%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP160^2 : 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 : n015.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 08:56:02 AM UTC 2026
% Result : Theorem 3.24s 1.67s
% Output : Refutation 3.24s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP160^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n015.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:28:31 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.26/0.46 ---- Original DTF file ---
% 0.26/0.46 thf(spec,logic,$$dhol).
% 0.26/0.46 %------------------------------------------------------------------------------
% 0.26/0.46 % File : ITP160^2 : TPTP v9.2.1. Released v7.5.0.
% 0.26/0.46 % Domain : Interactive Theorem Proving
% 0.26/0.46 % Problem : Sledgehammer Refine_Basic problem prob_1493__3600276_1
% 0.26/0.46 % Version : Especial.
% 0.26/0.46 % English :
% 0.26/0.46
% 0.26/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.26/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.26/0.46 % Source : [Des21]
% 0.26/0.46 % Names : Refine_Basic/prob_1493__3600276_1 [Des21]
% 0.26/0.46
% 0.26/0.46 % Status : Theorem
% 0.26/0.46 % Rating : 0.67 v8.2.0, 0.33 v8.1.0, 0.25 v7.5.0
% 0.26/0.46 % Syntax : Number of formulae : 335 ( 167 unt; 58 typ; 0 def)
% 0.26/0.46 % Number of atoms : 542 ( 239 equ; 8 cnn)
% 0.26/0.46 % Maximal formula atoms : 13 ( 1 avg)
% 0.26/0.46 % Number of connectives : 8965 ( 48 ~; 5 |; 25 &;8654 @)
% 0.26/0.46 % ( 0 <=>; 233 =>; 0 <=; 0 <~>)
% 0.26/0.46 % Maximal formula depth : 36 ( 10 avg)
% 0.26/0.46 % Number of types : 8 ( 7 usr)
% 0.26/0.46 % Number of type conns : 2280 (2280 >; 0 *; 0 +; 0 <<)
% 0.26/0.46 % Number of symbols : 58 ( 51 usr; 9 con; 0-7 aty)
% 0.26/0.46 % Number of variables : 1471 ( 70 ^;1311 !; 9 ?;1471 :)
% 0.26/0.46 % ( 81 !>; 0 ?*; 0 @-; 0 @+)
% 0.26/0.46 % SPC : TH1_THM_EQU_NAR
% 0.26/0.46
% 0.26/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:20:45.194
% 0.26/0.46 %------------------------------------------------------------------------------
% 0.26/0.46 % Could-be-implicit typings (10)
% 0.26/0.46 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
% 0.26/0.46 refine1665802226e_nres: $tType > $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_t_Autoref__Tagging_Oannot,type,
% 0.26/0.46 autoref_annot: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.26/0.46 product_prod: $tType > $tType > $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_t_Set_Oset,type,
% 0.26/0.46 set: $tType > $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_t_Nat_Onat,type,
% 0.26/0.46 nat: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_t_Int_Oint,type,
% 0.26/0.46 int: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_tf_d,type,
% 0.26/0.46 d: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_tf_c,type,
% 0.26/0.46 c: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_tf_b,type,
% 0.26/0.46 b: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_tf_a,type,
% 0.26/0.46 a: $tType ).
% 0.26/0.46
% 0.26/0.46 % Explicit typings (48)
% 0.26/0.46 thf(sy_cl_HOL_Otype,type,
% 0.26/0.46 type:
% 0.26/0.46 !>[A: $tType] : $o ).
% 0.26/0.46
% 0.26/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.26/0.46 ord:
% 0.26/0.46 !>[A: $tType] : $o ).
% 0.26/0.46
% 0.26/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.26/0.46 order:
% 0.26/0.46 !>[A: $tType] : $o ).
% 0.26/0.46
% 0.26/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.26/0.46 linorder:
% 0.26/0.46 !>[A: $tType] : $o ).
% 0.26/0.46
% 0.26/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.26/0.46 preorder:
% 0.26/0.46 !>[A: $tType] : $o ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Autoref__Tagging_OANNOT,type,
% 0.26/0.46 autoref_ANNOT:
% 0.26/0.46 !>[A: $tType] : ( A > autoref_annot > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Autoref__Tagging_OAPP,type,
% 0.26/0.46 autoref_APP:
% 0.26/0.46 !>[B: $tType,A: $tType] : ( ( B > A ) > B > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Autoref__Tagging_OOP,type,
% 0.26/0.46 autoref_OP:
% 0.26/0.46 !>[A: $tType] : ( A > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Autoref__Tagging_Orel__annot,type,
% 0.26/0.46 autoref_rel_annot:
% 0.26/0.46 !>[C: $tType,A: $tType] : ( ( set @ ( product_prod @ C @ A ) ) > autoref_annot ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Autoref__Translate_OREMOVE__INTERNAL,type,
% 0.26/0.46 autore2013370551TERNAL:
% 0.26/0.46 !>[A: $tType] : ( A > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Autoref__Translate_OREMOVE__INTERNAL__EQ,type,
% 0.26/0.46 autore813025987NAL_EQ:
% 0.26/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Fun_Ocomp,type,
% 0.26/0.46 comp:
% 0.26/0.46 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Fun_Ofun__upd,type,
% 0.26/0.46 fun_upd:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Fun_Oid,type,
% 0.26/0.46 id:
% 0.26/0.46 !>[A: $tType] : ( A > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_If,type,
% 0.26/0.46 if:
% 0.26/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_List_Oall__interval__nat,type,
% 0.26/0.46 all_interval_nat: ( nat > $o ) > nat > nat > $o ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Misc_Ouncurry,type,
% 0.26/0.46 uncurry:
% 0.26/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Nat_Onat_Ocase__nat,type,
% 0.26/0.46 case_nat:
% 0.26/0.46 !>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
% 0.26/0.46 rec_nat:
% 0.26/0.46 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.26/0.46 ord_less:
% 0.26/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.26/0.46 ord_less_eq:
% 0.26/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Product__Type_OPair,type,
% 0.26/0.46 product_Pair:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Product__Type_Oapfst,type,
% 0.26/0.46 product_apfst:
% 0.26/0.46 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Product__Type_Oapsnd,type,
% 0.26/0.46 product_apsnd:
% 0.26/0.46 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Product__Type_Obool_Ocase__bool,type,
% 0.26/0.46 product_case_bool:
% 0.26/0.46 !>[A: $tType] : ( A > A > $o > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Product__Type_Ocurry,type,
% 0.26/0.46 product_curry:
% 0.26/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.26/0.46 produc2004651681e_prod:
% 0.26/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Product__Type_Omap__prod,type,
% 0.26/0.46 product_map_prod:
% 0.26/0.46 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 0.26/0.46 product_rec_bool:
% 0.26/0.46 !>[T: $tType] : ( T > T > $o > T ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.26/0.46 product_rec_prod:
% 0.26/0.46 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.26/0.46 product_case_prod:
% 0.26/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.26/0.46 product_fst:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.26/0.46 product_snd:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.26/0.46 product_swap:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
% 0.26/0.46 refine1687780735RETURN:
% 0.26/0.46 !>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind,type,
% 0.26/0.46 refine463715084e_bind:
% 0.26/0.46 !>[B: $tType,A: $tType] : ( ( refine1665802226e_nres @ B ) > ( B > ( refine1665802226e_nres @ A ) ) > ( refine1665802226e_nres @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Olift__assn,type,
% 0.26/0.46 refine1580981607t_assn:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( B > $o ) > A > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres__rel,type,
% 0.26/0.46 refine476890328es_rel:
% 0.26/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oop__nres__ASSERT__bnd,type,
% 0.26/0.46 refine171938416RT_bnd:
% 0.26/0.46 !>[A: $tType] : ( $o > ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Refine__Misc_Oless__than__bool,type,
% 0.26/0.46 refine1361002633n_bool: set @ ( product_prod @ $o @ $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Relation_OId,type,
% 0.26/0.46 id2:
% 0.26/0.46 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Relators_Ofun__rel,type,
% 0.26/0.46 fun_rel:
% 0.26/0.46 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Relators_Oprod__rel,type,
% 0.26/0.46 prod_rel:
% 0.26/0.46 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Relators_OrelAPP,type,
% 0.26/0.46 relAPP:
% 0.26/0.46 !>[C1: $tType,A1: $tType,A: $tType] : ( ( ( set @ ( product_prod @ C1 @ A1 ) ) > A ) > ( set @ ( product_prod @ C1 @ A1 ) ) > A ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Set_OCollect,type,
% 0.26/0.46 collect:
% 0.26/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_member,type,
% 0.26/0.46 member:
% 0.26/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_v_R1,type,
% 0.26/0.46 r1: set @ ( product_prod @ a @ c ) ).
% 0.26/0.46
% 0.26/0.46 thf(sy_v_R2,type,
% 0.26/0.46 r2: set @ ( product_prod @ b @ d ) ).
% 0.26/0.46
% 0.26/0.46 % Relevant facts (252)
% 0.26/0.46 thf(fact_0_param__bind,axiom,
% 0.26/0.46 ! [C: $tType,A: $tType,D: $tType,B: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( ( refine1665802226e_nres @ A ) > ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( refine1665802226e_nres @ C ) > ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) @ ( product_Pair @ ( ( refine1665802226e_nres @ A ) > ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( refine1665802226e_nres @ C ) > ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) @ ( refine463715084e_bind @ A @ B ) @ ( refine463715084e_bind @ C @ D ) ) @ ( relAPP @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) @ ( set @ ( product_prod @ ( ( refine1665802226e_nres @ A ) > ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( refine1665802226e_nres @ C ) > ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) @ ( relAPP @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ C ) @ ( ( set @ ( product_prod @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) > ( set @ ( product_prod @ ( ( refine1665802226e_nres @ A ) > ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( refine1665802226e_nres @ C ) > ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) ) @ ( fun_rel @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ C ) @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) @ ( relAPP @ A @ C @ ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ C ) ) ) @ ( refine476890328es_rel @ A @ C ) @ Ra ) ) @ ( relAPP @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) @ ( set @ ( product_prod @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) @ ( relAPP @ ( A > ( refine1665802226e_nres @ B ) ) @ ( C > ( refine1665802226e_nres @ D ) ) @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) ) > ( set @ ( product_prod @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) ) @ ( fun_rel @ ( A > ( refine1665802226e_nres @ B ) ) @ ( C > ( refine1665802226e_nres @ D ) ) @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) @ ( relAPP @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) @ ( set @ ( product_prod @ ( A > ( refine1665802226e_nres @ B ) ) @ ( C > ( refine1665802226e_nres @ D ) ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) ) > ( set @ ( product_prod @ ( A > ( refine1665802226e_nres @ B ) ) @ ( C > ( refine1665802226e_nres @ D ) ) ) ) ) @ ( fun_rel @ A @ C @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) @ Ra ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) ) @ ( refine476890328es_rel @ B @ D ) @ Rb ) ) ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) ) @ ( refine476890328es_rel @ B @ D ) @ Rb ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % param_bind
% 0.26/0.46 thf(fact_1_fun__relI,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: set @ ( product_prod @ A @ B ),F: A > C,F2: B > D,B2: set @ ( product_prod @ C @ D )] :
% 0.26/0.46 ( ! [A3: A,A4: B] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ A4 ) @ A2 )
% 0.26/0.46 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( F2 @ A4 ) ) @ B2 ) )
% 0.26/0.46 => ( member @ ( product_prod @ ( A > C ) @ ( B > D ) ) @ ( product_Pair @ ( A > C ) @ ( B > D ) @ F @ F2 ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ A2 ) @ B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_relI
% 0.26/0.46 thf(fact_2_autoref__Let,axiom,
% 0.26/0.46 ! [C: $tType,A: $tType,D: $tType,B: $tType,Ra: set @ ( product_prod @ A @ C ),Rr: set @ ( product_prod @ B @ D )] :
% 0.26/0.46 ( member @ ( product_prod @ ( A > ( A > B ) > B ) @ ( C > ( C > D ) > D ) )
% 0.26/0.46 @ ( product_Pair @ ( A > ( A > B ) > B ) @ ( C > ( C > D ) > D )
% 0.26/0.46 @ ^ [S: A,F3: A > B] : ( F3 @ S )
% 0.26/0.46 @ ^ [S: C,F3: C > D] : ( F3 @ S ) )
% 0.26/0.46 @ ( relAPP @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) @ ( set @ ( product_prod @ ( A > ( A > B ) > B ) @ ( C > ( C > D ) > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) ) ) > ( set @ ( product_prod @ ( A > ( A > B ) > B ) @ ( C > ( C > D ) > D ) ) ) ) @ ( fun_rel @ A @ C @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) ) @ Ra ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) ) ) @ ( relAPP @ ( A > B ) @ ( C > D ) @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( C > D ) @ B @ D ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ Ra ) @ Rr ) ) @ Rr ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_Let
% 0.26/0.46 thf(fact_3_fun__relD1,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,Ra: set @ ( product_prod @ A @ C ),Rr: set @ ( product_prod @ B @ D ),X: A,R: B] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ Ra ) @ Rr ) )
% 0.26/0.46 => ( ( ( F @ X )
% 0.26/0.46 = R )
% 0.26/0.46 => ! [X2: C] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X2 ) @ Ra )
% 0.26/0.46 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ R @ ( F2 @ X2 ) ) @ Rr ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_relD1
% 0.26/0.46 thf(fact_4_fun__relD2,axiom,
% 0.26/0.46 ! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,Ra: set @ ( product_prod @ A @ C ),Rr: set @ ( product_prod @ B @ D ),X3: C,R2: D] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ Ra ) @ Rr ) )
% 0.26/0.46 => ( ( ( F2 @ X3 )
% 0.26/0.46 = R2 )
% 0.26/0.46 => ! [X4: A] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ X3 ) @ Ra )
% 0.26/0.46 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X4 ) @ R2 ) @ Rr ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_relD2
% 0.26/0.46 thf(fact_5_tagged__fun__relD__lhs,axiom,
% 0.26/0.46 ! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D ),X: A,X3: C] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ A2 )
% 0.26/0.46 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X ) @ ( F2 @ X3 ) ) @ B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % tagged_fun_relD_lhs
% 0.26/0.46 thf(fact_6_tagged__fun__relD__rhs,axiom,
% 0.26/0.46 ! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D ),X: A,X3: C] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ A2 )
% 0.26/0.46 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X ) @ ( F2 @ X3 ) ) @ B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % tagged_fun_relD_rhs
% 0.26/0.46 thf(fact_7_tagged__fun__relD__both,axiom,
% 0.26/0.46 ! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D ),X: A,X3: C] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ A2 )
% 0.26/0.46 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X ) @ ( F2 @ X3 ) ) @ B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % tagged_fun_relD_both
% 0.26/0.46 thf(fact_8_tagged__fun__relD__none,axiom,
% 0.26/0.46 ! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D ),X: A,X3: C] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ A2 )
% 0.26/0.46 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X ) @ ( F2 @ X3 ) ) @ B2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % tagged_fun_relD_none
% 0.26/0.46 thf(fact_9_prod_Oinject,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.26/0.46 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.26/0.46 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.26/0.46 = ( ( X1 = Y1 )
% 0.26/0.46 & ( X22 = Y2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod.inject
% 0.26/0.46 thf(fact_10_old_Oprod_Oinject,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,A5: A,B3: B,A6: A,B4: B] :
% 0.26/0.46 ( ( ( product_Pair @ A @ B @ A5 @ B3 )
% 0.26/0.46 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 0.26/0.46 = ( ( A5 = A6 )
% 0.26/0.46 & ( B3 = B4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % old.prod.inject
% 0.26/0.46 thf(fact_11_autoref__APP,axiom,
% 0.26/0.46 ! [C: $tType,A: $tType,D: $tType,B: $tType,C2: A > B,A5: C > D,Ra: set @ ( product_prod @ A @ C ),Rr: set @ ( product_prod @ B @ D ),X: A,X3: C] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ C2 @ A5 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ Ra ) @ Rr ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ Ra )
% 0.26/0.46 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( autoref_APP @ A @ B @ C2 @ X ) @ ( autoref_APP @ C @ D @ A5 @ X3 ) ) @ Rr ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_APP
% 0.26/0.46 thf(fact_12_lift__assnI,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,S2: A,S3: B,R3: set @ ( product_prod @ A @ B ),Phi: B > $o] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S2 @ S3 ) @ R3 )
% 0.26/0.46 => ( ( Phi @ S3 )
% 0.26/0.46 => ( refine1580981607t_assn @ A @ B @ R3 @ Phi @ S2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lift_assnI
% 0.26/0.46 thf(fact_13_old_Oprod_Oinducts,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.26/0.46 ( ! [A3: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A3 @ B5 ) )
% 0.26/0.46 => ( P @ Prod ) ) ).
% 0.26/0.46
% 0.26/0.46 % old.prod.inducts
% 0.26/0.46 thf(fact_14_old_Oprod_Oexhaust,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.26/0.46 ~ ! [A3: A,B5: B] :
% 0.26/0.46 ( Y
% 0.26/0.46 != ( product_Pair @ A @ B @ A3 @ B5 ) ) ).
% 0.26/0.46
% 0.26/0.46 % old.prod.exhaust
% 0.26/0.46 thf(fact_15_prod__induct7,axiom,
% 0.26/0.46 ! [G: $tType,F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) )] :
% 0.26/0.46 ( ! [A3: A,B5: B,C3: C,D2: D,E2: E,F5: F4,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G ) @ E2 @ ( product_Pair @ F4 @ G @ F5 @ G2 ) ) ) ) ) ) )
% 0.26/0.46 => ( P @ X ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_induct7
% 0.26/0.46 thf(fact_16_prod__induct6,axiom,
% 0.26/0.46 ! [F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
% 0.26/0.46 ( ! [A3: A,B5: B,C3: C,D2: D,E2: E,F5: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) )
% 0.26/0.46 => ( P @ X ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_induct6
% 0.26/0.46 thf(fact_17_prod__induct5,axiom,
% 0.26/0.46 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.26/0.46 ( ! [A3: A,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.26/0.46 => ( P @ X ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_induct5
% 0.26/0.46 thf(fact_18_prod__induct4,axiom,
% 0.26/0.46 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.26/0.46 ( ! [A3: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 0.26/0.46 => ( P @ X ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_induct4
% 0.26/0.46 thf(fact_19_prod__induct3,axiom,
% 0.26/0.46 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.26/0.46 ( ! [A3: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
% 0.26/0.46 => ( P @ X ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_induct3
% 0.26/0.46 thf(fact_20_prod__cases7,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) )] :
% 0.26/0.46 ~ ! [A3: A,B5: B,C3: C,D2: D,E2: E,F5: F4,G2: G] :
% 0.26/0.46 ( Y
% 0.26/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G ) @ E2 @ ( product_Pair @ F4 @ G @ F5 @ G2 ) ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_cases7
% 0.26/0.46 thf(fact_21_prod__cases6,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
% 0.26/0.46 ~ ! [A3: A,B5: B,C3: C,D2: D,E2: E,F5: F4] :
% 0.26/0.46 ( Y
% 0.26/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_cases6
% 0.26/0.46 thf(fact_22_prod__cases5,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.26/0.46 ~ ! [A3: A,B5: B,C3: C,D2: D,E2: E] :
% 0.26/0.46 ( Y
% 0.26/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_cases5
% 0.26/0.46 thf(fact_23_prod__cases4,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.26/0.46 ~ ! [A3: A,B5: B,C3: C,D2: D] :
% 0.26/0.46 ( Y
% 0.26/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_cases4
% 0.26/0.46 thf(fact_24_prod__cases3,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.26/0.46 ~ ! [A3: A,B5: B,C3: C] :
% 0.26/0.46 ( Y
% 0.26/0.46 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_cases3
% 0.26/0.46 thf(fact_25_Pair__inject,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,A5: A,B3: B,A6: A,B4: B] :
% 0.26/0.46 ( ( ( product_Pair @ A @ B @ A5 @ B3 )
% 0.26/0.46 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 0.26/0.46 => ~ ( ( A5 = A6 )
% 0.26/0.46 => ( B3 != B4 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Pair_inject
% 0.26/0.46 thf(fact_26_prod__cases,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.26/0.46 ( ! [A3: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A3 @ B5 ) )
% 0.26/0.46 => ( P @ P2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_cases
% 0.26/0.46 thf(fact_27_surj__pair,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.26/0.46 ? [X5: A,Y3: B] :
% 0.26/0.46 ( P2
% 0.26/0.46 = ( product_Pair @ A @ B @ X5 @ Y3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % surj_pair
% 0.26/0.46 thf(fact_28_relAPP__def,axiom,
% 0.26/0.46 ! [A: $tType,A1: $tType,C1: $tType] :
% 0.26/0.46 ( ( relAPP @ C1 @ A1 @ A )
% 0.26/0.46 = ( ^ [F3: ( set @ ( product_prod @ C1 @ A1 ) ) > A] : F3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % relAPP_def
% 0.26/0.46 thf(fact_29_lift__assn__def,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( refine1580981607t_assn @ A @ B )
% 0.26/0.46 = ( ^ [R4: set @ ( product_prod @ A @ B ),Phi2: B > $o,S: A] :
% 0.26/0.46 ? [S4: B] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S @ S4 ) @ R4 )
% 0.26/0.46 & ( Phi2 @ S4 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % lift_assn_def
% 0.26/0.46 thf(fact_30_autoref__RETURN,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > ( refine1665802226e_nres @ A ) ) @ ( B > ( refine1665802226e_nres @ B ) ) ) @ ( product_Pair @ ( A > ( refine1665802226e_nres @ A ) ) @ ( B > ( refine1665802226e_nres @ B ) ) @ ( refine1687780735RETURN @ A ) @ ( refine1687780735RETURN @ B ) ) @ ( relAPP @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) @ ( set @ ( product_prod @ ( A > ( refine1665802226e_nres @ A ) ) @ ( B > ( refine1665802226e_nres @ B ) ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) > ( set @ ( product_prod @ ( A > ( refine1665802226e_nres @ A ) ) @ ( B > ( refine1665802226e_nres @ B ) ) ) ) ) @ ( fun_rel @ A @ B @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) @ ( refine476890328es_rel @ A @ B ) @ R3 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_RETURN
% 0.26/0.46 thf(fact_31_param__op__nres__ASSERT__bnd,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,Phi3: $o,Phi: $o,M: refine1665802226e_nres @ A,M2: refine1665802226e_nres @ B,R3: set @ ( product_prod @ A @ B )] :
% 0.26/0.46 ( ( Phi3
% 0.26/0.46 => Phi )
% 0.26/0.46 => ( ( Phi3
% 0.26/0.46 => ( Phi
% 0.26/0.46 => ( member @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) @ M @ M2 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) @ ( refine476890328es_rel @ A @ B ) @ R3 ) ) ) )
% 0.26/0.46 => ( member @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) @ ( refine171938416RT_bnd @ A @ Phi @ M ) @ ( refine171938416RT_bnd @ B @ Phi3 @ M2 ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) @ ( refine476890328es_rel @ A @ B ) @ R3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % param_op_nres_ASSERT_bnd
% 0.26/0.46 thf(fact_32_autoref__If,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) @ ( product_Pair @ ( $o > A > A > A ) @ ( $o > B > B > B ) @ ( if @ A ) @ ( if @ B ) ) @ ( relAPP @ ( A > A > A ) @ ( B > B > B ) @ ( set @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) > ( set @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) ) ) @ ( fun_rel @ $o @ $o @ ( A > A > A ) @ ( B > B > B ) ) @ ( id2 @ $o ) ) @ ( relAPP @ ( A > A ) @ ( B > B ) @ ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) > ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) ) @ ( fun_rel @ A @ B @ ( A > A ) @ ( B > B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) ) @ ( fun_rel @ A @ B @ A @ B ) @ R3 ) @ R3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_If
% 0.26/0.46 thf(fact_33_fun__relE2,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,F: A > B,F2: A > C,Rv: set @ ( product_prod @ B @ C ),T2: B,X: A] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_Pair @ ( A > B ) @ ( A > C ) @ F @ F2 ) @ ( relAPP @ B @ C @ ( set @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( fun_rel @ A @ A @ B @ C ) @ ( id2 @ A ) ) @ Rv ) )
% 0.26/0.46 => ( ( T2
% 0.26/0.46 = ( F @ X ) )
% 0.26/0.46 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ T2 @ ( F2 @ X ) ) @ Rv ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_relE2
% 0.26/0.46 thf(fact_34_fun__relE1,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,B: $tType,F: A > B,F2: A > C,Rv: set @ ( product_prod @ B @ C ),T3: C,X: A] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_Pair @ ( A > B ) @ ( A > C ) @ F @ F2 ) @ ( relAPP @ B @ C @ ( set @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( fun_rel @ A @ A @ B @ C ) @ ( id2 @ A ) ) @ Rv ) )
% 0.26/0.46 => ( ( T3
% 0.26/0.46 = ( F2 @ X ) )
% 0.26/0.46 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ ( F @ X ) @ T3 ) @ Rv ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_relE1
% 0.26/0.46 thf(fact_35_autoref__Let__cong,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,D: $tType,C: $tType,X3: A,X: B,Ra: set @ ( product_prod @ A @ B ),F2: A > C,F: B > D,Rr: set @ ( product_prod @ C @ D )] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ X ) @ Ra )
% 0.26/0.46 => ( ! [Y3: B] :
% 0.26/0.46 ( ( autore2013370551TERNAL @ $o @ ( X = Y3 ) )
% 0.26/0.46 => ! [Y4: A] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y4 @ Y3 ) @ Ra )
% 0.26/0.46 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F2 @ Y4 ) @ ( autoref_APP @ B @ D @ F @ Y3 ) ) @ Rr ) ) )
% 0.26/0.46 => ( member @ ( product_prod @ C @ D )
% 0.26/0.46 @ ( product_Pair @ C @ D @ ( F2 @ X3 )
% 0.26/0.46 @ ( autoref_APP @ ( B > D ) @ D
% 0.26/0.46 @ ( autoref_APP @ B @ ( ( B > D ) > D )
% 0.26/0.46 @ ( autoref_ANNOT @ ( B > ( B > D ) > D )
% 0.26/0.46 @ ( autoref_OP @ ( B > ( B > D ) > D )
% 0.26/0.46 @ ^ [S: B,F3: B > D] : ( F3 @ S ) )
% 0.26/0.46 @ ( autoref_rel_annot @ ( A > ( A > C ) > C ) @ ( B > ( B > D ) > D ) @ ( relAPP @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ ( set @ ( product_prod @ ( A > ( A > C ) > C ) @ ( B > ( B > D ) > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) ) ) > ( set @ ( product_prod @ ( A > ( A > C ) > C ) @ ( B > ( B > D ) > D ) ) ) ) @ ( fun_rel @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) ) @ Ra ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) ) ) @ ( relAPP @ ( A > C ) @ ( B > D ) @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) ) ) ) @ ( fun_rel @ ( A > C ) @ ( B > D ) @ C @ D ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ Ra ) @ Rr ) ) @ Rr ) ) ) )
% 0.26/0.46 @ X )
% 0.26/0.46 @ F ) )
% 0.26/0.46 @ Rr ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_Let_cong
% 0.26/0.46 thf(fact_36_internal__case__prod__conv,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A5: B,B3: C] :
% 0.26/0.46 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A5 @ B3 ) )
% 0.26/0.46 = ( C2 @ A5 @ B3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % internal_case_prod_conv
% 0.26/0.46 thf(fact_37_old_Oprod_Orec,axiom,
% 0.26/0.46 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A5: A,B3: B] :
% 0.26/0.46 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.26/0.46 = ( F1 @ A5 @ B3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % old.prod.rec
% 0.26/0.46 thf(fact_38_APP__def,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType] :
% 0.26/0.46 ( ( autoref_APP @ B @ A )
% 0.26/0.46 = ( ^ [F3: B > A] : F3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % APP_def
% 0.26/0.46 thf(fact_39_autoref__id,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > A ) @ ( B > B ) ) @ ( product_Pair @ ( A > A ) @ ( B > B ) @ ( id @ A ) @ ( id @ B ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) ) @ ( fun_rel @ A @ B @ A @ B ) @ R3 ) @ R3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_id
% 0.26/0.46 thf(fact_40_autoref__REMOVE__INTERNAL__EQ,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,C2: A,A5: B,R3: set @ ( product_prod @ A @ B ),C4: A] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ C2 @ A5 ) @ R3 )
% 0.26/0.46 => ( ( autore813025987NAL_EQ @ A @ C2 @ C4 )
% 0.26/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ C4 @ A5 ) @ R3 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_REMOVE_INTERNAL_EQ
% 0.26/0.46 thf(fact_41_nres__more__simps_I6_J,axiom,
% 0.26/0.46 ! [A: $tType,X: A,Y: A] :
% 0.26/0.46 ( ( ( refine1687780735RETURN @ A @ X )
% 0.26/0.46 = ( refine1687780735RETURN @ A @ Y ) )
% 0.26/0.46 = ( X = Y ) ) ).
% 0.26/0.46
% 0.26/0.46 % nres_more_simps(6)
% 0.26/0.46 thf(fact_42_mem__Collect__eq,axiom,
% 0.26/0.46 ! [A: $tType,A5: A,P: A > $o] :
% 0.26/0.46 ( ( member @ A @ A5 @ ( collect @ A @ P ) )
% 0.26/0.46 = ( P @ A5 ) ) ).
% 0.26/0.46
% 0.26/0.46 % mem_Collect_eq
% 0.26/0.46 thf(fact_43_Collect__mem__eq,axiom,
% 0.26/0.46 ! [A: $tType,A2: set @ A] :
% 0.26/0.46 ( ( collect @ A
% 0.26/0.46 @ ^ [X6: A] : ( member @ A @ X6 @ A2 ) )
% 0.26/0.46 = A2 ) ).
% 0.26/0.46
% 0.26/0.46 % Collect_mem_eq
% 0.26/0.46 thf(fact_44_Collect__cong,axiom,
% 0.26/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.26/0.46 ( ! [X5: A] :
% 0.26/0.46 ( ( P @ X5 )
% 0.26/0.46 = ( Q @ X5 ) )
% 0.26/0.46 => ( ( collect @ A @ P )
% 0.26/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % Collect_cong
% 0.26/0.46 thf(fact_45_ext,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,F: A > B,G3: A > B] :
% 0.26/0.46 ( ! [X5: A] :
% 0.26/0.46 ( ( F @ X5 )
% 0.26/0.46 = ( G3 @ X5 ) )
% 0.26/0.46 => ( F = G3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % ext
% 0.26/0.46 thf(fact_46_nres__monad1,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,X: B,F: B > ( refine1665802226e_nres @ A )] :
% 0.26/0.46 ( ( refine463715084e_bind @ B @ A @ ( refine1687780735RETURN @ B @ X ) @ F )
% 0.26/0.46 = ( F @ X ) ) ).
% 0.26/0.46
% 0.26/0.46 % nres_monad1
% 0.26/0.46 thf(fact_47_nres__monad2,axiom,
% 0.26/0.46 ! [A: $tType,M3: refine1665802226e_nres @ A] :
% 0.26/0.46 ( ( refine463715084e_bind @ A @ A @ M3 @ ( refine1687780735RETURN @ A ) )
% 0.26/0.46 = M3 ) ).
% 0.26/0.46
% 0.26/0.46 % nres_monad2
% 0.26/0.46 thf(fact_48_fun__rel__id__simp,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( relAPP @ B @ B @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( fun_rel @ A @ A @ B @ B ) @ ( id2 @ A ) ) @ ( id2 @ B ) )
% 0.26/0.46 = ( id2 @ ( A > B ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_rel_id_simp
% 0.26/0.46 thf(fact_49_relprop__triggers_I3_J,axiom,
% 0.26/0.46 ! [D: $tType,R3: set @ ( product_prod @ D @ D )] :
% 0.26/0.46 ( ( R3
% 0.26/0.46 = ( id2 @ D ) )
% 0.26/0.46 => ( ( id2 @ D )
% 0.26/0.46 = R3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % relprop_triggers(3)
% 0.26/0.46 thf(fact_50_relprop__triggers_I2_J,axiom,
% 0.26/0.46 ! [C: $tType,R3: set @ ( product_prod @ C @ C )] :
% 0.26/0.46 ( ( R3
% 0.26/0.46 = ( id2 @ C ) )
% 0.26/0.46 => ( R3
% 0.26/0.46 = ( id2 @ C ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % relprop_triggers(2)
% 0.26/0.46 thf(fact_51_autoref__int_I6_J,axiom,
% 0.26/0.46 ( member @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) )
% 0.26/0.46 @ ( product_Pair @ ( int > int > $o ) @ ( int > int > $o )
% 0.26/0.46 @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 0.26/0.46 @ ^ [Y5: int,Z: int] : ( Y5 = Z ) )
% 0.26/0.46 @ ( relAPP @ ( int > $o ) @ ( int > $o ) @ ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) > ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) ) @ ( fun_rel @ int @ int @ ( int > $o ) @ ( int > $o ) ) @ ( id2 @ int ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) ) @ ( fun_rel @ int @ int @ $o @ $o ) @ ( id2 @ int ) ) @ ( id2 @ $o ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_int(6)
% 0.26/0.46 thf(fact_52_autoref__nat_I7_J,axiom,
% 0.26/0.46 ( member @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) )
% 0.26/0.46 @ ( product_Pair @ ( nat > nat > $o ) @ ( nat > nat > $o )
% 0.26/0.46 @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 0.26/0.46 @ ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 0.26/0.46 @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) > ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ ( nat > $o ) @ ( nat > $o ) ) @ ( id2 @ nat ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_nat(7)
% 0.26/0.46 thf(fact_53_autoref__bool_I8_J,axiom,
% 0.26/0.46 member @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) @ ( product_Pair @ ( $o > $o > $o ) @ ( $o > $o > $o ) @ (=>) @ (=>) ) @ ( relAPP @ ( $o > $o ) @ ( $o > $o ) @ ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) > ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ ( $o > $o ) @ ( $o > $o ) ) @ ( id2 @ $o ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ $o @ $o ) @ ( id2 @ $o ) ) @ ( id2 @ $o ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_bool(8)
% 0.26/0.46 thf(fact_54_autoref__bool_I4_J,axiom,
% 0.26/0.46 member @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) @ ( product_Pair @ ( $o > $o ) @ ( $o > $o ) @ (~) @ (~) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ $o @ $o ) @ ( id2 @ $o ) ) @ ( id2 @ $o ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_bool(4)
% 0.26/0.46 thf(fact_55_autoref__bool_I3_J,axiom,
% 0.26/0.46 member @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) @ ( product_Pair @ ( $o > $o > $o ) @ ( $o > $o > $o ) @ (|) @ (|) ) @ ( relAPP @ ( $o > $o ) @ ( $o > $o ) @ ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) > ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ ( $o > $o ) @ ( $o > $o ) ) @ ( id2 @ $o ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ $o @ $o ) @ ( id2 @ $o ) ) @ ( id2 @ $o ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_bool(3)
% 0.26/0.46 thf(fact_56_autoref__bool_I2_J,axiom,
% 0.26/0.46 member @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) @ ( product_Pair @ ( $o > $o > $o ) @ ( $o > $o > $o ) @ (&) @ (&) ) @ ( relAPP @ ( $o > $o ) @ ( $o > $o ) @ ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) > ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ ( $o > $o ) @ ( $o > $o ) ) @ ( id2 @ $o ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ $o @ $o ) @ ( id2 @ $o ) ) @ ( id2 @ $o ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_bool(2)
% 0.26/0.46 thf(fact_57_autoref__bool_I1_J,axiom,
% 0.26/0.46 ! [X: $o] : ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ X @ X ) @ ( id2 @ $o ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_bool(1)
% 0.26/0.46 thf(fact_58_relprop__id__orient,axiom,
% 0.26/0.46 ! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
% 0.26/0.46 ( ( R3
% 0.26/0.46 = ( id2 @ A ) )
% 0.26/0.46 => ( ( id2 @ A )
% 0.26/0.46 = R3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % relprop_id_orient
% 0.26/0.46 thf(fact_59_rel__ANNOT__eq,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,T2: A,R3: set @ ( product_prod @ B @ A )] :
% 0.26/0.46 ( T2
% 0.26/0.46 = ( autoref_ANNOT @ A @ T2 @ ( autoref_rel_annot @ B @ A @ R3 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % rel_ANNOT_eq
% 0.26/0.46 thf(fact_60_ANNOT__def,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( autoref_ANNOT @ A )
% 0.26/0.46 = ( ^ [X6: A,A7: autoref_annot] : X6 ) ) ).
