~ ((((ord_less @ (set @ A)) @ A2) @ B2)))))))))))))))).
% 3.60/1.76 thf(fact_141_less__nres_Oelims_I1_J, axiom, (! [A:$tType,X3:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A),Y3:$o]: ((((((ord_less @ (refine1665802226e_nres @ A)) @ X3) @ Xa) = Y3) => (((X3 = (refine1767639642_FAILi @ A)) => Y3) => (((? [Uv:(set @ A)]: ((X3 = ((refine605929679le_RES @ A) @ Uv)))) => ((Xa = (refine1767639642_FAILi @ A)) => (~ (Y3)))) => (~ ((! [A2:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ A2)) => (! [B2:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B2)) => (Y3 = (~ ((((ord_less @ (set @ A)) @ A2) @ B2)))))))))))))))))).
% 3.60/1.76 thf(fact_142_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X3:A]: (((((ord_less_eq @ A) @ Y3) @ X3) => (~ ((((ord_less @ A) @ X3) @ Y3)))))))))).
% 3.60/1.76 thf(fact_143_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((~ ((((ord_less @ A) @ X3) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 3.60/1.76 thf(fact_144_le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X4:A,Y6:A]: (((((ord_less @ A) @ X4) @ Y6) | (X4 = Y6))))))))).
% 3.60/1.76 thf(fact_145_less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X4:A,Y6:A]: (((((ord_less_eq @ A) @ X4) @ Y6) & (X4 != Y6))))))))).
% 3.60/1.76 thf(fact_146_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C3) => ((! [X5:B,Y5:B]: (((((ord_less @ B) @ X5) @ Y5) => (((ord_less @ A) @ (F @ X5)) @ (F @ Y5))))) => (((ord_less @ A) @ A3) @ (F @ C3))))))))))).
% 3.60/1.76 thf(fact_147_order__le__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B3:A,F:(A > C),C3:C]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less @ C) @ (F @ B3)) @ C3) => ((! [X5:A,Y5:A]: (((((ord_less_eq @ A) @ X5) @ Y5) => (((ord_less_eq @ C) @ (F @ X5)) @ (F @ Y5))))) => (((ord_less @ C) @ (F @ A3)) @ C3)))))))))).
% 3.60/1.76 thf(fact_148_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((((ord_less @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C3) => ((! [X5:B,Y5:B]: (((((ord_less_eq @ B) @ X5) @ Y5) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y5))))) => (((ord_less @ A) @ A3) @ (F @ C3))))))))))).
% 3.60/1.76 thf(fact_149_order__less__le__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B3:A,F:(A > C),C3:C]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C3) => ((! [X5:A,Y5:A]: (((((ord_less @ A) @ X5) @ Y5) => (((ord_less @ C) @ (F @ X5)) @ (F @ Y5))))) => (((ord_less @ C) @ (F @ A3)) @ C3)))))))))).
% 3.60/1.76 thf(fact_150_not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y3))) = (((ord_less @ A) @ Y3) @ X3)))))))).
% 3.60/1.76 thf(fact_151_not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((~ ((((ord_less @ A) @ X3) @ Y3))) = (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 3.60/1.76 thf(fact_152_le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((A3 != B3) => (((ord_less @ A) @ A3) @ B3))))))))).
% 3.60/1.76 thf(fact_153_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y3:A]: (((~ ((((ord_less @ A) @ X3) @ Y3))) => ((((ord_less_eq @ A) @ X3) @ Y3) = (X3 = Y3))))))))).
% 3.60/1.76 thf(fact_154_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((~ ((((ord_less @ A) @ X3) @ Y3))) = (X3 = Y3))))))))).
% 3.60/1.76 thf(fact_155_less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (((ord_less_eq @ A) @ X3) @ Y3)))))))).
% 3.60/1.76 thf(fact_156_le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less @ A) @ Y3) @ Z2) => (((ord_less @ A) @ X3) @ Z2))))))))).
% 3.60/1.76 thf(fact_157_less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A,Z2:A]: (((((ord_less @ A) @ X3) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z2) => (((ord_less @ A) @ X3) @ Z2))))))))).
% 3.60/1.76 thf(fact_158_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,Y3:A]: (((! [X5:A]: (((((ord_less @ A) @ Z2) @ X5) => (((ord_less_eq @ A) @ Y3) @ X5)))) => (((ord_less_eq @ A) @ Y3) @ Z2)))))))).
% 3.60/1.76 thf(fact_159_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y3:A,Z2:A]: (((! [X5:A]: (((((ord_less @ A) @ X5) @ Y3) => (((ord_less_eq @ A) @ X5) @ Z2)))) => (((ord_less_eq @ A) @ Y3) @ Z2)))))))).
% 3.60/1.76 thf(fact_160_le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) | (((ord_less @ A) @ Y3) @ X3)))))))).
% 3.60/1.76 thf(fact_161_le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less @ A) @ X3) @ Y3) | (X3 = Y3))))))))).
% 3.60/1.76 thf(fact_162_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X4:A,Y6:A]: (((((ord_less_eq @ A) @ X4) @ Y6) & (~ ((((ord_less_eq @ A) @ Y6) @ X4))))))))))).
% 3.60/1.76 thf(fact_163_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y3:A,X3:A]: (((~ ((((ord_less_eq @ A) @ Y3) @ X3))) => (((ord_less @ A) @ X3) @ Y3)))))))).
% 3.60/1.76 thf(fact_164_order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less @ A) @ B3) @ C3) => (((ord_less @ A) @ A3) @ C3))))))))).
% 3.60/1.76 thf(fact_165_order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C3:A]: (((((ord_less @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C3) => (((ord_less @ A) @ A3) @ C3))))))))).
% 3.60/1.76 thf(fact_166_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A6:A,B5:A]: (((((ord_less @ A) @ A6) @ B5) | (A6 = B5))))))))).
% 3.60/1.76 thf(fact_167_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A6:A,B5:A]: (((((ord_less_eq @ A) @ A6) @ B5) & (A6 != B5))))))))).
% 3.60/1.76 thf(fact_168_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less @ A) @ C3) @ B3) => (((ord_less @ A) @ C3) @ A3))))))))).
% 3.60/1.76 thf(fact_169_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C3:A]: (((((ord_less @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C3) @ B3) => (((ord_less @ A) @ C3) @ A3))))))))).
% 3.60/1.76 thf(fact_170_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,X3:A,Y3:A]: (((((ord_less @ A) @ Z2) @ X3) => ((! [W2:A]: (((((ord_less @ A) @ Z2) @ W2) => ((((ord_less @ A) @ W2) @ X3) => (((ord_less_eq @ A) @ Y3) @ W2))))) => (((ord_less_eq @ A) @ Y3) @ Z2))))))))).
% 3.60/1.76 thf(fact_171_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X3:A,Y3:A,Z2:A]: (((((ord_less @ A) @ X3) @ Y3) => ((! [W2:A]: (((((ord_less @ A) @ X3) @ W2) => ((((ord_less @ A) @ W2) @ Y3) => (((ord_less_eq @ A) @ W2) @ Z2))))) => (((ord_less_eq @ A) @ Y3) @ Z2))))))))).
% 3.60/1.76 thf(fact_172_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (~ ((((ord_less_eq @ A) @ T2) @ X7)))))))))))))).
% 3.60/1.76 thf(fact_173_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (((ord_less_eq @ A) @ X7) @ T2)))))))))))).
% 3.60/1.76 thf(fact_174_pinf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (((ord_less_eq @ A) @ T2) @ X7)))))))))))).
% 3.60/1.76 thf(fact_175_pinf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (~ ((((ord_less_eq @ A) @ X7) @ T2)))))))))))))).
% 3.60/1.76 thf(fact_176_verit__comp__simplify1_I3_J, axiom, (! [B:$tType]: (((linorder @ B) => (! [B4:B,A4:B]: (((~ ((((ord_less_eq @ B) @ B4) @ A4))) = (((ord_less @ B) @ A4) @ B4)))))))).
% 3.60/1.76 thf(fact_177_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A3:B,B3:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A3) @ B3)) = ((C3 @ A3) @ B3))))).
% 3.60/1.76 thf(fact_178_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A3:A,B3:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((F1 @ A3) @ B3))))).
% 3.60/1.76 thf(fact_179_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R) => (((ord_less_eq @ B) @ (As @ I)) @ (As @ J)))))))))))).
% 3.60/1.76 thf(fact_180_less__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F3) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F3))))))))))).
% 3.60/1.76 thf(fact_181_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B3:A]: (((A3 = B3) | ((~ ((((ord_less_eq @ A) @ A3) @ B3))) | (~ ((((ord_less_eq @ A) @ B3) @ A3))))))))))).
% 3.60/1.76 thf(fact_182_verit__comp__simplify1_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ A3))))))))).
% 3.60/1.76 thf(fact_183_pinf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P5:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((P @ X5) = (P5 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (((P @ X7) & (Q @ X7)) = ((P5 @ X7) & (Q2 @ X7)))))))))))))))).
% 3.60/1.76 thf(fact_184_pinf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P5:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((P @ X5) = (P5 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (((P @ X7) | (Q @ X7)) = ((P5 @ X7) | (Q2 @ X7)))))))))))))))).
% 3.60/1.76 thf(fact_185_pinf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (X7 != T2)))))))))))).
% 3.60/1.76 thf(fact_186_pinf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (X7 != T2)))))))))))).
% 3.60/1.76 thf(fact_187_pinf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (~ ((((ord_less @ A) @ X7) @ T2)))))))))))))).
% 3.60/1.76 thf(fact_188_pinf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ Z3) @ X7) => (((ord_less @ A) @ T2) @ X7)))))))))))).
% 3.60/1.76 thf(fact_189_pinf_I11_J, axiom, (! [C:$tType,D2:$tType]: (((ord @ C) => (! [F7:D2]: ((? [Z3:C]: ((! [X7:C]: (((((ord_less @ C) @ Z3) @ X7) => (F7 = F7)))))))))))).
% 3.60/1.76 thf(fact_190_minf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P5:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((P @ X5) = (P5 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (((P @ X7) & (Q @ X7)) = ((P5 @ X7) & (Q2 @ X7)))))))))))))))).
% 3.60/1.76 thf(fact_191_minf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P5:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((P @ X5) = (P5 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (((P @ X7) | (Q @ X7)) = ((P5 @ X7) | (Q2 @ X7)))))))))))))))).
% 3.60/1.76 thf(fact_192_minf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (X7 != T2)))))))))))).
% 3.60/1.76 thf(fact_193_minf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (X7 != T2)))))))))))).
% 3.60/1.76 thf(fact_194_minf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (((ord_less @ A) @ X7) @ T2)))))))))))).
% 3.60/1.76 thf(fact_195_minf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (~ ((((ord_less @ A) @ T2) @ X7)))))))))))))).
% 3.60/1.76 thf(fact_196_minf_I11_J, axiom, (! [C:$tType,D2:$tType]: (((ord @ C) => (! [F7:D2]: ((? [Z3:C]: ((! [X7:C]: (((((ord_less @ C) @ X7) @ Z3) => (F7 = F7)))))))))))).
% 3.60/1.76 thf(fact_197_psubsetI, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => ((A5 != B6) => (((ord_less @ (set @ A)) @ A5) @ B6)))))).
% 3.60/1.76 thf(fact_198_subset__antisym, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ A5) => (A5 = B6)))))).
% 3.60/1.76 thf(fact_199_subsetI, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ A5) => (((member @ A) @ X5) @ B6)))) => (((ord_less_eq @ (set @ A)) @ A5) @ B6))))).
% 3.60/1.76 thf(fact_200_in__mono, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => ((((member @ A) @ X3) @ A5) => (((member @ A) @ X3) @ B6)))))).
% 3.60/1.76 thf(fact_201_subsetD, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => ((((member @ A) @ C3) @ A5) => (((member @ A) @ C3) @ B6)))))).
% 3.60/1.76 thf(fact_202_equalityE, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((A5 = B6) => (~ (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A5)))))))))).
% 3.60/1.76 thf(fact_203_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A7) => (((member @ A) @ X4) @ B7)))))))))).
% 3.60/1.76 thf(fact_204_equalityD1, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((A5 = B6) => (((ord_less_eq @ (set @ A)) @ A5) @ B6))))).
% 3.60/1.76 thf(fact_205_equalityD2, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((A5 = B6) => (((ord_less_eq @ (set @ A)) @ B6) @ A5))))).
% 3.60/1.76 thf(fact_206_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((! [T4:A]: (((((member @ A) @ T4) @ A7) => (((member @ A) @ T4) @ B7)))))))))).
% 3.60/1.76 thf(fact_207_subset__refl, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A5) @ A5)))).
% 3.60/1.76 thf(fact_208_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) => (Q @ X5)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 3.60/1.76 thf(fact_209_subset__trans, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => (((ord_less_eq @ (set @ A)) @ A5) @ C4)))))).
% 3.60/1.76 thf(fact_210_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z:(set @ A)]: ((Y4 = Z))) = (^ [A7:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A7)))))))).
% 3.60/1.76 thf(fact_211_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X4:A]: (((P @ X4) => (Q @ X4)))))))).
% 3.60/1.76 thf(fact_212_psubsetD, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),C3:A]: (((((ord_less @ (set @ A)) @ A5) @ B6) => ((((member @ A) @ C3) @ A5) => (((member @ A) @ C3) @ B6)))))).
% 3.60/1.76 thf(fact_213_psubset__trans, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less @ (set @ A)) @ A5) @ B6) => ((((ord_less @ (set @ A)) @ B6) @ C4) => (((ord_less @ (set @ A)) @ A5) @ C4)))))).
% 3.60/1.76 thf(fact_214_psubsetE, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A5) @ B6) => (~ (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => (((ord_less_eq @ (set @ A)) @ B6) @ A5)))))))).
% 3.60/1.76 thf(fact_215_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) & (A7 != B7)))))))).
% 3.60/1.76 thf(fact_216_psubset__imp__subset, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A5) @ B6) => (((ord_less_eq @ (set @ A)) @ A5) @ B6))))).
% 3.60/1.76 thf(fact_217_psubset__subset__trans, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less @ (set @ A)) @ A5) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => (((ord_less @ (set @ A)) @ A5) @ C4)))))).
% 3.60/1.76 thf(fact_218_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) & (~ ((((ord_less_eq @ (set @ A)) @ B7) @ A7)))))))))).
% 3.60/1.76 thf(fact_219_subset__psubset__trans, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => ((((ord_less @ (set @ A)) @ B6) @ C4) => (((ord_less @ (set @ A)) @ A5) @ C4)))))).
% 3.60/1.76 thf(fact_220_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B7) | (A7 = B7)))))))).
% 3.60/1.76 thf(fact_221_less__nres_Opelims_I1_J, axiom, (! [A:$tType,X3:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A),Y3:$o]: ((((((ord_less @ (refine1665802226e_nres @ A)) @ X3) @ Xa) = Y3) => ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1378444575es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ X3) @ Xa)) => (((X3 = (refine1767639642_FAILi @ A)) => ((~ (Y3)) => (~ ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1378444575es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ (refine1767639642_FAILi @ A)) @ Xa)))))) => ((! [Uv:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ Uv)) => ((Xa = (refine1767639642_FAILi @ A)) => (Y3 => (~ ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1378444575es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ Uv)) @ (refine1767639642_FAILi @ A)))))))))) => (~ ((! [A2:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ A2)) => (! [B2:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B2)) => ((Y3 = (((ord_less @ (set @ A)) @ A2) @ B2)) => (~ ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1378444575es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A2)) @ ((refine605929679le_RES @ A) @ B2))))))))))))))))))))).
% 3.60/1.76 thf(fact_222_less__nres_Opelims_I2_J, axiom, (! [A:$tType,X3:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A)]: (((((ord_less @ (refine1665802226e_nres @ A)) @ X3) @ Xa) => ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1378444575es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ X3) @ Xa)) => ((! [Uv:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ Uv)) => ((Xa = (refine1767639642_FAILi @ A)) => (~ ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1378444575es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ Uv)) @ (refine1767639642_FAILi @ A))))))))) => (~ ((! [A2:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ A2)) => (! [B2:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B2)) => ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1378444575es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A2)) @ ((refine605929679le_RES @ A) @ B2))) => (~ ((((ord_less @ (set @ A)) @ A2) @ B2)))))))))))))))))).
% 3.60/1.76 thf(fact_223_less__nres_Opelims_I3_J, axiom, (! [A:$tType,X3:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A)]: (((~ ((((ord_less @ (refine1665802226e_nres @ A)) @ X3) @ Xa))) => ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1378444575es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ X3) @ Xa)) => (((X3 = (refine1767639642_FAILi @ A)) => (~ ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1378444575es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ (refine1767639642_FAILi @ A)) @ Xa))))) => (~ ((! [A2:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ A2)) => (! [B2:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B2)) => ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1378444575es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A2)) @ ((refine605929679le_RES @ A) @ B2))) => (((ord_less @ (set @ A)) @ A2) @ B2)))))))))))))))).
