%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP161^2 : 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 : n018.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 08:56:03 AM UTC 2026
% Result : Theorem 3.10s 1.72s
% Output : Refutation 3.10s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP161^2 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n018.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 08:28:16 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.45 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : ITP161^2 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.46 % Domain : Interactive Theorem Proving
% 0.30/0.46 % Problem : Sledgehammer Refine_Basic problem prob_1708__3602512_1
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Des21]
% 0.30/0.46 % Names : Refine_Basic/prob_1708__3602512_1 [Des21]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.33 v8.1.0, 0.25 v7.5.0
% 0.30/0.46 % Syntax : Number of formulae : 342 ( 129 unt; 51 typ; 0 def)
% 0.30/0.46 % Number of atoms : 783 ( 206 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.30/0.46 % Number of connectives : 4514 ( 83 ~; 4 |; 34 &;4008 @)
% 0.30/0.46 % ( 0 <=>; 385 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 28 ( 9 avg)
% 0.30/0.46 % Number of types : 4 ( 3 usr)
% 0.30/0.46 % Number of type conns : 225 ( 225 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 51 ( 48 usr; 4 con; 0-6 aty)
% 0.30/0.46 % Number of variables : 1297 ( 58 ^;1159 !; 20 ?;1297 :)
% 0.30/0.46 % ( 60 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.46 % SPC : TH1_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:21:09.061
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % Could-be-implicit typings (6)
% 0.30/0.46 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
% 0.30/0.46 refine1665802226e_nres: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Product__Type_Ounit,type,
% 0.30/0.46 product_unit: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.30/0.46 product_prod: $tType > $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Set_Oset,type,
% 0.30/0.46 set: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_tf_b,type,
% 0.30/0.46 b: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_tf_a,type,
% 0.30/0.46 a: $tType ).
% 0.30/0.46
% 0.30/0.46 % Explicit typings (45)
% 0.30/0.46 thf(sy_cl_HOL_Otype,type,
% 0.30/0.46 type:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Obot,type,
% 0.30/0.46 bot:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.30/0.46 ord:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Otop,type,
% 0.30/0.46 top:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.30/0.46 order:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/0.46 linorder:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/0.46 preorder:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oorder__top,type,
% 0.30/0.46 order_top:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.30/0.46 comple187826305attice:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.30/0.46 bNF_Ca1785829860lChain:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_If,type,
% 0.30/0.46 if:
% 0.30/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Misc_Ofun__of__rel,type,
% 0.30/0.46 fun_of_rel:
% 0.30/0.46 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Misc_Ouncurry,type,
% 0.30/0.46 uncurry:
% 0.30/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.30/0.46 bot_bot:
% 0.30/0.46 !>[A: $tType] : A ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.46 ord_less_eq:
% 0.30/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oorder_Oantimono,type,
% 0.30/0.46 antimono:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oorder_Omono,type,
% 0.30/0.46 mono:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.30/0.46 order_Greatest:
% 0.30/0.46 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.30/0.46 top_top:
% 0.30/0.46 !>[A: $tType] : A ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Partial__Function_Oflat__ord,type,
% 0.30/0.46 partial_flat_ord:
% 0.30/0.46 !>[A: $tType] : ( A > A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Product__Type_OPair,type,
% 0.30/0.46 product_Pair:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Product__Type_Ocurry,type,
% 0.30/0.46 product_curry:
% 0.30/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.30/0.46 produc2004651681e_prod:
% 0.30/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.30/0.46 product_rec_prod:
% 0.30/0.46 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_OASSERT,type,
% 0.30/0.46 refine1814851989ASSERT: $o > ( refine1665802226e_nres @ product_unit ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
% 0.30/0.46 refine1687780735RETURN:
% 0.30/0.46 !>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oabs__fun,type,
% 0.30/0.46 refine81118332bs_fun:
% 0.30/0.46 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind,type,
% 0.30/0.46 refine463715084e_bind:
% 0.30/0.46 !>[B: $tType,A: $tType] : ( ( refine1665802226e_nres @ B ) > ( B > ( refine1665802226e_nres @ A ) ) > ( refine1665802226e_nres @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oconc__fun,type,
% 0.30/0.46 refine1073749519nc_fun:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres,type,
% 0.30/0.46 refine1315500908_inres:
% 0.30/0.46 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Olift__assn,type,
% 0.30/0.46 refine1580981607t_assn:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( B > $o ) > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres,type,
% 0.30/0.46 refine406925620_inres:
% 0.30/0.46 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail,type,
% 0.30/0.46 refine1102455758nofail:
% 0.30/0.46 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_OFAILi,type,
% 0.30/0.46 refine1767639642_FAILi:
% 0.30/0.46 !>[A: $tType] : ( refine1665802226e_nres @ A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES,type,
% 0.30/0.46 refine605929679le_RES:
% 0.30/0.46 !>[A: $tType] : ( ( set @ A ) > ( refine1665802226e_nres @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_Orec__nres,type,
% 0.30/0.46 refine1442219249c_nres:
% 0.30/0.46 !>[C: $tType,A: $tType] : ( C > ( ( set @ A ) > C ) > ( refine1665802226e_nres @ A ) > C ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES,type,
% 0.30/0.46 refine1672542526he_RES:
% 0.30/0.46 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Relation_ODomain,type,
% 0.30/0.46 domain:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Relation_OImage,type,
% 0.30/0.46 image:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OCollect,type,
% 0.30/0.46 collect:
% 0.30/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_member,type,
% 0.30/0.46 member:
% 0.30/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v__092_060Phi_062,type,
% 0.30/0.46 phi: a > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_f,type,
% 0.30/0.46 f: a > ( refine1665802226e_nres @ b ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_m,type,
% 0.30/0.46 m: refine1665802226e_nres @ b ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_x,type,
% 0.30/0.46 x: a ).
% 0.30/0.46
% 0.30/0.46 % Relevant facts (250)
% 0.30/0.46 thf(fact_0_assms_I1_J,axiom,
% 0.30/0.46 phi @ x ).
% 0.30/0.46
% 0.30/0.46 % assms(1)
% 0.30/0.46 thf(fact_1_assms_I2_J,axiom,
% 0.30/0.46 ord_less_eq @ ( refine1665802226e_nres @ b ) @ m @ ( f @ x ) ).
% 0.30/0.46
% 0.30/0.46 % assms(2)
% 0.30/0.46 thf(fact_2_nres__more__simps_I4_J,axiom,
% 0.30/0.46 ! [A: $tType,X: set @ A,Y: set @ A] :
% 0.30/0.46 ( ( ( refine605929679le_RES @ A @ X )
% 0.30/0.46 = ( refine605929679le_RES @ A @ Y ) )
% 0.30/0.46 = ( X = Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres_more_simps(4)
% 0.30/0.46 thf(fact_3_nres_Oinject,axiom,
% 0.30/0.46 ! [A: $tType,X2: set @ A,Y2: set @ A] :
% 0.30/0.46 ( ( ( refine605929679le_RES @ A @ X2 )
% 0.30/0.46 = ( refine605929679le_RES @ A @ Y2 ) )
% 0.30/0.46 = ( X2 = Y2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres.inject
% 0.30/0.46 thf(fact_4_if__rule,axiom,
% 0.30/0.46 ! [A: $tType,B2: $o,S1: refine1665802226e_nres @ A,Phi: A > $o,S2: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( B2
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S1 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) )
% 0.30/0.46 => ( ( ~ B2
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( if @ ( refine1665802226e_nres @ A ) @ B2 @ S1 @ S2 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % if_rule
% 0.30/0.46 thf(fact_5_RES__rule,axiom,
% 0.30/0.46 ! [A: $tType,S: set @ A,Phi: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ S )
% 0.30/0.46 => ( Phi @ X3 ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ S ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % RES_rule
% 0.30/0.46 thf(fact_6_SPEC__rule,axiom,
% 0.30/0.46 ! [A: $tType,Phi: A > $o,Phi2: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( Phi @ X3 )
% 0.30/0.46 => ( Phi2 @ X3 ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SPEC_rule
% 0.30/0.46 thf(fact_7_SPEC__trans,axiom,
% 0.30/0.46 ! [A: $tType,X4: refine1665802226e_nres @ A,Y3: refine1665802226e_nres @ A,Postcond: A > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ Y3 )
% 0.30/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Y3 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SPEC_trans
% 0.30/0.46 thf(fact_8_lhs__step__If,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A )
% 0.30/0.46 => ! [B2: $o,T2: A,M: A,E: A] :
% 0.30/0.46 ( ( B2
% 0.30/0.46 => ( ord_less_eq @ A @ T2 @ M ) )
% 0.30/0.46 => ( ( ~ B2
% 0.30/0.46 => ( ord_less_eq @ A @ E @ M ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( if @ A @ B2 @ T2 @ E ) @ M ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % lhs_step_If
% 0.30/0.46 thf(fact_9_SPEC__cons__rule,axiom,
% 0.30/0.46 ! [A: $tType,M: refine1665802226e_nres @ A,Phi: A > $o,Psi: A > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( Phi @ X3 )
% 0.30/0.46 => ( Psi @ X3 ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SPEC_cons_rule
% 0.30/0.46 thf(fact_10_RES__bind__choose,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X4: A,X: set @ A,M: refine1665802226e_nres @ B,F: A > ( refine1665802226e_nres @ B )] :
% 0.30/0.46 ( ( member @ A @ X4 @ X )
% 0.30/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( F @ X4 ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine463715084e_bind @ A @ B @ ( refine605929679le_RES @ A @ X ) @ F ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % RES_bind_choose
% 0.30/0.46 thf(fact_11_order__mono__setup_Orefl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A )
% 0.30/0.46 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_mono_setup.refl
% 0.30/0.46 thf(fact_12_the__RES_Osimps,axiom,
% 0.30/0.46 ! [A: $tType,X: set @ A] :
% 0.30/0.46 ( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X ) )
% 0.30/0.46 = X ) ).
% 0.30/0.46
% 0.30/0.46 % the_RES.simps
% 0.30/0.46 thf(fact_13_nf__inres__RES,axiom,
% 0.30/0.46 ! [A: $tType,X: set @ A,X4: A] :
% 0.30/0.46 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ X ) @ X4 )
% 0.30/0.46 = ( member @ A @ X4 @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % nf_inres_RES
% 0.30/0.46 thf(fact_14_nf__inres__SPEC,axiom,
% 0.30/0.46 ! [A: $tType,Phi: A > $o,X4: A] :
% 0.30/0.46 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ X4 )
% 0.30/0.46 = ( Phi @ X4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % nf_inres_SPEC
% 0.30/0.46 thf(fact_15_nres_Osimps_I7_J,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,F1: C,F2: ( set @ A ) > C,X2: set @ A] :
% 0.30/0.46 ( ( refine1442219249c_nres @ C @ A @ F1 @ F2 @ ( refine605929679le_RES @ A @ X2 ) )
% 0.30/0.46 = ( F2 @ X2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres.simps(7)
% 0.30/0.46 thf(fact_16_le__funD,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B )
% 0.30/0.46 => ! [F: A > B,G: A > B,X4: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_funD
% 0.30/0.46 thf(fact_17_le__funE,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B )
% 0.30/0.46 => ! [F: A > B,G: A > B,X4: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_funE
% 0.30/0.46 thf(fact_18_le__funI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B )
% 0.30/0.46 => ! [F: A > B,G: A > B] :
% 0.30/0.46 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.30/0.46 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_funI
% 0.30/0.46 thf(fact_19_Refine__Misc_Oif__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A )
% 0.30/0.46 => ! [B2: $o,M1: A,M12: A,M2: A,M22: A] :
% 0.30/0.46 ( ( B2
% 0.30/0.46 => ( ord_less_eq @ A @ M1 @ M12 ) )
% 0.30/0.46 => ( ( ~ B2
% 0.30/0.46 => ( ord_less_eq @ A @ M2 @ M22 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( if @ A @ B2 @ M1 @ M2 ) @ ( if @ A @ B2 @ M12 @ M22 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Refine_Misc.if_mono
% 0.30/0.46 thf(fact_20_le__fun__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B )
% 0.30/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/0.46 = ( ^ [F3: A > B,G2: A > B] :
% 0.30/0.46 ! [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G2 @ X5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_fun_def
% 0.30/0.46 thf(fact_21_order__subst1,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( order @ B )
% 0.30/0.46 & ( order @ A ) )
% 0.30/0.46 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.30/0.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.30/0.46 => ( ! [X3: B,Y4: B] :
% 0.30/0.46 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.30/0.46 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_subst1
% 0.30/0.46 thf(fact_22_less__eq__nres_Osimps_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) )
% 0.30/0.46 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_eq_nres.simps(2)
% 0.30/0.46 thf(fact_23_nres__order__simps_I5_J,axiom,
% 0.30/0.46 ! [E2: $tType,X: set @ E2,Y: set @ E2] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ E2 ) @ ( refine605929679le_RES @ E2 @ X ) @ ( refine605929679le_RES @ E2 @ Y ) )
% 0.30/0.46 = ( ord_less_eq @ ( set @ E2 ) @ X @ Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres_order_simps(5)
% 0.30/0.46 thf(fact_24_order__mono__setup_Omono__let,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( preorder @ A )
% 0.30/0.46 => ! [F: B > A,F4: B > A,X4: B] :
% 0.30/0.46 ( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( F4 @ X3 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F4 @ X4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_mono_setup.mono_let
% 0.30/0.46 thf(fact_25_order__mono__setup_Omono__if,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A )
% 0.30/0.46 => ! [T2: A,T3: A,E: A,E3: A,B2: $o] :
% 0.30/0.46 ( ( ord_less_eq @ A @ T2 @ T3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ E @ E3 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( if @ A @ B2 @ T2 @ E ) @ ( if @ A @ B2 @ T3 @ E3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_mono_setup.mono_if
% 0.30/0.46 thf(fact_26_dual__order_Oantisym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [B2: A,A2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.antisym
% 0.30/0.46 thf(fact_27_dual__order_Oeq__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.30/0.46 = ( ^ [A3: A,B3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.46 & ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.eq_iff
% 0.30/0.46 thf(fact_28_dual__order_Otrans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [B2: A,A2: A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.trans
% 0.30/0.46 thf(fact_29_linorder__wlog,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A )
% 0.30/0.46 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.30/0.46 ( ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.46 => ( P @ A4 @ B4 ) )
% 0.30/0.46 => ( ! [A4: A,B4: A] :
% 0.30/0.46 ( ( P @ B4 @ A4 )
% 0.30/0.46 => ( P @ A4 @ B4 ) )
% 0.30/0.46 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % linorder_wlog
% 0.30/0.46 thf(fact_30_dual__order_Orefl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.refl
% 0.30/0.46 thf(fact_31_order__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A )
% 0.30/0.46 => ! [X4: A,Y3: A,Z2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.30/0.46 => ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_trans
% 0.30/0.46 thf(fact_32_order__class_Oorder_Oantisym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [A2: A,B2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_class.order.antisym
% 0.30/0.46 thf(fact_33_ord__le__eq__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A )
% 0.30/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.46 => ( ( B2 = C2 )
% 0.30/0.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_trans
% 0.30/0.46 thf(fact_34_ord__eq__le__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A )
% 0.30/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_trans
% 0.30/0.46 thf(fact_35_order__class_Oorder_Oeq__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.30/0.46 = ( ^ [A3: A,B3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.46 & ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_class.order.eq_iff
% 0.30/0.46 thf(fact_36_antisym__conv,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [Y3: A,X4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ Y3 @ X4 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.30/0.46 = ( X4 = Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym_conv
% 0.30/0.46 thf(fact_37_le__cases3,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A )
% 0.30/0.46 => ! [X4: A,Y3: A,Z2: A] :
% 0.30/0.46 ( ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ Y3 @ X4 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ X4 @ Z2 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Y3 @ X4 ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
% 0.30/0.46 => ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cases3
% 0.30/0.46 thf(fact_38_order_Otrans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order.trans
% 0.30/0.46 thf(fact_39_le__cases,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A )
% 0.30/0.46 => ! [X4: A,Y3: A] :
% 0.30/0.46 ( ~ ( ord_less_eq @ A @ X4 @ Y3 )
% 0.30/0.46 => ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cases
% 0.30/0.46 thf(fact_40_eq__refl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A )
% 0.30/0.46 => ! [X4: A,Y3: A] :
% 0.30/0.46 ( ( X4 = Y3 )
% 0.30/0.46 => ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_refl
% 0.30/0.46 thf(fact_41_linear,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A )
% 0.30/0.46 => ! [X4: A,Y3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.30/0.46 | ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % linear
% 0.30/0.46 thf(fact_42_antisym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [X4: A,Y3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ Y3 @ X4 )
% 0.30/0.46 => ( X4 = Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym
% 0.30/0.46 thf(fact_43_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,P: A > $o] :
% 0.30/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.30/0.46 = ( P @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_44_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ( collect @ A
% 0.30/0.46 @ ^ [X5: A] : ( member @ A @ X5 @ A5 ) )
% 0.30/0.46 = A5 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_45_Collect__cong,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( P @ X3 )
% 0.30/0.46 = ( Q @ X3 ) )
% 0.30/0.46 => ( ( collect @ A @ P )
% 0.30/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_cong
% 0.30/0.46 thf(fact_46_ext,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( F @ X3 )
% 0.30/0.46 = ( G @ X3 ) )
% 0.30/0.46 => ( F = G ) ) ).
% 0.30/0.46
% 0.30/0.46 % ext
% 0.30/0.46 thf(fact_47_eq__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.30/0.46 = ( ^ [X5: A,Y6: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X5 @ Y6 )
% 0.30/0.46 & ( ord_less_eq @ A @ Y6 @ X5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_iff
% 0.30/0.46 thf(fact_48_ord__le__eq__subst,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( ord @ B )
% 0.30/0.46 & ( ord @ A ) )
% 0.30/0.46 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.46 => ( ( ( F @ B2 )
% 0.30/0.46 = C2 )
% 0.30/0.46 => ( ! [X3: A,Y4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_subst
% 0.30/0.46 thf(fact_49_ord__eq__le__subst,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( ord @ B )
% 0.30/0.46 & ( ord @ A ) )
% 0.30/0.46 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.30/0.46 ( ( A2
% 0.30/0.46 = ( F @ B2 ) )
% 0.30/0.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.30/0.46 => ( ! [X3: B,Y4: B] :
% 0.30/0.46 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.30/0.46 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_subst
% 0.30/0.46 thf(fact_50_order__subst2,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType] :
% 0.30/0.46 ( ( ( order @ C )
% 0.30/0.46 & ( order @ A ) )
% 0.30/0.46 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.30/0.46 => ( ! [X3: A,Y4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.30/0.46 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.30/0.46 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_subst2
% 0.30/0.46 thf(fact_51_the__RES__inv,axiom,
% 0.30/0.46 ! [A: $tType,M: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( refine1102455758nofail @ A @ M )
% 0.30/0.46 => ( ( refine605929679le_RES @ A @ ( refine1672542526he_RES @ A @ M ) )
% 0.30/0.46 = M ) ) ).
% 0.30/0.46
% 0.30/0.46 % the_RES_inv
% 0.30/0.46 thf(fact_52_Greatest__equality,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [P: A > $o,X4: A] :
% 0.30/0.46 ( ( P @ X4 )
% 0.30/0.46 => ( ! [Y4: A] :
% 0.30/0.46 ( ( P @ Y4 )
% 0.30/0.46 => ( ord_less_eq @ A @ Y4 @ X4 ) )
% 0.30/0.46 => ( ( order_Greatest @ A @ P )
% 0.30/0.46 = X4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Greatest_equality
% 0.30/0.46 thf(fact_53_GreatestI2__order,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [P: A > $o,X4: A,Q: A > $o] :
% 0.30/0.46 ( ( P @ X4 )
% 0.30/0.46 => ( ! [Y4: A] :
% 0.30/0.46 ( ( P @ Y4 )
% 0.30/0.46 => ( ord_less_eq @ A @ Y4 @ X4 ) )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( P @ X3 )
% 0.30/0.46 => ( ! [Y7: A] :
% 0.30/0.46 ( ( P @ Y7 )
% 0.30/0.46 => ( ord_less_eq @ A @ Y7 @ X3 ) )
% 0.30/0.46 => ( Q @ X3 ) ) )
% 0.30/0.46 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % GreatestI2_order
% 0.30/0.46 thf(fact_54_transfer_Otransfer__Let,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [Alpha: C > A,F: B > C,F5: B > A,X4: B] :
% 0.30/0.46 ( ! [X3: B] : ( ord_less_eq @ A @ ( Alpha @ ( F @ X3 ) ) @ ( F5 @ X3 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( Alpha @ ( F @ X4 ) ) @ ( F5 @ X4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer.transfer_Let
% 0.30/0.46 thf(fact_55_transfer_Otransfer__if,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [B2: $o,Alpha: C > A,S12: C,S1: A,S22: C,S2: A] :
% 0.30/0.46 ( ( B2
% 0.30/0.46 => ( ord_less_eq @ A @ ( Alpha @ S12 ) @ S1 ) )
% 0.30/0.46 => ( ( ~ B2
% 0.30/0.46 => ( ord_less_eq @ A @ ( Alpha @ S22 ) @ S2 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( Alpha @ ( if @ C @ B2 @ S12 @ S22 ) ) @ ( if @ A @ B2 @ S1 @ S2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer.transfer_if
% 0.30/0.46 thf(fact_56_le__rel__bool__arg__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A )
% 0.30/0.46 => ( ( ord_less_eq @ ( $o > A ) )
% 0.30/0.46 = ( ^ [X6: $o > A,Y8: $o > A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( X6 @ $false ) @ ( Y8 @ $false ) )
% 0.30/0.46 & ( ord_less_eq @ A @ ( X6 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_rel_bool_arg_iff
% 0.30/0.46 thf(fact_57_verit__la__disequality,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A )
% 0.30/0.46 => ! [A2: A,B2: A] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 | ~ ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.46 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % verit_la_disequality
% 0.30/0.46 thf(fact_58_ord__eq__le__eq__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A )
% 0.30/0.46 => ! [A2: A,B2: A,C2: A,D: A] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.46 => ( ( C2 = D )
% 0.30/0.46 => ( ord_less_eq @ A @ A2 @ D ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_eq_trans
% 0.30/0.46 thf(fact_59_intro__spec__refine__iff,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType,X: set @ B,F: B > ( refine1665802226e_nres @ A ),R: set @ ( product_prod @ A @ C ),M3: refine1665802226e_nres @ C] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ ( refine605929679le_RES @ B @ X ) @ F ) @ ( refine1073749519nc_fun @ A @ C @ R @ M3 ) )
% 0.30/0.46 = ( ! [X5: B] :
% 0.30/0.46 ( ( member @ B @ X5 @ X )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X5 ) @ ( refine1073749519nc_fun @ A @ C @ R @ M3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % intro_spec_refine_iff
% 0.30/0.46 thf(fact_60_intro__bind__refine__id,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,M: refine1665802226e_nres @ A,M4: A,F: A > ( refine1665802226e_nres @ B ),R: set @ ( product_prod @ B @ C ),M5: refine1665802226e_nres @ C] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M
% 0.30/0.46 @ ( refine605929679le_RES @ A
% 0.30/0.46 @ ( collect @ A
% 0.30/0.46 @ ( ^ [Y5: A,Z: A] : ( Y5 = Z )
% 0.30/0.46 @ M4 ) ) ) )
% 0.30/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ M4 ) @ ( refine1073749519nc_fun @ B @ C @ R @ M5 ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ ( refine1073749519nc_fun @ B @ C @ R @ M5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % intro_bind_refine_id
% 0.30/0.46 thf(fact_61_nofail__simps_I2_J,axiom,
% 0.30/0.46 ! [B: $tType,X: set @ B] : ( refine1102455758nofail @ B @ ( refine605929679le_RES @ B @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % nofail_simps(2)
% 0.30/0.46 thf(fact_62_subset__Collect__conv,axiom,
% 0.30/0.46 ! [A: $tType,S: set @ A,P: A > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
% 0.30/0.46 = ( ! [X5: A] :
% 0.30/0.46 ( ( member @ A @ X5 @ S )
% 0.30/0.46 => ( P @ X5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_Collect_conv
% 0.30/0.46 thf(fact_63_nofail__antimono__fun,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > ( refine1665802226e_nres @ B ),G: A > ( refine1665802226e_nres @ B ),X4: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > ( refine1665802226e_nres @ B ) ) @ F @ G )
% 0.30/0.46 => ( ( refine1102455758nofail @ B @ ( G @ X4 ) )
% 0.30/0.46 => ( refine1102455758nofail @ B @ ( F @ X4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nofail_antimono_fun
% 0.30/0.46 thf(fact_64_pw__conc__nofail,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 0.30/0.46 ( ( refine1102455758nofail @ A @ ( refine1073749519nc_fun @ A @ B @ R @ S ) )
% 0.30/0.46 = ( refine1102455758nofail @ B @ S ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_conc_nofail
% 0.30/0.46 thf(fact_65_pwD1,axiom,
% 0.30/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S3 )
% 0.30/0.46 => ( ( refine1102455758nofail @ A @ S3 )
% 0.30/0.46 => ( refine1102455758nofail @ A @ S ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pwD1
% 0.30/0.46 thf(fact_66_le__nofailI,axiom,
% 0.30/0.46 ! [A: $tType,M6: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( ( refine1102455758nofail @ A @ M6 )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ M6 ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ M6 ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_nofailI
% 0.30/0.46 thf(fact_67_nofail__RES__conv,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( refine1102455758nofail @ A )
% 0.30/0.46 = ( ^ [M7: refine1665802226e_nres @ A] :
% 0.30/0.46 ? [M8: set @ A] :
% 0.30/0.46 ( M7
% 0.30/0.46 = ( refine605929679le_RES @ A @ M8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nofail_RES_conv
% 0.30/0.46 thf(fact_68_if__refine,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B2: $o,B5: $o,S1: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),S13: refine1665802226e_nres @ B,S2: refine1665802226e_nres @ A,S23: refine1665802226e_nres @ B] :
% 0.30/0.46 ( ( B2 = B5 )
% 0.30/0.46 => ( ( B2
% 0.30/0.46 => ( B5
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S1 @ ( refine1073749519nc_fun @ A @ B @ R @ S13 ) ) ) )
% 0.30/0.46 => ( ( ~ B2
% 0.30/0.46 => ( ~ B5
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S2 @ ( refine1073749519nc_fun @ A @ B @ R @ S23 ) ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( if @ ( refine1665802226e_nres @ A ) @ B2 @ S1 @ S2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( if @ ( refine1665802226e_nres @ B ) @ B5 @ S13 @ S23 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % if_refine
% 0.30/0.46 thf(fact_69_conc__trans,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,C3: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B6: refine1665802226e_nres @ B,R2: set @ ( product_prod @ B @ C ),A5: refine1665802226e_nres @ C] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C3 @ ( refine1073749519nc_fun @ A @ B @ R @ B6 ) )
% 0.30/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B6 @ ( refine1073749519nc_fun @ B @ C @ R2 @ A5 ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C3 @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1073749519nc_fun @ B @ C @ R2 @ A5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % conc_trans
% 0.30/0.46 thf(fact_70_intro__prgR,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),A2: refine1665802226e_nres @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ A2 ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % intro_prgR
% 0.30/0.46 thf(fact_71_ref__two__step,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A5: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B6: refine1665802226e_nres @ B,C3: refine1665802226e_nres @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A5 @ ( refine1073749519nc_fun @ A @ B @ R @ B6 ) )
% 0.30/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B6 @ C3 )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A5 @ ( refine1073749519nc_fun @ A @ B @ R @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ref_two_step
% 0.30/0.46 thf(fact_72_conc__fun__R__mono,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),M3: refine1665802226e_nres @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ R2 )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1073749519nc_fun @ A @ B @ R @ M3 ) @ ( refine1073749519nc_fun @ A @ B @ R2 @ M3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % conc_fun_R_mono
% 0.30/0.46 thf(fact_73_le__RES__nofailI,axiom,
% 0.30/0.46 ! [A: $tType,A2: refine1665802226e_nres @ A,X4: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine605929679le_RES @ A @ X4 ) )
% 0.30/0.46 => ( refine1102455758nofail @ A @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_RES_nofailI
% 0.30/0.46 thf(fact_74_pw__RES__bind__choose_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X: set @ B,F: B > ( refine1665802226e_nres @ A )] :
% 0.30/0.46 ( ( refine1102455758nofail @ A @ ( refine463715084e_bind @ B @ A @ ( refine605929679le_RES @ B @ X ) @ F ) )
% 0.30/0.46 = ( ! [X5: B] :
% 0.30/0.46 ( ( member @ B @ X5 @ X )
% 0.30/0.46 => ( refine1102455758nofail @ A @ ( F @ X5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_RES_bind_choose(1)
% 0.30/0.46 thf(fact_75_rhs__step__bind__RES,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,X7: A,X8: set @ A,M: refine1665802226e_nres @ B,R: set @ ( product_prod @ B @ C ),F4: A > ( refine1665802226e_nres @ C )] :
% 0.30/0.46 ( ( member @ A @ X7 @ X8 )
% 0.30/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine1073749519nc_fun @ B @ C @ R @ ( F4 @ X7 ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine1073749519nc_fun @ B @ C @ R @ ( refine463715084e_bind @ A @ C @ ( refine605929679le_RES @ A @ X8 ) @ F4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rhs_step_bind_RES
% 0.30/0.46 thf(fact_76_rhs__step__bind__SPEC,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,Phi: A > $o,X7: A,M: refine1665802226e_nres @ B,R: set @ ( product_prod @ B @ C ),F4: A > ( refine1665802226e_nres @ C )] :
% 0.30/0.46 ( ( Phi @ X7 )
% 0.30/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine1073749519nc_fun @ B @ C @ R @ ( F4 @ X7 ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine1073749519nc_fun @ B @ C @ R @ ( refine463715084e_bind @ A @ C @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ F4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rhs_step_bind_SPEC
% 0.30/0.46 thf(fact_77_subsetI,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ A5 )
% 0.30/0.46 => ( member @ A @ X3 @ B6 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetI
% 0.30/0.46 thf(fact_78_subset__antisym,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
% 0.30/0.46 => ( A5 = B6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_antisym
% 0.30/0.46 thf(fact_79_bind__sim__select__rule,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,M: refine1665802226e_nres @ B,F4: B > ( refine1665802226e_nres @ A ),Psi: A > $o,F: B > ( refine1665802226e_nres @ C ),Phi: C > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F4 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) )
% 0.30/0.46 => ( ! [X3: B] :
% 0.30/0.46 ( ( refine1102455758nofail @ B @ M )
% 0.30/0.46 => ( ( refine1315500908_inres @ B @ M @ X3 )
% 0.30/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F4 @ X3 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( F @ X3 ) @ ( refine605929679le_RES @ C @ ( collect @ C @ Phi ) ) ) ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine463715084e_bind @ B @ C @ M @ F ) @ ( refine605929679le_RES @ C @ ( collect @ C @ Phi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bind_sim_select_rule
% 0.30/0.46 thf(fact_80_relprop__triggers_I6_J,axiom,
% 0.30/0.46 ! [I: $tType,R: set @ I,R2: set @ I] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ I ) @ R @ R2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ I ) @ R @ R2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % relprop_triggers(6)
% 0.30/0.46 thf(fact_81_in__mono,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B6: set @ A,X4: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.46 => ( ( member @ A @ X4 @ A5 )
% 0.30/0.46 => ( member @ A @ X4 @ B6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % in_mono
% 0.30/0.46 thf(fact_82_subsetD,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B6: set @ A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.46 => ( ( member @ A @ C2 @ A5 )
% 0.30/0.46 => ( member @ A @ C2 @ B6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetD
% 0.30/0.46 thf(fact_83_equalityE,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.46 ( ( A5 = B6 )
% 0.30/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityE
% 0.30/0.46 thf(fact_84_inres__simps_I2_J,axiom,
% 0.30/0.46 ! [B: $tType,X: set @ B] :
% 0.30/0.46 ( ( refine1315500908_inres @ B @ ( refine605929679le_RES @ B @ X ) )
% 0.30/0.46 = ( ^ [X5: B] : ( member @ B @ X5 @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inres_simps(2)
% 0.30/0.46 thf(fact_85_pwD2,axiom,
% 0.30/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A,X4: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S3 )
% 0.30/0.46 => ( ( refine1315500908_inres @ A @ S @ X4 )
% 0.30/0.46 => ( refine1315500908_inres @ A @ S3 @ X4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pwD2
% 0.30/0.46 thf(fact_86_pw__eqI,axiom,
% 0.30/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( ( refine1102455758nofail @ A @ S )
% 0.30/0.46 = ( refine1102455758nofail @ A @ S3 ) )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( refine1315500908_inres @ A @ S @ X3 )
% 0.30/0.46 = ( refine1315500908_inres @ A @ S3 @ X3 ) )
% 0.30/0.46 => ( S = S3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_eqI
% 0.30/0.46 thf(fact_87_pw__eq__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ^ [Y5: refine1665802226e_nres @ A,Z: refine1665802226e_nres @ A] : ( Y5 = Z ) )
% 0.30/0.46 = ( ^ [S4: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( ( refine1102455758nofail @ A @ S4 )
% 0.30/0.46 = ( refine1102455758nofail @ A @ S5 ) )
% 0.30/0.46 & ! [X5: A] :
% 0.30/0.46 ( ( refine1315500908_inres @ A @ S4 @ X5 )
% 0.30/0.46 = ( refine1315500908_inres @ A @ S5 @ X5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_eq_iff
% 0.30/0.46 thf(fact_88_not__nofail__inres,axiom,
% 0.30/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,X4: A] :
% 0.30/0.46 ( ~ ( refine1102455758nofail @ A @ S )
% 0.30/0.46 => ( refine1315500908_inres @ A @ S @ X4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % not_nofail_inres
% 0.30/0.46 thf(fact_89_pw__RES__bind__choose_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X: set @ B,F: B > ( refine1665802226e_nres @ A ),Y3: A] :
% 0.30/0.46 ( ( refine1315500908_inres @ A @ ( refine463715084e_bind @ B @ A @ ( refine605929679le_RES @ B @ X ) @ F ) @ Y3 )
% 0.30/0.46 = ( ? [X5: B] :
% 0.30/0.46 ( ( member @ B @ X5 @ X )
% 0.30/0.46 & ( refine1315500908_inres @ A @ ( F @ X5 ) @ Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_RES_bind_choose(2)
% 0.30/0.46 thf(fact_90_pw__leI,axiom,
% 0.30/0.46 ! [A: $tType,S3: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( ( refine1102455758nofail @ A @ S3 )
% 0.30/0.46 => ( ( refine1102455758nofail @ A @ S )
% 0.30/0.46 & ! [X3: A] :
% 0.30/0.46 ( ( refine1315500908_inres @ A @ S @ X3 )
% 0.30/0.46 => ( refine1315500908_inres @ A @ S3 @ X3 ) ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_leI
% 0.30/0.46 thf(fact_91_pw__leI_H,axiom,
% 0.30/0.46 ! [A: $tType,S3: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( ( refine1102455758nofail @ A @ S3 )
% 0.30/0.46 => ( refine1102455758nofail @ A @ S ) )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( refine1102455758nofail @ A @ S3 )
% 0.30/0.46 => ( ( refine1315500908_inres @ A @ S @ X3 )
% 0.30/0.46 => ( refine1315500908_inres @ A @ S3 @ X3 ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_leI'
% 0.30/0.46 thf(fact_92_pw__le__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) )
% 0.30/0.46 = ( ^ [S4: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( refine1102455758nofail @ A @ S5 )
% 0.30/0.46 => ( ( refine1102455758nofail @ A @ S4 )
% 0.30/0.46 & ! [X5: A] :
% 0.30/0.46 ( ( refine1315500908_inres @ A @ S4 @ X5 )
% 0.30/0.46 => ( refine1315500908_inres @ A @ S5 @ X5 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_le_iff
% 0.30/0.46 thf(fact_93_pw__bind__nofail,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,M3: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
% 0.30/0.46 ( ( refine1102455758nofail @ A @ ( refine463715084e_bind @ B @ A @ M3 @ F ) )
% 0.30/0.46 = ( ( refine1102455758nofail @ B @ M3 )
% 0.30/0.46 & ! [X5: B] :
% 0.30/0.46 ( ( refine1315500908_inres @ B @ M3 @ X5 )
% 0.30/0.46 => ( refine1102455758nofail @ A @ ( F @ X5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_bind_nofail
% 0.30/0.46 thf(fact_94_nf__inres__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( refine406925620_inres @ A )
% 0.30/0.46 = ( ^ [M7: refine1665802226e_nres @ A,X5: A] :
% 0.30/0.46 ( ( refine1102455758nofail @ A @ M7 )
% 0.30/0.46 & ( refine1315500908_inres @ A @ M7 @ X5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nf_inres_def
% 0.30/0.46 thf(fact_95_pw__bind__le__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,M3: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A ),S: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M3 @ F ) @ S )
% 0.30/0.46 = ( ( ( refine1102455758nofail @ A @ S )
% 0.30/0.46 => ( refine1102455758nofail @ B @ M3 ) )
% 0.30/0.46 & ! [X5: B] :
% 0.30/0.46 ( ( ( refine1102455758nofail @ B @ M3 )
% 0.30/0.46 & ( refine1315500908_inres @ B @ M3 @ X5 ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X5 ) @ S ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_bind_le_iff
% 0.30/0.46 thf(fact_96_pw__bind__leI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,S: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
% 0.30/0.46 ( ( ( refine1102455758nofail @ A @ S )
% 0.30/0.46 => ( refine1102455758nofail @ B @ M3 ) )
% 0.30/0.46 => ( ! [X3: B] :
% 0.30/0.46 ( ( refine1102455758nofail @ B @ M3 )
% 0.30/0.46 => ( ( refine1315500908_inres @ B @ M3 @ X3 )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X3 ) @ S ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M3 @ F ) @ S ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_bind_leI
% 0.30/0.46 thf(fact_97_Collect__mono__iff,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.30/0.46 = ( ! [X5: A] :
% 0.30/0.46 ( ( P @ X5 )
% 0.30/0.46 => ( Q @ X5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mono_iff
% 0.30/0.46 thf(fact_98_set__eq__subset,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ^ [Y5: set @ A,Z: set @ A] : ( Y5 = Z ) )
% 0.30/0.46 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_eq_subset
% 0.30/0.46 thf(fact_99_subset__trans,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B6: set @ A,C3: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C3 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_trans
% 0.30/0.46 thf(fact_100_Collect__mono,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( P @ X3 )
% 0.30/0.46 => ( Q @ X3 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mono
% 0.30/0.46 thf(fact_101_subset__refl,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 0.30/0.46
% 0.30/0.46 % subset_refl
% 0.30/0.46 thf(fact_102_subset__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.30/0.46 ! [T4: A] :
% 0.30/0.46 ( ( member @ A @ T4 @ A6 )
% 0.30/0.46 => ( member @ A @ T4 @ B7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_iff
% 0.30/0.46 thf(fact_103_equalityD2,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.46 ( ( A5 = B6 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityD2
% 0.30/0.46 thf(fact_104_equalityD1,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 0.30/0.46 ( ( A5 = B6 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityD1
% 0.30/0.46 thf(fact_105_subset__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.30/0.46 ! [X5: A] :
% 0.30/0.46 ( ( member @ A @ X5 @ A6 )
% 0.30/0.46 => ( member @ A @ X5 @ B7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_eq
% 0.30/0.46 thf(fact_106_pw__ref__iff,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,S: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),S3: refine1665802226e_nres @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ ( refine1073749519nc_fun @ A @ B @ R @ S3 ) )
% 0.30/0.46 = ( ( refine1102455758nofail @ B @ S3 )
% 0.30/0.46 => ( ( refine1102455758nofail @ A @ S )
% 0.30/0.46 & ! [X5: A] :
% 0.30/0.46 ( ( refine1315500908_inres @ A @ S @ X5 )
% 0.30/0.46 => ? [S6: B] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ S6 ) @ R )
% 0.30/0.46 & ( refine1315500908_inres @ B @ S3 @ S6 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_ref_iff
% 0.30/0.46 thf(fact_107_pw__ref__I,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,S3: refine1665802226e_nres @ A,S: refine1665802226e_nres @ B,R: set @ ( product_prod @ B @ A )] :
% 0.30/0.46 ( ( ( refine1102455758nofail @ A @ S3 )
% 0.30/0.46 => ( ( refine1102455758nofail @ B @ S )
% 0.30/0.46 & ! [X3: B] :
% 0.30/0.46 ( ( refine1315500908_inres @ B @ S @ X3 )
% 0.30/0.46 => ? [S7: A] :
% 0.30/0.46 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ S7 ) @ R )
% 0.30/0.46 & ( refine1315500908_inres @ A @ S3 @ S7 ) ) ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ S @ ( refine1073749519nc_fun @ B @ A @ R @ S3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_ref_I
% 0.30/0.46 thf(fact_108_rhs__step__bind,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D2: $tType,B: $tType,M: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),M4: refine1665802226e_nres @ B,X4: A,Lhs: refine1665802226e_nres @ C,S: set @ ( product_prod @ C @ D2 ),F4: B > ( refine1665802226e_nres @ D2 )] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine1073749519nc_fun @ A @ B @ R @ M4 ) )
% 0.30/0.46 => ( ( refine1315500908_inres @ A @ M @ X4 )
% 0.30/0.46 => ( ! [X9: B] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ X9 ) @ R )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ Lhs @ ( refine1073749519nc_fun @ C @ D2 @ S @ ( F4 @ X9 ) ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ Lhs @ ( refine1073749519nc_fun @ C @ D2 @ S @ ( refine463715084e_bind @ B @ D2 @ M4 @ F4 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rhs_step_bind
% 0.30/0.46 thf(fact_109_RES__refine,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,S: set @ A,S3: set @ B,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ! [S8: A] :
% 0.30/0.46 ( ( member @ A @ S8 @ S )
% 0.30/0.46 => ? [X10: B] :
% 0.30/0.46 ( ( member @ B @ X10 @ S3 )
% 0.30/0.46 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S8 @ X10 ) @ R ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ S ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine605929679le_RES @ B @ S3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % RES_refine
% 0.30/0.46 thf(fact_110_less__eq__nres_Oelims_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,X4: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
% 0.30/0.46 ( ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ Xa )
% 0.30/0.46 = Y3 )
% 0.30/0.46 => ( ( ( Xa
% 0.30/0.46 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.46 => ~ Y3 )
% 0.30/0.46 => ( ! [A4: set @ A] :
% 0.30/0.46 ( ( X4
% 0.30/0.46 = ( refine605929679le_RES @ A @ A4 ) )
% 0.30/0.46 => ! [B4: set @ A] :
% 0.30/0.46 ( ( Xa
% 0.30/0.46 = ( refine605929679le_RES @ A @ B4 ) )
% 0.30/0.46 => ( Y3
% 0.30/0.46 = ( ~ ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ) )
% 0.30/0.46 => ~ ( ( X4
% 0.30/0.46 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.46 => ( ? [Uv: set @ A] :
% 0.30/0.46 ( Xa
% 0.30/0.46 = ( refine605929679le_RES @ A @ Uv ) )
% 0.30/0.46 => Y3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_eq_nres.elims(1)
% 0.30/0.46 thf(fact_111_less__eq__nres_Oelims_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,X4: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ Xa )
% 0.30/0.46 => ( ( Xa
% 0.30/0.46 != ( refine1767639642_FAILi @ A ) )
% 0.30/0.46 => ~ ! [A4: set @ A] :
% 0.30/0.46 ( ( X4
% 0.30/0.46 = ( refine605929679le_RES @ A @ A4 ) )
% 0.30/0.46 => ! [B4: set @ A] :
% 0.30/0.46 ( ( Xa
% 0.30/0.46 = ( refine605929679le_RES @ A @ B4 ) )
% 0.30/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_eq_nres.elims(2)
% 0.30/0.46 thf(fact_112_less__eq__nres_Ocases,axiom,
% 0.30/0.46 ! [A: $tType,X4: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
% 0.30/0.46 ( ! [Uu: refine1665802226e_nres @ A] :
% 0.30/0.46 ( X4
% 0.30/0.46 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ Uu @ ( refine1767639642_FAILi @ A ) ) )
% 0.30/0.46 => ( ! [A4: set @ A,B4: set @ A] :
% 0.30/0.46 ( X4
% 0.30/0.46 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) ) )
% 0.30/0.46 => ~ ! [Uv: set @ A] :
% 0.30/0.46 ( X4
% 0.30/0.46 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_eq_nres.cases
% 0.30/0.46 thf(fact_113_less__nres_Ocases,axiom,
% 0.30/0.46 ! [A: $tType,X4: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
% 0.30/0.46 ( ! [Uu: refine1665802226e_nres @ A] :
% 0.30/0.46 ( X4
% 0.30/0.46 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Uu ) )
% 0.30/0.46 => ( ! [Uv: set @ A] :
% 0.30/0.46 ( X4
% 0.30/0.46 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) ) )
% 0.30/0.46 => ~ ! [A4: set @ A,B4: set @ A] :
% 0.30/0.46 ( X4
% 0.30/0.46 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_nres.cases
% 0.30/0.46 thf(fact_114_sup__nres_Ocases,axiom,
% 0.30/0.46 ! [A: $tType,X4: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
% 0.30/0.46 ( ! [Uu: refine1665802226e_nres @ A] :
% 0.30/0.46 ( X4
% 0.30/0.46 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ Uu @ ( refine1767639642_FAILi @ A ) ) )
% 0.30/0.46 => ( ! [V: set @ A] :
% 0.30/0.46 ( X4
% 0.30/0.46 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V ) ) )
% 0.30/0.46 => ~ ! [A4: set @ A,B4: set @ A] :
% 0.30/0.46 ( X4
% 0.30/0.46 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sup_nres.cases
% 0.30/0.46 thf(fact_115_introR,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A,A7: B,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ A7 ) @ R )
% 0.30/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ A7 ) @ R ) ) ).