% 0.26/0.46
% 0.26/0.46 % ANNOT_def
% 0.26/0.46 thf(fact_61_OP__def,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( autoref_OP @ A )
% 0.26/0.46 = ( ^ [X6: A] : X6 ) ) ).
% 0.26/0.46
% 0.26/0.46 % OP_def
% 0.26/0.46 thf(fact_62_rel__cong,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,F: A > B,G3: A > B,X: A,Y: A] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( A > B ) @ ( A > B ) ) @ ( product_Pair @ ( A > B ) @ ( A > B ) @ F @ G3 ) @ ( id2 @ ( A > B ) ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id2 @ A ) )
% 0.26/0.46 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( G3 @ Y ) ) @ ( id2 @ B ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % rel_cong
% 0.26/0.46 thf(fact_63_rel__fun__cong,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,F: A > B,G3: A > B,X: A] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( A > B ) @ ( A > B ) ) @ ( product_Pair @ ( A > B ) @ ( A > B ) @ F @ G3 ) @ ( id2 @ ( A > B ) ) )
% 0.26/0.46 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( G3 @ X ) ) @ ( id2 @ B ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % rel_fun_cong
% 0.26/0.46 thf(fact_64_REMOVE__INTERNAL__EQ__def,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( autore813025987NAL_EQ @ A )
% 0.26/0.46 = ( ^ [Y5: A,Z: A] : ( Y5 = Z ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % REMOVE_INTERNAL_EQ_def
% 0.26/0.46 thf(fact_65_REMOVE__INTERNAL__def,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( autore2013370551TERNAL @ A )
% 0.26/0.46 = ( ^ [X6: A] : X6 ) ) ).
% 0.26/0.46
% 0.26/0.46 % REMOVE_INTERNAL_def
% 0.26/0.46 thf(fact_66_REMOVE__INTERNAL__EQI,axiom,
% 0.26/0.46 ! [A: $tType,A5: A] : ( autore813025987NAL_EQ @ A @ A5 @ A5 ) ).
% 0.26/0.46
% 0.26/0.46 % REMOVE_INTERNAL_EQI
% 0.26/0.46 thf(fact_67_rel__arg__cong,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,X: A,Y: A,F: A > B] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id2 @ A ) )
% 0.26/0.46 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y ) ) @ ( id2 @ B ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % rel_arg_cong
% 0.26/0.46 thf(fact_68_fun__rel__id,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 0.26/0.46 ( ( R1
% 0.26/0.46 = ( id2 @ A ) )
% 0.26/0.46 => ( ( R22
% 0.26/0.46 = ( id2 @ B ) )
% 0.26/0.46 => ( ( relAPP @ B @ B @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( fun_rel @ A @ A @ B @ B ) @ R1 ) @ R22 )
% 0.26/0.46 = ( id2 @ ( A > B ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_rel_id
% 0.26/0.46 thf(fact_69_IdI,axiom,
% 0.26/0.46 ! [A: $tType,A5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( id2 @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 % IdI
% 0.26/0.46 thf(fact_70_pair__in__Id__conv,axiom,
% 0.26/0.46 ! [A: $tType,A5: A,B3: A] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ ( id2 @ A ) )
% 0.26/0.46 = ( A5 = B3 ) ) ).
% 0.26/0.46
% 0.26/0.46 % pair_in_Id_conv
% 0.26/0.46 thf(fact_71_autoref__bool_I6_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) @ ( product_Pair @ ( A > A > $o > A ) @ ( B > B > $o > B ) @ ( product_rec_bool @ A ) @ ( product_rec_bool @ B ) ) @ ( relAPP @ ( A > $o > A ) @ ( B > $o > B ) @ ( set @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) > ( set @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) ) ) @ ( fun_rel @ A @ B @ ( A > $o > A ) @ ( B > $o > B ) ) @ R3 ) @ ( relAPP @ ( $o > A ) @ ( $o > B ) @ ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) > ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) ) @ ( fun_rel @ A @ B @ ( $o > A ) @ ( $o > B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) ) @ ( fun_rel @ $o @ $o @ A @ B ) @ ( id2 @ $o ) ) @ R3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_bool(6)
% 0.26/0.46 thf(fact_72_autoref__If__cong,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,C4: $o,C2: $o,T3: A,T2: B,R3: set @ ( product_prod @ A @ B ),E3: A,E4: B] :
% 0.26/0.46 ( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ C4 @ C2 ) @ ( id2 @ $o ) )
% 0.26/0.46 => ( ( ( autore2013370551TERNAL @ $o @ C2 )
% 0.26/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ T3 @ T2 ) @ R3 ) )
% 0.26/0.46 => ( ( ~ ( autore2013370551TERNAL @ $o @ C2 )
% 0.26/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ E3 @ E4 ) @ R3 ) )
% 0.26/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( if @ A @ C4 @ T3 @ E3 ) @ ( autoref_APP @ B @ B @ ( autoref_APP @ B @ ( B > B ) @ ( autoref_APP @ $o @ ( B > B > B ) @ ( autoref_ANNOT @ ( $o > B > B > B ) @ ( autoref_OP @ ( $o > B > B > B ) @ ( if @ B ) ) @ ( autoref_rel_annot @ ( $o > A > A > A ) @ ( $o > B > B > B ) @ ( relAPP @ ( A > A > A ) @ ( B > B > B ) @ ( set @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) > ( set @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) ) ) @ ( fun_rel @ $o @ $o @ ( A > A > A ) @ ( B > B > B ) ) @ ( id2 @ $o ) ) @ ( relAPP @ ( A > A ) @ ( B > B ) @ ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) > ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) ) @ ( fun_rel @ A @ B @ ( A > A ) @ ( B > B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) ) @ ( fun_rel @ A @ B @ A @ B ) @ R3 ) @ R3 ) ) ) ) ) @ C2 ) @ T2 ) @ E4 ) ) @ R3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_If_cong
% 0.26/0.46 thf(fact_73_param__fun__upd,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,Ra: set @ ( product_prod @ A @ B ),Rb: set @ ( product_prod @ C @ D )] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( A > A > $o ) @ ( B > B > $o ) )
% 0.26/0.46 @ ( product_Pair @ ( A > A > $o ) @ ( B > B > $o )
% 0.26/0.46 @ ^ [Y5: A,Z: A] : ( Y5 = Z )
% 0.26/0.46 @ ^ [Y5: B,Z: B] : ( Y5 = Z ) )
% 0.26/0.46 @ ( relAPP @ ( A > $o ) @ ( B > $o ) @ ( set @ ( product_prod @ ( A > A > $o ) @ ( B > B > $o ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) ) > ( set @ ( product_prod @ ( A > A > $o ) @ ( B > B > $o ) ) ) ) @ ( fun_rel @ A @ B @ ( A > $o ) @ ( B > $o ) ) @ Ra ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) ) ) @ ( fun_rel @ A @ B @ $o @ $o ) @ Ra ) @ ( id2 @ $o ) ) ) )
% 0.26/0.46 => ( member @ ( product_prod @ ( ( A > C ) > A > C > A > C ) @ ( ( B > D ) > B > D > B > D ) ) @ ( product_Pair @ ( ( A > C ) > A > C > A > C ) @ ( ( B > D ) > B > D > B > D ) @ ( fun_upd @ A @ C ) @ ( fun_upd @ B @ D ) ) @ ( relAPP @ ( A > C > A > C ) @ ( B > D > B > D ) @ ( set @ ( product_prod @ ( ( A > C ) > A > C > A > C ) @ ( ( B > D ) > B > D > B > D ) ) ) @ ( relAPP @ ( A > C ) @ ( B > D ) @ ( ( set @ ( product_prod @ ( A > C > A > C ) @ ( B > D > B > D ) ) ) > ( set @ ( product_prod @ ( ( A > C ) > A > C > A > C ) @ ( ( B > D ) > B > D > B > D ) ) ) ) @ ( fun_rel @ ( A > C ) @ ( B > D ) @ ( A > C > A > C ) @ ( B > D > B > D ) ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ Ra ) @ Rb ) ) @ ( relAPP @ ( C > A > C ) @ ( D > B > D ) @ ( set @ ( product_prod @ ( A > C > A > C ) @ ( B > D > B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( C > A > C ) @ ( D > B > D ) ) ) > ( set @ ( product_prod @ ( A > C > A > C ) @ ( B > D > B > D ) ) ) ) @ ( fun_rel @ A @ B @ ( C > A > C ) @ ( D > B > D ) ) @ Ra ) @ ( relAPP @ ( A > C ) @ ( B > D ) @ ( set @ ( product_prod @ ( C > A > C ) @ ( D > B > D ) ) ) @ ( relAPP @ C @ D @ ( ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) > ( set @ ( product_prod @ ( C > A > C ) @ ( D > B > D ) ) ) ) @ ( fun_rel @ C @ D @ ( A > C ) @ ( B > D ) ) @ Rb ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ Ra ) @ Rb ) ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % param_fun_upd
% 0.26/0.46 thf(fact_74_id__apply,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( id @ A )
% 0.26/0.46 = ( ^ [X6: A] : X6 ) ) ).
% 0.26/0.46
% 0.26/0.46 % id_apply
% 0.26/0.46 thf(fact_75_prod__refine_I3_J,axiom,
% 0.26/0.46 ! [B: $tType,D: $tType,C: $tType,A: $tType,F4: $tType,E: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D ),Rr: set @ ( product_prod @ E @ F4 )] : ( member @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) @ ( product_Pair @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) @ ( product_rec_prod @ A @ B @ E ) @ ( product_rec_prod @ C @ D @ F4 ) ) @ ( relAPP @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) @ ( set @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) ) @ ( relAPP @ ( A > B > E ) @ ( C > D > F4 ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) > ( set @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) ) ) @ ( fun_rel @ ( A > B > E ) @ ( C > D > F4 ) @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) @ ( relAPP @ ( B > E ) @ ( D > F4 ) @ ( set @ ( product_prod @ ( A > B > E ) @ ( C > D > F4 ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) > ( set @ ( product_prod @ ( A > B > E ) @ ( C > D > F4 ) ) ) ) @ ( fun_rel @ A @ C @ ( B > E ) @ ( D > F4 ) ) @ Ra ) @ ( relAPP @ E @ F4 @ ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) @ ( relAPP @ B @ D @ ( ( set @ ( product_prod @ E @ F4 ) ) > ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) ) @ ( fun_rel @ B @ D @ E @ F4 ) @ Rb ) @ Rr ) ) ) @ ( relAPP @ E @ F4 @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ E @ F4 ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ E @ F4 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) @ Rr ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_refine(3)
% 0.26/0.46 thf(fact_76_autoref__bool_I5_J,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) @ ( product_Pair @ ( A > A > $o > A ) @ ( B > B > $o > B ) @ ( product_case_bool @ A ) @ ( product_case_bool @ B ) ) @ ( relAPP @ ( A > $o > A ) @ ( B > $o > B ) @ ( set @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) > ( set @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) ) ) @ ( fun_rel @ A @ B @ ( A > $o > A ) @ ( B > $o > B ) ) @ R3 ) @ ( relAPP @ ( $o > A ) @ ( $o > B ) @ ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) > ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) ) @ ( fun_rel @ A @ B @ ( $o > A ) @ ( $o > B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) ) @ ( fun_rel @ $o @ $o @ A @ B ) @ ( id2 @ $o ) ) @ R3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % autoref_bool(5)
% 0.26/0.46 thf(fact_77_fun__upd__upd,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z2: B] :
% 0.26/0.46 ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z2 )
% 0.26/0.46 = ( fun_upd @ A @ B @ F @ X @ Z2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_upd_upd
% 0.26/0.46 thf(fact_78_fun__upd__triv,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 0.26/0.46 ( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
% 0.26/0.46 = F ) ).
% 0.26/0.46
% 0.26/0.46 % fun_upd_triv
% 0.26/0.46 thf(fact_79_fun__upd__apply,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType] :
% 0.26/0.46 ( ( fun_upd @ B @ A )
% 0.26/0.46 = ( ^ [F3: B > A,X6: B,Y6: A,Z3: B] : ( if @ A @ ( Z3 = X6 ) @ Y6 @ ( F3 @ Z3 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_upd_apply
% 0.26/0.46 thf(fact_80_old_Obool_Osimps_I6_J,axiom,
% 0.26/0.46 ! [T: $tType,F1: T,F22: T] :
% 0.26/0.46 ( ( product_rec_bool @ T @ F1 @ F22 @ $false )
% 0.26/0.46 = F22 ) ).
% 0.26/0.46
% 0.26/0.46 % old.bool.simps(6)
% 0.26/0.46 thf(fact_81_old_Obool_Osimps_I5_J,axiom,
% 0.26/0.46 ! [T: $tType,F1: T,F22: T] :
% 0.26/0.46 ( ( product_rec_bool @ T @ F1 @ F22 @ $true )
% 0.26/0.46 = F1 ) ).
% 0.26/0.46
% 0.26/0.46 % old.bool.simps(5)
% 0.26/0.46 thf(fact_82_prod__rel__simp,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A,B3: B,A6: C,B4: D,R1: set @ ( product_prod @ A @ C ),R22: set @ ( product_prod @ B @ D )] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ C @ D @ A6 @ B4 ) ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ R1 ) @ R22 ) )
% 0.26/0.46 = ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A5 @ A6 ) @ R1 )
% 0.26/0.46 & ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ B3 @ B4 ) @ R22 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_rel_simp
% 0.26/0.46 thf(fact_83_prod__rel__id__simp,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( relAPP @ B @ B @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) @ ( prod_rel @ A @ A @ B @ B ) @ ( id2 @ A ) ) @ ( id2 @ B ) )
% 0.26/0.46 = ( id2 @ ( product_prod @ A @ B ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_rel_id_simp
% 0.26/0.46 thf(fact_84_fun__upd__def,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType] :
% 0.26/0.46 ( ( fun_upd @ A @ B )
% 0.26/0.46 = ( ^ [F3: A > B,A7: A,B6: B,X6: A] : ( if @ B @ ( X6 = A7 ) @ B6 @ ( F3 @ X6 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_upd_def
% 0.26/0.46 thf(fact_85_fun__upd__eqD,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G3: A > B,Z2: B] :
% 0.26/0.46 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 0.26/0.46 = ( fun_upd @ A @ B @ G3 @ X @ Z2 ) )
% 0.26/0.46 => ( Y = Z2 ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_upd_eqD
% 0.26/0.46 thf(fact_86_fun__upd__idem,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
% 0.26/0.46 ( ( ( F @ X )
% 0.26/0.46 = Y )
% 0.26/0.46 => ( ( fun_upd @ B @ A @ F @ X @ Y )
% 0.26/0.46 = F ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_upd_idem
% 0.26/0.46 thf(fact_87_fun__upd__same,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
% 0.26/0.46 ( ( fun_upd @ B @ A @ F @ X @ Y @ X )
% 0.26/0.46 = Y ) ).
% 0.26/0.46
% 0.26/0.46 % fun_upd_same
% 0.26/0.46 thf(fact_88_fun__upd__other,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,Z2: A,X: A,F: A > B,Y: B] :
% 0.26/0.46 ( ( Z2 != X )
% 0.26/0.46 => ( ( fun_upd @ A @ B @ F @ X @ Y @ Z2 )
% 0.26/0.46 = ( F @ Z2 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_upd_other
% 0.26/0.46 thf(fact_89_fun__upd__twist,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,A5: A,C2: A,M: A > B,B3: B,D3: B] :
% 0.26/0.46 ( ( A5 != C2 )
% 0.26/0.46 => ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A5 @ B3 ) @ C2 @ D3 )
% 0.26/0.46 = ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C2 @ D3 ) @ A5 @ B3 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_upd_twist
% 0.26/0.46 thf(fact_90_fun__upd__idem__iff,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
% 0.26/0.46 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 0.26/0.46 = F )
% 0.26/0.46 = ( ( F @ X )
% 0.26/0.46 = Y ) ) ).
% 0.26/0.46
% 0.26/0.46 % fun_upd_idem_iff
% 0.26/0.46 thf(fact_91_rec__bool__is__case,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( product_rec_bool @ A )
% 0.26/0.46 = ( product_case_bool @ A ) ) ).
% 0.26/0.46
% 0.26/0.46 % rec_bool_is_case
% 0.26/0.46 thf(fact_92_bool_Osplit__sel__asm,axiom,
% 0.26/0.46 ! [A: $tType,P: A > $o,F1: A,F22: A,Bool: $o] :
% 0.26/0.46 ( ( P @ ( product_case_bool @ A @ F1 @ F22 @ Bool ) )
% 0.26/0.46 = ( ~ ( ( Bool
% 0.26/0.46 & ~ ( P @ F1 ) )
% 0.26/0.46 | ( ~ Bool
% 0.26/0.46 & ~ ( P @ F22 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % bool.split_sel_asm
% 0.26/0.46 thf(fact_93_bool_Ocase__distrib,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,H: A > B,F1: A,F22: A,Bool: $o] :
% 0.26/0.46 ( ( H @ ( product_case_bool @ A @ F1 @ F22 @ Bool ) )
% 0.26/0.46 = ( product_case_bool @ B @ ( H @ F1 ) @ ( H @ F22 ) @ Bool ) ) ).
% 0.26/0.46
% 0.26/0.46 % bool.case_distrib
% 0.26/0.46 thf(fact_94_bool_Ocase__eq__if,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( product_case_bool @ A )
% 0.26/0.46 = ( ^ [F12: A,F23: A,Bool2: $o] : ( if @ A @ Bool2 @ F12 @ F23 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % bool.case_eq_if
% 0.26/0.46 thf(fact_95_bool_Osplit__sel,axiom,
% 0.26/0.46 ! [A: $tType,P: A > $o,F1: A,F22: A,Bool: $o] :
% 0.26/0.46 ( ( P @ ( product_case_bool @ A @ F1 @ F22 @ Bool ) )
% 0.26/0.46 = ( ( Bool
% 0.26/0.46 => ( P @ F1 ) )
% 0.26/0.46 & ( ~ Bool
% 0.26/0.46 => ( P @ F22 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % bool.split_sel
% 0.26/0.46 thf(fact_96_old_Obool_Osimps_I3_J,axiom,
% 0.26/0.46 ! [A: $tType,F1: A,F22: A] :
% 0.26/0.46 ( ( product_case_bool @ A @ F1 @ F22 @ $true )
% 0.26/0.46 = F1 ) ).
% 0.26/0.46
% 0.26/0.46 % old.bool.simps(3)
% 0.26/0.46 thf(fact_97_old_Obool_Osimps_I4_J,axiom,
% 0.26/0.46 ! [A: $tType,F1: A,F22: A] :
% 0.26/0.46 ( ( product_case_bool @ A @ F1 @ F22 @ $false )
% 0.26/0.46 = F22 ) ).
% 0.26/0.46
% 0.26/0.46 % old.bool.simps(4)
% 0.26/0.46 thf(fact_98_prod__relI,axiom,
% 0.26/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,A5: A,A6: B,R1: set @ ( product_prod @ A @ B ),B3: C,B4: D,R22: set @ ( product_prod @ C @ D )] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ A6 ) @ R1 )
% 0.26/0.46 => ( ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ B3 @ B4 ) @ R22 )
% 0.26/0.46 => ( member @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_Pair @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_Pair @ A @ C @ A5 @ B3 ) @ ( product_Pair @ B @ D @ A6 @ B4 ) ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) @ ( prod_rel @ A @ B @ C @ D ) @ R1 ) @ R22 ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_relI
% 0.26/0.46 thf(fact_99_prod__relE,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,P2: product_prod @ A @ B,P3: product_prod @ C @ D,R1: set @ ( product_prod @ A @ C ),R22: set @ ( product_prod @ B @ D )] :
% 0.26/0.46 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ P2 @ P3 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ R1 ) @ R22 ) )
% 0.26/0.46 => ~ ! [A3: A,B5: B] :
% 0.26/0.46 ( ( P2
% 0.26/0.46 = ( product_Pair @ A @ B @ A3 @ B5 ) )
% 0.26/0.46 => ! [A4: C,B7: D] :
% 0.26/0.46 ( ( P3
% 0.26/0.46 = ( product_Pair @ C @ D @ A4 @ B7 ) )
% 0.26/0.46 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ A4 ) @ R1 )
% 0.26/0.46 => ~ ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ B5 @ B7 ) @ R22 ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_relE
% 0.26/0.46 thf(fact_100_prod__refine_I1_J,axiom,
% 0.26/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( A > B > ( product_prod @ A @ B ) ) @ ( C > D > ( product_prod @ C @ D ) ) ) @ ( product_Pair @ ( A > B > ( product_prod @ A @ B ) ) @ ( C > D > ( product_prod @ C @ D ) ) @ ( product_Pair @ A @ B ) @ ( product_Pair @ C @ D ) ) @ ( relAPP @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) @ ( set @ ( product_prod @ ( A > B > ( product_prod @ A @ B ) ) @ ( C > D > ( product_prod @ C @ D ) ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) ) ) > ( set @ ( product_prod @ ( A > B > ( product_prod @ A @ B ) ) @ ( C > D > ( product_prod @ C @ D ) ) ) ) ) @ ( fun_rel @ A @ C @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) ) @ Ra ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) ) ) @ ( relAPP @ B @ D @ ( ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) > ( set @ ( product_prod @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) ) ) ) @ ( fun_rel @ B @ D @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) @ Rb ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_refine(1)
% 0.26/0.46 thf(fact_101_prod__rel__id,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 0.26/0.46 ( ( R1
% 0.26/0.46 = ( id2 @ A ) )
% 0.26/0.46 => ( ( R22
% 0.26/0.46 = ( id2 @ B ) )
% 0.26/0.46 => ( ( relAPP @ B @ B @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) @ ( prod_rel @ A @ A @ B @ B ) @ R1 ) @ R22 )
% 0.26/0.46 = ( id2 @ ( product_prod @ A @ B ) ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % prod_rel_id
% 0.26/0.46 thf(fact_102_id__def,axiom,
% 0.26/0.46 ! [A: $tType] :
% 0.26/0.46 ( ( id @ A )
% 0.26/0.46 = ( ^ [X6: A] : X6 ) ) ).
% 0.26/0.46
% 0.26/0.46 % id_def
% 0.26/0.46 thf(fact_103_eq__id__iff,axiom,
% 0.26/0.46 ! [A: $tType,F: A > A] :
% 0.26/0.46 ( ( ! [X6: A] :
% 0.26/0.46 ( ( F @ X6 )
% 0.26/0.46 = X6 ) )
% 0.26/0.46 = ( F
% 0.26/0.46 = ( id @ A ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % eq_id_iff
% 0.26/0.46 thf(fact_104_param__and__cong2,axiom,
% 0.26/0.46 ! [A5: $o,A6: $o,B3: $o,B4: $o] :
% 0.26/0.46 ( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ A5 @ A6 ) @ ( id2 @ $o ) )
% 0.26/0.46 => ( ( A5
% 0.26/0.46 => ( A6
% 0.26/0.46 => ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ B3 @ B4 ) @ ( id2 @ $o ) ) ) )
% 0.26/0.46 => ( member @ ( product_prod @ $o @ $o )
% 0.26/0.46 @ ( product_Pair @ $o @ $o
% 0.26/0.46 @ ( B3
% 0.26/0.46 & A5 )
% 0.26/0.46 @ ( B4
% 0.26/0.46 & A6 ) )
% 0.26/0.46 @ ( id2 @ $o ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % param_and_cong2
% 0.26/0.46 thf(fact_105_param__and__cong1,axiom,
% 0.26/0.46 ! [A5: $o,A6: $o,B3: $o,B4: $o] :
% 0.26/0.46 ( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ A5 @ A6 ) @ ( id2 @ $o ) )
% 0.26/0.46 => ( ( A5
% 0.26/0.46 => ( A6
% 0.26/0.46 => ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ B3 @ B4 ) @ ( id2 @ $o ) ) ) )
% 0.26/0.46 => ( member @ ( product_prod @ $o @ $o )
% 0.26/0.46 @ ( product_Pair @ $o @ $o
% 0.26/0.46 @ ( A5
% 0.26/0.46 & B3 )
% 0.26/0.46 @ ( A6
% 0.26/0.46 & B4 ) )
% 0.26/0.46 @ ( id2 @ $o ) ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % param_and_cong1
% 0.26/0.46 thf(fact_106_param__bool_I1_J,axiom,
% 0.26/0.46 member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ $true @ $true ) @ ( id2 @ $o ) ).
% 0.26/0.46
% 0.26/0.46 % param_bool(1)
% 0.26/0.46 thf(fact_107_param__bool_I2_J,axiom,
% 0.26/0.46 member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ $false @ $false ) @ ( id2 @ $o ) ).
% 0.26/0.46
% 0.26/0.46 % param_bool(2)
% 0.26/0.46 thf(fact_108_IdE,axiom,
% 0.26/0.46 ! [A: $tType,P2: product_prod @ A @ A] :
% 0.26/0.46 ( ( member @ ( product_prod @ A @ A ) @ P2 @ ( id2 @ A ) )
% 0.26/0.46 => ~ ! [X5: A] :
% 0.26/0.46 ( P2
% 0.26/0.46 != ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ).
% 0.26/0.46
% 0.26/0.46 % IdE
% 0.26/0.46 thf(fact_109_param__if,axiom,
% 0.26/0.46 ! [B: $tType,A: $tType,C2: $o,C4: $o,T2: A,T3: B,R3: set @ ( product_prod @ A @ B ),E4: A,E3: B] :
% 0.26/0.46 ( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ C2 @ C4 ) @ ( id2 @ $o ) )
% 0.26/0.46 => ( ( C2
% 0.26/0.47 => ( C4
% 0.26/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ T2 @ T3 ) @ R3 ) ) )
% 0.26/0.47 => ( ( ~ C2
% 0.26/0.47 => ( ~ C4
% 0.26/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ E4 @ E3 ) @ R3 ) ) )
% 0.26/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( if @ A @ C2 @ T2 @ E4 ) @ ( if @ B @ C4 @ T3 @ E3 ) ) @ R3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % param_if
% 0.26/0.47 thf(fact_110_param__uncurry,axiom,
% 0.26/0.47 ! [B: $tType,E: $tType,D: $tType,A: $tType,F4: $tType,C: $tType,A2: set @ ( product_prod @ A @ D ),B2: set @ ( product_prod @ B @ E ),C5: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) @ ( ( D > E > F4 ) > ( product_prod @ D @ E ) > F4 ) ) @ ( product_Pair @ ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) @ ( ( D > E > F4 ) > ( product_prod @ D @ E ) > F4 ) @ ( uncurry @ A @ B @ C ) @ ( uncurry @ D @ E @ F4 ) ) @ ( relAPP @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) @ ( set @ ( product_prod @ ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) @ ( ( D > E > F4 ) > ( product_prod @ D @ E ) > F4 ) ) ) @ ( relAPP @ ( A > B > C ) @ ( D > E > F4 ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) > ( set @ ( product_prod @ ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) @ ( ( D > E > F4 ) > ( product_prod @ D @ E ) > F4 ) ) ) ) @ ( fun_rel @ ( A > B > C ) @ ( D > E > F4 ) @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) @ ( relAPP @ ( B > C ) @ ( E > F4 ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) > ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) ) @ ( fun_rel @ A @ D @ ( B > C ) @ ( E > F4 ) ) @ A2 ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) @ ( relAPP @ B @ E @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) ) @ ( fun_rel @ B @ E @ C @ F4 ) @ B2 ) @ C5 ) ) ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ C @ F4 ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) ) ) ) @ ( prod_rel @ A @ D @ B @ E ) @ A2 ) @ B2 ) ) @ C5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % param_uncurry
% 0.26/0.47 thf(fact_111_param__all__interval__nat,axiom,
% 0.26/0.47 member @ ( product_prod @ ( ( nat > $o ) > nat > nat > $o ) @ ( ( nat > $o ) > nat > nat > $o ) ) @ ( product_Pair @ ( ( nat > $o ) > nat > nat > $o ) @ ( ( nat > $o ) > nat > nat > $o ) @ all_interval_nat @ all_interval_nat ) @ ( relAPP @ ( nat > nat > $o ) @ ( nat > nat > $o ) @ ( set @ ( product_prod @ ( ( nat > $o ) > nat > nat > $o ) @ ( ( nat > $o ) > nat > nat > $o ) ) ) @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) > ( set @ ( product_prod @ ( ( nat > $o ) > nat > nat > $o ) @ ( ( nat > $o ) > nat > nat > $o ) ) ) ) @ ( fun_rel @ ( nat > $o ) @ ( nat > $o ) @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) > ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ ( nat > $o ) @ ( nat > $o ) ) @ ( id2 @ nat ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % param_all_interval_nat
% 0.26/0.47 thf(fact_112_param__curry,axiom,
% 0.26/0.47 ! [A: $tType,D: $tType,B: $tType,E: $tType,F4: $tType,C: $tType,Ra: set @ ( product_prod @ A @ D ),Rb: set @ ( product_prod @ B @ E ),Rc: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) @ ( ( ( product_prod @ D @ E ) > F4 ) > D > E > F4 ) ) @ ( product_Pair @ ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) @ ( ( ( product_prod @ D @ E ) > F4 ) > D > E > F4 ) @ ( product_curry @ A @ B @ C ) @ ( product_curry @ D @ E @ F4 ) ) @ ( relAPP @ ( A > B > C ) @ ( D > E > F4 ) @ ( set @ ( product_prod @ ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) @ ( ( ( product_prod @ D @ E ) > F4 ) > D > E > F4 ) ) ) @ ( relAPP @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) @ ( ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) > ( set @ ( product_prod @ ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) @ ( ( ( product_prod @ D @ E ) > F4 ) > D > E > F4 ) ) ) ) @ ( fun_rel @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) @ ( A > B > C ) @ ( D > E > F4 ) ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ C @ F4 ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) ) ) ) @ ( prod_rel @ A @ D @ B @ E ) @ Ra ) @ Rb ) ) @ Rc ) ) @ ( relAPP @ ( B > C ) @ ( E > F4 ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) > ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) ) @ ( fun_rel @ A @ D @ ( B > C ) @ ( E > F4 ) ) @ Ra ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) @ ( relAPP @ B @ E @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) ) @ ( fun_rel @ B @ E @ C @ F4 ) @ Rb ) @ Rc ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % param_curry
% 0.26/0.47 thf(fact_113_prod__refine_I2_J,axiom,
% 0.26/0.47 ! [B: $tType,D: $tType,C: $tType,A: $tType,F4: $tType,E: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D ),Rr: set @ ( product_prod @ E @ F4 )] : ( member @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) @ ( product_Pair @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) @ ( product_case_prod @ A @ B @ E ) @ ( product_case_prod @ C @ D @ F4 ) ) @ ( relAPP @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) @ ( set @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) ) @ ( relAPP @ ( A > B > E ) @ ( C > D > F4 ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) > ( set @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) ) ) @ ( fun_rel @ ( A > B > E ) @ ( C > D > F4 ) @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) @ ( relAPP @ ( B > E ) @ ( D > F4 ) @ ( set @ ( product_prod @ ( A > B > E ) @ ( C > D > F4 ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) > ( set @ ( product_prod @ ( A > B > E ) @ ( C > D > F4 ) ) ) ) @ ( fun_rel @ A @ C @ ( B > E ) @ ( D > F4 ) ) @ Ra ) @ ( relAPP @ E @ F4 @ ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) @ ( relAPP @ B @ D @ ( ( set @ ( product_prod @ E @ F4 ) ) > ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) ) @ ( fun_rel @ B @ D @ E @ F4 ) @ Rb ) @ Rr ) ) ) @ ( relAPP @ E @ F4 @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ E @ F4 ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ E @ F4 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) @ Rr ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod_refine(2)
% 0.26/0.47 thf(fact_114_autoref__rec__nat,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > ( nat > A > A ) > nat > A ) @ ( B > ( nat > B > B ) > nat > B ) ) @ ( product_Pair @ ( A > ( nat > A > A ) > nat > A ) @ ( B > ( nat > B > B ) > nat > B ) @ ( rec_nat @ A ) @ ( rec_nat @ B ) ) @ ( relAPP @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) @ ( set @ ( product_prod @ ( A > ( nat > A > A ) > nat > A ) @ ( B > ( nat > B > B ) > nat > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) ) ) > ( set @ ( product_prod @ ( A > ( nat > A > A ) > nat > A ) @ ( B > ( nat > B > B ) > nat > B ) ) ) ) @ ( fun_rel @ A @ B @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) ) @ R3 ) @ ( relAPP @ ( nat > A ) @ ( nat > B ) @ ( set @ ( product_prod @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) ) ) @ ( relAPP @ ( nat > A > A ) @ ( nat > B > B ) @ ( ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) > ( set @ ( product_prod @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) ) ) ) @ ( fun_rel @ ( nat > A > A ) @ ( nat > B > B ) @ ( nat > A ) @ ( nat > B ) ) @ ( relAPP @ ( A > A ) @ ( B > B ) @ ( set @ ( product_prod @ ( nat > A > A ) @ ( nat > B > B ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) > ( set @ ( product_prod @ ( nat > A > A ) @ ( nat > B > B ) ) ) ) @ ( fun_rel @ nat @ nat @ ( A > A ) @ ( B > B ) ) @ ( id2 @ nat ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) ) @ ( fun_rel @ A @ B @ A @ B ) @ R3 ) @ R3 ) ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) ) @ ( fun_rel @ nat @ nat @ A @ B ) @ ( id2 @ nat ) ) @ R3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % autoref_rec_nat
% 0.26/0.47 thf(fact_115_autoref__int_I5_J,axiom,
% 0.26/0.47 member @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) @ ( product_Pair @ ( int > int > $o ) @ ( int > int > $o ) @ ( ord_less_eq @ int ) @ ( ord_less_eq @ int ) ) @ ( relAPP @ ( int > $o ) @ ( int > $o ) @ ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) > ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) ) @ ( fun_rel @ int @ int @ ( int > $o ) @ ( int > $o ) ) @ ( id2 @ int ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) ) @ ( fun_rel @ int @ int @ $o @ $o ) @ ( id2 @ int ) ) @ ( id2 @ $o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % autoref_int(5)
% 0.26/0.47 thf(fact_116_case__prod__Pair,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType] :
% 0.26/0.47 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
% 0.26/0.47 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % case_prod_Pair
% 0.26/0.47 thf(fact_117_curryI,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A5: A,B3: B] :
% 0.26/0.47 ( ( F @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.26/0.47 => ( product_curry @ A @ B @ $o @ F @ A5 @ B3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % curryI
% 0.26/0.47 thf(fact_118_curry__conv,axiom,
% 0.26/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.26/0.47 ( ( product_curry @ B @ C @ A )
% 0.26/0.47 = ( ^ [F3: ( product_prod @ B @ C ) > A,A7: B,B6: C] : ( F3 @ ( product_Pair @ B @ C @ A7 @ B6 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % curry_conv
% 0.26/0.47 thf(fact_119_case__prod__curry,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.26/0.47 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 0.26/0.47 = F ) ).
% 0.26/0.47
% 0.26/0.47 % case_prod_curry
% 0.26/0.47 thf(fact_120_curry__case__prod,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 0.26/0.47 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
% 0.26/0.47 = F ) ).