% 3.60/1.76 thf(fact_224_less__eq__nres_Opelims_I3_J, axiom, (! [A:$tType,X3:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A)]: (((~ ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ X3) @ Xa))) => ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1554218259es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ X3) @ Xa)) => ((! [A2:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ A2)) => (! [B2:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B2)) => ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1554218259es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A2)) @ ((refine605929679le_RES @ A) @ B2))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2)))))))) => (~ (((X3 = (refine1767639642_FAILi @ A)) => (! [Uv:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ Uv)) => (~ ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1554218259es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ (refine1767639642_FAILi @ A)) @ ((refine605929679le_RES @ A) @ Uv))))))))))))))))).
% 3.60/1.76 thf(fact_225_less__eq__nres_Opelims_I2_J, axiom, (! [A:$tType,X3:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ X3) @ Xa) => ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1554218259es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ X3) @ Xa)) => (((Xa = (refine1767639642_FAILi @ A)) => (~ ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1554218259es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ X3) @ (refine1767639642_FAILi @ A)))))) => (~ ((! [A2:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ A2)) => (! [B2:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B2)) => ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1554218259es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A2)) @ ((refine605929679le_RES @ A) @ B2))) => (~ ((((ord_less_eq @ (set @ A)) @ A2) @ B2)))))))))))))))))).
% 3.60/1.76 thf(fact_226_less__eq__nres_Opelims_I1_J, axiom, (! [A:$tType,X3:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A),Y3:$o]: ((((((ord_less_eq @ (refine1665802226e_nres @ A)) @ X3) @ Xa) = Y3) => ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1554218259es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ X3) @ Xa)) => (((Xa = (refine1767639642_FAILi @ A)) => (Y3 => (~ ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1554218259es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ X3) @ (refine1767639642_FAILi @ A))))))) => ((! [A2:(set @ A)]: (((X3 = ((refine605929679le_RES @ A) @ A2)) => (! [B2:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B2)) => ((Y3 = (((ord_less_eq @ (set @ A)) @ A2) @ B2)) => (~ ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1554218259es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A2)) @ ((refine605929679le_RES @ A) @ B2)))))))))))) => (~ (((X3 = (refine1767639642_FAILi @ A)) => (! [Uv:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ Uv)) => ((~ (Y3)) => (~ ((((accp @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))) @ (refine1554218259es_rel @ A)) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ (refine1767639642_FAILi @ A)) @ ((refine605929679le_RES @ A) @ Uv))))))))))))))))))).
% 3.60/1.76 thf(fact_227_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X3:A,Q:(A > $o)]: (((P @ X3) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ Y5) @ X3)))) => ((! [X5:A]: (((P @ X5) => ((! [Y7:A]: (((P @ Y7) => (((ord_less_eq @ A) @ Y7) @ X5)))) => (Q @ X5))))) => (Q @ ((order_Greatest @ A) @ P))))))))))).
% 3.60/1.76 thf(fact_228_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X3:A]: (((P @ X3) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ Y5) @ X3)))) => (((order_Greatest @ A) @ P) = X3))))))))).
% 3.60/1.76 thf(fact_229_subset__Collect__conv, axiom, (! [A:$tType,S:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ S) @ ((collect @ A) @ P)) = (! [X4:A]: (((((member @ A) @ X4) @ S) => (P @ X4)))))))).
% 3.60/1.76 thf(fact_230_relprop__triggers_I6_J, axiom, (! [I2:$tType,R2:(set @ I2),R3:(set @ I2)]: (((((ord_less_eq @ (set @ I2)) @ R2) @ R3) => (((ord_less_eq @ (set @ I2)) @ R2) @ R3))))).
% 3.60/1.76 thf(fact_231_ord__eq__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A,D4:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C3) => ((C3 = D4) => (((ord_less_eq @ A) @ A3) @ D4)))))))))).
% 3.60/1.76 thf(fact_232_bex2I, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,S:(set @ ((product_prod @ A) @ B)),P:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ S) => (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ S) => ((P @ A3) @ B3)) => (? [A2:A,B2:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ S) & ((P @ A2) @ B2))))))))).
% 3.60/1.76 thf(fact_233_pairself_Ocases, axiom, (! [B:$tType,A:$tType,X3:((product_prod @ (A > B)) @ ((product_prod @ A) @ A))]: ((~ ((! [F6:(A > B),A2:A,B2:A]: ((X3 != ((((product_Pair @ (A > B)) @ ((product_prod @ A) @ A)) @ F6) @ ((((product_Pair @ A) @ A) @ A2) @ B2)))))))))).
% 3.60/1.76 thf(fact_234_pairself_Oinduct, axiom, (! [B:$tType,A:$tType,P:((A > B) > (((product_prod @ A) @ A) > $o)),A0:(A > B),A1:((product_prod @ A) @ A)]: (((! [F6:(A > B),A2:A,B2:A]: (((P @ F6) @ ((((product_Pair @ A) @ A) @ A2) @ B2)))) => ((P @ A0) @ A1))))).
% 3.60/1.76 thf(fact_235_uncurry__apply, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),A3:B,B3:C]: (((((((uncurry @ B) @ C) @ A) @ F) @ ((((product_Pair @ B) @ C) @ A3) @ B3)) = ((F @ A3) @ B3))))).
% 3.60/1.76 thf(fact_236_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A3:A]: (((! [X5:A]: (((! [Y7:A]: (((((ord_less @ B) @ (F @ Y7)) @ (F @ X5)) => (P @ Y7)))) => (P @ X5)))) => (P @ A3)))))))).
% 3.60/1.76 thf(fact_237_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A3:A]: (((! [X5:A]: (((! [Y7:A]: (((((ord_less @ B) @ (F @ Y7)) @ (F @ X5)) => (P @ Y7)))) => (P @ X5)))) => (P @ A3)))))))).
% 3.60/1.76 thf(fact_238_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,Y3:A]: (((X3 != Y3) => ((~ ((((ord_less @ A) @ X3) @ Y3))) => (((ord_less @ A) @ Y3) @ X3))))))))).
% 3.60/1.76 thf(fact_239_linordered__field__no__ub, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X7:A]: ((? [X_1:A]: ((((ord_less @ A) @ X7) @ X_1))))))))).
% 3.60/1.76 thf(fact_240_linordered__field__no__lb, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X7:A]: ((? [Y5:A]: ((((ord_less @ A) @ Y5) @ X7))))))))).
% 3.60/1.76 thf(fact_241_all__nat__split__at, axiom, (! [A:$tType]: (((linorder @ A) => (! [K:A,P:(A > $o)]: (((! [I3:A]: (((((ord_less @ A) @ I3) @ K) => (P @ I3)))) => ((P @ K) => ((! [I3:A]: (((((ord_less @ A) @ K) @ I3) => (P @ I3)))) => (! [X_12:A]: ((P @ X_12)))))))))))).
% 3.60/1.76 thf(fact_242_dependent__wellorder__choice, axiom, (! [B:$tType,A:$tType]: (((wellorder @ A) => (! [P:((A > B) > (A > (B > $o)))]: (((! [R4:B,F6:(A > B),G4:(A > B),X5:A]: (((! [Y7:A]: (((((ord_less @ A) @ Y7) @ X5) => ((F6 @ Y7) = (G4 @ Y7))))) => ((((P @ F6) @ X5) @ R4) = (((P @ G4) @ X5) @ R4))))) => ((! [X5:A,F6:(A > B)]: (((! [Y7:A]: (((((ord_less @ A) @ Y7) @ X5) => (((P @ F6) @ Y7) @ (F6 @ Y7))))) => (? [X_12:B]: ((((P @ F6) @ X5) @ X_12)))))) => (? [F6:(A > B)]: ((! [X7:A]: ((((P @ F6) @ X7) @ (F6 @ X7)))))))))))))).
% 3.60/1.76 thf(fact_243_Refine__Misc_Oif__mono, axiom, (! [A:$tType]: (((ord @ A) => (! [B3:$o,M1:A,M12:A,M22:A,M23:A]: (((B3 => (((ord_less_eq @ A) @ M1) @ M12)) => (((~ (B3)) => (((ord_less_eq @ A) @ M22) @ M23)) => (((ord_less_eq @ A) @ ((((if @ A) @ B3) @ M1) @ M22)) @ ((((if @ A) @ B3) @ M12) @ M23)))))))))).
% 3.60/1.76 thf(fact_244_chain__subset__def, axiom, (! [A:$tType]: (((chain_subset @ A) = (^ [C5:(set @ (set @ A))]: ((! [X4:(set @ A)]: (((((member @ (set @ A)) @ X4) @ C5) => (! [Y6:(set @ A)]: (((((member @ (set @ A)) @ Y6) @ C5) => ((((ord_less_eq @ (set @ A)) @ X4) @ Y6) | (((ord_less_eq @ (set @ A)) @ Y6) @ X4)))))))))))))).
% 3.60/1.76 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 3.60/1.76 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 3.60/1.76 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 3.60/1.76 thf(tcon_Set_Oset___Orderings_Opreorder_1, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 3.60/1.76 thf(tcon_Set_Oset___Orderings_Oorder_2, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 3.60/1.76 thf(tcon_Set_Oset___Orderings_Oord_3, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 3.60/1.76 thf(tcon_HOL_Obool___Orderings_Opreorder_4, axiom, (preorder @ $o)).
% 3.60/1.76 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 3.60/1.76 thf(tcon_HOL_Obool___Orderings_Oorder_5, axiom, (order @ $o)).
% 3.60/1.76 thf(tcon_HOL_Obool___Orderings_Oord_6, axiom, (ord @ $o)).
% 3.60/1.76 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_7, axiom, (! [A8:$tType]: ((preorder @ (refine1665802226e_nres @ A8))))).
% 3.60/1.76 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_8, axiom, (! [A8:$tType]: ((order @ (refine1665802226e_nres @ A8))))).
% 3.60/1.76 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_9, axiom, (! [A8:$tType]: ((ord @ (refine1665802226e_nres @ A8))))).
% 3.60/1.76 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 3.60/1.76 thf(help_If_2_1_T, axiom, (! [A:$tType,X3:A,Y3:A]: ((((((if @ A) @ $false) @ X3) @ Y3) = Y3)))).
% 3.60/1.76 thf(help_If_1_1_T, axiom, (! [A:$tType,X3:A,Y3:A]: ((((((if @ A) @ $true) @ X3) @ Y3) = X3)))).
% 3.60/1.76 thf(conj_0, conjecture, ((((refine406925620_inres @ a) @ ((refine605929679le_RES @ a) @ ((collect @ a) @ phi))) @ x) = (phi @ x))).
% 3.60/1.76 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.jdFRlkfssT/DTF2DTF14407.p
% 3.60/1.76 ------------------------
% 3.77/1.82 ---- Cleaned THF ---
% 3.77/1.82 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
% 3.77/1.82 refine1665802226e_nres: $tType > $tType ).
% 3.77/1.82
% 3.77/1.82 thf(ty_t_Product__Type_Oprod,type,
% 3.77/1.82 product_prod: $tType > $tType > $tType ).
% 3.77/1.82
% 3.77/1.82 thf(ty_t_Set_Oset,type,
% 3.77/1.82 set: $tType > $tType ).
% 3.77/1.82
% 3.77/1.82 thf(ty_tf_a,type,
% 3.77/1.82 a: $tType ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_HOL_Otype,type,
% 3.77/1.82 type:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_Orderings_Oord,type,
% 3.77/1.82 ord:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_Orderings_Oorder,type,
% 3.77/1.82 order:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_Orderings_Ono__bot,type,
% 3.77/1.82 no_bot:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_Orderings_Ono__top,type,
% 3.77/1.82 no_top:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_Orderings_Olinorder,type,
% 3.77/1.82 linorder:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_Orderings_Opreorder,type,
% 3.77/1.82 preorder:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_Orderings_Owellorder,type,
% 3.77/1.82 wellorder:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_Orderings_Odense__order,type,
% 3.77/1.82 dense_order:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_Rings_Olinordered__idom,type,
% 3.77/1.82 linordered_idom:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_Fields_Olinordered__field,type,
% 3.77/1.82 linordered_field:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_Orderings_Odense__linorder,type,
% 3.77/1.82 dense_linorder:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 3.77/1.82 condit1656338222tinuum:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 3.77/1.82 condit1037483654norder:
% 3.77/1.82 !>[A: $tType] : $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 3.77/1.82 bNF_Ca1785829860lChain:
% 3.77/1.82 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_If,type,
% 3.77/1.82 if:
% 3.77/1.82 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Misc_Ouncurry,type,
% 3.77/1.82 uncurry:
% 3.77/1.82 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Orderings_Oord__class_Oless,type,
% 3.77/1.82 ord_less:
% 3.77/1.82 !>[A: $tType] : ( A > A > $o ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.77/1.82 ord_less_eq:
% 3.77/1.82 !>[A: $tType] : ( A > A > $o ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 3.77/1.82 order_Greatest:
% 3.77/1.82 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Product__Type_OPair,type,
% 3.77/1.82 product_Pair:
% 3.77/1.82 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 3.77/1.82 produc2004651681e_prod:
% 3.77/1.82 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 3.77/1.82 product_rec_prod:
% 3.77/1.82 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
% 3.77/1.82 refine1687780735RETURN:
% 3.77/1.82 !>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres,type,
% 3.77/1.82 refine1315500908_inres:
% 3.77/1.82 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oless__eq__nres__rel,type,
% 3.77/1.82 refine1554218259es_rel:
% 3.77/1.82 !>[A: $tType] : ( ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) > ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) > $o ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oless__nres__rel,type,
% 3.77/1.82 refine1378444575es_rel:
% 3.77/1.82 !>[A: $tType] : ( ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) > ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) > $o ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres,type,
% 3.77/1.82 refine406925620_inres:
% 3.77/1.82 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail,type,
% 3.77/1.82 refine1102455758nofail:
% 3.77/1.82 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > $o ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_OFAILi,type,
% 3.77/1.82 refine1767639642_FAILi:
% 3.77/1.82 !>[A: $tType] : ( refine1665802226e_nres @ A ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES,type,
% 3.77/1.82 refine605929679le_RES:
% 3.77/1.82 !>[A: $tType] : ( ( set @ A ) > ( refine1665802226e_nres @ A ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_Ocase__nres,type,
% 3.77/1.82 refine60818195e_nres:
% 3.77/1.82 !>[B: $tType,A: $tType] : ( B > ( ( set @ A ) > B ) > ( refine1665802226e_nres @ A ) > B ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_Orec__nres,type,
% 3.77/1.82 refine1442219249c_nres:
% 3.77/1.82 !>[C: $tType,A: $tType] : ( C > ( ( set @ A ) > C ) > ( refine1665802226e_nres @ A ) > C ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES,type,
% 3.77/1.82 refine1672542526he_RES:
% 3.77/1.82 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > ( set @ A ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Set_OCollect,type,
% 3.77/1.82 collect:
% 3.77/1.82 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Wellfounded_Oaccp,type,
% 3.77/1.82 accp:
% 3.77/1.82 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_Zorn_Ochain__subset,type,
% 3.77/1.82 chain_subset:
% 3.77/1.82 !>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_c_member,type,
% 3.77/1.82 member:
% 3.77/1.82 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.77/1.82
% 3.77/1.82 thf(sy_v__092_060Phi_062,type,
% 3.77/1.82 phi: a > $o ).
% 3.77/1.82
% 3.77/1.82 thf(sy_v_x,type,
% 3.77/1.82 x: a ).