% 0.30/0.46
% 0.30/0.46 % introR
% 0.30/0.46 thf(fact_116_bex2I,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A,B2: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S )
% 0.30/0.46 => ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S )
% 0.30/0.46 => ( P @ A2 @ B2 ) )
% 0.30/0.46 => ? [A4: A,B4: B] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ S )
% 0.30/0.46 & ( P @ A4 @ B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bex2I
% 0.30/0.46 thf(fact_117_pairself_Ocases,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X4: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
% 0.30/0.46 ~ ! [F6: A > B,A4: A,B4: A] :
% 0.30/0.46 ( X4
% 0.30/0.46 != ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F6 @ ( product_Pair @ A @ A @ A4 @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pairself.cases
% 0.30/0.46 thf(fact_118_pairself_Oinduct,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,P: ( A > B ) > ( product_prod @ A @ A ) > $o,A0: A > B,A1: product_prod @ A @ A] :
% 0.30/0.46 ( ! [F6: A > B,A4: A,B4: A] : ( P @ F6 @ ( product_Pair @ A @ A @ A4 @ B4 ) )
% 0.30/0.46 => ( P @ A0 @ A1 ) ) ).
% 0.30/0.46
% 0.30/0.46 % pairself.induct
% 0.30/0.46 thf(fact_119_less__eq__nres_Osimps_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,Uu2: refine1665802226e_nres @ A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Uu2 @ ( refine1767639642_FAILi @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_eq_nres.simps(1)
% 0.30/0.46 thf(fact_120_nres_Odistinct_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,X2: set @ A] :
% 0.30/0.46 ( ( refine1767639642_FAILi @ A )
% 0.30/0.46 != ( refine605929679le_RES @ A @ X2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres.distinct(1)
% 0.30/0.46 thf(fact_121_nres_Oinduct,axiom,
% 0.30/0.46 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > $o,Nres: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( P @ ( refine1767639642_FAILi @ A ) )
% 0.30/0.46 => ( ! [X3: set @ A] : ( P @ ( refine605929679le_RES @ A @ X3 ) )
% 0.30/0.46 => ( P @ Nres ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres.induct
% 0.30/0.46 thf(fact_122_nres_Oexhaust,axiom,
% 0.30/0.46 ! [A: $tType,Y3: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( Y3
% 0.30/0.46 != ( refine1767639642_FAILi @ A ) )
% 0.30/0.46 => ~ ! [X22: set @ A] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( refine605929679le_RES @ A @ X22 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres.exhaust
% 0.30/0.46 thf(fact_123_sup__nres_Oinduct,axiom,
% 0.30/0.46 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ! [Uu: refine1665802226e_nres @ A] : ( P @ Uu @ ( refine1767639642_FAILi @ A ) )
% 0.30/0.46 => ( ! [V: set @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V ) )
% 0.30/0.46 => ( ! [A4: set @ A,B4: set @ A] : ( P @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) )
% 0.30/0.46 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sup_nres.induct
% 0.30/0.46 thf(fact_124_less__nres_Oinduct,axiom,
% 0.30/0.46 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ! [X_1: refine1665802226e_nres @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ X_1 )
% 0.30/0.46 => ( ! [Uv: set @ A] : ( P @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) )
% 0.30/0.46 => ( ! [A4: set @ A,B4: set @ A] : ( P @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) )
% 0.30/0.46 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_nres.induct
% 0.30/0.46 thf(fact_125_less__eq__nres_Oinduct,axiom,
% 0.30/0.46 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ! [Uu: refine1665802226e_nres @ A] : ( P @ Uu @ ( refine1767639642_FAILi @ A ) )
% 0.30/0.46 => ( ! [A4: set @ A,B4: set @ A] : ( P @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) )
% 0.30/0.46 => ( ! [Uv: set @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) )
% 0.30/0.46 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_eq_nres.induct
% 0.30/0.46 thf(fact_126_nres_Osimps_I6_J,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,F1: C,F2: ( set @ A ) > C] :
% 0.30/0.46 ( ( refine1442219249c_nres @ C @ A @ F1 @ F2 @ ( refine1767639642_FAILi @ A ) )
% 0.30/0.46 = F1 ) ).
% 0.30/0.46
% 0.30/0.46 % nres.simps(6)
% 0.30/0.46 thf(fact_127_less__eq__nres_Osimps_I3_J,axiom,
% 0.30/0.46 ! [A: $tType,Uv2: set @ A] :
% 0.30/0.46 ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_eq_nres.simps(3)
% 0.30/0.46 thf(fact_128_Let__refine_H,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D2: $tType,B: $tType,M: A,M4: B,R: set @ ( product_prod @ A @ B ),F: A > ( refine1665802226e_nres @ C ),S: set @ ( product_prod @ C @ D2 ),F4: B > ( refine1665802226e_nres @ D2 )] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ M @ M4 ) @ R )
% 0.30/0.46 => ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ M @ M4 ) @ R )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( F @ M ) @ ( refine1073749519nc_fun @ C @ D2 @ S @ ( F4 @ M4 ) ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( F @ M ) @ ( refine1073749519nc_fun @ C @ D2 @ S @ ( F4 @ M4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Let_refine'
% 0.30/0.46 thf(fact_129_less__eq__nres_Oelims_I3_J,axiom,
% 0.30/0.46 ! [A: $tType,X4: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ Xa )
% 0.30/0.46 => ( ! [A4: set @ A] :
% 0.30/0.46 ( ( X4
% 0.30/0.46 = ( refine605929679le_RES @ A @ A4 ) )
% 0.30/0.46 => ! [B4: set @ A] :
% 0.30/0.46 ( ( Xa
% 0.30/0.46 = ( refine605929679le_RES @ A @ B4 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) )
% 0.30/0.46 => ~ ( ( X4
% 0.30/0.46 = ( refine1767639642_FAILi @ A ) )
% 0.30/0.46 => ! [Uv: set @ A] :
% 0.30/0.46 ( Xa
% 0.30/0.46 != ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_eq_nres.elims(3)
% 0.30/0.46 thf(fact_130_old_Oprod_Oinject,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B5: B] :
% 0.30/0.46 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.30/0.46 = ( product_Pair @ A @ B @ A7 @ B5 ) )
% 0.30/0.46 = ( ( A2 = A7 )
% 0.30/0.46 & ( B2 = B5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.prod.inject
% 0.30/0.46 thf(fact_131_prod_Oinject,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 0.30/0.46 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 0.30/0.46 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.30/0.46 = ( ( X1 = Y1 )
% 0.30/0.46 & ( X2 = Y2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod.inject
% 0.30/0.46 thf(fact_132_subrelI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S9: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ! [X3: A,Y4: B] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R3 )
% 0.30/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S9 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S9 ) ) ).
% 0.30/0.46
% 0.30/0.46 % subrelI
% 0.30/0.46 thf(fact_133_lift__assn__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( refine1580981607t_assn @ A @ B )
% 0.30/0.46 = ( ^ [R4: set @ ( product_prod @ A @ B ),Phi3: B > $o,S10: A] :
% 0.30/0.46 ? [S6: B] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S10 @ S6 ) @ R4 )
% 0.30/0.46 & ( Phi3 @ S6 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % lift_assn_def
% 0.30/0.46 thf(fact_134_surj__pair,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.30/0.46 ? [X3: A,Y4: B] :
% 0.30/0.46 ( P2
% 0.30/0.46 = ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % surj_pair
% 0.30/0.46 thf(fact_135_prod__cases,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.30/0.46 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.30/0.46 => ( P @ P2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_cases
% 0.30/0.46 thf(fact_136_Pair__inject,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B5: B] :
% 0.30/0.46 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.30/0.46 = ( product_Pair @ A @ B @ A7 @ B5 ) )
% 0.30/0.46 => ~ ( ( A2 = A7 )
% 0.30/0.46 => ( B2 != B5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Pair_inject
% 0.30/0.46 thf(fact_137_prod__cases3,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.46 ~ ! [A4: A,B4: B,C4: C] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_cases3
% 0.30/0.46 thf(fact_138_prod__cases4,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.30/0.46 ~ ! [A4: A,B4: B,C4: C,D3: D2] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B4 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_cases4
% 0.30/0.46 thf(fact_139_prod__cases5,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
% 0.30/0.46 ~ ! [A4: A,B4: B,C4: C,D3: D2,E4: E2] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C4 @ ( product_Pair @ D2 @ E2 @ D3 @ E4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_cases5
% 0.30/0.46 thf(fact_140_prod__cases6,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F7: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) )] :
% 0.30/0.46 ~ ! [A4: A,B4: B,C4: C,D3: D2,E4: E2,F6: F7] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F7 ) @ D3 @ ( product_Pair @ E2 @ F7 @ E4 @ F6 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_cases6
% 0.30/0.46 thf(fact_141_prod__cases7,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F7: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) )] :
% 0.30/0.46 ~ ! [A4: A,B4: B,C4: C,D3: D2,E4: E2,F6: F7,G4: G3] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F7 @ G3 ) @ E4 @ ( product_Pair @ F7 @ G3 @ F6 @ G4 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_cases7
% 0.30/0.46 thf(fact_142_prod__induct3,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.46 ( ! [A4: A,B4: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) )
% 0.30/0.46 => ( P @ X4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_induct3
% 0.30/0.46 thf(fact_143_prod__induct4,axiom,
% 0.30/0.46 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.30/0.46 ( ! [A4: A,B4: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B4 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
% 0.30/0.46 => ( P @ X4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_induct4
% 0.30/0.46 thf(fact_144_prod__induct5,axiom,
% 0.30/0.46 ! [E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
% 0.30/0.46 ( ! [A4: A,B4: B,C4: C,D3: D2,E4: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C4 @ ( product_Pair @ D2 @ E2 @ D3 @ E4 ) ) ) ) )
% 0.30/0.46 => ( P @ X4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_induct5
% 0.30/0.46 thf(fact_145_prod__induct6,axiom,
% 0.30/0.46 ! [F7: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) )] :
% 0.30/0.46 ( ! [A4: A,B4: B,C4: C,D3: D2,E4: E2,F6: F7] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F7 ) @ D3 @ ( product_Pair @ E2 @ F7 @ E4 @ F6 ) ) ) ) ) )
% 0.30/0.46 => ( P @ X4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_induct6
% 0.30/0.46 thf(fact_146_prod__induct7,axiom,
% 0.30/0.46 ! [G3: $tType,F7: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) )] :
% 0.30/0.46 ( ! [A4: A,B4: B,C4: C,D3: D2,E4: E2,F6: F7,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F7 @ G3 ) @ E4 @ ( product_Pair @ F7 @ G3 @ F6 @ G4 ) ) ) ) ) ) )
% 0.30/0.46 => ( P @ X4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_induct7
% 0.30/0.46 thf(fact_147_old_Oprod_Oexhaust,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 0.30/0.46 ~ ! [A4: A,B4: B] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.prod.exhaust
% 0.30/0.46 thf(fact_148_old_Oprod_Oinducts,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.30/0.46 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.30/0.46 => ( P @ Prod ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.prod.inducts
% 0.30/0.46 thf(fact_149_lift__assnI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,S9: A,S11: B,R: set @ ( product_prod @ A @ B ),Phi: B > $o] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S9 @ S11 ) @ R )
% 0.30/0.46 => ( ( Phi @ S11 )
% 0.30/0.46 => ( refine1580981607t_assn @ A @ B @ R @ Phi @ S9 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % lift_assnI
% 0.30/0.46 thf(fact_150_internal__case__prod__conv,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
% 0.30/0.46 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.30/0.46 = ( C2 @ A2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % internal_case_prod_conv
% 0.30/0.46 thf(fact_151_return__refine__prop__return,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,X4: B,R: set @ ( product_prod @ B @ A )] :
% 0.30/0.46 ( ( refine1102455758nofail @ A @ M )
% 0.30/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine1687780735RETURN @ B @ X4 ) @ ( refine1073749519nc_fun @ B @ A @ R @ M ) )
% 0.30/0.46 => ~ ! [X9: A] :
% 0.30/0.46 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ X9 ) @ R )
% 0.30/0.46 => ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X9 ) @ M ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % return_refine_prop_return
% 0.30/0.46 thf(fact_152_ret__le__down__conv,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,C2: B,R: set @ ( product_prod @ B @ A )] :
% 0.30/0.46 ( ( refine1102455758nofail @ A @ M )
% 0.30/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine1687780735RETURN @ B @ C2 ) @ ( refine1073749519nc_fun @ B @ A @ R @ M ) )
% 0.30/0.46 = ( ? [A3: A] :
% 0.30/0.46 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ C2 @ A3 ) @ R )
% 0.30/0.46 & ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ A3 ) @ M ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ret_le_down_conv
% 0.30/0.46 thf(fact_153_old_Oprod_Orec,axiom,
% 0.30/0.46 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.30/0.46 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 = ( F1 @ A2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.prod.rec
% 0.30/0.46 thf(fact_154_nres__more__simps_I6_J,axiom,
% 0.30/0.46 ! [A: $tType,X4: A,Y3: A] :
% 0.30/0.46 ( ( ( refine1687780735RETURN @ A @ X4 )
% 0.30/0.46 = ( refine1687780735RETURN @ A @ Y3 ) )
% 0.30/0.46 = ( X4 = Y3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres_more_simps(6)
% 0.30/0.46 thf(fact_155_nres__order__simps_I20_J,axiom,
% 0.30/0.46 ! [W: $tType,X4: W,Y3: W] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X4 ) @ ( refine1687780735RETURN @ W @ Y3 ) )
% 0.30/0.46 = ( X4 = Y3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres_order_simps(20)
% 0.30/0.46 thf(fact_156_nres__monad1,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X4: B,F: B > ( refine1665802226e_nres @ A )] :
% 0.30/0.46 ( ( refine463715084e_bind @ B @ A @ ( refine1687780735RETURN @ B @ X4 ) @ F )
% 0.30/0.46 = ( F @ X4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres_monad1
% 0.30/0.46 thf(fact_157_nres__monad2,axiom,
% 0.30/0.46 ! [A: $tType,M3: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( refine463715084e_bind @ A @ A @ M3 @ ( refine1687780735RETURN @ A ) )
% 0.30/0.46 = M3 ) ).
% 0.30/0.46
% 0.30/0.46 % nres_monad2
% 0.30/0.46 thf(fact_158_nofail__simps_I3_J,axiom,
% 0.30/0.46 ! [C: $tType,X4: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % nofail_simps(3)
% 0.30/0.46 thf(fact_159_inres__simps_I3_J,axiom,
% 0.30/0.46 ! [C: $tType,X4: C] :
% 0.30/0.46 ( ( refine1315500908_inres @ C @ ( refine1687780735RETURN @ C @ X4 ) )
% 0.30/0.46 = ( ^ [Y5: C,Z: C] : ( Y5 = Z )
% 0.30/0.46 @ X4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % inres_simps(3)
% 0.30/0.46 thf(fact_160_nres__order__simps_I21_J,axiom,
% 0.30/0.46 ! [X11: $tType,X4: X11,Y: set @ X11] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ X11 ) @ ( refine1687780735RETURN @ X11 @ X4 ) @ ( refine605929679le_RES @ X11 @ Y ) )
% 0.30/0.46 = ( member @ X11 @ X4 @ Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres_order_simps(21)
% 0.30/0.46 thf(fact_161_SPEC__eq__is__RETURN_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,X4: A] :
% 0.30/0.46 ( ( refine605929679le_RES @ A
% 0.30/0.46 @ ( collect @ A
% 0.30/0.46 @ ( ^ [Y5: A,Z: A] : ( Y5 = Z )
% 0.30/0.46 @ X4 ) ) )
% 0.30/0.46 = ( refine1687780735RETURN @ A @ X4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % SPEC_eq_is_RETURN(1)
% 0.30/0.46 thf(fact_162_RETURN__rule,axiom,
% 0.30/0.46 ! [A: $tType,Phi: A > $o,X4: A] :
% 0.30/0.46 ( ( Phi @ X4 )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X4 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % RETURN_rule
% 0.30/0.46 thf(fact_163_lhs__step__RES,axiom,
% 0.30/0.46 ! [A: $tType,X: set @ A,M: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ X )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ X ) @ M ) ) ).
% 0.30/0.46
% 0.30/0.46 % lhs_step_RES
% 0.30/0.46 thf(fact_164_RETURN__to__SPEC__rule,axiom,
% 0.30/0.46 ! [A: $tType,M: refine1665802226e_nres @ A,V2: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M
% 0.30/0.46 @ ( refine605929679le_RES @ A
% 0.30/0.46 @ ( collect @ A
% 0.30/0.46 @ ( ^ [Y5: A,Z: A] : ( Y5 = Z )
% 0.30/0.46 @ V2 ) ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine1687780735RETURN @ A @ V2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % RETURN_to_SPEC_rule
% 0.30/0.46 thf(fact_165_if__RETURN__refine,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B2: $o,B5: $o,S1: A,R: set @ ( product_prod @ A @ B ),S13: refine1665802226e_nres @ B,S2: A,S23: refine1665802226e_nres @ B] :
% 0.30/0.46 ( ( B2 = B5 )
% 0.30/0.46 => ( ( B2
% 0.30/0.46 => ( B5
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ S1 ) @ ( refine1073749519nc_fun @ A @ B @ R @ S13 ) ) ) )
% 0.30/0.46 => ( ( ~ B2
% 0.30/0.46 => ( ~ B5
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ S2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ S23 ) ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ ( if @ A @ B2 @ S1 @ S2 ) ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( if @ ( refine1665802226e_nres @ B ) @ B5 @ S13 @ S23 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % if_RETURN_refine
% 0.30/0.46 thf(fact_166_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,M3: refine1665802226e_nres @ A,M6: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ M6 )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M3 )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X3 ) @ ( F4 @ X3 ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M3 @ F ) @ ( refine463715084e_bind @ A @ B @ M6 @ F4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(1)
% 0.30/0.46 thf(fact_167_bind__cong,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M4: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
% 0.30/0.46 ( ( M = M4 )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M4 )
% 0.30/0.46 => ( ( F @ X3 )
% 0.30/0.46 = ( F4 @ X3 ) ) )
% 0.30/0.46 => ( ( refine463715084e_bind @ A @ B @ M @ F )
% 0.30/0.46 = ( refine463715084e_bind @ A @ B @ M4 @ F4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bind_cong
% 0.30/0.46 thf(fact_168_inres__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( refine1315500908_inres @ A )
% 0.30/0.46 = ( ^ [S4: refine1665802226e_nres @ A,X5: A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X5 ) @ S4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inres_def
% 0.30/0.46 thf(fact_169_RETURN__refine,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X4: A,X7: B,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ X7 ) @ R )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X4 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1687780735RETURN @ B @ X7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % RETURN_refine
% 0.30/0.46 thf(fact_170_RETURN__ref__RETURND,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C2: A,R: set @ ( product_prod @ A @ B ),A2: B] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ C2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1687780735RETURN @ B @ A2 ) ) )
% 0.30/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ C2 @ A2 ) @ R ) ) ).
% 0.30/0.46
% 0.30/0.46 % RETURN_ref_RETURND
% 0.30/0.46 thf(fact_171_RETURN__ref__SPECD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C2: A,R: set @ ( product_prod @ A @ B ),Phi: B > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ C2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine605929679le_RES @ B @ ( collect @ B @ Phi ) ) ) )
% 0.30/0.46 => ~ ! [A4: B] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ C2 @ A4 ) @ R )
% 0.30/0.46 => ~ ( Phi @ A4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % RETURN_ref_SPECD
% 0.30/0.46 thf(fact_172_RETURN__SPEC__refine,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X4: B,R: set @ ( product_prod @ B @ A ),Phi: A > $o] :
% 0.30/0.46 ( ? [X12: A] :
% 0.30/0.46 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ X12 ) @ R )
% 0.30/0.46 & ( Phi @ X12 ) )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine1687780735RETURN @ B @ X4 ) @ ( refine1073749519nc_fun @ B @ A @ R @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % RETURN_SPEC_refine
% 0.30/0.46 thf(fact_173_pw__abs__inres,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M3: refine1665802226e_nres @ B,A2: A] :
% 0.30/0.46 ( ( refine1315500908_inres @ A @ ( refine81118332bs_fun @ B @ A @ R @ M3 ) @ A2 )
% 0.30/0.46 = ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M3 ) )
% 0.30/0.46 => ? [C5: B] :
% 0.30/0.46 ( ( refine1315500908_inres @ B @ M3 @ C5 )
% 0.30/0.46 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ C5 @ A2 ) @ R ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_abs_inres
% 0.30/0.46 thf(fact_174_relChain__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B )
% 0.30/0.46 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.30/0.46 = ( ^ [R5: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.30/0.46 ! [I2: A,J: A] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R5 )
% 0.30/0.46 => ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relChain_def
% 0.30/0.46 thf(fact_175_uncurry__apply,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
% 0.30/0.46 ( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.30/0.46 = ( F @ A2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % uncurry_apply
% 0.30/0.46 thf(fact_176_abs__trans__additional_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A5: refine1665802226e_nres @ A,B6: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C3: refine1665802226e_nres @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A5 @ B6 )
% 0.30/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ B6 ) @ C3 )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A5 ) @ C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_trans_additional(1)
% 0.30/0.46 thf(fact_177_abs__trans,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),C3: refine1665802226e_nres @ B,B6: refine1665802226e_nres @ A,R2: set @ ( product_prod @ A @ C ),A5: refine1665802226e_nres @ C] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine81118332bs_fun @ B @ A @ R @ C3 ) @ B6 )
% 0.30/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ B6 ) @ A5 )
% 0.30/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ ( refine81118332bs_fun @ B @ A @ R @ C3 ) ) @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_trans
% 0.30/0.46 thf(fact_178_uncurry__curry__id,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.30/0.46 ( ( uncurry @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 0.30/0.46 = F ) ).
% 0.30/0.46
% 0.30/0.46 % uncurry_curry_id
% 0.30/0.46 thf(fact_179_curry__uncurry__id,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 0.30/0.46 ( ( product_curry @ A @ B @ C @ ( uncurry @ A @ B @ C @ F ) )
% 0.30/0.46 = F ) ).
% 0.30/0.46
% 0.30/0.46 % curry_uncurry_id
% 0.30/0.46 thf(fact_180_pw__abs__nofail,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M3: refine1665802226e_nres @ B] :
% 0.30/0.46 ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M3 ) )
% 0.30/0.46 = ( ( refine1102455758nofail @ B @ M3 )
% 0.30/0.46 & ! [X5: B] :
% 0.30/0.46 ( ( refine1315500908_inres @ B @ M3 @ X5 )
% 0.30/0.46 => ( member @ B @ X5 @ ( domain @ B @ A @ R ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_abs_nofail
% 0.30/0.46 thf(fact_181_curryI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.30/0.46 ( ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 => ( product_curry @ A @ B @ $o @ F @ A2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % curryI
% 0.30/0.46 thf(fact_182_curry__conv,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.46 ( ( product_curry @ B @ C @ A )
% 0.30/0.46 = ( ^ [F3: ( product_prod @ B @ C ) > A,A3: B,B3: C] : ( F3 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % curry_conv
% 0.30/0.46 thf(fact_183_curryE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.30/0.46 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 0.30/0.46 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % curryE
% 0.30/0.46 thf(fact_184_curryD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.30/0.46 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 0.30/0.46 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % curryD
% 0.30/0.46 thf(fact_185_DomainE,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A,R3: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) )
% 0.30/0.46 => ~ ! [B4: B] :
% 0.30/0.46 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B4 ) @ R3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % DomainE
% 0.30/0.46 thf(fact_186_Domain__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A,R3: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) )
% 0.30/0.46 = ( ? [Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ Y6 ) @ R3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domain_iff
% 0.30/0.46 thf(fact_187_Domain_Ocases,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A,R3: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) )
% 0.30/0.46 => ~ ! [B4: B] :
% 0.30/0.46 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B4 ) @ R3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domain.cases
% 0.30/0.46 thf(fact_188_Domain_Osimps,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A,R3: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) )
% 0.30/0.46 = ( ? [A3: A,B3: B] :
% 0.30/0.46 ( ( A2 = A3 )
% 0.30/0.46 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domain.simps
% 0.30/0.46 thf(fact_189_Domain_ODomainI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A,B2: B,R3: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R3 )
% 0.30/0.46 => ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domain.DomainI
% 0.30/0.46 thf(fact_190_Domain_Oinducts,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X4: A,R3: set @ ( product_prod @ A @ B ),P: A > $o] :
% 0.30/0.46 ( ( member @ A @ X4 @ ( domain @ A @ B @ R3 ) )
% 0.30/0.46 => ( ! [A4: A,B4: B] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R3 )
% 0.30/0.46 => ( P @ A4 ) )
% 0.30/0.46 => ( P @ X4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domain.inducts
% 0.30/0.46 thf(fact_191_Domain__mono,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S9: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S9 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R3 ) @ ( domain @ A @ B @ S9 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domain_mono
% 0.30/0.46 thf(fact_192_for__in__RI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X4: A,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( member @ A @ X4 @ ( domain @ A @ B @ R ) )
% 0.30/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ ( fun_of_rel @ A @ B @ R @ X4 ) ) @ R ) ) ).
% 0.30/0.46
% 0.30/0.46 % for_in_RI
% 0.30/0.46 thf(fact_193_abs__fun__simps_I3_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: set @ B,R: set @ ( product_prod @ B @ A )] :
% 0.30/0.46 ( ~ ( ord_less_eq @ ( set @ B ) @ X @ ( domain @ B @ A @ R ) )
% 0.30/0.46 => ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X ) )
% 0.30/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_fun_simps(3)
% 0.30/0.46 thf(fact_194_abs__fun__simps_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X: set @ B,R: set @ ( product_prod @ B @ A )] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ B ) @ X @ ( domain @ B @ A @ R ) )
% 0.30/0.46 => ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X ) )
% 0.30/0.46 = ( refine605929679le_RES @ A @ ( image @ B @ A @ R @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_fun_simps(2)
% 0.30/0.46 thf(fact_195_top__apply,axiom,
% 0.30/0.46 ! [C: $tType,D2: $tType] :
% 0.30/0.46 ( ( top @ C )
% 0.30/0.46 => ( ( top_top @ ( D2 > C ) )
% 0.30/0.46 = ( ^ [X5: D2] : ( top_top @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_apply
% 0.30/0.46 thf(fact_196_ImageI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A,B2: B,R3: set @ ( product_prod @ A @ B ),A5: set @ A] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R3 )
% 0.30/0.46 => ( ( member @ A @ A2 @ A5 )
% 0.30/0.46 => ( member @ B @ B2 @ ( image @ A @ B @ R3 @ A5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ImageI
% 0.30/0.46 thf(fact_197_nres__order__simps_I4_J,axiom,
% 0.30/0.46 ! [D2: $tType,M3: refine1665802226e_nres @ D2] :
% 0.30/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ D2 ) @ ( top_top @ ( refine1665802226e_nres @ D2 ) ) @ M3 )
% 0.30/0.46 = ( M3
% 0.30/0.46 = ( top_top @ ( refine1665802226e_nres @ D2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres_order_simps(4)
% 0.30/0.46 thf(fact_198_conc__fun__FAIL,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( refine1073749519nc_fun @ A @ B @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
% 0.30/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % conc_fun_FAIL
% 0.30/0.46 thf(fact_199_conc__fun__fail__iff_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 0.30/0.46 ( ( ( refine1073749519nc_fun @ A @ B @ R @ S )
% 0.30/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 0.30/0.46 = ( S
% 0.30/0.46 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % conc_fun_fail_iff(1)
% 0.30/0.46 thf(fact_200_conc__fun__fail__iff_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 0.30/0.46 ( ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 0.30/0.46 = ( refine1073749519nc_fun @ A @ B @ R @ S ) )
% 0.30/0.46 = ( S
% 0.30/0.46 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % conc_fun_fail_iff(2)
% 0.30/0.46 thf(fact_201_bind__FAIL,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: B > ( refine1665802226e_nres @ A )] :
% 0.30/0.46 ( ( refine463715084e_bind @ B @ A @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ F )
% 0.30/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bind_FAIL
% 0.30/0.46 thf(fact_202_nofail__simps_I1_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ~ ( refine1102455758nofail @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nofail_simps(1)
% 0.30/0.46 thf(fact_203_inres__simps_I1_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( refine1315500908_inres @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 0.30/0.46 = ( ^ [Uu3: A] : $true ) ) ).
% 0.30/0.46
% 0.30/0.46 % inres_simps(1)
% 0.30/0.46 thf(fact_204_nres__simp__internals_I2_J,axiom,
% 0.30/0.46 ! [B: $tType] :
% 0.30/0.46 ( ( refine1767639642_FAILi @ B )
% 0.30/0.46 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres_simp_internals(2)
% 0.30/0.46 thf(fact_205_abs__fun__simps_I1_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
% 0.30/0.46 ( ( refine81118332bs_fun @ B @ A @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
% 0.30/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_fun_simps(1)
% 0.30/0.46 thf(fact_206_FAIL__refine,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X @ ( refine1073749519nc_fun @ A @ B @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % FAIL_refine
% 0.30/0.46 thf(fact_207_rev__ImageI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A,A5: set @ A,B2: B,R3: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( member @ A @ A2 @ A5 )
% 0.30/0.46 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R3 )
% 0.30/0.46 => ( member @ B @ B2 @ ( image @ A @ B @ R3 @ A5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_ImageI
% 0.30/0.46 thf(fact_208_Image__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A5: set @ B] :
% 0.30/0.46 ( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ A5 ) )
% 0.30/0.46 = ( ? [X5: B] :
% 0.30/0.46 ( ( member @ B @ X5 @ A5 )
% 0.30/0.46 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X5 @ B2 ) @ R3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Image_iff
% 0.30/0.46 thf(fact_209_ImageE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A5: set @ B] :
% 0.30/0.46 ( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ A5 ) )
% 0.30/0.46 => ~ ! [X3: B] :
% 0.30/0.46 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B2 ) @ R3 )
% 0.30/0.46 => ~ ( member @ B @ X3 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ImageE
% 0.30/0.46 thf(fact_210_nres__inequalities_I3_J,axiom,
% 0.30/0.46 ! [C: $tType,X4: C] :
% 0.30/0.46 ( ( top_top @ ( refine1665802226e_nres @ C ) )
% 0.30/0.46 != ( refine1687780735RETURN @ C @ X4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres_inequalities(3)
% 0.30/0.46 thf(fact_211_top__nres__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 0.30/0.46 = ( refine1767639642_FAILi @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_nres_def
% 0.30/0.46 thf(fact_212_top__greatest,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_top @ A )
% 0.30/0.46 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_greatest
% 0.30/0.46 thf(fact_213_top_Oextremum__unique,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_top @ A )
% 0.30/0.46 => ! [A2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.30/0.46 = ( A2
% 0.30/0.46 = ( top_top @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top.extremum_unique
% 0.30/0.46 thf(fact_214_top_Oextremum__uniqueI,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_top @ A )
% 0.30/0.46 => ! [A2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.30/0.46 => ( A2
% 0.30/0.46 = ( top_top @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top.extremum_uniqueI
% 0.30/0.46 thf(fact_215_nres__order__simps_I3_J,axiom,
% 0.30/0.46 ! [C: $tType,M3: refine1665802226e_nres @ C] : ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ M3 @ ( top_top @ ( refine1665802226e_nres @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres_order_simps(3)
% 0.30/0.46 thf(fact_216_nres__inequalities_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,X: set @ A] :
% 0.30/0.46 ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 0.30/0.46 != ( refine605929679le_RES @ A @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres_inequalities(1)
% 0.30/0.46 thf(fact_217_nres__cases,axiom,
% 0.30/0.46 ! [A: $tType,M3: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( M3
% 0.30/0.46 != ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 0.30/0.46 => ~ ! [X13: set @ A] :
% 0.30/0.46 ( M3
% 0.30/0.46 != ( refine605929679le_RES @ A @ X13 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nres_cases
% 0.30/0.46 thf(fact_218_not__nofail__iff,axiom,
% 0.30/0.46 ! [A: $tType,S: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( ~ ( refine1102455758nofail @ A @ S ) )
% 0.30/0.46 = ( S
% 0.30/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % not_nofail_iff
% 0.30/0.46 thf(fact_219_nofail__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( refine1102455758nofail @ A )
% 0.30/0.46 = ( ^ [S4: refine1665802226e_nres @ A] :
% 0.30/0.46 ( S4
% 0.30/0.46 != ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nofail_def
% 0.30/0.46 thf(fact_220_intro__nofail_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,S: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 0.30/0.46 != S )
% 0.30/0.46 = ( refine1102455758nofail @ A @ S ) ) ).