% 0.26/0.47
% 0.26/0.47 % curry_case_prod
% 0.26/0.47 thf(fact_121_lhs__step__If,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( ord @ A )
% 0.26/0.47 => ! [B3: $o,T2: A,M: A,E4: A] :
% 0.26/0.47 ( ( B3
% 0.26/0.47 => ( ord_less_eq @ A @ T2 @ M ) )
% 0.26/0.47 => ( ( ~ B3
% 0.26/0.47 => ( ord_less_eq @ A @ E4 @ M ) )
% 0.26/0.47 => ( ord_less_eq @ A @ ( if @ A @ B3 @ T2 @ E4 ) @ M ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % lhs_step_If
% 0.26/0.47 thf(fact_122_mem__case__prodE,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 0.26/0.47 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P2 ) )
% 0.26/0.47 => ~ ! [X5: B,Y3: C] :
% 0.26/0.47 ( ( P2
% 0.26/0.47 = ( product_Pair @ B @ C @ X5 @ Y3 ) )
% 0.26/0.47 => ~ ( member @ A @ Z2 @ ( C2 @ X5 @ Y3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % mem_case_prodE
% 0.26/0.47 thf(fact_123_old_Oprod_Ocase,axiom,
% 0.26/0.47 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 0.26/0.47 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.26/0.47 = ( F @ X1 @ X22 ) ) ).
% 0.26/0.47
% 0.26/0.47 % old.prod.case
% 0.26/0.47 thf(fact_124_param__case__prod_H_H,axiom,
% 0.26/0.47 ! [C: $tType,D: $tType,B: $tType,A: $tType,F4: $tType,E: $tType,P2: product_prod @ A @ B,P3: product_prod @ C @ D,F: A > B > E,F2: C > D > F4,R3: set @ ( product_prod @ E @ F4 )] :
% 0.26/0.47 ( ! [A3: A,B5: B,A4: C,B7: D] :
% 0.26/0.47 ( ( P2
% 0.26/0.47 = ( product_Pair @ A @ B @ A3 @ B5 ) )
% 0.26/0.47 => ( ( P3
% 0.26/0.47 = ( product_Pair @ C @ D @ A4 @ B7 ) )
% 0.26/0.47 => ( member @ ( product_prod @ E @ F4 ) @ ( product_Pair @ E @ F4 @ ( F @ A3 @ B5 ) @ ( F2 @ A4 @ B7 ) ) @ R3 ) ) )
% 0.26/0.47 => ( member @ ( product_prod @ E @ F4 ) @ ( product_Pair @ E @ F4 @ ( product_case_prod @ A @ B @ E @ F @ P2 ) @ ( product_case_prod @ C @ D @ F4 @ F2 @ P3 ) ) @ R3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % param_case_prod''
% 0.26/0.47 thf(fact_125_rec__prod__is__case,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.47 ( ( product_rec_prod @ A @ B @ C )
% 0.26/0.47 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.26/0.47
% 0.26/0.47 % rec_prod_is_case
% 0.26/0.47 thf(fact_126_curryD,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A5: A,B3: B] :
% 0.26/0.47 ( ( product_curry @ A @ B @ $o @ F @ A5 @ B3 )
% 0.26/0.47 => ( F @ ( product_Pair @ A @ B @ A5 @ B3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % curryD
% 0.26/0.47 thf(fact_127_curryE,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A5: A,B3: B] :
% 0.26/0.47 ( ( product_curry @ A @ B @ $o @ F @ A5 @ B3 )
% 0.26/0.47 => ( F @ ( product_Pair @ A @ B @ A5 @ B3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % curryE
% 0.26/0.47 thf(fact_128_internal__case__prod__def,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.47 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.26/0.47 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.26/0.47
% 0.26/0.47 % internal_case_prod_def
% 0.26/0.47 thf(fact_129_param__case__prod_H,axiom,
% 0.26/0.47 ! [C: $tType,D: $tType,B: $tType,A: $tType,F4: $tType,E: $tType,P2: product_prod @ A @ B,P3: product_prod @ C @ D,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D ),F: A > B > E,F2: C > D > F4,R3: set @ ( product_prod @ E @ F4 )] :
% 0.26/0.47 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ P2 @ P3 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) )
% 0.26/0.47 => ( ! [A3: A,B5: B,A4: C,B7: D] :
% 0.26/0.47 ( ( P2
% 0.26/0.47 = ( product_Pair @ A @ B @ A3 @ B5 ) )
% 0.26/0.47 => ( ( P3
% 0.26/0.47 = ( product_Pair @ C @ D @ A4 @ B7 ) )
% 0.26/0.47 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ A4 ) @ Ra )
% 0.26/0.47 => ( ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ B5 @ B7 ) @ Rb )
% 0.26/0.47 => ( member @ ( product_prod @ E @ F4 ) @ ( product_Pair @ E @ F4 @ ( F @ A3 @ B5 ) @ ( F2 @ A4 @ B7 ) ) @ R3 ) ) ) ) )
% 0.26/0.47 => ( member @ ( product_prod @ E @ F4 ) @ ( product_Pair @ E @ F4 @ ( product_case_prod @ A @ B @ E @ F @ P2 ) @ ( product_case_prod @ C @ D @ F4 @ F2 @ P3 ) ) @ R3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % param_case_prod'
% 0.26/0.47 thf(fact_130_autoref__case__nat,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,Ra: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > ( nat > A ) > nat > A ) @ ( B > ( nat > B ) > nat > B ) ) @ ( product_Pair @ ( A > ( nat > A ) > nat > A ) @ ( B > ( nat > B ) > nat > B ) @ ( case_nat @ A ) @ ( case_nat @ B ) ) @ ( relAPP @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) @ ( set @ ( product_prod @ ( A > ( nat > A ) > nat > A ) @ ( B > ( nat > B ) > nat > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) ) ) > ( set @ ( product_prod @ ( A > ( nat > A ) > nat > A ) @ ( B > ( nat > B ) > nat > B ) ) ) ) @ ( fun_rel @ A @ B @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) ) @ Ra ) @ ( relAPP @ ( nat > A ) @ ( nat > B ) @ ( set @ ( product_prod @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) ) ) @ ( relAPP @ ( nat > A ) @ ( nat > B ) @ ( ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) > ( set @ ( product_prod @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) ) ) ) @ ( fun_rel @ ( nat > A ) @ ( nat > B ) @ ( nat > A ) @ ( nat > B ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) ) @ ( fun_rel @ nat @ nat @ A @ B ) @ ( id2 @ nat ) ) @ Ra ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) ) @ ( fun_rel @ nat @ nat @ A @ B ) @ ( id2 @ nat ) ) @ Ra ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % autoref_case_nat
% 0.26/0.47 thf(fact_131_uncurry__curry__id,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.26/0.47 ( ( uncurry @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 0.26/0.47 = F ) ).
% 0.26/0.47
% 0.26/0.47 % uncurry_curry_id
% 0.26/0.47 thf(fact_132_curry__uncurry__id,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 0.26/0.47 ( ( product_curry @ A @ B @ C @ ( uncurry @ A @ B @ C @ F ) )
% 0.26/0.47 = F ) ).
% 0.26/0.47
% 0.26/0.47 % curry_uncurry_id
% 0.26/0.47 thf(fact_133_uncurry__apply,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A5: B,B3: C] :
% 0.26/0.47 ( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A5 @ B3 ) )
% 0.26/0.47 = ( F @ A5 @ B3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % uncurry_apply
% 0.26/0.47 thf(fact_134_nres__order__simps_I20_J,axiom,
% 0.26/0.47 ! [W: $tType,X: W,Y: W] :
% 0.26/0.47 ( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X ) @ ( refine1687780735RETURN @ W @ Y ) )
% 0.26/0.47 = ( X = Y ) ) ).
% 0.26/0.47
% 0.26/0.47 % nres_order_simps(20)
% 0.26/0.47 thf(fact_135_subrelI,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.26/0.47 ( ! [X5: A,Y3: B] :
% 0.26/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ R )
% 0.26/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ S2 ) )
% 0.26/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % subrelI
% 0.26/0.47 thf(fact_136_fun__rel__mono,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,D: $tType,C: $tType,R1: set @ ( product_prod @ A @ B ),R22: set @ ( product_prod @ A @ B ),R32: set @ ( product_prod @ C @ D ),R42: set @ ( product_prod @ C @ D )] :
% 0.26/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R1 @ R22 )
% 0.26/0.47 => ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ R32 @ R42 )
% 0.26/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ R22 ) @ R32 ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ R1 ) @ R42 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % fun_rel_mono
% 0.26/0.47 thf(fact_137_prod__rel__mono,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,D: $tType,C: $tType,R22: set @ ( product_prod @ A @ B ),R1: set @ ( product_prod @ A @ B ),R32: set @ ( product_prod @ C @ D ),R42: set @ ( product_prod @ C @ D )] :
% 0.26/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R22 @ R1 )
% 0.26/0.47 => ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ R32 @ R42 )
% 0.26/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) @ ( prod_rel @ A @ B @ C @ D ) @ R22 ) @ R32 ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) @ ( prod_rel @ A @ B @ C @ D ) @ R1 ) @ R42 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod_rel_mono
% 0.26/0.47 thf(fact_138_bind__cong,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M2: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F2: A > ( refine1665802226e_nres @ B )] :
% 0.26/0.47 ( ( M = M2 )
% 0.26/0.47 => ( ! [X5: A] :
% 0.26/0.47 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X5 ) @ M2 )
% 0.26/0.47 => ( ( F @ X5 )
% 0.26/0.47 = ( F2 @ X5 ) ) )
% 0.26/0.47 => ( ( refine463715084e_bind @ A @ B @ M @ F )
% 0.26/0.47 = ( refine463715084e_bind @ A @ B @ M2 @ F2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % bind_cong
% 0.26/0.47 thf(fact_139_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,M3: refine1665802226e_nres @ A,M4: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F2: A > ( refine1665802226e_nres @ B )] :
% 0.26/0.47 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ M4 )
% 0.26/0.47 => ( ! [X5: A] :
% 0.26/0.47 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X5 ) @ M3 )
% 0.26/0.47 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X5 ) @ ( F2 @ X5 ) ) )
% 0.26/0.47 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M3 @ F ) @ ( refine463715084e_bind @ A @ B @ M4 @ F2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(1)
% 0.26/0.47 thf(fact_140_ord__eq__le__eq__trans,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( ord @ A )
% 0.26/0.47 => ! [A5: A,B3: A,C2: A,D3: A] :
% 0.26/0.47 ( ( A5 = B3 )
% 0.26/0.47 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.26/0.47 => ( ( C2 = D3 )
% 0.26/0.47 => ( ord_less_eq @ A @ A5 @ D3 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ord_eq_le_eq_trans
% 0.26/0.47 thf(fact_141_pairself_Oinduct,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,P: ( A > B ) > ( product_prod @ A @ A ) > $o,A0: A > B,A12: product_prod @ A @ A] :
% 0.26/0.47 ( ! [F5: A > B,A3: A,B5: A] : ( P @ F5 @ ( product_Pair @ A @ A @ A3 @ B5 ) )
% 0.26/0.47 => ( P @ A0 @ A12 ) ) ).
% 0.26/0.47
% 0.26/0.47 % pairself.induct
% 0.26/0.47 thf(fact_142_pairself_Ocases,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
% 0.26/0.47 ~ ! [F5: A > B,A3: A,B5: A] :
% 0.26/0.47 ( X
% 0.26/0.47 != ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F5 @ ( product_Pair @ A @ A @ A3 @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % pairself.cases
% 0.26/0.47 thf(fact_143_bex2I,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,A5: A,B3: B,S5: set @ ( product_prod @ A @ B ),P: A > B > $o] :
% 0.26/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ S5 )
% 0.26/0.47 => ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ S5 )
% 0.26/0.47 => ( P @ A5 @ B3 ) )
% 0.26/0.47 => ? [A3: A,B5: B] :
% 0.26/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ S5 )
% 0.26/0.47 & ( P @ A3 @ B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % bex2I
% 0.26/0.47 thf(fact_144_autoref__nat_I6_J,axiom,
% 0.26/0.47 member @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) @ ( product_Pair @ ( nat > nat > $o ) @ ( nat > nat > $o ) @ ( ord_less_eq @ nat ) @ ( ord_less_eq @ nat ) ) @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) > ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ ( nat > $o ) @ ( nat > $o ) ) @ ( id2 @ nat ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % autoref_nat(6)
% 0.26/0.47 thf(fact_145_autoref__int_I4_J,axiom,
% 0.26/0.47 member @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) @ ( product_Pair @ ( int > int > $o ) @ ( int > int > $o ) @ ( ord_less @ int ) @ ( ord_less @ int ) ) @ ( relAPP @ ( int > $o ) @ ( int > $o ) @ ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) > ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) ) @ ( fun_rel @ int @ int @ ( int > $o ) @ ( int > $o ) ) @ ( id2 @ int ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) ) @ ( fun_rel @ int @ int @ $o @ $o ) @ ( id2 @ int ) ) @ ( id2 @ $o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % autoref_int(4)
% 0.26/0.47 thf(fact_146_autoref__nat_I5_J,axiom,
% 0.26/0.47 member @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) @ ( product_Pair @ ( nat > nat > $o ) @ ( nat > nat > $o ) @ ( ord_less @ nat ) @ ( ord_less @ nat ) ) @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) > ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ ( nat > $o ) @ ( nat > $o ) ) @ ( id2 @ nat ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % autoref_nat(5)
% 0.26/0.47 thf(fact_147_case__prod__mono,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.47 ( ( ord @ C )
% 0.26/0.47 => ! [P2: product_prod @ A @ B,F: A > B > C,F2: A > B > C] :
% 0.26/0.47 ( ! [A3: A,B5: B] :
% 0.26/0.47 ( ( P2
% 0.26/0.47 = ( product_Pair @ A @ B @ A3 @ B5 ) )
% 0.26/0.47 => ( ord_less_eq @ C @ ( F @ A3 @ B5 ) @ ( F2 @ A3 @ B5 ) ) )
% 0.26/0.47 => ( ord_less_eq @ C @ ( product_case_prod @ A @ B @ C @ F @ P2 ) @ ( product_case_prod @ A @ B @ C @ F2 @ P2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % case_prod_mono
% 0.26/0.47 thf(fact_148_param__apsnd,axiom,
% 0.26/0.47 ! [A: $tType,D: $tType,C: $tType,F4: $tType,E: $tType,B: $tType,Rb: set @ ( product_prod @ A @ D ),Rc: set @ ( product_prod @ B @ E ),Ra: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( A > B ) > ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( D > E ) > ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) @ ( product_Pair @ ( ( A > B ) > ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( D > E ) > ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) @ ( product_apsnd @ A @ B @ C ) @ ( product_apsnd @ D @ E @ F4 ) ) @ ( relAPP @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) @ ( set @ ( product_prod @ ( ( A > B ) > ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( D > E ) > ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) @ ( relAPP @ ( A > B ) @ ( D > E ) @ ( ( set @ ( product_prod @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) > ( set @ ( product_prod @ ( ( A > B ) > ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( D > E ) > ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( D > E ) @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) ) @ ( fun_rel @ A @ D @ B @ E ) @ Rb ) @ Rc ) ) @ ( relAPP @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) @ ( relAPP @ ( product_prod @ C @ A ) @ ( product_prod @ F4 @ D ) @ ( ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) ) ) > ( set @ ( product_prod @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) ) @ ( fun_rel @ ( product_prod @ C @ A ) @ ( product_prod @ F4 @ D ) @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) ) @ ( relAPP @ A @ D @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ F4 @ D ) ) ) @ ( relAPP @ C @ F4 @ ( ( set @ ( product_prod @ A @ D ) ) > ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ F4 @ D ) ) ) ) @ ( prod_rel @ C @ F4 @ A @ D ) @ Ra ) @ Rb ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) ) ) @ ( relAPP @ C @ F4 @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) ) ) ) @ ( prod_rel @ C @ F4 @ B @ E ) @ Ra ) @ Rc ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % param_apsnd
% 0.26/0.47 thf(fact_149_apsnd__conv,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y: C] :
% 0.26/0.47 ( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y ) )
% 0.26/0.47 = ( product_Pair @ A @ B @ X @ ( F @ Y ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % apsnd_conv
% 0.26/0.47 thf(fact_150_apsnd__id,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType] :
% 0.26/0.47 ( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
% 0.26/0.47 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % apsnd_id
% 0.26/0.47 thf(fact_151_relprop__triggers_I6_J,axiom,
% 0.26/0.47 ! [I: $tType,R3: set @ I,R5: set @ I] :
% 0.26/0.47 ( ( ord_less_eq @ ( set @ I ) @ R3 @ R5 )
% 0.26/0.47 => ( ord_less_eq @ ( set @ I ) @ R3 @ R5 ) ) ).
% 0.26/0.47
% 0.26/0.47 % relprop_triggers(6)
% 0.26/0.47 thf(fact_152_Collect__case__prod__mono,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,A2: A > B > $o,B2: A > B > $o] :
% 0.26/0.47 ( ( ord_less_eq @ ( A > B > $o ) @ A2 @ B2 )
% 0.26/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A2 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Collect_case_prod_mono
% 0.26/0.47 thf(fact_153_subset__Collect__conv,axiom,
% 0.26/0.47 ! [A: $tType,S5: set @ A,P: A > $o] :
% 0.26/0.47 ( ( ord_less_eq @ ( set @ A ) @ S5 @ ( collect @ A @ P ) )
% 0.26/0.47 = ( ! [X6: A] :
% 0.26/0.47 ( ( member @ A @ X6 @ S5 )
% 0.26/0.47 => ( P @ X6 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_Collect_conv
% 0.26/0.47 thf(fact_154_exists__leI,axiom,
% 0.26/0.47 ! [N: nat,P: nat > $o] :
% 0.26/0.47 ( ( ! [N2: nat] :
% 0.26/0.47 ( ( ord_less @ nat @ N2 @ N )
% 0.26/0.47 => ~ ( P @ N2 ) )
% 0.26/0.47 => ( P @ N ) )
% 0.26/0.47 => ? [N3: nat] :
% 0.26/0.47 ( ( ord_less_eq @ nat @ N3 @ N )
% 0.26/0.47 & ( P @ N3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % exists_leI
% 0.26/0.47 thf(fact_155_Refine__Misc_Oif__mono,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( ord @ A )
% 0.26/0.47 => ! [B3: $o,M1: A,M12: A,M22: A,M23: A] :
% 0.26/0.47 ( ( B3
% 0.26/0.47 => ( ord_less_eq @ A @ M1 @ M12 ) )
% 0.26/0.47 => ( ( ~ B3
% 0.26/0.47 => ( ord_less_eq @ A @ M22 @ M23 ) )
% 0.26/0.47 => ( ord_less_eq @ A @ ( if @ A @ B3 @ M1 @ M22 ) @ ( if @ A @ B3 @ M12 @ M23 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Refine_Misc.if_mono
% 0.26/0.47 thf(fact_156_param__apfst,axiom,
% 0.26/0.47 ! [D: $tType,A: $tType,B: $tType,E: $tType,F4: $tType,C: $tType,Ra: set @ ( product_prod @ A @ D ),Rb: set @ ( product_prod @ B @ E ),Rc: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( A > B ) > ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( D > E ) > ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) @ ( product_Pair @ ( ( A > B ) > ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( D > E ) > ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) @ ( product_apfst @ A @ B @ C ) @ ( product_apfst @ D @ E @ F4 ) ) @ ( relAPP @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) @ ( set @ ( product_prod @ ( ( A > B ) > ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( D > E ) > ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) @ ( relAPP @ ( A > B ) @ ( D > E ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) > ( set @ ( product_prod @ ( ( A > B ) > ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( D > E ) > ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( D > E ) @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) ) @ ( fun_rel @ A @ D @ B @ E ) @ Ra ) @ Rb ) ) @ ( relAPP @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) @ ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) @ ( relAPP @ ( product_prod @ A @ C ) @ ( product_prod @ D @ F4 ) @ ( ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ C ) @ ( product_prod @ D @ F4 ) @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ D @ F4 ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ D @ F4 ) ) ) ) @ ( prod_rel @ A @ D @ C @ F4 ) @ Ra ) @ Rc ) ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) ) ) @ ( relAPP @ B @ E @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) ) ) ) @ ( prod_rel @ B @ E @ C @ F4 ) @ Rb ) @ Rc ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % param_apfst
% 0.26/0.47 thf(fact_157_param__map__prod,axiom,
% 0.26/0.47 ! [C: $tType,G: $tType,E: $tType,A: $tType,B: $tType,F4: $tType,H2: $tType,D: $tType,Ra: set @ ( product_prod @ A @ E ),Rb: set @ ( product_prod @ B @ F4 ),Rc: set @ ( product_prod @ C @ G ),Rd: set @ ( product_prod @ D @ H2 )] : ( member @ ( product_prod @ ( ( A > B ) > ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( E > F4 ) > ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) @ ( product_Pair @ ( ( A > B ) > ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( E > F4 ) > ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) @ ( product_map_prod @ A @ B @ C @ D ) @ ( product_map_prod @ E @ F4 @ G @ H2 ) ) @ ( relAPP @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) @ ( set @ ( product_prod @ ( ( A > B ) > ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( E > F4 ) > ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) @ ( relAPP @ ( A > B ) @ ( E > F4 ) @ ( ( set @ ( product_prod @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) > ( set @ ( product_prod @ ( ( A > B ) > ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( E > F4 ) > ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( E > F4 ) @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) @ ( relAPP @ B @ F4 @ ( set @ ( product_prod @ ( A > B ) @ ( E > F4 ) ) ) @ ( relAPP @ A @ E @ ( ( set @ ( product_prod @ B @ F4 ) ) > ( set @ ( product_prod @ ( A > B ) @ ( E > F4 ) ) ) ) @ ( fun_rel @ A @ E @ B @ F4 ) @ Ra ) @ Rb ) ) @ ( relAPP @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) @ ( set @ ( product_prod @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) @ ( relAPP @ ( C > D ) @ ( G > H2 ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) > ( set @ ( product_prod @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) ) @ ( fun_rel @ ( C > D ) @ ( G > H2 ) @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) @ ( relAPP @ D @ H2 @ ( set @ ( product_prod @ ( C > D ) @ ( G > H2 ) ) ) @ ( relAPP @ C @ G @ ( ( set @ ( product_prod @ D @ H2 ) ) > ( set @ ( product_prod @ ( C > D ) @ ( G > H2 ) ) ) ) @ ( fun_rel @ C @ G @ D @ H2 ) @ Rc ) @ Rd ) ) @ ( relAPP @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) @ ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) @ ( relAPP @ ( product_prod @ A @ C ) @ ( product_prod @ E @ G ) @ ( ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ C ) @ ( product_prod @ E @ G ) @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) ) @ ( relAPP @ C @ G @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ E @ G ) ) ) @ ( relAPP @ A @ E @ ( ( set @ ( product_prod @ C @ G ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ E @ G ) ) ) ) @ ( prod_rel @ A @ E @ C @ G ) @ Ra ) @ Rc ) ) @ ( relAPP @ D @ H2 @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) ) ) @ ( relAPP @ B @ F4 @ ( ( set @ ( product_prod @ D @ H2 ) ) > ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) ) ) ) @ ( prod_rel @ B @ F4 @ D @ H2 ) @ Rb ) @ Rd ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % param_map_prod
% 0.26/0.47 thf(fact_158_param__prod__swap,axiom,
% 0.26/0.47 ! [B: $tType,D: $tType,C: $tType,A: $tType,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) @ ( ( product_prod @ C @ D ) > ( product_prod @ D @ C ) ) ) @ ( product_Pair @ ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) @ ( ( product_prod @ C @ D ) > ( product_prod @ D @ C ) ) @ ( product_swap @ A @ B ) @ ( product_swap @ C @ D ) ) @ ( relAPP @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) @ ( ( product_prod @ C @ D ) > ( product_prod @ D @ C ) ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) @ ( ( product_prod @ C @ D ) > ( product_prod @ D @ C ) ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) ) @ ( relAPP @ A @ C @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) ) ) @ ( relAPP @ B @ D @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) ) ) ) @ ( prod_rel @ B @ D @ A @ C ) @ B2 ) @ A2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % param_prod_swap
% 0.26/0.47 thf(fact_159_swap__swap,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.26/0.47 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 0.26/0.47 = P2 ) ).
% 0.26/0.47
% 0.26/0.47 % swap_swap
% 0.26/0.47 thf(fact_160_map__prod__simp,axiom,
% 0.26/0.47 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G3: D > B,A5: C,B3: D] :
% 0.26/0.47 ( ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ ( product_Pair @ C @ D @ A5 @ B3 ) )
% 0.26/0.47 = ( product_Pair @ A @ B @ ( F @ A5 ) @ ( G3 @ B3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % map_prod_simp
% 0.26/0.47 thf(fact_161_apfst__conv,axiom,
% 0.26/0.47 ! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y: B] :
% 0.26/0.47 ( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y ) )
% 0.26/0.47 = ( product_Pair @ A @ B @ ( F @ X ) @ Y ) ) ).
% 0.26/0.47
% 0.26/0.47 % apfst_conv
% 0.26/0.47 thf(fact_162_swap__simp,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,X: B,Y: A] :
% 0.26/0.47 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.26/0.47 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 0.26/0.47
% 0.26/0.47 % swap_simp
% 0.26/0.47 thf(fact_163_apfst__id,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType] :
% 0.26/0.47 ( ( product_apfst @ A @ A @ B @ ( id @ A ) )
% 0.26/0.47 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % apfst_id
% 0.26/0.47 thf(fact_164_apfst__def,axiom,
% 0.26/0.47 ! [B: $tType,C: $tType,A: $tType] :
% 0.26/0.47 ( ( product_apfst @ A @ C @ B )
% 0.26/0.47 = ( ^ [F3: A > C] : ( product_map_prod @ A @ C @ B @ B @ F3 @ ( id @ B ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % apfst_def
% 0.26/0.47 thf(fact_165_apsnd__apfst__commute,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G3: D > A,P2: product_prod @ D @ C] :
% 0.26/0.47 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G3 @ P2 ) )
% 0.26/0.47 = ( product_apfst @ D @ A @ B @ G3 @ ( product_apsnd @ C @ B @ D @ F @ P2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % apsnd_apfst_commute
% 0.26/0.47 thf(fact_166_apsnd__def,axiom,
% 0.26/0.47 ! [B: $tType,C: $tType,A: $tType] :
% 0.26/0.47 ( ( product_apsnd @ B @ C @ A )
% 0.26/0.47 = ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % apsnd_def
% 0.26/0.47 thf(fact_167_prod_Omap__id0,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType] :
% 0.26/0.47 ( ( product_map_prod @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) )
% 0.26/0.47 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod.map_id0
% 0.26/0.47 thf(fact_168_autoref__comp,axiom,
% 0.26/0.47 ! [A: $tType,D: $tType,C: $tType,F4: $tType,E: $tType,B: $tType,Ra: set @ ( product_prod @ A @ D ),Rb: set @ ( product_prod @ B @ E ),Rc: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( A > B ) > ( C > A ) > C > B ) @ ( ( D > E ) > ( F4 > D ) > F4 > E ) ) @ ( product_Pair @ ( ( A > B ) > ( C > A ) > C > B ) @ ( ( D > E ) > ( F4 > D ) > F4 > E ) @ ( comp @ A @ B @ C ) @ ( comp @ D @ E @ F4 ) ) @ ( relAPP @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) @ ( set @ ( product_prod @ ( ( A > B ) > ( C > A ) > C > B ) @ ( ( D > E ) > ( F4 > D ) > F4 > E ) ) ) @ ( relAPP @ ( A > B ) @ ( D > E ) @ ( ( set @ ( product_prod @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) ) ) > ( set @ ( product_prod @ ( ( A > B ) > ( C > A ) > C > B ) @ ( ( D > E ) > ( F4 > D ) > F4 > E ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( D > E ) @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) ) @ ( fun_rel @ A @ D @ B @ E ) @ Ra ) @ Rb ) ) @ ( relAPP @ ( C > B ) @ ( F4 > E ) @ ( set @ ( product_prod @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) ) ) @ ( relAPP @ ( C > A ) @ ( F4 > D ) @ ( ( set @ ( product_prod @ ( C > B ) @ ( F4 > E ) ) ) > ( set @ ( product_prod @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) ) ) ) @ ( fun_rel @ ( C > A ) @ ( F4 > D ) @ ( C > B ) @ ( F4 > E ) ) @ ( relAPP @ A @ D @ ( set @ ( product_prod @ ( C > A ) @ ( F4 > D ) ) ) @ ( relAPP @ C @ F4 @ ( ( set @ ( product_prod @ A @ D ) ) > ( set @ ( product_prod @ ( C > A ) @ ( F4 > D ) ) ) ) @ ( fun_rel @ C @ F4 @ A @ D ) @ Rc ) @ Ra ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( C > B ) @ ( F4 > E ) ) ) @ ( relAPP @ C @ F4 @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( C > B ) @ ( F4 > E ) ) ) ) @ ( fun_rel @ C @ F4 @ B @ E ) @ Rc ) @ Rb ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % autoref_comp
% 0.26/0.47 thf(fact_169_less__than__bool__iff,axiom,
% 0.26/0.47 ! [X: $o,Y: $o] :
% 0.26/0.47 ( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ X @ Y ) @ refine1361002633n_bool )
% 0.26/0.47 = ( ~ X
% 0.26/0.47 & Y ) ) ).
% 0.26/0.47
% 0.26/0.47 % less_than_bool_iff
% 0.26/0.47 thf(fact_170_comp__apply,axiom,
% 0.26/0.47 ! [C: $tType,A: $tType,B: $tType] :
% 0.26/0.47 ( ( comp @ B @ A @ C )
% 0.26/0.47 = ( ^ [F3: B > A,G4: C > B,X6: C] : ( F3 @ ( G4 @ X6 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % comp_apply
% 0.26/0.47 thf(fact_171_comp__id,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,F: A > B] :
% 0.26/0.47 ( ( comp @ A @ B @ A @ F @ ( id @ A ) )
% 0.26/0.47 = F ) ).
% 0.26/0.47
% 0.26/0.47 % comp_id
% 0.26/0.47 thf(fact_172_id__comp,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,G3: A > B] :
% 0.26/0.47 ( ( comp @ B @ B @ A @ ( id @ B ) @ G3 )
% 0.26/0.47 = G3 ) ).
% 0.26/0.47
% 0.26/0.47 % id_comp
% 0.26/0.47 thf(fact_173_fun_Omap__id,axiom,
% 0.26/0.47 ! [A: $tType,D: $tType,T2: D > A] :
% 0.26/0.47 ( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
% 0.26/0.47 = T2 ) ).
% 0.26/0.47
% 0.26/0.47 % fun.map_id
% 0.26/0.47 thf(fact_174_swap__comp__swap,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType] :
% 0.26/0.47 ( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
% 0.26/0.47 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % swap_comp_swap
% 0.26/0.47 thf(fact_175_map__prod_Ocompositionality,axiom,
% 0.26/0.47 ! [D: $tType,F4: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G3: D > F4,H: A > C,I2: B > D,Prod: product_prod @ A @ B] :
% 0.26/0.47 ( ( product_map_prod @ C @ E @ D @ F4 @ F @ G3 @ ( product_map_prod @ A @ C @ B @ D @ H @ I2 @ Prod ) )
% 0.26/0.47 = ( product_map_prod @ A @ E @ B @ F4 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F4 @ B @ G3 @ I2 ) @ Prod ) ) ).
% 0.26/0.47
% 0.26/0.47 % map_prod.compositionality
% 0.26/0.47 thf(fact_176_map__prod__compose,axiom,
% 0.26/0.47 ! [D: $tType,C: $tType,A: $tType,E: $tType,F4: $tType,B: $tType,F1: E > C,F22: A > E,G1: F4 > D,G22: B > F4] :
% 0.26/0.47 ( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F4 @ D @ B @ G1 @ G22 ) )
% 0.26/0.47 = ( comp @ ( product_prod @ E @ F4 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F4 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F4 @ F22 @ G22 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % map_prod_compose
% 0.26/0.47 thf(fact_177_map__prod_Ocomp,axiom,
% 0.26/0.47 ! [A: $tType,C: $tType,E: $tType,F4: $tType,D: $tType,B: $tType,F: C > E,G3: D > F4,H: A > C,I2: B > D] :
% 0.26/0.47 ( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F4 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F4 @ F @ G3 ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I2 ) )
% 0.26/0.47 = ( product_map_prod @ A @ E @ B @ F4 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F4 @ B @ G3 @ I2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % map_prod.comp
% 0.26/0.47 thf(fact_178_apfst__compose,axiom,
% 0.26/0.47 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G3: D > C,X: product_prod @ D @ B] :
% 0.26/0.47 ( ( product_apfst @ C @ A @ B @ F @ ( product_apfst @ D @ C @ B @ G3 @ X ) )
% 0.26/0.47 = ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F @ G3 ) @ X ) ) ).
% 0.26/0.47
% 0.26/0.47 % apfst_compose
% 0.26/0.47 thf(fact_179_comp__eq__id__dest,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,A5: C > B,B3: A > C,C2: A > B,V: A] :
% 0.26/0.47 ( ( ( comp @ C @ B @ A @ A5 @ B3 )
% 0.26/0.47 = ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
% 0.26/0.47 => ( ( A5 @ ( B3 @ V ) )
% 0.26/0.47 = ( C2 @ V ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % comp_eq_id_dest
% 0.26/0.47 thf(fact_180_fun_Omap__id0,axiom,
% 0.26/0.47 ! [A: $tType,D: $tType] :
% 0.26/0.47 ( ( comp @ A @ A @ D @ ( id @ A ) )
% 0.26/0.47 = ( id @ ( D > A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % fun.map_id0
% 0.26/0.47 thf(fact_181_comp__cong__left,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,C: $tType,X: A > B,Y: A > B,F: C > A] :
% 0.26/0.47 ( ( X = Y )
% 0.26/0.47 => ( ( comp @ A @ B @ C @ X @ F )
% 0.26/0.47 = ( comp @ A @ B @ C @ Y @ F ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % comp_cong_left
% 0.26/0.47 thf(fact_182_comp__cong__right,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,X: A > B,Y: A > B,F: B > C] :
% 0.26/0.47 ( ( X = Y )
% 0.26/0.47 => ( ( comp @ B @ C @ A @ F @ X )
% 0.26/0.47 = ( comp @ B @ C @ A @ F @ Y ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % comp_cong_right
% 0.26/0.47 thf(fact_183_fun__comp__eq__conv,axiom,
% 0.26/0.47 ! [B: $tType,C: $tType,A: $tType,F: C > B,G3: A > C,Fg: A > B] :
% 0.26/0.47 ( ( ( comp @ C @ B @ A @ F @ G3 )
% 0.26/0.47 = Fg )
% 0.26/0.47 = ( ! [X6: A] :
% 0.26/0.47 ( ( F @ ( G3 @ X6 ) )
% 0.26/0.47 = ( Fg @ X6 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % fun_comp_eq_conv
% 0.26/0.47 thf(fact_184_comp__def,axiom,
% 0.26/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.26/0.47 ( ( comp @ B @ C @ A )
% 0.26/0.47 = ( ^ [F3: B > C,G4: A > B,X6: A] : ( F3 @ ( G4 @ X6 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % comp_def
% 0.26/0.47 thf(fact_185_comp__assoc,axiom,
% 0.26/0.47 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G3: C > D,H: A > C] :
% 0.26/0.47 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G3 ) @ H )
% 0.26/0.47 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G3 @ H ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % comp_assoc
% 0.26/0.47 thf(fact_186_comp__eq__dest,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,D: $tType,A: $tType,A5: C > B,B3: A > C,C2: D > B,D3: A > D,V: A] :
% 0.26/0.47 ( ( ( comp @ C @ B @ A @ A5 @ B3 )
% 0.26/0.47 = ( comp @ D @ B @ A @ C2 @ D3 ) )
% 0.26/0.47 => ( ( A5 @ ( B3 @ V ) )
% 0.26/0.47 = ( C2 @ ( D3 @ V ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % comp_eq_dest
% 0.26/0.47 thf(fact_187_comp__eq__elim,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,D: $tType,A: $tType,A5: C > B,B3: A > C,C2: D > B,D3: A > D] :
% 0.26/0.47 ( ( ( comp @ C @ B @ A @ A5 @ B3 )
% 0.26/0.47 = ( comp @ D @ B @ A @ C2 @ D3 ) )
% 0.26/0.47 => ! [V2: A] :
% 0.26/0.47 ( ( A5 @ ( B3 @ V2 ) )
% 0.26/0.47 = ( C2 @ ( D3 @ V2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % comp_eq_elim
% 0.26/0.47 thf(fact_188_comp__eq__dest__lhs,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,A5: C > B,B3: A > C,C2: A > B,V: A] :
% 0.26/0.47 ( ( ( comp @ C @ B @ A @ A5 @ B3 )
% 0.26/0.47 = C2 )
% 0.26/0.47 => ( ( A5 @ ( B3 @ V ) )
% 0.26/0.47 = ( C2 @ V ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % comp_eq_dest_lhs
% 0.26/0.47 thf(fact_189_fun__upd__comp,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,C: $tType,F: C > B,G3: A > C,X: A,Y: C] :
% 0.26/0.47 ( ( comp @ C @ B @ A @ F @ ( fun_upd @ A @ C @ G3 @ X @ Y ) )
% 0.26/0.47 = ( fun_upd @ A @ B @ ( comp @ C @ B @ A @ F @ G3 ) @ X @ ( F @ Y ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % fun_upd_comp
% 0.26/0.47 thf(fact_190_apsnd__compose,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G3: D > C,X: product_prod @ A @ D] :
% 0.26/0.47 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G3 @ X ) )
% 0.26/0.47 = ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G3 ) @ X ) ) ).