% 3.77/1.82
% 3.77/1.82 thf(fact_0_nres__more__simps_I4_J,axiom,
% 3.77/1.82 ! [A: $tType,X: set @ A,Y: set @ A] :
% 3.77/1.82 ( ( ( refine605929679le_RES @ A @ X )
% 3.77/1.82 = ( refine605929679le_RES @ A @ Y ) )
% 3.77/1.82 = ( X = Y ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_1_nres_Oinject,axiom,
% 3.77/1.82 ! [A: $tType,X2: set @ A,Y2: set @ A] :
% 3.77/1.82 ( ( ( refine605929679le_RES @ A @ X2 )
% 3.77/1.82 = ( refine605929679le_RES @ A @ Y2 ) )
% 3.77/1.82 = ( X2 = Y2 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_2_nf__inres__RES,axiom,
% 3.77/1.82 ! [A: $tType,X: set @ A,X3: A] :
% 3.77/1.82 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ X ) @ X3 )
% 3.77/1.82 = ( member @ A @ X3 @ X ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_3_the__RES_Osimps,axiom,
% 3.77/1.82 ! [A: $tType,X: set @ A] :
% 3.77/1.82 ( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X ) )
% 3.77/1.82 = X ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_4_nres_Osimps_I7_J,axiom,
% 3.77/1.82 ! [C: $tType,A: $tType,F1: C,F2: ( set @ A ) > C,X2: set @ A] :
% 3.77/1.82 ( ( refine1442219249c_nres @ C @ A @ F1 @ F2 @ ( refine605929679le_RES @ A @ X2 ) )
% 3.77/1.82 = ( F2 @ X2 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_5_nres_Osimps_I5_J,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType,F1: B,F2: ( set @ A ) > B,X2: set @ A] :
% 3.77/1.82 ( ( refine60818195e_nres @ B @ A @ F1 @ F2 @ ( refine605929679le_RES @ A @ X2 ) )
% 3.77/1.82 = ( F2 @ X2 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_6_inres__simps_I2_J,axiom,
% 3.77/1.82 ! [B: $tType,X: set @ B] :
% 3.77/1.82 ( ( refine1315500908_inres @ B @ ( refine605929679le_RES @ B @ X ) )
% 3.77/1.82 = ( ^ [X4: B] : ( member @ B @ X4 @ X ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_7_nofail__simps_I2_J,axiom,
% 3.77/1.82 ! [B: $tType,X: set @ B] : ( refine1102455758nofail @ B @ ( refine605929679le_RES @ B @ X ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_8_nofail__RES__conv,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( refine1102455758nofail @ A )
% 3.77/1.82 = ( ^ [M: refine1665802226e_nres @ A] :
% 3.77/1.82 ? [M2: set @ A] :
% 3.77/1.82 ( M
% 3.77/1.82 = ( refine605929679le_RES @ A @ M2 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_9_nres_Odistinct_I1_J,axiom,
% 3.77/1.82 ! [A: $tType,X2: set @ A] :
% 3.77/1.82 ( ( refine1767639642_FAILi @ A )
% 3.77/1.82 != ( refine605929679le_RES @ A @ X2 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_10_nres_Oinduct,axiom,
% 3.77/1.82 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > $o,Nres: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( P @ ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ( ! [X5: set @ A] : ( P @ ( refine605929679le_RES @ A @ X5 ) )
% 3.77/1.82 => ( P @ Nres ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_11_nres_Oexhaust,axiom,
% 3.77/1.82 ! [A: $tType,Y3: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( Y3
% 3.77/1.82 != ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ~ ! [X22: set @ A] :
% 3.77/1.82 ( Y3
% 3.77/1.82 != ( refine605929679le_RES @ A @ X22 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_12_sup__nres_Oinduct,axiom,
% 3.77/1.82 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ! [Uu: refine1665802226e_nres @ A] : ( P @ Uu @ ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ( ! [V: set @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V ) )
% 3.77/1.82 => ( ! [A2: set @ A,B2: set @ A] : ( P @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ( P @ A0 @ A1 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_13_less__nres_Oinduct,axiom,
% 3.77/1.82 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ! [X_1: refine1665802226e_nres @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ X_1 )
% 3.77/1.82 => ( ! [Uv: set @ A] : ( P @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ( ! [A2: set @ A,B2: set @ A] : ( P @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ( P @ A0 @ A1 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_14_less__eq__nres_Oinduct,axiom,
% 3.77/1.82 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ! [Uu: refine1665802226e_nres @ A] : ( P @ Uu @ ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ( ! [A2: set @ A,B2: set @ A] : ( P @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ( ! [Uv: set @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) )
% 3.77/1.82 => ( P @ A0 @ A1 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_15_the__RES__inv,axiom,
% 3.77/1.82 ! [A: $tType,M3: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( refine1102455758nofail @ A @ M3 )
% 3.77/1.82 => ( ( refine605929679le_RES @ A @ ( refine1672542526he_RES @ A @ M3 ) )
% 3.77/1.82 = M3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_16_nres_Osimps_I6_J,axiom,
% 3.77/1.82 ! [A: $tType,C: $tType,F1: C,F2: ( set @ A ) > C] :
% 3.77/1.82 ( ( refine1442219249c_nres @ C @ A @ F1 @ F2 @ ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 = F1 ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_17_nres_Osimps_I4_J,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType,F1: B,F2: ( set @ A ) > B] :
% 3.77/1.82 ( ( refine60818195e_nres @ B @ A @ F1 @ F2 @ ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 = F1 ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_18_pw__eqI,axiom,
% 3.77/1.82 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( ( refine1102455758nofail @ A @ S )
% 3.77/1.82 = ( refine1102455758nofail @ A @ S2 ) )
% 3.77/1.82 => ( ! [X5: A] :
% 3.77/1.82 ( ( refine1315500908_inres @ A @ S @ X5 )
% 3.77/1.82 = ( refine1315500908_inres @ A @ S2 @ X5 ) )
% 3.77/1.82 => ( S = S2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_19_pw__eq__iff,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ^ [Y4: refine1665802226e_nres @ A,Z: refine1665802226e_nres @ A] : ( Y4 = Z ) )
% 3.77/1.82 = ( ^ [S3: refine1665802226e_nres @ A,S4: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( ( refine1102455758nofail @ A @ S3 )
% 3.77/1.82 = ( refine1102455758nofail @ A @ S4 ) )
% 3.77/1.82 & ! [X4: A] :
% 3.77/1.82 ( ( refine1315500908_inres @ A @ S3 @ X4 )
% 3.77/1.82 = ( refine1315500908_inres @ A @ S4 @ X4 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_20_not__nofail__inres,axiom,
% 3.77/1.82 ! [A: $tType,S: refine1665802226e_nres @ A,X3: A] :
% 3.77/1.82 ( ~ ( refine1102455758nofail @ A @ S )
% 3.77/1.82 => ( refine1315500908_inres @ A @ S @ X3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_21_nf__inres__def,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( refine406925620_inres @ A )
% 3.77/1.82 = ( ^ [M: refine1665802226e_nres @ A,X4: A] :
% 3.77/1.82 ( ( refine1102455758nofail @ A @ M )
% 3.77/1.82 & ( refine1315500908_inres @ A @ M @ X4 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_22_inres__simps_I3_J,axiom,
% 3.77/1.82 ! [C: $tType,X3: C] :
% 3.77/1.82 ( ( refine1315500908_inres @ C @ ( refine1687780735RETURN @ C @ X3 ) )
% 3.77/1.82 = ( ^ [Y4: C,Z: C] : ( Y4 = Z )
% 3.77/1.82 @ X3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_23_nofail__simps_I3_J,axiom,
% 3.77/1.82 ! [C: $tType,X3: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_24_pw__leI,axiom,
% 3.77/1.82 ! [A: $tType,S2: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( ( refine1102455758nofail @ A @ S2 )
% 3.77/1.82 => ( ( refine1102455758nofail @ A @ S )
% 3.77/1.82 & ! [X5: A] :
% 3.77/1.82 ( ( refine1315500908_inres @ A @ S @ X5 )
% 3.77/1.82 => ( refine1315500908_inres @ A @ S2 @ X5 ) ) ) )
% 3.77/1.82 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_25_pw__leI_H,axiom,
% 3.77/1.82 ! [A: $tType,S2: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( ( refine1102455758nofail @ A @ S2 )
% 3.77/1.82 => ( refine1102455758nofail @ A @ S ) )
% 3.77/1.82 => ( ! [X5: A] :
% 3.77/1.82 ( ( refine1102455758nofail @ A @ S2 )
% 3.77/1.82 => ( ( refine1315500908_inres @ A @ S @ X5 )
% 3.77/1.82 => ( refine1315500908_inres @ A @ S2 @ X5 ) ) )
% 3.77/1.82 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_26_pw__le__iff,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) )
% 3.77/1.82 = ( ^ [S3: refine1665802226e_nres @ A,S4: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( refine1102455758nofail @ A @ S4 )
% 3.77/1.82 => ( ( refine1102455758nofail @ A @ S3 )
% 3.77/1.82 & ! [X4: A] :
% 3.77/1.82 ( ( refine1315500908_inres @ A @ S3 @ X4 )
% 3.77/1.82 => ( refine1315500908_inres @ A @ S4 @ X4 ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_27_less__eq__nres_Osimps_I3_J,axiom,
% 3.77/1.82 ! [A: $tType,Uv2: set @ A] :
% 3.77/1.82 ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv2 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_28_less__nres_Osimps_I2_J,axiom,
% 3.77/1.82 ! [A: $tType,Uv2: set @ A] : ( ord_less @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv2 ) @ ( refine1767639642_FAILi @ A ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_29_less__eq__nres_Ocases,axiom,
% 3.77/1.82 ! [A: $tType,X3: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
% 3.77/1.82 ( ! [Uu: refine1665802226e_nres @ A] :
% 3.77/1.82 ( X3
% 3.77/1.82 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ Uu @ ( refine1767639642_FAILi @ A ) ) )
% 3.77/1.82 => ( ! [A2: set @ A,B2: set @ A] :
% 3.77/1.82 ( X3
% 3.77/1.82 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
% 3.77/1.82 => ~ ! [Uv: set @ A] :
% 3.77/1.82 ( X3
% 3.77/1.82 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_30_less__nres_Ocases,axiom,
% 3.77/1.82 ! [A: $tType,X3: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
% 3.77/1.82 ( ! [Uu: refine1665802226e_nres @ A] :
% 3.77/1.82 ( X3
% 3.77/1.82 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Uu ) )
% 3.77/1.82 => ( ! [Uv: set @ A] :
% 3.77/1.82 ( X3
% 3.77/1.82 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) ) )
% 3.77/1.82 => ~ ! [A2: set @ A,B2: set @ A] :
% 3.77/1.82 ( X3
% 3.77/1.82 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_31_sup__nres_Ocases,axiom,
% 3.77/1.82 ! [A: $tType,X3: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
% 3.77/1.82 ( ! [Uu: refine1665802226e_nres @ A] :
% 3.77/1.82 ( X3
% 3.77/1.82 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ Uu @ ( refine1767639642_FAILi @ A ) ) )
% 3.77/1.82 => ( ! [V: set @ A] :
% 3.77/1.82 ( X3
% 3.77/1.82 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V ) ) )
% 3.77/1.82 => ~ ! [A2: set @ A,B2: set @ A] :
% 3.77/1.82 ( X3
% 3.77/1.82 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_32_nres__more__simps_I6_J,axiom,
% 3.77/1.82 ! [A: $tType,X3: A,Y3: A] :
% 3.77/1.82 ( ( ( refine1687780735RETURN @ A @ X3 )
% 3.77/1.82 = ( refine1687780735RETURN @ A @ Y3 ) )
% 3.77/1.82 = ( X3 = Y3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_33_nres__order__simps_I20_J,axiom,
% 3.77/1.82 ! [W: $tType,X3: W,Y3: W] :
% 3.77/1.82 ( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X3 ) @ ( refine1687780735RETURN @ W @ Y3 ) )
% 3.77/1.82 = ( X3 = Y3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_34_nres__order__simps_I21_J,axiom,
% 3.77/1.82 ! [X6: $tType,X3: X6,Y: set @ X6] :
% 3.77/1.82 ( ( ord_less_eq @ ( refine1665802226e_nres @ X6 ) @ ( refine1687780735RETURN @ X6 @ X3 ) @ ( refine605929679le_RES @ X6 @ Y ) )
% 3.77/1.82 = ( member @ X6 @ X3 @ Y ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_35_inres__def,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( refine1315500908_inres @ A )
% 3.77/1.82 = ( ^ [S3: refine1665802226e_nres @ A,X4: A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X4 ) @ S3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_36_less__nres_Osimps_I1_J,axiom,
% 3.77/1.82 ! [A: $tType,Uu2: refine1665802226e_nres @ A] :
% 3.77/1.82 ~ ( ord_less @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Uu2 ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_37_less__eq__nres_Osimps_I1_J,axiom,
% 3.77/1.82 ! [A: $tType,Uu2: refine1665802226e_nres @ A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Uu2 @ ( refine1767639642_FAILi @ A ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_38_le__nofailI,axiom,
% 3.77/1.82 ! [A: $tType,M4: refine1665802226e_nres @ A,M5: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( ( refine1102455758nofail @ A @ M4 )
% 3.77/1.82 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M5 @ M4 ) )
% 3.77/1.82 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M5 @ M4 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_39_pwD1,axiom,
% 3.77/1.82 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 )
% 3.77/1.82 => ( ( refine1102455758nofail @ A @ S2 )
% 3.77/1.82 => ( refine1102455758nofail @ A @ S ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_40_pwD2,axiom,
% 3.77/1.82 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A,X3: A] :
% 3.77/1.82 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 )
% 3.77/1.82 => ( ( refine1315500908_inres @ A @ S @ X3 )
% 3.77/1.82 => ( refine1315500908_inres @ A @ S2 @ X3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_41_old_Oprod_Oinject,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 3.77/1.82 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 3.77/1.82 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.77/1.82 = ( ( A3 = A4 )
% 3.77/1.82 & ( B3 = B4 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_42_prod_Oinject,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 3.77/1.82 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 3.77/1.82 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 3.77/1.82 = ( ( X1 = Y1 )
% 3.77/1.82 & ( X2 = Y2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_43_order__mono__setup_Orefl,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_44_complete__interval,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( condit1037483654norder @ A )
% 3.77/1.82 => ! [A3: A,B3: A,P: A > $o] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( P @ A3 )
% 3.77/1.82 => ( ~ ( P @ B3 )
% 3.77/1.82 => ? [C2: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A3 @ C2 )
% 3.77/1.82 & ( ord_less_eq @ A @ C2 @ B3 )
% 3.77/1.82 & ! [X7: A] :
% 3.77/1.82 ( ( ( ord_less_eq @ A @ A3 @ X7 )
% 3.77/1.82 & ( ord_less @ A @ X7 @ C2 ) )
% 3.77/1.82 => ( P @ X7 ) )
% 3.77/1.82 & ! [D: A] :
% 3.77/1.82 ( ! [X5: A] :
% 3.77/1.82 ( ( ( ord_less_eq @ A @ A3 @ X5 )
% 3.77/1.82 & ( ord_less @ A @ X5 @ D ) )
% 3.77/1.82 => ( P @ X5 ) )
% 3.77/1.82 => ( ord_less_eq @ A @ D @ C2 ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_45_mem__Collect__eq,axiom,
% 3.77/1.82 ! [A: $tType,A3: A,P: A > $o] :
% 3.77/1.82 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 3.77/1.82 = ( P @ A3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_46_Collect__mem__eq,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A] :
% 3.77/1.82 ( ( collect @ A
% 3.77/1.82 @ ^ [X4: A] : ( member @ A @ X4 @ A5 ) )
% 3.77/1.82 = A5 ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_47_Collect__cong,axiom,
% 3.77/1.82 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.77/1.82 ( ! [X5: A] :
% 3.77/1.82 ( ( P @ X5 )
% 3.77/1.82 = ( Q @ X5 ) )
% 3.77/1.82 => ( ( collect @ A @ P )
% 3.77/1.82 = ( collect @ A @ Q ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_48_ext,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.77/1.82 ( ! [X5: A] :
% 3.77/1.82 ( ( F @ X5 )
% 3.77/1.82 = ( G @ X5 ) )
% 3.77/1.82 => ( F = G ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_49_order_Onot__eq__order__implies__strict,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [A3: A,B3: A] :
% 3.77/1.82 ( ( A3 != B3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.77/1.82 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_50_dual__order_Ostrict__implies__order,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [B3: A,A3: A] :
% 3.77/1.82 ( ( ord_less @ A @ B3 @ A3 )
% 3.77/1.82 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_51_dual__order_Ostrict__iff__order,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ( ( ord_less @ A )
% 3.77/1.82 = ( ^ [B5: A,A6: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ B5 @ A6 )
% 3.77/1.82 & ( A6 != B5 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_52_dual__order_Oorder__iff__strict,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ( ( ord_less_eq @ A )
% 3.77/1.82 = ( ^ [B5: A,A6: A] :
% 3.77/1.82 ( ( ord_less @ A @ B5 @ A6 )
% 3.77/1.82 | ( A6 = B5 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_53_order_Ostrict__implies__order,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [A3: A,B3: A] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ B3 )
% 3.77/1.82 => ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_54_less__eq__nres_Osimps_I2_J,axiom,
% 3.77/1.82 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.77/1.82 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A3 ) @ ( refine605929679le_RES @ A @ B3 ) )
% 3.77/1.82 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_55_nres__order__simps_I5_J,axiom,
% 3.77/1.82 ! [E: $tType,X: set @ E,Y: set @ E] :
% 3.77/1.82 ( ( ord_less_eq @ ( refine1665802226e_nres @ E ) @ ( refine605929679le_RES @ E @ X ) @ ( refine605929679le_RES @ E @ Y ) )
% 3.77/1.82 = ( ord_less_eq @ ( set @ E ) @ X @ Y ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_56_less__nres_Osimps_I3_J,axiom,
% 3.77/1.82 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.77/1.82 ( ( ord_less @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A3 ) @ ( refine605929679le_RES @ A @ B3 ) )
% 3.77/1.82 = ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_57_le__funD,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType] :
% 3.77/1.82 ( ( ord @ B )
% 3.77/1.82 => ! [F: A > B,G: A > B,X3: A] :
% 3.77/1.82 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.77/1.82 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_58_le__funE,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType] :
% 3.77/1.82 ( ( ord @ B )
% 3.77/1.82 => ! [F: A > B,G: A > B,X3: A] :
% 3.77/1.82 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.77/1.82 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_59_le__funI,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType] :
% 3.77/1.82 ( ( ord @ B )
% 3.77/1.82 => ! [F: A > B,G: A > B] :
% 3.77/1.82 ( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
% 3.77/1.82 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_60_le__fun__def,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType] :
% 3.77/1.82 ( ( ord @ B )
% 3.77/1.82 => ( ( ord_less_eq @ ( A > B ) )
% 3.77/1.82 = ( ^ [F3: A > B,G2: A > B] :
% 3.77/1.82 ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_61_order__subst1,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType] :
% 3.77/1.82 ( ( ( order @ B )
% 3.77/1.82 & ( order @ A ) )