% 0.30/0.46
% 0.30/0.46 % intro_nofail(2)
% 0.30/0.46 thf(fact_221_ibind__strict_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,F: product_unit > ( refine1665802226e_nres @ A )] :
% 0.30/0.46 ( ( refine463715084e_bind @ product_unit @ A @ ( top_top @ ( refine1665802226e_nres @ product_unit ) ) @ F )
% 0.30/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ibind_strict(2)
% 0.30/0.46 thf(fact_222_Image__mono,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R6: set @ ( product_prod @ A @ B ),R3: set @ ( product_prod @ A @ B ),A8: set @ A,A5: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R6 @ R3 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A8 @ A5 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R6 @ A8 ) @ ( image @ A @ B @ R3 @ A5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Image_mono
% 0.30/0.46 thf(fact_223_meta__le__everything__if__top,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_top @ A )
% 0.30/0.46 => ! [M: A,X4: A] :
% 0.30/0.46 ( ( M
% 0.30/0.46 = ( top_top @ A ) )
% 0.30/0.46 => ( ord_less_eq @ A @ X4 @ M ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % meta_le_everything_if_top
% 0.30/0.46 thf(fact_224_pw__flat__ge__iff,axiom,
% 0.30/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( top_top @ ( refine1665802226e_nres @ A ) ) @ S @ S3 )
% 0.30/0.46 = ( ( refine1102455758nofail @ A @ S )
% 0.30/0.46 => ( ( refine1102455758nofail @ A @ S3 )
% 0.30/0.46 & ! [X5: A] :
% 0.30/0.46 ( ( refine1315500908_inres @ A @ S @ X5 )
% 0.30/0.46 = ( refine1315500908_inres @ A @ S3 @ X5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_flat_ge_iff
% 0.30/0.46 thf(fact_225_subset__UNIV,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_UNIV
% 0.30/0.46 thf(fact_226_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I2_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,M3: refine1665802226e_nres @ A,M6: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
% 0.30/0.46 ( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( top_top @ ( refine1665802226e_nres @ A ) ) @ M3 @ M6 )
% 0.30/0.46 => ( ! [X3: A] : ( partial_flat_ord @ ( refine1665802226e_nres @ B ) @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ ( F @ X3 ) @ ( F4 @ X3 ) )
% 0.30/0.46 => ( partial_flat_ord @ ( refine1665802226e_nres @ B ) @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ ( refine463715084e_bind @ A @ B @ M3 @ F ) @ ( refine463715084e_bind @ A @ B @ M6 @ F4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(2)
% 0.30/0.46 thf(fact_227_flat__ord__compat_I2_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [X4: A,Y3: A] :
% 0.30/0.46 ( ( partial_flat_ord @ A @ ( top_top @ A ) @ X4 @ Y3 )
% 0.30/0.46 => ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % flat_ord_compat(2)
% 0.30/0.46 thf(fact_228_imonad1,axiom,
% 0.30/0.46 ! [A: $tType,U: product_unit,F: product_unit > ( refine1665802226e_nres @ A )] :
% 0.30/0.46 ( ( refine463715084e_bind @ product_unit @ A @ ( refine1687780735RETURN @ product_unit @ U ) @ F )
% 0.30/0.46 = ( F @ U ) ) ).
% 0.30/0.46
% 0.30/0.46 % imonad1
% 0.30/0.46 thf(fact_229_flat__ord_OantimonoD,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( order @ B )
% 0.30/0.46 => ! [B2: A,F: A > B,X4: A,Y3: A] :
% 0.30/0.46 ( ( antimono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
% 0.30/0.46 => ( ( partial_flat_ord @ A @ B2 @ X4 @ Y3 )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % flat_ord.antimonoD
% 0.30/0.46 thf(fact_230_flat__ord_OantimonoE,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( order @ B )
% 0.30/0.46 => ! [B2: A,F: A > B,X4: A,Y3: A] :
% 0.30/0.46 ( ( antimono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
% 0.30/0.46 => ( ( partial_flat_ord @ A @ B2 @ X4 @ Y3 )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % flat_ord.antimonoE
% 0.30/0.46 thf(fact_231_UNIV__I,axiom,
% 0.30/0.46 ! [A: $tType,X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UNIV_I
% 0.30/0.46 thf(fact_232_relprop__UNIV__orient,axiom,
% 0.30/0.46 ! [A: $tType,R: set @ A] :
% 0.30/0.46 ( ( R
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 => ( ( top_top @ ( set @ A ) )
% 0.30/0.46 = R ) ) ).
% 0.30/0.46
% 0.30/0.46 % relprop_UNIV_orient
% 0.30/0.46 thf(fact_233_eq__UNIV__iff,axiom,
% 0.30/0.46 ! [A: $tType,S: set @ A] :
% 0.30/0.46 ( ( S
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 = ( ! [X5: A] : ( member @ A @ X5 @ S ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_UNIV_iff
% 0.30/0.46 thf(fact_234_UNIV__witness,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UNIV_witness
% 0.30/0.46 thf(fact_235_top__set__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( top_top @ ( set @ A ) )
% 0.30/0.46 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_set_def
% 0.30/0.46 thf(fact_236_UNIV__eq__I,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ! [X3: A] : ( member @ A @ X3 @ A5 )
% 0.30/0.46 => ( ( top_top @ ( set @ A ) )
% 0.30/0.46 = A5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % UNIV_eq_I
% 0.30/0.46 thf(fact_237_top__empty__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( top_top @ ( A > $o ) )
% 0.30/0.46 = ( ^ [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_empty_eq
% 0.30/0.46 thf(fact_238_order_Oantimono_Ocong,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( order @ B )
% 0.30/0.46 => ( ( antimono @ A @ B )
% 0.30/0.46 = ( antimono @ A @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order.antimono.cong
% 0.30/0.46 thf(fact_239_flat__ord_Oantimono__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( order @ B )
% 0.30/0.46 => ! [B2: A,F: A > B] :
% 0.30/0.46 ( ( antimono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
% 0.30/0.46 = ( ! [X5: A,Y6: A] :
% 0.30/0.46 ( ( partial_flat_ord @ A @ B2 @ X5 @ Y6 )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ Y6 ) @ ( F @ X5 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % flat_ord.antimono_def
% 0.30/0.46 thf(fact_240_flat__ord_OantimonoI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( order @ B )
% 0.30/0.46 => ! [B2: A,F: A > B] :
% 0.30/0.46 ( ! [X3: A,Y4: A] :
% 0.30/0.46 ( ( partial_flat_ord @ A @ B2 @ X3 @ Y4 )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X3 ) ) )
% 0.30/0.46 => ( antimono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % flat_ord.antimonoI
% 0.30/0.46 thf(fact_241_flat__ord_Omono__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( order @ B )
% 0.30/0.46 => ! [B2: A,F: A > B] :
% 0.30/0.46 ( ( mono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
% 0.30/0.46 = ( ! [X5: A,Y6: A] :
% 0.30/0.46 ( ( partial_flat_ord @ A @ B2 @ X5 @ Y6 )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y6 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % flat_ord.mono_def
% 0.30/0.46 thf(fact_242_flat__ord_OmonoI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( order @ B )
% 0.30/0.46 => ! [B2: A,F: A > B] :
% 0.30/0.46 ( ! [X3: A,Y4: A] :
% 0.30/0.46 ( ( partial_flat_ord @ A @ B2 @ X3 @ Y4 )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.30/0.46 => ( mono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % flat_ord.monoI
% 0.30/0.46 thf(fact_243_top1I,axiom,
% 0.30/0.46 ! [A: $tType,X4: A] : ( top_top @ ( A > $o ) @ X4 ) ).
% 0.30/0.46
% 0.30/0.46 % top1I
% 0.30/0.46 thf(fact_244_order_Omono_Ocong,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( order @ B )
% 0.30/0.46 => ( ( mono @ A @ B )
% 0.30/0.46 = ( mono @ A @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order.mono.cong
% 0.30/0.46 thf(fact_245_flat__ord_OmonoD,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( order @ B )
% 0.30/0.46 => ! [B2: A,F: A > B,X4: A,Y3: A] :
% 0.30/0.46 ( ( mono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
% 0.30/0.46 => ( ( partial_flat_ord @ A @ B2 @ X4 @ Y3 )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % flat_ord.monoD
% 0.30/0.46 thf(fact_246_flat__ord_OmonoE,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( order @ B )
% 0.30/0.46 => ! [B2: A,F: A > B,X4: A,Y3: A] :
% 0.30/0.46 ( ( mono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
% 0.30/0.46 => ( ( partial_flat_ord @ A @ B2 @ X4 @ Y3 )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % flat_ord.monoE
% 0.30/0.46 thf(fact_247_pw__flat__le__iff,axiom,
% 0.30/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A] :
% 0.30/0.46 ( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( bot_bot @ ( refine1665802226e_nres @ A ) ) @ S @ S3 )
% 0.30/0.46 = ( ? [X14: A] : ( refine1315500908_inres @ A @ S @ X14 )
% 0.30/0.46 => ( ( ( refine1102455758nofail @ A @ S )
% 0.30/0.46 = ( refine1102455758nofail @ A @ S3 ) )
% 0.30/0.46 & ! [X5: A] :
% 0.30/0.46 ( ( refine1315500908_inres @ A @ S @ X5 )
% 0.30/0.46 = ( refine1315500908_inres @ A @ S3 @ X5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pw_flat_le_iff
% 0.30/0.46 thf(fact_248_ASSERT__simps_I2_J,axiom,
% 0.30/0.46 ( ( refine1814851989ASSERT @ $false )
% 0.30/0.46 = ( top_top @ ( refine1665802226e_nres @ product_unit ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ASSERT_simps(2)
% 0.30/0.46 thf(fact_249_bot__apply,axiom,
% 0.30/0.46 ! [C: $tType,D2: $tType] :
% 0.30/0.46 ( ( bot @ C )
% 0.30/0.46 => ( ( bot_bot @ ( D2 > C ) )
% 0.30/0.46 = ( ^ [X5: D2] : ( bot_bot @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot_apply
% 0.30/0.46
% 0.30/0.46 % Type constructors (37)
% 0.30/0.46 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A10 )
% 0.30/0.46 => ( comple187826305attice @ ( A9 > A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( order_top @ A10 )
% 0.30/0.46 => ( order_top @ ( A9 > A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( preorder @ A10 )
% 0.30/0.46 => ( preorder @ ( A9 > A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( order @ A10 )
% 0.30/0.46 => ( order @ ( A9 > A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Otop,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( top @ A10 )
% 0.30/0.46 => ( top @ ( A9 > A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( ord @ A10 )
% 0.30/0.46 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Obot,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( bot @ A10 )
% 0.30/0.46 => ( bot @ ( A9 > A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_1,axiom,
% 0.30/0.46 ! [A9: $tType] : ( comple187826305attice @ ( set @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
% 0.30/0.46 ! [A9: $tType] : ( order_top @ ( set @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
% 0.30/0.46 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
% 0.30/0.46 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Otop_5,axiom,
% 0.30/0.46 ! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 0.30/0.46 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Obot_7,axiom,
% 0.30/0.46 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_8,axiom,
% 0.30/0.46 comple187826305attice @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__top_9,axiom,
% 0.30/0.46 order_top @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 0.30/0.46 preorder @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.30/0.46 linorder @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 0.30/0.46 order @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Otop_12,axiom,
% 0.30/0.46 top @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
% 0.30/0.46 ord @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Obot_14,axiom,
% 0.30/0.46 bot @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_15,axiom,
% 0.30/0.46 comple187826305attice @ product_unit ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_16,axiom,
% 0.30/0.46 order_top @ product_unit ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_17,axiom,
% 0.30/0.46 preorder @ product_unit ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_18,axiom,
% 0.30/0.46 linorder @ product_unit ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder_19,axiom,
% 0.30/0.46 order @ product_unit ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Product__Type_Ounit___Orderings_Otop_20,axiom,
% 0.30/0.46 top @ product_unit ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oord_21,axiom,
% 0.30/0.46 ord @ product_unit ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Product__Type_Ounit___Orderings_Obot_22,axiom,
% 0.30/0.46 bot @ product_unit ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Complete__Lattices_Ocomplete__lattice_23,axiom,
% 0.30/0.46 ! [A9: $tType] : ( comple187826305attice @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__top_24,axiom,
% 0.30/0.46 ! [A9: $tType] : ( order_top @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_25,axiom,
% 0.30/0.46 ! [A9: $tType] : ( preorder @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_26,axiom,
% 0.30/0.46 ! [A9: $tType] : ( order @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_27,axiom,
% 0.30/0.46 ! [A9: $tType] : ( top @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_28,axiom,
% 0.30/0.46 ! [A9: $tType] : ( ord @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Obot_29,axiom,
% 0.30/0.46 ! [A9: $tType] : ( bot @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Helper facts (3)
% 0.30/0.46 thf(help_If_3_1_T,axiom,
% 0.30/0.46 ! [P: $o] :
% 0.30/0.46 ( ( P = $true )
% 0.30/0.46 | ( P = $false ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(help_If_2_1_T,axiom,
% 0.30/0.46 ! [A: $tType,X4: A,Y3: A] :
% 0.30/0.46 ( ( if @ A @ $false @ X4 @ Y3 )
% 0.30/0.46 = Y3 ) ).
% 0.30/0.46
% 0.30/0.46 thf(help_If_1_1_T,axiom,
% 0.30/0.46 ! [A: $tType,X4: A,Y3: A] :
% 0.30/0.46 ( ( if @ A @ $true @ X4 @ Y3 )
% 0.30/0.46 = X4 ) ).
% 0.30/0.46
% 0.30/0.46 % Conjectures (1)
% 0.30/0.46 thf(conj_0,conjecture,
% 0.30/0.46 ord_less_eq @ ( refine1665802226e_nres @ b ) @ m @ ( refine463715084e_bind @ a @ b @ ( refine605929679le_RES @ a @ ( collect @ a @ phi ) ) @ f ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.90/1.56 ---- Embedded in THF ---
% 2.90/1.56 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.90/1.56 %%% Generated on Tue Feb 24 08:28:17 EST 2026
% 2.90/1.56 %%% using '$$dhol' embedding, version 1.3.0.
% 2.90/1.56 %%% Logic specification used:
% 2.90/1.56 %%% thf(spec, logic, $$dhol).
% 2.90/1.56
% 2.90/1.56 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.8D84XgAAWH/DTF2DTF13304.p
% 2.90/1.56 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres, type, refine1665802226e_nres: ($tType > $tType)).
% 2.90/1.56 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 2.90/1.56 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.90/1.56 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.90/1.56 thf(ty_tf_b, type, b: $tType).
% 2.90/1.56 thf(ty_tf_a, type, a: $tType).
% 2.90/1.56 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.90/1.56 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 2.90/1.56 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.90/1.56 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.90/1.56 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.90/1.56 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.90/1.56 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.90/1.56 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 2.90/1.56 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: ($o))).
% 2.90/1.56 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 2.90/1.56 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.90/1.56 thf(sy_c_Misc_Ofun__of__rel, type, fun_of_rel: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ A)) > (B > A))))).
% 2.90/1.56 thf(sy_c_Misc_Ouncurry, type, uncurry: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.90/1.56 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.90/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.90/1.56 thf(sy_c_Orderings_Oorder_Oantimono, type, antimono: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > $o))))).
% 2.90/1.56 thf(sy_c_Orderings_Oorder_Omono, type, mono: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > $o))))).
% 2.90/1.56 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 2.90/1.56 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.90/1.56 thf(sy_c_Partial__Function_Oflat__ord, type, partial_flat_ord: (!> [A:$tType]: ((A > (A > (A > $o)))))).
% 2.90/1.56 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.90/1.56 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 2.90/1.56 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.90/1.56 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.90/1.56 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_OASSERT, type, refine1814851989ASSERT: ($o > (refine1665802226e_nres @ product_unit))).
% 2.90/1.56 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN, type, refine1687780735RETURN: (!> [A:$tType]: ((A > (refine1665802226e_nres @ A))))).
% 2.90/1.56 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oabs__fun, type, refine81118332bs_fun: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ A)) > ((refine1665802226e_nres @ B) > (refine1665802226e_nres @ A)))))).
% 2.90/1.56 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind, type, refine463715084e_bind: (!> [B:$tType,A:$tType]: (((refine1665802226e_nres @ B) > ((B > (refine1665802226e_nres @ A)) > (refine1665802226e_nres @ A)))))).
% 2.90/1.56 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oconc__fun, type, refine1073749519nc_fun: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > ((refine1665802226e_nres @ B) > (refine1665802226e_nres @ A)))))).
% 2.90/1.56 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres, type, refine1315500908_inres: (!> [A:$tType]: (((refine1665802226e_nres @ A) > (A > $o))))).
% 2.90/1.56 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Olift__assn, type, refine1580981607t_assn: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > ((B > $o) > (A > $o)))))).
% 2.90/1.56 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres, type, refine406925620_inres: (!> [A:$tType]: (((refine1665802226e_nres @ A) > (A > $o))))).
% 2.90/1.56 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail, type, refine1102455758nofail: (!> [A:$tType]: (((refine1665802226e_nres @ A) > $o)))).
% 2.90/1.56 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_OFAILi, type, refine1767639642_FAILi: (!> [A:$tType]: ((refine1665802226e_nres @ A)))).
% 2.90/1.56 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES, type, refine605929679le_RES: (!> [A:$tType]: (((set @ A) > (refine1665802226e_nres @ A))))).
% 2.90/1.56 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_Orec__nres, type, refine1442219249c_nres: (!> [C:$tType,A:$tType]: ((C > (((set @ A) > C) > ((refine1665802226e_nres @ A) > C)))))).
% 2.90/1.56 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES, type, refine1672542526he_RES: (!> [A:$tType]: (((refine1665802226e_nres @ A) > (set @ A))))).
% 2.90/1.56 thf(sy_c_Relation_ODomain, type, domain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ A))))).
% 2.90/1.56 thf(sy_c_Relation_OImage, type, image: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > ((set @ A) > (set @ B)))))).
% 2.90/1.56 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.90/1.56 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.90/1.56 thf(sy_v__092_060Phi_062, type, phi: (a > $o)).
% 2.90/1.56 thf(sy_v_f, type, f: (a > (refine1665802226e_nres @ b))).
% 2.90/1.56 thf(sy_v_m, type, m: (refine1665802226e_nres @ b)).
% 2.90/1.56 thf(sy_v_x, type, x: a).
% 2.90/1.56 thf(fact_0_assms_I1_J, axiom, (phi @ x)).
% 2.90/1.56 thf(fact_1_assms_I2_J, axiom, (((ord_less_eq @ (refine1665802226e_nres @ b)) @ m) @ (f @ x))).
% 2.90/1.56 thf(fact_2_nres__more__simps_I4_J, axiom, (! [A:$tType,X:(set @ A),Y:(set @ A)]: (((((refine605929679le_RES @ A) @ X) = ((refine605929679le_RES @ A) @ Y)) = (X = Y))))).
% 2.90/1.56 thf(fact_3_nres_Oinject, axiom, (! [A:$tType,X2:(set @ A),Y2:(set @ A)]: (((((refine605929679le_RES @ A) @ X2) = ((refine605929679le_RES @ A) @ Y2)) = (X2 = Y2))))).
% 2.90/1.56 thf(fact_4_if__rule, axiom, (! [A:$tType,B2:$o,S1:(refine1665802226e_nres @ A),Phi:(A > $o),S2:(refine1665802226e_nres @ A)]: (((B2 => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S1) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi)))) => (((~ (B2)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S2) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((if @ (refine1665802226e_nres @ A)) @ B2) @ S1) @ S2)) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi)))))))).
% 2.90/1.56 thf(fact_5_RES__rule, axiom, (! [A:$tType,S:(set @ A),Phi:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ S) => (Phi @ X3)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ S)) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))))))).
% 2.90/1.56 thf(fact_6_SPEC__rule, axiom, (! [A:$tType,Phi:(A > $o),Phi2:(A > $o)]: (((! [X3:A]: (((Phi @ X3) => (Phi2 @ X3)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi2))))))).
% 2.90/1.56 thf(fact_7_SPEC__trans, axiom, (! [A:$tType,X4:(refine1665802226e_nres @ A),Y3:(refine1665802226e_nres @ A),Postcond:(A > $o)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ X4) @ Y3) => ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ Y3) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Postcond))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ X4) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Postcond)))))))).
% 2.90/1.56 thf(fact_8_lhs__step__If, axiom, (! [A:$tType]: (((ord @ A) => (! [B2:$o,T2:A,M:A,E:A]: (((B2 => (((ord_less_eq @ A) @ T2) @ M)) => (((~ (B2)) => (((ord_less_eq @ A) @ E) @ M)) => (((ord_less_eq @ A) @ ((((if @ A) @ B2) @ T2) @ E)) @ M))))))))).
% 2.90/1.56 thf(fact_9_SPEC__cons__rule, axiom, (! [A:$tType,M:(refine1665802226e_nres @ A),Phi:(A > $o),Psi:(A > $o)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ M) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))) => ((! [X3:A]: (((Phi @ X3) => (Psi @ X3)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ M) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Psi)))))))).
% 2.90/1.56 thf(fact_10_RES__bind__choose, axiom, (! [B:$tType,A:$tType,X4:A,X:(set @ A),M:(refine1665802226e_nres @ B),F:(A > (refine1665802226e_nres @ B))]: (((((member @ A) @ X4) @ X) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ M) @ (F @ X4)) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ M) @ ((((refine463715084e_bind @ A) @ B) @ ((refine605929679le_RES @ A) @ X)) @ F))))))).
% 2.90/1.56 thf(fact_11_order__mono__setup_Orefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A]: ((((ord_less_eq @ A) @ X4) @ X4))))))).
% 2.90/1.56 thf(fact_12_the__RES_Osimps, axiom, (! [A:$tType,X:(set @ A)]: ((((refine1672542526he_RES @ A) @ ((refine605929679le_RES @ A) @ X)) = X)))).
% 2.90/1.56 thf(fact_13_nf__inres__RES, axiom, (! [A:$tType,X:(set @ A),X4:A]: (((((refine406925620_inres @ A) @ ((refine605929679le_RES @ A) @ X)) @ X4) = (((member @ A) @ X4) @ X))))).
% 2.90/1.56 thf(fact_14_nf__inres__SPEC, axiom, (! [A:$tType,Phi:(A > $o),X4:A]: (((((refine406925620_inres @ A) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))) @ X4) = (Phi @ X4))))).
% 2.90/1.56 thf(fact_15_nres_Osimps_I7_J, axiom, (! [C:$tType,A:$tType,F1:C,F2:((set @ A) > C),X2:(set @ A)]: (((((((refine1442219249c_nres @ C) @ A) @ F1) @ F2) @ ((refine605929679le_RES @ A) @ X2)) = (F2 @ X2))))).
% 2.90/1.56 thf(fact_16_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X4)) @ (G @ X4))))))))).
% 2.90/1.56 thf(fact_17_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X4)) @ (G @ X4))))))))).
% 2.90/1.56 thf(fact_18_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.90/1.56 thf(fact_19_Refine__Misc_Oif__mono, axiom, (! [A:$tType]: (((ord @ A) => (! [B2:$o,M1:A,M12:A,M2:A,M22:A]: (((B2 => (((ord_less_eq @ A) @ M1) @ M12)) => (((~ (B2)) => (((ord_less_eq @ A) @ M2) @ M22)) => (((ord_less_eq @ A) @ ((((if @ A) @ B2) @ M1) @ M2)) @ ((((if @ A) @ B2) @ M12) @ M22)))))))))).
% 2.90/1.56 thf(fact_20_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: ((! [X5:A]: ((((ord_less_eq @ B) @ (F3 @ X5)) @ (G2 @ X5))))))))))).
% 2.90/1.56 thf(fact_21_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X3:B,Y4:B]: (((((ord_less_eq @ B) @ X3) @ Y4) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.90/1.56 thf(fact_22_less__eq__nres_Osimps_I2_J, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A2)) @ ((refine605929679le_RES @ A) @ B2)) = (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.90/1.56 thf(fact_23_nres__order__simps_I5_J, axiom, (! [E2:$tType,X:(set @ E2),Y:(set @ E2)]: (((((ord_less_eq @ (refine1665802226e_nres @ E2)) @ ((refine605929679le_RES @ E2) @ X)) @ ((refine605929679le_RES @ E2) @ Y)) = (((ord_less_eq @ (set @ E2)) @ X) @ Y))))).
% 2.90/1.56 thf(fact_24_order__mono__setup_Omono__let, axiom, (! [A:$tType,B:$tType]: (((preorder @ A) => (! [F:(B > A),F4:(B > A),X4:B]: (((! [X3:B]: ((((ord_less_eq @ A) @ (F @ X3)) @ (F4 @ X3)))) => (((ord_less_eq @ A) @ (F @ X4)) @ (F4 @ X4))))))))).
% 2.90/1.56 thf(fact_25_order__mono__setup_Omono__if, axiom, (! [A:$tType]: (((preorder @ A) => (! [T2:A,T3:A,E:A,E3:A,B2:$o]: (((((ord_less_eq @ A) @ T2) @ T3) => ((((ord_less_eq @ A) @ E) @ E3) => (((ord_less_eq @ A) @ ((((if @ A) @ B2) @ T2) @ E)) @ ((((if @ A) @ B2) @ T3) @ E3)))))))))).
% 2.90/1.56 thf(fact_26_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.90/1.56 thf(fact_27_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [A3:A,B3:A]: (((((ord_less_eq @ A) @ B3) @ A3) & (((ord_less_eq @ A) @ A3) @ B3))))))))).
% 2.90/1.56 thf(fact_28_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 2.90/1.56 thf(fact_29_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((P @ A4) @ B4)))) => ((! [A4:A,B4:A]: ((((P @ B4) @ A4) => ((P @ A4) @ B4)))) => ((P @ A2) @ B2))))))))).
% 2.90/1.56 thf(fact_30_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.90/1.56 thf(fact_31_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z2) => (((ord_less_eq @ A) @ X4) @ Z2))))))))).
% 2.90/1.56 thf(fact_32_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.90/1.56 thf(fact_33_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.90/1.56 thf(fact_34_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.90/1.56 thf(fact_35_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) & (((ord_less_eq @ A) @ B3) @ A3))))))))).
% 2.90/1.56 thf(fact_36_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X4:A]: (((((ord_less_eq @ A) @ Y3) @ X4) => ((((ord_less_eq @ A) @ X4) @ Y3) = (X4 = Y3))))))))).
% 2.90/1.56 thf(fact_37_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A,Z2:A]: ((((((ord_less_eq @ A) @ X4) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z2)))) => (((((ord_less_eq @ A) @ Y3) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Z2)))) => (((((ord_less_eq @ A) @ X4) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y3)))) => (((((ord_less_eq @ A) @ Z2) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X4)))) => (((((ord_less_eq @ A) @ Y3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X4)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Y3))))))))))))))))).
% 2.90/1.56 thf(fact_38_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.90/1.56 thf(fact_39_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X4) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X4)))))))).
% 2.90/1.56 thf(fact_40_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y3:A]: (((X4 = Y3) => (((ord_less_eq @ A) @ X4) @ Y3)))))))).
% 2.90/1.56 thf(fact_41_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X4)))))))).
% 2.90/1.56 thf(fact_42_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X4) => (X4 = Y3))))))))).
% 2.90/1.56 thf(fact_43_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.90/1.56 thf(fact_44_Collect__mem__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((collect @ A) @ (^ [X5:A]: ((((member @ A) @ X5) @ A5)))) = A5)))).
% 2.90/1.56 thf(fact_45_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.90/1.56 thf(fact_46_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.90/1.56 thf(fact_47_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [X5:A,Y6:A]: (((((ord_less_eq @ A) @ X5) @ Y6) & (((ord_less_eq @ A) @ Y6) @ X5))))))))).
% 2.90/1.56 thf(fact_48_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C2) => ((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C2)))))))))).
% 2.90/1.56 thf(fact_49_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X3:B,Y4:B]: (((((ord_less_eq @ B) @ X3) @ Y4) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.90/1.56 thf(fact_50_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ C) @ (F @ A2)) @ C2)))))))))).
% 2.90/1.56 thf(fact_51_the__RES__inv, axiom, (! [A:$tType,M:(refine1665802226e_nres @ A)]: ((((refine1102455758nofail @ A) @ M) => (((refine605929679le_RES @ A) @ ((refine1672542526he_RES @ A) @ M)) = M))))).
% 2.90/1.56 thf(fact_52_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X4:A]: (((P @ X4) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ A) @ Y4) @ X4)))) => (((order_Greatest @ A) @ P) = X4))))))))).
% 2.90/1.56 thf(fact_53_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X4:A,Q:(A > $o)]: (((P @ X4) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ A) @ Y4) @ X4)))) => ((! [X3:A]: (((P @ X3) => ((! [Y7:A]: (((P @ Y7) => (((ord_less_eq @ A) @ Y7) @ X3)))) => (Q @ X3))))) => (Q @ ((order_Greatest @ A) @ P))))))))))).
% 2.90/1.56 thf(fact_54_transfer_Otransfer__Let, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comple187826305attice @ A) => (! [Alpha:(C > A),F:(B > C),F5:(B > A),X4:B]: (((! [X3:B]: ((((ord_less_eq @ A) @ (Alpha @ (F @ X3))) @ (F5 @ X3)))) => (((ord_less_eq @ A) @ (Alpha @ (F @ X4))) @ (F5 @ X4))))))))).
% 2.90/1.56 thf(fact_55_transfer_Otransfer__if, axiom, (! [C:$tType,A:$tType]: (((comple187826305attice @ A) => (! [B2:$o,Alpha:(C > A),S12:C,S1:A,S22:C,S2:A]: (((B2 => (((ord_less_eq @ A) @ (Alpha @ S12)) @ S1)) => (((~ (B2)) => (((ord_less_eq @ A) @ (Alpha @ S22)) @ S2)) => (((ord_less_eq @ A) @ (Alpha @ ((((if @ C) @ B2) @ S12) @ S22))) @ ((((if @ A) @ B2) @ S1) @ S2)))))))))).
% 2.90/1.56 thf(fact_56_le__rel__bool__arg__iff, axiom, (! [A:$tType]: (((ord @ A) => ((ord_less_eq @ ($o > A)) = (^ [X6:($o > A),Y8:($o > A)]: (((((ord_less_eq @ A) @ (X6 @ $false)) @ (Y8 @ $false)) & (((ord_less_eq @ A) @ (X6 @ $true)) @ (Y8 @ $true)))))))))).
% 2.90/1.56 thf(fact_57_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((A2 = B2) | ((~ ((((ord_less_eq @ A) @ A2) @ B2))) | (~ ((((ord_less_eq @ A) @ B2) @ A2))))))))))).
% 2.90/1.56 thf(fact_58_ord__eq__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A,D:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => ((C2 = D) => (((ord_less_eq @ A) @ A2) @ D)))))))))).
% 2.90/1.56 thf(fact_59_intro__spec__refine__iff, axiom, (! [C:$tType,A:$tType,B:$tType,X:(set @ B),F:(B > (refine1665802226e_nres @ A)),R:(set @ ((product_prod @ A) @ C)),M3:(refine1665802226e_nres @ C)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine463715084e_bind @ B) @ A) @ ((refine605929679le_RES @ B) @ X)) @ F)) @ ((((refine1073749519nc_fun @ A) @ C) @ R) @ M3)) = (! [X5:B]: (((((member @ B) @ X5) @ X) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ (F @ X5)) @ ((((refine1073749519nc_fun @ A) @ C) @ R) @ M3))))))))).
% 2.90/1.56 thf(fact_60_intro__bind__refine__id, axiom, (! [A:$tType,B:$tType,C:$tType,M:(refine1665802226e_nres @ A),M4:A,F:(A > (refine1665802226e_nres @ B)),R:(set @ ((product_prod @ B) @ C)),M5:(refine1665802226e_nres @ C)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ M) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ ((^ [Y5:A,Z:A]: ((Y5 = Z))) @ M4)))) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ (F @ M4)) @ ((((refine1073749519nc_fun @ B) @ C) @ R) @ M5)) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine463715084e_bind @ A) @ B) @ M) @ F)) @ ((((refine1073749519nc_fun @ B) @ C) @ R) @ M5))))))).
% 2.90/1.56 thf(fact_61_nofail__simps_I2_J, axiom, (! [B:$tType,X:(set @ B)]: (((refine1102455758nofail @ B) @ ((refine605929679le_RES @ B) @ X))))).
% 2.90/1.56 thf(fact_62_subset__Collect__conv, axiom, (! [A:$tType,S:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ S) @ ((collect @ A) @ P)) = (! [X5:A]: (((((member @ A) @ X5) @ S) => (P @ X5)))))))).
% 2.90/1.56 thf(fact_63_nofail__antimono__fun, axiom, (! [B:$tType,A:$tType,F:(A > (refine1665802226e_nres @ B)),G:(A > (refine1665802226e_nres @ B)),X4:A]: (((((ord_less_eq @ (A > (refine1665802226e_nres @ B))) @ F) @ G) => (((refine1102455758nofail @ B) @ (G @ X4)) => ((refine1102455758nofail @ B) @ (F @ X4))))))).
% 2.90/1.56 thf(fact_64_pw__conc__nofail, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),S:(refine1665802226e_nres @ B)]: ((((refine1102455758nofail @ A) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ S)) = ((refine1102455758nofail @ B) @ S))))).
% 2.90/1.56 thf(fact_65_pwD1, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S3:(refine1665802226e_nres @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S3) => (((refine1102455758nofail @ A) @ S3) => ((refine1102455758nofail @ A) @ S)))))).
% 2.90/1.56 thf(fact_66_le__nofailI, axiom, (! [A:$tType,M6:(refine1665802226e_nres @ A),M3:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ M6) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ M3) @ M6)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ M3) @ M6))))).
% 2.90/1.56 thf(fact_67_nofail__RES__conv, axiom, (! [A:$tType]: (((refine1102455758nofail @ A) = (^ [M7:(refine1665802226e_nres @ A)]: ((? [M8:(set @ A)]: ((M7 = ((refine605929679le_RES @ A) @ M8)))))))))).
% 2.90/1.56 thf(fact_68_if__refine, axiom, (! [A:$tType,B:$tType,B2:$o,B5:$o,S1:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),S13:(refine1665802226e_nres @ B),S2:(refine1665802226e_nres @ A),S23:(refine1665802226e_nres @ B)]: (((B2 = B5) => ((B2 => (B5 => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S1) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ S13)))) => (((~ (B2)) => ((~ (B5)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ S23)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((if @ (refine1665802226e_nres @ A)) @ B2) @ S1) @ S2)) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ ((((if @ (refine1665802226e_nres @ B)) @ B5) @ S13) @ S23))))))))).
% 2.90/1.56 thf(fact_69_conc__trans, axiom, (! [A:$tType,B:$tType,C:$tType,C3:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),B6:(refine1665802226e_nres @ B),R2:(set @ ((product_prod @ B) @ C)),A5:(refine1665802226e_nres @ C)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ C3) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ B6)) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ B6) @ ((((refine1073749519nc_fun @ B) @ C) @ R2) @ A5)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ C3) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ ((((refine1073749519nc_fun @ B) @ C) @ R2) @ A5)))))))).
% 2.90/1.56 thf(fact_70_intro__prgR, axiom, (! [A:$tType,B:$tType,C2:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),A2:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ C2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ A2)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ C2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ A2)))))).
% 2.90/1.56 thf(fact_71_ref__two__step, axiom, (! [A:$tType,B:$tType,A5:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),B6:(refine1665802226e_nres @ B),C3:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ A5) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ B6)) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ B6) @ C3) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ A5) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ C3))))))).
% 2.90/1.56 thf(fact_72_conc__fun__R__mono, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),R2:(set @ ((product_prod @ A) @ B)),M3:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ R2) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ M3)) @ ((((refine1073749519nc_fun @ A) @ B) @ R2) @ M3)))))).
% 2.90/1.56 thf(fact_73_le__RES__nofailI, axiom, (! [A:$tType,A2:(refine1665802226e_nres @ A),X4:(set @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ A2) @ ((refine605929679le_RES @ A) @ X4)) => ((refine1102455758nofail @ A) @ A2))))).
% 2.90/1.56 thf(fact_74_pw__RES__bind__choose_I1_J, axiom, (! [A:$tType,B:$tType,X:(set @ B),F:(B > (refine1665802226e_nres @ A))]: ((((refine1102455758nofail @ A) @ ((((refine463715084e_bind @ B) @ A) @ ((refine605929679le_RES @ B) @ X)) @ F)) = (! [X5:B]: (((((member @ B) @ X5) @ X) => ((refine1102455758nofail @ A) @ (F @ X5))))))))).
% 2.90/1.56 thf(fact_75_rhs__step__bind__RES, axiom, (! [B:$tType,C:$tType,A:$tType,X7:A,X8:(set @ A),M:(refine1665802226e_nres @ B),R:(set @ ((product_prod @ B) @ C)),F4:(A > (refine1665802226e_nres @ C))]: (((((member @ A) @ X7) @ X8) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ M) @ ((((refine1073749519nc_fun @ B) @ C) @ R) @ (F4 @ X7))) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ M) @ ((((refine1073749519nc_fun @ B) @ C) @ R) @ ((((refine463715084e_bind @ A) @ C) @ ((refine605929679le_RES @ A) @ X8)) @ F4)))))))).