% 0.26/0.47
% 0.26/0.47 % apsnd_compose
% 0.26/0.47 thf(fact_191_prod_Omap__id,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.26/0.47 ( ( product_map_prod @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) @ T2 )
% 0.26/0.47 = T2 ) ).
% 0.26/0.47
% 0.26/0.47 % prod.map_id
% 0.26/0.47 thf(fact_192_predicate2I,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 0.26/0.47 ( ! [X5: A,Y3: B] :
% 0.26/0.47 ( ( P @ X5 @ Y3 )
% 0.26/0.47 => ( Q @ X5 @ Y3 ) )
% 0.26/0.47 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 0.26/0.47
% 0.26/0.47 % predicate2I
% 0.26/0.47 thf(fact_193_prod__refine_I4_J,axiom,
% 0.26/0.47 ! [B: $tType,D: $tType,C: $tType,A: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( ( product_prod @ A @ B ) > A ) @ ( ( product_prod @ C @ D ) > C ) ) @ ( product_Pair @ ( ( product_prod @ A @ B ) > A ) @ ( ( product_prod @ C @ D ) > C ) @ ( product_fst @ A @ B ) @ ( product_fst @ C @ D ) ) @ ( relAPP @ A @ C @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > A ) @ ( ( product_prod @ C @ D ) > C ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > A ) @ ( ( product_prod @ C @ D ) > C ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ A @ C ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) @ Ra ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod_refine(4)
% 0.26/0.47 thf(fact_194_prod__refine_I5_J,axiom,
% 0.26/0.47 ! [C: $tType,A: $tType,D: $tType,B: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( ( product_prod @ A @ B ) > B ) @ ( ( product_prod @ C @ D ) > D ) ) @ ( product_Pair @ ( ( product_prod @ A @ B ) > B ) @ ( ( product_prod @ C @ D ) > D ) @ ( product_snd @ A @ B ) @ ( product_snd @ C @ D ) ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > B ) @ ( ( product_prod @ C @ D ) > D ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > B ) @ ( ( product_prod @ C @ D ) > D ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ B @ D ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) @ Rb ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod_refine(5)
% 0.26/0.47 thf(fact_195_fst__map__prod,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G3: D > B,X: product_prod @ C @ D] :
% 0.26/0.47 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ X ) )
% 0.26/0.47 = ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % fst_map_prod
% 0.26/0.47 thf(fact_196_snd__map__prod,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G3: D > A,X: product_prod @ C @ D] :
% 0.26/0.47 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G3 @ X ) )
% 0.26/0.47 = ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % snd_map_prod
% 0.26/0.47 thf(fact_197_fst__apfst,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
% 0.26/0.47 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
% 0.26/0.47 = ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % fst_apfst
% 0.26/0.47 thf(fact_198_apfst__eq__conv,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G3: C > A] :
% 0.26/0.47 ( ( ( product_apfst @ C @ A @ B @ F @ X )
% 0.26/0.47 = ( product_apfst @ C @ A @ B @ G3 @ X ) )
% 0.26/0.47 = ( ( F @ ( product_fst @ C @ B @ X ) )
% 0.26/0.47 = ( G3 @ ( product_fst @ C @ B @ X ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % apfst_eq_conv
% 0.26/0.47 thf(fact_199_fst__apsnd,axiom,
% 0.26/0.47 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
% 0.26/0.47 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
% 0.26/0.47 = ( product_fst @ A @ C @ X ) ) ).
% 0.26/0.47
% 0.26/0.47 % fst_apsnd
% 0.26/0.47 thf(fact_200_snd__apfst,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
% 0.26/0.47 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
% 0.26/0.47 = ( product_snd @ C @ A @ X ) ) ).
% 0.26/0.47
% 0.26/0.47 % snd_apfst
% 0.26/0.47 thf(fact_201_snd__apsnd,axiom,
% 0.26/0.47 ! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
% 0.26/0.47 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
% 0.26/0.47 = ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % snd_apsnd
% 0.26/0.47 thf(fact_202_apsnd__eq__conv,axiom,
% 0.26/0.47 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G3: C > B] :
% 0.26/0.47 ( ( ( product_apsnd @ C @ B @ A @ F @ X )
% 0.26/0.47 = ( product_apsnd @ C @ B @ A @ G3 @ X ) )
% 0.26/0.47 = ( ( F @ ( product_snd @ A @ C @ X ) )
% 0.26/0.47 = ( G3 @ ( product_snd @ A @ C @ X ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % apsnd_eq_conv
% 0.26/0.47 thf(fact_203_prod_Ocollapse,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.26/0.47 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.26/0.47 = Prod ) ).
% 0.26/0.47
% 0.26/0.47 % prod.collapse
% 0.26/0.47 thf(fact_204_fst__swap,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 0.26/0.47 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 0.26/0.47 = ( product_snd @ B @ A @ X ) ) ).
% 0.26/0.47
% 0.26/0.47 % fst_swap
% 0.26/0.47 thf(fact_205_snd__swap,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 0.26/0.47 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 0.26/0.47 = ( product_fst @ A @ B @ X ) ) ).
% 0.26/0.47
% 0.26/0.47 % snd_swap
% 0.26/0.47 thf(fact_206_fst__comp__apsnd,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,F: B > C] :
% 0.26/0.47 ( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
% 0.26/0.47 = ( product_fst @ A @ B ) ) ).
% 0.26/0.47
% 0.26/0.47 % fst_comp_apsnd
% 0.26/0.47 thf(fact_207_snd__comp__apfst,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,F: A > C] :
% 0.26/0.47 ( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
% 0.26/0.47 = ( product_snd @ A @ B ) ) ).
% 0.26/0.47
% 0.26/0.47 % snd_comp_apfst
% 0.26/0.47 thf(fact_208_fst__comp__map__prod,axiom,
% 0.26/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > C,G3: B > D] :
% 0.26/0.47 ( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G3 ) )
% 0.26/0.47 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % fst_comp_map_prod
% 0.26/0.47 thf(fact_209_snd__comp__map__prod,axiom,
% 0.26/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > D,G3: B > C] :
% 0.26/0.47 ( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F @ G3 ) )
% 0.26/0.47 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ G3 @ ( product_snd @ A @ B ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % snd_comp_map_prod
% 0.26/0.47 thf(fact_210_fst__comp__apfst,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,F: A > C] :
% 0.26/0.47 ( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
% 0.26/0.47 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % fst_comp_apfst
% 0.26/0.47 thf(fact_211_snd__comp__apsnd,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,F: B > C] :
% 0.26/0.47 ( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
% 0.26/0.47 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % snd_comp_apsnd
% 0.26/0.47 thf(fact_212_apfst__apsnd,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G3: D > B,X: product_prod @ C @ D] :
% 0.26/0.47 ( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G3 @ X ) )
% 0.26/0.47 = ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G3 @ ( product_snd @ C @ D @ X ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % apfst_apsnd
% 0.26/0.47 thf(fact_213_apsnd__apfst,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G3: D > A,X: product_prod @ D @ C] :
% 0.26/0.47 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G3 @ X ) )
% 0.26/0.47 = ( product_Pair @ A @ B @ ( G3 @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % apsnd_apfst
% 0.26/0.47 thf(fact_214_prod_Oswap__def,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType] :
% 0.26/0.47 ( ( product_swap @ A @ B )
% 0.26/0.47 = ( ^ [P4: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P4 ) @ ( product_fst @ A @ B @ P4 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod.swap_def
% 0.26/0.47 thf(fact_215_prod_Osplit__sel,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.26/0.47 ( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.26/0.47 = ( ( Prod
% 0.26/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.26/0.47 => ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod.split_sel
% 0.26/0.47 thf(fact_216_prod_Oexhaust__sel,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.26/0.47 ( Prod
% 0.26/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod.exhaust_sel
% 0.26/0.47 thf(fact_217_prod_Osplit__sel__asm,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.26/0.47 ( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.26/0.47 = ( ~ ( ( Prod
% 0.26/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.26/0.47 & ~ ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod.split_sel_asm
% 0.26/0.47 thf(fact_218_surjective__pairing,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.26/0.47 ( T2
% 0.26/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % surjective_pairing
% 0.26/0.47 thf(fact_219_sndE,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,X: product_prod @ A @ B,A5: A,B3: B,P: B > $o] :
% 0.26/0.47 ( ( X
% 0.26/0.47 = ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.26/0.47 => ( ( P @ ( product_snd @ A @ B @ X ) )
% 0.26/0.47 => ( P @ B3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sndE
% 0.26/0.47 thf(fact_220_fstE,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A5: A,B3: B,P: A > $o] :
% 0.26/0.47 ( ( X
% 0.26/0.47 = ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.26/0.47 => ( ( P @ ( product_fst @ A @ B @ X ) )
% 0.26/0.47 => ( P @ A5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % fstE
% 0.26/0.47 thf(fact_221_snd__eqD,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,X: B,Y: A,A5: A] :
% 0.26/0.47 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.26/0.47 = A5 )
% 0.26/0.47 => ( Y = A5 ) ) ).
% 0.26/0.47
% 0.26/0.47 % snd_eqD
% 0.26/0.47 thf(fact_222_snd__conv,axiom,
% 0.26/0.47 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 0.26/0.47 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 0.26/0.47 = X22 ) ).
% 0.26/0.47
% 0.26/0.47 % snd_conv
% 0.26/0.47 thf(fact_223_fst__eqD,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,X: A,Y: B,A5: A] :
% 0.26/0.47 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 0.26/0.47 = A5 )
% 0.26/0.47 => ( X = A5 ) ) ).
% 0.26/0.47
% 0.26/0.47 % fst_eqD
% 0.26/0.47 thf(fact_224_fst__conv,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 0.26/0.47 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.26/0.47 = X1 ) ).
% 0.26/0.47
% 0.26/0.47 % fst_conv
% 0.26/0.47 thf(fact_225_prod__eq__iff,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType] :
% 0.26/0.47 ( ( ^ [Y5: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y5 = Z ) )
% 0.26/0.47 = ( ^ [S: product_prod @ A @ B,T4: product_prod @ A @ B] :
% 0.26/0.47 ( ( ( product_fst @ A @ B @ S )
% 0.26/0.47 = ( product_fst @ A @ B @ T4 ) )
% 0.26/0.47 & ( ( product_snd @ A @ B @ S )
% 0.26/0.47 = ( product_snd @ A @ B @ T4 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod_eq_iff
% 0.26/0.47 thf(fact_226_prod_Oexpand,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.26/0.47 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.26/0.47 = ( product_fst @ A @ B @ Prod2 ) )
% 0.26/0.47 & ( ( product_snd @ A @ B @ Prod )
% 0.26/0.47 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.26/0.47 => ( Prod = Prod2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod.expand
% 0.26/0.47 thf(fact_227_prod__eqI,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.26/0.47 ( ( ( product_fst @ A @ B @ P2 )
% 0.26/0.47 = ( product_fst @ A @ B @ Q2 ) )
% 0.26/0.47 => ( ( ( product_snd @ A @ B @ P2 )
% 0.26/0.47 = ( product_snd @ A @ B @ Q2 ) )
% 0.26/0.47 => ( P2 = Q2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod_eqI
% 0.26/0.47 thf(fact_228_All__prod__contract,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,P: A > B > $o] :
% 0.26/0.47 ( ( ! [A7: A,X7: B] : ( P @ A7 @ X7 ) )
% 0.26/0.47 = ( ! [Z3: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z3 ) @ ( product_snd @ A @ B @ Z3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % All_prod_contract
% 0.26/0.47 thf(fact_229_Ex__prod__contract,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,P: A > B > $o] :
% 0.26/0.47 ( ( ? [A7: A,X7: B] : ( P @ A7 @ X7 ) )
% 0.26/0.47 = ( ? [Z3: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z3 ) @ ( product_snd @ A @ B @ Z3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Ex_prod_contract
% 0.26/0.47 thf(fact_230_case__prod__beta,axiom,
% 0.26/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.26/0.47 ( ( product_case_prod @ B @ C @ A )
% 0.26/0.47 = ( ^ [F3: B > C > A,P4: product_prod @ B @ C] : ( F3 @ ( product_fst @ B @ C @ P4 ) @ ( product_snd @ B @ C @ P4 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % case_prod_beta
% 0.26/0.47 thf(fact_231_prod_Ocase__eq__if,axiom,
% 0.26/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.47 ( ( product_case_prod @ A @ B @ C )
% 0.26/0.47 = ( ^ [F3: A > B > C,Prod3: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod.case_eq_if
% 0.26/0.47 thf(fact_232_Product__Type_OCollect__case__prodD,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A2: A > B > $o] :
% 0.26/0.47 ( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A2 ) ) )
% 0.26/0.47 => ( A2 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Product_Type.Collect_case_prodD
% 0.26/0.47 thf(fact_233_order__subst1,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType] :
% 0.26/0.47 ( ( ( order @ B )
% 0.26/0.47 & ( order @ A ) )
% 0.26/0.47 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 0.26/0.47 ( ( ord_less_eq @ A @ A5 @ ( F @ B3 ) )
% 0.26/0.47 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.26/0.47 => ( ! [X5: B,Y3: B] :
% 0.26/0.47 ( ( ord_less_eq @ B @ X5 @ Y3 )
% 0.26/0.47 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.26/0.47 => ( ord_less_eq @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % order_subst1
% 0.26/0.47 thf(fact_234_order__subst2,axiom,
% 0.26/0.47 ! [A: $tType,C: $tType] :
% 0.26/0.47 ( ( ( order @ C )
% 0.26/0.47 & ( order @ A ) )
% 0.26/0.47 => ! [A5: A,B3: A,F: A > C,C2: C] :
% 0.26/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.26/0.47 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 0.26/0.47 => ( ! [X5: A,Y3: A] :
% 0.26/0.47 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 0.26/0.47 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.26/0.47 => ( ord_less_eq @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % order_subst2
% 0.26/0.47 thf(fact_235_ord__eq__le__subst,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType] :
% 0.26/0.47 ( ( ( ord @ B )
% 0.26/0.47 & ( ord @ A ) )
% 0.26/0.47 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 0.26/0.47 ( ( A5
% 0.26/0.47 = ( F @ B3 ) )
% 0.26/0.47 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.26/0.47 => ( ! [X5: B,Y3: B] :
% 0.26/0.47 ( ( ord_less_eq @ B @ X5 @ Y3 )
% 0.26/0.47 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.26/0.47 => ( ord_less_eq @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ord_eq_le_subst
% 0.26/0.47 thf(fact_236_ord__le__eq__subst,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType] :
% 0.26/0.47 ( ( ( ord @ B )
% 0.26/0.47 & ( ord @ A ) )
% 0.26/0.47 => ! [A5: A,B3: A,F: A > B,C2: B] :
% 0.26/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.26/0.47 => ( ( ( F @ B3 )
% 0.26/0.47 = C2 )
% 0.26/0.47 => ( ! [X5: A,Y3: A] :
% 0.26/0.47 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 0.26/0.47 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.26/0.47 => ( ord_less_eq @ B @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ord_le_eq_subst
% 0.26/0.47 thf(fact_237_eq__iff,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( order @ A )
% 0.26/0.47 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.26/0.47 = ( ^ [X6: A,Y6: A] :
% 0.26/0.47 ( ( ord_less_eq @ A @ X6 @ Y6 )
% 0.26/0.47 & ( ord_less_eq @ A @ Y6 @ X6 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % eq_iff
% 0.26/0.47 thf(fact_238_antisym,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( order @ A )
% 0.26/0.47 => ! [X: A,Y: A] :
% 0.26/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.26/0.47 => ( ( ord_less_eq @ A @ Y @ X )
% 0.26/0.47 => ( X = Y ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % antisym
% 0.26/0.47 thf(fact_239_linear,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( linorder @ A )
% 0.26/0.47 => ! [X: A,Y: A] :
% 0.26/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.26/0.47 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % linear
% 0.26/0.47 thf(fact_240_eq__refl,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( preorder @ A )
% 0.26/0.47 => ! [X: A,Y: A] :
% 0.26/0.47 ( ( X = Y )
% 0.26/0.47 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % eq_refl
% 0.26/0.47 thf(fact_241_le__cases,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( linorder @ A )
% 0.26/0.47 => ! [X: A,Y: A] :
% 0.26/0.47 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.26/0.47 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % le_cases
% 0.26/0.47 thf(fact_242_order_Otrans,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( order @ A )
% 0.26/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.26/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.26/0.47 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.26/0.47 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % order.trans
% 0.26/0.47 thf(fact_243_le__cases3,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( linorder @ A )
% 0.26/0.47 => ! [X: A,Y: A,Z2: A] :
% 0.26/0.47 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.26/0.47 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 0.26/0.47 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.26/0.47 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.26/0.47 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.26/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 0.26/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 0.26/0.47 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.26/0.47 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.26/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.26/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.26/0.47 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % le_cases3
% 0.26/0.47 thf(fact_244_antisym__conv,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( order @ A )
% 0.26/0.47 => ! [Y: A,X: A] :
% 0.26/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.26/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.26/0.47 = ( X = Y ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % antisym_conv
% 0.26/0.47 thf(fact_245_order__class_Oorder_Oeq__iff,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( order @ A )
% 0.26/0.47 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.26/0.47 = ( ^ [A7: A,B6: A] :
% 0.26/0.47 ( ( ord_less_eq @ A @ A7 @ B6 )
% 0.26/0.47 & ( ord_less_eq @ A @ B6 @ A7 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % order_class.order.eq_iff
% 0.26/0.47 thf(fact_246_ord__eq__le__trans,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( ord @ A )
% 0.26/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.26/0.47 ( ( A5 = B3 )
% 0.26/0.47 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.26/0.47 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ord_eq_le_trans
% 0.26/0.47 thf(fact_247_ord__le__eq__trans,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( ord @ A )
% 0.26/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.26/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.26/0.47 => ( ( B3 = C2 )
% 0.26/0.47 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ord_le_eq_trans
% 0.26/0.47 thf(fact_248_order__class_Oorder_Oantisym,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( order @ A )
% 0.26/0.47 => ! [A5: A,B3: A] :
% 0.26/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.26/0.47 => ( ( ord_less_eq @ A @ B3 @ A5 )
% 0.26/0.47 => ( A5 = B3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % order_class.order.antisym
% 0.26/0.47 thf(fact_249_order__trans,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( preorder @ A )
% 0.26/0.47 => ! [X: A,Y: A,Z2: A] :
% 0.26/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.26/0.47 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.26/0.47 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % order_trans
% 0.26/0.47 thf(fact_250_dual__order_Orefl,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( order @ A )
% 0.26/0.47 => ! [A5: A] : ( ord_less_eq @ A @ A5 @ A5 ) ) ).
% 0.26/0.47
% 0.26/0.47 % dual_order.refl
% 0.26/0.47 thf(fact_251_linorder__wlog,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( linorder @ A )
% 0.26/0.47 => ! [P: A > A > $o,A5: A,B3: A] :
% 0.26/0.47 ( ! [A3: A,B5: A] :
% 0.26/0.47 ( ( ord_less_eq @ A @ A3 @ B5 )
% 0.26/0.47 => ( P @ A3 @ B5 ) )
% 0.26/0.47 => ( ! [A3: A,B5: A] :
% 0.26/0.47 ( ( P @ B5 @ A3 )
% 0.26/0.47 => ( P @ A3 @ B5 ) )
% 0.26/0.47 => ( P @ A5 @ B3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % linorder_wlog
% 0.26/0.47
% 0.26/0.47 % Type constructors (21)
% 0.26/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.26/0.47 ! [A8: $tType,A9: $tType] :
% 0.26/0.47 ( ( preorder @ A9 )
% 0.26/0.47 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.26/0.47 ! [A8: $tType,A9: $tType] :
% 0.26/0.47 ( ( order @ A9 )
% 0.26/0.47 => ( order @ ( A8 > A9 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.26/0.47 ! [A8: $tType,A9: $tType] :
% 0.26/0.47 ( ( ord @ A9 )
% 0.26/0.47 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 0.26/0.47 preorder @ int ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.26/0.47 linorder @ int ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 0.26/0.47 order @ int ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 0.26/0.47 ord @ int ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
% 0.26/0.47 preorder @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder_5,axiom,
% 0.26/0.47 linorder @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_6,axiom,
% 0.26/0.47 order @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Orderings_Oord_7,axiom,
% 0.26/0.47 ord @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_8,axiom,
% 0.26/0.47 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 0.26/0.47 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
% 0.26/0.47 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_11,axiom,
% 0.26/0.47 preorder @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_12,axiom,
% 0.26/0.47 linorder @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
% 0.26/0.47 order @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
% 0.26/0.47 ord @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_15,axiom,
% 0.26/0.47 ! [A8: $tType] : ( preorder @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_16,axiom,
% 0.26/0.47 ! [A8: $tType] : ( order @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_17,axiom,
% 0.26/0.47 ! [A8: $tType] : ( ord @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 % Helper facts (3)
% 0.26/0.47 thf(help_If_3_1_T,axiom,
% 0.26/0.47 ! [P: $o] :
% 0.26/0.47 ( ( P = $true )
% 0.26/0.47 | ( P = $false ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(help_If_2_1_T,axiom,
% 0.26/0.47 ! [A: $tType,X: A,Y: A] :
% 0.26/0.47 ( ( if @ A @ $false @ X @ Y )
% 0.26/0.47 = Y ) ).
% 0.26/0.47
% 0.26/0.47 thf(help_If_1_1_T,axiom,
% 0.26/0.47 ! [A: $tType,X: A,Y: A] :
% 0.26/0.47 ( ( if @ A @ $true @ X @ Y )
% 0.26/0.47 = X ) ).
% 0.26/0.47
% 0.26/0.47 % Conjectures (1)
% 0.26/0.47 thf(conj_0,conjecture,
% 0.26/0.47 member @ ( product_prod @ ( ( refine1665802226e_nres @ a ) > ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( refine1665802226e_nres @ c ) > ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) @ ( product_Pair @ ( ( refine1665802226e_nres @ a ) > ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( refine1665802226e_nres @ c ) > ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) @ ( refine463715084e_bind @ a @ b ) @ ( refine463715084e_bind @ c @ d ) ) @ ( relAPP @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) @ ( set @ ( product_prod @ ( ( refine1665802226e_nres @ a ) > ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( refine1665802226e_nres @ c ) > ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) @ ( relAPP @ ( refine1665802226e_nres @ a ) @ ( refine1665802226e_nres @ c ) @ ( ( set @ ( product_prod @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) > ( set @ ( product_prod @ ( ( refine1665802226e_nres @ a ) > ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( refine1665802226e_nres @ c ) > ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) ) @ ( fun_rel @ ( refine1665802226e_nres @ a ) @ ( refine1665802226e_nres @ c ) @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) @ ( relAPP @ a @ c @ ( set @ ( product_prod @ ( refine1665802226e_nres @ a ) @ ( refine1665802226e_nres @ c ) ) ) @ ( refine476890328es_rel @ a @ c ) @ r1 ) ) @ ( relAPP @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) @ ( set @ ( product_prod @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) @ ( relAPP @ ( a > ( refine1665802226e_nres @ b ) ) @ ( c > ( refine1665802226e_nres @ d ) ) @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) ) > ( set @ ( product_prod @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) ) @ ( fun_rel @ ( a > ( refine1665802226e_nres @ b ) ) @ ( c > ( refine1665802226e_nres @ d ) ) @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) @ ( relAPP @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) @ ( set @ ( product_prod @ ( a > ( refine1665802226e_nres @ b ) ) @ ( c > ( refine1665802226e_nres @ d ) ) ) ) @ ( relAPP @ a @ c @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) ) > ( set @ ( product_prod @ ( a > ( refine1665802226e_nres @ b ) ) @ ( c > ( refine1665802226e_nres @ d ) ) ) ) ) @ ( fun_rel @ a @ c @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) @ r1 ) @ ( relAPP @ b @ d @ ( set @ ( product_prod @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) ) @ ( refine476890328es_rel @ b @ d ) @ r2 ) ) ) @ ( relAPP @ b @ d @ ( set @ ( product_prod @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) ) @ ( refine476890328es_rel @ b @ d ) @ r2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 ------------------------
% 3.09/1.54 ---- Embedded in THF ---
% 3.09/1.54 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.09/1.54 %%% Generated on Tue Feb 24 08:28:32 EST 2026
% 3.09/1.54 %%% using '$$dhol' embedding, version 1.3.0.
% 3.09/1.54 %%% Logic specification used:
% 3.09/1.54 %%% thf(spec, logic, $$dhol).
% 3.09/1.54
% 3.09/1.54 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.LLLxuS3oP5/DTF2DTF2166.p
% 3.09/1.54 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres, type, refine1665802226e_nres: ($tType > $tType)).
% 3.09/1.54 thf(ty_t_Autoref__Tagging_Oannot, type, autoref_annot: $tType).
% 3.09/1.54 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 3.09/1.54 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.09/1.54 thf(ty_t_Nat_Onat, type, nat: $tType).
% 3.09/1.54 thf(ty_t_Int_Oint, type, int: $tType).
% 3.09/1.54 thf(ty_tf_d, type, d: $tType).
% 3.09/1.54 thf(ty_tf_c, type, c: $tType).
% 3.09/1.54 thf(ty_tf_b, type, b: $tType).
% 3.09/1.54 thf(ty_tf_a, type, a: $tType).
% 3.09/1.54 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 3.09/1.54 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 3.09/1.54 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 3.09/1.54 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 3.09/1.54 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 3.09/1.54 thf(sy_c_Autoref__Tagging_OANNOT, type, autoref_ANNOT: (!> [A:$tType]: ((A > (autoref_annot > A))))).
% 3.09/1.54 thf(sy_c_Autoref__Tagging_OAPP, type, autoref_APP: (!> [B:$tType,A:$tType]: (((B > A) > (B > A))))).
% 3.09/1.54 thf(sy_c_Autoref__Tagging_OOP, type, autoref_OP: (!> [A:$tType]: ((A > A)))).
% 3.09/1.54 thf(sy_c_Autoref__Tagging_Orel__annot, type, autoref_rel_annot: (!> [C:$tType,A:$tType]: (((set @ ((product_prod @ C) @ A)) > autoref_annot)))).
% 3.09/1.54 thf(sy_c_Autoref__Translate_OREMOVE__INTERNAL, type, autore2013370551TERNAL: (!> [A:$tType]: ((A > A)))).
% 3.09/1.54 thf(sy_c_Autoref__Translate_OREMOVE__INTERNAL__EQ, type, autore813025987NAL_EQ: (!> [A:$tType]: ((A > (A > $o))))).
% 3.09/1.54 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 3.09/1.54 thf(sy_c_Fun_Ofun__upd, type, fun_upd: (!> [A:$tType,B:$tType]: (((A > B) > (A > (B > (A > B))))))).
% 3.09/1.54 thf(sy_c_Fun_Oid, type, id: (!> [A:$tType]: ((A > A)))).
% 3.09/1.54 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 3.09/1.54 thf(sy_c_List_Oall__interval__nat, type, all_interval_nat: ((nat > $o) > (nat > (nat > $o)))).
% 3.09/1.54 thf(sy_c_Misc_Ouncurry, type, uncurry: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.09/1.54 thf(sy_c_Nat_Onat_Ocase__nat, type, case_nat: (!> [A:$tType]: ((A > ((nat > A) > (nat > A)))))).
% 3.09/1.54 thf(sy_c_Nat_Oold_Onat_Orec__nat, type, rec_nat: (!> [T:$tType]: ((T > ((nat > (T > T)) > (nat > T)))))).
% 3.09/1.54 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 3.09/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.09/1.54 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 3.09/1.54 thf(sy_c_Product__Type_Oapfst, type, product_apfst: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > (((product_prod @ A) @ B) > ((product_prod @ C) @ B)))))).
% 3.09/1.54 thf(sy_c_Product__Type_Oapsnd, type, product_apsnd: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((product_prod @ A) @ B) > ((product_prod @ A) @ C)))))).
% 3.09/1.54 thf(sy_c_Product__Type_Obool_Ocase__bool, type, product_case_bool: (!> [A:$tType]: ((A > (A > ($o > A)))))).
% 3.09/1.54 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 3.09/1.54 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.09/1.54 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 3.09/1.54 thf(sy_c_Product__Type_Oold_Obool_Orec__bool, type, product_rec_bool: (!> [T:$tType]: ((T > (T > ($o > T)))))).
% 3.09/1.54 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 3.09/1.54 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.09/1.54 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 3.09/1.54 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 3.09/1.54 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 3.09/1.54 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN, type, refine1687780735RETURN: (!> [A:$tType]: ((A > (refine1665802226e_nres @ A))))).
% 3.09/1.54 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)))))).
% 3.09/1.54 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Olift__assn, type, refine1580981607t_assn: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > ((B > $o) > (A > $o)))))).
% 3.09/1.54 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres__rel, type, refine476890328es_rel: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ B))))))).
% 3.09/1.54 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oop__nres__ASSERT__bnd, type, refine171938416RT_bnd: (!> [A:$tType]: (($o > ((refine1665802226e_nres @ A) > (refine1665802226e_nres @ A)))))).
% 3.09/1.54 thf(sy_c_Refine__Misc_Oless__than__bool, type, refine1361002633n_bool: (set @ ((product_prod @ $o) @ $o))).
% 3.09/1.54 thf(sy_c_Relation_OId, type, id2: (!> [A:$tType]: ((set @ ((product_prod @ A) @ A))))).
% 3.09/1.54 thf(sy_c_Relators_Ofun__rel, type, fun_rel: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((set @ ((product_prod @ A) @ C)) > ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ (A > B)) @ (C > D)))))))).
% 3.09/1.54 thf(sy_c_Relators_Oprod__rel, type, prod_rel: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((set @ ((product_prod @ A) @ C)) > ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)))))))).
% 3.09/1.54 thf(sy_c_Relators_OrelAPP, type, relAPP: (!> [C1:$tType,A1:$tType,A:$tType]: ((((set @ ((product_prod @ C1) @ A1)) > A) > ((set @ ((product_prod @ C1) @ A1)) > A))))).
% 3.09/1.54 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.09/1.54 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.09/1.54 thf(sy_v_R1, type, r1: (set @ ((product_prod @ a) @ c))).
% 3.09/1.54 thf(sy_v_R2, type, r2: (set @ ((product_prod @ b) @ d))).
% 3.09/1.54 thf(fact_0_param__bind, axiom, (! [C:$tType,A:$tType,D:$tType,B:$tType,Ra:(set @ ((product_prod @ A) @ C)),Rb:(set @ ((product_prod @ B) @ D))]: ((((member @ ((product_prod @ ((refine1665802226e_nres @ A) > ((A > (refine1665802226e_nres @ B)) > (refine1665802226e_nres @ B)))) @ ((refine1665802226e_nres @ C) > ((C > (refine1665802226e_nres @ D)) > (refine1665802226e_nres @ D))))) @ ((((product_Pair @ ((refine1665802226e_nres @ A) > ((A > (refine1665802226e_nres @ B)) > (refine1665802226e_nres @ B)))) @ ((refine1665802226e_nres @ C) > ((C > (refine1665802226e_nres @ D)) > (refine1665802226e_nres @ D)))) @ ((refine463715084e_bind @ A) @ B)) @ ((refine463715084e_bind @ C) @ D))) @ (((((relAPP @ ((A > (refine1665802226e_nres @ B)) > (refine1665802226e_nres @ B))) @ ((C > (refine1665802226e_nres @ D)) > (refine1665802226e_nres @ D))) @ (set @ ((product_prod @ ((refine1665802226e_nres @ A) > ((A > (refine1665802226e_nres @ B)) > (refine1665802226e_nres @ B)))) @ ((refine1665802226e_nres @ C) > ((C > (refine1665802226e_nres @ D)) > (refine1665802226e_nres @ D)))))) @ (((((relAPP @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ C)) @ ((set @ ((product_prod @ ((A > (refine1665802226e_nres @ B)) > (refine1665802226e_nres @ B))) @ ((C > (refine1665802226e_nres @ D)) > (refine1665802226e_nres @ D)))) > (set @ ((product_prod @ ((refine1665802226e_nres @ A) > ((A > (refine1665802226e_nres @ B)) > (refine1665802226e_nres @ B)))) @ ((refine1665802226e_nres @ C) > ((C > (refine1665802226e_nres @ D)) > (refine1665802226e_nres @ D))))))) @ ((((fun_rel @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ C)) @ ((A > (refine1665802226e_nres @ B)) > (refine1665802226e_nres @ B))) @ ((C > (refine1665802226e_nres @ D)) > (refine1665802226e_nres @ D)))) @ (((((relAPP @ A) @ C) @ (set @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ C)))) @ ((refine476890328es_rel @ A) @ C)) @ Ra))) @ (((((relAPP @ (refine1665802226e_nres @ B)) @ (refine1665802226e_nres @ D)) @ (set @ ((product_prod @ ((A > (refine1665802226e_nres @ B)) > (refine1665802226e_nres @ B))) @ ((C > (refine1665802226e_nres @ D)) > (refine1665802226e_nres @ D))))) @ (((((relAPP @ (A > (refine1665802226e_nres @ B))) @ (C > (refine1665802226e_nres @ D))) @ ((set @ ((product_prod @ (refine1665802226e_nres @ B)) @ (refine1665802226e_nres @ D))) > (set @ ((product_prod @ ((A > (refine1665802226e_nres @ B)) > (refine1665802226e_nres @ B))) @ ((C > (refine1665802226e_nres @ D)) > (refine1665802226e_nres @ D)))))) @ ((((fun_rel @ (A > (refine1665802226e_nres @ B))) @ (C > (refine1665802226e_nres @ D))) @ (refine1665802226e_nres @ B)) @ (refine1665802226e_nres @ D))) @ (((((relAPP @ (refine1665802226e_nres @ B)) @ (refine1665802226e_nres @ D)) @ (set @ ((product_prod @ (A > (refine1665802226e_nres @ B))) @ (C > (refine1665802226e_nres @ D))))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ (refine1665802226e_nres @ B)) @ (refine1665802226e_nres @ D))) > (set @ ((product_prod @ (A > (refine1665802226e_nres @ B))) @ (C > (refine1665802226e_nres @ D)))))) @ ((((fun_rel @ A) @ C) @ (refine1665802226e_nres @ B)) @ (refine1665802226e_nres @ D))) @ Ra)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ (refine1665802226e_nres @ B)) @ (refine1665802226e_nres @ D)))) @ ((refine476890328es_rel @ B) @ D)) @ Rb)))) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ (refine1665802226e_nres @ B)) @ (refine1665802226e_nres @ D)))) @ ((refine476890328es_rel @ B) @ D)) @ Rb))))))).
% 3.09/1.54 thf(fact_1_fun__relI, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,A2:(set @ ((product_prod @ A) @ B)),F:(A > C),F2:(B > D),B2:(set @ ((product_prod @ C) @ D))]: (((! [A3:A,A4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ A4)) @ A2) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F @ A3)) @ (F2 @ A4))) @ B2)))) => (((member @ ((product_prod @ (A > C)) @ (B > D))) @ ((((product_Pair @ (A > C)) @ (B > D)) @ F) @ F2)) @ (((((relAPP @ C) @ D) @ (set @ ((product_prod @ (A > C)) @ (B > D)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ C) @ D)) > (set @ ((product_prod @ (A > C)) @ (B > D))))) @ ((((fun_rel @ A) @ B) @ C) @ D)) @ A2)) @ B2)))))).