% 3.77/1.82 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 3.77/1.82 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 3.77/1.82 => ( ! [X5: B,Y5: B] :
% 3.77/1.82 ( ( ord_less_eq @ B @ X5 @ Y5 )
% 3.77/1.82 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 3.77/1.82 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_62_order__subst2,axiom,
% 3.77/1.82 ! [A: $tType,C: $tType] :
% 3.77/1.82 ( ( ( order @ C )
% 3.77/1.82 & ( order @ A ) )
% 3.77/1.82 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 3.77/1.82 => ( ! [X5: A,Y5: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ X5 @ Y5 )
% 3.77/1.82 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 3.77/1.82 => ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_63_ord__eq__le__subst,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType] :
% 3.77/1.82 ( ( ( ord @ B )
% 3.77/1.82 & ( ord @ A ) )
% 3.77/1.82 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 3.77/1.82 ( ( A3
% 3.77/1.82 = ( F @ B3 ) )
% 3.77/1.82 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 3.77/1.82 => ( ! [X5: B,Y5: B] :
% 3.77/1.82 ( ( ord_less_eq @ B @ X5 @ Y5 )
% 3.77/1.82 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 3.77/1.82 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_64_ord__le__eq__subst,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType] :
% 3.77/1.82 ( ( ( ord @ B )
% 3.77/1.82 & ( ord @ A ) )
% 3.77/1.82 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( ( F @ B3 )
% 3.77/1.82 = C3 )
% 3.77/1.82 => ( ! [X5: A,Y5: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ X5 @ Y5 )
% 3.77/1.82 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 3.77/1.82 => ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_65_eq__iff,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 3.77/1.82 = ( ^ [X4: A,Y6: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ X4 @ Y6 )
% 3.77/1.82 & ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_66_antisym,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ Y3 @ X3 )
% 3.77/1.82 => ( X3 = Y3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_67_linear,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.77/1.82 | ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_68_eq__refl,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( X3 = Y3 )
% 3.77/1.82 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_69_le__cases,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ~ ( ord_less_eq @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_70_order_Otrans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [A3: A,B3: A,C3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 3.77/1.82 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_71_le__cases3,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A,Z2: A] :
% 3.77/1.82 ( ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.77/1.82 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 3.77/1.82 => ( ( ( ord_less_eq @ A @ Y3 @ X3 )
% 3.77/1.82 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 3.77/1.82 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 3.77/1.82 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 3.77/1.82 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 3.77/1.82 => ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
% 3.77/1.82 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 3.77/1.82 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 3.77/1.82 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 3.77/1.82 => ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_72_antisym__conv,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [Y3: A,X3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.77/1.82 = ( X3 = Y3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_73_order__class_Oorder_Oeq__iff,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 3.77/1.82 = ( ^ [A6: A,B5: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A6 @ B5 )
% 3.77/1.82 & ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_74_ord__eq__le__trans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ord @ A )
% 3.77/1.82 => ! [A3: A,B3: A,C3: A] :
% 3.77/1.82 ( ( A3 = B3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 3.77/1.82 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_75_ord__le__eq__trans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ord @ A )
% 3.77/1.82 => ! [A3: A,B3: A,C3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( B3 = C3 )
% 3.77/1.82 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_76_order__class_Oorder_Oantisym,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [A3: A,B3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.77/1.82 => ( A3 = B3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_77_order__trans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A,Z2: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 3.77/1.82 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_78_dual__order_Orefl,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_79_linorder__wlog,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [P: A > A > $o,A3: A,B3: A] :
% 3.77/1.82 ( ! [A2: A,B2: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.77/1.82 => ( P @ A2 @ B2 ) )
% 3.77/1.82 => ( ! [A2: A,B2: A] :
% 3.77/1.82 ( ( P @ B2 @ A2 )
% 3.77/1.82 => ( P @ A2 @ B2 ) )
% 3.77/1.82 => ( P @ A3 @ B3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_80_dual__order_Otrans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [B3: A,A3: A,C3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 3.77/1.82 => ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_81_dual__order_Oeq__iff,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 3.77/1.82 = ( ^ [A6: A,B5: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ B5 @ A6 )
% 3.77/1.82 & ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_82_dual__order_Oantisym,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [B3: A,A3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.77/1.82 => ( A3 = B3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_83_order__mono__setup_Omono__if,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [T2: A,T3: A,E2: A,E3: A,B3: $o] :
% 3.77/1.82 ( ( ord_less_eq @ A @ T2 @ T3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ E2 @ E3 )
% 3.77/1.82 => ( ord_less_eq @ A @ ( if @ A @ B3 @ T2 @ E2 ) @ ( if @ A @ B3 @ T3 @ E3 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_84_order__mono__setup_Omono__let,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [F: B > A,F4: B > A,X3: B] :
% 3.77/1.82 ( ! [X5: B] : ( ord_less_eq @ A @ ( F @ X5 ) @ ( F4 @ X5 ) )
% 3.77/1.82 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F4 @ X3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_85_surj__pair,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 3.77/1.82 ? [X5: A,Y5: B] :
% 3.77/1.82 ( P2
% 3.77/1.82 = ( product_Pair @ A @ B @ X5 @ Y5 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_86_prod__cases,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 3.77/1.82 ( ! [A2: A,B2: B] : ( P @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.77/1.82 => ( P @ P2 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_87_Pair__inject,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 3.77/1.82 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 3.77/1.82 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.77/1.82 => ~ ( ( A3 = A4 )
% 3.77/1.82 => ( B3 != B4 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_88_prod__cases3,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.77/1.82 ~ ! [A2: A,B2: B,C2: C] :
% 3.77/1.82 ( Y3
% 3.77/1.82 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_89_prod__cases4,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 3.77/1.82 ~ ! [A2: A,B2: B,C2: C,D3: D2] :
% 3.77/1.82 ( Y3
% 3.77/1.82 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B2 @ ( product_Pair @ C @ D2 @ C2 @ D3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_90_prod__cases5,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 3.77/1.82 ~ ! [A2: A,B2: B,C2: C,D3: D2,E4: E] :
% 3.77/1.82 ( Y3
% 3.77/1.82 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C2 @ ( product_Pair @ D2 @ E @ D3 @ E4 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_91_prod__cases6,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
% 3.77/1.82 ~ ! [A2: A,B2: B,C2: C,D3: D2,E4: E,F6: F5] :
% 3.77/1.82 ( Y3
% 3.77/1.82 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E4 @ F6 ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_92_prod__cases7,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
% 3.77/1.82 ~ ! [A2: A,B2: B,C2: C,D3: D2,E4: E,F6: F5,G4: G3] :
% 3.77/1.82 ( Y3
% 3.77/1.82 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E4 @ ( product_Pair @ F5 @ G3 @ F6 @ G4 ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_93_prod__induct3,axiom,
% 3.77/1.82 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.77/1.82 ( ! [A2: A,B2: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C2 ) ) )
% 3.77/1.82 => ( P @ X3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_94_prod__induct4,axiom,
% 3.77/1.82 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 3.77/1.82 ( ! [A2: A,B2: B,C2: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B2 @ ( product_Pair @ C @ D2 @ C2 @ D3 ) ) ) )
% 3.77/1.82 => ( P @ X3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_95_prod__induct5,axiom,
% 3.77/1.82 ! [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,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 3.77/1.82 ( ! [A2: A,B2: B,C2: C,D3: D2,E4: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C2 @ ( product_Pair @ D2 @ E @ D3 @ E4 ) ) ) ) )
% 3.77/1.82 => ( P @ X3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_96_prod__induct6,axiom,
% 3.77/1.82 ! [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,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
% 3.77/1.82 ( ! [A2: A,B2: B,C2: C,D3: D2,E4: E,F6: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E4 @ F6 ) ) ) ) ) )
% 3.77/1.82 => ( P @ X3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_97_prod__induct7,axiom,
% 3.77/1.82 ! [G3: $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 @ G3 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
% 3.77/1.82 ( ! [A2: A,B2: B,C2: C,D3: D2,E4: E,F6: F5,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E4 @ ( product_Pair @ F5 @ G3 @ F6 @ G4 ) ) ) ) ) ) )
% 3.77/1.82 => ( P @ X3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_98_old_Oprod_Oexhaust,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 3.77/1.82 ~ ! [A2: A,B2: B] :
% 3.77/1.82 ( Y3
% 3.77/1.82 != ( product_Pair @ A @ B @ A2 @ B2 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_99_old_Oprod_Oinducts,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 3.77/1.82 ( ! [A2: A,B2: B] : ( P @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.77/1.82 => ( P @ Prod ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_100_ord__eq__less__subst,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType] :
% 3.77/1.82 ( ( ( ord @ B )
% 3.77/1.82 & ( ord @ A ) )
% 3.77/1.82 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 3.77/1.82 ( ( A3
% 3.77/1.82 = ( F @ B3 ) )
% 3.77/1.82 => ( ( ord_less @ B @ B3 @ C3 )
% 3.77/1.82 => ( ! [X5: B,Y5: B] :
% 3.77/1.82 ( ( ord_less @ B @ X5 @ Y5 )
% 3.77/1.82 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 3.77/1.82 => ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_101_ord__less__eq__subst,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType] :
% 3.77/1.82 ( ( ( ord @ B )
% 3.77/1.82 & ( ord @ A ) )
% 3.77/1.82 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( ( F @ B3 )
% 3.77/1.82 = C3 )
% 3.77/1.82 => ( ! [X5: A,Y5: A] :
% 3.77/1.82 ( ( ord_less @ A @ X5 @ Y5 )
% 3.77/1.82 => ( ord_less @ B @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 3.77/1.82 => ( ord_less @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_102_order__less__subst1,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType] :
% 3.77/1.82 ( ( ( order @ B )
% 3.77/1.82 & ( order @ A ) )
% 3.77/1.82 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 3.77/1.82 => ( ( ord_less @ B @ B3 @ C3 )
% 3.77/1.82 => ( ! [X5: B,Y5: B] :
% 3.77/1.82 ( ( ord_less @ B @ X5 @ Y5 )
% 3.77/1.82 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 3.77/1.82 => ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_103_order__less__subst2,axiom,
% 3.77/1.82 ! [A: $tType,C: $tType] :
% 3.77/1.82 ( ( ( order @ C )
% 3.77/1.82 & ( order @ A ) )
% 3.77/1.82 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( ord_less @ C @ ( F @ B3 ) @ C3 )
% 3.77/1.82 => ( ! [X5: A,Y5: A] :
% 3.77/1.82 ( ( ord_less @ A @ X5 @ Y5 )
% 3.77/1.82 => ( ord_less @ C @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 3.77/1.82 => ( ord_less @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_104_lt__ex,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( no_bot @ A )
% 3.77/1.82 => ! [X3: A] :
% 3.77/1.82 ? [Y5: A] : ( ord_less @ A @ Y5 @ X3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_105_gt__ex,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( no_top @ A )
% 3.77/1.82 => ! [X3: A] :
% 3.77/1.82 ? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_106_neqE,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( X3 != Y3 )
% 3.77/1.82 => ( ~ ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_107_neq__iff,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( X3 != Y3 )
% 3.77/1.82 = ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 | ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_108_order_Oasym,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [A3: A,B3: A] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ B3 )
% 3.77/1.82 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_109_dense,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( dense_order @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ? [Z3: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Z3 )
% 3.77/1.82 & ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_110_less__imp__neq,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ( X3 != Y3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_111_less__asym,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_112_less__asym_H,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [A3: A,B3: A] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ B3 )
% 3.77/1.82 => ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_113_less__trans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A,Z2: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ( ord_less @ A @ Y3 @ Z2 )
% 3.77/1.82 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_114_less__linear,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 | ( X3 = Y3 )
% 3.77/1.82 | ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_115_less__irrefl,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [X3: A] :
% 3.77/1.82 ~ ( ord_less @ A @ X3 @ X3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_116_ord__eq__less__trans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ord @ A )
% 3.77/1.82 => ! [A3: A,B3: A,C3: A] :
% 3.77/1.82 ( ( A3 = B3 )
% 3.77/1.82 => ( ( ord_less @ A @ B3 @ C3 )
% 3.77/1.82 => ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_117_ord__less__eq__trans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ord @ A )
% 3.77/1.82 => ! [A3: A,B3: A,C3: A] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( B3 = C3 )
% 3.77/1.82 => ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_118_dual__order_Oasym,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [B3: A,A3: A] :
% 3.77/1.82 ( ( ord_less @ A @ B3 @ A3 )
% 3.77/1.82 => ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_119_less__imp__not__eq,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ( X3 != Y3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_120_less__not__sym,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_121_less__induct,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( wellorder @ A )
% 3.77/1.82 => ! [P: A > $o,A3: A] :
% 3.77/1.82 ( ! [X5: A] :
% 3.77/1.82 ( ! [Y7: A] :
% 3.77/1.82 ( ( ord_less @ A @ Y7 @ X5 )
% 3.77/1.82 => ( P @ Y7 ) )
% 3.77/1.82 => ( P @ X5 ) )
% 3.77/1.82 => ( P @ A3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_122_antisym__conv3,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [Y3: A,X3: A] :
% 3.77/1.82 ( ~ ( ord_less @ A @ Y3 @ X3 )
% 3.77/1.82 => ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 3.77/1.82 = ( X3 = Y3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_123_less__imp__not__eq2,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ( Y3 != X3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_124_less__imp__triv,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A,P: $o] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ( ord_less @ A @ Y3 @ X3 )
% 3.77/1.82 => P ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_125_linorder__cases,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ~ ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ( X3 != Y3 )
% 3.77/1.82 => ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_126_dual__order_Oirrefl,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [A3: A] :
% 3.77/1.82 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_127_order_Ostrict__trans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [A3: A,B3: A,C3: A] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( ord_less @ A @ B3 @ C3 )
% 3.77/1.82 => ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_128_less__imp__not__less,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_129_exists__least__iff,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( wellorder @ A )
% 3.77/1.82 => ( ( ^ [P3: A > $o] :
% 3.77/1.82 ? [X8: A] : ( P3 @ X8 ) )
% 3.77/1.82 = ( ^ [P4: A > $o] :
% 3.77/1.82 ? [N: A] :
% 3.77/1.82 ( ( P4 @ N )
% 3.77/1.82 & ! [M: A] :
% 3.77/1.82 ( ( ord_less @ A @ M @ N )
% 3.77/1.82 => ~ ( P4 @ M ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_130_linorder__less__wlog,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [P: A > A > $o,A3: A,B3: A] :
% 3.77/1.82 ( ! [A2: A,B2: A] :
% 3.77/1.82 ( ( ord_less @ A @ A2 @ B2 )
% 3.77/1.82 => ( P @ A2 @ B2 ) )
% 3.77/1.82 => ( ! [A2: A] : ( P @ A2 @ A2 )
% 3.77/1.82 => ( ! [A2: A,B2: A] :
% 3.77/1.82 ( ( P @ B2 @ A2 )
% 3.77/1.82 => ( P @ A2 @ B2 ) )
% 3.77/1.82 => ( P @ A3 @ B3 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_131_dual__order_Ostrict__trans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [B3: A,A3: A,C3: A] :
% 3.77/1.82 ( ( ord_less @ A @ B3 @ A3 )
% 3.77/1.82 => ( ( ord_less @ A @ C3 @ B3 )