% 2.90/1.56 thf(fact_76_rhs__step__bind__SPEC, axiom, (! [B:$tType,C:$tType,A:$tType,Phi:(A > $o),X7:A,M:(refine1665802226e_nres @ B),R:(set @ ((product_prod @ B) @ C)),F4:(A > (refine1665802226e_nres @ C))]: (((Phi @ X7) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ M) @ ((((refine1073749519nc_fun @ B) @ C) @ R) @ (F4 @ X7))) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ M) @ ((((refine1073749519nc_fun @ B) @ C) @ R) @ ((((refine463715084e_bind @ A) @ C) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))) @ F4)))))))).
% 2.90/1.56 thf(fact_77_subsetI, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A5) => (((member @ A) @ X3) @ B6)))) => (((ord_less_eq @ (set @ A)) @ A5) @ B6))))).
% 2.90/1.56 thf(fact_78_subset__antisym, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ A5) => (A5 = B6)))))).
% 2.90/1.56 thf(fact_79_bind__sim__select__rule, axiom, (! [A:$tType,B:$tType,C:$tType,M:(refine1665802226e_nres @ B),F4:(B > (refine1665802226e_nres @ A)),Psi:(A > $o),F:(B > (refine1665802226e_nres @ C)),Phi:(C > $o)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine463715084e_bind @ B) @ A) @ M) @ F4)) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Psi))) => ((! [X3:B]: ((((refine1102455758nofail @ B) @ M) => ((((refine1315500908_inres @ B) @ M) @ X3) => ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ (F4 @ X3)) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Psi))) => (((ord_less_eq @ (refine1665802226e_nres @ C)) @ (F @ X3)) @ ((refine605929679le_RES @ C) @ ((collect @ C) @ Phi)))))))) => (((ord_less_eq @ (refine1665802226e_nres @ C)) @ ((((refine463715084e_bind @ B) @ C) @ M) @ F)) @ ((refine605929679le_RES @ C) @ ((collect @ C) @ Phi)))))))).
% 2.90/1.56 thf(fact_80_relprop__triggers_I6_J, axiom, (! [I:$tType,R:(set @ I),R2:(set @ I)]: (((((ord_less_eq @ (set @ I)) @ R) @ R2) => (((ord_less_eq @ (set @ I)) @ R) @ R2))))).
% 2.90/1.56 thf(fact_81_in__mono, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),X4:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => ((((member @ A) @ X4) @ A5) => (((member @ A) @ X4) @ B6)))))).
% 2.90/1.56 thf(fact_82_subsetD, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => ((((member @ A) @ C2) @ A5) => (((member @ A) @ C2) @ B6)))))).
% 2.90/1.56 thf(fact_83_equalityE, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((A5 = B6) => (~ (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A5)))))))))).
% 2.90/1.56 thf(fact_84_inres__simps_I2_J, axiom, (! [B:$tType,X:(set @ B)]: ((((refine1315500908_inres @ B) @ ((refine605929679le_RES @ B) @ X)) = (^ [X5:B]: ((((member @ B) @ X5) @ X))))))).
% 2.90/1.56 thf(fact_85_pwD2, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S3:(refine1665802226e_nres @ A),X4:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S3) => ((((refine1315500908_inres @ A) @ S) @ X4) => (((refine1315500908_inres @ A) @ S3) @ X4)))))).
% 2.90/1.56 thf(fact_86_pw__eqI, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S3:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S) = ((refine1102455758nofail @ A) @ S3)) => ((! [X3:A]: (((((refine1315500908_inres @ A) @ S) @ X3) = (((refine1315500908_inres @ A) @ S3) @ X3)))) => (S = S3)))))).
% 2.90/1.56 thf(fact_87_pw__eq__iff, axiom, (! [A:$tType]: (((^ [Y5:(refine1665802226e_nres @ A),Z:(refine1665802226e_nres @ A)]: ((Y5 = Z))) = (^ [S4:(refine1665802226e_nres @ A),S5:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S4) = ((refine1102455758nofail @ A) @ S5)) & (! [X5:A]: (((((refine1315500908_inres @ A) @ S4) @ X5) = (((refine1315500908_inres @ A) @ S5) @ X5))))))))))).
% 2.90/1.56 thf(fact_88_not__nofail__inres, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),X4:A]: (((~ (((refine1102455758nofail @ A) @ S))) => (((refine1315500908_inres @ A) @ S) @ X4))))).
% 2.90/1.56 thf(fact_89_pw__RES__bind__choose_I2_J, axiom, (! [A:$tType,B:$tType,X:(set @ B),F:(B > (refine1665802226e_nres @ A)),Y3:A]: (((((refine1315500908_inres @ A) @ ((((refine463715084e_bind @ B) @ A) @ ((refine605929679le_RES @ B) @ X)) @ F)) @ Y3) = (? [X5:B]: (((((member @ B) @ X5) @ X) & (((refine1315500908_inres @ A) @ (F @ X5)) @ Y3)))))))).
% 2.90/1.56 thf(fact_90_pw__leI, axiom, (! [A:$tType,S3:(refine1665802226e_nres @ A),S:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S3) => (((refine1102455758nofail @ A) @ S) & (! [X3:A]: (((((refine1315500908_inres @ A) @ S) @ X3) => (((refine1315500908_inres @ A) @ S3) @ X3)))))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S3))))).
% 2.90/1.56 thf(fact_91_pw__leI_H, axiom, (! [A:$tType,S3:(refine1665802226e_nres @ A),S:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S3) => ((refine1102455758nofail @ A) @ S)) => ((! [X3:A]: ((((refine1102455758nofail @ A) @ S3) => ((((refine1315500908_inres @ A) @ S) @ X3) => (((refine1315500908_inres @ A) @ S3) @ X3))))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S3)))))).
% 2.90/1.56 thf(fact_92_pw__le__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (refine1665802226e_nres @ A)) = (^ [S4:(refine1665802226e_nres @ A),S5:(refine1665802226e_nres @ A)]: ((((refine1102455758nofail @ A) @ S5) => (((refine1102455758nofail @ A) @ S4) & (! [X5:A]: (((((refine1315500908_inres @ A) @ S4) @ X5) => (((refine1315500908_inres @ A) @ S5) @ X5)))))))))))).
% 2.90/1.56 thf(fact_93_pw__bind__nofail, axiom, (! [A:$tType,B:$tType,M3:(refine1665802226e_nres @ B),F:(B > (refine1665802226e_nres @ A))]: ((((refine1102455758nofail @ A) @ ((((refine463715084e_bind @ B) @ A) @ M3) @ F)) = (((refine1102455758nofail @ B) @ M3) & (! [X5:B]: (((((refine1315500908_inres @ B) @ M3) @ X5) => ((refine1102455758nofail @ A) @ (F @ X5)))))))))).
% 2.90/1.56 thf(fact_94_nf__inres__def, axiom, (! [A:$tType]: (((refine406925620_inres @ A) = (^ [M7:(refine1665802226e_nres @ A),X5:A]: ((((refine1102455758nofail @ A) @ M7) & (((refine1315500908_inres @ A) @ M7) @ X5)))))))).
% 2.90/1.56 thf(fact_95_pw__bind__le__iff, axiom, (! [A:$tType,B:$tType,M3:(refine1665802226e_nres @ B),F:(B > (refine1665802226e_nres @ A)),S:(refine1665802226e_nres @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine463715084e_bind @ B) @ A) @ M3) @ F)) @ S) = ((((refine1102455758nofail @ A) @ S) => ((refine1102455758nofail @ B) @ M3)) & (! [X5:B]: (((((refine1102455758nofail @ B) @ M3) & (((refine1315500908_inres @ B) @ M3) @ X5)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ (F @ X5)) @ S))))))))).
% 2.90/1.56 thf(fact_96_pw__bind__leI, axiom, (! [B:$tType,A:$tType,S:(refine1665802226e_nres @ A),M3:(refine1665802226e_nres @ B),F:(B > (refine1665802226e_nres @ A))]: (((((refine1102455758nofail @ A) @ S) => ((refine1102455758nofail @ B) @ M3)) => ((! [X3:B]: ((((refine1102455758nofail @ B) @ M3) => ((((refine1315500908_inres @ B) @ M3) @ X3) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ (F @ X3)) @ S))))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine463715084e_bind @ B) @ A) @ M3) @ F)) @ S)))))).
% 2.90/1.56 thf(fact_97_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X5:A]: (((P @ X5) => (Q @ X5)))))))).
% 2.90/1.56 thf(fact_98_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z:(set @ A)]: ((Y5 = Z))) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A6)))))))).
% 2.90/1.56 thf(fact_99_subset__trans, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ C3) => (((ord_less_eq @ (set @ A)) @ A5) @ C3)))))).
% 2.90/1.56 thf(fact_100_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.90/1.56 thf(fact_101_subset__refl, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A5) @ A5)))).
% 2.90/1.56 thf(fact_102_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: ((! [T4:A]: (((((member @ A) @ T4) @ A6) => (((member @ A) @ T4) @ B7)))))))))).
% 2.90/1.56 thf(fact_103_equalityD2, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((A5 = B6) => (((ord_less_eq @ (set @ A)) @ B6) @ A5))))).
% 2.90/1.56 thf(fact_104_equalityD1, axiom, (! [A:$tType,A5:(set @ A),B6:(set @ A)]: (((A5 = B6) => (((ord_less_eq @ (set @ A)) @ A5) @ B6))))).
% 2.90/1.56 thf(fact_105_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: ((! [X5:A]: (((((member @ A) @ X5) @ A6) => (((member @ A) @ X5) @ B7)))))))))).
% 2.90/1.56 thf(fact_106_pw__ref__iff, axiom, (! [B:$tType,A:$tType,S:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),S3:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ S3)) = (((refine1102455758nofail @ B) @ S3) => (((refine1102455758nofail @ A) @ S) & (! [X5:A]: (((((refine1315500908_inres @ A) @ S) @ X5) => (? [S6:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ S6)) @ R) & (((refine1315500908_inres @ B) @ S3) @ S6))))))))))))).
% 2.90/1.56 thf(fact_107_pw__ref__I, axiom, (! [B:$tType,A:$tType,S3:(refine1665802226e_nres @ A),S:(refine1665802226e_nres @ B),R:(set @ ((product_prod @ B) @ A))]: (((((refine1102455758nofail @ A) @ S3) => (((refine1102455758nofail @ B) @ S) & (! [X3:B]: (((((refine1315500908_inres @ B) @ S) @ X3) => (? [S7:A]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X3) @ S7)) @ R) & (((refine1315500908_inres @ A) @ S3) @ S7))))))))) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ S) @ ((((refine1073749519nc_fun @ B) @ A) @ R) @ S3)))))).
% 2.90/1.56 thf(fact_108_rhs__step__bind, axiom, (! [A:$tType,C:$tType,D2:$tType,B:$tType,M:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),M4:(refine1665802226e_nres @ B),X4:A,Lhs:(refine1665802226e_nres @ C),S:(set @ ((product_prod @ C) @ D2)),F4:(B > (refine1665802226e_nres @ D2))]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ M) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ M4)) => ((((refine1315500908_inres @ A) @ M) @ X4) => ((! [X9:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ X9)) @ R) => (((ord_less_eq @ (refine1665802226e_nres @ C)) @ Lhs) @ ((((refine1073749519nc_fun @ C) @ D2) @ S) @ (F4 @ X9)))))) => (((ord_less_eq @ (refine1665802226e_nres @ C)) @ Lhs) @ ((((refine1073749519nc_fun @ C) @ D2) @ S) @ ((((refine463715084e_bind @ B) @ D2) @ M4) @ F4))))))))).
% 2.90/1.56 thf(fact_109_RES__refine, axiom, (! [A:$tType,B:$tType,S:(set @ A),S3:(set @ B),R:(set @ ((product_prod @ A) @ B))]: (((! [S8:A]: (((((member @ A) @ S8) @ S) => (? [X10:B]: (((((member @ B) @ X10) @ S3) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ S8) @ X10)) @ R))))))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ S)) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ ((refine605929679le_RES @ B) @ S3))))))).
% 2.90/1.56 thf(fact_110_less__eq__nres_Oelims_I1_J, axiom, (! [A:$tType,X4:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A),Y3:$o]: ((((((ord_less_eq @ (refine1665802226e_nres @ A)) @ X4) @ Xa) = Y3) => (((Xa = (refine1767639642_FAILi @ A)) => (~ (Y3))) => ((! [A4:(set @ A)]: (((X4 = ((refine605929679le_RES @ A) @ A4)) => (! [B4:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B4)) => (Y3 = (~ ((((ord_less_eq @ (set @ A)) @ A4) @ B4)))))))))) => (~ (((X4 = (refine1767639642_FAILi @ A)) => ((? [Uv:(set @ A)]: ((Xa = ((refine605929679le_RES @ A) @ Uv)))) => Y3)))))))))).
% 2.90/1.56 thf(fact_111_less__eq__nres_Oelims_I2_J, axiom, (! [A:$tType,X4:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ X4) @ Xa) => ((Xa != (refine1767639642_FAILi @ A)) => (~ ((! [A4:(set @ A)]: (((X4 = ((refine605929679le_RES @ A) @ A4)) => (! [B4:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B4)) => (~ ((((ord_less_eq @ (set @ A)) @ A4) @ B4)))))))))))))))).
% 2.90/1.56 thf(fact_112_less__eq__nres_Ocases, axiom, (! [A:$tType,X4:((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))]: (((! [Uu:(refine1665802226e_nres @ A)]: ((X4 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ Uu) @ (refine1767639642_FAILi @ A))))) => ((! [A4:(set @ A),B4:(set @ A)]: ((X4 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A4)) @ ((refine605929679le_RES @ A) @ B4))))) => (~ ((! [Uv:(set @ A)]: ((X4 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ (refine1767639642_FAILi @ A)) @ ((refine605929679le_RES @ A) @ Uv)))))))))))).
% 2.90/1.56 thf(fact_113_less__nres_Ocases, axiom, (! [A:$tType,X4:((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))]: (((! [Uu:(refine1665802226e_nres @ A)]: ((X4 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ (refine1767639642_FAILi @ A)) @ Uu)))) => ((! [Uv:(set @ A)]: ((X4 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ Uv)) @ (refine1767639642_FAILi @ A))))) => (~ ((! [A4:(set @ A),B4:(set @ A)]: ((X4 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A4)) @ ((refine605929679le_RES @ A) @ B4)))))))))))).
% 2.90/1.56 thf(fact_114_sup__nres_Ocases, axiom, (! [A:$tType,X4:((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A))]: (((! [Uu:(refine1665802226e_nres @ A)]: ((X4 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ Uu) @ (refine1767639642_FAILi @ A))))) => ((! [V:(set @ A)]: ((X4 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ (refine1767639642_FAILi @ A)) @ ((refine605929679le_RES @ A) @ V))))) => (~ ((! [A4:(set @ A),B4:(set @ A)]: ((X4 != ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A4)) @ ((refine605929679le_RES @ A) @ B4)))))))))))).
% 2.90/1.56 thf(fact_115_introR, axiom, (! [B:$tType,A:$tType,A2:A,A7:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ A7)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ A7)) @ R))))).
% 2.90/1.56 thf(fact_116_bex2I, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,S:(set @ ((product_prod @ A) @ B)),P:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ S) => (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ S) => ((P @ A2) @ B2)) => (? [A4:A,B4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ S) & ((P @ A4) @ B4))))))))).
% 2.90/1.56 thf(fact_117_pairself_Ocases, axiom, (! [B:$tType,A:$tType,X4:((product_prod @ (A > B)) @ ((product_prod @ A) @ A))]: ((~ ((! [F6:(A > B),A4:A,B4:A]: ((X4 != ((((product_Pair @ (A > B)) @ ((product_prod @ A) @ A)) @ F6) @ ((((product_Pair @ A) @ A) @ A4) @ B4)))))))))).
% 2.90/1.56 thf(fact_118_pairself_Oinduct, axiom, (! [B:$tType,A:$tType,P:((A > B) > (((product_prod @ A) @ A) > $o)),A0:(A > B),A1:((product_prod @ A) @ A)]: (((! [F6:(A > B),A4:A,B4:A]: (((P @ F6) @ ((((product_Pair @ A) @ A) @ A4) @ B4)))) => ((P @ A0) @ A1))))).
% 2.90/1.56 thf(fact_119_less__eq__nres_Osimps_I1_J, axiom, (! [A:$tType,Uu2:(refine1665802226e_nres @ A)]: ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ Uu2) @ (refine1767639642_FAILi @ A))))).
% 2.90/1.56 thf(fact_120_nres_Odistinct_I1_J, axiom, (! [A:$tType,X2:(set @ A)]: (((refine1767639642_FAILi @ A) != ((refine605929679le_RES @ A) @ X2))))).
% 2.90/1.56 thf(fact_121_nres_Oinduct, axiom, (! [A:$tType,P:((refine1665802226e_nres @ A) > $o),Nres:(refine1665802226e_nres @ A)]: (((P @ (refine1767639642_FAILi @ A)) => ((! [X3:(set @ A)]: ((P @ ((refine605929679le_RES @ A) @ X3)))) => (P @ Nres)))))).
% 2.90/1.56 thf(fact_122_nres_Oexhaust, axiom, (! [A:$tType,Y3:(refine1665802226e_nres @ A)]: (((Y3 != (refine1767639642_FAILi @ A)) => (~ ((! [X22:(set @ A)]: ((Y3 != ((refine605929679le_RES @ A) @ X22)))))))))).
% 2.90/1.56 thf(fact_123_sup__nres_Oinduct, axiom, (! [A:$tType,P:((refine1665802226e_nres @ A) > ((refine1665802226e_nres @ A) > $o)),A0:(refine1665802226e_nres @ A),A1:(refine1665802226e_nres @ A)]: (((! [Uu:(refine1665802226e_nres @ A)]: (((P @ Uu) @ (refine1767639642_FAILi @ A)))) => ((! [V:(set @ A)]: (((P @ (refine1767639642_FAILi @ A)) @ ((refine605929679le_RES @ A) @ V)))) => ((! [A4:(set @ A),B4:(set @ A)]: (((P @ ((refine605929679le_RES @ A) @ A4)) @ ((refine605929679le_RES @ A) @ B4)))) => ((P @ A0) @ A1))))))).
% 2.90/1.56 thf(fact_124_less__nres_Oinduct, axiom, (! [A:$tType,P:((refine1665802226e_nres @ A) > ((refine1665802226e_nres @ A) > $o)),A0:(refine1665802226e_nres @ A),A1:(refine1665802226e_nres @ A)]: (((! [X_1:(refine1665802226e_nres @ A)]: (((P @ (refine1767639642_FAILi @ A)) @ X_1))) => ((! [Uv:(set @ A)]: (((P @ ((refine605929679le_RES @ A) @ Uv)) @ (refine1767639642_FAILi @ A)))) => ((! [A4:(set @ A),B4:(set @ A)]: (((P @ ((refine605929679le_RES @ A) @ A4)) @ ((refine605929679le_RES @ A) @ B4)))) => ((P @ A0) @ A1))))))).
% 2.90/1.56 thf(fact_125_less__eq__nres_Oinduct, axiom, (! [A:$tType,P:((refine1665802226e_nres @ A) > ((refine1665802226e_nres @ A) > $o)),A0:(refine1665802226e_nres @ A),A1:(refine1665802226e_nres @ A)]: (((! [Uu:(refine1665802226e_nres @ A)]: (((P @ Uu) @ (refine1767639642_FAILi @ A)))) => ((! [A4:(set @ A),B4:(set @ A)]: (((P @ ((refine605929679le_RES @ A) @ A4)) @ ((refine605929679le_RES @ A) @ B4)))) => ((! [Uv:(set @ A)]: (((P @ (refine1767639642_FAILi @ A)) @ ((refine605929679le_RES @ A) @ Uv)))) => ((P @ A0) @ A1))))))).
% 2.90/1.56 thf(fact_126_nres_Osimps_I6_J, axiom, (! [A:$tType,C:$tType,F1:C,F2:((set @ A) > C)]: (((((((refine1442219249c_nres @ C) @ A) @ F1) @ F2) @ (refine1767639642_FAILi @ A)) = F1)))).
% 2.90/1.56 thf(fact_127_less__eq__nres_Osimps_I3_J, axiom, (! [A:$tType,Uv2:(set @ A)]: ((~ ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ (refine1767639642_FAILi @ A)) @ ((refine605929679le_RES @ A) @ Uv2))))))).
% 2.90/1.56 thf(fact_128_Let__refine_H, axiom, (! [A:$tType,C:$tType,D2:$tType,B:$tType,M:A,M4:B,R:(set @ ((product_prod @ A) @ B)),F:(A > (refine1665802226e_nres @ C)),S:(set @ ((product_prod @ C) @ D2)),F4:(B > (refine1665802226e_nres @ D2))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ M) @ M4)) @ R) => (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ M) @ M4)) @ R) => (((ord_less_eq @ (refine1665802226e_nres @ C)) @ (F @ M)) @ ((((refine1073749519nc_fun @ C) @ D2) @ S) @ (F4 @ M4)))) => (((ord_less_eq @ (refine1665802226e_nres @ C)) @ (F @ M)) @ ((((refine1073749519nc_fun @ C) @ D2) @ S) @ (F4 @ M4)))))))).
% 2.90/1.56 thf(fact_129_less__eq__nres_Oelims_I3_J, axiom, (! [A:$tType,X4:(refine1665802226e_nres @ A),Xa:(refine1665802226e_nres @ A)]: (((~ ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ X4) @ Xa))) => ((! [A4:(set @ A)]: (((X4 = ((refine605929679le_RES @ A) @ A4)) => (! [B4:(set @ A)]: (((Xa = ((refine605929679le_RES @ A) @ B4)) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))))) => (~ (((X4 = (refine1767639642_FAILi @ A)) => (! [Uv:(set @ A)]: ((Xa != ((refine605929679le_RES @ A) @ Uv)))))))))))).
% 2.90/1.56 thf(fact_130_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A7:A,B5:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A7) @ B5)) = ((A2 = A7) & (B2 = B5)))))).
% 2.90/1.56 thf(fact_131_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X2:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X2) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 2.90/1.56 thf(fact_132_subrelI, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B)),S9:(set @ ((product_prod @ A) @ B))]: (((! [X3:A,Y4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y4)) @ R3) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y4)) @ S9)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R3) @ S9))))).
% 2.90/1.56 thf(fact_133_lift__assn__def, axiom, (! [B:$tType,A:$tType]: ((((refine1580981607t_assn @ A) @ B) = (^ [R4:(set @ ((product_prod @ A) @ B)),Phi3:(B > $o),S10:A]: ((? [S6:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ S10) @ S6)) @ R4) & (Phi3 @ S6)))))))))).
% 2.90/1.56 thf(fact_134_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X3:A,Y4:B]: ((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y4))))))).
% 2.90/1.56 thf(fact_135_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P @ P2))))).
% 2.90/1.56 thf(fact_136_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A7:A,B5:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A7) @ B5)) => (~ (((A2 = A7) => (B2 != B5)))))))).
% 2.90/1.56 thf(fact_137_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A4:A,B4:B,C4:C]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C4)))))))))).
% 2.90/1.56 thf(fact_138_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: ((~ ((! [A4:A,B4:B,C4:C,D3:D2]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B4) @ ((((product_Pair @ C) @ D2) @ C4) @ D3))))))))))).
% 2.90/1.56 thf(fact_139_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E2:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2))))]: ((~ ((! [A4:A,B4:B,C4:C,D3:D2,E4:E2]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E2)) @ C4) @ ((((product_Pair @ D2) @ E2) @ D3) @ E4)))))))))))).
% 2.90/1.56 thf(fact_140_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E2:$tType,F7:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F7)))))]: ((~ ((! [A4:A,B4:B,C4:C,D3:D2,E4:E2,F6:F7]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F7))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F7)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F7))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E2) @ F7)) @ D3) @ ((((product_Pair @ E2) @ F7) @ E4) @ F6))))))))))))).
% 2.90/1.56 thf(fact_141_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E2:$tType,F7:$tType,G3:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F7) @ G3))))))]: ((~ ((! [A4:A,B4:B,C4:C,D3:D2,E4:E2,F6:F7,G4:G3]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F7) @ G3)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F7) @ G3))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F7) @ G3)))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E2) @ ((product_prod @ F7) @ G3))) @ D3) @ ((((product_Pair @ E2) @ ((product_prod @ F7) @ G3)) @ E4) @ ((((product_Pair @ F7) @ G3) @ F6) @ G4)))))))))))))).
% 2.90/1.56 thf(fact_142_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A4:A,B4:B,C4:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C4))))) => (P @ X4))))).
% 2.90/1.56 thf(fact_143_prod__induct4, axiom, (! [D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: (((! [A4:A,B4:B,C4:C,D3:D2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B4) @ ((((product_Pair @ C) @ D2) @ C4) @ D3)))))) => (P @ X4))))).
% 2.90/1.56 thf(fact_144_prod__induct5, axiom, (! [E2:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2)))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2))))]: (((! [A4:A,B4:B,C4:C,D3:D2,E4:E2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E2)) @ C4) @ ((((product_Pair @ D2) @ E2) @ D3) @ E4))))))) => (P @ X4))))).
% 2.90/1.56 thf(fact_145_prod__induct6, axiom, (! [F7:$tType,E2:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F7))))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F7)))))]: (((! [A4:A,B4:B,C4:C,D3:D2,E4:E2,F6:F7]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F7))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F7)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F7))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E2) @ F7)) @ D3) @ ((((product_Pair @ E2) @ F7) @ E4) @ F6)))))))) => (P @ X4))))).
% 2.90/1.56 thf(fact_146_prod__induct7, axiom, (! [G3:$tType,F7:$tType,E2:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F7) @ G3)))))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F7) @ G3))))))]: (((! [A4:A,B4:B,C4:C,D3:D2,E4:E2,F6:F7,G4:G3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F7) @ G3)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F7) @ G3))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F7) @ G3)))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E2) @ ((product_prod @ F7) @ G3))) @ D3) @ ((((product_Pair @ E2) @ ((product_prod @ F7) @ G3)) @ E4) @ ((((product_Pair @ F7) @ G3) @ F6) @ G4))))))))) => (P @ X4))))).
% 2.90/1.56 thf(fact_147_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y3:((product_prod @ A) @ B)]: ((~ ((! [A4:A,B4:B]: ((Y3 != ((((product_Pair @ A) @ B) @ A4) @ B4))))))))).
% 2.90/1.56 thf(fact_148_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P @ Prod))))).
% 2.90/1.56 thf(fact_149_lift__assnI, axiom, (! [B:$tType,A:$tType,S9:A,S11:B,R:(set @ ((product_prod @ A) @ B)),Phi:(B > $o)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ S9) @ S11)) @ R) => ((Phi @ S11) => (((((refine1580981607t_assn @ A) @ B) @ R) @ Phi) @ S9)))))).
% 2.90/1.56 thf(fact_150_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C2 @ A2) @ B2))))).
% 2.90/1.56 thf(fact_151_return__refine__prop__return, axiom, (! [B:$tType,A:$tType,M:(refine1665802226e_nres @ A),X4:B,R:(set @ ((product_prod @ B) @ A))]: ((((refine1102455758nofail @ A) @ M) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((refine1687780735RETURN @ B) @ X4)) @ ((((refine1073749519nc_fun @ B) @ A) @ R) @ M)) => (~ ((! [X9:A]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X4) @ X9)) @ R) => (~ ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X9)) @ M))))))))))))).
% 2.90/1.56 thf(fact_152_ret__le__down__conv, axiom, (! [B:$tType,A:$tType,M:(refine1665802226e_nres @ A),C2:B,R:(set @ ((product_prod @ B) @ A))]: ((((refine1102455758nofail @ A) @ M) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((refine1687780735RETURN @ B) @ C2)) @ ((((refine1073749519nc_fun @ B) @ A) @ R) @ M)) = (? [A3:A]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ C2) @ A3)) @ R) & (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ A3)) @ M))))))))).
% 2.90/1.56 thf(fact_153_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 2.90/1.56 thf(fact_154_nres__more__simps_I6_J, axiom, (! [A:$tType,X4:A,Y3:A]: (((((refine1687780735RETURN @ A) @ X4) = ((refine1687780735RETURN @ A) @ Y3)) = (X4 = Y3))))).
% 2.90/1.56 thf(fact_155_nres__order__simps_I20_J, axiom, (! [W:$tType,X4:W,Y3:W]: (((((ord_less_eq @ (refine1665802226e_nres @ W)) @ ((refine1687780735RETURN @ W) @ X4)) @ ((refine1687780735RETURN @ W) @ Y3)) = (X4 = Y3))))).
% 2.90/1.56 thf(fact_156_nres__monad1, axiom, (! [A:$tType,B:$tType,X4:B,F:(B > (refine1665802226e_nres @ A))]: ((((((refine463715084e_bind @ B) @ A) @ ((refine1687780735RETURN @ B) @ X4)) @ F) = (F @ X4))))).
% 2.90/1.56 thf(fact_157_nres__monad2, axiom, (! [A:$tType,M3:(refine1665802226e_nres @ A)]: ((((((refine463715084e_bind @ A) @ A) @ M3) @ (refine1687780735RETURN @ A)) = M3)))).
% 2.90/1.56 thf(fact_158_nofail__simps_I3_J, axiom, (! [C:$tType,X4:C]: (((refine1102455758nofail @ C) @ ((refine1687780735RETURN @ C) @ X4))))).
% 2.90/1.56 thf(fact_159_inres__simps_I3_J, axiom, (! [C:$tType,X4:C]: ((((refine1315500908_inres @ C) @ ((refine1687780735RETURN @ C) @ X4)) = ((^ [Y5:C,Z:C]: ((Y5 = Z))) @ X4))))).
% 2.90/1.56 thf(fact_160_nres__order__simps_I21_J, axiom, (! [X11:$tType,X4:X11,Y:(set @ X11)]: (((((ord_less_eq @ (refine1665802226e_nres @ X11)) @ ((refine1687780735RETURN @ X11) @ X4)) @ ((refine605929679le_RES @ X11) @ Y)) = (((member @ X11) @ X4) @ Y))))).
% 2.90/1.56 thf(fact_161_SPEC__eq__is__RETURN_I1_J, axiom, (! [A:$tType,X4:A]: ((((refine605929679le_RES @ A) @ ((collect @ A) @ ((^ [Y5:A,Z:A]: ((Y5 = Z))) @ X4))) = ((refine1687780735RETURN @ A) @ X4))))).
% 2.90/1.56 thf(fact_162_RETURN__rule, axiom, (! [A:$tType,Phi:(A > $o),X4:A]: (((Phi @ X4) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X4)) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))))))).
% 2.90/1.56 thf(fact_163_lhs__step__RES, axiom, (! [A:$tType,X:(set @ A),M:(refine1665802226e_nres @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ X) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X3)) @ M)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ X)) @ M))))).
% 2.90/1.56 thf(fact_164_RETURN__to__SPEC__rule, axiom, (! [A:$tType,M:(refine1665802226e_nres @ A),V2:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ M) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ ((^ [Y5:A,Z:A]: ((Y5 = Z))) @ V2)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ M) @ ((refine1687780735RETURN @ A) @ V2)))))).
% 2.90/1.56 thf(fact_165_if__RETURN__refine, axiom, (! [A:$tType,B:$tType,B2:$o,B5:$o,S1:A,R:(set @ ((product_prod @ A) @ B)),S13:(refine1665802226e_nres @ B),S2:A,S23:(refine1665802226e_nres @ B)]: (((B2 = B5) => ((B2 => (B5 => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ S1)) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ S13)))) => (((~ (B2)) => ((~ (B5)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ S2)) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ S23)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ ((((if @ A) @ B2) @ S1) @ S2))) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ ((((if @ (refine1665802226e_nres @ B)) @ B5) @ S13) @ S23))))))))).
% 2.90/1.56 thf(fact_166_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J, axiom, (! [B:$tType,A:$tType,M3:(refine1665802226e_nres @ A),M6:(refine1665802226e_nres @ A),F:(A > (refine1665802226e_nres @ B)),F4:(A > (refine1665802226e_nres @ B))]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ M3) @ M6) => ((! [X3:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X3)) @ M3) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ (F @ X3)) @ (F4 @ X3))))) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine463715084e_bind @ A) @ B) @ M3) @ F)) @ ((((refine463715084e_bind @ A) @ B) @ M6) @ F4))))))).
% 2.90/1.56 thf(fact_167_bind__cong, axiom, (! [B:$tType,A:$tType,M:(refine1665802226e_nres @ A),M4:(refine1665802226e_nres @ A),F:(A > (refine1665802226e_nres @ B)),F4:(A > (refine1665802226e_nres @ B))]: (((M = M4) => ((! [X3:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X3)) @ M4) => ((F @ X3) = (F4 @ X3))))) => (((((refine463715084e_bind @ A) @ B) @ M) @ F) = ((((refine463715084e_bind @ A) @ B) @ M4) @ F4))))))).
% 2.90/1.56 thf(fact_168_inres__def, axiom, (! [A:$tType]: (((refine1315500908_inres @ A) = (^ [S4:(refine1665802226e_nres @ A),X5:A]: ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X5)) @ S4))))))).
% 2.90/1.57 thf(fact_169_RETURN__refine, axiom, (! [A:$tType,B:$tType,X4:A,X7:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ X7)) @ R) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X4)) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ ((refine1687780735RETURN @ B) @ X7))))))).
% 2.90/1.57 thf(fact_170_RETURN__ref__RETURND, axiom, (! [B:$tType,A:$tType,C2:A,R:(set @ ((product_prod @ A) @ B)),A2:B]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ C2)) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ ((refine1687780735RETURN @ B) @ A2))) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ C2) @ A2)) @ R))))).
% 2.90/1.57 thf(fact_171_RETURN__ref__SPECD, axiom, (! [A:$tType,B:$tType,C2:A,R:(set @ ((product_prod @ A) @ B)),Phi:(B > $o)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ C2)) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ ((refine605929679le_RES @ B) @ ((collect @ B) @ Phi)))) => (~ ((! [A4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ C2) @ A4)) @ R) => (~ ((Phi @ A4)))))))))))).
% 2.90/1.57 thf(fact_172_RETURN__SPEC__refine, axiom, (! [B:$tType,A:$tType,X4:B,R:(set @ ((product_prod @ B) @ A)),Phi:(A > $o)]: (((? [X12:A]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X4) @ X12)) @ R) & (Phi @ X12)))) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((refine1687780735RETURN @ B) @ X4)) @ ((((refine1073749519nc_fun @ B) @ A) @ R) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi)))))))).
% 2.90/1.57 thf(fact_173_pw__abs__inres, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ B) @ A)),M3:(refine1665802226e_nres @ B),A2:A]: (((((refine1315500908_inres @ A) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ M3)) @ A2) = (((refine1102455758nofail @ A) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ M3)) => (? [C5:B]: (((((refine1315500908_inres @ B) @ M3) @ C5) & (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ C5) @ A2)) @ R))))))))).
% 2.90/1.57 thf(fact_174_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R5:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I2:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I2) @ J)) @ R5) => (((ord_less_eq @ B) @ (As @ I2)) @ (As @ J)))))))))))).
% 2.90/1.57 thf(fact_175_uncurry__apply, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),A2:B,B2:C]: (((((((uncurry @ B) @ C) @ A) @ F) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((F @ A2) @ B2))))).
% 2.90/1.57 thf(fact_176_abs__trans__additional_I1_J, axiom, (! [A:$tType,B:$tType,A5:(refine1665802226e_nres @ A),B6:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),C3:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ A5) @ B6) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine81118332bs_fun @ A) @ B) @ R) @ B6)) @ C3) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine81118332bs_fun @ A) @ B) @ R) @ A5)) @ C3)))))).
% 2.90/1.57 thf(fact_177_abs__trans, axiom, (! [B:$tType,A:$tType,C:$tType,R:(set @ ((product_prod @ B) @ A)),C3:(refine1665802226e_nres @ B),B6:(refine1665802226e_nres @ A),R2:(set @ ((product_prod @ A) @ C)),A5:(refine1665802226e_nres @ C)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ C3)) @ B6) => ((((ord_less_eq @ (refine1665802226e_nres @ C)) @ ((((refine81118332bs_fun @ A) @ C) @ R2) @ B6)) @ A5) => (((ord_less_eq @ (refine1665802226e_nres @ C)) @ ((((refine81118332bs_fun @ A) @ C) @ R2) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ C3))) @ A5)))))).
% 2.90/1.57 thf(fact_178_uncurry__curry__id, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((uncurry @ A) @ B) @ C) @ ((((product_curry @ A) @ B) @ C) @ F)) = F)))).
% 2.90/1.57 thf(fact_179_curry__uncurry__id, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C))]: ((((((product_curry @ A) @ B) @ C) @ ((((uncurry @ A) @ B) @ C) @ F)) = F)))).
% 2.90/1.57 thf(fact_180_pw__abs__nofail, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ B) @ A)),M3:(refine1665802226e_nres @ B)]: ((((refine1102455758nofail @ A) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ M3)) = (((refine1102455758nofail @ B) @ M3) & (! [X5:B]: (((((refine1315500908_inres @ B) @ M3) @ X5) => (((member @ B) @ X5) @ (((domain @ B) @ A) @ R)))))))))).
% 2.90/1.57 thf(fact_181_curryI, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: (((F @ ((((product_Pair @ A) @ B) @ A2) @ B2)) => ((((((product_curry @ A) @ B) @ $o) @ F) @ A2) @ B2))))).
% 2.90/1.57 thf(fact_182_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F3:(((product_prod @ B) @ C) > A),A3:B,B3:C]: ((F3 @ ((((product_Pair @ B) @ C) @ A3) @ B3)))))))).
% 2.90/1.57 thf(fact_183_curryE, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A2) @ B2) => (F @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 2.90/1.57 thf(fact_184_curryD, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A2) @ B2) => (F @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 2.90/1.57 thf(fact_185_DomainE, axiom, (! [B:$tType,A:$tType,A2:A,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ (((domain @ A) @ B) @ R3)) => (~ ((! [B4:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B4)) @ R3))))))))))).
% 2.90/1.57 thf(fact_186_Domain__iff, axiom, (! [A:$tType,B:$tType,A2:A,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ (((domain @ A) @ B) @ R3)) = (? [Y6:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ Y6)) @ R3))))))).
% 2.90/1.57 thf(fact_187_Domain_Ocases, axiom, (! [B:$tType,A:$tType,A2:A,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ (((domain @ A) @ B) @ R3)) => (~ ((! [B4:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B4)) @ R3))))))))))).