% 3.09/1.54 thf(fact_2_autoref__Let, axiom, (! [C:$tType,A:$tType,D:$tType,B:$tType,Ra:(set @ ((product_prod @ A) @ C)),Rr:(set @ ((product_prod @ B) @ D))]: ((((member @ ((product_prod @ (A > ((A > B) > B))) @ (C > ((C > D) > D)))) @ ((((product_Pair @ (A > ((A > B) > B))) @ (C > ((C > D) > D))) @ (^ [S:A,F3:(A > B)]: ((F3 @ S)))) @ (^ [S:C,F3:(C > D)]: ((F3 @ S))))) @ (((((relAPP @ ((A > B) > B)) @ ((C > D) > D)) @ (set @ ((product_prod @ (A > ((A > B) > B))) @ (C > ((C > D) > D))))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ ((A > B) > B)) @ ((C > D) > D))) > (set @ ((product_prod @ (A > ((A > B) > B))) @ (C > ((C > D) > D)))))) @ ((((fun_rel @ A) @ C) @ ((A > B) > B)) @ ((C > D) > D))) @ Ra)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ ((A > B) > B)) @ ((C > D) > D)))) @ (((((relAPP @ (A > B)) @ (C > D)) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ ((A > B) > B)) @ ((C > D) > D))))) @ ((((fun_rel @ (A > B)) @ (C > D)) @ B) @ D)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ (A > B)) @ (C > D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ (A > B)) @ (C > D))))) @ ((((fun_rel @ A) @ C) @ B) @ D)) @ Ra)) @ Rr))) @ Rr)))))).
% 3.09/1.54 thf(fact_3_fun__relD1, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F:(A > B),F2:(C > D),Ra:(set @ ((product_prod @ A) @ C)),Rr:(set @ ((product_prod @ B) @ D)),X:A,R:B]: (((((member @ ((product_prod @ (A > B)) @ (C > D))) @ ((((product_Pair @ (A > B)) @ (C > D)) @ F) @ F2)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ (A > B)) @ (C > D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ (A > B)) @ (C > D))))) @ ((((fun_rel @ A) @ C) @ B) @ D)) @ Ra)) @ Rr)) => (((F @ X) = R) => (! [X2:C]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X) @ X2)) @ Ra) => (((member @ ((product_prod @ B) @ D)) @ ((((product_Pair @ B) @ D) @ R) @ (F2 @ X2))) @ Rr))))))))).
% 3.09/1.54 thf(fact_4_fun__relD2, axiom, (! [C:$tType,A:$tType,D:$tType,B:$tType,F:(A > B),F2:(C > D),Ra:(set @ ((product_prod @ A) @ C)),Rr:(set @ ((product_prod @ B) @ D)),X3:C,R2:D]: (((((member @ ((product_prod @ (A > B)) @ (C > D))) @ ((((product_Pair @ (A > B)) @ (C > D)) @ F) @ F2)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ (A > B)) @ (C > D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ (A > B)) @ (C > D))))) @ ((((fun_rel @ A) @ C) @ B) @ D)) @ Ra)) @ Rr)) => (((F2 @ X3) = R2) => (! [X4:A]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X4) @ X3)) @ Ra) => (((member @ ((product_prod @ B) @ D)) @ ((((product_Pair @ B) @ D) @ (F @ X4)) @ R2)) @ Rr))))))))).
% 3.09/1.54 thf(fact_5_tagged__fun__relD__lhs, axiom, (! [C:$tType,A:$tType,D:$tType,B:$tType,F:(A > B),F2:(C > D),A2:(set @ ((product_prod @ A) @ C)),B2:(set @ ((product_prod @ B) @ D)),X:A,X3:C]: (((((member @ ((product_prod @ (A > B)) @ (C > D))) @ ((((product_Pair @ (A > B)) @ (C > D)) @ F) @ F2)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ (A > B)) @ (C > D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ (A > B)) @ (C > D))))) @ ((((fun_rel @ A) @ C) @ B) @ D)) @ A2)) @ B2)) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X) @ X3)) @ A2) => (((member @ ((product_prod @ B) @ D)) @ ((((product_Pair @ B) @ D) @ (F @ X)) @ (F2 @ X3))) @ B2)))))).
% 3.09/1.54 thf(fact_6_tagged__fun__relD__rhs, axiom, (! [C:$tType,A:$tType,D:$tType,B:$tType,F:(A > B),F2:(C > D),A2:(set @ ((product_prod @ A) @ C)),B2:(set @ ((product_prod @ B) @ D)),X:A,X3:C]: (((((member @ ((product_prod @ (A > B)) @ (C > D))) @ ((((product_Pair @ (A > B)) @ (C > D)) @ F) @ F2)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ (A > B)) @ (C > D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ (A > B)) @ (C > D))))) @ ((((fun_rel @ A) @ C) @ B) @ D)) @ A2)) @ B2)) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X) @ X3)) @ A2) => (((member @ ((product_prod @ B) @ D)) @ ((((product_Pair @ B) @ D) @ (F @ X)) @ (F2 @ X3))) @ B2)))))).
% 3.09/1.54 thf(fact_7_tagged__fun__relD__both, axiom, (! [C:$tType,A:$tType,D:$tType,B:$tType,F:(A > B),F2:(C > D),A2:(set @ ((product_prod @ A) @ C)),B2:(set @ ((product_prod @ B) @ D)),X:A,X3:C]: (((((member @ ((product_prod @ (A > B)) @ (C > D))) @ ((((product_Pair @ (A > B)) @ (C > D)) @ F) @ F2)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ (A > B)) @ (C > D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ (A > B)) @ (C > D))))) @ ((((fun_rel @ A) @ C) @ B) @ D)) @ A2)) @ B2)) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X) @ X3)) @ A2) => (((member @ ((product_prod @ B) @ D)) @ ((((product_Pair @ B) @ D) @ (F @ X)) @ (F2 @ X3))) @ B2)))))).
% 3.09/1.54 thf(fact_8_tagged__fun__relD__none, axiom, (! [C:$tType,A:$tType,D:$tType,B:$tType,F:(A > B),F2:(C > D),A2:(set @ ((product_prod @ A) @ C)),B2:(set @ ((product_prod @ B) @ D)),X:A,X3:C]: (((((member @ ((product_prod @ (A > B)) @ (C > D))) @ ((((product_Pair @ (A > B)) @ (C > D)) @ F) @ F2)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ (A > B)) @ (C > D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ (A > B)) @ (C > D))))) @ ((((fun_rel @ A) @ C) @ B) @ D)) @ A2)) @ B2)) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X) @ X3)) @ A2) => (((member @ ((product_prod @ B) @ D)) @ ((((product_Pair @ B) @ D) @ (F @ X)) @ (F2 @ X3))) @ B2)))))).
% 3.09/1.54 thf(fact_9_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 3.09/1.54 thf(fact_10_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A5:A,B3:B,A6:A,B4:B]: (((((((product_Pair @ A) @ B) @ A5) @ B3) = ((((product_Pair @ A) @ B) @ A6) @ B4)) = ((A5 = A6) & (B3 = B4)))))).
% 3.09/1.54 thf(fact_11_autoref__APP, axiom, (! [C:$tType,A:$tType,D:$tType,B:$tType,C2:(A > B),A5:(C > D),Ra:(set @ ((product_prod @ A) @ C)),Rr:(set @ ((product_prod @ B) @ D)),X:A,X3:C]: (((((member @ ((product_prod @ (A > B)) @ (C > D))) @ ((((product_Pair @ (A > B)) @ (C > D)) @ C2) @ A5)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ (A > B)) @ (C > D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ (A > B)) @ (C > D))))) @ ((((fun_rel @ A) @ C) @ B) @ D)) @ Ra)) @ Rr)) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X) @ X3)) @ Ra) => (((member @ ((product_prod @ B) @ D)) @ ((((product_Pair @ B) @ D) @ ((((autoref_APP @ A) @ B) @ C2) @ X)) @ ((((autoref_APP @ C) @ D) @ A5) @ X3))) @ Rr)))))).
% 3.09/1.54 thf(fact_12_lift__assnI, axiom, (! [B:$tType,A:$tType,S2:A,S3:B,R3:(set @ ((product_prod @ A) @ B)),Phi:(B > $o)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ S2) @ S3)) @ R3) => ((Phi @ S3) => (((((refine1580981607t_assn @ A) @ B) @ R3) @ Phi) @ S2)))))).
% 3.09/1.54 thf(fact_13_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A3:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A3) @ B5)))) => (P @ Prod))))).
% 3.09/1.54 thf(fact_14_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A3:A,B5:B]: ((Y != ((((product_Pair @ A) @ B) @ A3) @ B5))))))))).
% 3.09/1.54 thf(fact_15_prod__induct7, axiom, (! [G:$tType,F4:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F4) @ G)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F4) @ G))))))]: (((! [A3:A,B5:B,C3:C,D2:D,E2:E,F5:F4,G2:G]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F4) @ G)))))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F4) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F4) @ G)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F4) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F4) @ G)) @ E2) @ ((((product_Pair @ F4) @ G) @ F5) @ G2))))))))) => (P @ X))))).
% 3.09/1.54 thf(fact_16_prod__induct6, axiom, (! [F4:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F4))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F4)))))]: (((! [A3:A,B5:B,C3:C,D2:D,E2:E,F5:F4]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F4))))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F4)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F4))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F4)) @ D2) @ ((((product_Pair @ E) @ F4) @ E2) @ F5)))))))) => (P @ X))))).
% 3.09/1.54 thf(fact_17_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A3:A,B5:B,C3:C,D2:D,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P @ X))))).
% 3.09/1.54 thf(fact_18_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A3:A,B5:B,C3:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C3) @ D2)))))) => (P @ X))))).
% 3.09/1.54 thf(fact_19_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A3:A,B5:B,C3:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A3) @ ((((product_Pair @ B) @ C) @ B5) @ C3))))) => (P @ X))))).
% 3.09/1.54 thf(fact_20_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F4:$tType,G:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F4) @ G))))))]: ((~ ((! [A3:A,B5:B,C3:C,D2:D,E2:E,F5:F4,G2:G]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F4) @ G)))))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F4) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F4) @ G)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F4) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F4) @ G)) @ E2) @ ((((product_Pair @ F4) @ G) @ F5) @ G2)))))))))))))).
% 3.09/1.54 thf(fact_21_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F4:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F4)))))]: ((~ ((! [A3:A,B5:B,C3:C,D2:D,E2:E,F5:F4]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F4))))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F4)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F4))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F4)) @ D2) @ ((((product_Pair @ E) @ F4) @ E2) @ F5))))))))))))).
% 3.09/1.54 thf(fact_22_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A3:A,B5:B,C3:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 3.09/1.54 thf(fact_23_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A3:A,B5:B,C3:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C3) @ D2))))))))))).
% 3.09/1.54 thf(fact_24_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A3:A,B5:B,C3:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A3) @ ((((product_Pair @ B) @ C) @ B5) @ C3)))))))))).
% 3.09/1.54 thf(fact_25_Pair__inject, axiom, (! [A:$tType,B:$tType,A5:A,B3:B,A6:A,B4:B]: (((((((product_Pair @ A) @ B) @ A5) @ B3) = ((((product_Pair @ A) @ B) @ A6) @ B4)) => (~ (((A5 = A6) => (B3 != B4)))))))).
% 3.09/1.54 thf(fact_26_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A3:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A3) @ B5)))) => (P @ P2))))).
% 3.09/1.54 thf(fact_27_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X5:A,Y3:B]: ((P2 = ((((product_Pair @ A) @ B) @ X5) @ Y3))))))).
% 3.09/1.54 thf(fact_28_relAPP__def, axiom, (! [A:$tType,A1:$tType,C1:$tType]: (((((relAPP @ C1) @ A1) @ A) = (^ [F3:((set @ ((product_prod @ C1) @ A1)) > A)]: (F3)))))).
% 3.09/1.54 thf(fact_29_lift__assn__def, axiom, (! [B:$tType,A:$tType]: ((((refine1580981607t_assn @ A) @ B) = (^ [R4:(set @ ((product_prod @ A) @ B)),Phi2:(B > $o),S:A]: ((? [S4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ S) @ S4)) @ R4) & (Phi2 @ S4)))))))))).
% 3.09/1.54 thf(fact_30_autoref__RETURN, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B))]: ((((member @ ((product_prod @ (A > (refine1665802226e_nres @ A))) @ (B > (refine1665802226e_nres @ B)))) @ ((((product_Pair @ (A > (refine1665802226e_nres @ A))) @ (B > (refine1665802226e_nres @ B))) @ (refine1687780735RETURN @ A)) @ (refine1687780735RETURN @ B))) @ (((((relAPP @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ B)) @ (set @ ((product_prod @ (A > (refine1665802226e_nres @ A))) @ (B > (refine1665802226e_nres @ B))))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ B))) > (set @ ((product_prod @ (A > (refine1665802226e_nres @ A))) @ (B > (refine1665802226e_nres @ B)))))) @ ((((fun_rel @ A) @ B) @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ B))) @ R3)) @ (((((relAPP @ A) @ B) @ (set @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ B)))) @ ((refine476890328es_rel @ A) @ B)) @ R3)))))).
% 3.09/1.54 thf(fact_31_param__op__nres__ASSERT__bnd, axiom, (! [B:$tType,A:$tType,Phi3:$o,Phi:$o,M:(refine1665802226e_nres @ A),M2:(refine1665802226e_nres @ B),R3:(set @ ((product_prod @ A) @ B))]: (((Phi3 => Phi) => ((Phi3 => (Phi => (((member @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ B))) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ B)) @ M) @ M2)) @ (((((relAPP @ A) @ B) @ (set @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ B)))) @ ((refine476890328es_rel @ A) @ B)) @ R3)))) => (((member @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ B))) @ ((((product_Pair @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ B)) @ (((refine171938416RT_bnd @ A) @ Phi) @ M)) @ (((refine171938416RT_bnd @ B) @ Phi3) @ M2))) @ (((((relAPP @ A) @ B) @ (set @ ((product_prod @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ B)))) @ ((refine476890328es_rel @ A) @ B)) @ R3))))))).
% 3.09/1.54 thf(fact_32_autoref__If, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B))]: ((((member @ ((product_prod @ ($o > (A > (A > A)))) @ ($o > (B > (B > B))))) @ ((((product_Pair @ ($o > (A > (A > A)))) @ ($o > (B > (B > B)))) @ (if @ A)) @ (if @ B))) @ (((((relAPP @ (A > (A > A))) @ (B > (B > B))) @ (set @ ((product_prod @ ($o > (A > (A > A)))) @ ($o > (B > (B > B)))))) @ (((((relAPP @ $o) @ $o) @ ((set @ ((product_prod @ (A > (A > A))) @ (B > (B > B)))) > (set @ ((product_prod @ ($o > (A > (A > A)))) @ ($o > (B > (B > B))))))) @ ((((fun_rel @ $o) @ $o) @ (A > (A > A))) @ (B > (B > B)))) @ (id2 @ $o))) @ (((((relAPP @ (A > A)) @ (B > B)) @ (set @ ((product_prod @ (A > (A > A))) @ (B > (B > B))))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ (A > A)) @ (B > B))) > (set @ ((product_prod @ (A > (A > A))) @ (B > (B > B)))))) @ ((((fun_rel @ A) @ B) @ (A > A)) @ (B > B))) @ R3)) @ (((((relAPP @ A) @ B) @ (set @ ((product_prod @ (A > A)) @ (B > B)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ (A > A)) @ (B > B))))) @ ((((fun_rel @ A) @ B) @ A) @ B)) @ R3)) @ R3))))))).
% 3.09/1.54 thf(fact_33_fun__relE2, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > B),F2:(A > C),Rv:(set @ ((product_prod @ B) @ C)),T2:B,X:A]: (((((member @ ((product_prod @ (A > B)) @ (A > C))) @ ((((product_Pair @ (A > B)) @ (A > C)) @ F) @ F2)) @ (((((relAPP @ B) @ C) @ (set @ ((product_prod @ (A > B)) @ (A > C)))) @ (((((relAPP @ A) @ A) @ ((set @ ((product_prod @ B) @ C)) > (set @ ((product_prod @ (A > B)) @ (A > C))))) @ ((((fun_rel @ A) @ A) @ B) @ C)) @ (id2 @ A))) @ Rv)) => ((T2 = (F @ X)) => (((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ T2) @ (F2 @ X))) @ Rv)))))).
% 3.09/1.54 thf(fact_34_fun__relE1, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > B),F2:(A > C),Rv:(set @ ((product_prod @ B) @ C)),T3:C,X:A]: (((((member @ ((product_prod @ (A > B)) @ (A > C))) @ ((((product_Pair @ (A > B)) @ (A > C)) @ F) @ F2)) @ (((((relAPP @ B) @ C) @ (set @ ((product_prod @ (A > B)) @ (A > C)))) @ (((((relAPP @ A) @ A) @ ((set @ ((product_prod @ B) @ C)) > (set @ ((product_prod @ (A > B)) @ (A > C))))) @ ((((fun_rel @ A) @ A) @ B) @ C)) @ (id2 @ A))) @ Rv)) => ((T3 = (F2 @ X)) => (((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ (F @ X)) @ T3)) @ Rv)))))).
% 3.09/1.54 thf(fact_35_autoref__Let__cong, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,X3:A,X:B,Ra:(set @ ((product_prod @ A) @ B)),F2:(A > C),F:(B > D),Rr:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ X)) @ Ra) => ((! [Y3:B]: ((((autore2013370551TERNAL @ $o) @ (X = Y3)) => (! [Y4:A]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y4) @ Y3)) @ Ra) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F2 @ Y4)) @ ((((autoref_APP @ B) @ D) @ F) @ Y3))) @ Rr))))))) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F2 @ X3)) @ ((((autoref_APP @ (B > D)) @ D) @ ((((autoref_APP @ B) @ ((B > D) > D)) @ (((autoref_ANNOT @ (B > ((B > D) > D))) @ ((autoref_OP @ (B > ((B > D) > D))) @ (^ [S:B,F3:(B > D)]: ((F3 @ S))))) @ (((autoref_rel_annot @ (A > ((A > C) > C))) @ (B > ((B > D) > D))) @ (((((relAPP @ ((A > C) > C)) @ ((B > D) > D)) @ (set @ ((product_prod @ (A > ((A > C) > C))) @ (B > ((B > D) > D))))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ ((A > C) > C)) @ ((B > D) > D))) > (set @ ((product_prod @ (A > ((A > C) > C))) @ (B > ((B > D) > D)))))) @ ((((fun_rel @ A) @ B) @ ((A > C) > C)) @ ((B > D) > D))) @ Ra)) @ (((((relAPP @ C) @ D) @ (set @ ((product_prod @ ((A > C) > C)) @ ((B > D) > D)))) @ (((((relAPP @ (A > C)) @ (B > D)) @ ((set @ ((product_prod @ C) @ D)) > (set @ ((product_prod @ ((A > C) > C)) @ ((B > D) > D))))) @ ((((fun_rel @ (A > C)) @ (B > D)) @ C) @ D)) @ (((((relAPP @ C) @ D) @ (set @ ((product_prod @ (A > C)) @ (B > D)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ C) @ D)) > (set @ ((product_prod @ (A > C)) @ (B > D))))) @ ((((fun_rel @ A) @ B) @ C) @ D)) @ Ra)) @ Rr))) @ Rr))))) @ X)) @ F))) @ Rr)))))).
% 3.09/1.54 thf(fact_36_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A5:B,B3:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A5) @ B3)) = ((C2 @ A5) @ B3))))).
% 3.09/1.54 thf(fact_37_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A5:A,B3:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) = ((F1 @ A5) @ B3))))).
% 3.09/1.54 thf(fact_38_APP__def, axiom, (! [A:$tType,B:$tType]: ((((autoref_APP @ B) @ A) = (^ [F3:(B > A)]: (F3)))))).
% 3.09/1.54 thf(fact_39_autoref__id, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B))]: ((((member @ ((product_prod @ (A > A)) @ (B > B))) @ ((((product_Pair @ (A > A)) @ (B > B)) @ (id @ A)) @ (id @ B))) @ (((((relAPP @ A) @ B) @ (set @ ((product_prod @ (A > A)) @ (B > B)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ (A > A)) @ (B > B))))) @ ((((fun_rel @ A) @ B) @ A) @ B)) @ R3)) @ R3))))).
% 3.09/1.54 thf(fact_40_autoref__REMOVE__INTERNAL__EQ, axiom, (! [B:$tType,A:$tType,C2:A,A5:B,R3:(set @ ((product_prod @ A) @ B)),C4:A]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ C2) @ A5)) @ R3) => ((((autore813025987NAL_EQ @ A) @ C2) @ C4) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ C4) @ A5)) @ R3)))))).
% 3.09/1.54 thf(fact_41_nres__more__simps_I6_J, axiom, (! [A:$tType,X:A,Y:A]: (((((refine1687780735RETURN @ A) @ X) = ((refine1687780735RETURN @ A) @ Y)) = (X = Y))))).
% 3.09/1.54 thf(fact_42_mem__Collect__eq, axiom, (! [A:$tType,A5:A,P:(A > $o)]: (((((member @ A) @ A5) @ ((collect @ A) @ P)) = (P @ A5))))).
% 3.09/1.54 thf(fact_43_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X6:A]: ((((member @ A) @ X6) @ A2)))) = A2)))).
% 3.09/1.54 thf(fact_44_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) = (Q @ X5)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.09/1.54 thf(fact_45_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G3:(A > B)]: (((! [X5:A]: (((F @ X5) = (G3 @ X5)))) => (F = G3))))).
% 3.09/1.54 thf(fact_46_nres__monad1, axiom, (! [A:$tType,B:$tType,X:B,F:(B > (refine1665802226e_nres @ A))]: ((((((refine463715084e_bind @ B) @ A) @ ((refine1687780735RETURN @ B) @ X)) @ F) = (F @ X))))).
% 3.09/1.54 thf(fact_47_nres__monad2, axiom, (! [A:$tType,M3:(refine1665802226e_nres @ A)]: ((((((refine463715084e_bind @ A) @ A) @ M3) @ (refine1687780735RETURN @ A)) = M3)))).
% 3.09/1.54 thf(fact_48_fun__rel__id__simp, axiom, (! [B:$tType,A:$tType]: (((((((relAPP @ B) @ B) @ (set @ ((product_prod @ (A > B)) @ (A > B)))) @ (((((relAPP @ A) @ A) @ ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ (A > B)) @ (A > B))))) @ ((((fun_rel @ A) @ A) @ B) @ B)) @ (id2 @ A))) @ (id2 @ B)) = (id2 @ (A > B)))))).
% 3.09/1.54 thf(fact_49_relprop__triggers_I3_J, axiom, (! [D:$tType,R3:(set @ ((product_prod @ D) @ D))]: (((R3 = (id2 @ D)) => ((id2 @ D) = R3))))).
% 3.09/1.54 thf(fact_50_relprop__triggers_I2_J, axiom, (! [C:$tType,R3:(set @ ((product_prod @ C) @ C))]: (((R3 = (id2 @ C)) => (R3 = (id2 @ C)))))).
% 3.09/1.54 thf(fact_51_autoref__int_I6_J, axiom, (((member @ ((product_prod @ (int > (int > $o))) @ (int > (int > $o)))) @ ((((product_Pair @ (int > (int > $o))) @ (int > (int > $o))) @ (^ [Y5:int,Z:int]: ((Y5 = Z)))) @ (^ [Y5:int,Z:int]: ((Y5 = Z))))) @ (((((relAPP @ (int > $o)) @ (int > $o)) @ (set @ ((product_prod @ (int > (int > $o))) @ (int > (int > $o))))) @ (((((relAPP @ int) @ int) @ ((set @ ((product_prod @ (int > $o)) @ (int > $o))) > (set @ ((product_prod @ (int > (int > $o))) @ (int > (int > $o)))))) @ ((((fun_rel @ int) @ int) @ (int > $o)) @ (int > $o))) @ (id2 @ int))) @ (((((relAPP @ $o) @ $o) @ (set @ ((product_prod @ (int > $o)) @ (int > $o)))) @ (((((relAPP @ int) @ int) @ ((set @ ((product_prod @ $o) @ $o)) > (set @ ((product_prod @ (int > $o)) @ (int > $o))))) @ ((((fun_rel @ int) @ int) @ $o) @ $o)) @ (id2 @ int))) @ (id2 @ $o))))).
% 3.09/1.54 thf(fact_52_autoref__nat_I7_J, axiom, (((member @ ((product_prod @ (nat > (nat > $o))) @ (nat > (nat > $o)))) @ ((((product_Pair @ (nat > (nat > $o))) @ (nat > (nat > $o))) @ (^ [Y5:nat,Z:nat]: ((Y5 = Z)))) @ (^ [Y5:nat,Z:nat]: ((Y5 = Z))))) @ (((((relAPP @ (nat > $o)) @ (nat > $o)) @ (set @ ((product_prod @ (nat > (nat > $o))) @ (nat > (nat > $o))))) @ (((((relAPP @ nat) @ nat) @ ((set @ ((product_prod @ (nat > $o)) @ (nat > $o))) > (set @ ((product_prod @ (nat > (nat > $o))) @ (nat > (nat > $o)))))) @ ((((fun_rel @ nat) @ nat) @ (nat > $o)) @ (nat > $o))) @ (id2 @ nat))) @ (((((relAPP @ $o) @ $o) @ (set @ ((product_prod @ (nat > $o)) @ (nat > $o)))) @ (((((relAPP @ nat) @ nat) @ ((set @ ((product_prod @ $o) @ $o)) > (set @ ((product_prod @ (nat > $o)) @ (nat > $o))))) @ ((((fun_rel @ nat) @ nat) @ $o) @ $o)) @ (id2 @ nat))) @ (id2 @ $o))))).
% 3.09/1.54 thf(fact_53_autoref__bool_I8_J, axiom, (((member @ ((product_prod @ ($o > ($o > $o))) @ ($o > ($o > $o)))) @ ((((product_Pair @ ($o > ($o > $o))) @ ($o > ($o > $o))) @ (=>)) @ (=>))) @ (((((relAPP @ ($o > $o)) @ ($o > $o)) @ (set @ ((product_prod @ ($o > ($o > $o))) @ ($o > ($o > $o))))) @ (((((relAPP @ $o) @ $o) @ ((set @ ((product_prod @ ($o > $o)) @ ($o > $o))) > (set @ ((product_prod @ ($o > ($o > $o))) @ ($o > ($o > $o)))))) @ ((((fun_rel @ $o) @ $o) @ ($o > $o)) @ ($o > $o))) @ (id2 @ $o))) @ (((((relAPP @ $o) @ $o) @ (set @ ((product_prod @ ($o > $o)) @ ($o > $o)))) @ (((((relAPP @ $o) @ $o) @ ((set @ ((product_prod @ $o) @ $o)) > (set @ ((product_prod @ ($o > $o)) @ ($o > $o))))) @ ((((fun_rel @ $o) @ $o) @ $o) @ $o)) @ (id2 @ $o))) @ (id2 @ $o))))).
% 3.09/1.54 thf(fact_54_autoref__bool_I4_J, axiom, (((member @ ((product_prod @ ($o > $o)) @ ($o > $o))) @ ((((product_Pair @ ($o > $o)) @ ($o > $o)) @ (~)) @ (~))) @ (((((relAPP @ $o) @ $o) @ (set @ ((product_prod @ ($o > $o)) @ ($o > $o)))) @ (((((relAPP @ $o) @ $o) @ ((set @ ((product_prod @ $o) @ $o)) > (set @ ((product_prod @ ($o > $o)) @ ($o > $o))))) @ ((((fun_rel @ $o) @ $o) @ $o) @ $o)) @ (id2 @ $o))) @ (id2 @ $o)))).
% 3.09/1.54 thf(fact_55_autoref__bool_I3_J, axiom, (((member @ ((product_prod @ ($o > ($o > $o))) @ ($o > ($o > $o)))) @ ((((product_Pair @ ($o > ($o > $o))) @ ($o > ($o > $o))) @ (|)) @ (|))) @ (((((relAPP @ ($o > $o)) @ ($o > $o)) @ (set @ ((product_prod @ ($o > ($o > $o))) @ ($o > ($o > $o))))) @ (((((relAPP @ $o) @ $o) @ ((set @ ((product_prod @ ($o > $o)) @ ($o > $o))) > (set @ ((product_prod @ ($o > ($o > $o))) @ ($o > ($o > $o)))))) @ ((((fun_rel @ $o) @ $o) @ ($o > $o)) @ ($o > $o))) @ (id2 @ $o))) @ (((((relAPP @ $o) @ $o) @ (set @ ((product_prod @ ($o > $o)) @ ($o > $o)))) @ (((((relAPP @ $o) @ $o) @ ((set @ ((product_prod @ $o) @ $o)) > (set @ ((product_prod @ ($o > $o)) @ ($o > $o))))) @ ((((fun_rel @ $o) @ $o) @ $o) @ $o)) @ (id2 @ $o))) @ (id2 @ $o))))).
% 3.09/1.54 thf(fact_56_autoref__bool_I2_J, axiom, (((member @ ((product_prod @ ($o > ($o > $o))) @ ($o > ($o > $o)))) @ ((((product_Pair @ ($o > ($o > $o))) @ ($o > ($o > $o))) @ (&)) @ (&))) @ (((((relAPP @ ($o > $o)) @ ($o > $o)) @ (set @ ((product_prod @ ($o > ($o > $o))) @ ($o > ($o > $o))))) @ (((((relAPP @ $o) @ $o) @ ((set @ ((product_prod @ ($o > $o)) @ ($o > $o))) > (set @ ((product_prod @ ($o > ($o > $o))) @ ($o > ($o > $o)))))) @ ((((fun_rel @ $o) @ $o) @ ($o > $o)) @ ($o > $o))) @ (id2 @ $o))) @ (((((relAPP @ $o) @ $o) @ (set @ ((product_prod @ ($o > $o)) @ ($o > $o)))) @ (((((relAPP @ $o) @ $o) @ ((set @ ((product_prod @ $o) @ $o)) > (set @ ((product_prod @ ($o > $o)) @ ($o > $o))))) @ ((((fun_rel @ $o) @ $o) @ $o) @ $o)) @ (id2 @ $o))) @ (id2 @ $o))))).
% 3.09/1.54 thf(fact_57_autoref__bool_I1_J, axiom, (! [X:$o]: ((((member @ ((product_prod @ $o) @ $o)) @ ((((product_Pair @ $o) @ $o) @ X) @ X)) @ (id2 @ $o))))).
% 3.09/1.54 thf(fact_58_relprop__id__orient, axiom, (! [A:$tType,R3:(set @ ((product_prod @ A) @ A))]: (((R3 = (id2 @ A)) => ((id2 @ A) = R3))))).
% 3.09/1.54 thf(fact_59_rel__ANNOT__eq, axiom, (! [A:$tType,B:$tType,T2:A,R3:(set @ ((product_prod @ B) @ A))]: ((T2 = (((autoref_ANNOT @ A) @ T2) @ (((autoref_rel_annot @ B) @ A) @ R3)))))).
% 3.09/1.54 thf(fact_60_ANNOT__def, axiom, (! [A:$tType]: (((autoref_ANNOT @ A) = (^ [X6:A,A7:autoref_annot]: (X6)))))).
% 3.09/1.54 thf(fact_61_OP__def, axiom, (! [A:$tType]: (((autoref_OP @ A) = (^ [X6:A]: (X6)))))).
% 3.09/1.54 thf(fact_62_rel__cong, axiom, (! [A:$tType,B:$tType,F:(A > B),G3:(A > B),X:A,Y:A]: (((((member @ ((product_prod @ (A > B)) @ (A > B))) @ ((((product_Pair @ (A > B)) @ (A > B)) @ F) @ G3)) @ (id2 @ (A > B))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ (id2 @ A)) => (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X)) @ (G3 @ Y))) @ (id2 @ B))))))).
% 3.09/1.54 thf(fact_63_rel__fun__cong, axiom, (! [A:$tType,B:$tType,F:(A > B),G3:(A > B),X:A]: (((((member @ ((product_prod @ (A > B)) @ (A > B))) @ ((((product_Pair @ (A > B)) @ (A > B)) @ F) @ G3)) @ (id2 @ (A > B))) => (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X)) @ (G3 @ X))) @ (id2 @ B)))))).
% 3.09/1.54 thf(fact_64_REMOVE__INTERNAL__EQ__def, axiom, (! [A:$tType]: (((autore813025987NAL_EQ @ A) = (^ [Y5:A,Z:A]: ((Y5 = Z))))))).
% 3.09/1.54 thf(fact_65_REMOVE__INTERNAL__def, axiom, (! [A:$tType]: (((autore2013370551TERNAL @ A) = (^ [X6:A]: (X6)))))).
% 3.09/1.54 thf(fact_66_REMOVE__INTERNAL__EQI, axiom, (! [A:$tType,A5:A]: ((((autore813025987NAL_EQ @ A) @ A5) @ A5)))).
% 3.09/1.54 thf(fact_67_rel__arg__cong, axiom, (! [A:$tType,B:$tType,X:A,Y:A,F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ (id2 @ A)) => (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X)) @ (F @ Y))) @ (id2 @ B)))))).
% 3.09/1.54 thf(fact_68_fun__rel__id, axiom, (! [B:$tType,A:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B))]: (((R1 = (id2 @ A)) => ((R22 = (id2 @ B)) => ((((((relAPP @ B) @ B) @ (set @ ((product_prod @ (A > B)) @ (A > B)))) @ (((((relAPP @ A) @ A) @ ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ (A > B)) @ (A > B))))) @ ((((fun_rel @ A) @ A) @ B) @ B)) @ R1)) @ R22) = (id2 @ (A > B)))))))).
% 3.09/1.54 thf(fact_69_IdI, axiom, (! [A:$tType,A5:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ A5)) @ (id2 @ A))))).
% 3.09/1.54 thf(fact_70_pair__in__Id__conv, axiom, (! [A:$tType,A5:A,B3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ B3)) @ (id2 @ A)) = (A5 = B3))))).
% 3.09/1.54 thf(fact_71_autoref__bool_I6_J, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B))]: ((((member @ ((product_prod @ (A > (A > ($o > A)))) @ (B > (B > ($o > B))))) @ ((((product_Pair @ (A > (A > ($o > A)))) @ (B > (B > ($o > B)))) @ (product_rec_bool @ A)) @ (product_rec_bool @ B))) @ (((((relAPP @ (A > ($o > A))) @ (B > ($o > B))) @ (set @ ((product_prod @ (A > (A > ($o > A)))) @ (B > (B > ($o > B)))))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ (A > ($o > A))) @ (B > ($o > B)))) > (set @ ((product_prod @ (A > (A > ($o > A)))) @ (B > (B > ($o > B))))))) @ ((((fun_rel @ A) @ B) @ (A > ($o > A))) @ (B > ($o > B)))) @ R3)) @ (((((relAPP @ ($o > A)) @ ($o > B)) @ (set @ ((product_prod @ (A > ($o > A))) @ (B > ($o > B))))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ ($o > A)) @ ($o > B))) > (set @ ((product_prod @ (A > ($o > A))) @ (B > ($o > B)))))) @ ((((fun_rel @ A) @ B) @ ($o > A)) @ ($o > B))) @ R3)) @ (((((relAPP @ A) @ B) @ (set @ ((product_prod @ ($o > A)) @ ($o > B)))) @ (((((relAPP @ $o) @ $o) @ ((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ ($o > A)) @ ($o > B))))) @ ((((fun_rel @ $o) @ $o) @ A) @ B)) @ (id2 @ $o))) @ R3))))))).