% 3.77/1.82 => ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_132_not__less__iff__gr__or__eq,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 3.77/1.82 = ( ( ord_less @ A @ Y3 @ X3 )
% 3.77/1.82 | ( X3 = Y3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_133_order_Ostrict__implies__not__eq,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [A3: A,B3: A] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ B3 )
% 3.77/1.82 => ( A3 != B3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_134_dual__order_Ostrict__implies__not__eq,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [B3: A,A3: A] :
% 3.77/1.82 ( ( ord_less @ A @ B3 @ A3 )
% 3.77/1.82 => ( A3 != B3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_135_ex__gt__or__lt,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( condit1656338222tinuum @ A )
% 3.77/1.82 => ! [A3: A] :
% 3.77/1.82 ? [B2: A] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ B2 )
% 3.77/1.82 | ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_136_less__eq__nres_Oelims_I3_J,axiom,
% 3.77/1.82 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 3.77/1.82 => ( ! [A2: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ A2 ) )
% 3.77/1.82 => ! [B2: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) )
% 3.77/1.82 => ~ ( ( X3
% 3.77/1.82 = ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ! [Uv: set @ A] :
% 3.77/1.82 ( Xa
% 3.77/1.82 != ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_137_less__eq__nres_Oelims_I2_J,axiom,
% 3.77/1.82 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 3.77/1.82 => ( ( Xa
% 3.77/1.82 != ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ~ ! [A2: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ A2 ) )
% 3.77/1.82 => ! [B2: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_138_less__eq__nres_Oelims_I1_J,axiom,
% 3.77/1.82 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
% 3.77/1.82 ( ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 3.77/1.82 = Y3 )
% 3.77/1.82 => ( ( ( Xa
% 3.77/1.82 = ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ~ Y3 )
% 3.77/1.82 => ( ! [A2: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ A2 ) )
% 3.77/1.82 => ! [B2: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ( Y3
% 3.77/1.82 = ( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) )
% 3.77/1.82 => ~ ( ( X3
% 3.77/1.82 = ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ( ? [Uv: set @ A] :
% 3.77/1.82 ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ Uv ) )
% 3.77/1.82 => Y3 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_139_less__nres_Oelims_I3_J,axiom,
% 3.77/1.82 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ~ ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 3.77/1.82 => ( ( X3
% 3.77/1.82 != ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ~ ! [A2: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ A2 ) )
% 3.77/1.82 => ! [B2: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_140_less__nres_Oelims_I2_J,axiom,
% 3.77/1.82 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 3.77/1.82 => ( ( ? [Uv: set @ A] :
% 3.77/1.82 ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ Uv ) )
% 3.77/1.82 => ( Xa
% 3.77/1.82 != ( refine1767639642_FAILi @ A ) ) )
% 3.77/1.82 => ~ ! [A2: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ A2 ) )
% 3.77/1.82 => ! [B2: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ~ ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_141_less__nres_Oelims_I1_J,axiom,
% 3.77/1.82 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
% 3.77/1.82 ( ( ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 3.77/1.82 = Y3 )
% 3.77/1.82 => ( ( ( X3
% 3.77/1.82 = ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => Y3 )
% 3.77/1.82 => ( ( ? [Uv: set @ A] :
% 3.77/1.82 ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ Uv ) )
% 3.77/1.82 => ( ( Xa
% 3.77/1.82 = ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ~ Y3 ) )
% 3.77/1.82 => ~ ! [A2: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ A2 ) )
% 3.77/1.82 => ! [B2: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ( Y3
% 3.77/1.82 = ( ~ ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_142_leD,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [Y3: A,X3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 3.77/1.82 => ~ ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_143_leI,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ~ ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_144_le__less,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ( ( ord_less_eq @ A )
% 3.77/1.82 = ( ^ [X4: A,Y6: A] :
% 3.77/1.82 ( ( ord_less @ A @ X4 @ Y6 )
% 3.77/1.82 | ( X4 = Y6 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_145_less__le,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ( ( ord_less @ A )
% 3.77/1.82 = ( ^ [X4: A,Y6: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ X4 @ Y6 )
% 3.77/1.82 & ( X4 != Y6 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_146_order__le__less__subst1,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType] :
% 3.77/1.82 ( ( ( order @ B )
% 3.77/1.82 & ( order @ A ) )
% 3.77/1.82 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 3.77/1.82 => ( ( ord_less @ B @ B3 @ C3 )
% 3.77/1.82 => ( ! [X5: B,Y5: B] :
% 3.77/1.82 ( ( ord_less @ B @ X5 @ Y5 )
% 3.77/1.82 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 3.77/1.82 => ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_147_order__le__less__subst2,axiom,
% 3.77/1.82 ! [A: $tType,C: $tType] :
% 3.77/1.82 ( ( ( order @ C )
% 3.77/1.82 & ( order @ A ) )
% 3.77/1.82 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( ord_less @ C @ ( F @ B3 ) @ C3 )
% 3.77/1.82 => ( ! [X5: A,Y5: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ X5 @ Y5 )
% 3.77/1.82 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 3.77/1.82 => ( ord_less @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_148_order__less__le__subst1,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType] :
% 3.77/1.82 ( ( ( order @ B )
% 3.77/1.82 & ( order @ A ) )
% 3.77/1.82 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ ( F @ B3 ) )
% 3.77/1.82 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 3.77/1.82 => ( ! [X5: B,Y5: B] :
% 3.77/1.82 ( ( ord_less_eq @ B @ X5 @ Y5 )
% 3.77/1.82 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 3.77/1.82 => ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_149_order__less__le__subst2,axiom,
% 3.77/1.82 ! [A: $tType,C: $tType] :
% 3.77/1.82 ( ( ( order @ C )
% 3.77/1.82 & ( order @ A ) )
% 3.77/1.82 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 3.77/1.82 => ( ! [X5: A,Y5: A] :
% 3.77/1.82 ( ( ord_less @ A @ X5 @ Y5 )
% 3.77/1.82 => ( ord_less @ C @ ( F @ X5 ) @ ( F @ Y5 ) ) )
% 3.77/1.82 => ( ord_less @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_150_not__le,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ~ ( ord_less_eq @ A @ X3 @ Y3 ) )
% 3.77/1.82 = ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_151_not__less,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 3.77/1.82 = ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_152_le__neq__trans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [A3: A,B3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( A3 != B3 )
% 3.77/1.82 => ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_153_antisym__conv1,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ~ ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.77/1.82 = ( X3 = Y3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_154_antisym__conv2,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
% 3.77/1.82 = ( X3 = Y3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_155_less__imp__le,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_156_le__less__trans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A,Z2: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ( ord_less @ A @ Y3 @ Z2 )
% 3.77/1.82 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_157_less__le__trans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A,Z2: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 3.77/1.82 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_158_dense__ge,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( dense_linorder @ A )
% 3.77/1.82 => ! [Z2: A,Y3: A] :
% 3.77/1.82 ( ! [X5: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z2 @ X5 )
% 3.77/1.82 => ( ord_less_eq @ A @ Y3 @ X5 ) )
% 3.77/1.82 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_159_dense__le,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( dense_linorder @ A )
% 3.77/1.82 => ! [Y3: A,Z2: A] :
% 3.77/1.82 ( ! [X5: A] :
% 3.77/1.82 ( ( ord_less @ A @ X5 @ Y3 )
% 3.77/1.82 => ( ord_less_eq @ A @ X5 @ Z2 ) )
% 3.77/1.82 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_160_le__less__linear,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.77/1.82 | ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_161_le__imp__less__or__eq,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 | ( X3 = Y3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_162_less__le__not__le,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( preorder @ A )
% 3.77/1.82 => ( ( ord_less @ A )
% 3.77/1.82 = ( ^ [X4: A,Y6: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ X4 @ Y6 )
% 3.77/1.82 & ~ ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_163_not__le__imp__less,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [Y3: A,X3: A] :
% 3.77/1.82 ( ~ ( ord_less_eq @ A @ Y3 @ X3 )
% 3.77/1.82 => ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_164_order_Ostrict__trans1,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [A3: A,B3: A,C3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( ord_less @ A @ B3 @ C3 )
% 3.77/1.82 => ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_165_order_Ostrict__trans2,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [A3: A,B3: A,C3: A] :
% 3.77/1.82 ( ( ord_less @ A @ A3 @ B3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 3.77/1.82 => ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_166_order_Oorder__iff__strict,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ( ( ord_less_eq @ A )
% 3.77/1.82 = ( ^ [A6: A,B5: A] :
% 3.77/1.82 ( ( ord_less @ A @ A6 @ B5 )
% 3.77/1.82 | ( A6 = B5 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_167_order_Ostrict__iff__order,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ( ( ord_less @ A )
% 3.77/1.82 = ( ^ [A6: A,B5: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ A6 @ B5 )
% 3.77/1.82 & ( A6 != B5 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_168_dual__order_Ostrict__trans1,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [B3: A,A3: A,C3: A] :
% 3.77/1.82 ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.77/1.82 => ( ( ord_less @ A @ C3 @ B3 )
% 3.77/1.82 => ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_169_dual__order_Ostrict__trans2,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [B3: A,A3: A,C3: A] :
% 3.77/1.82 ( ( ord_less @ A @ B3 @ A3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 3.77/1.82 => ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_170_dense__ge__bounded,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( dense_linorder @ A )
% 3.77/1.82 => ! [Z2: A,X3: A,Y3: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z2 @ X3 )
% 3.77/1.82 => ( ! [W2: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z2 @ W2 )
% 3.77/1.82 => ( ( ord_less @ A @ W2 @ X3 )
% 3.77/1.82 => ( ord_less_eq @ A @ Y3 @ W2 ) ) )
% 3.77/1.82 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_171_dense__le__bounded,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( dense_linorder @ A )
% 3.77/1.82 => ! [X3: A,Y3: A,Z2: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ! [W2: A] :
% 3.77/1.82 ( ( ord_less @ A @ X3 @ W2 )
% 3.77/1.82 => ( ( ord_less @ A @ W2 @ Y3 )
% 3.77/1.82 => ( ord_less_eq @ A @ W2 @ Z2 ) ) )
% 3.77/1.82 => ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_172_minf_I8_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [T2: A] :
% 3.77/1.82 ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ X7 @ Z3 )
% 3.77/1.82 => ~ ( ord_less_eq @ A @ T2 @ X7 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_173_minf_I6_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [T2: A] :
% 3.77/1.82 ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ X7 @ Z3 )
% 3.77/1.82 => ( ord_less_eq @ A @ X7 @ T2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_174_pinf_I8_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [T2: A] :
% 3.77/1.82 ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z3 @ X7 )
% 3.77/1.82 => ( ord_less_eq @ A @ T2 @ X7 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_175_pinf_I6_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [T2: A] :
% 3.77/1.82 ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z3 @ X7 )
% 3.77/1.82 => ~ ( ord_less_eq @ A @ X7 @ T2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_176_verit__comp__simplify1_I3_J,axiom,
% 3.77/1.82 ! [B: $tType] :
% 3.77/1.82 ( ( linorder @ B )
% 3.77/1.82 => ! [B4: B,A4: B] :
% 3.77/1.82 ( ( ~ ( ord_less_eq @ B @ B4 @ A4 ) )
% 3.77/1.82 = ( ord_less @ B @ A4 @ B4 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_177_internal__case__prod__conv,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B3: C] :
% 3.77/1.82 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 3.77/1.82 = ( C3 @ A3 @ B3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_178_old_Oprod_Orec,axiom,
% 3.77/1.82 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 3.77/1.82 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 3.77/1.82 = ( F1 @ A3 @ B3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_179_relChain__def,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType] :
% 3.77/1.82 ( ( ord @ B )
% 3.77/1.82 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 3.77/1.82 = ( ^ [R: set @ ( product_prod @ A @ A ),As: A > B] :
% 3.77/1.82 ! [I: A,J: A] :
% 3.77/1.82 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
% 3.77/1.82 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_180_less__fun__def,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType] :
% 3.77/1.82 ( ( ord @ B )
% 3.77/1.82 => ( ( ord_less @ ( A > B ) )
% 3.77/1.82 = ( ^ [F3: A > B,G2: A > B] :
% 3.77/1.82 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 3.77/1.82 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_181_verit__la__disequality,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [A3: A,B3: A] :
% 3.77/1.82 ( ( A3 = B3 )
% 3.77/1.82 | ~ ( ord_less_eq @ A @ A3 @ B3 )
% 3.77/1.82 | ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_182_verit__comp__simplify1_I1_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [A3: A] :
% 3.77/1.82 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_183_pinf_I1_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
% 3.77/1.82 ( ? [Z4: A] :
% 3.77/1.82 ! [X5: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z4 @ X5 )
% 3.77/1.82 => ( ( P @ X5 )
% 3.77/1.82 = ( P5 @ X5 ) ) )
% 3.77/1.82 => ( ? [Z4: A] :
% 3.77/1.82 ! [X5: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z4 @ X5 )
% 3.77/1.82 => ( ( Q @ X5 )
% 3.77/1.82 = ( Q2 @ X5 ) ) )
% 3.77/1.82 => ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z3 @ X7 )
% 3.77/1.82 => ( ( ( P @ X7 )
% 3.77/1.82 & ( Q @ X7 ) )
% 3.77/1.82 = ( ( P5 @ X7 )
% 3.77/1.82 & ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_184_pinf_I2_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
% 3.77/1.82 ( ? [Z4: A] :
% 3.77/1.82 ! [X5: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z4 @ X5 )
% 3.77/1.82 => ( ( P @ X5 )
% 3.77/1.82 = ( P5 @ X5 ) ) )
% 3.77/1.82 => ( ? [Z4: A] :
% 3.77/1.82 ! [X5: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z4 @ X5 )
% 3.77/1.82 => ( ( Q @ X5 )
% 3.77/1.82 = ( Q2 @ X5 ) ) )
% 3.77/1.82 => ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z3 @ X7 )
% 3.77/1.82 => ( ( ( P @ X7 )
% 3.77/1.82 | ( Q @ X7 ) )
% 3.77/1.82 = ( ( P5 @ X7 )
% 3.77/1.82 | ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_185_pinf_I3_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [T2: A] :
% 3.77/1.82 ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z3 @ X7 )
% 3.77/1.82 => ( X7 != T2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_186_pinf_I4_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [T2: A] :
% 3.77/1.82 ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z3 @ X7 )
% 3.77/1.82 => ( X7 != T2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_187_pinf_I5_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [T2: A] :
% 3.77/1.82 ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z3 @ X7 )
% 3.77/1.82 => ~ ( ord_less @ A @ X7 @ T2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_188_pinf_I7_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [T2: A] :
% 3.77/1.82 ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ Z3 @ X7 )
% 3.77/1.82 => ( ord_less @ A @ T2 @ X7 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_189_pinf_I11_J,axiom,
% 3.77/1.82 ! [C: $tType,D2: $tType] :
% 3.77/1.82 ( ( ord @ C )
% 3.77/1.82 => ! [F7: D2] :
% 3.77/1.82 ? [Z3: C] :
% 3.77/1.82 ! [X7: C] :
% 3.77/1.82 ( ( ord_less @ C @ Z3 @ X7 )
% 3.77/1.82 => ( F7 = F7 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_190_minf_I1_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
% 3.77/1.82 ( ? [Z4: A] :
% 3.77/1.82 ! [X5: A] :
% 3.77/1.82 ( ( ord_less @ A @ X5 @ Z4 )
% 3.77/1.82 => ( ( P @ X5 )
% 3.77/1.82 = ( P5 @ X5 ) ) )
% 3.77/1.82 => ( ? [Z4: A] :
% 3.77/1.82 ! [X5: A] :
% 3.77/1.82 ( ( ord_less @ A @ X5 @ Z4 )
% 3.77/1.82 => ( ( Q @ X5 )
% 3.77/1.82 = ( Q2 @ X5 ) ) )
% 3.77/1.82 => ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ X7 @ Z3 )
% 3.77/1.82 => ( ( ( P @ X7 )
% 3.77/1.82 & ( Q @ X7 ) )
% 3.77/1.82 = ( ( P5 @ X7 )
% 3.77/1.82 & ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_191_minf_I2_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
% 3.77/1.82 ( ? [Z4: A] :
% 3.77/1.82 ! [X5: A] :
% 3.77/1.82 ( ( ord_less @ A @ X5 @ Z4 )
% 3.77/1.82 => ( ( P @ X5 )
% 3.77/1.82 = ( P5 @ X5 ) ) )
% 3.77/1.82 => ( ? [Z4: A] :
% 3.77/1.82 ! [X5: A] :
% 3.77/1.82 ( ( ord_less @ A @ X5 @ Z4 )
% 3.77/1.82 => ( ( Q @ X5 )
% 3.77/1.82 = ( Q2 @ X5 ) ) )
% 3.77/1.82 => ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ X7 @ Z3 )
% 3.77/1.82 => ( ( ( P @ X7 )
% 3.77/1.82 | ( Q @ X7 ) )
% 3.77/1.82 = ( ( P5 @ X7 )
% 3.77/1.82 | ( Q2 @ X7 ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_192_minf_I3_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [T2: A] :
% 3.77/1.82 ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ X7 @ Z3 )