% 2.90/1.57 thf(fact_188_Domain_Osimps, axiom, (! [B:$tType,A:$tType,A2:A,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ (((domain @ A) @ B) @ R3)) = (? [A3:A,B3:B]: (((A2 = A3) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R3)))))))).
% 2.90/1.57 thf(fact_189_Domain_ODomainI, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R3) => (((member @ A) @ A2) @ (((domain @ A) @ B) @ R3)))))).
% 2.90/1.57 thf(fact_190_Domain_Oinducts, axiom, (! [B:$tType,A:$tType,X4:A,R3:(set @ ((product_prod @ A) @ B)),P:(A > $o)]: (((((member @ A) @ X4) @ (((domain @ A) @ B) @ R3)) => ((! [A4:A,B4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ R3) => (P @ A4)))) => (P @ X4)))))).
% 2.90/1.57 thf(fact_191_Domain__mono, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B)),S9:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R3) @ S9) => (((ord_less_eq @ (set @ A)) @ (((domain @ A) @ B) @ R3)) @ (((domain @ A) @ B) @ S9)))))).
% 2.90/1.57 thf(fact_192_for__in__RI, axiom, (! [B:$tType,A:$tType,X4:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ X4) @ (((domain @ A) @ B) @ R)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ ((((fun_of_rel @ A) @ B) @ R) @ X4))) @ R))))).
% 2.90/1.57 thf(fact_193_abs__fun__simps_I3_J, axiom, (! [B:$tType,A:$tType,X:(set @ B),R:(set @ ((product_prod @ B) @ A))]: (((~ ((((ord_less_eq @ (set @ B)) @ X) @ (((domain @ B) @ A) @ R)))) => (((((refine81118332bs_fun @ B) @ A) @ R) @ ((refine605929679le_RES @ B) @ X)) = (top_top @ (refine1665802226e_nres @ A))))))).
% 2.90/1.57 thf(fact_194_abs__fun__simps_I2_J, axiom, (! [A:$tType,B:$tType,X:(set @ B),R:(set @ ((product_prod @ B) @ A))]: (((((ord_less_eq @ (set @ B)) @ X) @ (((domain @ B) @ A) @ R)) => (((((refine81118332bs_fun @ B) @ A) @ R) @ ((refine605929679le_RES @ B) @ X)) = ((refine605929679le_RES @ A) @ ((((image @ B) @ A) @ R) @ X))))))).
% 2.90/1.57 thf(fact_195_top__apply, axiom, (! [C:$tType,D2:$tType]: (((top @ C) => ((top_top @ (D2 > C)) = (^ [X5:D2]: ((top_top @ C)))))))).
% 2.90/1.57 thf(fact_196_ImageI, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,R3:(set @ ((product_prod @ A) @ B)),A5:(set @ A)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R3) => ((((member @ A) @ A2) @ A5) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ R3) @ A5))))))).
% 2.90/1.57 thf(fact_197_nres__order__simps_I4_J, axiom, (! [D2:$tType,M3:(refine1665802226e_nres @ D2)]: (((((ord_less_eq @ (refine1665802226e_nres @ D2)) @ (top_top @ (refine1665802226e_nres @ D2))) @ M3) = (M3 = (top_top @ (refine1665802226e_nres @ D2))))))).
% 2.90/1.57 thf(fact_198_conc__fun__FAIL, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: ((((((refine1073749519nc_fun @ A) @ B) @ R) @ (top_top @ (refine1665802226e_nres @ B))) = (top_top @ (refine1665802226e_nres @ A)))))).
% 2.90/1.57 thf(fact_199_conc__fun__fail__iff_I1_J, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),S:(refine1665802226e_nres @ B)]: (((((((refine1073749519nc_fun @ A) @ B) @ R) @ S) = (top_top @ (refine1665802226e_nres @ A))) = (S = (top_top @ (refine1665802226e_nres @ B))))))).
% 2.90/1.57 thf(fact_200_conc__fun__fail__iff_I2_J, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),S:(refine1665802226e_nres @ B)]: ((((top_top @ (refine1665802226e_nres @ A)) = ((((refine1073749519nc_fun @ A) @ B) @ R) @ S)) = (S = (top_top @ (refine1665802226e_nres @ B))))))).
% 2.90/1.57 thf(fact_201_bind__FAIL, axiom, (! [B:$tType,A:$tType,F:(B > (refine1665802226e_nres @ A))]: ((((((refine463715084e_bind @ B) @ A) @ (top_top @ (refine1665802226e_nres @ B))) @ F) = (top_top @ (refine1665802226e_nres @ A)))))).
% 2.90/1.57 thf(fact_202_nofail__simps_I1_J, axiom, (! [A:$tType]: ((~ (((refine1102455758nofail @ A) @ (top_top @ (refine1665802226e_nres @ A)))))))).
% 2.90/1.57 thf(fact_203_inres__simps_I1_J, axiom, (! [A:$tType]: ((((refine1315500908_inres @ A) @ (top_top @ (refine1665802226e_nres @ A))) = (^ [Uu3:A]: ($true)))))).
% 2.90/1.57 thf(fact_204_nres__simp__internals_I2_J, axiom, (! [B:$tType]: (((refine1767639642_FAILi @ B) = (top_top @ (refine1665802226e_nres @ B)))))).
% 2.90/1.57 thf(fact_205_abs__fun__simps_I1_J, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ B) @ A))]: ((((((refine81118332bs_fun @ B) @ A) @ R) @ (top_top @ (refine1665802226e_nres @ B))) = (top_top @ (refine1665802226e_nres @ A)))))).
% 2.90/1.57 thf(fact_206_FAIL__refine, axiom, (! [A:$tType,B:$tType,X:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B))]: ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ X) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ (top_top @ (refine1665802226e_nres @ B))))))).
% 2.90/1.57 thf(fact_207_rev__ImageI, axiom, (! [B:$tType,A:$tType,A2:A,A5:(set @ A),B2:B,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ A5) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R3) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ R3) @ A5))))))).
% 2.90/1.57 thf(fact_208_Image__iff, axiom, (! [A:$tType,B:$tType,B2:A,R3:(set @ ((product_prod @ B) @ A)),A5:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ R3) @ A5)) = (? [X5:B]: (((((member @ B) @ X5) @ A5) & (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X5) @ B2)) @ R3)))))))).
% 2.90/1.57 thf(fact_209_ImageE, axiom, (! [A:$tType,B:$tType,B2:A,R3:(set @ ((product_prod @ B) @ A)),A5:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ R3) @ A5)) => (~ ((! [X3:B]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X3) @ B2)) @ R3) => (~ ((((member @ B) @ X3) @ A5)))))))))))).
% 2.90/1.57 thf(fact_210_nres__inequalities_I3_J, axiom, (! [C:$tType,X4:C]: (((top_top @ (refine1665802226e_nres @ C)) != ((refine1687780735RETURN @ C) @ X4))))).
% 2.90/1.57 thf(fact_211_top__nres__def, axiom, (! [A:$tType]: (((top_top @ (refine1665802226e_nres @ A)) = (refine1767639642_FAILi @ A))))).
% 2.90/1.57 thf(fact_212_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ (top_top @ A)))))))).
% 2.90/1.57 thf(fact_213_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) = (A2 = (top_top @ A))))))))).
% 2.90/1.57 thf(fact_214_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) => (A2 = (top_top @ A))))))))).
% 2.90/1.57 thf(fact_215_nres__order__simps_I3_J, axiom, (! [C:$tType,M3:(refine1665802226e_nres @ C)]: ((((ord_less_eq @ (refine1665802226e_nres @ C)) @ M3) @ (top_top @ (refine1665802226e_nres @ C)))))).
% 2.90/1.57 thf(fact_216_nres__inequalities_I1_J, axiom, (! [A:$tType,X:(set @ A)]: (((top_top @ (refine1665802226e_nres @ A)) != ((refine605929679le_RES @ A) @ X))))).
% 2.90/1.57 thf(fact_217_nres__cases, axiom, (! [A:$tType,M3:(refine1665802226e_nres @ A)]: (((M3 != (top_top @ (refine1665802226e_nres @ A))) => (~ ((! [X13:(set @ A)]: ((M3 != ((refine605929679le_RES @ A) @ X13)))))))))).
% 2.90/1.57 thf(fact_218_not__nofail__iff, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A)]: (((~ (((refine1102455758nofail @ A) @ S))) = (S = (top_top @ (refine1665802226e_nres @ A))))))).
% 2.90/1.57 thf(fact_219_nofail__def, axiom, (! [A:$tType]: (((refine1102455758nofail @ A) = (^ [S4:(refine1665802226e_nres @ A)]: ((S4 != (top_top @ (refine1665802226e_nres @ A))))))))).
% 2.90/1.57 thf(fact_220_intro__nofail_I2_J, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A)]: ((((top_top @ (refine1665802226e_nres @ A)) != S) = ((refine1102455758nofail @ A) @ S))))).
% 2.90/1.57 thf(fact_221_ibind__strict_I2_J, axiom, (! [A:$tType,F:(product_unit > (refine1665802226e_nres @ A))]: ((((((refine463715084e_bind @ product_unit) @ A) @ (top_top @ (refine1665802226e_nres @ product_unit))) @ F) = (top_top @ (refine1665802226e_nres @ A)))))).
% 2.90/1.57 thf(fact_222_Image__mono, axiom, (! [B:$tType,A:$tType,R6:(set @ ((product_prod @ A) @ B)),R3:(set @ ((product_prod @ A) @ B)),A8:(set @ A),A5:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R6) @ R3) => ((((ord_less_eq @ (set @ A)) @ A8) @ A5) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ R6) @ A8)) @ ((((image @ A) @ B) @ R3) @ A5))))))).
% 2.90/1.57 thf(fact_223_meta__le__everything__if__top, axiom, (! [A:$tType]: (((order_top @ A) => (! [M:A,X4:A]: (((M = (top_top @ A)) => (((ord_less_eq @ A) @ X4) @ M)))))))).
% 2.90/1.57 thf(fact_224_pw__flat__ge__iff, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S3:(refine1665802226e_nres @ A)]: ((((((partial_flat_ord @ (refine1665802226e_nres @ A)) @ (top_top @ (refine1665802226e_nres @ A))) @ S) @ S3) = (((refine1102455758nofail @ A) @ S) => (((refine1102455758nofail @ A) @ S3) & (! [X5:A]: (((((refine1315500908_inres @ A) @ S) @ X5) = (((refine1315500908_inres @ A) @ S3) @ X5)))))))))).
% 2.90/1.57 thf(fact_225_subset__UNIV, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A5) @ (top_top @ (set @ A)))))).
% 2.90/1.57 thf(fact_226_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I2_J, axiom, (! [B:$tType,A:$tType,M3:(refine1665802226e_nres @ A),M6:(refine1665802226e_nres @ A),F:(A > (refine1665802226e_nres @ B)),F4:(A > (refine1665802226e_nres @ B))]: ((((((partial_flat_ord @ (refine1665802226e_nres @ A)) @ (top_top @ (refine1665802226e_nres @ A))) @ M3) @ M6) => ((! [X3:A]: (((((partial_flat_ord @ (refine1665802226e_nres @ B)) @ (top_top @ (refine1665802226e_nres @ B))) @ (F @ X3)) @ (F4 @ X3)))) => ((((partial_flat_ord @ (refine1665802226e_nres @ B)) @ (top_top @ (refine1665802226e_nres @ B))) @ ((((refine463715084e_bind @ A) @ B) @ M3) @ F)) @ ((((refine463715084e_bind @ A) @ B) @ M6) @ F4))))))).
% 2.90/1.57 thf(fact_227_flat__ord__compat_I2_J, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [X4:A,Y3:A]: ((((((partial_flat_ord @ A) @ (top_top @ A)) @ X4) @ Y3) => (((ord_less_eq @ A) @ Y3) @ X4)))))))).
% 2.90/1.57 thf(fact_228_imonad1, axiom, (! [A:$tType,U:product_unit,F:(product_unit > (refine1665802226e_nres @ A))]: ((((((refine463715084e_bind @ product_unit) @ A) @ ((refine1687780735RETURN @ product_unit) @ U)) @ F) = (F @ U))))).
% 2.90/1.57 thf(fact_229_flat__ord_OantimonoD, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [B2:A,F:(A > B),X4:A,Y3:A]: ((((((antimono @ A) @ B) @ ((partial_flat_ord @ A) @ B2)) @ F) => (((((partial_flat_ord @ A) @ B2) @ X4) @ Y3) => (((ord_less_eq @ B) @ (F @ Y3)) @ (F @ X4)))))))))).
% 2.90/1.57 thf(fact_230_flat__ord_OantimonoE, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [B2:A,F:(A > B),X4:A,Y3:A]: ((((((antimono @ A) @ B) @ ((partial_flat_ord @ A) @ B2)) @ F) => (((((partial_flat_ord @ A) @ B2) @ X4) @ Y3) => (((ord_less_eq @ B) @ (F @ Y3)) @ (F @ X4)))))))))).
% 2.90/1.57 thf(fact_231_UNIV__I, axiom, (! [A:$tType,X4:A]: ((((member @ A) @ X4) @ (top_top @ (set @ A)))))).
% 2.90/1.57 thf(fact_232_relprop__UNIV__orient, axiom, (! [A:$tType,R:(set @ A)]: (((R = (top_top @ (set @ A))) => ((top_top @ (set @ A)) = R))))).
% 2.90/1.57 thf(fact_233_eq__UNIV__iff, axiom, (! [A:$tType,S:(set @ A)]: (((S = (top_top @ (set @ A))) = (! [X5:A]: ((((member @ A) @ X5) @ S))))))).
% 2.90/1.57 thf(fact_234_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 2.90/1.57 thf(fact_235_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.90/1.57 thf(fact_236_UNIV__eq__I, axiom, (! [A:$tType,A5:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A5))) => ((top_top @ (set @ A)) = A5))))).
% 2.90/1.57 thf(fact_237_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X5:A]: ((((member @ A) @ X5) @ (top_top @ (set @ A))))))))).
% 2.90/1.57 thf(fact_238_order_Oantimono_Ocong, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (((antimono @ A) @ B) = ((antimono @ A) @ B)))))).
% 2.90/1.57 thf(fact_239_flat__ord_Oantimono__def, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [B2:A,F:(A > B)]: ((((((antimono @ A) @ B) @ ((partial_flat_ord @ A) @ B2)) @ F) = (! [X5:A,Y6:A]: ((((((partial_flat_ord @ A) @ B2) @ X5) @ Y6) => (((ord_less_eq @ B) @ (F @ Y6)) @ (F @ X5)))))))))))).
% 2.90/1.57 thf(fact_240_flat__ord_OantimonoI, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [B2:A,F:(A > B)]: (((! [X3:A,Y4:A]: ((((((partial_flat_ord @ A) @ B2) @ X3) @ Y4) => (((ord_less_eq @ B) @ (F @ Y4)) @ (F @ X3))))) => ((((antimono @ A) @ B) @ ((partial_flat_ord @ A) @ B2)) @ F)))))))).
% 2.90/1.57 thf(fact_241_flat__ord_Omono__def, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [B2:A,F:(A > B)]: ((((((mono @ A) @ B) @ ((partial_flat_ord @ A) @ B2)) @ F) = (! [X5:A,Y6:A]: ((((((partial_flat_ord @ A) @ B2) @ X5) @ Y6) => (((ord_less_eq @ B) @ (F @ X5)) @ (F @ Y6)))))))))))).
% 2.90/1.57 thf(fact_242_flat__ord_OmonoI, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [B2:A,F:(A > B)]: (((! [X3:A,Y4:A]: ((((((partial_flat_ord @ A) @ B2) @ X3) @ Y4) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y4))))) => ((((mono @ A) @ B) @ ((partial_flat_ord @ A) @ B2)) @ F)))))))).
% 2.90/1.57 thf(fact_243_top1I, axiom, (! [A:$tType,X4:A]: (((top_top @ (A > $o)) @ X4)))).
% 2.90/1.57 thf(fact_244_order_Omono_Ocong, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (((mono @ A) @ B) = ((mono @ A) @ B)))))).
% 2.90/1.57 thf(fact_245_flat__ord_OmonoD, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [B2:A,F:(A > B),X4:A,Y3:A]: ((((((mono @ A) @ B) @ ((partial_flat_ord @ A) @ B2)) @ F) => (((((partial_flat_ord @ A) @ B2) @ X4) @ Y3) => (((ord_less_eq @ B) @ (F @ X4)) @ (F @ Y3)))))))))).
% 2.90/1.57 thf(fact_246_flat__ord_OmonoE, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [B2:A,F:(A > B),X4:A,Y3:A]: ((((((mono @ A) @ B) @ ((partial_flat_ord @ A) @ B2)) @ F) => (((((partial_flat_ord @ A) @ B2) @ X4) @ Y3) => (((ord_less_eq @ B) @ (F @ X4)) @ (F @ Y3)))))))))).
% 2.90/1.57 thf(fact_247_pw__flat__le__iff, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S3:(refine1665802226e_nres @ A)]: ((((((partial_flat_ord @ (refine1665802226e_nres @ A)) @ (bot_bot @ (refine1665802226e_nres @ A))) @ S) @ S3) = ((? [X14:A]: ((((refine1315500908_inres @ A) @ S) @ X14))) => ((((refine1102455758nofail @ A) @ S) = ((refine1102455758nofail @ A) @ S3)) & (! [X5:A]: (((((refine1315500908_inres @ A) @ S) @ X5) = (((refine1315500908_inres @ A) @ S3) @ X5)))))))))).
% 2.90/1.57 thf(fact_248_ASSERT__simps_I2_J, axiom, ((refine1814851989ASSERT @ $false) = (top_top @ (refine1665802226e_nres @ product_unit)))).
% 2.90/1.57 thf(fact_249_bot__apply, axiom, (! [C:$tType,D2:$tType]: (((bot @ C) => ((bot_bot @ (D2 > C)) = (^ [X5:D2]: ((bot_bot @ C)))))))).
% 2.90/1.57 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A9:$tType,A10:$tType]: (((comple187826305attice @ A10) => (comple187826305attice @ (A9 > A10)))))).
% 2.90/1.57 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A9:$tType,A10:$tType]: (((order_top @ A10) => (order_top @ (A9 > A10)))))).
% 2.90/1.57 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A10:$tType]: (((preorder @ A10) => (preorder @ (A9 > A10)))))).
% 2.90/1.57 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: (((order @ A10) => (order @ (A9 > A10)))))).
% 2.90/1.57 thf(tcon_fun___Orderings_Otop, axiom, (! [A9:$tType,A10:$tType]: (((top @ A10) => (top @ (A9 > A10)))))).
% 2.90/1.57 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 2.90/1.57 thf(tcon_fun___Orderings_Obot, axiom, (! [A9:$tType,A10:$tType]: (((bot @ A10) => (bot @ (A9 > A10)))))).
% 2.90/1.57 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_1, axiom, (! [A9:$tType]: ((comple187826305attice @ (set @ A9))))).
% 2.90/1.57 thf(tcon_Set_Oset___Orderings_Oorder__top_2, axiom, (! [A9:$tType]: ((order_top @ (set @ A9))))).
% 2.90/1.57 thf(tcon_Set_Oset___Orderings_Opreorder_3, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 2.90/1.57 thf(tcon_Set_Oset___Orderings_Oorder_4, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 2.90/1.57 thf(tcon_Set_Oset___Orderings_Otop_5, axiom, (! [A9:$tType]: ((top @ (set @ A9))))).
% 2.90/1.57 thf(tcon_Set_Oset___Orderings_Oord_6, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 2.90/1.57 thf(tcon_Set_Oset___Orderings_Obot_7, axiom, (! [A9:$tType]: ((bot @ (set @ A9))))).
% 2.90/1.57 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_8, axiom, (comple187826305attice @ $o)).
% 2.90/1.57 thf(tcon_HOL_Obool___Orderings_Oorder__top_9, axiom, (order_top @ $o)).
% 2.90/1.57 thf(tcon_HOL_Obool___Orderings_Opreorder_10, axiom, (preorder @ $o)).
% 2.90/1.57 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.90/1.57 thf(tcon_HOL_Obool___Orderings_Oorder_11, axiom, (order @ $o)).
% 2.90/1.57 thf(tcon_HOL_Obool___Orderings_Otop_12, axiom, (top @ $o)).
% 2.90/1.57 thf(tcon_HOL_Obool___Orderings_Oord_13, axiom, (ord @ $o)).
% 2.90/1.57 thf(tcon_HOL_Obool___Orderings_Obot_14, axiom, (bot @ $o)).
% 2.90/1.57 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_15, axiom, (comple187826305attice @ product_unit)).
% 2.90/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_16, axiom, (order_top @ product_unit)).
% 2.90/1.57 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_17, axiom, (preorder @ product_unit)).
% 2.90/1.57 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_18, axiom, (linorder @ product_unit)).
% 2.90/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oorder_19, axiom, (order @ product_unit)).
% 2.90/1.57 thf(tcon_Product__Type_Ounit___Orderings_Otop_20, axiom, (top @ product_unit)).
% 2.90/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oord_21, axiom, (ord @ product_unit)).
% 2.90/1.57 thf(tcon_Product__Type_Ounit___Orderings_Obot_22, axiom, (bot @ product_unit)).
% 2.90/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Complete__Lattices_Ocomplete__lattice_23, axiom, (! [A9:$tType]: ((comple187826305attice @ (refine1665802226e_nres @ A9))))).
% 2.90/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__top_24, axiom, (! [A9:$tType]: ((order_top @ (refine1665802226e_nres @ A9))))).
% 2.90/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_25, axiom, (! [A9:$tType]: ((preorder @ (refine1665802226e_nres @ A9))))).
% 2.90/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_26, axiom, (! [A9:$tType]: ((order @ (refine1665802226e_nres @ A9))))).
% 2.90/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_27, axiom, (! [A9:$tType]: ((top @ (refine1665802226e_nres @ A9))))).
% 2.90/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_28, axiom, (! [A9:$tType]: ((ord @ (refine1665802226e_nres @ A9))))).
% 2.90/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Obot_29, axiom, (! [A9:$tType]: ((bot @ (refine1665802226e_nres @ A9))))).
% 2.90/1.57 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.90/1.57 thf(help_If_2_1_T, axiom, (! [A:$tType,X4:A,Y3:A]: ((((((if @ A) @ $false) @ X4) @ Y3) = Y3)))).
% 2.90/1.57 thf(help_If_1_1_T, axiom, (! [A:$tType,X4:A,Y3:A]: ((((((if @ A) @ $true) @ X4) @ Y3) = X4)))).
% 2.90/1.57 thf(conj_0, conjecture, (((ord_less_eq @ (refine1665802226e_nres @ b)) @ m) @ ((((refine463715084e_bind @ a) @ b) @ ((refine605929679le_RES @ a) @ ((collect @ a) @ phi))) @ f))).
% 2.90/1.57 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.8D84XgAAWH/DTF2DTF13304.p
% 2.90/1.57 ------------------------
% 3.06/1.61 ---- Cleaned THF ---
% 3.06/1.61 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
% 3.06/1.61 refine1665802226e_nres: $tType > $tType ).
% 3.06/1.61
% 3.06/1.61 thf(ty_t_Product__Type_Ounit,type,
% 3.06/1.61 product_unit: $tType ).
% 3.06/1.61
% 3.06/1.61 thf(ty_t_Product__Type_Oprod,type,
% 3.06/1.61 product_prod: $tType > $tType > $tType ).
% 3.06/1.61
% 3.06/1.61 thf(ty_t_Set_Oset,type,
% 3.06/1.61 set: $tType > $tType ).
% 3.06/1.61
% 3.06/1.61 thf(ty_tf_b,type,
% 3.06/1.61 b: $tType ).
% 3.06/1.61
% 3.06/1.61 thf(ty_tf_a,type,
% 3.06/1.61 a: $tType ).
% 3.06/1.61
% 3.06/1.61 thf(sy_cl_HOL_Otype,type,
% 3.06/1.61 type:
% 3.06/1.61 !>[A: $tType] : $o ).
% 3.06/1.61
% 3.06/1.61 thf(sy_cl_Orderings_Obot,type,
% 3.06/1.61 bot:
% 3.06/1.61 !>[A: $tType] : $o ).
% 3.06/1.61
% 3.06/1.61 thf(sy_cl_Orderings_Oord,type,
% 3.06/1.61 ord:
% 3.06/1.61 !>[A: $tType] : $o ).
% 3.06/1.61
% 3.06/1.61 thf(sy_cl_Orderings_Otop,type,
% 3.06/1.61 top:
% 3.06/1.61 !>[A: $tType] : $o ).
% 3.06/1.61
% 3.06/1.61 thf(sy_cl_Orderings_Oorder,type,
% 3.06/1.61 order:
% 3.06/1.61 !>[A: $tType] : $o ).
% 3.06/1.61
% 3.06/1.61 thf(sy_cl_Orderings_Olinorder,type,
% 3.06/1.61 linorder:
% 3.06/1.61 !>[A: $tType] : $o ).
% 3.06/1.61
% 3.06/1.61 thf(sy_cl_Orderings_Opreorder,type,
% 3.06/1.61 preorder:
% 3.06/1.61 !>[A: $tType] : $o ).
% 3.06/1.61
% 3.06/1.61 thf(sy_cl_Orderings_Oorder__top,type,
% 3.06/1.61 order_top:
% 3.06/1.61 !>[A: $tType] : $o ).
% 3.06/1.61
% 3.06/1.61 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 3.06/1.61 comple187826305attice:
% 3.06/1.61 !>[A: $tType] : $o ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 3.06/1.61 bNF_Ca1785829860lChain:
% 3.06/1.61 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_If,type,
% 3.06/1.61 if:
% 3.06/1.61 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Misc_Ofun__of__rel,type,
% 3.06/1.61 fun_of_rel:
% 3.06/1.61 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Misc_Ouncurry,type,
% 3.06/1.61 uncurry:
% 3.06/1.61 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Orderings_Obot__class_Obot,type,
% 3.06/1.61 bot_bot:
% 3.06/1.61 !>[A: $tType] : A ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.06/1.61 ord_less_eq:
% 3.06/1.61 !>[A: $tType] : ( A > A > $o ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Orderings_Oorder_Oantimono,type,
% 3.06/1.61 antimono:
% 3.06/1.61 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Orderings_Oorder_Omono,type,
% 3.06/1.61 mono:
% 3.06/1.61 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 3.06/1.61 order_Greatest:
% 3.06/1.61 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Orderings_Otop__class_Otop,type,
% 3.06/1.61 top_top:
% 3.06/1.61 !>[A: $tType] : A ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Partial__Function_Oflat__ord,type,
% 3.06/1.61 partial_flat_ord:
% 3.06/1.61 !>[A: $tType] : ( A > A > A > $o ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Product__Type_OPair,type,
% 3.06/1.61 product_Pair:
% 3.06/1.61 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Product__Type_Ocurry,type,
% 3.06/1.61 product_curry:
% 3.06/1.61 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 3.06/1.61 produc2004651681e_prod:
% 3.06/1.61 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 3.06/1.61 product_rec_prod:
% 3.06/1.61 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_OASSERT,type,
% 3.06/1.61 refine1814851989ASSERT: $o > ( refine1665802226e_nres @ product_unit ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
% 3.06/1.61 refine1687780735RETURN:
% 3.06/1.61 !>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oabs__fun,type,
% 3.06/1.61 refine81118332bs_fun:
% 3.06/1.61 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind,type,
% 3.06/1.61 refine463715084e_bind:
% 3.06/1.61 !>[B: $tType,A: $tType] : ( ( refine1665802226e_nres @ B ) > ( B > ( refine1665802226e_nres @ A ) ) > ( refine1665802226e_nres @ A ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oconc__fun,type,
% 3.06/1.61 refine1073749519nc_fun:
% 3.06/1.61 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres,type,
% 3.06/1.61 refine1315500908_inres:
% 3.06/1.61 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Olift__assn,type,
% 3.06/1.61 refine1580981607t_assn:
% 3.06/1.61 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( B > $o ) > A > $o ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres,type,
% 3.06/1.61 refine406925620_inres:
% 3.06/1.61 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail,type,
% 3.06/1.61 refine1102455758nofail:
% 3.06/1.61 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > $o ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_OFAILi,type,
% 3.06/1.61 refine1767639642_FAILi:
% 3.06/1.61 !>[A: $tType] : ( refine1665802226e_nres @ A ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES,type,
% 3.06/1.61 refine605929679le_RES:
% 3.06/1.61 !>[A: $tType] : ( ( set @ A ) > ( refine1665802226e_nres @ A ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_Orec__nres,type,
% 3.06/1.61 refine1442219249c_nres:
% 3.06/1.61 !>[C: $tType,A: $tType] : ( C > ( ( set @ A ) > C ) > ( refine1665802226e_nres @ A ) > C ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES,type,
% 3.06/1.61 refine1672542526he_RES:
% 3.06/1.61 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > ( set @ A ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Relation_ODomain,type,
% 3.06/1.61 domain:
% 3.06/1.61 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Relation_OImage,type,
% 3.06/1.61 image:
% 3.06/1.61 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_Set_OCollect,type,
% 3.06/1.61 collect:
% 3.06/1.61 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_c_member,type,
% 3.06/1.61 member:
% 3.06/1.61 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_v__092_060Phi_062,type,
% 3.06/1.61 phi: a > $o ).
% 3.06/1.61
% 3.06/1.61 thf(sy_v_f,type,
% 3.06/1.61 f: a > ( refine1665802226e_nres @ b ) ).
% 3.06/1.61
% 3.06/1.61 thf(sy_v_m,type,
% 3.06/1.61 m: refine1665802226e_nres @ b ).
% 3.06/1.61
% 3.06/1.61 thf(sy_v_x,type,
% 3.06/1.61 x: a ).
% 3.06/1.61
% 3.06/1.61 thf(fact_0_assms_I1_J,axiom,
% 3.06/1.61 phi @ x ).