% 3.09/1.54 thf(fact_72_autoref__If__cong, axiom, (! [B:$tType,A:$tType,C4:$o,C2:$o,T3:A,T2:B,R3:(set @ ((product_prod @ A) @ B)),E3:A,E4:B]: (((((member @ ((product_prod @ $o) @ $o)) @ ((((product_Pair @ $o) @ $o) @ C4) @ C2)) @ (id2 @ $o)) => ((((autore2013370551TERNAL @ $o) @ C2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ T3) @ T2)) @ R3)) => (((~ (((autore2013370551TERNAL @ $o) @ C2))) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ E3) @ E4)) @ R3)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ ((((if @ A) @ C4) @ T3) @ E3)) @ ((((autoref_APP @ B) @ B) @ ((((autoref_APP @ B) @ (B > B)) @ ((((autoref_APP @ $o) @ (B > (B > B))) @ (((autoref_ANNOT @ ($o > (B > (B > B)))) @ ((autoref_OP @ ($o > (B > (B > B)))) @ (if @ B))) @ (((autoref_rel_annot @ ($o > (A > (A > A)))) @ ($o > (B > (B > B)))) @ (((((relAPP @ (A > (A > A))) @ (B > (B > B))) @ (set @ ((product_prod @ ($o > (A > (A > A)))) @ ($o > (B > (B > B)))))) @ (((((relAPP @ $o) @ $o) @ ((set @ ((product_prod @ (A > (A > A))) @ (B > (B > B)))) > (set @ ((product_prod @ ($o > (A > (A > A)))) @ ($o > (B > (B > B))))))) @ ((((fun_rel @ $o) @ $o) @ (A > (A > A))) @ (B > (B > B)))) @ (id2 @ $o))) @ (((((relAPP @ (A > A)) @ (B > B)) @ (set @ ((product_prod @ (A > (A > A))) @ (B > (B > B))))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ (A > A)) @ (B > B))) > (set @ ((product_prod @ (A > (A > A))) @ (B > (B > B)))))) @ ((((fun_rel @ A) @ B) @ (A > A)) @ (B > B))) @ R3)) @ (((((relAPP @ A) @ B) @ (set @ ((product_prod @ (A > A)) @ (B > B)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ (A > A)) @ (B > B))))) @ ((((fun_rel @ A) @ B) @ A) @ B)) @ R3)) @ R3)))))) @ C2)) @ T2)) @ E4))) @ R3))))))).
% 3.09/1.54 thf(fact_73_param__fun__upd, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,Ra:(set @ ((product_prod @ A) @ B)),Rb:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ (A > (A > $o))) @ (B > (B > $o)))) @ ((((product_Pair @ (A > (A > $o))) @ (B > (B > $o))) @ (^ [Y5:A,Z:A]: ((Y5 = Z)))) @ (^ [Y5:B,Z:B]: ((Y5 = Z))))) @ (((((relAPP @ (A > $o)) @ (B > $o)) @ (set @ ((product_prod @ (A > (A > $o))) @ (B > (B > $o))))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ (A > $o)) @ (B > $o))) > (set @ ((product_prod @ (A > (A > $o))) @ (B > (B > $o)))))) @ ((((fun_rel @ A) @ B) @ (A > $o)) @ (B > $o))) @ Ra)) @ (((((relAPP @ $o) @ $o) @ (set @ ((product_prod @ (A > $o)) @ (B > $o)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ $o) @ $o)) > (set @ ((product_prod @ (A > $o)) @ (B > $o))))) @ ((((fun_rel @ A) @ B) @ $o) @ $o)) @ Ra)) @ (id2 @ $o)))) => (((member @ ((product_prod @ ((A > C) > (A > (C > (A > C))))) @ ((B > D) > (B > (D > (B > D)))))) @ ((((product_Pair @ ((A > C) > (A > (C > (A > C))))) @ ((B > D) > (B > (D > (B > D))))) @ ((fun_upd @ A) @ C)) @ ((fun_upd @ B) @ D))) @ (((((relAPP @ (A > (C > (A > C)))) @ (B > (D > (B > D)))) @ (set @ ((product_prod @ ((A > C) > (A > (C > (A > C))))) @ ((B > D) > (B > (D > (B > D))))))) @ (((((relAPP @ (A > C)) @ (B > D)) @ ((set @ ((product_prod @ (A > (C > (A > C)))) @ (B > (D > (B > D))))) > (set @ ((product_prod @ ((A > C) > (A > (C > (A > C))))) @ ((B > D) > (B > (D > (B > D)))))))) @ ((((fun_rel @ (A > C)) @ (B > D)) @ (A > (C > (A > C)))) @ (B > (D > (B > D))))) @ (((((relAPP @ C) @ D) @ (set @ ((product_prod @ (A > C)) @ (B > D)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ C) @ D)) > (set @ ((product_prod @ (A > C)) @ (B > D))))) @ ((((fun_rel @ A) @ B) @ C) @ D)) @ Ra)) @ Rb))) @ (((((relAPP @ (C > (A > C))) @ (D > (B > D))) @ (set @ ((product_prod @ (A > (C > (A > C)))) @ (B > (D > (B > D)))))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ (C > (A > C))) @ (D > (B > D)))) > (set @ ((product_prod @ (A > (C > (A > C)))) @ (B > (D > (B > D))))))) @ ((((fun_rel @ A) @ B) @ (C > (A > C))) @ (D > (B > D)))) @ Ra)) @ (((((relAPP @ (A > C)) @ (B > D)) @ (set @ ((product_prod @ (C > (A > C))) @ (D > (B > D))))) @ (((((relAPP @ C) @ D) @ ((set @ ((product_prod @ (A > C)) @ (B > D))) > (set @ ((product_prod @ (C > (A > C))) @ (D > (B > D)))))) @ ((((fun_rel @ C) @ D) @ (A > C)) @ (B > D))) @ Rb)) @ (((((relAPP @ C) @ D) @ (set @ ((product_prod @ (A > C)) @ (B > D)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ C) @ D)) > (set @ ((product_prod @ (A > C)) @ (B > D))))) @ ((((fun_rel @ A) @ B) @ C) @ D)) @ Ra)) @ Rb))))))))).
% 3.09/1.54 thf(fact_74_id__apply, axiom, (! [A:$tType]: (((id @ A) = (^ [X6:A]: (X6)))))).
% 3.09/1.54 thf(fact_75_prod__refine_I3_J, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F4:$tType,E:$tType,Ra:(set @ ((product_prod @ A) @ C)),Rb:(set @ ((product_prod @ B) @ D)),Rr:(set @ ((product_prod @ E) @ F4))]: ((((member @ ((product_prod @ ((A > (B > E)) > (((product_prod @ A) @ B) > E))) @ ((C > (D > F4)) > (((product_prod @ C) @ D) > F4)))) @ ((((product_Pair @ ((A > (B > E)) > (((product_prod @ A) @ B) > E))) @ ((C > (D > F4)) > (((product_prod @ C) @ D) > F4))) @ (((product_rec_prod @ A) @ B) @ E)) @ (((product_rec_prod @ C) @ D) @ F4))) @ (((((relAPP @ (((product_prod @ A) @ B) > E)) @ (((product_prod @ C) @ D) > F4)) @ (set @ ((product_prod @ ((A > (B > E)) > (((product_prod @ A) @ B) > E))) @ ((C > (D > F4)) > (((product_prod @ C) @ D) > F4))))) @ (((((relAPP @ (A > (B > E))) @ (C > (D > F4))) @ ((set @ ((product_prod @ (((product_prod @ A) @ B) > E)) @ (((product_prod @ C) @ D) > F4))) > (set @ ((product_prod @ ((A > (B > E)) > (((product_prod @ A) @ B) > E))) @ ((C > (D > F4)) > (((product_prod @ C) @ D) > F4)))))) @ ((((fun_rel @ (A > (B > E))) @ (C > (D > F4))) @ (((product_prod @ A) @ B) > E)) @ (((product_prod @ C) @ D) > F4))) @ (((((relAPP @ (B > E)) @ (D > F4)) @ (set @ ((product_prod @ (A > (B > E))) @ (C > (D > F4))))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ (B > E)) @ (D > F4))) > (set @ ((product_prod @ (A > (B > E))) @ (C > (D > F4)))))) @ ((((fun_rel @ A) @ C) @ (B > E)) @ (D > F4))) @ Ra)) @ (((((relAPP @ E) @ F4) @ (set @ ((product_prod @ (B > E)) @ (D > F4)))) @ (((((relAPP @ B) @ D) @ ((set @ ((product_prod @ E) @ F4)) > (set @ ((product_prod @ (B > E)) @ (D > F4))))) @ ((((fun_rel @ B) @ D) @ E) @ F4)) @ Rb)) @ Rr)))) @ (((((relAPP @ E) @ F4) @ (set @ ((product_prod @ (((product_prod @ A) @ B) > E)) @ (((product_prod @ C) @ D) > F4)))) @ (((((relAPP @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((set @ ((product_prod @ E) @ F4)) > (set @ ((product_prod @ (((product_prod @ A) @ B) > E)) @ (((product_prod @ C) @ D) > F4))))) @ ((((fun_rel @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ E) @ F4)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))))) @ ((((prod_rel @ A) @ C) @ B) @ D)) @ Ra)) @ Rb))) @ Rr)))))).
% 3.09/1.54 thf(fact_76_autoref__bool_I5_J, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B))]: ((((member @ ((product_prod @ (A > (A > ($o > A)))) @ (B > (B > ($o > B))))) @ ((((product_Pair @ (A > (A > ($o > A)))) @ (B > (B > ($o > B)))) @ (product_case_bool @ A)) @ (product_case_bool @ B))) @ (((((relAPP @ (A > ($o > A))) @ (B > ($o > B))) @ (set @ ((product_prod @ (A > (A > ($o > A)))) @ (B > (B > ($o > B)))))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ (A > ($o > A))) @ (B > ($o > B)))) > (set @ ((product_prod @ (A > (A > ($o > A)))) @ (B > (B > ($o > B))))))) @ ((((fun_rel @ A) @ B) @ (A > ($o > A))) @ (B > ($o > B)))) @ R3)) @ (((((relAPP @ ($o > A)) @ ($o > B)) @ (set @ ((product_prod @ (A > ($o > A))) @ (B > ($o > B))))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ ($o > A)) @ ($o > B))) > (set @ ((product_prod @ (A > ($o > A))) @ (B > ($o > B)))))) @ ((((fun_rel @ A) @ B) @ ($o > A)) @ ($o > B))) @ R3)) @ (((((relAPP @ A) @ B) @ (set @ ((product_prod @ ($o > A)) @ ($o > B)))) @ (((((relAPP @ $o) @ $o) @ ((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ ($o > A)) @ ($o > B))))) @ ((((fun_rel @ $o) @ $o) @ A) @ B)) @ (id2 @ $o))) @ R3))))))).
% 3.09/1.54 thf(fact_77_fun__upd__upd, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B,Z2:B]: (((((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ F) @ X) @ Y)) @ X) @ Z2) = (((((fun_upd @ A) @ B) @ F) @ X) @ Z2))))).
% 3.09/1.54 thf(fact_78_fun__upd__triv, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A]: (((((((fun_upd @ A) @ B) @ F) @ X) @ (F @ X)) = F)))).
% 3.09/1.54 thf(fact_79_fun__upd__apply, axiom, (! [A:$tType,B:$tType]: ((((fun_upd @ B) @ A) = (^ [F3:(B > A),X6:B,Y6:A,Z3:B]: (((((if @ A) @ (Z3 = X6)) @ Y6) @ (F3 @ Z3)))))))).
% 3.09/1.54 thf(fact_80_old_Obool_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F22:T]: ((((((product_rec_bool @ T) @ F1) @ F22) @ $false) = F22)))).
% 3.09/1.54 thf(fact_81_old_Obool_Osimps_I5_J, axiom, (! [T:$tType,F1:T,F22:T]: ((((((product_rec_bool @ T) @ F1) @ F22) @ $true) = F1)))).
% 3.09/1.54 thf(fact_82_prod__rel__simp, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A5:A,B3:B,A6:C,B4:D,R1:(set @ ((product_prod @ A) @ C)),R22:(set @ ((product_prod @ B) @ D))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) @ ((((product_Pair @ C) @ D) @ A6) @ B4))) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))))) @ ((((prod_rel @ A) @ C) @ B) @ D)) @ R1)) @ R22)) = ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A5) @ A6)) @ R1) & (((member @ ((product_prod @ B) @ D)) @ ((((product_Pair @ B) @ D) @ B3) @ B4)) @ R22)))))).
% 3.09/1.54 thf(fact_83_prod__rel__id__simp, axiom, (! [B:$tType,A:$tType]: (((((((relAPP @ B) @ B) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (((((relAPP @ A) @ A) @ ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))))) @ ((((prod_rel @ A) @ A) @ B) @ B)) @ (id2 @ A))) @ (id2 @ B)) = (id2 @ ((product_prod @ A) @ B)))))).
% 3.09/1.54 thf(fact_84_fun__upd__def, axiom, (! [B:$tType,A:$tType]: ((((fun_upd @ A) @ B) = (^ [F3:(A > B),A7:A,B6:B,X6:A]: (((((if @ B) @ (X6 = A7)) @ B6) @ (F3 @ X6)))))))).
% 3.09/1.54 thf(fact_85_fun__upd__eqD, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B,G3:(A > B),Z2:B]: ((((((((fun_upd @ A) @ B) @ F) @ X) @ Y) = (((((fun_upd @ A) @ B) @ G3) @ X) @ Z2)) => (Y = Z2))))).
% 3.09/1.54 thf(fact_86_fun__upd__idem, axiom, (! [A:$tType,B:$tType,F:(B > A),X:B,Y:A]: ((((F @ X) = Y) => ((((((fun_upd @ B) @ A) @ F) @ X) @ Y) = F))))).
% 3.09/1.54 thf(fact_87_fun__upd__same, axiom, (! [B:$tType,A:$tType,F:(B > A),X:B,Y:A]: ((((((((fun_upd @ B) @ A) @ F) @ X) @ Y) @ X) = Y)))).
% 3.09/1.54 thf(fact_88_fun__upd__other, axiom, (! [B:$tType,A:$tType,Z2:A,X:A,F:(A > B),Y:B]: (((Z2 != X) => (((((((fun_upd @ A) @ B) @ F) @ X) @ Y) @ Z2) = (F @ Z2)))))).
% 3.09/1.54 thf(fact_89_fun__upd__twist, axiom, (! [A:$tType,B:$tType,A5:A,C2:A,M:(A > B),B3:B,D3:B]: (((A5 != C2) => ((((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ M) @ A5) @ B3)) @ C2) @ D3) = (((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ M) @ C2) @ D3)) @ A5) @ B3)))))).
% 3.09/1.54 thf(fact_90_fun__upd__idem__iff, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B]: ((((((((fun_upd @ A) @ B) @ F) @ X) @ Y) = F) = ((F @ X) = Y))))).
% 3.09/1.54 thf(fact_91_rec__bool__is__case, axiom, (! [A:$tType]: (((product_rec_bool @ A) = (product_case_bool @ A))))).
% 3.09/1.54 thf(fact_92_bool_Osplit__sel__asm, axiom, (! [A:$tType,P:(A > $o),F1:A,F22:A,Bool:$o]: (((P @ ((((product_case_bool @ A) @ F1) @ F22) @ Bool)) = (~ (((Bool & (~ ((P @ F1)))) | ((~ (Bool)) & (~ ((P @ F22))))))))))).
% 3.09/1.54 thf(fact_93_bool_Ocase__distrib, axiom, (! [A:$tType,B:$tType,H:(A > B),F1:A,F22:A,Bool:$o]: (((H @ ((((product_case_bool @ A) @ F1) @ F22) @ Bool)) = ((((product_case_bool @ B) @ (H @ F1)) @ (H @ F22)) @ Bool))))).
% 3.09/1.54 thf(fact_94_bool_Ocase__eq__if, axiom, (! [A:$tType]: (((product_case_bool @ A) = (^ [F12:A,F23:A,Bool2:$o]: (((((if @ A) @ Bool2) @ F12) @ F23))))))).
% 3.09/1.54 thf(fact_95_bool_Osplit__sel, axiom, (! [A:$tType,P:(A > $o),F1:A,F22:A,Bool:$o]: (((P @ ((((product_case_bool @ A) @ F1) @ F22) @ Bool)) = ((Bool => (P @ F1)) & ((~ (Bool)) => (P @ F22))))))).
% 3.09/1.54 thf(fact_96_old_Obool_Osimps_I3_J, axiom, (! [A:$tType,F1:A,F22:A]: ((((((product_case_bool @ A) @ F1) @ F22) @ $true) = F1)))).
% 3.09/1.54 thf(fact_97_old_Obool_Osimps_I4_J, axiom, (! [A:$tType,F1:A,F22:A]: ((((((product_case_bool @ A) @ F1) @ F22) @ $false) = F22)))).
% 3.09/1.54 thf(fact_98_prod__relI, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,A5:A,A6:B,R1:(set @ ((product_prod @ A) @ B)),B3:C,B4:D,R22:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ A6)) @ R1) => ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ B3) @ B4)) @ R22) => (((member @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D))) @ ((((product_Pair @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((product_Pair @ A) @ C) @ A5) @ B3)) @ ((((product_Pair @ B) @ D) @ A6) @ B4))) @ (((((relAPP @ C) @ D) @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ C) @ D)) > (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D))))) @ ((((prod_rel @ A) @ B) @ C) @ D)) @ R1)) @ R22))))))).
% 3.09/1.54 thf(fact_99_prod__relE, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,P2:((product_prod @ A) @ B),P3:((product_prod @ C) @ D),R1:(set @ ((product_prod @ A) @ C)),R22:(set @ ((product_prod @ B) @ D))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ P2) @ P3)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))))) @ ((((prod_rel @ A) @ C) @ B) @ D)) @ R1)) @ R22)) => (~ ((! [A3:A,B5:B]: (((P2 = ((((product_Pair @ A) @ B) @ A3) @ B5)) => (! [A4:C,B7:D]: (((P3 = ((((product_Pair @ C) @ D) @ A4) @ B7)) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A3) @ A4)) @ R1) => (~ ((((member @ ((product_prod @ B) @ D)) @ ((((product_Pair @ B) @ D) @ B5) @ B7)) @ R22)))))))))))))))).
% 3.09/1.54 thf(fact_100_prod__refine_I1_J, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,Ra:(set @ ((product_prod @ A) @ C)),Rb:(set @ ((product_prod @ B) @ D))]: ((((member @ ((product_prod @ (A > (B > ((product_prod @ A) @ B)))) @ (C > (D > ((product_prod @ C) @ D))))) @ ((((product_Pair @ (A > (B > ((product_prod @ A) @ B)))) @ (C > (D > ((product_prod @ C) @ D)))) @ ((product_Pair @ A) @ B)) @ ((product_Pair @ C) @ D))) @ (((((relAPP @ (B > ((product_prod @ A) @ B))) @ (D > ((product_prod @ C) @ D))) @ (set @ ((product_prod @ (A > (B > ((product_prod @ A) @ B)))) @ (C > (D > ((product_prod @ C) @ D)))))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ (B > ((product_prod @ A) @ B))) @ (D > ((product_prod @ C) @ D)))) > (set @ ((product_prod @ (A > (B > ((product_prod @ A) @ B)))) @ (C > (D > ((product_prod @ C) @ D))))))) @ ((((fun_rel @ A) @ C) @ (B > ((product_prod @ A) @ B))) @ (D > ((product_prod @ C) @ D)))) @ Ra)) @ (((((relAPP @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ (set @ ((product_prod @ (B > ((product_prod @ A) @ B))) @ (D > ((product_prod @ C) @ D))))) @ (((((relAPP @ B) @ D) @ ((set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))) > (set @ ((product_prod @ (B > ((product_prod @ A) @ B))) @ (D > ((product_prod @ C) @ D)))))) @ ((((fun_rel @ B) @ D) @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))) @ Rb)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))))) @ ((((prod_rel @ A) @ C) @ B) @ D)) @ Ra)) @ Rb))))))).
% 3.09/1.54 thf(fact_101_prod__rel__id, axiom, (! [B:$tType,A:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B))]: (((R1 = (id2 @ A)) => ((R22 = (id2 @ B)) => ((((((relAPP @ B) @ B) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (((((relAPP @ A) @ A) @ ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))))) @ ((((prod_rel @ A) @ A) @ B) @ B)) @ R1)) @ R22) = (id2 @ ((product_prod @ A) @ B)))))))).
% 3.09/1.54 thf(fact_102_id__def, axiom, (! [A:$tType]: (((id @ A) = (^ [X6:A]: (X6)))))).
% 3.09/1.54 thf(fact_103_eq__id__iff, axiom, (! [A:$tType,F:(A > A)]: (((! [X6:A]: (((F @ X6) = X6))) = (F = (id @ A)))))).
% 3.09/1.54 thf(fact_104_param__and__cong2, axiom, (! [A5:$o,A6:$o,B3:$o,B4:$o]: (((((member @ ((product_prod @ $o) @ $o)) @ ((((product_Pair @ $o) @ $o) @ A5) @ A6)) @ (id2 @ $o)) => ((A5 => (A6 => (((member @ ((product_prod @ $o) @ $o)) @ ((((product_Pair @ $o) @ $o) @ B3) @ B4)) @ (id2 @ $o)))) => (((member @ ((product_prod @ $o) @ $o)) @ ((((product_Pair @ $o) @ $o) @ (B3 & A5)) @ (B4 & A6))) @ (id2 @ $o))))))).
% 3.09/1.54 thf(fact_105_param__and__cong1, axiom, (! [A5:$o,A6:$o,B3:$o,B4:$o]: (((((member @ ((product_prod @ $o) @ $o)) @ ((((product_Pair @ $o) @ $o) @ A5) @ A6)) @ (id2 @ $o)) => ((A5 => (A6 => (((member @ ((product_prod @ $o) @ $o)) @ ((((product_Pair @ $o) @ $o) @ B3) @ B4)) @ (id2 @ $o)))) => (((member @ ((product_prod @ $o) @ $o)) @ ((((product_Pair @ $o) @ $o) @ (A5 & B3)) @ (A6 & B4))) @ (id2 @ $o))))))).
% 3.09/1.54 thf(fact_106_param__bool_I1_J, axiom, (((member @ ((product_prod @ $o) @ $o)) @ ((((product_Pair @ $o) @ $o) @ $true) @ $true)) @ (id2 @ $o))).
% 3.09/1.54 thf(fact_107_param__bool_I2_J, axiom, (((member @ ((product_prod @ $o) @ $o)) @ ((((product_Pair @ $o) @ $o) @ $false) @ $false)) @ (id2 @ $o))).
% 3.09/1.54 thf(fact_108_IdE, axiom, (! [A:$tType,P2:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ P2) @ (id2 @ A)) => (~ ((! [X5:A]: ((P2 != ((((product_Pair @ A) @ A) @ X5) @ X5)))))))))).
% 3.09/1.54 thf(fact_109_param__if, axiom, (! [B:$tType,A:$tType,C2:$o,C4:$o,T2:A,T3:B,R3:(set @ ((product_prod @ A) @ B)),E4:A,E3:B]: (((((member @ ((product_prod @ $o) @ $o)) @ ((((product_Pair @ $o) @ $o) @ C2) @ C4)) @ (id2 @ $o)) => ((C2 => (C4 => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ T2) @ T3)) @ R3))) => (((~ (C2)) => ((~ (C4)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ E4) @ E3)) @ R3))) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ ((((if @ A) @ C2) @ T2) @ E4)) @ ((((if @ B) @ C4) @ T3) @ E3))) @ R3))))))).
% 3.09/1.54 thf(fact_110_param__uncurry, axiom, (! [B:$tType,E:$tType,D:$tType,A:$tType,F4:$tType,C:$tType,A2:(set @ ((product_prod @ A) @ D)),B2:(set @ ((product_prod @ B) @ E)),C5:(set @ ((product_prod @ C) @ F4))]: ((((member @ ((product_prod @ ((A > (B > C)) > (((product_prod @ A) @ B) > C))) @ ((D > (E > F4)) > (((product_prod @ D) @ E) > F4)))) @ ((((product_Pair @ ((A > (B > C)) > (((product_prod @ A) @ B) > C))) @ ((D > (E > F4)) > (((product_prod @ D) @ E) > F4))) @ (((uncurry @ A) @ B) @ C)) @ (((uncurry @ D) @ E) @ F4))) @ (((((relAPP @ (((product_prod @ A) @ B) > C)) @ (((product_prod @ D) @ E) > F4)) @ (set @ ((product_prod @ ((A > (B > C)) > (((product_prod @ A) @ B) > C))) @ ((D > (E > F4)) > (((product_prod @ D) @ E) > F4))))) @ (((((relAPP @ (A > (B > C))) @ (D > (E > F4))) @ ((set @ ((product_prod @ (((product_prod @ A) @ B) > C)) @ (((product_prod @ D) @ E) > F4))) > (set @ ((product_prod @ ((A > (B > C)) > (((product_prod @ A) @ B) > C))) @ ((D > (E > F4)) > (((product_prod @ D) @ E) > F4)))))) @ ((((fun_rel @ (A > (B > C))) @ (D > (E > F4))) @ (((product_prod @ A) @ B) > C)) @ (((product_prod @ D) @ E) > F4))) @ (((((relAPP @ (B > C)) @ (E > F4)) @ (set @ ((product_prod @ (A > (B > C))) @ (D > (E > F4))))) @ (((((relAPP @ A) @ D) @ ((set @ ((product_prod @ (B > C)) @ (E > F4))) > (set @ ((product_prod @ (A > (B > C))) @ (D > (E > F4)))))) @ ((((fun_rel @ A) @ D) @ (B > C)) @ (E > F4))) @ A2)) @ (((((relAPP @ C) @ F4) @ (set @ ((product_prod @ (B > C)) @ (E > F4)))) @ (((((relAPP @ B) @ E) @ ((set @ ((product_prod @ C) @ F4)) > (set @ ((product_prod @ (B > C)) @ (E > F4))))) @ ((((fun_rel @ B) @ E) @ C) @ F4)) @ B2)) @ C5)))) @ (((((relAPP @ C) @ F4) @ (set @ ((product_prod @ (((product_prod @ A) @ B) > C)) @ (((product_prod @ D) @ E) > F4)))) @ (((((relAPP @ ((product_prod @ A) @ B)) @ ((product_prod @ D) @ E)) @ ((set @ ((product_prod @ C) @ F4)) > (set @ ((product_prod @ (((product_prod @ A) @ B) > C)) @ (((product_prod @ D) @ E) > F4))))) @ ((((fun_rel @ ((product_prod @ A) @ B)) @ ((product_prod @ D) @ E)) @ C) @ F4)) @ (((((relAPP @ B) @ E) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ D) @ E)))) @ (((((relAPP @ A) @ D) @ ((set @ ((product_prod @ B) @ E)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ D) @ E))))) @ ((((prod_rel @ A) @ D) @ B) @ E)) @ A2)) @ B2))) @ C5)))))).
% 3.09/1.54 thf(fact_111_param__all__interval__nat, axiom, (((member @ ((product_prod @ ((nat > $o) > (nat > (nat > $o)))) @ ((nat > $o) > (nat > (nat > $o))))) @ ((((product_Pair @ ((nat > $o) > (nat > (nat > $o)))) @ ((nat > $o) > (nat > (nat > $o)))) @ all_interval_nat) @ all_interval_nat)) @ (((((relAPP @ (nat > (nat > $o))) @ (nat > (nat > $o))) @ (set @ ((product_prod @ ((nat > $o) > (nat > (nat > $o)))) @ ((nat > $o) > (nat > (nat > $o)))))) @ (((((relAPP @ (nat > $o)) @ (nat > $o)) @ ((set @ ((product_prod @ (nat > (nat > $o))) @ (nat > (nat > $o)))) > (set @ ((product_prod @ ((nat > $o) > (nat > (nat > $o)))) @ ((nat > $o) > (nat > (nat > $o))))))) @ ((((fun_rel @ (nat > $o)) @ (nat > $o)) @ (nat > (nat > $o))) @ (nat > (nat > $o)))) @ (((((relAPP @ $o) @ $o) @ (set @ ((product_prod @ (nat > $o)) @ (nat > $o)))) @ (((((relAPP @ nat) @ nat) @ ((set @ ((product_prod @ $o) @ $o)) > (set @ ((product_prod @ (nat > $o)) @ (nat > $o))))) @ ((((fun_rel @ nat) @ nat) @ $o) @ $o)) @ (id2 @ nat))) @ (id2 @ $o)))) @ (((((relAPP @ (nat > $o)) @ (nat > $o)) @ (set @ ((product_prod @ (nat > (nat > $o))) @ (nat > (nat > $o))))) @ (((((relAPP @ nat) @ nat) @ ((set @ ((product_prod @ (nat > $o)) @ (nat > $o))) > (set @ ((product_prod @ (nat > (nat > $o))) @ (nat > (nat > $o)))))) @ ((((fun_rel @ nat) @ nat) @ (nat > $o)) @ (nat > $o))) @ (id2 @ nat))) @ (((((relAPP @ $o) @ $o) @ (set @ ((product_prod @ (nat > $o)) @ (nat > $o)))) @ (((((relAPP @ nat) @ nat) @ ((set @ ((product_prod @ $o) @ $o)) > (set @ ((product_prod @ (nat > $o)) @ (nat > $o))))) @ ((((fun_rel @ nat) @ nat) @ $o) @ $o)) @ (id2 @ nat))) @ (id2 @ $o)))))).
% 3.09/1.54 thf(fact_112_param__curry, axiom, (! [A:$tType,D:$tType,B:$tType,E:$tType,F4:$tType,C:$tType,Ra:(set @ ((product_prod @ A) @ D)),Rb:(set @ ((product_prod @ B) @ E)),Rc:(set @ ((product_prod @ C) @ F4))]: ((((member @ ((product_prod @ ((((product_prod @ A) @ B) > C) > (A > (B > C)))) @ ((((product_prod @ D) @ E) > F4) > (D > (E > F4))))) @ ((((product_Pair @ ((((product_prod @ A) @ B) > C) > (A > (B > C)))) @ ((((product_prod @ D) @ E) > F4) > (D > (E > F4)))) @ (((product_curry @ A) @ B) @ C)) @ (((product_curry @ D) @ E) @ F4))) @ (((((relAPP @ (A > (B > C))) @ (D > (E > F4))) @ (set @ ((product_prod @ ((((product_prod @ A) @ B) > C) > (A > (B > C)))) @ ((((product_prod @ D) @ E) > F4) > (D > (E > F4)))))) @ (((((relAPP @ (((product_prod @ A) @ B) > C)) @ (((product_prod @ D) @ E) > F4)) @ ((set @ ((product_prod @ (A > (B > C))) @ (D > (E > F4)))) > (set @ ((product_prod @ ((((product_prod @ A) @ B) > C) > (A > (B > C)))) @ ((((product_prod @ D) @ E) > F4) > (D > (E > F4))))))) @ ((((fun_rel @ (((product_prod @ A) @ B) > C)) @ (((product_prod @ D) @ E) > F4)) @ (A > (B > C))) @ (D > (E > F4)))) @ (((((relAPP @ C) @ F4) @ (set @ ((product_prod @ (((product_prod @ A) @ B) > C)) @ (((product_prod @ D) @ E) > F4)))) @ (((((relAPP @ ((product_prod @ A) @ B)) @ ((product_prod @ D) @ E)) @ ((set @ ((product_prod @ C) @ F4)) > (set @ ((product_prod @ (((product_prod @ A) @ B) > C)) @ (((product_prod @ D) @ E) > F4))))) @ ((((fun_rel @ ((product_prod @ A) @ B)) @ ((product_prod @ D) @ E)) @ C) @ F4)) @ (((((relAPP @ B) @ E) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ D) @ E)))) @ (((((relAPP @ A) @ D) @ ((set @ ((product_prod @ B) @ E)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ D) @ E))))) @ ((((prod_rel @ A) @ D) @ B) @ E)) @ Ra)) @ Rb))) @ Rc))) @ (((((relAPP @ (B > C)) @ (E > F4)) @ (set @ ((product_prod @ (A > (B > C))) @ (D > (E > F4))))) @ (((((relAPP @ A) @ D) @ ((set @ ((product_prod @ (B > C)) @ (E > F4))) > (set @ ((product_prod @ (A > (B > C))) @ (D > (E > F4)))))) @ ((((fun_rel @ A) @ D) @ (B > C)) @ (E > F4))) @ Ra)) @ (((((relAPP @ C) @ F4) @ (set @ ((product_prod @ (B > C)) @ (E > F4)))) @ (((((relAPP @ B) @ E) @ ((set @ ((product_prod @ C) @ F4)) > (set @ ((product_prod @ (B > C)) @ (E > F4))))) @ ((((fun_rel @ B) @ E) @ C) @ F4)) @ Rb)) @ Rc))))))).
% 3.09/1.54 thf(fact_113_prod__refine_I2_J, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F4:$tType,E:$tType,Ra:(set @ ((product_prod @ A) @ C)),Rb:(set @ ((product_prod @ B) @ D)),Rr:(set @ ((product_prod @ E) @ F4))]: ((((member @ ((product_prod @ ((A > (B > E)) > (((product_prod @ A) @ B) > E))) @ ((C > (D > F4)) > (((product_prod @ C) @ D) > F4)))) @ ((((product_Pair @ ((A > (B > E)) > (((product_prod @ A) @ B) > E))) @ ((C > (D > F4)) > (((product_prod @ C) @ D) > F4))) @ (((product_case_prod @ A) @ B) @ E)) @ (((product_case_prod @ C) @ D) @ F4))) @ (((((relAPP @ (((product_prod @ A) @ B) > E)) @ (((product_prod @ C) @ D) > F4)) @ (set @ ((product_prod @ ((A > (B > E)) > (((product_prod @ A) @ B) > E))) @ ((C > (D > F4)) > (((product_prod @ C) @ D) > F4))))) @ (((((relAPP @ (A > (B > E))) @ (C > (D > F4))) @ ((set @ ((product_prod @ (((product_prod @ A) @ B) > E)) @ (((product_prod @ C) @ D) > F4))) > (set @ ((product_prod @ ((A > (B > E)) > (((product_prod @ A) @ B) > E))) @ ((C > (D > F4)) > (((product_prod @ C) @ D) > F4)))))) @ ((((fun_rel @ (A > (B > E))) @ (C > (D > F4))) @ (((product_prod @ A) @ B) > E)) @ (((product_prod @ C) @ D) > F4))) @ (((((relAPP @ (B > E)) @ (D > F4)) @ (set @ ((product_prod @ (A > (B > E))) @ (C > (D > F4))))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ (B > E)) @ (D > F4))) > (set @ ((product_prod @ (A > (B > E))) @ (C > (D > F4)))))) @ ((((fun_rel @ A) @ C) @ (B > E)) @ (D > F4))) @ Ra)) @ (((((relAPP @ E) @ F4) @ (set @ ((product_prod @ (B > E)) @ (D > F4)))) @ (((((relAPP @ B) @ D) @ ((set @ ((product_prod @ E) @ F4)) > (set @ ((product_prod @ (B > E)) @ (D > F4))))) @ ((((fun_rel @ B) @ D) @ E) @ F4)) @ Rb)) @ Rr)))) @ (((((relAPP @ E) @ F4) @ (set @ ((product_prod @ (((product_prod @ A) @ B) > E)) @ (((product_prod @ C) @ D) > F4)))) @ (((((relAPP @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((set @ ((product_prod @ E) @ F4)) > (set @ ((product_prod @ (((product_prod @ A) @ B) > E)) @ (((product_prod @ C) @ D) > F4))))) @ ((((fun_rel @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ E) @ F4)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))))) @ ((((prod_rel @ A) @ C) @ B) @ D)) @ Ra)) @ Rb))) @ Rr)))))).