% 3.77/1.82 => ( X7 != T2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_193_minf_I4_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [T2: A] :
% 3.77/1.82 ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ X7 @ Z3 )
% 3.77/1.82 => ( X7 != T2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_194_minf_I5_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [T2: A] :
% 3.77/1.82 ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ X7 @ Z3 )
% 3.77/1.82 => ( ord_less @ A @ X7 @ T2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_195_minf_I7_J,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [T2: A] :
% 3.77/1.82 ? [Z3: A] :
% 3.77/1.82 ! [X7: A] :
% 3.77/1.82 ( ( ord_less @ A @ X7 @ Z3 )
% 3.77/1.82 => ~ ( ord_less @ A @ T2 @ X7 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_196_minf_I11_J,axiom,
% 3.77/1.82 ! [C: $tType,D2: $tType] :
% 3.77/1.82 ( ( ord @ C )
% 3.77/1.82 => ! [F7: D2] :
% 3.77/1.82 ? [Z3: C] :
% 3.77/1.82 ! [X7: C] :
% 3.77/1.82 ( ( ord_less @ C @ X7 @ Z3 )
% 3.77/1.82 => ( F7 = F7 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_197_psubsetI,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 3.77/1.82 => ( ( A5 != B6 )
% 3.77/1.82 => ( ord_less @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_198_subset__antisym,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 3.77/1.82 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
% 3.77/1.82 => ( A5 = B6 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_199_subsetI,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 3.77/1.82 ( ! [X5: A] :
% 3.77/1.82 ( ( member @ A @ X5 @ A5 )
% 3.77/1.82 => ( member @ A @ X5 @ B6 ) )
% 3.77/1.82 => ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_200_in__mono,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A,X3: A] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 3.77/1.82 => ( ( member @ A @ X3 @ A5 )
% 3.77/1.82 => ( member @ A @ X3 @ B6 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_201_subsetD,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A,C3: A] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 3.77/1.82 => ( ( member @ A @ C3 @ A5 )
% 3.77/1.82 => ( member @ A @ C3 @ B6 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_202_equalityE,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 3.77/1.82 ( ( A5 = B6 )
% 3.77/1.82 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 3.77/1.82 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_203_subset__eq,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) )
% 3.77/1.82 = ( ^ [A7: set @ A,B7: set @ A] :
% 3.77/1.82 ! [X4: A] :
% 3.77/1.82 ( ( member @ A @ X4 @ A7 )
% 3.77/1.82 => ( member @ A @ X4 @ B7 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_204_equalityD1,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 3.77/1.82 ( ( A5 = B6 )
% 3.77/1.82 => ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_205_equalityD2,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 3.77/1.82 ( ( A5 = B6 )
% 3.77/1.82 => ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_206_subset__iff,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) )
% 3.77/1.82 = ( ^ [A7: set @ A,B7: set @ A] :
% 3.77/1.82 ! [T4: A] :
% 3.77/1.82 ( ( member @ A @ T4 @ A7 )
% 3.77/1.82 => ( member @ A @ T4 @ B7 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_207_subset__refl,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_208_Collect__mono,axiom,
% 3.77/1.82 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.77/1.82 ( ! [X5: A] :
% 3.77/1.82 ( ( P @ X5 )
% 3.77/1.82 => ( Q @ X5 ) )
% 3.77/1.82 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_209_subset__trans,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A,C4: set @ A] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 3.77/1.82 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 3.77/1.82 => ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_210_set__eq__subset,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ^ [Y4: set @ A,Z: set @ A] : ( Y4 = Z ) )
% 3.77/1.82 = ( ^ [A7: set @ A,B7: set @ A] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 3.77/1.82 & ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_211_Collect__mono__iff,axiom,
% 3.77/1.82 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 3.77/1.82 = ( ! [X4: A] :
% 3.77/1.82 ( ( P @ X4 )
% 3.77/1.82 => ( Q @ X4 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_212_psubsetD,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A,C3: A] :
% 3.77/1.82 ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
% 3.77/1.82 => ( ( member @ A @ C3 @ A5 )
% 3.77/1.82 => ( member @ A @ C3 @ B6 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_213_psubset__trans,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A,C4: set @ A] :
% 3.77/1.82 ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
% 3.77/1.82 => ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
% 3.77/1.82 => ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_214_psubsetE,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 3.77/1.82 ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
% 3.77/1.82 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 3.77/1.82 => ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_215_psubset__eq,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ord_less @ ( set @ A ) )
% 3.77/1.82 = ( ^ [A7: set @ A,B7: set @ A] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 3.77/1.82 & ( A7 != B7 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_216_psubset__imp__subset,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 3.77/1.82 ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
% 3.77/1.82 => ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_217_psubset__subset__trans,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A,C4: set @ A] :
% 3.77/1.82 ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
% 3.77/1.82 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 3.77/1.82 => ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_218_subset__not__subset__eq,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ord_less @ ( set @ A ) )
% 3.77/1.82 = ( ^ [A7: set @ A,B7: set @ A] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 3.77/1.82 & ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_219_subset__psubset__trans,axiom,
% 3.77/1.82 ! [A: $tType,A5: set @ A,B6: set @ A,C4: set @ A] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 3.77/1.82 => ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
% 3.77/1.82 => ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_220_subset__iff__psubset__eq,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) )
% 3.77/1.82 = ( ^ [A7: set @ A,B7: set @ A] :
% 3.77/1.82 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 3.77/1.82 | ( A7 = B7 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_221_less__nres_Opelims_I1_J,axiom,
% 3.77/1.82 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
% 3.77/1.82 ( ( ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 3.77/1.82 = Y3 )
% 3.77/1.82 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
% 3.77/1.82 => ( ( ( X3
% 3.77/1.82 = ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ( ~ Y3
% 3.77/1.82 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Xa ) ) ) )
% 3.77/1.82 => ( ! [Uv: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ Uv ) )
% 3.77/1.82 => ( ( Xa
% 3.77/1.82 = ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ( Y3
% 3.77/1.82 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) ) ) ) ) )
% 3.77/1.82 => ~ ! [A2: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ A2 ) )
% 3.77/1.82 => ! [B2: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ( ( Y3
% 3.77/1.82 = ( ord_less @ ( set @ A ) @ A2 @ B2 ) )
% 3.77/1.82 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) ) ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_222_less__nres_Opelims_I2_J,axiom,
% 3.77/1.82 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 3.77/1.82 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
% 3.77/1.82 => ( ! [Uv: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ Uv ) )
% 3.77/1.82 => ( ( Xa
% 3.77/1.82 = ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) ) ) ) )
% 3.77/1.82 => ~ ! [A2: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ A2 ) )
% 3.77/1.82 => ! [B2: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
% 3.77/1.82 => ~ ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_223_less__nres_Opelims_I3_J,axiom,
% 3.77/1.82 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ~ ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 3.77/1.82 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
% 3.77/1.82 => ( ( ( X3
% 3.77/1.82 = ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Xa ) ) )
% 3.77/1.82 => ~ ! [A2: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ A2 ) )
% 3.77/1.82 => ! [B2: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
% 3.77/1.82 => ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_224_less__eq__nres_Opelims_I3_J,axiom,
% 3.77/1.82 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 3.77/1.82 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
% 3.77/1.82 => ( ! [A2: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ A2 ) )
% 3.77/1.82 => ! [B2: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
% 3.77/1.82 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) )
% 3.77/1.82 => ~ ( ( X3
% 3.77/1.82 = ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ! [Uv: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ Uv ) )
% 3.77/1.82 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_225_less__eq__nres_Opelims_I2_J,axiom,
% 3.77/1.82 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 3.77/1.82 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 3.77/1.82 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
% 3.77/1.82 => ( ( ( Xa
% 3.77/1.82 = ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ ( refine1767639642_FAILi @ A ) ) ) )
% 3.77/1.82 => ~ ! [A2: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ A2 ) )
% 3.77/1.82 => ! [B2: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
% 3.77/1.82 => ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_226_less__eq__nres_Opelims_I1_J,axiom,
% 3.77/1.82 ! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
% 3.77/1.82 ( ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
% 3.77/1.82 = Y3 )
% 3.77/1.82 => ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
% 3.77/1.82 => ( ( ( Xa
% 3.77/1.82 = ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ( Y3
% 3.77/1.82 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ ( refine1767639642_FAILi @ A ) ) ) ) )
% 3.77/1.82 => ( ! [A2: set @ A] :
% 3.77/1.82 ( ( X3
% 3.77/1.82 = ( refine605929679le_RES @ A @ A2 ) )
% 3.77/1.82 => ! [B2: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ B2 ) )
% 3.77/1.82 => ( ( Y3
% 3.77/1.82 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) )
% 3.77/1.82 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) ) ) ) )
% 3.77/1.82 => ~ ( ( X3
% 3.77/1.82 = ( refine1767639642_FAILi @ A ) )
% 3.77/1.82 => ! [Uv: set @ A] :
% 3.77/1.82 ( ( Xa
% 3.77/1.82 = ( refine605929679le_RES @ A @ Uv ) )
% 3.77/1.82 => ( ~ Y3
% 3.77/1.82 => ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_227_GreatestI2__order,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [P: A > $o,X3: A,Q: A > $o] :
% 3.77/1.82 ( ( P @ X3 )
% 3.77/1.82 => ( ! [Y5: A] :
% 3.77/1.82 ( ( P @ Y5 )
% 3.77/1.82 => ( ord_less_eq @ A @ Y5 @ X3 ) )
% 3.77/1.82 => ( ! [X5: A] :
% 3.77/1.82 ( ( P @ X5 )
% 3.77/1.82 => ( ! [Y7: A] :
% 3.77/1.82 ( ( P @ Y7 )
% 3.77/1.82 => ( ord_less_eq @ A @ Y7 @ X5 ) )
% 3.77/1.82 => ( Q @ X5 ) ) )
% 3.77/1.82 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_228_Greatest__equality,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( order @ A )
% 3.77/1.82 => ! [P: A > $o,X3: A] :
% 3.77/1.82 ( ( P @ X3 )
% 3.77/1.82 => ( ! [Y5: A] :
% 3.77/1.82 ( ( P @ Y5 )
% 3.77/1.82 => ( ord_less_eq @ A @ Y5 @ X3 ) )
% 3.77/1.82 => ( ( order_Greatest @ A @ P )
% 3.77/1.82 = X3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_229_subset__Collect__conv,axiom,
% 3.77/1.82 ! [A: $tType,S: set @ A,P: A > $o] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
% 3.77/1.82 = ( ! [X4: A] :
% 3.77/1.82 ( ( member @ A @ X4 @ S )
% 3.77/1.82 => ( P @ X4 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_230_relprop__triggers_I6_J,axiom,
% 3.77/1.82 ! [I2: $tType,R2: set @ I2,R3: set @ I2] :
% 3.77/1.82 ( ( ord_less_eq @ ( set @ I2 ) @ R2 @ R3 )
% 3.77/1.82 => ( ord_less_eq @ ( set @ I2 ) @ R2 @ R3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_231_ord__eq__le__eq__trans,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ord @ A )
% 3.77/1.82 => ! [A3: A,B3: A,C3: A,D4: A] :
% 3.77/1.82 ( ( A3 = B3 )
% 3.77/1.82 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 3.77/1.82 => ( ( C3 = D4 )
% 3.77/1.82 => ( ord_less_eq @ A @ A3 @ D4 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_232_bex2I,axiom,
% 3.77/1.82 ! [A: $tType,B: $tType,A3: A,B3: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
% 3.77/1.82 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ S )
% 3.77/1.82 => ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ S )
% 3.77/1.82 => ( P @ A3 @ B3 ) )
% 3.77/1.82 => ? [A2: A,B2: B] :
% 3.77/1.82 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S )
% 3.77/1.82 & ( P @ A2 @ B2 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_233_pairself_Ocases,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType,X3: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
% 3.77/1.82 ~ ! [F6: A > B,A2: A,B2: A] :
% 3.77/1.82 ( X3
% 3.77/1.82 != ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F6 @ ( product_Pair @ A @ A @ A2 @ B2 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_234_pairself_Oinduct,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType,P: ( A > B ) > ( product_prod @ A @ A ) > $o,A0: A > B,A1: product_prod @ A @ A] :
% 3.77/1.82 ( ! [F6: A > B,A2: A,B2: A] : ( P @ F6 @ ( product_Pair @ A @ A @ A2 @ B2 ) )
% 3.77/1.82 => ( P @ A0 @ A1 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_235_uncurry__apply,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B3: C] :
% 3.77/1.82 ( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 3.77/1.82 = ( F @ A3 @ B3 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_236_measure__induct,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType] :
% 3.77/1.82 ( ( wellorder @ B )
% 3.77/1.82 => ! [F: A > B,P: A > $o,A3: A] :
% 3.77/1.82 ( ! [X5: A] :
% 3.77/1.82 ( ! [Y7: A] :
% 3.77/1.82 ( ( ord_less @ B @ ( F @ Y7 ) @ ( F @ X5 ) )
% 3.77/1.82 => ( P @ Y7 ) )
% 3.77/1.82 => ( P @ X5 ) )
% 3.77/1.82 => ( P @ A3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_237_measure__induct__rule,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType] :
% 3.77/1.82 ( ( wellorder @ B )
% 3.77/1.82 => ! [F: A > B,P: A > $o,A3: A] :
% 3.77/1.82 ( ! [X5: A] :
% 3.77/1.82 ( ! [Y7: A] :
% 3.77/1.82 ( ( ord_less @ B @ ( F @ Y7 ) @ ( F @ X5 ) )
% 3.77/1.82 => ( P @ Y7 ) )
% 3.77/1.82 => ( P @ X5 ) )
% 3.77/1.82 => ( P @ A3 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_238_linorder__neqE__linordered__idom,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linordered_idom @ A )
% 3.77/1.82 => ! [X3: A,Y3: A] :
% 3.77/1.82 ( ( X3 != Y3 )
% 3.77/1.82 => ( ~ ( ord_less @ A @ X3 @ Y3 )
% 3.77/1.82 => ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_239_linordered__field__no__ub,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linordered_field @ A )
% 3.77/1.82 => ! [X7: A] :
% 3.77/1.82 ? [X_1: A] : ( ord_less @ A @ X7 @ X_1 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_240_linordered__field__no__lb,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linordered_field @ A )
% 3.77/1.82 => ! [X7: A] :
% 3.77/1.82 ? [Y5: A] : ( ord_less @ A @ Y5 @ X7 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_241_all__nat__split__at,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( linorder @ A )
% 3.77/1.82 => ! [K: A,P: A > $o] :
% 3.77/1.82 ( ! [I3: A] :
% 3.77/1.82 ( ( ord_less @ A @ I3 @ K )
% 3.77/1.82 => ( P @ I3 ) )
% 3.77/1.82 => ( ( P @ K )
% 3.77/1.82 => ( ! [I3: A] :
% 3.77/1.82 ( ( ord_less @ A @ K @ I3 )
% 3.77/1.82 => ( P @ I3 ) )
% 3.77/1.82 => ! [X_12: A] : ( P @ X_12 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_242_dependent__wellorder__choice,axiom,
% 3.77/1.82 ! [B: $tType,A: $tType] :
% 3.77/1.82 ( ( wellorder @ A )
% 3.77/1.82 => ! [P: ( A > B ) > A > B > $o] :
% 3.77/1.82 ( ! [R4: B,F6: A > B,G4: A > B,X5: A] :
% 3.77/1.82 ( ! [Y7: A] :
% 3.77/1.82 ( ( ord_less @ A @ Y7 @ X5 )
% 3.77/1.82 => ( ( F6 @ Y7 )
% 3.77/1.82 = ( G4 @ Y7 ) ) )
% 3.77/1.82 => ( ( P @ F6 @ X5 @ R4 )
% 3.77/1.82 = ( P @ G4 @ X5 @ R4 ) ) )
% 3.77/1.82 => ( ! [X5: A,F6: A > B] :
% 3.77/1.82 ( ! [Y7: A] :
% 3.77/1.82 ( ( ord_less @ A @ Y7 @ X5 )
% 3.77/1.82 => ( P @ F6 @ Y7 @ ( F6 @ Y7 ) ) )
% 3.77/1.82 => ? [X_12: B] : ( P @ F6 @ X5 @ X_12 ) )
% 3.77/1.82 => ? [F6: A > B] :
% 3.77/1.82 ! [X7: A] : ( P @ F6 @ X7 @ ( F6 @ X7 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_243_Refine__Misc_Oif__mono,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( ord @ A )
% 3.77/1.82 => ! [B3: $o,M1: A,M12: A,M22: A,M23: A] :
% 3.77/1.82 ( ( B3
% 3.77/1.82 => ( ord_less_eq @ A @ M1 @ M12 ) )
% 3.77/1.82 => ( ( ~ B3
% 3.77/1.82 => ( ord_less_eq @ A @ M22 @ M23 ) )
% 3.77/1.82 => ( ord_less_eq @ A @ ( if @ A @ B3 @ M1 @ M22 ) @ ( if @ A @ B3 @ M12 @ M23 ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(fact_244_chain__subset__def,axiom,
% 3.77/1.82 ! [A: $tType] :
% 3.77/1.82 ( ( chain_subset @ A )
% 3.77/1.82 = ( ^ [C5: set @ ( set @ A )] :
% 3.77/1.82 ! [X4: set @ A] :
% 3.77/1.82 ( ( member @ ( set @ A ) @ X4 @ C5 )
% 3.77/1.82 => ! [Y6: set @ A] :
% 3.77/1.82 ( ( member @ ( set @ A ) @ Y6 @ C5 )
% 3.77/1.82 => ( ( ord_less_eq @ ( set @ A ) @ X4 @ Y6 )
% 3.77/1.82 | ( ord_less_eq @ ( set @ A ) @ Y6 @ X4 ) ) ) ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.77/1.82 ! [A8: $tType,A9: $tType] :
% 3.77/1.82 ( ( preorder @ A9 )
% 3.77/1.82 => ( preorder @ ( A8 > A9 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.77/1.82 ! [A8: $tType,A9: $tType] :
% 3.77/1.82 ( ( order @ A9 )
% 3.77/1.82 => ( order @ ( A8 > A9 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(tcon_fun___Orderings_Oord,axiom,
% 3.77/1.82 ! [A8: $tType,A9: $tType] :
% 3.77/1.82 ( ( ord @ A9 )
% 3.77/1.82 => ( ord @ ( A8 > A9 ) ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
% 3.77/1.82 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 3.77/1.82 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
% 3.77/1.82 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
% 3.77/1.82 preorder @ $o ).