% 3.06/1.61
% 3.06/1.61 thf(fact_1_assms_I2_J,axiom,
% 3.06/1.61 ord_less_eq @ ( refine1665802226e_nres @ b ) @ m @ ( f @ x ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_2_nres__more__simps_I4_J,axiom,
% 3.06/1.61 ! [A: $tType,X: set @ A,Y: set @ A] :
% 3.06/1.61 ( ( ( refine605929679le_RES @ A @ X )
% 3.06/1.61 = ( refine605929679le_RES @ A @ Y ) )
% 3.06/1.61 = ( X = Y ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_3_nres_Oinject,axiom,
% 3.06/1.61 ! [A: $tType,X2: set @ A,Y2: set @ A] :
% 3.06/1.61 ( ( ( refine605929679le_RES @ A @ X2 )
% 3.06/1.61 = ( refine605929679le_RES @ A @ Y2 ) )
% 3.06/1.61 = ( X2 = Y2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_4_if__rule,axiom,
% 3.06/1.61 ! [A: $tType,B2: $o,S1: refine1665802226e_nres @ A,Phi: A > $o,S2: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( B2
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S1 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) )
% 3.06/1.61 => ( ( ~ B2
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( if @ ( refine1665802226e_nres @ A ) @ B2 @ S1 @ S2 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_5_RES__rule,axiom,
% 3.06/1.61 ! [A: $tType,S: set @ A,Phi: A > $o] :
% 3.06/1.61 ( ! [X3: A] :
% 3.06/1.61 ( ( member @ A @ X3 @ S )
% 3.06/1.61 => ( Phi @ X3 ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ S ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_6_SPEC__rule,axiom,
% 3.06/1.61 ! [A: $tType,Phi: A > $o,Phi2: A > $o] :
% 3.06/1.61 ( ! [X3: A] :
% 3.06/1.61 ( ( Phi @ X3 )
% 3.06/1.61 => ( Phi2 @ X3 ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi2 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_7_SPEC__trans,axiom,
% 3.06/1.61 ! [A: $tType,X4: refine1665802226e_nres @ A,Y3: refine1665802226e_nres @ A,Postcond: A > $o] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ Y3 )
% 3.06/1.61 => ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Y3 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_8_lhs__step__If,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( ord @ A )
% 3.06/1.61 => ! [B2: $o,T2: A,M: A,E: A] :
% 3.06/1.61 ( ( B2
% 3.06/1.61 => ( ord_less_eq @ A @ T2 @ M ) )
% 3.06/1.61 => ( ( ~ B2
% 3.06/1.61 => ( ord_less_eq @ A @ E @ M ) )
% 3.06/1.61 => ( ord_less_eq @ A @ ( if @ A @ B2 @ T2 @ E ) @ M ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_9_SPEC__cons__rule,axiom,
% 3.06/1.61 ! [A: $tType,M: refine1665802226e_nres @ A,Phi: A > $o,Psi: A > $o] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
% 3.06/1.61 => ( ! [X3: A] :
% 3.06/1.61 ( ( Phi @ X3 )
% 3.06/1.61 => ( Psi @ X3 ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_10_RES__bind__choose,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,X4: A,X: set @ A,M: refine1665802226e_nres @ B,F: A > ( refine1665802226e_nres @ B )] :
% 3.06/1.61 ( ( member @ A @ X4 @ X )
% 3.06/1.61 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( F @ X4 ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine463715084e_bind @ A @ B @ ( refine605929679le_RES @ A @ X ) @ F ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_11_order__mono__setup_Orefl,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( preorder @ A )
% 3.06/1.61 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_12_the__RES_Osimps,axiom,
% 3.06/1.61 ! [A: $tType,X: set @ A] :
% 3.06/1.61 ( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X ) )
% 3.06/1.61 = X ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_13_nf__inres__RES,axiom,
% 3.06/1.61 ! [A: $tType,X: set @ A,X4: A] :
% 3.06/1.61 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ X ) @ X4 )
% 3.06/1.61 = ( member @ A @ X4 @ X ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_14_nf__inres__SPEC,axiom,
% 3.06/1.61 ! [A: $tType,Phi: A > $o,X4: A] :
% 3.06/1.61 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ X4 )
% 3.06/1.61 = ( Phi @ X4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_15_nres_Osimps_I7_J,axiom,
% 3.06/1.61 ! [C: $tType,A: $tType,F1: C,F2: ( set @ A ) > C,X2: set @ A] :
% 3.06/1.61 ( ( refine1442219249c_nres @ C @ A @ F1 @ F2 @ ( refine605929679le_RES @ A @ X2 ) )
% 3.06/1.61 = ( F2 @ X2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_16_le__funD,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( ord @ B )
% 3.06/1.61 => ! [F: A > B,G: A > B,X4: A] :
% 3.06/1.61 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.06/1.61 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_17_le__funE,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( ord @ B )
% 3.06/1.61 => ! [F: A > B,G: A > B,X4: A] :
% 3.06/1.61 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.06/1.61 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_18_le__funI,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( ord @ B )
% 3.06/1.61 => ! [F: A > B,G: A > B] :
% 3.06/1.61 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 3.06/1.61 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_19_Refine__Misc_Oif__mono,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( ord @ A )
% 3.06/1.61 => ! [B2: $o,M1: A,M12: A,M2: A,M22: A] :
% 3.06/1.61 ( ( B2
% 3.06/1.61 => ( ord_less_eq @ A @ M1 @ M12 ) )
% 3.06/1.61 => ( ( ~ B2
% 3.06/1.61 => ( ord_less_eq @ A @ M2 @ M22 ) )
% 3.06/1.61 => ( ord_less_eq @ A @ ( if @ A @ B2 @ M1 @ M2 ) @ ( if @ A @ B2 @ M12 @ M22 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_20_le__fun__def,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( ord @ B )
% 3.06/1.61 => ( ( ord_less_eq @ ( A > B ) )
% 3.06/1.61 = ( ^ [F3: A > B,G2: A > B] :
% 3.06/1.61 ! [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G2 @ X5 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_21_order__subst1,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType] :
% 3.06/1.61 ( ( ( order @ B )
% 3.06/1.61 & ( order @ A ) )
% 3.06/1.61 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 3.06/1.61 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 3.06/1.61 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 3.06/1.61 => ( ! [X3: B,Y4: B] :
% 3.06/1.61 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 3.06/1.61 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 3.06/1.61 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_22_less__eq__nres_Osimps_I2_J,axiom,
% 3.06/1.61 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) )
% 3.06/1.61 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_23_nres__order__simps_I5_J,axiom,
% 3.06/1.61 ! [E2: $tType,X: set @ E2,Y: set @ E2] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ E2 ) @ ( refine605929679le_RES @ E2 @ X ) @ ( refine605929679le_RES @ E2 @ Y ) )
% 3.06/1.61 = ( ord_less_eq @ ( set @ E2 ) @ X @ Y ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_24_order__mono__setup_Omono__let,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType] :
% 3.06/1.61 ( ( preorder @ A )
% 3.06/1.61 => ! [F: B > A,F4: B > A,X4: B] :
% 3.06/1.61 ( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( F4 @ X3 ) )
% 3.06/1.61 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F4 @ X4 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_25_order__mono__setup_Omono__if,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( preorder @ A )
% 3.06/1.61 => ! [T2: A,T3: A,E: A,E3: A,B2: $o] :
% 3.06/1.61 ( ( ord_less_eq @ A @ T2 @ T3 )
% 3.06/1.61 => ( ( ord_less_eq @ A @ E @ E3 )
% 3.06/1.61 => ( ord_less_eq @ A @ ( if @ A @ B2 @ T2 @ E ) @ ( if @ A @ B2 @ T3 @ E3 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_26_dual__order_Oantisym,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order @ A )
% 3.06/1.61 => ! [B2: A,A2: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.06/1.61 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.06/1.61 => ( A2 = B2 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_27_dual__order_Oeq__iff,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order @ A )
% 3.06/1.61 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 3.06/1.61 = ( ^ [A3: A,B3: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.06/1.61 & ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_28_dual__order_Otrans,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order @ A )
% 3.06/1.61 => ! [B2: A,A2: A,C2: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.06/1.61 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 3.06/1.61 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_29_linorder__wlog,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( linorder @ A )
% 3.06/1.61 => ! [P: A > A > $o,A2: A,B2: A] :
% 3.06/1.61 ( ! [A4: A,B4: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.06/1.61 => ( P @ A4 @ B4 ) )
% 3.06/1.61 => ( ! [A4: A,B4: A] :
% 3.06/1.61 ( ( P @ B4 @ A4 )
% 3.06/1.61 => ( P @ A4 @ B4 ) )
% 3.06/1.61 => ( P @ A2 @ B2 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_30_dual__order_Orefl,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order @ A )
% 3.06/1.61 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_31_order__trans,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( preorder @ A )
% 3.06/1.61 => ! [X4: A,Y3: A,Z2: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 3.06/1.61 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 3.06/1.61 => ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_32_order__class_Oorder_Oantisym,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order @ A )
% 3.06/1.61 => ! [A2: A,B2: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.06/1.61 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.06/1.61 => ( A2 = B2 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_33_ord__le__eq__trans,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( ord @ A )
% 3.06/1.61 => ! [A2: A,B2: A,C2: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.06/1.61 => ( ( B2 = C2 )
% 3.06/1.61 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_34_ord__eq__le__trans,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( ord @ A )
% 3.06/1.61 => ! [A2: A,B2: A,C2: A] :
% 3.06/1.61 ( ( A2 = B2 )
% 3.06/1.61 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 3.06/1.61 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_35_order__class_Oorder_Oeq__iff,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order @ A )
% 3.06/1.61 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 3.06/1.61 = ( ^ [A3: A,B3: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.06/1.61 & ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_36_antisym__conv,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order @ A )
% 3.06/1.61 => ! [Y3: A,X4: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ Y3 @ X4 )
% 3.06/1.61 => ( ( ord_less_eq @ A @ X4 @ Y3 )
% 3.06/1.61 = ( X4 = Y3 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_37_le__cases3,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( linorder @ A )
% 3.06/1.61 => ! [X4: A,Y3: A,Z2: A] :
% 3.06/1.61 ( ( ( ord_less_eq @ A @ X4 @ Y3 )
% 3.06/1.61 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 3.06/1.61 => ( ( ( ord_less_eq @ A @ Y3 @ X4 )
% 3.06/1.61 => ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
% 3.06/1.61 => ( ( ( ord_less_eq @ A @ X4 @ Z2 )
% 3.06/1.61 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 3.06/1.61 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 3.06/1.61 => ~ ( ord_less_eq @ A @ Y3 @ X4 ) )
% 3.06/1.61 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 3.06/1.61 => ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
% 3.06/1.61 => ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
% 3.06/1.61 => ~ ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_38_order_Otrans,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order @ A )
% 3.06/1.61 => ! [A2: A,B2: A,C2: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.06/1.61 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 3.06/1.61 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_39_le__cases,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( linorder @ A )
% 3.06/1.61 => ! [X4: A,Y3: A] :
% 3.06/1.61 ( ~ ( ord_less_eq @ A @ X4 @ Y3 )
% 3.06/1.61 => ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_40_eq__refl,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( preorder @ A )
% 3.06/1.61 => ! [X4: A,Y3: A] :
% 3.06/1.61 ( ( X4 = Y3 )
% 3.06/1.61 => ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_41_linear,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( linorder @ A )
% 3.06/1.61 => ! [X4: A,Y3: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 3.06/1.61 | ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_42_antisym,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order @ A )
% 3.06/1.61 => ! [X4: A,Y3: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 3.06/1.61 => ( ( ord_less_eq @ A @ Y3 @ X4 )
% 3.06/1.61 => ( X4 = Y3 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_43_mem__Collect__eq,axiom,
% 3.06/1.61 ! [A: $tType,A2: A,P: A > $o] :
% 3.06/1.61 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 3.06/1.61 = ( P @ A2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_44_Collect__mem__eq,axiom,
% 3.06/1.61 ! [A: $tType,A5: set @ A] :
% 3.06/1.61 ( ( collect @ A
% 3.06/1.61 @ ^ [X5: A] : ( member @ A @ X5 @ A5 ) )
% 3.06/1.61 = A5 ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_45_Collect__cong,axiom,
% 3.06/1.61 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.06/1.61 ( ! [X3: A] :
% 3.06/1.61 ( ( P @ X3 )
% 3.06/1.61 = ( Q @ X3 ) )
% 3.06/1.61 => ( ( collect @ A @ P )
% 3.06/1.61 = ( collect @ A @ Q ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_46_ext,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.06/1.61 ( ! [X3: A] :
% 3.06/1.61 ( ( F @ X3 )
% 3.06/1.61 = ( G @ X3 ) )
% 3.06/1.61 => ( F = G ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_47_eq__iff,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order @ A )
% 3.06/1.61 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 3.06/1.61 = ( ^ [X5: A,Y6: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ X5 @ Y6 )
% 3.06/1.61 & ( ord_less_eq @ A @ Y6 @ X5 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_48_ord__le__eq__subst,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType] :
% 3.06/1.61 ( ( ( ord @ B )
% 3.06/1.61 & ( ord @ A ) )
% 3.06/1.61 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 3.06/1.61 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.06/1.61 => ( ( ( F @ B2 )
% 3.06/1.61 = C2 )
% 3.06/1.61 => ( ! [X3: A,Y4: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 3.06/1.61 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 3.06/1.61 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_49_ord__eq__le__subst,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType] :
% 3.06/1.61 ( ( ( ord @ B )
% 3.06/1.61 & ( ord @ A ) )
% 3.06/1.61 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 3.06/1.61 ( ( A2
% 3.06/1.61 = ( F @ B2 ) )
% 3.06/1.61 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 3.06/1.61 => ( ! [X3: B,Y4: B] :
% 3.06/1.61 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 3.06/1.61 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 3.06/1.61 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_50_order__subst2,axiom,
% 3.06/1.61 ! [A: $tType,C: $tType] :
% 3.06/1.61 ( ( ( order @ C )
% 3.06/1.61 & ( order @ A ) )
% 3.06/1.61 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 3.06/1.61 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.06/1.61 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 3.06/1.61 => ( ! [X3: A,Y4: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 3.06/1.61 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 3.06/1.61 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_51_the__RES__inv,axiom,
% 3.06/1.61 ! [A: $tType,M: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( refine1102455758nofail @ A @ M )
% 3.06/1.61 => ( ( refine605929679le_RES @ A @ ( refine1672542526he_RES @ A @ M ) )
% 3.06/1.61 = M ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_52_Greatest__equality,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order @ A )
% 3.06/1.61 => ! [P: A > $o,X4: A] :
% 3.06/1.61 ( ( P @ X4 )
% 3.06/1.61 => ( ! [Y4: A] :
% 3.06/1.61 ( ( P @ Y4 )
% 3.06/1.61 => ( ord_less_eq @ A @ Y4 @ X4 ) )
% 3.06/1.61 => ( ( order_Greatest @ A @ P )
% 3.06/1.61 = X4 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_53_GreatestI2__order,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order @ A )
% 3.06/1.61 => ! [P: A > $o,X4: A,Q: A > $o] :
% 3.06/1.61 ( ( P @ X4 )
% 3.06/1.61 => ( ! [Y4: A] :
% 3.06/1.61 ( ( P @ Y4 )
% 3.06/1.61 => ( ord_less_eq @ A @ Y4 @ X4 ) )
% 3.06/1.61 => ( ! [X3: A] :
% 3.06/1.61 ( ( P @ X3 )
% 3.06/1.61 => ( ! [Y7: A] :
% 3.06/1.61 ( ( P @ Y7 )
% 3.06/1.61 => ( ord_less_eq @ A @ Y7 @ X3 ) )
% 3.06/1.61 => ( Q @ X3 ) ) )
% 3.06/1.61 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_54_transfer_Otransfer__Let,axiom,
% 3.06/1.61 ! [C: $tType,A: $tType,B: $tType] :
% 3.06/1.61 ( ( comple187826305attice @ A )
% 3.06/1.61 => ! [Alpha: C > A,F: B > C,F5: B > A,X4: B] :
% 3.06/1.61 ( ! [X3: B] : ( ord_less_eq @ A @ ( Alpha @ ( F @ X3 ) ) @ ( F5 @ X3 ) )
% 3.06/1.61 => ( ord_less_eq @ A @ ( Alpha @ ( F @ X4 ) ) @ ( F5 @ X4 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_55_transfer_Otransfer__if,axiom,
% 3.06/1.61 ! [C: $tType,A: $tType] :
% 3.06/1.61 ( ( comple187826305attice @ A )
% 3.06/1.61 => ! [B2: $o,Alpha: C > A,S12: C,S1: A,S22: C,S2: A] :
% 3.06/1.61 ( ( B2
% 3.06/1.61 => ( ord_less_eq @ A @ ( Alpha @ S12 ) @ S1 ) )
% 3.06/1.61 => ( ( ~ B2
% 3.06/1.61 => ( ord_less_eq @ A @ ( Alpha @ S22 ) @ S2 ) )
% 3.06/1.61 => ( ord_less_eq @ A @ ( Alpha @ ( if @ C @ B2 @ S12 @ S22 ) ) @ ( if @ A @ B2 @ S1 @ S2 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_56_le__rel__bool__arg__iff,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( ord @ A )
% 3.06/1.61 => ( ( ord_less_eq @ ( $o > A ) )
% 3.06/1.61 = ( ^ [X6: $o > A,Y8: $o > A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ ( X6 @ $false ) @ ( Y8 @ $false ) )
% 3.06/1.61 & ( ord_less_eq @ A @ ( X6 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_57_verit__la__disequality,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( linorder @ A )
% 3.06/1.61 => ! [A2: A,B2: A] :
% 3.06/1.61 ( ( A2 = B2 )
% 3.06/1.61 | ~ ( ord_less_eq @ A @ A2 @ B2 )
% 3.06/1.61 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_58_ord__eq__le__eq__trans,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( ord @ A )
% 3.06/1.61 => ! [A2: A,B2: A,C2: A,D: A] :
% 3.06/1.61 ( ( A2 = B2 )
% 3.06/1.61 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 3.06/1.61 => ( ( C2 = D )
% 3.06/1.61 => ( ord_less_eq @ A @ A2 @ D ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_59_intro__spec__refine__iff,axiom,
% 3.06/1.61 ! [C: $tType,A: $tType,B: $tType,X: set @ B,F: B > ( refine1665802226e_nres @ A ),R: set @ ( product_prod @ A @ C ),M3: refine1665802226e_nres @ C] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ ( refine605929679le_RES @ B @ X ) @ F ) @ ( refine1073749519nc_fun @ A @ C @ R @ M3 ) )
% 3.06/1.61 = ( ! [X5: B] :
% 3.06/1.61 ( ( member @ B @ X5 @ X )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X5 ) @ ( refine1073749519nc_fun @ A @ C @ R @ M3 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_60_intro__bind__refine__id,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,C: $tType,M: refine1665802226e_nres @ A,M4: A,F: A > ( refine1665802226e_nres @ B ),R: set @ ( product_prod @ B @ C ),M5: refine1665802226e_nres @ C] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M
% 3.06/1.61 @ ( refine605929679le_RES @ A
% 3.06/1.61 @ ( collect @ A
% 3.06/1.61 @ ( ^ [Y5: A,Z: A] : ( Y5 = Z )
% 3.06/1.61 @ M4 ) ) ) )
% 3.06/1.61 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ M4 ) @ ( refine1073749519nc_fun @ B @ C @ R @ M5 ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ ( refine1073749519nc_fun @ B @ C @ R @ M5 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_61_nofail__simps_I2_J,axiom,
% 3.06/1.61 ! [B: $tType,X: set @ B] : ( refine1102455758nofail @ B @ ( refine605929679le_RES @ B @ X ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_62_subset__Collect__conv,axiom,
% 3.06/1.61 ! [A: $tType,S: set @ A,P: A > $o] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
% 3.06/1.61 = ( ! [X5: A] :
% 3.06/1.61 ( ( member @ A @ X5 @ S )
% 3.06/1.61 => ( P @ X5 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_63_nofail__antimono__fun,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,F: A > ( refine1665802226e_nres @ B ),G: A > ( refine1665802226e_nres @ B ),X4: A] :
% 3.06/1.61 ( ( ord_less_eq @ ( A > ( refine1665802226e_nres @ B ) ) @ F @ G )
% 3.06/1.61 => ( ( refine1102455758nofail @ B @ ( G @ X4 ) )
% 3.06/1.61 => ( refine1102455758nofail @ B @ ( F @ X4 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_64_pw__conc__nofail,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 3.06/1.61 ( ( refine1102455758nofail @ A @ ( refine1073749519nc_fun @ A @ B @ R @ S ) )
% 3.06/1.61 = ( refine1102455758nofail @ B @ S ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_65_pwD1,axiom,
% 3.06/1.61 ! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S3 )
% 3.06/1.61 => ( ( refine1102455758nofail @ A @ S3 )
% 3.06/1.61 => ( refine1102455758nofail @ A @ S ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_66_le__nofailI,axiom,
% 3.06/1.61 ! [A: $tType,M6: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( ( refine1102455758nofail @ A @ M6 )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ M6 ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ M6 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_67_nofail__RES__conv,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( refine1102455758nofail @ A )
% 3.06/1.61 = ( ^ [M7: refine1665802226e_nres @ A] :
% 3.06/1.61 ? [M8: set @ A] :
% 3.06/1.61 ( M7
% 3.06/1.61 = ( refine605929679le_RES @ A @ M8 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_68_if__refine,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,B2: $o,B5: $o,S1: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),S13: refine1665802226e_nres @ B,S2: refine1665802226e_nres @ A,S23: refine1665802226e_nres @ B] :
% 3.06/1.61 ( ( B2 = B5 )
% 3.06/1.61 => ( ( B2
% 3.06/1.61 => ( B5
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S1 @ ( refine1073749519nc_fun @ A @ B @ R @ S13 ) ) ) )
% 3.06/1.61 => ( ( ~ B2
% 3.06/1.61 => ( ~ B5
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S2 @ ( refine1073749519nc_fun @ A @ B @ R @ S23 ) ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( if @ ( refine1665802226e_nres @ A ) @ B2 @ S1 @ S2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( if @ ( refine1665802226e_nres @ B ) @ B5 @ S13 @ S23 ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_69_conc__trans,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,C: $tType,C3: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B6: refine1665802226e_nres @ B,R2: set @ ( product_prod @ B @ C ),A5: refine1665802226e_nres @ C] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C3 @ ( refine1073749519nc_fun @ A @ B @ R @ B6 ) )
% 3.06/1.61 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B6 @ ( refine1073749519nc_fun @ B @ C @ R2 @ A5 ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C3 @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1073749519nc_fun @ B @ C @ R2 @ A5 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_70_intro__prgR,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,C2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),A2: refine1665802226e_nres @ B] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ A2 ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ A2 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_71_ref__two__step,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,A5: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B6: refine1665802226e_nres @ B,C3: refine1665802226e_nres @ B] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A5 @ ( refine1073749519nc_fun @ A @ B @ R @ B6 ) )
% 3.06/1.61 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B6 @ C3 )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A5 @ ( refine1073749519nc_fun @ A @ B @ R @ C3 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_72_conc__fun__R__mono,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),M3: refine1665802226e_nres @ B] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ R2 )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1073749519nc_fun @ A @ B @ R @ M3 ) @ ( refine1073749519nc_fun @ A @ B @ R2 @ M3 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_73_le__RES__nofailI,axiom,
% 3.06/1.61 ! [A: $tType,A2: refine1665802226e_nres @ A,X4: set @ A] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine605929679le_RES @ A @ X4 ) )
% 3.06/1.61 => ( refine1102455758nofail @ A @ A2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_74_pw__RES__bind__choose_I1_J,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,X: set @ B,F: B > ( refine1665802226e_nres @ A )] :
% 3.06/1.61 ( ( refine1102455758nofail @ A @ ( refine463715084e_bind @ B @ A @ ( refine605929679le_RES @ B @ X ) @ F ) )
% 3.06/1.61 = ( ! [X5: B] :
% 3.06/1.61 ( ( member @ B @ X5 @ X )
% 3.06/1.61 => ( refine1102455758nofail @ A @ ( F @ X5 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_75_rhs__step__bind__RES,axiom,
% 3.06/1.61 ! [B: $tType,C: $tType,A: $tType,X7: A,X8: set @ A,M: refine1665802226e_nres @ B,R: set @ ( product_prod @ B @ C ),F4: A > ( refine1665802226e_nres @ C )] :
% 3.06/1.61 ( ( member @ A @ X7 @ X8 )
% 3.06/1.61 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine1073749519nc_fun @ B @ C @ R @ ( F4 @ X7 ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine1073749519nc_fun @ B @ C @ R @ ( refine463715084e_bind @ A @ C @ ( refine605929679le_RES @ A @ X8 ) @ F4 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_76_rhs__step__bind__SPEC,axiom,
% 3.06/1.61 ! [B: $tType,C: $tType,A: $tType,Phi: A > $o,X7: A,M: refine1665802226e_nres @ B,R: set @ ( product_prod @ B @ C ),F4: A > ( refine1665802226e_nres @ C )] :
% 3.06/1.61 ( ( Phi @ X7 )
% 3.06/1.61 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine1073749519nc_fun @ B @ C @ R @ ( F4 @ X7 ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine1073749519nc_fun @ B @ C @ R @ ( refine463715084e_bind @ A @ C @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ F4 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_77_subsetI,axiom,
% 3.06/1.61 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 3.06/1.61 ( ! [X3: A] :
% 3.06/1.61 ( ( member @ A @ X3 @ A5 )
% 3.06/1.61 => ( member @ A @ X3 @ B6 ) )
% 3.06/1.61 => ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_78_subset__antisym,axiom,
% 3.06/1.61 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 3.06/1.61 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
% 3.06/1.61 => ( A5 = B6 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_79_bind__sim__select__rule,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,C: $tType,M: refine1665802226e_nres @ B,F4: B > ( refine1665802226e_nres @ A ),Psi: A > $o,F: B > ( refine1665802226e_nres @ C ),Phi: C > $o] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F4 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) )
% 3.06/1.61 => ( ! [X3: B] :
% 3.06/1.61 ( ( refine1102455758nofail @ B @ M )
% 3.06/1.61 => ( ( refine1315500908_inres @ B @ M @ X3 )
% 3.06/1.61 => ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F4 @ X3 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( F @ X3 ) @ ( refine605929679le_RES @ C @ ( collect @ C @ Phi ) ) ) ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine463715084e_bind @ B @ C @ M @ F ) @ ( refine605929679le_RES @ C @ ( collect @ C @ Phi ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_80_relprop__triggers_I6_J,axiom,
% 3.06/1.61 ! [I: $tType,R: set @ I,R2: set @ I] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ I ) @ R @ R2 )
% 3.06/1.61 => ( ord_less_eq @ ( set @ I ) @ R @ R2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_81_in__mono,axiom,
% 3.06/1.61 ! [A: $tType,A5: set @ A,B6: set @ A,X4: A] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 3.06/1.61 => ( ( member @ A @ X4 @ A5 )
% 3.06/1.61 => ( member @ A @ X4 @ B6 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_82_subsetD,axiom,
% 3.06/1.61 ! [A: $tType,A5: set @ A,B6: set @ A,C2: A] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 3.06/1.61 => ( ( member @ A @ C2 @ A5 )
% 3.06/1.61 => ( member @ A @ C2 @ B6 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_83_equalityE,axiom,
% 3.06/1.61 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 3.06/1.61 ( ( A5 = B6 )
% 3.06/1.61 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 3.06/1.61 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_84_inres__simps_I2_J,axiom,
% 3.06/1.61 ! [B: $tType,X: set @ B] :
% 3.06/1.61 ( ( refine1315500908_inres @ B @ ( refine605929679le_RES @ B @ X ) )
% 3.06/1.61 = ( ^ [X5: B] : ( member @ B @ X5 @ X ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_85_pwD2,axiom,
% 3.06/1.61 ! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A,X4: A] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S3 )
% 3.06/1.61 => ( ( refine1315500908_inres @ A @ S @ X4 )
% 3.06/1.61 => ( refine1315500908_inres @ A @ S3 @ X4 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_86_pw__eqI,axiom,
% 3.06/1.61 ! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( ( refine1102455758nofail @ A @ S )
% 3.06/1.61 = ( refine1102455758nofail @ A @ S3 ) )
% 3.06/1.61 => ( ! [X3: A] :
% 3.06/1.61 ( ( refine1315500908_inres @ A @ S @ X3 )
% 3.06/1.61 = ( refine1315500908_inres @ A @ S3 @ X3 ) )
% 3.06/1.61 => ( S = S3 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_87_pw__eq__iff,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( ^ [Y5: refine1665802226e_nres @ A,Z: refine1665802226e_nres @ A] : ( Y5 = Z ) )
% 3.06/1.61 = ( ^ [S4: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( ( refine1102455758nofail @ A @ S4 )
% 3.06/1.61 = ( refine1102455758nofail @ A @ S5 ) )
% 3.06/1.61 & ! [X5: A] :
% 3.06/1.61 ( ( refine1315500908_inres @ A @ S4 @ X5 )
% 3.06/1.61 = ( refine1315500908_inres @ A @ S5 @ X5 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_88_not__nofail__inres,axiom,
% 3.06/1.61 ! [A: $tType,S: refine1665802226e_nres @ A,X4: A] :
% 3.06/1.61 ( ~ ( refine1102455758nofail @ A @ S )
% 3.06/1.61 => ( refine1315500908_inres @ A @ S @ X4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_89_pw__RES__bind__choose_I2_J,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,X: set @ B,F: B > ( refine1665802226e_nres @ A ),Y3: A] :
% 3.06/1.61 ( ( refine1315500908_inres @ A @ ( refine463715084e_bind @ B @ A @ ( refine605929679le_RES @ B @ X ) @ F ) @ Y3 )
% 3.06/1.61 = ( ? [X5: B] :
% 3.06/1.61 ( ( member @ B @ X5 @ X )
% 3.06/1.61 & ( refine1315500908_inres @ A @ ( F @ X5 ) @ Y3 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_90_pw__leI,axiom,
% 3.06/1.61 ! [A: $tType,S3: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( ( refine1102455758nofail @ A @ S3 )
% 3.06/1.61 => ( ( refine1102455758nofail @ A @ S )
% 3.06/1.61 & ! [X3: A] :
% 3.06/1.61 ( ( refine1315500908_inres @ A @ S @ X3 )
% 3.06/1.61 => ( refine1315500908_inres @ A @ S3 @ X3 ) ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S3 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_91_pw__leI_H,axiom,
% 3.06/1.61 ! [A: $tType,S3: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( ( refine1102455758nofail @ A @ S3 )
% 3.06/1.61 => ( refine1102455758nofail @ A @ S ) )
% 3.06/1.61 => ( ! [X3: A] :
% 3.06/1.61 ( ( refine1102455758nofail @ A @ S3 )
% 3.06/1.61 => ( ( refine1315500908_inres @ A @ S @ X3 )
% 3.06/1.61 => ( refine1315500908_inres @ A @ S3 @ X3 ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S3 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_92_pw__le__iff,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) )
% 3.06/1.61 = ( ^ [S4: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( refine1102455758nofail @ A @ S5 )
% 3.06/1.61 => ( ( refine1102455758nofail @ A @ S4 )
% 3.06/1.61 & ! [X5: A] :
% 3.06/1.61 ( ( refine1315500908_inres @ A @ S4 @ X5 )
% 3.06/1.61 => ( refine1315500908_inres @ A @ S5 @ X5 ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_93_pw__bind__nofail,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,M3: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
% 3.06/1.61 ( ( refine1102455758nofail @ A @ ( refine463715084e_bind @ B @ A @ M3 @ F ) )
% 3.06/1.61 = ( ( refine1102455758nofail @ B @ M3 )
% 3.06/1.61 & ! [X5: B] :
% 3.06/1.61 ( ( refine1315500908_inres @ B @ M3 @ X5 )
% 3.06/1.61 => ( refine1102455758nofail @ A @ ( F @ X5 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_94_nf__inres__def,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( refine406925620_inres @ A )
% 3.06/1.61 = ( ^ [M7: refine1665802226e_nres @ A,X5: A] :
% 3.06/1.61 ( ( refine1102455758nofail @ A @ M7 )
% 3.06/1.61 & ( refine1315500908_inres @ A @ M7 @ X5 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_95_pw__bind__le__iff,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,M3: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A ),S: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M3 @ F ) @ S )
% 3.06/1.61 = ( ( ( refine1102455758nofail @ A @ S )
% 3.06/1.61 => ( refine1102455758nofail @ B @ M3 ) )
% 3.06/1.61 & ! [X5: B] :
% 3.06/1.61 ( ( ( refine1102455758nofail @ B @ M3 )
% 3.06/1.61 & ( refine1315500908_inres @ B @ M3 @ X5 ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X5 ) @ S ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_96_pw__bind__leI,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,S: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
% 3.06/1.61 ( ( ( refine1102455758nofail @ A @ S )
% 3.06/1.61 => ( refine1102455758nofail @ B @ M3 ) )
% 3.06/1.61 => ( ! [X3: B] :
% 3.06/1.61 ( ( refine1102455758nofail @ B @ M3 )
% 3.06/1.61 => ( ( refine1315500908_inres @ B @ M3 @ X3 )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X3 ) @ S ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M3 @ F ) @ S ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_97_Collect__mono__iff,axiom,
% 3.06/1.61 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 3.06/1.61 = ( ! [X5: A] :
% 3.06/1.61 ( ( P @ X5 )
% 3.06/1.61 => ( Q @ X5 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_98_set__eq__subset,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( ^ [Y5: set @ A,Z: set @ A] : ( Y5 = Z ) )
% 3.06/1.61 = ( ^ [A6: set @ A,B7: set @ A] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 3.06/1.61 & ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_99_subset__trans,axiom,
% 3.06/1.61 ! [A: $tType,A5: set @ A,B6: set @ A,C3: set @ A] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 3.06/1.61 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C3 )
% 3.06/1.61 => ( ord_less_eq @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_100_Collect__mono,axiom,
% 3.06/1.61 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.06/1.61 ( ! [X3: A] :
% 3.06/1.61 ( ( P @ X3 )
% 3.06/1.61 => ( Q @ X3 ) )
% 3.06/1.61 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_101_subset__refl,axiom,
% 3.06/1.61 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_102_subset__iff,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ A ) )
% 3.06/1.61 = ( ^ [A6: set @ A,B7: set @ A] :
% 3.06/1.61 ! [T4: A] :
% 3.06/1.61 ( ( member @ A @ T4 @ A6 )
% 3.06/1.61 => ( member @ A @ T4 @ B7 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_103_equalityD2,axiom,
% 3.06/1.61 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 3.06/1.61 ( ( A5 = B6 )
% 3.06/1.61 => ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_104_equalityD1,axiom,
% 3.06/1.61 ! [A: $tType,A5: set @ A,B6: set @ A] :
% 3.06/1.61 ( ( A5 = B6 )
% 3.06/1.61 => ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_105_subset__eq,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ A ) )
% 3.06/1.61 = ( ^ [A6: set @ A,B7: set @ A] :
% 3.06/1.61 ! [X5: A] :
% 3.06/1.61 ( ( member @ A @ X5 @ A6 )
% 3.06/1.61 => ( member @ A @ X5 @ B7 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_106_pw__ref__iff,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,S: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),S3: refine1665802226e_nres @ B] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ ( refine1073749519nc_fun @ A @ B @ R @ S3 ) )
% 3.06/1.61 = ( ( refine1102455758nofail @ B @ S3 )
% 3.06/1.61 => ( ( refine1102455758nofail @ A @ S )
% 3.06/1.61 & ! [X5: A] :
% 3.06/1.61 ( ( refine1315500908_inres @ A @ S @ X5 )
% 3.06/1.61 => ? [S6: B] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ S6 ) @ R )
% 3.06/1.61 & ( refine1315500908_inres @ B @ S3 @ S6 ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_107_pw__ref__I,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,S3: refine1665802226e_nres @ A,S: refine1665802226e_nres @ B,R: set @ ( product_prod @ B @ A )] :
% 3.06/1.61 ( ( ( refine1102455758nofail @ A @ S3 )
% 3.06/1.61 => ( ( refine1102455758nofail @ B @ S )
% 3.06/1.61 & ! [X3: B] :
% 3.06/1.61 ( ( refine1315500908_inres @ B @ S @ X3 )
% 3.06/1.61 => ? [S7: A] :
% 3.06/1.61 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ S7 ) @ R )
% 3.06/1.61 & ( refine1315500908_inres @ A @ S3 @ S7 ) ) ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ S @ ( refine1073749519nc_fun @ B @ A @ R @ S3 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_108_rhs__step__bind,axiom,
% 3.06/1.61 ! [A: $tType,C: $tType,D2: $tType,B: $tType,M: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),M4: refine1665802226e_nres @ B,X4: A,Lhs: refine1665802226e_nres @ C,S: set @ ( product_prod @ C @ D2 ),F4: B > ( refine1665802226e_nres @ D2 )] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine1073749519nc_fun @ A @ B @ R @ M4 ) )
% 3.06/1.61 => ( ( refine1315500908_inres @ A @ M @ X4 )
% 3.06/1.61 => ( ! [X9: B] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ X9 ) @ R )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ Lhs @ ( refine1073749519nc_fun @ C @ D2 @ S @ ( F4 @ X9 ) ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ Lhs @ ( refine1073749519nc_fun @ C @ D2 @ S @ ( refine463715084e_bind @ B @ D2 @ M4 @ F4 ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_109_RES__refine,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,S: set @ A,S3: set @ B,R: set @ ( product_prod @ A @ B )] :
% 3.06/1.61 ( ! [S8: A] :
% 3.06/1.61 ( ( member @ A @ S8 @ S )
% 3.06/1.61 => ? [X10: B] :
% 3.06/1.61 ( ( member @ B @ X10 @ S3 )
% 3.06/1.61 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S8 @ X10 ) @ R ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ S ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine605929679le_RES @ B @ S3 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_110_less__eq__nres_Oelims_I1_J,axiom,
% 3.06/1.61 ! [A: $tType,X4: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
% 3.06/1.61 ( ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ Xa )
% 3.06/1.61 = Y3 )
% 3.06/1.61 => ( ( ( Xa
% 3.06/1.61 = ( refine1767639642_FAILi @ A ) )
% 3.06/1.61 => ~ Y3 )
% 3.06/1.61 => ( ! [A4: set @ A] :
% 3.06/1.61 ( ( X4
% 3.06/1.61 = ( refine605929679le_RES @ A @ A4 ) )
% 3.06/1.61 => ! [B4: set @ A] :
% 3.06/1.61 ( ( Xa
% 3.06/1.61 = ( refine605929679le_RES @ A @ B4 ) )
% 3.06/1.61 => ( Y3
% 3.06/1.61 = ( ~ ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ) )
% 3.06/1.61 => ~ ( ( X4
% 3.06/1.61 = ( refine1767639642_FAILi @ A ) )
% 3.06/1.61 => ( ? [Uv: set @ A] :
% 3.06/1.61 ( Xa
% 3.06/1.61 = ( refine605929679le_RES @ A @ Uv ) )
% 3.06/1.61 => Y3 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_111_less__eq__nres_Oelims_I2_J,axiom,
% 3.06/1.61 ! [A: $tType,X4: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ Xa )
% 3.06/1.61 => ( ( Xa
% 3.06/1.61 != ( refine1767639642_FAILi @ A ) )
% 3.06/1.61 => ~ ! [A4: set @ A] :
% 3.06/1.61 ( ( X4
% 3.06/1.61 = ( refine605929679le_RES @ A @ A4 ) )
% 3.06/1.61 => ! [B4: set @ A] :
% 3.06/1.61 ( ( Xa
% 3.06/1.61 = ( refine605929679le_RES @ A @ B4 ) )
% 3.06/1.61 => ~ ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_112_less__eq__nres_Ocases,axiom,
% 3.06/1.61 ! [A: $tType,X4: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
% 3.06/1.61 ( ! [Uu: refine1665802226e_nres @ A] :
% 3.06/1.61 ( X4
% 3.06/1.61 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ Uu @ ( refine1767639642_FAILi @ A ) ) )
% 3.06/1.61 => ( ! [A4: set @ A,B4: set @ A] :
% 3.06/1.61 ( X4
% 3.06/1.61 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) ) )
% 3.06/1.61 => ~ ! [Uv: set @ A] :
% 3.06/1.61 ( X4
% 3.06/1.61 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_113_less__nres_Ocases,axiom,
% 3.06/1.61 ! [A: $tType,X4: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
% 3.06/1.61 ( ! [Uu: refine1665802226e_nres @ A] :
% 3.06/1.61 ( X4
% 3.06/1.61 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Uu ) )
% 3.06/1.61 => ( ! [Uv: set @ A] :
% 3.06/1.61 ( X4
% 3.06/1.61 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) ) )
% 3.06/1.61 => ~ ! [A4: set @ A,B4: set @ A] :
% 3.06/1.61 ( X4
% 3.06/1.61 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_114_sup__nres_Ocases,axiom,
% 3.06/1.61 ! [A: $tType,X4: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
% 3.06/1.61 ( ! [Uu: refine1665802226e_nres @ A] :
% 3.06/1.61 ( X4
% 3.06/1.61 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ Uu @ ( refine1767639642_FAILi @ A ) ) )
% 3.06/1.61 => ( ! [V: set @ A] :
% 3.06/1.61 ( X4
% 3.06/1.61 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V ) ) )
% 3.06/1.61 => ~ ! [A4: set @ A,B4: set @ A] :
% 3.06/1.61 ( X4
% 3.06/1.61 != ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_115_introR,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,A2: A,A7: B,R: set @ ( product_prod @ A @ B )] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ A7 ) @ R )
% 3.06/1.61 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ A7 ) @ R ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_116_bex2I,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,A2: A,B2: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S )
% 3.06/1.61 => ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S )
% 3.06/1.61 => ( P @ A2 @ B2 ) )
% 3.06/1.61 => ? [A4: A,B4: B] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ S )
% 3.06/1.61 & ( P @ A4 @ B4 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_117_pairself_Ocases,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,X4: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
% 3.06/1.61 ~ ! [F6: A > B,A4: A,B4: A] :
% 3.06/1.61 ( X4
% 3.06/1.61 != ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F6 @ ( product_Pair @ A @ A @ A4 @ B4 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_118_pairself_Oinduct,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,P: ( A > B ) > ( product_prod @ A @ A ) > $o,A0: A > B,A1: product_prod @ A @ A] :
% 3.06/1.61 ( ! [F6: A > B,A4: A,B4: A] : ( P @ F6 @ ( product_Pair @ A @ A @ A4 @ B4 ) )
% 3.06/1.61 => ( P @ A0 @ A1 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_119_less__eq__nres_Osimps_I1_J,axiom,
% 3.06/1.61 ! [A: $tType,Uu2: refine1665802226e_nres @ A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Uu2 @ ( refine1767639642_FAILi @ A ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_120_nres_Odistinct_I1_J,axiom,
% 3.06/1.61 ! [A: $tType,X2: set @ A] :
% 3.06/1.61 ( ( refine1767639642_FAILi @ A )
% 3.06/1.61 != ( refine605929679le_RES @ A @ X2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_121_nres_Oinduct,axiom,
% 3.06/1.61 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > $o,Nres: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( P @ ( refine1767639642_FAILi @ A ) )
% 3.06/1.61 => ( ! [X3: set @ A] : ( P @ ( refine605929679le_RES @ A @ X3 ) )
% 3.06/1.61 => ( P @ Nres ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_122_nres_Oexhaust,axiom,
% 3.06/1.61 ! [A: $tType,Y3: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( Y3
% 3.06/1.61 != ( refine1767639642_FAILi @ A ) )
% 3.06/1.61 => ~ ! [X22: set @ A] :
% 3.06/1.61 ( Y3
% 3.06/1.61 != ( refine605929679le_RES @ A @ X22 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_123_sup__nres_Oinduct,axiom,
% 3.06/1.61 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ! [Uu: refine1665802226e_nres @ A] : ( P @ Uu @ ( refine1767639642_FAILi @ A ) )
% 3.06/1.61 => ( ! [V: set @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V ) )
% 3.06/1.61 => ( ! [A4: set @ A,B4: set @ A] : ( P @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) )
% 3.06/1.61 => ( P @ A0 @ A1 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_124_less__nres_Oinduct,axiom,
% 3.06/1.61 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ! [X_1: refine1665802226e_nres @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ X_1 )