% 3.09/1.54 thf(fact_114_autoref__rec__nat, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B))]: ((((member @ ((product_prod @ (A > ((nat > (A > A)) > (nat > A)))) @ (B > ((nat > (B > B)) > (nat > B))))) @ ((((product_Pair @ (A > ((nat > (A > A)) > (nat > A)))) @ (B > ((nat > (B > B)) > (nat > B)))) @ (rec_nat @ A)) @ (rec_nat @ B))) @ (((((relAPP @ ((nat > (A > A)) > (nat > A))) @ ((nat > (B > B)) > (nat > B))) @ (set @ ((product_prod @ (A > ((nat > (A > A)) > (nat > A)))) @ (B > ((nat > (B > B)) > (nat > B)))))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ ((nat > (A > A)) > (nat > A))) @ ((nat > (B > B)) > (nat > B)))) > (set @ ((product_prod @ (A > ((nat > (A > A)) > (nat > A)))) @ (B > ((nat > (B > B)) > (nat > B))))))) @ ((((fun_rel @ A) @ B) @ ((nat > (A > A)) > (nat > A))) @ ((nat > (B > B)) > (nat > B)))) @ R3)) @ (((((relAPP @ (nat > A)) @ (nat > B)) @ (set @ ((product_prod @ ((nat > (A > A)) > (nat > A))) @ ((nat > (B > B)) > (nat > B))))) @ (((((relAPP @ (nat > (A > A))) @ (nat > (B > B))) @ ((set @ ((product_prod @ (nat > A)) @ (nat > B))) > (set @ ((product_prod @ ((nat > (A > A)) > (nat > A))) @ ((nat > (B > B)) > (nat > B)))))) @ ((((fun_rel @ (nat > (A > A))) @ (nat > (B > B))) @ (nat > A)) @ (nat > B))) @ (((((relAPP @ (A > A)) @ (B > B)) @ (set @ ((product_prod @ (nat > (A > A))) @ (nat > (B > B))))) @ (((((relAPP @ nat) @ nat) @ ((set @ ((product_prod @ (A > A)) @ (B > B))) > (set @ ((product_prod @ (nat > (A > A))) @ (nat > (B > B)))))) @ ((((fun_rel @ nat) @ nat) @ (A > A)) @ (B > B))) @ (id2 @ nat))) @ (((((relAPP @ A) @ B) @ (set @ ((product_prod @ (A > A)) @ (B > B)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ (A > A)) @ (B > B))))) @ ((((fun_rel @ A) @ B) @ A) @ B)) @ R3)) @ R3)))) @ (((((relAPP @ A) @ B) @ (set @ ((product_prod @ (nat > A)) @ (nat > B)))) @ (((((relAPP @ nat) @ nat) @ ((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ (nat > A)) @ (nat > B))))) @ ((((fun_rel @ nat) @ nat) @ A) @ B)) @ (id2 @ nat))) @ R3))))))).
% 3.09/1.54 thf(fact_115_autoref__int_I5_J, axiom, (((member @ ((product_prod @ (int > (int > $o))) @ (int > (int > $o)))) @ ((((product_Pair @ (int > (int > $o))) @ (int > (int > $o))) @ (ord_less_eq @ int)) @ (ord_less_eq @ int))) @ (((((relAPP @ (int > $o)) @ (int > $o)) @ (set @ ((product_prod @ (int > (int > $o))) @ (int > (int > $o))))) @ (((((relAPP @ int) @ int) @ ((set @ ((product_prod @ (int > $o)) @ (int > $o))) > (set @ ((product_prod @ (int > (int > $o))) @ (int > (int > $o)))))) @ ((((fun_rel @ int) @ int) @ (int > $o)) @ (int > $o))) @ (id2 @ int))) @ (((((relAPP @ $o) @ $o) @ (set @ ((product_prod @ (int > $o)) @ (int > $o)))) @ (((((relAPP @ int) @ int) @ ((set @ ((product_prod @ $o) @ $o)) > (set @ ((product_prod @ (int > $o)) @ (int > $o))))) @ ((((fun_rel @ int) @ int) @ $o) @ $o)) @ (id2 @ int))) @ (id2 @ $o))))).
% 3.09/1.54 thf(fact_116_case__prod__Pair, axiom, (! [B:$tType,A:$tType]: ((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) = (id @ ((product_prod @ A) @ B)))))).
% 3.09/1.54 thf(fact_117_curryI, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A5:A,B3:B]: (((F @ ((((product_Pair @ A) @ B) @ A5) @ B3)) => ((((((product_curry @ A) @ B) @ $o) @ F) @ A5) @ B3))))).
% 3.09/1.54 thf(fact_118_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F3:(((product_prod @ B) @ C) > A),A7:B,B6:C]: ((F3 @ ((((product_Pair @ B) @ C) @ A7) @ B6)))))))).
% 3.09/1.54 thf(fact_119_case__prod__curry, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ ((((product_curry @ A) @ B) @ C) @ F)) = F)))).
% 3.09/1.54 thf(fact_120_curry__case__prod, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C))]: ((((((product_curry @ A) @ B) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F)) = F)))).
% 3.09/1.54 thf(fact_121_lhs__step__If, axiom, (! [A:$tType]: (((ord @ A) => (! [B3:$o,T2:A,M:A,E4:A]: (((B3 => (((ord_less_eq @ A) @ T2) @ M)) => (((~ (B3)) => (((ord_less_eq @ A) @ E4) @ M)) => (((ord_less_eq @ A) @ ((((if @ A) @ B3) @ T2) @ E4)) @ M))))))))).
% 3.09/1.54 thf(fact_122_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z2:A,C2:(B > (C > (set @ A))),P2:((product_prod @ B) @ C)]: (((((member @ A) @ Z2) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ P2)) => (~ ((! [X5:B,Y3:C]: (((P2 = ((((product_Pair @ B) @ C) @ X5) @ Y3)) => (~ ((((member @ A) @ Z2) @ ((C2 @ X5) @ Y3))))))))))))).
% 3.09/1.54 thf(fact_123_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > (B > C)),X1:A,X22:B]: (((((((product_case_prod @ A) @ B) @ C) @ F) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = ((F @ X1) @ X22))))).
% 3.09/1.54 thf(fact_124_param__case__prod_H_H, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,F4:$tType,E:$tType,P2:((product_prod @ A) @ B),P3:((product_prod @ C) @ D),F:(A > (B > E)),F2:(C > (D > F4)),R3:(set @ ((product_prod @ E) @ F4))]: (((! [A3:A,B5:B,A4:C,B7:D]: (((P2 = ((((product_Pair @ A) @ B) @ A3) @ B5)) => ((P3 = ((((product_Pair @ C) @ D) @ A4) @ B7)) => (((member @ ((product_prod @ E) @ F4)) @ ((((product_Pair @ E) @ F4) @ ((F @ A3) @ B5)) @ ((F2 @ A4) @ B7))) @ R3))))) => (((member @ ((product_prod @ E) @ F4)) @ ((((product_Pair @ E) @ F4) @ (((((product_case_prod @ A) @ B) @ E) @ F) @ P2)) @ (((((product_case_prod @ C) @ D) @ F4) @ F2) @ P3))) @ R3))))).
% 3.09/1.54 thf(fact_125_rec__prod__is__case, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_rec_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 3.09/1.54 thf(fact_126_curryD, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A5:A,B3:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A5) @ B3) => (F @ ((((product_Pair @ A) @ B) @ A5) @ B3)))))).
% 3.09/1.54 thf(fact_127_curryE, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A5:A,B3:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A5) @ B3) => (F @ ((((product_Pair @ A) @ B) @ A5) @ B3)))))).
% 3.09/1.54 thf(fact_128_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc2004651681e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 3.09/1.54 thf(fact_129_param__case__prod_H, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,F4:$tType,E:$tType,P2:((product_prod @ A) @ B),P3:((product_prod @ C) @ D),Ra:(set @ ((product_prod @ A) @ C)),Rb:(set @ ((product_prod @ B) @ D)),F:(A > (B > E)),F2:(C > (D > F4)),R3:(set @ ((product_prod @ E) @ F4))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ P2) @ P3)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))))) @ ((((prod_rel @ A) @ C) @ B) @ D)) @ Ra)) @ Rb)) => ((! [A3:A,B5:B,A4:C,B7:D]: (((P2 = ((((product_Pair @ A) @ B) @ A3) @ B5)) => ((P3 = ((((product_Pair @ C) @ D) @ A4) @ B7)) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A3) @ A4)) @ Ra) => ((((member @ ((product_prod @ B) @ D)) @ ((((product_Pair @ B) @ D) @ B5) @ B7)) @ Rb) => (((member @ ((product_prod @ E) @ F4)) @ ((((product_Pair @ E) @ F4) @ ((F @ A3) @ B5)) @ ((F2 @ A4) @ B7))) @ R3))))))) => (((member @ ((product_prod @ E) @ F4)) @ ((((product_Pair @ E) @ F4) @ (((((product_case_prod @ A) @ B) @ E) @ F) @ P2)) @ (((((product_case_prod @ C) @ D) @ F4) @ F2) @ P3))) @ R3)))))).
% 3.09/1.54 thf(fact_130_autoref__case__nat, axiom, (! [B:$tType,A:$tType,Ra:(set @ ((product_prod @ A) @ B))]: ((((member @ ((product_prod @ (A > ((nat > A) > (nat > A)))) @ (B > ((nat > B) > (nat > B))))) @ ((((product_Pair @ (A > ((nat > A) > (nat > A)))) @ (B > ((nat > B) > (nat > B)))) @ (case_nat @ A)) @ (case_nat @ B))) @ (((((relAPP @ ((nat > A) > (nat > A))) @ ((nat > B) > (nat > B))) @ (set @ ((product_prod @ (A > ((nat > A) > (nat > A)))) @ (B > ((nat > B) > (nat > B)))))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ ((nat > A) > (nat > A))) @ ((nat > B) > (nat > B)))) > (set @ ((product_prod @ (A > ((nat > A) > (nat > A)))) @ (B > ((nat > B) > (nat > B))))))) @ ((((fun_rel @ A) @ B) @ ((nat > A) > (nat > A))) @ ((nat > B) > (nat > B)))) @ Ra)) @ (((((relAPP @ (nat > A)) @ (nat > B)) @ (set @ ((product_prod @ ((nat > A) > (nat > A))) @ ((nat > B) > (nat > B))))) @ (((((relAPP @ (nat > A)) @ (nat > B)) @ ((set @ ((product_prod @ (nat > A)) @ (nat > B))) > (set @ ((product_prod @ ((nat > A) > (nat > A))) @ ((nat > B) > (nat > B)))))) @ ((((fun_rel @ (nat > A)) @ (nat > B)) @ (nat > A)) @ (nat > B))) @ (((((relAPP @ A) @ B) @ (set @ ((product_prod @ (nat > A)) @ (nat > B)))) @ (((((relAPP @ nat) @ nat) @ ((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ (nat > A)) @ (nat > B))))) @ ((((fun_rel @ nat) @ nat) @ A) @ B)) @ (id2 @ nat))) @ Ra))) @ (((((relAPP @ A) @ B) @ (set @ ((product_prod @ (nat > A)) @ (nat > B)))) @ (((((relAPP @ nat) @ nat) @ ((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ (nat > A)) @ (nat > B))))) @ ((((fun_rel @ nat) @ nat) @ A) @ B)) @ (id2 @ nat))) @ Ra))))))).
% 3.09/1.54 thf(fact_131_uncurry__curry__id, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((uncurry @ A) @ B) @ C) @ ((((product_curry @ A) @ B) @ C) @ F)) = F)))).
% 3.09/1.54 thf(fact_132_curry__uncurry__id, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C))]: ((((((product_curry @ A) @ B) @ C) @ ((((uncurry @ A) @ B) @ C) @ F)) = F)))).
% 3.09/1.54 thf(fact_133_uncurry__apply, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),A5:B,B3:C]: (((((((uncurry @ B) @ C) @ A) @ F) @ ((((product_Pair @ B) @ C) @ A5) @ B3)) = ((F @ A5) @ B3))))).
% 3.09/1.54 thf(fact_134_nres__order__simps_I20_J, axiom, (! [W:$tType,X:W,Y:W]: (((((ord_less_eq @ (refine1665802226e_nres @ W)) @ ((refine1687780735RETURN @ W) @ X)) @ ((refine1687780735RETURN @ W) @ Y)) = (X = Y))))).
% 3.09/1.54 thf(fact_135_subrelI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: (((! [X5:A,Y3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y3)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y3)) @ S2)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S2))))).
% 3.09/1.54 thf(fact_136_fun__rel__mono, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,R1:(set @ ((product_prod @ A) @ B)),R22:(set @ ((product_prod @ A) @ B)),R32:(set @ ((product_prod @ C) @ D)),R42:(set @ ((product_prod @ C) @ D))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R1) @ R22) => ((((ord_less_eq @ (set @ ((product_prod @ C) @ D))) @ R32) @ R42) => (((ord_less_eq @ (set @ ((product_prod @ (A > C)) @ (B > D)))) @ (((((relAPP @ C) @ D) @ (set @ ((product_prod @ (A > C)) @ (B > D)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ C) @ D)) > (set @ ((product_prod @ (A > C)) @ (B > D))))) @ ((((fun_rel @ A) @ B) @ C) @ D)) @ R22)) @ R32)) @ (((((relAPP @ C) @ D) @ (set @ ((product_prod @ (A > C)) @ (B > D)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ C) @ D)) > (set @ ((product_prod @ (A > C)) @ (B > D))))) @ ((((fun_rel @ A) @ B) @ C) @ D)) @ R1)) @ R42))))))).
% 3.09/1.54 thf(fact_137_prod__rel__mono, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,R22:(set @ ((product_prod @ A) @ B)),R1:(set @ ((product_prod @ A) @ B)),R32:(set @ ((product_prod @ C) @ D)),R42:(set @ ((product_prod @ C) @ D))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R22) @ R1) => ((((ord_less_eq @ (set @ ((product_prod @ C) @ D))) @ R32) @ R42) => (((ord_less_eq @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)))) @ (((((relAPP @ C) @ D) @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ C) @ D)) > (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D))))) @ ((((prod_rel @ A) @ B) @ C) @ D)) @ R22)) @ R32)) @ (((((relAPP @ C) @ D) @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)))) @ (((((relAPP @ A) @ B) @ ((set @ ((product_prod @ C) @ D)) > (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D))))) @ ((((prod_rel @ A) @ B) @ C) @ D)) @ R1)) @ R42))))))).
% 3.09/1.54 thf(fact_138_bind__cong, axiom, (! [B:$tType,A:$tType,M:(refine1665802226e_nres @ A),M2:(refine1665802226e_nres @ A),F:(A > (refine1665802226e_nres @ B)),F2:(A > (refine1665802226e_nres @ B))]: (((M = M2) => ((! [X5:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X5)) @ M2) => ((F @ X5) = (F2 @ X5))))) => (((((refine463715084e_bind @ A) @ B) @ M) @ F) = ((((refine463715084e_bind @ A) @ B) @ M2) @ F2))))))).
% 3.09/1.54 thf(fact_139_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J, axiom, (! [B:$tType,A:$tType,M3:(refine1665802226e_nres @ A),M4:(refine1665802226e_nres @ A),F:(A > (refine1665802226e_nres @ B)),F2:(A > (refine1665802226e_nres @ B))]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ M3) @ M4) => ((! [X5:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X5)) @ M3) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ (F @ X5)) @ (F2 @ X5))))) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine463715084e_bind @ A) @ B) @ M3) @ F)) @ ((((refine463715084e_bind @ A) @ B) @ M4) @ F2))))))).
% 3.09/1.54 thf(fact_140_ord__eq__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A5:A,B3:A,C2:A,D3:A]: (((A5 = B3) => ((((ord_less_eq @ A) @ B3) @ C2) => ((C2 = D3) => (((ord_less_eq @ A) @ A5) @ D3)))))))))).
% 3.09/1.54 thf(fact_141_pairself_Oinduct, axiom, (! [B:$tType,A:$tType,P:((A > B) > (((product_prod @ A) @ A) > $o)),A0:(A > B),A12:((product_prod @ A) @ A)]: (((! [F5:(A > B),A3:A,B5:A]: (((P @ F5) @ ((((product_Pair @ A) @ A) @ A3) @ B5)))) => ((P @ A0) @ A12))))).
% 3.09/1.54 thf(fact_142_pairself_Ocases, axiom, (! [B:$tType,A:$tType,X:((product_prod @ (A > B)) @ ((product_prod @ A) @ A))]: ((~ ((! [F5:(A > B),A3:A,B5:A]: ((X != ((((product_Pair @ (A > B)) @ ((product_prod @ A) @ A)) @ F5) @ ((((product_Pair @ A) @ A) @ A3) @ B5)))))))))).
% 3.09/1.54 thf(fact_143_bex2I, axiom, (! [A:$tType,B:$tType,A5:A,B3:B,S5:(set @ ((product_prod @ A) @ B)),P:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) @ S5) => (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) @ S5) => ((P @ A5) @ B3)) => (? [A3:A,B5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B5)) @ S5) & ((P @ A3) @ B5))))))))).
% 3.09/1.54 thf(fact_144_autoref__nat_I6_J, axiom, (((member @ ((product_prod @ (nat > (nat > $o))) @ (nat > (nat > $o)))) @ ((((product_Pair @ (nat > (nat > $o))) @ (nat > (nat > $o))) @ (ord_less_eq @ nat)) @ (ord_less_eq @ nat))) @ (((((relAPP @ (nat > $o)) @ (nat > $o)) @ (set @ ((product_prod @ (nat > (nat > $o))) @ (nat > (nat > $o))))) @ (((((relAPP @ nat) @ nat) @ ((set @ ((product_prod @ (nat > $o)) @ (nat > $o))) > (set @ ((product_prod @ (nat > (nat > $o))) @ (nat > (nat > $o)))))) @ ((((fun_rel @ nat) @ nat) @ (nat > $o)) @ (nat > $o))) @ (id2 @ nat))) @ (((((relAPP @ $o) @ $o) @ (set @ ((product_prod @ (nat > $o)) @ (nat > $o)))) @ (((((relAPP @ nat) @ nat) @ ((set @ ((product_prod @ $o) @ $o)) > (set @ ((product_prod @ (nat > $o)) @ (nat > $o))))) @ ((((fun_rel @ nat) @ nat) @ $o) @ $o)) @ (id2 @ nat))) @ (id2 @ $o))))).
% 3.09/1.54 thf(fact_145_autoref__int_I4_J, axiom, (((member @ ((product_prod @ (int > (int > $o))) @ (int > (int > $o)))) @ ((((product_Pair @ (int > (int > $o))) @ (int > (int > $o))) @ (ord_less @ int)) @ (ord_less @ int))) @ (((((relAPP @ (int > $o)) @ (int > $o)) @ (set @ ((product_prod @ (int > (int > $o))) @ (int > (int > $o))))) @ (((((relAPP @ int) @ int) @ ((set @ ((product_prod @ (int > $o)) @ (int > $o))) > (set @ ((product_prod @ (int > (int > $o))) @ (int > (int > $o)))))) @ ((((fun_rel @ int) @ int) @ (int > $o)) @ (int > $o))) @ (id2 @ int))) @ (((((relAPP @ $o) @ $o) @ (set @ ((product_prod @ (int > $o)) @ (int > $o)))) @ (((((relAPP @ int) @ int) @ ((set @ ((product_prod @ $o) @ $o)) > (set @ ((product_prod @ (int > $o)) @ (int > $o))))) @ ((((fun_rel @ int) @ int) @ $o) @ $o)) @ (id2 @ int))) @ (id2 @ $o))))).
% 3.09/1.54 thf(fact_146_autoref__nat_I5_J, axiom, (((member @ ((product_prod @ (nat > (nat > $o))) @ (nat > (nat > $o)))) @ ((((product_Pair @ (nat > (nat > $o))) @ (nat > (nat > $o))) @ (ord_less @ nat)) @ (ord_less @ nat))) @ (((((relAPP @ (nat > $o)) @ (nat > $o)) @ (set @ ((product_prod @ (nat > (nat > $o))) @ (nat > (nat > $o))))) @ (((((relAPP @ nat) @ nat) @ ((set @ ((product_prod @ (nat > $o)) @ (nat > $o))) > (set @ ((product_prod @ (nat > (nat > $o))) @ (nat > (nat > $o)))))) @ ((((fun_rel @ nat) @ nat) @ (nat > $o)) @ (nat > $o))) @ (id2 @ nat))) @ (((((relAPP @ $o) @ $o) @ (set @ ((product_prod @ (nat > $o)) @ (nat > $o)))) @ (((((relAPP @ nat) @ nat) @ ((set @ ((product_prod @ $o) @ $o)) > (set @ ((product_prod @ (nat > $o)) @ (nat > $o))))) @ ((((fun_rel @ nat) @ nat) @ $o) @ $o)) @ (id2 @ nat))) @ (id2 @ $o))))).
% 3.09/1.54 thf(fact_147_case__prod__mono, axiom, (! [C:$tType,B:$tType,A:$tType]: (((ord @ C) => (! [P2:((product_prod @ A) @ B),F:(A > (B > C)),F2:(A > (B > C))]: (((! [A3:A,B5:B]: (((P2 = ((((product_Pair @ A) @ B) @ A3) @ B5)) => (((ord_less_eq @ C) @ ((F @ A3) @ B5)) @ ((F2 @ A3) @ B5))))) => (((ord_less_eq @ C) @ (((((product_case_prod @ A) @ B) @ C) @ F) @ P2)) @ (((((product_case_prod @ A) @ B) @ C) @ F2) @ P2))))))))).
% 3.09/1.54 thf(fact_148_param__apsnd, axiom, (! [A:$tType,D:$tType,C:$tType,F4:$tType,E:$tType,B:$tType,Rb:(set @ ((product_prod @ A) @ D)),Rc:(set @ ((product_prod @ B) @ E)),Ra:(set @ ((product_prod @ C) @ F4))]: ((((member @ ((product_prod @ ((A > B) > (((product_prod @ C) @ A) > ((product_prod @ C) @ B)))) @ ((D > E) > (((product_prod @ F4) @ D) > ((product_prod @ F4) @ E))))) @ ((((product_Pair @ ((A > B) > (((product_prod @ C) @ A) > ((product_prod @ C) @ B)))) @ ((D > E) > (((product_prod @ F4) @ D) > ((product_prod @ F4) @ E)))) @ (((product_apsnd @ A) @ B) @ C)) @ (((product_apsnd @ D) @ E) @ F4))) @ (((((relAPP @ (((product_prod @ C) @ A) > ((product_prod @ C) @ B))) @ (((product_prod @ F4) @ D) > ((product_prod @ F4) @ E))) @ (set @ ((product_prod @ ((A > B) > (((product_prod @ C) @ A) > ((product_prod @ C) @ B)))) @ ((D > E) > (((product_prod @ F4) @ D) > ((product_prod @ F4) @ E)))))) @ (((((relAPP @ (A > B)) @ (D > E)) @ ((set @ ((product_prod @ (((product_prod @ C) @ A) > ((product_prod @ C) @ B))) @ (((product_prod @ F4) @ D) > ((product_prod @ F4) @ E)))) > (set @ ((product_prod @ ((A > B) > (((product_prod @ C) @ A) > ((product_prod @ C) @ B)))) @ ((D > E) > (((product_prod @ F4) @ D) > ((product_prod @ F4) @ E))))))) @ ((((fun_rel @ (A > B)) @ (D > E)) @ (((product_prod @ C) @ A) > ((product_prod @ C) @ B))) @ (((product_prod @ F4) @ D) > ((product_prod @ F4) @ E)))) @ (((((relAPP @ B) @ E) @ (set @ ((product_prod @ (A > B)) @ (D > E)))) @ (((((relAPP @ A) @ D) @ ((set @ ((product_prod @ B) @ E)) > (set @ ((product_prod @ (A > B)) @ (D > E))))) @ ((((fun_rel @ A) @ D) @ B) @ E)) @ Rb)) @ Rc))) @ (((((relAPP @ ((product_prod @ C) @ B)) @ ((product_prod @ F4) @ E)) @ (set @ ((product_prod @ (((product_prod @ C) @ A) > ((product_prod @ C) @ B))) @ (((product_prod @ F4) @ D) > ((product_prod @ F4) @ E))))) @ (((((relAPP @ ((product_prod @ C) @ A)) @ ((product_prod @ F4) @ D)) @ ((set @ ((product_prod @ ((product_prod @ C) @ B)) @ ((product_prod @ F4) @ E))) > (set @ ((product_prod @ (((product_prod @ C) @ A) > ((product_prod @ C) @ B))) @ (((product_prod @ F4) @ D) > ((product_prod @ F4) @ E)))))) @ ((((fun_rel @ ((product_prod @ C) @ A)) @ ((product_prod @ F4) @ D)) @ ((product_prod @ C) @ B)) @ ((product_prod @ F4) @ E))) @ (((((relAPP @ A) @ D) @ (set @ ((product_prod @ ((product_prod @ C) @ A)) @ ((product_prod @ F4) @ D)))) @ (((((relAPP @ C) @ F4) @ ((set @ ((product_prod @ A) @ D)) > (set @ ((product_prod @ ((product_prod @ C) @ A)) @ ((product_prod @ F4) @ D))))) @ ((((prod_rel @ C) @ F4) @ A) @ D)) @ Ra)) @ Rb))) @ (((((relAPP @ B) @ E) @ (set @ ((product_prod @ ((product_prod @ C) @ B)) @ ((product_prod @ F4) @ E)))) @ (((((relAPP @ C) @ F4) @ ((set @ ((product_prod @ B) @ E)) > (set @ ((product_prod @ ((product_prod @ C) @ B)) @ ((product_prod @ F4) @ E))))) @ ((((prod_rel @ C) @ F4) @ B) @ E)) @ Ra)) @ Rc))))))).
% 3.09/1.54 thf(fact_149_apsnd__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > B),X:A,Y:C]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ ((((product_Pair @ A) @ C) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ X) @ (F @ Y)))))).
% 3.09/1.54 thf(fact_150_apsnd__id, axiom, (! [B:$tType,A:$tType]: ((((((product_apsnd @ B) @ B) @ A) @ (id @ B)) = (id @ ((product_prod @ A) @ B)))))).
% 3.09/1.54 thf(fact_151_relprop__triggers_I6_J, axiom, (! [I:$tType,R3:(set @ I),R5:(set @ I)]: (((((ord_less_eq @ (set @ I)) @ R3) @ R5) => (((ord_less_eq @ (set @ I)) @ R3) @ R5))))).
% 3.09/1.54 thf(fact_152_Collect__case__prod__mono, axiom, (! [B:$tType,A:$tType,A2:(A > (B > $o)),B2:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ A2) @ B2) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A2))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ B2))))))).
% 3.09/1.54 thf(fact_153_subset__Collect__conv, axiom, (! [A:$tType,S5:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ S5) @ ((collect @ A) @ P)) = (! [X6:A]: (((((member @ A) @ X6) @ S5) => (P @ X6)))))))).
% 3.09/1.54 thf(fact_154_exists__leI, axiom, (! [N:nat,P:(nat > $o)]: ((((! [N2:nat]: (((((ord_less @ nat) @ N2) @ N) => (~ ((P @ N2)))))) => (P @ N)) => (? [N3:nat]: (((((ord_less_eq @ nat) @ N3) @ N) & (P @ N3)))))))).
% 3.09/1.54 thf(fact_155_Refine__Misc_Oif__mono, axiom, (! [A:$tType]: (((ord @ A) => (! [B3:$o,M1:A,M12:A,M22:A,M23:A]: (((B3 => (((ord_less_eq @ A) @ M1) @ M12)) => (((~ (B3)) => (((ord_less_eq @ A) @ M22) @ M23)) => (((ord_less_eq @ A) @ ((((if @ A) @ B3) @ M1) @ M22)) @ ((((if @ A) @ B3) @ M12) @ M23)))))))))).
% 3.09/1.54 thf(fact_156_param__apfst, axiom, (! [D:$tType,A:$tType,B:$tType,E:$tType,F4:$tType,C:$tType,Ra:(set @ ((product_prod @ A) @ D)),Rb:(set @ ((product_prod @ B) @ E)),Rc:(set @ ((product_prod @ C) @ F4))]: ((((member @ ((product_prod @ ((A > B) > (((product_prod @ A) @ C) > ((product_prod @ B) @ C)))) @ ((D > E) > (((product_prod @ D) @ F4) > ((product_prod @ E) @ F4))))) @ ((((product_Pair @ ((A > B) > (((product_prod @ A) @ C) > ((product_prod @ B) @ C)))) @ ((D > E) > (((product_prod @ D) @ F4) > ((product_prod @ E) @ F4)))) @ (((product_apfst @ A) @ B) @ C)) @ (((product_apfst @ D) @ E) @ F4))) @ (((((relAPP @ (((product_prod @ A) @ C) > ((product_prod @ B) @ C))) @ (((product_prod @ D) @ F4) > ((product_prod @ E) @ F4))) @ (set @ ((product_prod @ ((A > B) > (((product_prod @ A) @ C) > ((product_prod @ B) @ C)))) @ ((D > E) > (((product_prod @ D) @ F4) > ((product_prod @ E) @ F4)))))) @ (((((relAPP @ (A > B)) @ (D > E)) @ ((set @ ((product_prod @ (((product_prod @ A) @ C) > ((product_prod @ B) @ C))) @ (((product_prod @ D) @ F4) > ((product_prod @ E) @ F4)))) > (set @ ((product_prod @ ((A > B) > (((product_prod @ A) @ C) > ((product_prod @ B) @ C)))) @ ((D > E) > (((product_prod @ D) @ F4) > ((product_prod @ E) @ F4))))))) @ ((((fun_rel @ (A > B)) @ (D > E)) @ (((product_prod @ A) @ C) > ((product_prod @ B) @ C))) @ (((product_prod @ D) @ F4) > ((product_prod @ E) @ F4)))) @ (((((relAPP @ B) @ E) @ (set @ ((product_prod @ (A > B)) @ (D > E)))) @ (((((relAPP @ A) @ D) @ ((set @ ((product_prod @ B) @ E)) > (set @ ((product_prod @ (A > B)) @ (D > E))))) @ ((((fun_rel @ A) @ D) @ B) @ E)) @ Ra)) @ Rb))) @ (((((relAPP @ ((product_prod @ B) @ C)) @ ((product_prod @ E) @ F4)) @ (set @ ((product_prod @ (((product_prod @ A) @ C) > ((product_prod @ B) @ C))) @ (((product_prod @ D) @ F4) > ((product_prod @ E) @ F4))))) @ (((((relAPP @ ((product_prod @ A) @ C)) @ ((product_prod @ D) @ F4)) @ ((set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ E) @ F4))) > (set @ ((product_prod @ (((product_prod @ A) @ C) > ((product_prod @ B) @ C))) @ (((product_prod @ D) @ F4) > ((product_prod @ E) @ F4)))))) @ ((((fun_rel @ ((product_prod @ A) @ C)) @ ((product_prod @ D) @ F4)) @ ((product_prod @ B) @ C)) @ ((product_prod @ E) @ F4))) @ (((((relAPP @ C) @ F4) @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ D) @ F4)))) @ (((((relAPP @ A) @ D) @ ((set @ ((product_prod @ C) @ F4)) > (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ D) @ F4))))) @ ((((prod_rel @ A) @ D) @ C) @ F4)) @ Ra)) @ Rc))) @ (((((relAPP @ C) @ F4) @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ E) @ F4)))) @ (((((relAPP @ B) @ E) @ ((set @ ((product_prod @ C) @ F4)) > (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ E) @ F4))))) @ ((((prod_rel @ B) @ E) @ C) @ F4)) @ Rb)) @ Rc))))))).
% 3.09/1.54 thf(fact_157_param__map__prod, axiom, (! [C:$tType,G:$tType,E:$tType,A:$tType,B:$tType,F4:$tType,H2:$tType,D:$tType,Ra:(set @ ((product_prod @ A) @ E)),Rb:(set @ ((product_prod @ B) @ F4)),Rc:(set @ ((product_prod @ C) @ G)),Rd:(set @ ((product_prod @ D) @ H2))]: ((((member @ ((product_prod @ ((A > B) > ((C > D) > (((product_prod @ A) @ C) > ((product_prod @ B) @ D))))) @ ((E > F4) > ((G > H2) > (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2)))))) @ ((((product_Pair @ ((A > B) > ((C > D) > (((product_prod @ A) @ C) > ((product_prod @ B) @ D))))) @ ((E > F4) > ((G > H2) > (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2))))) @ ((((product_map_prod @ A) @ B) @ C) @ D)) @ ((((product_map_prod @ E) @ F4) @ G) @ H2))) @ (((((relAPP @ ((C > D) > (((product_prod @ A) @ C) > ((product_prod @ B) @ D)))) @ ((G > H2) > (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2)))) @ (set @ ((product_prod @ ((A > B) > ((C > D) > (((product_prod @ A) @ C) > ((product_prod @ B) @ D))))) @ ((E > F4) > ((G > H2) > (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2))))))) @ (((((relAPP @ (A > B)) @ (E > F4)) @ ((set @ ((product_prod @ ((C > D) > (((product_prod @ A) @ C) > ((product_prod @ B) @ D)))) @ ((G > H2) > (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2))))) > (set @ ((product_prod @ ((A > B) > ((C > D) > (((product_prod @ A) @ C) > ((product_prod @ B) @ D))))) @ ((E > F4) > ((G > H2) > (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2)))))))) @ ((((fun_rel @ (A > B)) @ (E > F4)) @ ((C > D) > (((product_prod @ A) @ C) > ((product_prod @ B) @ D)))) @ ((G > H2) > (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2))))) @ (((((relAPP @ B) @ F4) @ (set @ ((product_prod @ (A > B)) @ (E > F4)))) @ (((((relAPP @ A) @ E) @ ((set @ ((product_prod @ B) @ F4)) > (set @ ((product_prod @ (A > B)) @ (E > F4))))) @ ((((fun_rel @ A) @ E) @ B) @ F4)) @ Ra)) @ Rb))) @ (((((relAPP @ (((product_prod @ A) @ C) > ((product_prod @ B) @ D))) @ (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2))) @ (set @ ((product_prod @ ((C > D) > (((product_prod @ A) @ C) > ((product_prod @ B) @ D)))) @ ((G > H2) > (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2)))))) @ (((((relAPP @ (C > D)) @ (G > H2)) @ ((set @ ((product_prod @ (((product_prod @ A) @ C) > ((product_prod @ B) @ D))) @ (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2)))) > (set @ ((product_prod @ ((C > D) > (((product_prod @ A) @ C) > ((product_prod @ B) @ D)))) @ ((G > H2) > (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2))))))) @ ((((fun_rel @ (C > D)) @ (G > H2)) @ (((product_prod @ A) @ C) > ((product_prod @ B) @ D))) @ (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2)))) @ (((((relAPP @ D) @ H2) @ (set @ ((product_prod @ (C > D)) @ (G > H2)))) @ (((((relAPP @ C) @ G) @ ((set @ ((product_prod @ D) @ H2)) > (set @ ((product_prod @ (C > D)) @ (G > H2))))) @ ((((fun_rel @ C) @ G) @ D) @ H2)) @ Rc)) @ Rd))) @ (((((relAPP @ ((product_prod @ B) @ D)) @ ((product_prod @ F4) @ H2)) @ (set @ ((product_prod @ (((product_prod @ A) @ C) > ((product_prod @ B) @ D))) @ (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2))))) @ (((((relAPP @ ((product_prod @ A) @ C)) @ ((product_prod @ E) @ G)) @ ((set @ ((product_prod @ ((product_prod @ B) @ D)) @ ((product_prod @ F4) @ H2))) > (set @ ((product_prod @ (((product_prod @ A) @ C) > ((product_prod @ B) @ D))) @ (((product_prod @ E) @ G) > ((product_prod @ F4) @ H2)))))) @ ((((fun_rel @ ((product_prod @ A) @ C)) @ ((product_prod @ E) @ G)) @ ((product_prod @ B) @ D)) @ ((product_prod @ F4) @ H2))) @ (((((relAPP @ C) @ G) @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ E) @ G)))) @ (((((relAPP @ A) @ E) @ ((set @ ((product_prod @ C) @ G)) > (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ E) @ G))))) @ ((((prod_rel @ A) @ E) @ C) @ G)) @ Ra)) @ Rc))) @ (((((relAPP @ D) @ H2) @ (set @ ((product_prod @ ((product_prod @ B) @ D)) @ ((product_prod @ F4) @ H2)))) @ (((((relAPP @ B) @ F4) @ ((set @ ((product_prod @ D) @ H2)) > (set @ ((product_prod @ ((product_prod @ B) @ D)) @ ((product_prod @ F4) @ H2))))) @ ((((prod_rel @ B) @ F4) @ D) @ H2)) @ Rb)) @ Rd)))))))).