% 3.77/1.82
% 3.77/1.82 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 3.77/1.82 linorder @ $o ).
% 3.77/1.82
% 3.77/1.82 thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
% 3.77/1.82 order @ $o ).
% 3.77/1.82
% 3.77/1.82 thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
% 3.77/1.82 ord @ $o ).
% 3.77/1.82
% 3.77/1.82 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_7,axiom,
% 3.77/1.82 ! [A8: $tType] : ( preorder @ ( refine1665802226e_nres @ A8 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_8,axiom,
% 3.77/1.82 ! [A8: $tType] : ( order @ ( refine1665802226e_nres @ A8 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_9,axiom,
% 3.77/1.82 ! [A8: $tType] : ( ord @ ( refine1665802226e_nres @ A8 ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(help_If_3_1_T,axiom,
% 3.77/1.82 ! [P: $o] :
% 3.77/1.82 ( ( P = $true )
% 3.77/1.82 | ( P = $false ) ) ).
% 3.77/1.82
% 3.77/1.82 thf(help_If_2_1_T,axiom,
% 3.77/1.82 ! [A: $tType,X3: A,Y3: A] :
% 3.77/1.82 ( ( if @ A @ $false @ X3 @ Y3 )
% 3.77/1.82 = Y3 ) ).
% 3.77/1.82
% 3.77/1.82 thf(help_If_1_1_T,axiom,
% 3.77/1.82 ! [A: $tType,X3: A,Y3: A] :
% 3.77/1.82 ( ( if @ A @ $true @ X3 @ Y3 )
% 3.77/1.82 = X3 ) ).
% 3.77/1.82
% 3.77/1.82 thf(conj_0,conjecture,
% 3.77/1.82 ( ( refine406925620_inres @ a @ ( refine605929679le_RES @ a @ ( collect @ a @ phi ) ) @ x )
% 3.77/1.82 = ( phi @ x ) ) ).
% 3.77/1.82 ------------------------
% 3.77/1.86 % (14552)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_14407 for (2999ds/275Mi)
% 3.77/1.86 % (14549)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_14407 for (2999ds/27Mi)
% 3.77/1.86 % (14553)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_14407 for (2999ds/18Mi)
% 3.77/1.86 % (14551)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_14407 for (2999ds/2Mi)
% 3.77/1.86 % (14548)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_14407 for (2999ds/4Mi)
% 3.77/1.86 % (14547)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_14407 for (2999ds/183Mi)
% 3.77/1.86 % (14550)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_14407 for (2999ds/2Mi)
% 3.77/1.86 % (14550)Instruction limit reached!
% 3.77/1.86 % (14550)------------------------------
% 3.77/1.86 % (14550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.77/1.86 % (14550)Termination reason: Unknown
% 3.77/1.86 % (14551)Instruction limit reached!
% 3.77/1.86 % (14551)------------------------------
% 3.77/1.86 % (14551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.77/1.86 % (14550)Termination phase: shuffling
% 3.77/1.86
% 3.77/1.86 % (14550)Memory used [KB]: 1407
% 3.77/1.86 % (14550)Time elapsed: 0.003 s
% 3.77/1.86 % (14550)Instructions burned: 2 (million)
% 3.77/1.86 % (14550)------------------------------
% 3.77/1.86 % (14550)------------------------------
% 3.77/1.86 % (14551)Termination reason: Unknown
% 3.77/1.86 % (14551)Termination phase: shuffling
% 3.77/1.86
% 3.77/1.86 % (14551)Memory used [KB]: 1407
% 3.77/1.86 % (14551)Time elapsed: 0.003 s
% 3.77/1.86 % (14551)Instructions burned: 2 (million)
% 3.77/1.86 % (14551)------------------------------
% 3.77/1.86 % (14551)------------------------------
% 3.77/1.86 % (14548)Instruction limit reached!
% 3.77/1.86 % (14548)------------------------------
% 3.77/1.86 % (14548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.77/1.86 % (14548)Termination reason: Unknown
% 3.77/1.86 % (14548)Termination phase: shuffling
% 3.77/1.86
% 3.77/1.86 % (14548)Memory used [KB]: 1407
% 3.77/1.86 % (14548)Time elapsed: 0.004 s
% 3.77/1.86 % (14548)Instructions burned: 4 (million)
% 3.77/1.86 % (14548)------------------------------
% 3.77/1.86 % (14548)------------------------------
% 3.77/1.87 % (14553)Instruction limit reached!
% 3.77/1.87 % (14553)------------------------------
% 3.77/1.87 % (14553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.77/1.87 % (14553)Termination reason: Unknown
% 3.77/1.87 % (14553)Termination phase: shuffling
% 3.77/1.87
% 3.77/1.87 % (14553)Memory used [KB]: 1791
% 3.77/1.87 % (14553)Time elapsed: 0.012 s
% 3.77/1.87 % (14553)Instructions burned: 19 (million)
% 3.77/1.87 % (14553)------------------------------
% 3.77/1.87 % (14553)------------------------------
% 3.77/1.88 % (14549)Instruction limit reached!
% 3.77/1.88 % (14549)------------------------------
% 3.77/1.88 % (14549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.77/1.88 % (14549)Termination reason: Unknown
% 3.77/1.88 % (14549)Termination phase: Property scanning
% 3.77/1.88
% 3.77/1.88 % (14549)Memory used [KB]: 1918
% 3.77/1.88 % (14549)Time elapsed: 0.017 s
% 3.77/1.88 % (14549)Instructions burned: 28 (million)
% 3.77/1.88 % (14549)------------------------------
% 3.77/1.88 % (14549)------------------------------
% 3.77/1.88 % (14555)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_14407 for (2999ds/37Mi)
% 3.77/1.88 % (14556)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_14407 for (2999ds/15Mi)
% 3.77/1.88 % (14554)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_14407 for (2999ds/3Mi)
% 3.77/1.88 % (14554)Instruction limit reached!
% 3.77/1.88 % (14554)------------------------------
% 3.77/1.88 % (14554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.77/1.88 % (14554)Termination reason: Unknown
% 3.77/1.88 % (14554)Termination phase: shuffling
% 3.77/1.88
% 3.77/1.88 % (14554)Memory used [KB]: 1407
% 3.77/1.88 % (14554)Time elapsed: 0.003 s
% 3.77/1.88 % (14554)Instructions burned: 3 (million)
% 3.77/1.88 % (14554)------------------------------
% 3.77/1.88 % (14554)------------------------------
% 3.77/1.89 % (14556)Instruction limit reached!
% 3.77/1.89 % (14556)------------------------------
% 3.77/1.89 % (14556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.77/1.89 % (14556)Termination reason: Unknown
% 3.77/1.89 % (14556)Termination phase: shuffling
% 3.77/1.89
% 3.77/1.89 % (14556)Memory used [KB]: 1663
% 3.77/1.89 % (14556)Time elapsed: 0.010 s
% 3.77/1.89 % (14556)Instructions burned: 15 (million)
% 3.77/1.89 % (14556)------------------------------
% 3.77/1.89 % (14556)------------------------------
% 3.77/1.89 % (14557)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_14407 for (2999ds/3Mi)
% 3.77/1.89 % (14557)Instruction limit reached!
% 3.77/1.89 % (14557)------------------------------
% 3.77/1.89 % (14557)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.77/1.89 % (14557)Termination reason: Unknown
% 3.77/1.89 % (14557)Termination phase: shuffling
% 3.77/1.89
% 3.77/1.89 % (14557)Memory used [KB]: 1407
% 3.77/1.89 % (14557)Time elapsed: 0.004 s
% 3.77/1.89 % (14557)Instructions burned: 4 (million)
% 3.77/1.89 % (14557)------------------------------
% 3.77/1.89 % (14557)------------------------------
% 3.82/1.90 % (14558)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_14407 for (2999ds/1041Mi)
% 3.82/1.90 % (14552)First to succeed.
% 3.82/1.90 % (14555)Instruction limit reached!
% 3.82/1.90 % (14555)------------------------------
% 3.82/1.90 % (14555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.82/1.90 % (14555)Termination reason: Unknown
% 3.82/1.90 % (14555)Termination phase: shuffling
% 3.82/1.90
% 3.82/1.90 % (14555)Memory used [KB]: 2174
% 3.82/1.90 % (14555)Time elapsed: 0.022 s
% 3.82/1.90 % (14555)Instructions burned: 38 (million)
% 3.82/1.90 % (14555)------------------------------
% 3.82/1.90 % (14555)------------------------------
% 3.82/1.90 % (14559)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_14407 for (2999ds/7Mi)
% 3.82/1.90 % (14552)Refutation found. Thanks to Tanya!
% 3.82/1.90 % SZS status Theorem for DTF2THF_14407
% 3.82/1.90 % SZS output start Proof for DTF2THF_14407
% 3.82/1.90 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 3.82/1.90 thf(type_def_6, type, set: ($tType) > $tType).
% 3.82/1.90 thf(type_def_8, type, refine1665802226e_nres: ($tType) > $tType).
% 3.82/1.90 thf(type_def_9, type, a: $tType).
% 3.82/1.90 thf(func_def_0, type, refine1665802226e_nres: ($tType) > $tType).
% 3.82/1.90 thf(func_def_1, type, product_prod: ($tType * $tType) > $tType).
% 3.82/1.90 thf(func_def_2, type, set: ($tType) > $tType).
% 3.82/1.90 thf(func_def_3, type, a: $tType).
% 3.82/1.90 thf(func_def_4, type, type: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_5, type, ord: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_6, type, order: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_7, type, no_bot: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_8, type, no_top: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_9, type, linorder: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_10, type, preorder: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_11, type, wellorder: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_12, type, dense_order: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_13, type, linordered_idom: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_14, type, linordered_field: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_15, type, dense_linorder: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_16, type, condit1656338222tinuum: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_17, type, condit1037483654norder: !>[X0: $tType]:($o)).
% 3.82/1.90 thf(func_def_18, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 3.82/1.90 thf(func_def_19, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.82/1.90 thf(func_def_20, type, uncurry: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.82/1.90 thf(func_def_21, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.82/1.90 thf(func_def_22, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.82/1.90 thf(func_def_23, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 3.82/1.90 thf(func_def_24, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.82/1.90 thf(func_def_25, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.82/1.90 thf(func_def_26, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.82/1.90 thf(func_def_27, type, refine1687780735RETURN: !>[X0: $tType]:(X0 > refine1665802226e_nres @ X0)).
% 3.82/1.90 thf(func_def_28, type, refine1315500908_inres: !>[X0: $tType]:(refine1665802226e_nres @ X0 > X0 > $o)).
% 3.82/1.90 thf(func_def_29, type, refine1554218259es_rel: !>[X0: $tType]:(product_prod @ refine1665802226e_nres @ X0 @ refine1665802226e_nres @ X0 > product_prod @ refine1665802226e_nres @ X0 @ refine1665802226e_nres @ X0 > $o)).
% 3.82/1.90 thf(func_def_30, type, refine1378444575es_rel: !>[X0: $tType]:(product_prod @ refine1665802226e_nres @ X0 @ refine1665802226e_nres @ X0 > product_prod @ refine1665802226e_nres @ X0 @ refine1665802226e_nres @ X0 > $o)).
% 3.82/1.90 thf(func_def_31, type, refine406925620_inres: !>[X0: $tType]:(refine1665802226e_nres @ X0 > X0 > $o)).
% 3.82/1.90 thf(func_def_32, type, refine1102455758nofail: !>[X0: $tType]:(refine1665802226e_nres @ X0 > $o)).
% 3.82/1.90 thf(func_def_33, type, refine1767639642_FAILi: !>[X0: $tType]:(refine1665802226e_nres @ X0)).
% 3.82/1.90 thf(func_def_34, type, refine605929679le_RES: !>[X0: $tType]:(set @ X0 > refine1665802226e_nres @ X0)).
% 3.82/1.90 thf(func_def_35, type, refine60818195e_nres: !>[X0: $tType, X1: $tType]:(X0 > (set @ X1 > X0) > refine1665802226e_nres @ X1 > X0)).
% 3.82/1.90 thf(func_def_36, type, refine1442219249c_nres: !>[X0: $tType, X1: $tType]:(X0 > (set @ X1 > X0) > refine1665802226e_nres @ X1 > X0)).
% 3.82/1.90 thf(func_def_37, type, refine1672542526he_RES: !>[X0: $tType]:(refine1665802226e_nres @ X0 > set @ X0)).
% 3.82/1.90 thf(func_def_38, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.82/1.90 thf(func_def_39, type, accp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > $o)).
% 3.82/1.90 thf(func_def_40, type, chain_subset: !>[X0: $tType]:(set @ set @ X0 > $o)).
% 3.82/1.90 thf(func_def_41, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.82/1.90 thf(func_def_42, type, phi: a > $o).
% 3.82/1.90 thf(func_def_43, type, x: a).
% 3.82/1.90 thf(func_def_59, type, sK0: !>[X0: $tType]:(refine1665802226e_nres @ X0 > refine1665802226e_nres @ X0 > X0)).
% 3.82/1.90 thf(func_def_60, type, sK1: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.82/1.90 thf(func_def_61, type, sK2: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.82/1.90 thf(func_def_62, type, sK3: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.82/1.90 thf(func_def_63, type, sK4: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 3.82/1.90 thf(func_def_64, type, sK5: !>[X0: $tType]:(refine1665802226e_nres @ X0 > refine1665802226e_nres @ X0 > X0)).
% 3.82/1.90 thf(func_def_65, type, sK6: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > set @ product_prod @ X0 @ X1 > X0)).
% 3.82/1.90 thf(func_def_66, type, sK7: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > set @ product_prod @ X0 @ X1 > X1)).
% 3.82/1.90 thf(func_def_67, type, sK8: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.82/1.90 thf(func_def_68, type, sK9: !>[X0: $tType]:(refine1665802226e_nres @ X0 > refine1665802226e_nres @ X0 > X0)).
% 3.82/1.90 thf(func_def_69, type, sK10: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 3.82/1.90 thf(func_def_71, type, ph12: !>[X0: $tType]:(X0)).
% 3.82/1.90 thf(f999,plain,(
% 3.82/1.90 $false),
% 3.82/1.90 inference(avatar_sat_refutation,[status(thm)],[f907,f918,f940,f946,f995])).