% 3.06/1.61 => ( ! [Uv: set @ A] : ( P @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) )
% 3.06/1.61 => ( ! [A4: set @ A,B4: set @ A] : ( P @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) )
% 3.06/1.61 => ( P @ A0 @ A1 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_125_less__eq__nres_Oinduct,axiom,
% 3.06/1.61 ! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ! [Uu: refine1665802226e_nres @ A] : ( P @ Uu @ ( refine1767639642_FAILi @ A ) )
% 3.06/1.61 => ( ! [A4: set @ A,B4: set @ A] : ( P @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) )
% 3.06/1.61 => ( ! [Uv: set @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) )
% 3.06/1.61 => ( P @ A0 @ A1 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_126_nres_Osimps_I6_J,axiom,
% 3.06/1.61 ! [A: $tType,C: $tType,F1: C,F2: ( set @ A ) > C] :
% 3.06/1.61 ( ( refine1442219249c_nres @ C @ A @ F1 @ F2 @ ( refine1767639642_FAILi @ A ) )
% 3.06/1.61 = F1 ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_127_less__eq__nres_Osimps_I3_J,axiom,
% 3.06/1.61 ! [A: $tType,Uv2: set @ A] :
% 3.06/1.61 ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_128_Let__refine_H,axiom,
% 3.06/1.61 ! [A: $tType,C: $tType,D2: $tType,B: $tType,M: A,M4: B,R: set @ ( product_prod @ A @ B ),F: A > ( refine1665802226e_nres @ C ),S: set @ ( product_prod @ C @ D2 ),F4: B > ( refine1665802226e_nres @ D2 )] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ M @ M4 ) @ R )
% 3.06/1.61 => ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ M @ M4 ) @ R )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( F @ M ) @ ( refine1073749519nc_fun @ C @ D2 @ S @ ( F4 @ M4 ) ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( F @ M ) @ ( refine1073749519nc_fun @ C @ D2 @ S @ ( F4 @ M4 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_129_less__eq__nres_Oelims_I3_J,axiom,
% 3.06/1.61 ! [A: $tType,X4: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ Xa )
% 3.06/1.61 => ( ! [A4: set @ A] :
% 3.06/1.61 ( ( X4
% 3.06/1.61 = ( refine605929679le_RES @ A @ A4 ) )
% 3.06/1.61 => ! [B4: set @ A] :
% 3.06/1.61 ( ( Xa
% 3.06/1.61 = ( refine605929679le_RES @ A @ B4 ) )
% 3.06/1.61 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) )
% 3.06/1.61 => ~ ( ( X4
% 3.06/1.61 = ( refine1767639642_FAILi @ A ) )
% 3.06/1.61 => ! [Uv: set @ A] :
% 3.06/1.61 ( Xa
% 3.06/1.61 != ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_130_old_Oprod_Oinject,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B5: B] :
% 3.06/1.61 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.06/1.61 = ( product_Pair @ A @ B @ A7 @ B5 ) )
% 3.06/1.61 = ( ( A2 = A7 )
% 3.06/1.61 & ( B2 = B5 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_131_prod_Oinject,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 3.06/1.61 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 3.06/1.61 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 3.06/1.61 = ( ( X1 = Y1 )
% 3.06/1.61 & ( X2 = Y2 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_132_subrelI,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S9: set @ ( product_prod @ A @ B )] :
% 3.06/1.61 ( ! [X3: A,Y4: B] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R3 )
% 3.06/1.61 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S9 ) )
% 3.06/1.61 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_133_lift__assn__def,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( refine1580981607t_assn @ A @ B )
% 3.06/1.61 = ( ^ [R4: set @ ( product_prod @ A @ B ),Phi3: B > $o,S10: A] :
% 3.06/1.61 ? [S6: B] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S10 @ S6 ) @ R4 )
% 3.06/1.61 & ( Phi3 @ S6 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_134_surj__pair,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 3.06/1.61 ? [X3: A,Y4: B] :
% 3.06/1.61 ( P2
% 3.06/1.61 = ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_135_prod__cases,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 3.06/1.61 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.06/1.61 => ( P @ P2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_136_Pair__inject,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B5: B] :
% 3.06/1.61 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.06/1.61 = ( product_Pair @ A @ B @ A7 @ B5 ) )
% 3.06/1.61 => ~ ( ( A2 = A7 )
% 3.06/1.61 => ( B2 != B5 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_137_prod__cases3,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.06/1.61 ~ ! [A4: A,B4: B,C4: C] :
% 3.06/1.61 ( Y3
% 3.06/1.61 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_138_prod__cases4,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 3.06/1.61 ~ ! [A4: A,B4: B,C4: C,D3: D2] :
% 3.06/1.61 ( Y3
% 3.06/1.61 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B4 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_139_prod__cases5,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
% 3.06/1.61 ~ ! [A4: A,B4: B,C4: C,D3: D2,E4: E2] :
% 3.06/1.61 ( Y3
% 3.06/1.61 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C4 @ ( product_Pair @ D2 @ E2 @ D3 @ E4 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_140_prod__cases6,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F7: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) )] :
% 3.06/1.61 ~ ! [A4: A,B4: B,C4: C,D3: D2,E4: E2,F6: F7] :
% 3.06/1.61 ( Y3
% 3.06/1.61 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F7 ) @ D3 @ ( product_Pair @ E2 @ F7 @ E4 @ F6 ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_141_prod__cases7,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F7: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) )] :
% 3.06/1.61 ~ ! [A4: A,B4: B,C4: C,D3: D2,E4: E2,F6: F7,G4: G3] :
% 3.06/1.61 ( Y3
% 3.06/1.61 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F7 @ G3 ) @ E4 @ ( product_Pair @ F7 @ G3 @ F6 @ G4 ) ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_142_prod__induct3,axiom,
% 3.06/1.61 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.06/1.61 ( ! [A4: A,B4: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) )
% 3.06/1.61 => ( P @ X4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_143_prod__induct4,axiom,
% 3.06/1.61 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 3.06/1.61 ( ! [A4: A,B4: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B4 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
% 3.06/1.61 => ( P @ X4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_144_prod__induct5,axiom,
% 3.06/1.61 ! [E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
% 3.06/1.61 ( ! [A4: A,B4: B,C4: C,D3: D2,E4: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C4 @ ( product_Pair @ D2 @ E2 @ D3 @ E4 ) ) ) ) )
% 3.06/1.61 => ( P @ X4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_145_prod__induct6,axiom,
% 3.06/1.61 ! [F7: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) )] :
% 3.06/1.61 ( ! [A4: A,B4: B,C4: C,D3: D2,E4: E2,F6: F7] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F7 ) @ D3 @ ( product_Pair @ E2 @ F7 @ E4 @ F6 ) ) ) ) ) )
% 3.06/1.61 => ( P @ X4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_146_prod__induct7,axiom,
% 3.06/1.61 ! [G3: $tType,F7: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) )] :
% 3.06/1.61 ( ! [A4: A,B4: B,C4: C,D3: D2,E4: E2,F6: F7,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F7 @ G3 ) @ E4 @ ( product_Pair @ F7 @ G3 @ F6 @ G4 ) ) ) ) ) ) )
% 3.06/1.61 => ( P @ X4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_147_old_Oprod_Oexhaust,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 3.06/1.61 ~ ! [A4: A,B4: B] :
% 3.06/1.61 ( Y3
% 3.06/1.61 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_148_old_Oprod_Oinducts,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 3.06/1.61 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.06/1.61 => ( P @ Prod ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_149_lift__assnI,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,S9: A,S11: B,R: set @ ( product_prod @ A @ B ),Phi: B > $o] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S9 @ S11 ) @ R )
% 3.06/1.61 => ( ( Phi @ S11 )
% 3.06/1.61 => ( refine1580981607t_assn @ A @ B @ R @ Phi @ S9 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_150_internal__case__prod__conv,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
% 3.06/1.61 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 3.06/1.61 = ( C2 @ A2 @ B2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_151_return__refine__prop__return,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,X4: B,R: set @ ( product_prod @ B @ A )] :
% 3.06/1.61 ( ( refine1102455758nofail @ A @ M )
% 3.06/1.61 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine1687780735RETURN @ B @ X4 ) @ ( refine1073749519nc_fun @ B @ A @ R @ M ) )
% 3.06/1.61 => ~ ! [X9: A] :
% 3.06/1.61 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ X9 ) @ R )
% 3.06/1.61 => ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X9 ) @ M ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_152_ret__le__down__conv,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,C2: B,R: set @ ( product_prod @ B @ A )] :
% 3.06/1.61 ( ( refine1102455758nofail @ A @ M )
% 3.06/1.61 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine1687780735RETURN @ B @ C2 ) @ ( refine1073749519nc_fun @ B @ A @ R @ M ) )
% 3.06/1.61 = ( ? [A3: A] :
% 3.06/1.61 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ C2 @ A3 ) @ R )
% 3.06/1.61 & ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ A3 ) @ M ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_153_old_Oprod_Orec,axiom,
% 3.06/1.61 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 3.06/1.61 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.06/1.61 = ( F1 @ A2 @ B2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_154_nres__more__simps_I6_J,axiom,
% 3.06/1.61 ! [A: $tType,X4: A,Y3: A] :
% 3.06/1.61 ( ( ( refine1687780735RETURN @ A @ X4 )
% 3.06/1.61 = ( refine1687780735RETURN @ A @ Y3 ) )
% 3.06/1.61 = ( X4 = Y3 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_155_nres__order__simps_I20_J,axiom,
% 3.06/1.61 ! [W: $tType,X4: W,Y3: W] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X4 ) @ ( refine1687780735RETURN @ W @ Y3 ) )
% 3.06/1.61 = ( X4 = Y3 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_156_nres__monad1,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,X4: B,F: B > ( refine1665802226e_nres @ A )] :
% 3.06/1.61 ( ( refine463715084e_bind @ B @ A @ ( refine1687780735RETURN @ B @ X4 ) @ F )
% 3.06/1.61 = ( F @ X4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_157_nres__monad2,axiom,
% 3.06/1.61 ! [A: $tType,M3: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( refine463715084e_bind @ A @ A @ M3 @ ( refine1687780735RETURN @ A ) )
% 3.06/1.61 = M3 ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_158_nofail__simps_I3_J,axiom,
% 3.06/1.61 ! [C: $tType,X4: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_159_inres__simps_I3_J,axiom,
% 3.06/1.61 ! [C: $tType,X4: C] :
% 3.06/1.61 ( ( refine1315500908_inres @ C @ ( refine1687780735RETURN @ C @ X4 ) )
% 3.06/1.61 = ( ^ [Y5: C,Z: C] : ( Y5 = Z )
% 3.06/1.61 @ X4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_160_nres__order__simps_I21_J,axiom,
% 3.06/1.61 ! [X11: $tType,X4: X11,Y: set @ X11] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ X11 ) @ ( refine1687780735RETURN @ X11 @ X4 ) @ ( refine605929679le_RES @ X11 @ Y ) )
% 3.06/1.61 = ( member @ X11 @ X4 @ Y ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_161_SPEC__eq__is__RETURN_I1_J,axiom,
% 3.06/1.61 ! [A: $tType,X4: A] :
% 3.06/1.61 ( ( refine605929679le_RES @ A
% 3.06/1.61 @ ( collect @ A
% 3.06/1.61 @ ( ^ [Y5: A,Z: A] : ( Y5 = Z )
% 3.06/1.61 @ X4 ) ) )
% 3.06/1.61 = ( refine1687780735RETURN @ A @ X4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_162_RETURN__rule,axiom,
% 3.06/1.61 ! [A: $tType,Phi: A > $o,X4: A] :
% 3.06/1.61 ( ( Phi @ X4 )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X4 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_163_lhs__step__RES,axiom,
% 3.06/1.61 ! [A: $tType,X: set @ A,M: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ! [X3: A] :
% 3.06/1.61 ( ( member @ A @ X3 @ X )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ X ) @ M ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_164_RETURN__to__SPEC__rule,axiom,
% 3.06/1.61 ! [A: $tType,M: refine1665802226e_nres @ A,V2: A] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M
% 3.06/1.61 @ ( refine605929679le_RES @ A
% 3.06/1.61 @ ( collect @ A
% 3.06/1.61 @ ( ^ [Y5: A,Z: A] : ( Y5 = Z )
% 3.06/1.61 @ V2 ) ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine1687780735RETURN @ A @ V2 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_165_if__RETURN__refine,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,B2: $o,B5: $o,S1: A,R: set @ ( product_prod @ A @ B ),S13: refine1665802226e_nres @ B,S2: A,S23: refine1665802226e_nres @ B] :
% 3.06/1.61 ( ( B2 = B5 )
% 3.06/1.61 => ( ( B2
% 3.06/1.61 => ( B5
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ S1 ) @ ( refine1073749519nc_fun @ A @ B @ R @ S13 ) ) ) )
% 3.06/1.61 => ( ( ~ B2
% 3.06/1.61 => ( ~ B5
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ S2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ S23 ) ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ ( if @ A @ B2 @ S1 @ S2 ) ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( if @ ( refine1665802226e_nres @ B ) @ B5 @ S13 @ S23 ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_166_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,M3: refine1665802226e_nres @ A,M6: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ M6 )
% 3.06/1.61 => ( ! [X3: A] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M3 )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X3 ) @ ( F4 @ X3 ) ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M3 @ F ) @ ( refine463715084e_bind @ A @ B @ M6 @ F4 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_167_bind__cong,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M4: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
% 3.06/1.61 ( ( M = M4 )
% 3.06/1.61 => ( ! [X3: A] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M4 )
% 3.06/1.61 => ( ( F @ X3 )
% 3.06/1.61 = ( F4 @ X3 ) ) )
% 3.06/1.61 => ( ( refine463715084e_bind @ A @ B @ M @ F )
% 3.06/1.61 = ( refine463715084e_bind @ A @ B @ M4 @ F4 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_168_inres__def,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( refine1315500908_inres @ A )
% 3.06/1.61 = ( ^ [S4: refine1665802226e_nres @ A,X5: A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X5 ) @ S4 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_169_RETURN__refine,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,X4: A,X7: B,R: set @ ( product_prod @ A @ B )] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ X7 ) @ R )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X4 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1687780735RETURN @ B @ X7 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_170_RETURN__ref__RETURND,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,C2: A,R: set @ ( product_prod @ A @ B ),A2: B] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ C2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1687780735RETURN @ B @ A2 ) ) )
% 3.06/1.61 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ C2 @ A2 ) @ R ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_171_RETURN__ref__SPECD,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,C2: A,R: set @ ( product_prod @ A @ B ),Phi: B > $o] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ C2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine605929679le_RES @ B @ ( collect @ B @ Phi ) ) ) )
% 3.06/1.61 => ~ ! [A4: B] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ C2 @ A4 ) @ R )
% 3.06/1.61 => ~ ( Phi @ A4 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_172_RETURN__SPEC__refine,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,X4: B,R: set @ ( product_prod @ B @ A ),Phi: A > $o] :
% 3.06/1.61 ( ? [X12: A] :
% 3.06/1.61 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ X12 ) @ R )
% 3.06/1.61 & ( Phi @ X12 ) )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine1687780735RETURN @ B @ X4 ) @ ( refine1073749519nc_fun @ B @ A @ R @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_173_pw__abs__inres,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M3: refine1665802226e_nres @ B,A2: A] :
% 3.06/1.61 ( ( refine1315500908_inres @ A @ ( refine81118332bs_fun @ B @ A @ R @ M3 ) @ A2 )
% 3.06/1.61 = ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M3 ) )
% 3.06/1.61 => ? [C5: B] :
% 3.06/1.61 ( ( refine1315500908_inres @ B @ M3 @ C5 )
% 3.06/1.61 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ C5 @ A2 ) @ R ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_174_relChain__def,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( ord @ B )
% 3.06/1.61 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 3.06/1.61 = ( ^ [R5: set @ ( product_prod @ A @ A ),As: A > B] :
% 3.06/1.61 ! [I2: A,J: A] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R5 )
% 3.06/1.61 => ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_175_uncurry__apply,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
% 3.06/1.61 ( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 3.06/1.61 = ( F @ A2 @ B2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_176_abs__trans__additional_I1_J,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,A5: refine1665802226e_nres @ A,B6: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C3: refine1665802226e_nres @ B] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A5 @ B6 )
% 3.06/1.61 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ B6 ) @ C3 )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A5 ) @ C3 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_177_abs__trans,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),C3: refine1665802226e_nres @ B,B6: refine1665802226e_nres @ A,R2: set @ ( product_prod @ A @ C ),A5: refine1665802226e_nres @ C] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine81118332bs_fun @ B @ A @ R @ C3 ) @ B6 )
% 3.06/1.61 => ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ B6 ) @ A5 )
% 3.06/1.61 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ ( refine81118332bs_fun @ B @ A @ R @ C3 ) ) @ A5 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_178_uncurry__curry__id,axiom,
% 3.06/1.61 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 3.06/1.61 ( ( uncurry @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 3.06/1.61 = F ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_179_curry__uncurry__id,axiom,
% 3.06/1.61 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 3.06/1.61 ( ( product_curry @ A @ B @ C @ ( uncurry @ A @ B @ C @ F ) )
% 3.06/1.61 = F ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_180_pw__abs__nofail,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M3: refine1665802226e_nres @ B] :
% 3.06/1.61 ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M3 ) )
% 3.06/1.61 = ( ( refine1102455758nofail @ B @ M3 )
% 3.06/1.61 & ! [X5: B] :
% 3.06/1.61 ( ( refine1315500908_inres @ B @ M3 @ X5 )
% 3.06/1.61 => ( member @ B @ X5 @ ( domain @ B @ A @ R ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_181_curryI,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 3.06/1.61 ( ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.06/1.61 => ( product_curry @ A @ B @ $o @ F @ A2 @ B2 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_182_curry__conv,axiom,
% 3.06/1.61 ! [A: $tType,C: $tType,B: $tType] :
% 3.06/1.61 ( ( product_curry @ B @ C @ A )
% 3.06/1.61 = ( ^ [F3: ( product_prod @ B @ C ) > A,A3: B,B3: C] : ( F3 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_183_curryE,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 3.06/1.61 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 3.06/1.61 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_184_curryD,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 3.06/1.61 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 3.06/1.61 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_185_DomainE,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,A2: A,R3: set @ ( product_prod @ A @ B )] :
% 3.06/1.61 ( ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) )
% 3.06/1.61 => ~ ! [B4: B] :
% 3.06/1.61 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B4 ) @ R3 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_186_Domain__iff,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,A2: A,R3: set @ ( product_prod @ A @ B )] :
% 3.06/1.61 ( ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) )
% 3.06/1.61 = ( ? [Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ Y6 ) @ R3 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_187_Domain_Ocases,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,A2: A,R3: set @ ( product_prod @ A @ B )] :
% 3.06/1.61 ( ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) )
% 3.06/1.61 => ~ ! [B4: B] :
% 3.06/1.61 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B4 ) @ R3 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_188_Domain_Osimps,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,A2: A,R3: set @ ( product_prod @ A @ B )] :
% 3.06/1.61 ( ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) )
% 3.06/1.61 = ( ? [A3: A,B3: B] :
% 3.06/1.61 ( ( A2 = A3 )
% 3.06/1.61 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_189_Domain_ODomainI,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,A2: A,B2: B,R3: set @ ( product_prod @ A @ B )] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R3 )
% 3.06/1.61 => ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_190_Domain_Oinducts,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,X4: A,R3: set @ ( product_prod @ A @ B ),P: A > $o] :
% 3.06/1.61 ( ( member @ A @ X4 @ ( domain @ A @ B @ R3 ) )
% 3.06/1.61 => ( ! [A4: A,B4: B] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R3 )
% 3.06/1.61 => ( P @ A4 ) )
% 3.06/1.61 => ( P @ X4 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_191_Domain__mono,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S9: set @ ( product_prod @ A @ B )] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S9 )
% 3.06/1.61 => ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R3 ) @ ( domain @ A @ B @ S9 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_192_for__in__RI,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,X4: A,R: set @ ( product_prod @ A @ B )] :
% 3.06/1.61 ( ( member @ A @ X4 @ ( domain @ A @ B @ R ) )
% 3.06/1.61 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ ( fun_of_rel @ A @ B @ R @ X4 ) ) @ R ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_193_abs__fun__simps_I3_J,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,X: set @ B,R: set @ ( product_prod @ B @ A )] :
% 3.06/1.61 ( ~ ( ord_less_eq @ ( set @ B ) @ X @ ( domain @ B @ A @ R ) )
% 3.06/1.61 => ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X ) )
% 3.06/1.61 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_194_abs__fun__simps_I2_J,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,X: set @ B,R: set @ ( product_prod @ B @ A )] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ B ) @ X @ ( domain @ B @ A @ R ) )
% 3.06/1.61 => ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X ) )
% 3.06/1.61 = ( refine605929679le_RES @ A @ ( image @ B @ A @ R @ X ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_195_top__apply,axiom,
% 3.06/1.61 ! [C: $tType,D2: $tType] :
% 3.06/1.61 ( ( top @ C )
% 3.06/1.61 => ( ( top_top @ ( D2 > C ) )
% 3.06/1.61 = ( ^ [X5: D2] : ( top_top @ C ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_196_ImageI,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,A2: A,B2: B,R3: set @ ( product_prod @ A @ B ),A5: set @ A] :
% 3.06/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R3 )
% 3.06/1.61 => ( ( member @ A @ A2 @ A5 )
% 3.06/1.61 => ( member @ B @ B2 @ ( image @ A @ B @ R3 @ A5 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_197_nres__order__simps_I4_J,axiom,
% 3.06/1.61 ! [D2: $tType,M3: refine1665802226e_nres @ D2] :
% 3.06/1.61 ( ( ord_less_eq @ ( refine1665802226e_nres @ D2 ) @ ( top_top @ ( refine1665802226e_nres @ D2 ) ) @ M3 )
% 3.06/1.61 = ( M3
% 3.06/1.61 = ( top_top @ ( refine1665802226e_nres @ D2 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_198_conc__fun__FAIL,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 3.06/1.61 ( ( refine1073749519nc_fun @ A @ B @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
% 3.06/1.61 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_199_conc__fun__fail__iff_I1_J,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 3.06/1.61 ( ( ( refine1073749519nc_fun @ A @ B @ R @ S )
% 3.06/1.61 = ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 3.06/1.61 = ( S
% 3.06/1.61 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_200_conc__fun__fail__iff_I2_J,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 3.06/1.61 ( ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 3.06/1.61 = ( refine1073749519nc_fun @ A @ B @ R @ S ) )
% 3.06/1.61 = ( S
% 3.06/1.61 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_201_bind__FAIL,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,F: B > ( refine1665802226e_nres @ A )] :
% 3.06/1.61 ( ( refine463715084e_bind @ B @ A @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ F )
% 3.06/1.61 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_202_nofail__simps_I1_J,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ~ ( refine1102455758nofail @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_203_inres__simps_I1_J,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( refine1315500908_inres @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 3.06/1.61 = ( ^ [Uu3: A] : $true ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_204_nres__simp__internals_I2_J,axiom,
% 3.06/1.61 ! [B: $tType] :
% 3.06/1.61 ( ( refine1767639642_FAILi @ B )
% 3.06/1.61 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_205_abs__fun__simps_I1_J,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
% 3.06/1.61 ( ( refine81118332bs_fun @ B @ A @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
% 3.06/1.61 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_206_FAIL__refine,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,X: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X @ ( refine1073749519nc_fun @ A @ B @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_207_rev__ImageI,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,A2: A,A5: set @ A,B2: B,R3: set @ ( product_prod @ A @ B )] :
% 3.06/1.61 ( ( member @ A @ A2 @ A5 )
% 3.06/1.61 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R3 )
% 3.06/1.61 => ( member @ B @ B2 @ ( image @ A @ B @ R3 @ A5 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_208_Image__iff,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A5: set @ B] :
% 3.06/1.61 ( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ A5 ) )
% 3.06/1.61 = ( ? [X5: B] :
% 3.06/1.61 ( ( member @ B @ X5 @ A5 )
% 3.06/1.61 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X5 @ B2 ) @ R3 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_209_ImageE,axiom,
% 3.06/1.61 ! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A5: set @ B] :
% 3.06/1.61 ( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ A5 ) )
% 3.06/1.61 => ~ ! [X3: B] :
% 3.06/1.61 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B2 ) @ R3 )
% 3.06/1.61 => ~ ( member @ B @ X3 @ A5 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_210_nres__inequalities_I3_J,axiom,
% 3.06/1.61 ! [C: $tType,X4: C] :
% 3.06/1.61 ( ( top_top @ ( refine1665802226e_nres @ C ) )
% 3.06/1.61 != ( refine1687780735RETURN @ C @ X4 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_211_top__nres__def,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 3.06/1.61 = ( refine1767639642_FAILi @ A ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_212_top__greatest,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order_top @ A )
% 3.06/1.61 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_213_top_Oextremum__unique,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order_top @ A )
% 3.06/1.61 => ! [A2: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 3.06/1.61 = ( A2
% 3.06/1.61 = ( top_top @ A ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_214_top_Oextremum__uniqueI,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order_top @ A )
% 3.06/1.61 => ! [A2: A] :
% 3.06/1.61 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 3.06/1.61 => ( A2
% 3.06/1.61 = ( top_top @ A ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_215_nres__order__simps_I3_J,axiom,
% 3.06/1.61 ! [C: $tType,M3: refine1665802226e_nres @ C] : ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ M3 @ ( top_top @ ( refine1665802226e_nres @ C ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_216_nres__inequalities_I1_J,axiom,
% 3.06/1.61 ! [A: $tType,X: set @ A] :
% 3.06/1.61 ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 3.06/1.61 != ( refine605929679le_RES @ A @ X ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_217_nres__cases,axiom,
% 3.06/1.61 ! [A: $tType,M3: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( M3
% 3.06/1.61 != ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 3.06/1.61 => ~ ! [X13: set @ A] :
% 3.06/1.61 ( M3
% 3.06/1.61 != ( refine605929679le_RES @ A @ X13 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_218_not__nofail__iff,axiom,
% 3.06/1.61 ! [A: $tType,S: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( ~ ( refine1102455758nofail @ A @ S ) )
% 3.06/1.61 = ( S
% 3.06/1.61 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_219_nofail__def,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( refine1102455758nofail @ A )
% 3.06/1.61 = ( ^ [S4: refine1665802226e_nres @ A] :
% 3.06/1.61 ( S4
% 3.06/1.61 != ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_220_intro__nofail_I2_J,axiom,
% 3.06/1.61 ! [A: $tType,S: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 3.06/1.61 != S )
% 3.06/1.61 = ( refine1102455758nofail @ A @ S ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_221_ibind__strict_I2_J,axiom,
% 3.06/1.61 ! [A: $tType,F: product_unit > ( refine1665802226e_nres @ A )] :
% 3.06/1.61 ( ( refine463715084e_bind @ product_unit @ A @ ( top_top @ ( refine1665802226e_nres @ product_unit ) ) @ F )
% 3.06/1.61 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_222_Image__mono,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,R6: set @ ( product_prod @ A @ B ),R3: set @ ( product_prod @ A @ B ),A8: set @ A,A5: set @ A] :
% 3.06/1.61 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R6 @ R3 )
% 3.06/1.61 => ( ( ord_less_eq @ ( set @ A ) @ A8 @ A5 )
% 3.06/1.61 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R6 @ A8 ) @ ( image @ A @ B @ R3 @ A5 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_223_meta__le__everything__if__top,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( order_top @ A )
% 3.06/1.61 => ! [M: A,X4: A] :
% 3.06/1.61 ( ( M
% 3.06/1.61 = ( top_top @ A ) )
% 3.06/1.61 => ( ord_less_eq @ A @ X4 @ M ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_224_pw__flat__ge__iff,axiom,
% 3.06/1.61 ! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( top_top @ ( refine1665802226e_nres @ A ) ) @ S @ S3 )
% 3.06/1.61 = ( ( refine1102455758nofail @ A @ S )
% 3.06/1.61 => ( ( refine1102455758nofail @ A @ S3 )
% 3.06/1.61 & ! [X5: A] :
% 3.06/1.61 ( ( refine1315500908_inres @ A @ S @ X5 )
% 3.06/1.61 = ( refine1315500908_inres @ A @ S3 @ X5 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_225_subset__UNIV,axiom,
% 3.06/1.61 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_226_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I2_J,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType,M3: refine1665802226e_nres @ A,M6: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
% 3.06/1.61 ( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( top_top @ ( refine1665802226e_nres @ A ) ) @ M3 @ M6 )
% 3.06/1.61 => ( ! [X3: A] : ( partial_flat_ord @ ( refine1665802226e_nres @ B ) @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ ( F @ X3 ) @ ( F4 @ X3 ) )
% 3.06/1.61 => ( partial_flat_ord @ ( refine1665802226e_nres @ B ) @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ ( refine463715084e_bind @ A @ B @ M3 @ F ) @ ( refine463715084e_bind @ A @ B @ M6 @ F4 ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_227_flat__ord__compat_I2_J,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( comple187826305attice @ A )
% 3.06/1.61 => ! [X4: A,Y3: A] :
% 3.06/1.61 ( ( partial_flat_ord @ A @ ( top_top @ A ) @ X4 @ Y3 )
% 3.06/1.61 => ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_228_imonad1,axiom,
% 3.06/1.61 ! [A: $tType,U: product_unit,F: product_unit > ( refine1665802226e_nres @ A )] :
% 3.06/1.61 ( ( refine463715084e_bind @ product_unit @ A @ ( refine1687780735RETURN @ product_unit @ U ) @ F )
% 3.06/1.61 = ( F @ U ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_229_flat__ord_OantimonoD,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( order @ B )
% 3.06/1.61 => ! [B2: A,F: A > B,X4: A,Y3: A] :
% 3.06/1.61 ( ( antimono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
% 3.06/1.61 => ( ( partial_flat_ord @ A @ B2 @ X4 @ Y3 )
% 3.06/1.61 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X4 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_230_flat__ord_OantimonoE,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( order @ B )
% 3.06/1.61 => ! [B2: A,F: A > B,X4: A,Y3: A] :
% 3.06/1.61 ( ( antimono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
% 3.06/1.61 => ( ( partial_flat_ord @ A @ B2 @ X4 @ Y3 )
% 3.06/1.61 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X4 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_231_UNIV__I,axiom,
% 3.06/1.61 ! [A: $tType,X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_232_relprop__UNIV__orient,axiom,
% 3.06/1.61 ! [A: $tType,R: set @ A] :
% 3.06/1.61 ( ( R
% 3.06/1.61 = ( top_top @ ( set @ A ) ) )
% 3.06/1.61 => ( ( top_top @ ( set @ A ) )
% 3.06/1.61 = R ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_233_eq__UNIV__iff,axiom,
% 3.06/1.61 ! [A: $tType,S: set @ A] :
% 3.06/1.61 ( ( S
% 3.06/1.61 = ( top_top @ ( set @ A ) ) )
% 3.06/1.61 = ( ! [X5: A] : ( member @ A @ X5 @ S ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_234_UNIV__witness,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_235_top__set__def,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( top_top @ ( set @ A ) )
% 3.06/1.61 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_236_UNIV__eq__I,axiom,
% 3.06/1.61 ! [A: $tType,A5: set @ A] :
% 3.06/1.61 ( ! [X3: A] : ( member @ A @ X3 @ A5 )
% 3.06/1.61 => ( ( top_top @ ( set @ A ) )
% 3.06/1.61 = A5 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_237_top__empty__eq,axiom,
% 3.06/1.61 ! [A: $tType] :
% 3.06/1.61 ( ( top_top @ ( A > $o ) )
% 3.06/1.61 = ( ^ [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_238_order_Oantimono_Ocong,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( order @ B )
% 3.06/1.61 => ( ( antimono @ A @ B )
% 3.06/1.61 = ( antimono @ A @ B ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_239_flat__ord_Oantimono__def,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( order @ B )
% 3.06/1.61 => ! [B2: A,F: A > B] :
% 3.06/1.61 ( ( antimono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
% 3.06/1.61 = ( ! [X5: A,Y6: A] :
% 3.06/1.61 ( ( partial_flat_ord @ A @ B2 @ X5 @ Y6 )
% 3.06/1.61 => ( ord_less_eq @ B @ ( F @ Y6 ) @ ( F @ X5 ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_240_flat__ord_OantimonoI,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( order @ B )
% 3.06/1.61 => ! [B2: A,F: A > B] :
% 3.06/1.61 ( ! [X3: A,Y4: A] :
% 3.06/1.61 ( ( partial_flat_ord @ A @ B2 @ X3 @ Y4 )
% 3.06/1.61 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X3 ) ) )
% 3.06/1.61 => ( antimono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_241_flat__ord_Omono__def,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( order @ B )
% 3.06/1.61 => ! [B2: A,F: A > B] :
% 3.06/1.61 ( ( mono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
% 3.06/1.61 = ( ! [X5: A,Y6: A] :
% 3.06/1.61 ( ( partial_flat_ord @ A @ B2 @ X5 @ Y6 )
% 3.06/1.61 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y6 ) ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_242_flat__ord_OmonoI,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( order @ B )
% 3.06/1.61 => ! [B2: A,F: A > B] :
% 3.06/1.61 ( ! [X3: A,Y4: A] :
% 3.06/1.61 ( ( partial_flat_ord @ A @ B2 @ X3 @ Y4 )
% 3.06/1.61 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 3.06/1.61 => ( mono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_243_top1I,axiom,
% 3.06/1.61 ! [A: $tType,X4: A] : ( top_top @ ( A > $o ) @ X4 ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_244_order_Omono_Ocong,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( order @ B )
% 3.06/1.61 => ( ( mono @ A @ B )
% 3.06/1.61 = ( mono @ A @ B ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_245_flat__ord_OmonoD,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( order @ B )
% 3.06/1.61 => ! [B2: A,F: A > B,X4: A,Y3: A] :
% 3.06/1.61 ( ( mono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
% 3.06/1.61 => ( ( partial_flat_ord @ A @ B2 @ X4 @ Y3 )
% 3.06/1.61 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_246_flat__ord_OmonoE,axiom,
% 3.06/1.61 ! [B: $tType,A: $tType] :
% 3.06/1.61 ( ( order @ B )
% 3.06/1.61 => ! [B2: A,F: A > B,X4: A,Y3: A] :
% 3.06/1.61 ( ( mono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
% 3.06/1.61 => ( ( partial_flat_ord @ A @ B2 @ X4 @ Y3 )
% 3.06/1.61 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_247_pw__flat__le__iff,axiom,
% 3.06/1.61 ! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A] :
% 3.06/1.61 ( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( bot_bot @ ( refine1665802226e_nres @ A ) ) @ S @ S3 )
% 3.06/1.61 = ( ? [X14: A] : ( refine1315500908_inres @ A @ S @ X14 )
% 3.06/1.61 => ( ( ( refine1102455758nofail @ A @ S )
% 3.06/1.61 = ( refine1102455758nofail @ A @ S3 ) )
% 3.06/1.61 & ! [X5: A] :
% 3.06/1.61 ( ( refine1315500908_inres @ A @ S @ X5 )
% 3.06/1.61 = ( refine1315500908_inres @ A @ S3 @ X5 ) ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_248_ASSERT__simps_I2_J,axiom,
% 3.06/1.61 ( ( refine1814851989ASSERT @ $false )
% 3.06/1.61 = ( top_top @ ( refine1665802226e_nres @ product_unit ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(fact_249_bot__apply,axiom,
% 3.06/1.61 ! [C: $tType,D2: $tType] :
% 3.06/1.61 ( ( bot @ C )
% 3.06/1.61 => ( ( bot_bot @ ( D2 > C ) )
% 3.06/1.61 = ( ^ [X5: D2] : ( bot_bot @ C ) ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 3.06/1.61 ! [A9: $tType,A10: $tType] :
% 3.06/1.61 ( ( comple187826305attice @ A10 )
% 3.06/1.61 => ( comple187826305attice @ ( A9 > A10 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 3.06/1.61 ! [A9: $tType,A10: $tType] :
% 3.06/1.61 ( ( order_top @ A10 )
% 3.06/1.61 => ( order_top @ ( A9 > A10 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.06/1.61 ! [A9: $tType,A10: $tType] :
% 3.06/1.61 ( ( preorder @ A10 )
% 3.06/1.61 => ( preorder @ ( A9 > A10 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.06/1.61 ! [A9: $tType,A10: $tType] :
% 3.06/1.61 ( ( order @ A10 )
% 3.06/1.61 => ( order @ ( A9 > A10 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_fun___Orderings_Otop,axiom,
% 3.06/1.61 ! [A9: $tType,A10: $tType] :
% 3.06/1.61 ( ( top @ A10 )
% 3.06/1.61 => ( top @ ( A9 > A10 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_fun___Orderings_Oord,axiom,
% 3.06/1.61 ! [A9: $tType,A10: $tType] :
% 3.06/1.61 ( ( ord @ A10 )
% 3.06/1.61 => ( ord @ ( A9 > A10 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_fun___Orderings_Obot,axiom,
% 3.06/1.61 ! [A9: $tType,A10: $tType] :
% 3.06/1.61 ( ( bot @ A10 )
% 3.06/1.61 => ( bot @ ( A9 > A10 ) ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_1,axiom,
% 3.06/1.61 ! [A9: $tType] : ( comple187826305attice @ ( set @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
% 3.06/1.61 ! [A9: $tType] : ( order_top @ ( set @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
% 3.06/1.61 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
% 3.06/1.61 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Set_Oset___Orderings_Otop_5,axiom,
% 3.06/1.61 ! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 3.06/1.61 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Set_Oset___Orderings_Obot_7,axiom,
% 3.06/1.61 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_8,axiom,
% 3.06/1.61 comple187826305attice @ $o ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_HOL_Obool___Orderings_Oorder__top_9,axiom,
% 3.06/1.61 order_top @ $o ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 3.06/1.61 preorder @ $o ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 3.06/1.61 linorder @ $o ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 3.06/1.61 order @ $o ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_HOL_Obool___Orderings_Otop_12,axiom,
% 3.06/1.61 top @ $o ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
% 3.06/1.61 ord @ $o ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_HOL_Obool___Orderings_Obot_14,axiom,
% 3.06/1.61 bot @ $o ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_15,axiom,
% 3.06/1.61 comple187826305attice @ product_unit ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_16,axiom,
% 3.06/1.61 order_top @ product_unit ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_17,axiom,
% 3.06/1.61 preorder @ product_unit ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_18,axiom,
% 3.06/1.61 linorder @ product_unit ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Product__Type_Ounit___Orderings_Oorder_19,axiom,
% 3.06/1.61 order @ product_unit ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Product__Type_Ounit___Orderings_Otop_20,axiom,
% 3.06/1.61 top @ product_unit ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Product__Type_Ounit___Orderings_Oord_21,axiom,
% 3.06/1.61 ord @ product_unit ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Product__Type_Ounit___Orderings_Obot_22,axiom,
% 3.06/1.61 bot @ product_unit ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Complete__Lattices_Ocomplete__lattice_23,axiom,
% 3.06/1.61 ! [A9: $tType] : ( comple187826305attice @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__top_24,axiom,
% 3.06/1.61 ! [A9: $tType] : ( order_top @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_25,axiom,
% 3.06/1.61 ! [A9: $tType] : ( preorder @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_26,axiom,
% 3.06/1.61 ! [A9: $tType] : ( order @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_27,axiom,
% 3.06/1.61 ! [A9: $tType] : ( top @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_28,axiom,
% 3.06/1.61 ! [A9: $tType] : ( ord @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Obot_29,axiom,
% 3.06/1.61 ! [A9: $tType] : ( bot @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(help_If_3_1_T,axiom,
% 3.06/1.61 ! [P: $o] :
% 3.06/1.61 ( ( P = $true )
% 3.06/1.61 | ( P = $false ) ) ).
% 3.06/1.61
% 3.06/1.61 thf(help_If_2_1_T,axiom,
% 3.06/1.61 ! [A: $tType,X4: A,Y3: A] :
% 3.06/1.61 ( ( if @ A @ $false @ X4 @ Y3 )
% 3.06/1.61 = Y3 ) ).
% 3.06/1.61
% 3.06/1.61 thf(help_If_1_1_T,axiom,
% 3.06/1.61 ! [A: $tType,X4: A,Y3: A] :
% 3.06/1.61 ( ( if @ A @ $true @ X4 @ Y3 )
% 3.06/1.61 = X4 ) ).
% 3.06/1.61
% 3.06/1.61 thf(conj_0,conjecture,
% 3.06/1.61 ord_less_eq @ ( refine1665802226e_nres @ b ) @ m @ ( refine463715084e_bind @ a @ b @ ( refine605929679le_RES @ a @ ( collect @ a @ phi ) ) @ f ) ).
% 3.06/1.61 ------------------------
% 3.06/1.64 % (13595)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_13304 for (2999ds/27Mi)
% 3.06/1.64 % (13593)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/183Mi)
% 3.06/1.64 % (13596)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/2Mi)
% 3.06/1.64 % (13596)Instruction limit reached!
% 3.06/1.64 % (13596)------------------------------
% 3.06/1.64 % (13596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.64 % (13596)Termination reason: Unknown
% 3.06/1.64 % (13596)Termination phase: shuffling
% 3.06/1.64
% 3.06/1.64 % (13596)Memory used [KB]: 1407
% 3.06/1.64 % (13596)Time elapsed: 0.002 s
% 3.06/1.64 % (13596)Instructions burned: 2 (million)
% 3.06/1.64 % (13596)------------------------------
% 3.06/1.64 % (13596)------------------------------
% 3.06/1.65 % (13594)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_13304 for (2999ds/4Mi)
% 3.06/1.65 % (13594)Instruction limit reached!