% 3.82/1.90 thf(f995,plain,(
% 3.82/1.90 ~spl11_1 | ~spl11_4),
% 3.82/1.90 inference(avatar_contradiction_clause,[status(thm)],[f994])).
% 3.82/1.90 thf(f994,plain,(
% 3.82/1.90 $false | (~spl11_1 | ~spl11_4)),
% 3.82/1.90 inference(trivial_inequality_removal,[status(thm)],[f989])).
% 3.82/1.90 thf(f989,plain,(
% 3.82/1.90 ($false = $true) | (~spl11_1 | ~spl11_4)),
% 3.82/1.90 inference(superposition,[status(thm)],[f863,f971])).
% 3.82/1.90 thf(f971,plain,(
% 3.82/1.90 ($false = (refine1102455758nofail @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)))) | (~spl11_1 | ~spl11_4)),
% 3.82/1.90 inference(trivial_inequality_removal,[status(thm)],[f970])).
% 3.82/1.90 thf(f970,plain,(
% 3.82/1.90 ($false = (refine1102455758nofail @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)))) | ($false = $true) | (~spl11_1 | ~spl11_4)),
% 3.82/1.90 inference(forward_demodulation,[status(thm)],[f969,f917])).
% 3.82/1.90 thf(f917,plain,(
% 3.82/1.90 ((phi @ x) = $false) | ~spl11_4),
% 3.82/1.90 inference(avatar_component_clause,[status(thm)],[f915])).
% 3.82/1.90 thf(f915,definition,(
% 3.82/1.90 spl11_4 <=> ((phi @ x) = $false)),
% 3.82/1.90 introduced(definition,[new_symbols(naming,[spl11_4])],[avatar_definition])).
% 3.82/1.90 thf(f969,plain,(
% 3.82/1.90 ((phi @ x) = $true) | ($false = (refine1102455758nofail @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)))) | ~spl11_1),
% 3.82/1.90 inference(forward_demodulation,[status(thm)],[f963,f833])).
% 3.82/1.90 thf(f833,plain,(
% 3.82/1.90 ( ! [X0 : $tType,X2 : X0 > $o,X1 : X0] : (((X2 @ X1) = (member @ X0 @ X1 @ (collect @ X0 @ X2)))) )),
% 3.82/1.90 inference(cnf_transformation,[status(thm)],[f731])).
% 3.82/1.90 thf(f731,plain,(
% 3.82/1.90 ! [X0 : $tType,X1 : X0,X2 : X0 > $o] : ((X2 @ X1) = (member @ X0 @ X1 @ (collect @ X0 @ X2)))),
% 3.82/1.90 inference(rectify,[status(thm)],[f46])).
% 3.82/1.90 thf(f46,axiom,(
% 3.82/1.90 ! [X0 : $tType,X40 : X0,X14 : X0 > $o] : ((X14 @ X40) = (member @ X0 @ X40 @ (collect @ X0 @ X14)))),
% 3.82/1.90 file('/export/starexec/sandbox/tmp/tmp.jdFRlkfssT/DTF2THF_14407.p',fact_45_mem__Collect__eq)).
% 3.82/1.90 thf(f963,plain,(
% 3.82/1.90 ($false = (refine1102455758nofail @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)))) | ((member @ a @ x @ (collect @ a @ phi)) = $true) | ~spl11_1),
% 3.82/1.90 inference(binary_proxy_clausification,[status(thm)],[f962])).
% 3.82/1.90 thf(f962,plain,(
% 3.82/1.90 ((member @ a @ x @ (collect @ a @ phi)) = (refine1102455758nofail @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)))) | ~spl11_1),
% 3.82/1.90 inference(boolean_simplification,[status(thm)],[f961])).
% 3.82/1.90 thf(f961,plain,(
% 3.82/1.90 (($true & (refine1102455758nofail @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)))) = (member @ a @ x @ (collect @ a @ phi))) | ~spl11_1),
% 3.82/1.90 inference(superposition,[status(thm)],[f891,f902])).
% 3.82/1.90 thf(f902,plain,(
% 3.82/1.90 ((refine1315500908_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x) = $true) | ~spl11_1),
% 3.82/1.90 inference(avatar_component_clause,[status(thm)],[f900])).
% 3.82/1.90 thf(f900,definition,(
% 3.82/1.90 spl11_1 <=> ((refine1315500908_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x) = $true)),
% 3.82/1.90 introduced(definition,[new_symbols(naming,[spl11_1])],[avatar_definition])).
% 3.82/1.90 thf(f891,plain,(
% 3.82/1.90 ( ! [X0 : $tType,X2 : X0,X1 : set @ X0] : (((member @ X0 @ X2 @ X1) = ((refine1315500908_inres @ X0 @ (refine605929679le_RES @ X0 @ X1) @ X2) & (refine1102455758nofail @ X0 @ (refine605929679le_RES @ X0 @ X1))))) )),
% 3.82/1.90 inference(beta_eta_normalization,[status(thm)],[f878])).
% 3.82/1.90 thf(f878,plain,(
% 3.82/1.90 ( ! [X0 : $tType,X2 : X0,X1 : set @ X0] : ((((^[Y0 : refine1665802226e_nres @ X0]: ((^[Y1 : X0]: ((refine1315500908_inres @ X0 @ Y0 @ Y1) & (refine1102455758nofail @ X0 @ Y0))))) @ (refine605929679le_RES @ X0 @ X1) @ X2) = (member @ X0 @ X2 @ X1))) )),
% 3.82/1.90 inference(definition_unfolding,[status(thm)],[f844,f874])).
% 3.82/1.90 thf(f874,plain,(
% 3.82/1.90 ( ! [X0 : $tType] : ((refine406925620_inres @ X0 = (^[Y0 : refine1665802226e_nres @ X0]: ((^[Y1 : X0]: ((refine1315500908_inres @ X0 @ Y0 @ Y1) & (refine1102455758nofail @ X0 @ Y0))))))) )),
% 3.82/1.90 inference(cnf_transformation,[status(thm)],[f354])).
% 3.82/1.90 thf(f354,plain,(
% 3.82/1.90 ! [X0 : $tType] : (refine406925620_inres @ X0 = (^[Y0 : refine1665802226e_nres @ X0]: ((^[Y1 : X0]: ((refine1315500908_inres @ X0 @ Y0 @ Y1) & (refine1102455758nofail @ X0 @ Y0))))))),
% 3.82/1.90 inference(fool_elimination,[status(thm)],[f353])).
% 3.82/1.90 thf(f353,plain,(
% 3.82/1.90 ! [X0 : $tType] : ((^[X1 : refine1665802226e_nres @ X0, X2 : X0] : ((refine1102455758nofail @ X0 @ X1) & (refine1315500908_inres @ X0 @ X1 @ X2))) = refine406925620_inres @ X0)),
% 3.82/1.91 inference(rectify,[status(thm)],[f22])).
% 3.82/1.91 thf(f22,axiom,(
% 3.82/1.91 ! [X0 : $tType] : ((^[X12 : refine1665802226e_nres @ X0, X11 : X0] : ((refine1102455758nofail @ X0 @ X12) & (refine1315500908_inres @ X0 @ X12 @ X11))) = refine406925620_inres @ X0)),
% 3.82/1.91 file('/export/starexec/sandbox/tmp/tmp.jdFRlkfssT/DTF2THF_14407.p',fact_21_nf__inres__def)).
% 3.82/1.91 thf(f844,plain,(
% 3.82/1.91 ( ! [X0 : $tType,X2 : X0,X1 : set @ X0] : (((refine406925620_inres @ X0 @ (refine605929679le_RES @ X0 @ X1) @ X2) = (member @ X0 @ X2 @ X1))) )),
% 3.82/1.91 inference(cnf_transformation,[status(thm)],[f727])).
% 3.82/1.91 thf(f727,plain,(
% 3.82/1.91 ! [X0 : $tType,X1 : set @ X0,X2 : X0] : ((refine406925620_inres @ X0 @ (refine605929679le_RES @ X0 @ X1) @ X2) = (member @ X0 @ X2 @ X1))),
% 3.82/1.91 inference(rectify,[status(thm)],[f3])).
% 3.82/1.91 thf(f3,axiom,(
% 3.82/1.91 ! [X0 : $tType,X4 : set @ X0,X8 : X0] : ((refine406925620_inres @ X0 @ (refine605929679le_RES @ X0 @ X4) @ X8) = (member @ X0 @ X8 @ X4))),
% 3.82/1.91 file('/export/starexec/sandbox/tmp/tmp.jdFRlkfssT/DTF2THF_14407.p',fact_2_nf__inres__RES)).
% 3.82/1.91 thf(f863,plain,(
% 3.82/1.91 ( ! [X0 : $tType,X1 : set @ X0] : (((refine1102455758nofail @ X0 @ (refine605929679le_RES @ X0 @ X1)) = $true)) )),
% 3.82/1.91 inference(cnf_transformation,[status(thm)],[f502])).
% 3.82/1.91 thf(f502,plain,(
% 3.82/1.91 ! [X0 : $tType,X1 : set @ X0] : ((refine1102455758nofail @ X0 @ (refine605929679le_RES @ X0 @ X1)) = $true)),
% 3.82/1.91 inference(fool_elimination,[status(thm)],[f501])).
% 3.82/1.91 thf(f501,plain,(
% 3.82/1.91 ! [X0 : $tType,X1 : set @ X0] : (refine1102455758nofail @ X0 @ (refine605929679le_RES @ X0 @ X1))),
% 3.82/1.91 inference(rectify,[status(thm)],[f8])).
% 3.82/1.91 thf(f8,axiom,(
% 3.82/1.91 ! [X1 : $tType,X4 : set @ X1] : (refine1102455758nofail @ X1 @ (refine605929679le_RES @ X1 @ X4))),
% 3.82/1.91 file('/export/starexec/sandbox/tmp/tmp.jdFRlkfssT/DTF2THF_14407.p',fact_7_nofail__simps_I2_J)).
% 3.82/1.91 thf(f946,plain,(
% 3.82/1.91 ~spl11_2 | ~spl11_4),
% 3.82/1.91 inference(avatar_contradiction_clause,[status(thm)],[f945])).
% 3.82/1.91 thf(f945,plain,(
% 3.82/1.91 $false | (~spl11_2 | ~spl11_4)),
% 3.82/1.91 inference(trivial_inequality_removal,[status(thm)],[f941])).
% 3.82/1.91 thf(f941,plain,(
% 3.82/1.91 ($false = $true) | (~spl11_2 | ~spl11_4)),
% 3.82/1.91 inference(superposition,[status(thm)],[f917,f906])).
% 3.82/1.91 thf(f906,plain,(
% 3.82/1.91 ((phi @ x) = $true) | ~spl11_2),
% 3.82/1.91 inference(avatar_component_clause,[status(thm)],[f904])).
% 3.82/1.91 thf(f904,definition,(
% 3.82/1.91 spl11_2 <=> ((phi @ x) = $true)),
% 3.82/1.91 introduced(definition,[new_symbols(naming,[spl11_2])],[avatar_definition])).
% 3.82/1.91 thf(f940,plain,(
% 3.82/1.91 spl11_4 | ~spl11_3),
% 3.82/1.91 inference(avatar_split_clause,[status(thm)],[f934,f911,f915])).
% 3.82/1.91 thf(f911,definition,(
% 3.82/1.91 spl11_3 <=> ($false = (refine1315500908_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x))),
% 3.82/1.91 introduced(definition,[new_symbols(naming,[spl11_3])],[avatar_definition])).
% 3.82/1.91 thf(f934,plain,(
% 3.82/1.91 ((phi @ x) = $false) | ~spl11_3),
% 3.82/1.91 inference(superposition,[status(thm)],[f833,f926])).
% 3.82/1.91 thf(f926,plain,(
% 3.82/1.91 ($false = (member @ a @ x @ (collect @ a @ phi))) | ~spl11_3),
% 3.82/1.91 inference(boolean_simplification,[status(thm)],[f922])).
% 3.82/1.91 thf(f922,plain,(
% 3.82/1.91 ((member @ a @ x @ (collect @ a @ phi)) = ($false & (refine1102455758nofail @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi))))) | ~spl11_3),
% 3.82/1.91 inference(superposition,[status(thm)],[f891,f913])).
% 3.82/1.91 thf(f913,plain,(
% 3.82/1.91 ($false = (refine1315500908_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x)) | ~spl11_3),
% 3.82/1.91 inference(avatar_component_clause,[status(thm)],[f911])).
% 3.82/1.91 thf(f918,plain,(
% 3.82/1.91 spl11_3 | spl11_4),
% 3.82/1.91 inference(avatar_split_clause,[status(thm)],[f909,f915,f911])).
% 3.82/1.91 thf(f909,plain,(
% 3.82/1.91 ($false = (refine1315500908_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x)) | ((phi @ x) = $false)),
% 3.82/1.91 inference(trivial_inequality_removal,[status(thm)],[f908])).
% 3.82/1.91 thf(f908,plain,(
% 3.82/1.91 ((phi @ x) = $false) | ($false = (refine1315500908_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x)) | ($false = $true)),
% 3.82/1.91 inference(forward_demodulation,[status(thm)],[f896,f863])).
% 3.82/1.91 thf(f896,plain,(
% 3.82/1.91 ((phi @ x) = $false) | ($false = (refine1315500908_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x)) | ($false = (refine1102455758nofail @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi))))),
% 3.82/1.91 inference(binary_proxy_clausification,[status(thm)],[f895])).
% 3.82/1.91 thf(f895,plain,(
% 3.82/1.91 ($false = ((refine1315500908_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x) & (refine1102455758nofail @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi))))) | ((phi @ x) = $false)),
% 3.82/1.91 inference(binary_proxy_clausification,[status(thm)],[f893])).
% 3.82/1.91 thf(f893,plain,(
% 3.82/1.91 ((phi @ x) != ((refine1315500908_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x) & (refine1102455758nofail @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)))))),
% 3.82/1.91 inference(beta_eta_normalization,[status(thm)],[f879])).
% 3.82/1.91 thf(f879,plain,(
% 3.82/1.91 ((phi @ x) != ((^[Y0 : refine1665802226e_nres @ a]: ((^[Y1 : a]: ((refine1315500908_inres @ a @ Y0 @ Y1) & (refine1102455758nofail @ a @ Y0))))) @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x))),
% 3.82/1.91 inference(definition_unfolding,[status(thm)],[f862,f874])).
% 3.82/1.91 thf(f862,plain,(
% 3.82/1.91 ((refine406925620_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x) != (phi @ x))),
% 3.82/1.91 inference(cnf_transformation,[status(thm)],[f725])).
% 3.82/1.91 thf(f725,plain,(
% 3.82/1.91 ((refine406925620_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x) != (phi @ x))),
% 3.82/1.91 inference(flattening,[status(thm)],[f263])).
% 3.82/1.91 thf(f263,negated_conjecture,(
% 3.82/1.91 ~((refine406925620_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x) = (phi @ x))),
% 3.82/1.91 inference(negated_conjecture,[status(cth)],[f262])).
% 3.82/1.91 thf(f262,conjecture,(
% 3.82/1.91 ((refine406925620_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x) = (phi @ x))),
% 3.82/1.91 file('/export/starexec/sandbox/tmp/tmp.jdFRlkfssT/DTF2THF_14407.p',conj_0)).
% 3.82/1.91 thf(f907,plain,(
% 3.82/1.91 spl11_1 | spl11_2),
% 3.82/1.91 inference(avatar_split_clause,[status(thm)],[f898,f904,f900])).
% 3.82/1.91 thf(f898,plain,(
% 3.82/1.91 ((phi @ x) = $true) | ((refine1315500908_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x) = $true)),
% 3.82/1.91 inference(binary_proxy_clausification,[status(thm)],[f894])).
% 3.82/1.91 thf(f894,plain,(
% 3.82/1.91 (((refine1315500908_inres @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ x) & (refine1102455758nofail @ a @ (refine605929679le_RES @ a @ (collect @ a @ phi)))) = $true) | ((phi @ x) = $true)),
% 3.82/1.91 inference(binary_proxy_clausification,[status(thm)],[f893])).
% 3.82/1.91 % SZS output end Proof for DTF2THF_14407
% 3.82/1.91 % (14552)------------------------------
% 3.82/1.91 % (14552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.82/1.91 % (14552)Termination reason: Refutation
% 3.82/1.91
% 3.82/1.91 % (14552)Memory used [KB]: 6652
% 3.82/1.91 % (14552)Time elapsed: 0.042 s
% 3.82/1.91 % (14552)Instructions burned: 69 (million)
% 3.82/1.91 % (14552)------------------------------
% 3.82/1.91 % (14552)------------------------------
% 3.82/1.91 % (14546)Success in time 0.075 s
%------------------------------------------------------------------------------