% 3.06/1.65 % (13594)------------------------------
% 3.06/1.65 % (13594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.65 % (13594)Termination reason: Unknown
% 3.06/1.65 % (13594)Termination phase: shuffling
% 3.06/1.65
% 3.06/1.65 % (13594)Memory used [KB]: 1407
% 3.06/1.65 % (13594)Time elapsed: 0.004 s
% 3.06/1.65 % (13594)Instructions burned: 4 (million)
% 3.06/1.65 % (13594)------------------------------
% 3.06/1.65 % (13594)------------------------------
% 3.06/1.65 % (13597)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_13304 for (2999ds/2Mi)
% 3.06/1.65 % (13598)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_13304 for (2999ds/275Mi)
% 3.06/1.65 % (13599)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/18Mi)
% 3.06/1.65 % (13595)Instruction limit reached!
% 3.06/1.65 % (13595)------------------------------
% 3.06/1.65 % (13595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.65 % (13595)Termination reason: Unknown
% 3.06/1.65 % (13595)Termination phase: Property scanning
% 3.06/1.65
% 3.06/1.65 % (13595)Memory used [KB]: 1918
% 3.06/1.65 % (13595)Time elapsed: 0.011 s
% 3.06/1.65 % (13595)Instructions burned: 27 (million)
% 3.06/1.65 % (13595)------------------------------
% 3.06/1.65 % (13595)------------------------------
% 3.06/1.65 % (13597)Instruction limit reached!
% 3.06/1.65 % (13597)------------------------------
% 3.06/1.65 % (13597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.65 % (13597)Termination reason: Unknown
% 3.06/1.65 % (13597)Termination phase: shuffling
% 3.06/1.65
% 3.06/1.65 % (13597)Memory used [KB]: 1407
% 3.06/1.65 % (13597)Time elapsed: 0.004 s
% 3.06/1.65 % (13597)Instructions burned: 3 (million)
% 3.06/1.65 % (13597)------------------------------
% 3.06/1.65 % (13597)------------------------------
% 3.06/1.66 % (13600)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/3Mi)
% 3.06/1.66 % (13600)Instruction limit reached!
% 3.06/1.66 % (13600)------------------------------
% 3.06/1.66 % (13600)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.66 % (13600)Termination reason: Unknown
% 3.06/1.66 % (13600)Termination phase: shuffling
% 3.06/1.66
% 3.06/1.66 % (13600)Memory used [KB]: 1407
% 3.06/1.66 % (13600)Time elapsed: 0.002 s
% 3.06/1.66 % (13600)Instructions burned: 3 (million)
% 3.06/1.66 % (13600)------------------------------
% 3.06/1.66 % (13600)------------------------------
% 3.06/1.66 % (13602)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_13304 for (2999ds/15Mi)
% 3.06/1.66 % (13599)Instruction limit reached!
% 3.06/1.66 % (13599)------------------------------
% 3.06/1.66 % (13599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.66 % (13599)Termination reason: Unknown
% 3.06/1.66 % (13599)Termination phase: shuffling
% 3.06/1.66
% 3.06/1.66 % (13599)Memory used [KB]: 1791
% 3.06/1.66 % (13599)Time elapsed: 0.014 s
% 3.06/1.66 % (13599)Instructions burned: 18 (million)
% 3.06/1.66 % (13599)------------------------------
% 3.06/1.66 % (13599)------------------------------
% 3.06/1.66 % (13601)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_13304 for (2999ds/37Mi)
% 3.06/1.67 % (13604)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_13304 for (2999ds/1041Mi)
% 3.06/1.67 % (13602)Instruction limit reached!
% 3.06/1.67 % (13602)------------------------------
% 3.06/1.67 % (13602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.67 % (13602)Termination reason: Unknown
% 3.06/1.67 % (13602)Termination phase: shuffling
% 3.06/1.67
% 3.06/1.67 % (13602)Memory used [KB]: 1791
% 3.06/1.67 % (13602)Time elapsed: 0.007 s
% 3.06/1.67 % (13602)Instructions burned: 16 (million)
% 3.06/1.67 % (13602)------------------------------
% 3.06/1.67 % (13602)------------------------------
% 3.06/1.67 % (13603)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_13304 for (2999ds/3Mi)
% 3.06/1.67 % (13603)Instruction limit reached!
% 3.06/1.67 % (13603)------------------------------
% 3.06/1.67 % (13603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.67 % (13603)Termination reason: Unknown
% 3.06/1.67 % (13603)Termination phase: shuffling
% 3.06/1.67
% 3.06/1.67 % (13603)Memory used [KB]: 1407
% 3.06/1.67 % (13603)Time elapsed: 0.004 s
% 3.06/1.67 % (13603)Instructions burned: 3 (million)
% 3.06/1.67 % (13603)------------------------------
% 3.06/1.67 % (13603)------------------------------
% 3.06/1.68 % (13601)Instruction limit reached!
% 3.06/1.68 % (13601)------------------------------
% 3.06/1.68 % (13601)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.68 % (13601)Termination reason: Unknown
% 3.06/1.68 % (13601)Termination phase: shuffling
% 3.06/1.68
% 3.06/1.68 % (13601)Memory used [KB]: 2302
% 3.06/1.68 % (13601)Time elapsed: 0.014 s
% 3.06/1.68 % (13601)Instructions burned: 39 (million)
% 3.06/1.68 % (13601)------------------------------
% 3.06/1.68 % (13601)------------------------------
% 3.06/1.68 % (13606)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/16Mi)
% 3.06/1.68 % (13605)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/7Mi)
% 3.06/1.68 % (13606)Instruction limit reached!
% 3.06/1.68 % (13606)------------------------------
% 3.06/1.68 % (13606)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.68 % (13606)Termination reason: Unknown
% 3.06/1.68 % (13606)Termination phase: shuffling
% 3.06/1.68
% 3.06/1.68 % (13606)Memory used [KB]: 1791
% 3.06/1.68 % (13606)Time elapsed: 0.007 s
% 3.06/1.68 % (13606)Instructions burned: 17 (million)
% 3.06/1.68 % (13606)------------------------------
% 3.06/1.68 % (13606)------------------------------
% 3.06/1.69 % (13605)Instruction limit reached!
% 3.06/1.69 % (13605)------------------------------
% 3.06/1.69 % (13605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.69 % (13605)Termination reason: Unknown
% 3.06/1.69 % (13605)Termination phase: shuffling
% 3.06/1.69
% 3.06/1.69 % (13605)Memory used [KB]: 1535
% 3.06/1.69 % (13605)Time elapsed: 0.007 s
% 3.06/1.69 % (13605)Instructions burned: 8 (million)
% 3.06/1.69 % (13605)------------------------------
% 3.06/1.69 % (13605)------------------------------
% 3.06/1.69 % (13608)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/3Mi)
% 3.06/1.69 % (13608)Instruction limit reached!
% 3.06/1.69 % (13608)------------------------------
% 3.06/1.69 % (13608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.69 % (13608)Termination reason: Unknown
% 3.06/1.69 % (13608)Termination phase: shuffling
% 3.06/1.69
% 3.06/1.69 % (13608)Memory used [KB]: 1407
% 3.06/1.69 % (13608)Time elapsed: 0.003 s
% 3.06/1.69 % (13608)Instructions burned: 3 (million)
% 3.06/1.69 % (13608)------------------------------
% 3.06/1.69 % (13608)------------------------------
% 3.06/1.69 % (13607)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/3Mi)
% 3.06/1.69 % (13607)Instruction limit reached!
% 3.06/1.69 % (13607)------------------------------
% 3.06/1.69 % (13607)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.69 % (13607)Termination reason: Unknown
% 3.06/1.69 % (13607)Termination phase: shuffling
% 3.06/1.69
% 3.06/1.69 % (13607)Memory used [KB]: 1407
% 3.06/1.69 % (13607)Time elapsed: 0.004 s
% 3.06/1.69 % (13607)Instructions burned: 3 (million)
% 3.06/1.69 % (13607)------------------------------
% 3.06/1.69 % (13607)------------------------------
% 3.06/1.69 % (13609)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/7Mi)
% 3.06/1.70 % (13609)Instruction limit reached!
% 3.06/1.70 % (13609)------------------------------
% 3.06/1.70 % (13609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.70 % (13609)Termination reason: Unknown
% 3.06/1.70 % (13609)Termination phase: shuffling
% 3.06/1.70
% 3.06/1.70 % (13609)Memory used [KB]: 1535
% 3.06/1.70 % (13609)Time elapsed: 0.004 s
% 3.06/1.70 % (13609)Instructions burned: 8 (million)
% 3.06/1.70 % (13609)------------------------------
% 3.06/1.70 % (13609)------------------------------
% 3.10/1.70 % (13611)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/4Mi)
% 3.10/1.70 % (13611)Instruction limit reached!
% 3.10/1.70 % (13611)------------------------------
% 3.10/1.70 % (13611)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.70 % (13611)Termination reason: Unknown
% 3.10/1.70 % (13611)Termination phase: shuffling
% 3.10/1.70
% 3.10/1.70 % (13611)Memory used [KB]: 1535
% 3.10/1.70 % (13611)Time elapsed: 0.004 s
% 3.10/1.70 % (13611)Instructions burned: 6 (million)
% 3.10/1.70 % (13611)------------------------------
% 3.10/1.70 % (13611)------------------------------
% 3.10/1.70 % (13593)Instruction limit reached!
% 3.10/1.70 % (13593)------------------------------
% 3.10/1.70 % (13593)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.70 % (13593)Termination reason: Unknown
% 3.10/1.70 % (13593)Termination phase: Saturation
% 3.10/1.70
% 3.10/1.70 % (13593)Memory used [KB]: 7547
% 3.10/1.70 % (13593)Time elapsed: 0.060 s
% 3.10/1.70 % (13593)Instructions burned: 183 (million)
% 3.10/1.70 % (13593)------------------------------
% 3.10/1.70 % (13593)------------------------------
% 3.10/1.70 % (13610)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/3Mi)
% 3.10/1.71 % (13610)Instruction limit reached!
% 3.10/1.71 % (13610)------------------------------
% 3.10/1.71 % (13610)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.71 % (13610)Termination reason: Unknown
% 3.10/1.71 % (13610)Termination phase: shuffling
% 3.10/1.71
% 3.10/1.71 % (13610)Memory used [KB]: 1407
% 3.10/1.71 % (13610)Time elapsed: 0.004 s
% 3.10/1.71 % (13610)Instructions burned: 3 (million)
% 3.10/1.71 % (13610)------------------------------
% 3.10/1.71 % (13610)------------------------------
% 3.10/1.71 % (13613)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/710Mi)
% 3.10/1.71 % (13612)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/18Mi)
% 3.10/1.71 % (13614)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/6Mi)
% 3.10/1.71 % (13614)Instruction limit reached!
% 3.10/1.71 % (13614)------------------------------
% 3.10/1.71 % (13614)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.71 % (13614)Termination reason: Unknown
% 3.10/1.71 % (13614)Termination phase: shuffling
% 3.10/1.71
% 3.10/1.71 % (13614)Memory used [KB]: 1535
% 3.10/1.71 % (13614)Time elapsed: 0.004 s
% 3.10/1.71 % (13614)Instructions burned: 6 (million)
% 3.10/1.71 % (13614)------------------------------
% 3.10/1.71 % (13614)------------------------------
% 3.10/1.71 % (13615)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_13304 for (2999ds/902Mi)
% 3.10/1.72 % (13598)First to succeed.
% 3.10/1.72 % (13612)Instruction limit reached!
% 3.10/1.72 % (13612)------------------------------
% 3.10/1.72 % (13612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.72 % (13612)Termination reason: Unknown
% 3.10/1.72 % (13612)Termination phase: shuffling
% 3.10/1.72
% 3.10/1.72 % (13612)Memory used [KB]: 1791
% 3.10/1.72 % (13612)Time elapsed: 0.014 s
% 3.10/1.72 % (13612)Instructions burned: 19 (million)
% 3.10/1.72 % (13612)------------------------------
% 3.10/1.72 % (13612)------------------------------
% 3.10/1.72 % (13598)Refutation found. Thanks to Tanya!
% 3.10/1.72 % SZS status Theorem for DTF2THF_13304
% 3.10/1.72 % SZS output start Proof for DTF2THF_13304
% 3.10/1.72 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 3.10/1.72 thf(type_def_6, type, set: ($tType) > $tType).
% 3.10/1.72 thf(type_def_8, type, product_unit: $tType).
% 3.10/1.72 thf(type_def_9, type, refine1665802226e_nres: ($tType) > $tType).
% 3.10/1.72 thf(type_def_10, type, a: $tType).
% 3.10/1.72 thf(type_def_11, type, b: $tType).
% 3.10/1.72 thf(func_def_0, type, refine1665802226e_nres: ($tType) > $tType).
% 3.10/1.72 thf(func_def_1, type, product_unit: $tType).
% 3.10/1.72 thf(func_def_2, type, product_prod: ($tType * $tType) > $tType).
% 3.10/1.72 thf(func_def_3, type, set: ($tType) > $tType).
% 3.10/1.72 thf(func_def_4, type, b: $tType).
% 3.10/1.72 thf(func_def_5, type, a: $tType).
% 3.10/1.72 thf(func_def_6, type, type: !>[X0: $tType]:($o)).
% 3.10/1.72 thf(func_def_7, type, bot: !>[X0: $tType]:($o)).
% 3.10/1.72 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 3.10/1.72 thf(func_def_9, type, top: !>[X0: $tType]:($o)).
% 3.10/1.72 thf(func_def_10, type, order: !>[X0: $tType]:($o)).
% 3.10/1.72 thf(func_def_11, type, linorder: !>[X0: $tType]:($o)).
% 3.10/1.72 thf(func_def_12, type, preorder: !>[X0: $tType]:($o)).
% 3.10/1.72 thf(func_def_13, type, order_top: !>[X0: $tType]:($o)).
% 3.10/1.72 thf(func_def_14, type, comple187826305attice: !>[X0: $tType]:($o)).
% 3.10/1.72 thf(func_def_15, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 3.10/1.72 thf(func_def_16, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.10/1.72 thf(func_def_17, type, fun_of_rel: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > X0 > X1)).
% 3.10/1.72 thf(func_def_18, type, uncurry: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.10/1.72 thf(func_def_19, type, bot_bot: !>[X0: $tType]:(X0)).
% 3.10/1.72 thf(func_def_20, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.10/1.72 thf(func_def_21, type, antimono: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > $o)).
% 3.10/1.72 thf(func_def_22, type, mono: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > $o)).
% 3.10/1.72 thf(func_def_23, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 3.10/1.72 thf(func_def_24, type, top_top: !>[X0: $tType]:(X0)).
% 3.10/1.72 thf(func_def_25, type, partial_flat_ord: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 3.10/1.72 thf(func_def_26, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.10/1.72 thf(func_def_27, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 3.10/1.72 thf(func_def_28, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.10/1.72 thf(func_def_29, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.10/1.72 thf(func_def_30, type, refine1814851989ASSERT: $o > refine1665802226e_nres @ product_unit).
% 3.10/1.72 thf(func_def_31, type, refine1687780735RETURN: !>[X0: $tType]:(X0 > refine1665802226e_nres @ X0)).
% 3.10/1.72 thf(func_def_32, type, refine81118332bs_fun: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > refine1665802226e_nres @ X0 > refine1665802226e_nres @ X1)).
% 3.10/1.72 thf(func_def_33, type, refine463715084e_bind: !>[X0: $tType, X1: $tType]:(refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X1)).
% 3.10/1.72 thf(func_def_34, type, refine1073749519nc_fun: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > refine1665802226e_nres @ X1 > refine1665802226e_nres @ X0)).
% 3.10/1.72 thf(func_def_35, type, refine1315500908_inres: !>[X0: $tType]:(refine1665802226e_nres @ X0 > X0 > $o)).
% 3.10/1.72 thf(func_def_36, type, refine1580981607t_assn: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > (X1 > $o) > X0 > $o)).
% 3.10/1.72 thf(func_def_37, type, refine406925620_inres: !>[X0: $tType]:(refine1665802226e_nres @ X0 > X0 > $o)).
% 3.10/1.72 thf(func_def_38, type, refine1102455758nofail: !>[X0: $tType]:(refine1665802226e_nres @ X0 > $o)).
% 3.10/1.72 thf(func_def_39, type, refine1767639642_FAILi: !>[X0: $tType]:(refine1665802226e_nres @ X0)).
% 3.10/1.72 thf(func_def_40, type, refine605929679le_RES: !>[X0: $tType]:(set @ X0 > refine1665802226e_nres @ X0)).
% 3.10/1.72 thf(func_def_41, type, refine1442219249c_nres: !>[X0: $tType, X1: $tType]:(X0 > (set @ X1 > X0) > refine1665802226e_nres @ X1 > X0)).
% 3.10/1.72 thf(func_def_42, type, refine1672542526he_RES: !>[X0: $tType]:(refine1665802226e_nres @ X0 > set @ X0)).
% 3.10/1.72 thf(func_def_43, type, domain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0)).
% 3.10/1.72 thf(func_def_44, type, image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0 > set @ X1)).
% 3.10/1.72 thf(func_def_45, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.10/1.72 thf(func_def_46, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.10/1.72 thf(func_def_47, type, phi: a > $o).
% 3.10/1.72 thf(func_def_48, type, f: a > refine1665802226e_nres @ b).
% 3.10/1.72 thf(func_def_49, type, m: refine1665802226e_nres @ b).
% 3.10/1.72 thf(func_def_50, type, x: a).
% 3.10/1.72 thf(func_def_65, type, sK0: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.10/1.72 thf(func_def_66, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > refine1665802226e_nres @ X0) > refine1665802226e_nres @ X1 > X1)).
% 3.10/1.72 thf(func_def_67, type, sK2: !>[X0: $tType]:(refine1665802226e_nres @ X0 > set @ X0 > X0)).
% 3.10/1.72 thf(func_def_68, type, sK3: !>[X0: $tType]:(set @ X0 > X0)).
% 3.10/1.72 thf(func_def_69, type, sK4: !>[X0: $tType, X1: $tType]:(product_prod @ (X1 > X0) @ product_prod @ X1 @ X1 > X1 > X0)).
% 3.10/1.72 thf(func_def_70, type, sK5: !>[X0: $tType, X1: $tType]:(product_prod @ (X1 > X0) @ product_prod @ X1 @ X1 > X1)).
% 3.10/1.72 thf(func_def_71, type, sK6: !>[X0: $tType, X1: $tType]:(product_prod @ (X1 > X0) @ product_prod @ X1 @ X1 > X1)).
% 3.10/1.72 thf(func_def_72, type, sK7: !>[X0: $tType, X1: $tType]:((X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X0 > refine1665802226e_nres @ X1 > X0)).
% 3.10/1.72 thf(func_def_73, type, sK8: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > refine1665802226e_nres @ X0 > refine1665802226e_nres @ X1 > X0)).
% 3.10/1.72 thf(func_def_74, type, sK9: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 3.10/1.72 thf(func_def_75, type, sK10: !>[X0: $tType, X1: $tType]:(((X1 > X0) > product_prod @ X1 @ X1 > $o) > X1 > X0)).
% 3.10/1.72 thf(func_def_76, type, sK11: !>[X0: $tType, X1: $tType]:(((X1 > X0) > product_prod @ X1 @ X1 > $o) > X1)).
% 3.10/1.72 thf(func_def_77, type, sK12: !>[X0: $tType, X1: $tType]:(((X1 > X0) > product_prod @ X1 @ X1 > $o) > X1)).
% 3.10/1.72 thf(func_def_78, type, sK13: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.10/1.72 thf(func_def_79, type, sK14: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 3.10/1.72 thf(func_def_80, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 3.10/1.72 thf(func_def_81, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 3.10/1.72 thf(func_def_82, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 3.10/1.72 thf(func_def_83, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 3.10/1.72 thf(func_def_84, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 3.10/1.72 thf(func_def_85, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 3.10/1.72 thf(func_def_86, type, sK21: !>[X0: $tType, X1: $tType]:((X1 > refine1665802226e_nres @ X0) > set @ X1 > X1)).
% 3.10/1.72 thf(func_def_87, type, sK22: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.10/1.72 thf(func_def_88, type, sK23: !>[X0: $tType, X1: $tType]:(X1 > refine1665802226e_nres @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 3.10/1.72 thf(func_def_89, type, sK24: !>[X0: $tType, X1: $tType]:(refine1665802226e_nres @ X0 > set @ product_prod @ X1 @ X0 > refine1665802226e_nres @ X1 > X1)).
% 3.10/1.72 thf(func_def_90, type, sK25: !>[X0: $tType]:(set @ X0 > X0)).
% 3.10/1.72 thf(func_def_91, type, sK26: !>[X0: $tType]:(X0)).
% 3.10/1.72 thf(func_def_92, type, sK27: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.10/1.72 thf(func_def_93, type, sK28: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.10/1.72 thf(func_def_94, type, sK29: !>[X0: $tType, X1: $tType]:((X1 > refine1665802226e_nres @ X0) > (X1 > refine1665802226e_nres @ X0) > refine1665802226e_nres @ X1 > X1)).
% 3.10/1.72 thf(func_def_95, type, sK30: !>[X0: $tType]:(refine1665802226e_nres @ X0 > refine1665802226e_nres @ X0 > X0)).
% 3.10/1.72 thf(func_def_96, type, sK31: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.10/1.72 thf(func_def_97, type, sK32: !>[X0: $tType, X1: $tType]:(refine1665802226e_nres @ X0 > (X1 > refine1665802226e_nres @ X0) > refine1665802226e_nres @ X1 > X1)).
% 3.10/1.72 thf(func_def_98, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.10/1.72 thf(func_def_99, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.10/1.72 thf(func_def_100, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.72 thf(func_def_101, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.72 thf(func_def_102, type, sK37: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.10/1.72 thf(func_def_103, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.10/1.72 thf(func_def_104, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.10/1.72 thf(func_def_105, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.10/1.72 thf(func_def_106, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.72 thf(func_def_107, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.72 thf(func_def_108, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 3.10/1.72 thf(func_def_109, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 3.10/1.72 thf(func_def_110, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 3.10/1.72 thf(func_def_111, type, sK46: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > X0 > refine1665802226e_nres @ X1 > X1)).
% 3.10/1.72 thf(func_def_112, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 3.10/1.72 thf(func_def_113, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.10/1.72 thf(func_def_114, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.10/1.72 thf(func_def_115, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.10/1.72 thf(func_def_116, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.72 thf(func_def_117, type, sK52: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.72 thf(func_def_118, type, sK53: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 3.10/1.72 thf(func_def_119, type, sK54: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 3.10/1.72 thf(func_def_120, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 3.10/1.72 thf(func_def_121, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 3.10/1.72 thf(func_def_122, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 3.10/1.72 thf(func_def_123, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 3.10/1.72 thf(func_def_124, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 3.10/1.72 thf(func_def_125, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 3.10/1.72 thf(func_def_126, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 3.10/1.72 thf(func_def_127, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 3.10/1.72 thf(func_def_128, type, sK63: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 3.10/1.72 thf(func_def_129, type, sK64: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 3.10/1.72 thf(func_def_130, type, sK65: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 3.10/1.72 thf(func_def_131, type, sK66: !>[X0: $tType, X1: $tType]:((X1 > $o) > set @ product_prod @ X0 @ X1 > X0 > X1)).
% 3.10/1.72 thf(func_def_132, type, sK67: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.10/1.72 thf(func_def_133, type, sK68: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.72 thf(func_def_134, type, sK69: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.72 thf(func_def_135, type, sK70: !>[X0: $tType]:(refine1665802226e_nres @ X0 > set @ X0)).
% 3.10/1.72 thf(func_def_136, type, sK71: !>[X0: $tType, X1: $tType]:(X0 > (X1 > refine1665802226e_nres @ X0) > set @ X1 > X1)).
% 3.10/1.72 thf(func_def_137, type, sK72: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.72 thf(func_def_138, type, sK73: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.72 thf(func_def_139, type, sK74: !>[X0: $tType, X1: $tType]:((X1 > refine1665802226e_nres @ X0) > (X1 > refine1665802226e_nres @ X0) > refine1665802226e_nres @ X1 > X1)).
% 3.10/1.72 thf(func_def_140, type, sK75: !>[X0: $tType]:(refine1665802226e_nres @ X0 > refine1665802226e_nres @ X0 > X0)).
% 3.10/1.72 thf(func_def_141, type, sK76: !>[X0: $tType]:(refine1665802226e_nres @ X0 > refine1665802226e_nres @ X0 > X0)).
% 3.10/1.72 thf(func_def_142, type, sK77: !>[X0: $tType, X1: $tType, X2: $tType]:(refine1665802226e_nres @ X0 > set @ product_prod @ X1 @ X0 > (X2 > refine1665802226e_nres @ X1) > set @ X2 > X2)).
% 3.10/1.72 thf(func_def_143, type, sK78: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 3.10/1.72 thf(func_def_144, type, sK79: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 3.10/1.72 thf(func_def_145, type, sK80: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 3.10/1.72 thf(func_def_146, type, sK81: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 3.10/1.72 thf(func_def_147, type, sK82: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 3.10/1.72 thf(func_def_148, type, sK83: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 3.10/1.72 thf(func_def_149, type, sK84: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > $o) > (X1 > refine1665802226e_nres @ X2) > (X0 > $o) > (X1 > refine1665802226e_nres @ X0) > refine1665802226e_nres @ X1 > X1)).
% 3.10/1.72 thf(func_def_150, type, sK85: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X3 > refine1665802226e_nres @ X2) > set @ product_prod @ X1 @ X2 > refine1665802226e_nres @ X1 > X0 > set @ product_prod @ X0 @ X3 > X3)).
% 3.10/1.72 thf(func_def_151, type, sK86: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X1 > set @ X0 > X0)).
% 3.10/1.72 thf(func_def_152, type, sK87: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.10/1.72 thf(func_def_153, type, sK88: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.10/1.72 thf(func_def_154, type, sK89: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 3.10/1.72 thf(func_def_155, type, sK90: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 3.10/1.72 thf(func_def_156, type, sK91: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.10/1.72 thf(func_def_157, type, sK92: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.10/1.72 thf(func_def_158, type, sK93: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.10/1.72 thf(func_def_159, type, sK94: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.72 thf(func_def_160, type, sK95: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.72 thf(func_def_161, type, sK96: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > set @ product_prod @ X0 @ X1 > X0)).
% 3.10/1.72 thf(func_def_162, type, sK97: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > set @ product_prod @ X0 @ X1 > X1)).
% 3.10/1.72 thf(func_def_163, type, sK98: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > X0 > refine1665802226e_nres @ X1 > X1)).
% 3.10/1.72 thf(func_def_164, type, sK99: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.10/1.72 thf(func_def_165, type, sK100: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.10/1.72 thf(func_def_166, type, sK101: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.10/1.72 thf(func_def_168, type, ph103: !>[X0: $tType]:(X0)).
% 3.10/1.72 thf(f1365,plain,(
% 3.10/1.72 $false),
% 3.10/1.72 inference(subsumption_resolution,[status(thm)],[f1363,f1158])).
% 3.10/1.72 thf(f1158,plain,(
% 3.10/1.72 ((phi @ x) = $true)),
% 3.10/1.72 inference(cnf_transformation,[status(thm)],[f648])).
% 3.10/1.72 thf(f648,plain,(
% 3.10/1.72 ((phi @ x) = $true)),
% 3.10/1.72 inference(fool_elimination,[status(thm)],[f647])).
% 3.10/1.72 thf(f647,plain,(
% 3.10/1.72 (phi @ x)),
% 3.10/1.72 inference(rectify,[status(thm)],[f1])).
% 3.10/1.72 thf(f1,axiom,(
% 3.10/1.72 (phi @ x)),
% 3.10/1.72 file('/export/starexec/sandbox/tmp/tmp.8D84XgAAWH/DTF2THF_13304.p',fact_0_assms_I1_J)).
% 3.10/1.72 thf(f1363,plain,(
% 3.10/1.72 ((phi @ x) != $true)),
% 3.10/1.72 inference(trivial_inequality_removal,[status(thm)],[f1355])).
% 3.10/1.72 thf(f1355,plain,(
% 3.10/1.72 ($true != $true) | ((phi @ x) != $true)),
% 3.10/1.72 inference(superposition,[status(thm)],[f1336,f1179])).
% 3.10/1.72 thf(f1179,plain,(
% 3.10/1.72 ((ord_less_eq @ refine1665802226e_nres @ b @ m @ (f @ x)) = $true)),
% 3.10/1.72 inference(cnf_transformation,[status(thm)],[f668])).
% 3.10/1.72 thf(f668,plain,(
% 3.10/1.72 ((ord_less_eq @ refine1665802226e_nres @ b @ m @ (f @ x)) = $true)),
% 3.10/1.72 inference(fool_elimination,[status(thm)],[f667])).
% 3.10/1.72 thf(f667,plain,(
% 3.10/1.72 (ord_less_eq @ refine1665802226e_nres @ b @ m @ (f @ x))),
% 3.10/1.72 inference(rectify,[status(thm)],[f2])).
% 3.10/1.72 thf(f2,axiom,(
% 3.10/1.72 (ord_less_eq @ refine1665802226e_nres @ b @ m @ (f @ x))),
% 3.10/1.72 file('/export/starexec/sandbox/tmp/tmp.8D84XgAAWH/DTF2THF_13304.p',fact_1_assms_I2_J)).
% 3.10/1.72 thf(f1336,plain,(
% 3.10/1.72 ( ! [X0 : a] : (((ord_less_eq @ refine1665802226e_nres @ b @ m @ (f @ X0)) != $true) | ($true != (phi @ X0))) )),
% 3.10/1.72 inference(superposition,[status(thm)],[f1297,f1247])).
% 3.10/1.72 thf(f1247,plain,(
% 3.10/1.72 ( ! [X0 : $tType,X2 : X0 > $o,X1 : X0] : (((X2 @ X1) = (member @ X0 @ X1 @ (collect @ X0 @ X2)))) )),
% 3.10/1.72 inference(cnf_transformation,[status(thm)],[f792])).
% 3.10/1.72 thf(f792,plain,(
% 3.10/1.72 ! [X0 : $tType,X1 : X0,X2 : X0 > $o] : ((X2 @ X1) = (member @ X0 @ X1 @ (collect @ X0 @ X2)))),
% 3.10/1.72 inference(rectify,[status(thm)],[f44])).
% 3.10/1.72 thf(f44,axiom,(
% 3.10/1.72 ! [X0 : $tType,X33 : X0,X44 : X0 > $o] : ((member @ X0 @ X33 @ (collect @ X0 @ X44)) = (X44 @ X33))),
% 3.10/1.72 file('/export/starexec/sandbox/tmp/tmp.8D84XgAAWH/DTF2THF_13304.p',fact_43_mem__Collect__eq)).
% 3.10/1.72 thf(f1297,plain,(
% 3.10/1.72 ( ! [X0 : a] : (((member @ a @ X0 @ (collect @ a @ phi)) != $true) | ((ord_less_eq @ refine1665802226e_nres @ b @ m @ (f @ X0)) != $true)) )),
% 3.10/1.72 inference(trivial_inequality_removal,[status(thm)],[f1284])).
% 3.10/1.72 thf(f1284,plain,(
% 3.10/1.72 ( ! [X0 : a] : (((ord_less_eq @ refine1665802226e_nres @ b @ m @ (f @ X0)) != $true) | ($true != $true) | ((member @ a @ X0 @ (collect @ a @ phi)) != $true)) )),
% 3.10/1.72 inference(superposition,[status(thm)],[f1178,f1212])).
% 3.10/1.72 thf(f1212,plain,(
% 3.10/1.72 ( ! [X1 : $tType,X0 : $tType,X2 : X1,X3 : set @ X1,X4 : refine1665802226e_nres @ X0,X5 : X1 > refine1665802226e_nres @ X0] : (($true = (ord_less_eq @ refine1665802226e_nres @ X0 @ X4 @ (refine463715084e_bind @ X1 @ X0 @ (refine605929679le_RES @ X1 @ X3) @ X5))) | ($true != (member @ X1 @ X2 @ X3)) | ($true != (ord_less_eq @ refine1665802226e_nres @ X0 @ X4 @ (X5 @ X2)))) )),
% 3.10/1.72 inference(cnf_transformation,[status(thm)],[f818])).
% 3.10/1.72 thf(f818,plain,(
% 3.10/1.72 ! [X0 : $tType,X1 : $tType,X2 : X1,X3 : set @ X1,X4 : refine1665802226e_nres @ X0,X5 : X1 > refine1665802226e_nres @ X0] : (($true != (ord_less_eq @ refine1665802226e_nres @ X0 @ X4 @ (X5 @ X2))) | ($true = (ord_less_eq @ refine1665802226e_nres @ X0 @ X4 @ (refine463715084e_bind @ X1 @ X0 @ (refine605929679le_RES @ X1 @ X3) @ X5))) | ($true != (member @ X1 @ X2 @ X3)))),
% 3.10/1.72 inference(flattening,[status(thm)],[f817])).
% 3.10/1.72 thf(f817,plain,(
% 3.10/1.72 ! [X0 : $tType,X1 : $tType,X2 : X1,X3 : set @ X1,X4 : refine1665802226e_nres @ X0,X5 : X1 > refine1665802226e_nres @ X0] : ((($true = (ord_less_eq @ refine1665802226e_nres @ X0 @ X4 @ (refine463715084e_bind @ X1 @ X0 @ (refine605929679le_RES @ X1 @ X3) @ X5))) | ($true != (ord_less_eq @ refine1665802226e_nres @ X0 @ X4 @ (X5 @ X2)))) | ($true != (member @ X1 @ X2 @ X3)))),
% 3.10/1.72 inference(ennf_transformation,[status(thm)],[f359])).
% 3.10/1.72 thf(f359,plain,(
% 3.10/1.72 ! [X0 : $tType,X1 : $tType,X2 : X1,X3 : set @ X1,X4 : refine1665802226e_nres @ X0,X5 : X1 > refine1665802226e_nres @ X0] : (($true = (member @ X1 @ X2 @ X3)) => (($true = (ord_less_eq @ refine1665802226e_nres @ X0 @ X4 @ (X5 @ X2))) => ($true = (ord_less_eq @ refine1665802226e_nres @ X0 @ X4 @ (refine463715084e_bind @ X1 @ X0 @ (refine605929679le_RES @ X1 @ X3) @ X5)))))),
% 3.10/1.72 inference(fool_elimination,[status(thm)],[f358])).
% 3.10/1.72 thf(f358,plain,(
% 3.10/1.72 ! [X0 : $tType,X1 : $tType,X2 : X1,X3 : set @ X1,X4 : refine1665802226e_nres @ X0,X5 : X1 > refine1665802226e_nres @ X0] : ((member @ X1 @ X2 @ X3) => ((ord_less_eq @ refine1665802226e_nres @ X0 @ X4 @ (X5 @ X2)) => (ord_less_eq @ refine1665802226e_nres @ X0 @ X4 @ (refine463715084e_bind @ X1 @ X0 @ (refine605929679le_RES @ X1 @ X3) @ X5))))),
% 3.10/1.72 inference(rectify,[status(thm)],[f11])).
% 3.10/1.72 thf(f11,axiom,(
% 3.10/1.72 ! [X1 : $tType,X0 : $tType,X15 : X0,X4 : set @ X0,X19 : refine1665802226e_nres @ X1,X22 : X0 > refine1665802226e_nres @ X1] : ((member @ X0 @ X15 @ X4) => ((ord_less_eq @ refine1665802226e_nres @ X1 @ X19 @ (X22 @ X15)) => (ord_less_eq @ refine1665802226e_nres @ X1 @ X19 @ (refine463715084e_bind @ X0 @ X1 @ (refine605929679le_RES @ X0 @ X4) @ X22))))),
% 3.10/1.72 file('/export/starexec/sandbox/tmp/tmp.8D84XgAAWH/DTF2THF_13304.p',fact_10_RES__bind__choose)).
% 3.10/1.72 thf(f1178,plain,(
% 3.10/1.72 ((ord_less_eq @ refine1665802226e_nres @ b @ m @ (refine463715084e_bind @ a @ b @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ f)) != $true)),
% 3.10/1.72 inference(cnf_transformation,[status(thm)],[f783])).
% 3.10/1.72 thf(f783,plain,(
% 3.10/1.72 ((ord_less_eq @ refine1665802226e_nres @ b @ m @ (refine463715084e_bind @ a @ b @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ f)) != $true)),
% 3.10/1.72 inference(flattening,[status(thm)],[f335])).
% 3.10/1.72 thf(f335,plain,(
% 3.10/1.72 ~((ord_less_eq @ refine1665802226e_nres @ b @ m @ (refine463715084e_bind @ a @ b @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ f)) = $true)),
% 3.10/1.72 inference(fool_elimination,[status(thm)],[f334])).
% 3.10/1.72 thf(f334,plain,(
% 3.10/1.72 ~(ord_less_eq @ refine1665802226e_nres @ b @ m @ (refine463715084e_bind @ a @ b @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ f))),
% 3.10/1.72 inference(rectify,[status(thm)],[f292])).
% 3.10/1.72 thf(f292,negated_conjecture,(
% 3.10/1.72 ~(ord_less_eq @ refine1665802226e_nres @ b @ m @ (refine463715084e_bind @ a @ b @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ f))),
% 3.10/1.72 inference(negated_conjecture,[status(cth)],[f291])).
% 3.10/1.72 thf(f291,conjecture,(
% 3.10/1.72 (ord_less_eq @ refine1665802226e_nres @ b @ m @ (refine463715084e_bind @ a @ b @ (refine605929679le_RES @ a @ (collect @ a @ phi)) @ f))),
% 3.10/1.72 file('/export/starexec/sandbox/tmp/tmp.8D84XgAAWH/DTF2THF_13304.p',conj_0)).
% 3.10/1.72 % SZS output end Proof for DTF2THF_13304
% 3.10/1.72 % (13598)------------------------------
% 3.10/1.72 % (13598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.72 % (13598)Termination reason: Refutation
% 3.10/1.72
% 3.10/1.72 % (13598)Memory used [KB]: 7419
% 3.10/1.72 % (13598)Time elapsed: 0.072 s
% 3.10/1.72 % (13598)Instructions burned: 143 (million)
% 3.10/1.72 % (13598)------------------------------
% 3.10/1.72 % (13598)------------------------------
% 3.10/1.72 % (13592)Success in time 0.096 s
%------------------------------------------------------------------------------