% 3.09/1.54 thf(fact_158_param__prod__swap, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,A2:(set @ ((product_prod @ A) @ C)),B2:(set @ ((product_prod @ B) @ D))]: ((((member @ ((product_prod @ (((product_prod @ A) @ B) > ((product_prod @ B) @ A))) @ (((product_prod @ C) @ D) > ((product_prod @ D) @ C)))) @ ((((product_Pair @ (((product_prod @ A) @ B) > ((product_prod @ B) @ A))) @ (((product_prod @ C) @ D) > ((product_prod @ D) @ C))) @ ((product_swap @ A) @ B)) @ ((product_swap @ C) @ D))) @ (((((relAPP @ ((product_prod @ B) @ A)) @ ((product_prod @ D) @ C)) @ (set @ ((product_prod @ (((product_prod @ A) @ B) > ((product_prod @ B) @ A))) @ (((product_prod @ C) @ D) > ((product_prod @ D) @ C))))) @ (((((relAPP @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ D) @ C))) > (set @ ((product_prod @ (((product_prod @ A) @ B) > ((product_prod @ B) @ A))) @ (((product_prod @ C) @ D) > ((product_prod @ D) @ C)))))) @ ((((fun_rel @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((product_prod @ B) @ A)) @ ((product_prod @ D) @ C))) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))))) @ ((((prod_rel @ A) @ C) @ B) @ D)) @ A2)) @ B2))) @ (((((relAPP @ A) @ C) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ D) @ C)))) @ (((((relAPP @ B) @ D) @ ((set @ ((product_prod @ A) @ C)) > (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ D) @ C))))) @ ((((prod_rel @ B) @ D) @ A) @ C)) @ B2)) @ A2)))))).
% 3.09/1.54 thf(fact_159_swap__swap, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P2)) = P2)))).
% 3.09/1.54 thf(fact_160_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G3:(D > B),A5:C,B3:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G3) @ ((((product_Pair @ C) @ D) @ A5) @ B3)) = ((((product_Pair @ A) @ B) @ (F @ A5)) @ (G3 @ B3)))))).
% 3.09/1.54 thf(fact_161_apfst__conv, axiom, (! [C:$tType,A:$tType,B:$tType,F:(C > A),X:C,Y:B]: (((((((product_apfst @ C) @ A) @ B) @ F) @ ((((product_Pair @ C) @ B) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ (F @ X)) @ Y))))).
% 3.09/1.54 thf(fact_162_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X))))).
% 3.09/1.54 thf(fact_163_apfst__id, axiom, (! [B:$tType,A:$tType]: ((((((product_apfst @ A) @ A) @ B) @ (id @ A)) = (id @ ((product_prod @ A) @ B)))))).
% 3.09/1.54 thf(fact_164_apfst__def, axiom, (! [B:$tType,C:$tType,A:$tType]: (((((product_apfst @ A) @ C) @ B) = (^ [F3:(A > C)]: (((((((product_map_prod @ A) @ C) @ B) @ B) @ F3) @ (id @ B)))))))).
% 3.09/1.54 thf(fact_165_apsnd__apfst__commute, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F:(C > B),G3:(D > A),P2:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ (((((product_apfst @ D) @ A) @ C) @ G3) @ P2)) = (((((product_apfst @ D) @ A) @ B) @ G3) @ (((((product_apsnd @ C) @ B) @ D) @ F) @ P2)))))).
% 3.09/1.54 thf(fact_166_apsnd__def, axiom, (! [B:$tType,C:$tType,A:$tType]: (((((product_apsnd @ B) @ C) @ A) = (((((product_map_prod @ A) @ A) @ B) @ C) @ (id @ A)))))).
% 3.09/1.54 thf(fact_167_prod_Omap__id0, axiom, (! [B:$tType,A:$tType]: ((((((((product_map_prod @ A) @ A) @ B) @ B) @ (id @ A)) @ (id @ B)) = (id @ ((product_prod @ A) @ B)))))).
% 3.09/1.54 thf(fact_168_autoref__comp, axiom, (! [A:$tType,D:$tType,C:$tType,F4:$tType,E:$tType,B:$tType,Ra:(set @ ((product_prod @ A) @ D)),Rb:(set @ ((product_prod @ B) @ E)),Rc:(set @ ((product_prod @ C) @ F4))]: ((((member @ ((product_prod @ ((A > B) > ((C > A) > (C > B)))) @ ((D > E) > ((F4 > D) > (F4 > E))))) @ ((((product_Pair @ ((A > B) > ((C > A) > (C > B)))) @ ((D > E) > ((F4 > D) > (F4 > E)))) @ (((comp @ A) @ B) @ C)) @ (((comp @ D) @ E) @ F4))) @ (((((relAPP @ ((C > A) > (C > B))) @ ((F4 > D) > (F4 > E))) @ (set @ ((product_prod @ ((A > B) > ((C > A) > (C > B)))) @ ((D > E) > ((F4 > D) > (F4 > E)))))) @ (((((relAPP @ (A > B)) @ (D > E)) @ ((set @ ((product_prod @ ((C > A) > (C > B))) @ ((F4 > D) > (F4 > E)))) > (set @ ((product_prod @ ((A > B) > ((C > A) > (C > B)))) @ ((D > E) > ((F4 > D) > (F4 > E))))))) @ ((((fun_rel @ (A > B)) @ (D > E)) @ ((C > A) > (C > B))) @ ((F4 > D) > (F4 > E)))) @ (((((relAPP @ B) @ E) @ (set @ ((product_prod @ (A > B)) @ (D > E)))) @ (((((relAPP @ A) @ D) @ ((set @ ((product_prod @ B) @ E)) > (set @ ((product_prod @ (A > B)) @ (D > E))))) @ ((((fun_rel @ A) @ D) @ B) @ E)) @ Ra)) @ Rb))) @ (((((relAPP @ (C > B)) @ (F4 > E)) @ (set @ ((product_prod @ ((C > A) > (C > B))) @ ((F4 > D) > (F4 > E))))) @ (((((relAPP @ (C > A)) @ (F4 > D)) @ ((set @ ((product_prod @ (C > B)) @ (F4 > E))) > (set @ ((product_prod @ ((C > A) > (C > B))) @ ((F4 > D) > (F4 > E)))))) @ ((((fun_rel @ (C > A)) @ (F4 > D)) @ (C > B)) @ (F4 > E))) @ (((((relAPP @ A) @ D) @ (set @ ((product_prod @ (C > A)) @ (F4 > D)))) @ (((((relAPP @ C) @ F4) @ ((set @ ((product_prod @ A) @ D)) > (set @ ((product_prod @ (C > A)) @ (F4 > D))))) @ ((((fun_rel @ C) @ F4) @ A) @ D)) @ Rc)) @ Ra))) @ (((((relAPP @ B) @ E) @ (set @ ((product_prod @ (C > B)) @ (F4 > E)))) @ (((((relAPP @ C) @ F4) @ ((set @ ((product_prod @ B) @ E)) > (set @ ((product_prod @ (C > B)) @ (F4 > E))))) @ ((((fun_rel @ C) @ F4) @ B) @ E)) @ Rc)) @ Rb))))))).
% 3.09/1.54 thf(fact_169_less__than__bool__iff, axiom, (! [X:$o,Y:$o]: (((((member @ ((product_prod @ $o) @ $o)) @ ((((product_Pair @ $o) @ $o) @ X) @ Y)) @ refine1361002633n_bool) = ((~ (X)) & Y))))).
% 3.09/1.54 thf(fact_170_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F3:(B > A),G4:(C > B),X6:C]: ((F3 @ (G4 @ X6)))))))).
% 3.09/1.54 thf(fact_171_comp__id, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((((((comp @ A) @ B) @ A) @ F) @ (id @ A)) = F)))).
% 3.09/1.54 thf(fact_172_id__comp, axiom, (! [B:$tType,A:$tType,G3:(A > B)]: (((((((comp @ B) @ B) @ A) @ (id @ B)) @ G3) = G3)))).
% 3.09/1.54 thf(fact_173_fun_Omap__id, axiom, (! [A:$tType,D:$tType,T2:(D > A)]: (((((((comp @ A) @ A) @ D) @ (id @ A)) @ T2) = T2)))).
% 3.09/1.54 thf(fact_174_swap__comp__swap, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ ((product_swap @ A) @ B)) = (id @ ((product_prod @ A) @ B)))))).
% 3.09/1.54 thf(fact_175_map__prod_Ocompositionality, axiom, (! [D:$tType,F4:$tType,E:$tType,C:$tType,B:$tType,A:$tType,F:(C > E),G3:(D > F4),H:(A > C),I2:(B > D),Prod:((product_prod @ A) @ B)]: (((((((((product_map_prod @ C) @ E) @ D) @ F4) @ F) @ G3) @ (((((((product_map_prod @ A) @ C) @ B) @ D) @ H) @ I2) @ Prod)) = (((((((product_map_prod @ A) @ E) @ B) @ F4) @ (((((comp @ C) @ E) @ A) @ F) @ H)) @ (((((comp @ D) @ F4) @ B) @ G3) @ I2)) @ Prod))))).
% 3.09/1.54 thf(fact_176_map__prod__compose, axiom, (! [D:$tType,C:$tType,A:$tType,E:$tType,F4:$tType,B:$tType,F1:(E > C),F22:(A > E),G1:(F4 > D),G22:(B > F4)]: ((((((((product_map_prod @ A) @ C) @ B) @ D) @ (((((comp @ E) @ C) @ A) @ F1) @ F22)) @ (((((comp @ F4) @ D) @ B) @ G1) @ G22)) = (((((comp @ ((product_prod @ E) @ F4)) @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ E) @ C) @ F4) @ D) @ F1) @ G1)) @ ((((((product_map_prod @ A) @ E) @ B) @ F4) @ F22) @ G22)))))).
% 3.09/1.54 thf(fact_177_map__prod_Ocomp, axiom, (! [A:$tType,C:$tType,E:$tType,F4:$tType,D:$tType,B:$tType,F:(C > E),G3:(D > F4),H:(A > C),I2:(B > D)]: (((((((comp @ ((product_prod @ C) @ D)) @ ((product_prod @ E) @ F4)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ E) @ D) @ F4) @ F) @ G3)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ H) @ I2)) = ((((((product_map_prod @ A) @ E) @ B) @ F4) @ (((((comp @ C) @ E) @ A) @ F) @ H)) @ (((((comp @ D) @ F4) @ B) @ G3) @ I2)))))).
% 3.09/1.54 thf(fact_178_apfst__compose, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G3:(D > C),X:((product_prod @ D) @ B)]: (((((((product_apfst @ C) @ A) @ B) @ F) @ (((((product_apfst @ D) @ C) @ B) @ G3) @ X)) = (((((product_apfst @ D) @ A) @ B) @ (((((comp @ C) @ A) @ D) @ F) @ G3)) @ X))))).
% 3.09/1.54 thf(fact_179_comp__eq__id__dest, axiom, (! [C:$tType,B:$tType,A:$tType,A5:(C > B),B3:(A > C),C2:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A5) @ B3) = (((((comp @ B) @ B) @ A) @ (id @ B)) @ C2)) => ((A5 @ (B3 @ V)) = (C2 @ V)))))).
% 3.09/1.54 thf(fact_180_fun_Omap__id0, axiom, (! [A:$tType,D:$tType]: ((((((comp @ A) @ A) @ D) @ (id @ A)) = (id @ (D > A)))))).
% 3.09/1.54 thf(fact_181_comp__cong__left, axiom, (! [B:$tType,A:$tType,C:$tType,X:(A > B),Y:(A > B),F:(C > A)]: (((X = Y) => ((((((comp @ A) @ B) @ C) @ X) @ F) = (((((comp @ A) @ B) @ C) @ Y) @ F)))))).
% 3.09/1.54 thf(fact_182_comp__cong__right, axiom, (! [C:$tType,B:$tType,A:$tType,X:(A > B),Y:(A > B),F:(B > C)]: (((X = Y) => ((((((comp @ B) @ C) @ A) @ F) @ X) = (((((comp @ B) @ C) @ A) @ F) @ Y)))))).
% 3.09/1.54 thf(fact_183_fun__comp__eq__conv, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),G3:(A > C),Fg:(A > B)]: ((((((((comp @ C) @ B) @ A) @ F) @ G3) = Fg) = (! [X6:A]: (((F @ (G3 @ X6)) = (Fg @ X6)))))))).
% 3.09/1.54 thf(fact_184_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F3:(B > C),G4:(A > B),X6:A]: ((F3 @ (G4 @ X6)))))))).
% 3.09/1.54 thf(fact_185_comp__assoc, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F:(D > B),G3:(C > D),H:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F) @ G3)) @ H) = (((((comp @ D) @ B) @ A) @ F) @ (((((comp @ C) @ D) @ A) @ G3) @ H)))))).
% 3.09/1.54 thf(fact_186_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A5:(C > B),B3:(A > C),C2:(D > B),D3:(A > D),V:A]: ((((((((comp @ C) @ B) @ A) @ A5) @ B3) = (((((comp @ D) @ B) @ A) @ C2) @ D3)) => ((A5 @ (B3 @ V)) = (C2 @ (D3 @ V))))))).
% 3.09/1.54 thf(fact_187_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A5:(C > B),B3:(A > C),C2:(D > B),D3:(A > D)]: ((((((((comp @ C) @ B) @ A) @ A5) @ B3) = (((((comp @ D) @ B) @ A) @ C2) @ D3)) => (! [V2:A]: (((A5 @ (B3 @ V2)) = (C2 @ (D3 @ V2))))))))).
% 3.09/1.54 thf(fact_188_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A5:(C > B),B3:(A > C),C2:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A5) @ B3) = C2) => ((A5 @ (B3 @ V)) = (C2 @ V)))))).
% 3.09/1.54 thf(fact_189_fun__upd__comp, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > B),G3:(A > C),X:A,Y:C]: (((((((comp @ C) @ B) @ A) @ F) @ (((((fun_upd @ A) @ C) @ G3) @ X) @ Y)) = (((((fun_upd @ A) @ B) @ (((((comp @ C) @ B) @ A) @ F) @ G3)) @ X) @ (F @ Y)))))).
% 3.09/1.54 thf(fact_190_apsnd__compose, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,F:(C > B),G3:(D > C),X:((product_prod @ A) @ D)]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ (((((product_apsnd @ D) @ C) @ A) @ G3) @ X)) = (((((product_apsnd @ D) @ B) @ A) @ (((((comp @ C) @ B) @ D) @ F) @ G3)) @ X))))).
% 3.09/1.54 thf(fact_191_prod_Omap__id, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: (((((((((product_map_prod @ A) @ A) @ B) @ B) @ (id @ A)) @ (id @ B)) @ T2) = T2)))).
% 3.09/1.54 thf(fact_192_predicate2I, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Q:(A > (B > $o))]: (((! [X5:A,Y3:B]: ((((P @ X5) @ Y3) => ((Q @ X5) @ Y3)))) => (((ord_less_eq @ (A > (B > $o))) @ P) @ Q))))).
% 3.09/1.54 thf(fact_193_prod__refine_I4_J, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,Ra:(set @ ((product_prod @ A) @ C)),Rb:(set @ ((product_prod @ B) @ D))]: ((((member @ ((product_prod @ (((product_prod @ A) @ B) > A)) @ (((product_prod @ C) @ D) > C))) @ ((((product_Pair @ (((product_prod @ A) @ B) > A)) @ (((product_prod @ C) @ D) > C)) @ ((product_fst @ A) @ B)) @ ((product_fst @ C) @ D))) @ (((((relAPP @ A) @ C) @ (set @ ((product_prod @ (((product_prod @ A) @ B) > A)) @ (((product_prod @ C) @ D) > C)))) @ (((((relAPP @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((set @ ((product_prod @ A) @ C)) > (set @ ((product_prod @ (((product_prod @ A) @ B) > A)) @ (((product_prod @ C) @ D) > C))))) @ ((((fun_rel @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ A) @ C)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))))) @ ((((prod_rel @ A) @ C) @ B) @ D)) @ Ra)) @ Rb))) @ Ra))))).
% 3.09/1.54 thf(fact_194_prod__refine_I5_J, axiom, (! [C:$tType,A:$tType,D:$tType,B:$tType,Ra:(set @ ((product_prod @ A) @ C)),Rb:(set @ ((product_prod @ B) @ D))]: ((((member @ ((product_prod @ (((product_prod @ A) @ B) > B)) @ (((product_prod @ C) @ D) > D))) @ ((((product_Pair @ (((product_prod @ A) @ B) > B)) @ (((product_prod @ C) @ D) > D)) @ ((product_snd @ A) @ B)) @ ((product_snd @ C) @ D))) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ (((product_prod @ A) @ B) > B)) @ (((product_prod @ C) @ D) > D)))) @ (((((relAPP @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ (((product_prod @ A) @ B) > B)) @ (((product_prod @ C) @ D) > D))))) @ ((((fun_rel @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ B) @ D)) @ (((((relAPP @ B) @ D) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)))) @ (((((relAPP @ A) @ C) @ ((set @ ((product_prod @ B) @ D)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D))))) @ ((((prod_rel @ A) @ C) @ B) @ D)) @ Ra)) @ Rb))) @ Rb))))).
% 3.09/1.54 thf(fact_195_fst__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > A),G3:(D > B),X:((product_prod @ C) @ D)]: (((((product_fst @ A) @ B) @ (((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G3) @ X)) = (F @ (((product_fst @ C) @ D) @ X)))))).
% 3.09/1.54 thf(fact_196_snd__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > B),G3:(D > A),X:((product_prod @ C) @ D)]: (((((product_snd @ B) @ A) @ (((((((product_map_prod @ C) @ B) @ D) @ A) @ F) @ G3) @ X)) = (G3 @ (((product_snd @ C) @ D) @ X)))))).
% 3.09/1.54 thf(fact_197_fst__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > A),X:((product_prod @ C) @ B)]: (((((product_fst @ A) @ B) @ (((((product_apfst @ C) @ A) @ B) @ F) @ X)) = (F @ (((product_fst @ C) @ B) @ X)))))).
% 3.09/1.54 thf(fact_198_apfst__eq__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > A),X:((product_prod @ C) @ B),G3:(C > A)]: ((((((((product_apfst @ C) @ A) @ B) @ F) @ X) = (((((product_apfst @ C) @ A) @ B) @ G3) @ X)) = ((F @ (((product_fst @ C) @ B) @ X)) = (G3 @ (((product_fst @ C) @ B) @ X))))))).
% 3.09/1.54 thf(fact_199_fst__apsnd, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),X:((product_prod @ A) @ C)]: (((((product_fst @ A) @ B) @ (((((product_apsnd @ C) @ B) @ A) @ F) @ X)) = (((product_fst @ A) @ C) @ X))))).
% 3.09/1.54 thf(fact_200_snd__apfst, axiom, (! [B:$tType,A:$tType,C:$tType,F:(C > B),X:((product_prod @ C) @ A)]: (((((product_snd @ B) @ A) @ (((((product_apfst @ C) @ B) @ A) @ F) @ X)) = (((product_snd @ C) @ A) @ X))))).
% 3.09/1.54 thf(fact_201_snd__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F:(C > A),X:((product_prod @ B) @ C)]: (((((product_snd @ B) @ A) @ (((((product_apsnd @ C) @ A) @ B) @ F) @ X)) = (F @ (((product_snd @ B) @ C) @ X)))))).
% 3.09/1.54 thf(fact_202_apsnd__eq__conv, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),X:((product_prod @ A) @ C),G3:(C > B)]: ((((((((product_apsnd @ C) @ B) @ A) @ F) @ X) = (((((product_apsnd @ C) @ B) @ A) @ G3) @ X)) = ((F @ (((product_snd @ A) @ C) @ X)) = (G3 @ (((product_snd @ A) @ C) @ X))))))).
% 3.09/1.54 thf(fact_203_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 3.09/1.54 thf(fact_204_fst__swap, axiom, (! [A:$tType,B:$tType,X:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X)) = (((product_snd @ B) @ A) @ X))))).
% 3.09/1.54 thf(fact_205_snd__swap, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X)) = (((product_fst @ A) @ B) @ X))))).
% 3.09/1.54 thf(fact_206_fst__comp__apsnd, axiom, (! [C:$tType,B:$tType,A:$tType,F:(B > C)]: (((((((comp @ ((product_prod @ A) @ C)) @ A) @ ((product_prod @ A) @ B)) @ ((product_fst @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F)) = ((product_fst @ A) @ B))))).
% 3.09/1.54 thf(fact_207_snd__comp__apfst, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > C)]: (((((((comp @ ((product_prod @ C) @ B)) @ B) @ ((product_prod @ A) @ B)) @ ((product_snd @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F)) = ((product_snd @ A) @ B))))).
% 3.09/1.54 thf(fact_208_fst__comp__map__prod, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,F:(A > C),G3:(B > D)]: (((((((comp @ ((product_prod @ C) @ D)) @ C) @ ((product_prod @ A) @ B)) @ ((product_fst @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F) @ G3)) = (((((comp @ A) @ C) @ ((product_prod @ A) @ B)) @ F) @ ((product_fst @ A) @ B)))))).
% 3.09/1.54 thf(fact_209_snd__comp__map__prod, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,F:(A > D),G3:(B > C)]: (((((((comp @ ((product_prod @ D) @ C)) @ C) @ ((product_prod @ A) @ B)) @ ((product_snd @ D) @ C)) @ ((((((product_map_prod @ A) @ D) @ B) @ C) @ F) @ G3)) = (((((comp @ B) @ C) @ ((product_prod @ A) @ B)) @ G3) @ ((product_snd @ A) @ B)))))).
% 3.09/1.54 thf(fact_210_fst__comp__apfst, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > C)]: (((((((comp @ ((product_prod @ C) @ B)) @ C) @ ((product_prod @ A) @ B)) @ ((product_fst @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F)) = (((((comp @ A) @ C) @ ((product_prod @ A) @ B)) @ F) @ ((product_fst @ A) @ B)))))).
% 3.09/1.54 thf(fact_211_snd__comp__apsnd, axiom, (! [C:$tType,B:$tType,A:$tType,F:(B > C)]: (((((((comp @ ((product_prod @ A) @ C)) @ C) @ ((product_prod @ A) @ B)) @ ((product_snd @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F)) = (((((comp @ B) @ C) @ ((product_prod @ A) @ B)) @ F) @ ((product_snd @ A) @ B)))))).
% 3.09/1.54 thf(fact_212_apfst__apsnd, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,F:(C > A),G3:(D > B),X:((product_prod @ C) @ D)]: (((((((product_apfst @ C) @ A) @ B) @ F) @ (((((product_apsnd @ D) @ B) @ C) @ G3) @ X)) = ((((product_Pair @ A) @ B) @ (F @ (((product_fst @ C) @ D) @ X))) @ (G3 @ (((product_snd @ C) @ D) @ X))))))).
% 3.09/1.54 thf(fact_213_apsnd__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F:(C > B),G3:(D > A),X:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ (((((product_apfst @ D) @ A) @ C) @ G3) @ X)) = ((((product_Pair @ A) @ B) @ (G3 @ (((product_fst @ D) @ C) @ X))) @ (F @ (((product_snd @ D) @ C) @ X))))))).
% 3.09/1.54 thf(fact_214_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P4:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P4)) @ (((product_fst @ A) @ B) @ P4)))))))).
% 3.09/1.54 thf(fact_215_prod_Osplit__sel, axiom, (! [C:$tType,B:$tType,A:$tType,P:(C > $o),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) => (P @ ((F @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))).
% 3.09/1.54 thf(fact_216_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 3.09/1.54 thf(fact_217_prod_Osplit__sel__asm, axiom, (! [C:$tType,B:$tType,A:$tType,P:(C > $o),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = (~ (((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) & (~ ((P @ ((F @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))))))).
% 3.09/1.54 thf(fact_218_surjective__pairing, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: ((T2 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T2)) @ (((product_snd @ A) @ B) @ T2)))))).
% 3.09/1.54 thf(fact_219_sndE, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),A5:A,B3:B,P:(B > $o)]: (((X = ((((product_Pair @ A) @ B) @ A5) @ B3)) => ((P @ (((product_snd @ A) @ B) @ X)) => (P @ B3)))))).
% 3.09/1.54 thf(fact_220_fstE, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A5:A,B3:B,P:(A > $o)]: (((X = ((((product_Pair @ A) @ B) @ A5) @ B3)) => ((P @ (((product_fst @ A) @ B) @ X)) => (P @ A5)))))).
% 3.09/1.54 thf(fact_221_snd__eqD, axiom, (! [B:$tType,A:$tType,X:B,Y:A,A5:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = A5) => (Y = A5))))).
% 3.09/1.54 thf(fact_222_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X22:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X22)) = X22)))).
% 3.09/1.54 thf(fact_223_fst__eqD, axiom, (! [B:$tType,A:$tType,X:A,Y:B,A5:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X) @ Y)) = A5) => (X = A5))))).
% 3.09/1.54 thf(fact_224_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X22:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = X1)))).
% 3.09/1.54 thf(fact_225_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y5:((product_prod @ A) @ B),Z:((product_prod @ A) @ B)]: ((Y5 = Z))) = (^ [S:((product_prod @ A) @ B),T4:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S) = (((product_fst @ A) @ B) @ T4)) & ((((product_snd @ A) @ B) @ S) = (((product_snd @ A) @ B) @ T4))))))))).
% 3.09/1.54 thf(fact_226_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))).
% 3.09/1.54 thf(fact_227_prod__eqI, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),Q2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P2) = (((product_fst @ A) @ B) @ Q2)) => (((((product_snd @ A) @ B) @ P2) = (((product_snd @ A) @ B) @ Q2)) => (P2 = Q2)))))).
% 3.09/1.54 thf(fact_228_All__prod__contract, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o))]: (((! [A7:A,X7:B]: (((P @ A7) @ X7))) = (! [Z3:((product_prod @ A) @ B)]: (((P @ (((product_fst @ A) @ B) @ Z3)) @ (((product_snd @ A) @ B) @ Z3)))))))).
% 3.09/1.54 thf(fact_229_Ex__prod__contract, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o))]: (((? [A7:A,X7:B]: (((P @ A7) @ X7))) = (? [Z3:((product_prod @ A) @ B)]: (((P @ (((product_fst @ A) @ B) @ Z3)) @ (((product_snd @ A) @ B) @ Z3)))))))).
% 3.09/1.54 thf(fact_230_case__prod__beta, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ A) = (^ [F3:(B > (C > A)),P4:((product_prod @ B) @ C)]: (((F3 @ (((product_fst @ B) @ C) @ P4)) @ (((product_snd @ B) @ C) @ P4)))))))).
% 3.09/1.54 thf(fact_231_prod_Ocase__eq__if, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F3:(A > (B > C)),Prod3:((product_prod @ A) @ B)]: (((F3 @ (((product_fst @ A) @ B) @ Prod3)) @ (((product_snd @ A) @ B) @ Prod3)))))))).
% 3.09/1.54 thf(fact_232_Product__Type_OCollect__case__prodD, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A2:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A2))) => ((A2 @ (((product_fst @ A) @ B) @ X)) @ (((product_snd @ A) @ B) @ X)))))).
% 3.09/1.54 thf(fact_233_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A5:A,F:(B > A),B3:B,C2:B]: (((((ord_less_eq @ A) @ A5) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X5:B,Y3:B]: (((((ord_less_eq @ B) @ X5) @ Y3) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A5) @ (F @ C2))))))))))).
% 3.09/1.54 thf(fact_234_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A5:A,B3:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A5) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C2) => ((! [X5:A,Y3:A]: (((((ord_less_eq @ A) @ X5) @ Y3) => (((ord_less_eq @ C) @ (F @ X5)) @ (F @ Y3))))) => (((ord_less_eq @ C) @ (F @ A5)) @ C2)))))))))).
% 3.09/1.54 thf(fact_235_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A5:A,F:(B > A),B3:B,C2:B]: (((A5 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X5:B,Y3:B]: (((((ord_less_eq @ B) @ X5) @ Y3) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A5) @ (F @ C2))))))))))).
% 3.09/1.54 thf(fact_236_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A5:A,B3:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A5) @ B3) => (((F @ B3) = C2) => ((! [X5:A,Y3:A]: (((((ord_less_eq @ A) @ X5) @ Y3) => (((ord_less_eq @ B) @ (F @ X5)) @ (F @ Y3))))) => (((ord_less_eq @ B) @ (F @ A5)) @ C2)))))))))).
% 3.09/1.54 thf(fact_237_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [X6:A,Y6:A]: (((((ord_less_eq @ A) @ X6) @ Y6) & (((ord_less_eq @ A) @ Y6) @ X6))))))))).
% 3.09/1.54 thf(fact_238_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 3.09/1.54 thf(fact_239_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.09/1.54 thf(fact_240_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 3.09/1.54 thf(fact_241_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.09/1.54 thf(fact_242_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A5) @ C2))))))))).
% 3.09/1.54 thf(fact_243_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A,Z2:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z2)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z2)))) => (((((ord_less_eq @ A) @ X) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y)))) => (((((ord_less_eq @ A) @ Z2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 3.09/1.54 thf(fact_244_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 3.09/1.54 thf(fact_245_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [A7:A,B6:A]: (((((ord_less_eq @ A) @ A7) @ B6) & (((ord_less_eq @ A) @ B6) @ A7))))))))).
% 3.09/1.54 thf(fact_246_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A5:A,B3:A,C2:A]: (((A5 = B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A5) @ C2))))))))).
% 3.09/1.54 thf(fact_247_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A5:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((B3 = C2) => (((ord_less_eq @ A) @ A5) @ C2))))))))).
% 3.09/1.54 thf(fact_248_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B3:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((((ord_less_eq @ A) @ B3) @ A5) => (A5 = B3))))))))).
% 3.09/1.54 thf(fact_249_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 3.09/1.54 thf(fact_250_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A]: ((((ord_less_eq @ A) @ A5) @ A5))))))).
% 3.09/1.54 thf(fact_251_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A5:A,B3:A]: (((! [A3:A,B5:A]: (((((ord_less_eq @ A) @ A3) @ B5) => ((P @ A3) @ B5)))) => ((! [A3:A,B5:A]: ((((P @ B5) @ A3) => ((P @ A3) @ B5)))) => ((P @ A5) @ B3))))))))).
% 3.09/1.54 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 3.09/1.54 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 3.09/1.54 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 3.09/1.54 thf(tcon_Int_Oint___Orderings_Opreorder_1, axiom, (preorder @ int)).
% 3.09/1.54 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 3.09/1.54 thf(tcon_Int_Oint___Orderings_Oorder_2, axiom, (order @ int)).
% 3.09/1.54 thf(tcon_Int_Oint___Orderings_Oord_3, axiom, (ord @ int)).
% 3.09/1.54 thf(tcon_Nat_Onat___Orderings_Opreorder_4, axiom, (preorder @ nat)).
% 3.09/1.54 thf(tcon_Nat_Onat___Orderings_Olinorder_5, axiom, (linorder @ nat)).
% 3.09/1.54 thf(tcon_Nat_Onat___Orderings_Oorder_6, axiom, (order @ nat)).
% 3.09/1.54 thf(tcon_Nat_Onat___Orderings_Oord_7, axiom, (ord @ nat)).
% 3.09/1.54 thf(tcon_Set_Oset___Orderings_Opreorder_8, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 3.09/1.54 thf(tcon_Set_Oset___Orderings_Oorder_9, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 3.09/1.54 thf(tcon_Set_Oset___Orderings_Oord_10, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 3.09/1.54 thf(tcon_HOL_Obool___Orderings_Opreorder_11, axiom, (preorder @ $o)).
% 3.09/1.54 thf(tcon_HOL_Obool___Orderings_Olinorder_12, axiom, (linorder @ $o)).
% 3.09/1.54 thf(tcon_HOL_Obool___Orderings_Oorder_13, axiom, (order @ $o)).
% 3.09/1.54 thf(tcon_HOL_Obool___Orderings_Oord_14, axiom, (ord @ $o)).
% 3.09/1.54 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_15, axiom, (! [A8:$tType]: ((preorder @ (refine1665802226e_nres @ A8))))).
% 3.09/1.54 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_16, axiom, (! [A8:$tType]: ((order @ (refine1665802226e_nres @ A8))))).
% 3.09/1.54 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_17, axiom, (! [A8:$tType]: ((ord @ (refine1665802226e_nres @ A8))))).
% 3.09/1.54 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 3.09/1.54 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 3.09/1.54 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 3.09/1.54 thf(conj_0, conjecture, (((member @ ((product_prod @ ((refine1665802226e_nres @ a) > ((a > (refine1665802226e_nres @ b)) > (refine1665802226e_nres @ b)))) @ ((refine1665802226e_nres @ c) > ((c > (refine1665802226e_nres @ d)) > (refine1665802226e_nres @ d))))) @ ((((product_Pair @ ((refine1665802226e_nres @ a) > ((a > (refine1665802226e_nres @ b)) > (refine1665802226e_nres @ b)))) @ ((refine1665802226e_nres @ c) > ((c > (refine1665802226e_nres @ d)) > (refine1665802226e_nres @ d)))) @ ((refine463715084e_bind @ a) @ b)) @ ((refine463715084e_bind @ c) @ d))) @ (((((relAPP @ ((a > (refine1665802226e_nres @ b)) > (refine1665802226e_nres @ b))) @ ((c > (refine1665802226e_nres @ d)) > (refine1665802226e_nres @ d))) @ (set @ ((product_prod @ ((refine1665802226e_nres @ a) > ((a > (refine1665802226e_nres @ b)) > (refine1665802226e_nres @ b)))) @ ((refine1665802226e_nres @ c) > ((c > (refine1665802226e_nres @ d)) > (refine1665802226e_nres @ d)))))) @ (((((relAPP @ (refine1665802226e_nres @ a)) @ (refine1665802226e_nres @ c)) @ ((set @ ((product_prod @ ((a > (refine1665802226e_nres @ b)) > (refine1665802226e_nres @ b))) @ ((c > (refine1665802226e_nres @ d)) > (refine1665802226e_nres @ d)))) > (set @ ((product_prod @ ((refine1665802226e_nres @ a) > ((a > (refine1665802226e_nres @ b)) > (refine1665802226e_nres @ b)))) @ ((refine1665802226e_nres @ c) > ((c > (refine1665802226e_nres @ d)) > (refine1665802226e_nres @ d))))))) @ ((((fun_rel @ (refine1665802226e_nres @ a)) @ (refine1665802226e_nres @ c)) @ ((a > (refine1665802226e_nres @ b)) > (refine1665802226e_nres @ b))) @ ((c > (refine1665802226e_nres @ d)) > (refine1665802226e_nres @ d)))) @ (((((relAPP @ a) @ c) @ (set @ ((product_prod @ (refine1665802226e_nres @ a)) @ (refine1665802226e_nres @ c)))) @ ((refine476890328es_rel @ a) @ c)) @ r1))) @ (((((relAPP @ (refine1665802226e_nres @ b)) @ (refine1665802226e_nres @ d)) @ (set @ ((product_prod @ ((a > (refine1665802226e_nres @ b)) > (refine1665802226e_nres @ b))) @ ((c > (refine1665802226e_nres @ d)) > (refine1665802226e_nres @ d))))) @ (((((relAPP @ (a > (refine1665802226e_nres @ b))) @ (c > (refine1665802226e_nres @ d))) @ ((set @ ((product_prod @ (refine1665802226e_nres @ b)) @ (refine1665802226e_nres @ d))) > (set @ ((product_prod @ ((a > (refine1665802226e_nres @ b)) > (refine1665802226e_nres @ b))) @ ((c > (refine1665802226e_nres @ d)) > (refine1665802226e_nres @ d)))))) @ ((((fun_rel @ (a > (refine1665802226e_nres @ b))) @ (c > (refine1665802226e_nres @ d))) @ (refine1665802226e_nres @ b)) @ (refine1665802226e_nres @ d))) @ (((((relAPP @ (refine1665802226e_nres @ b)) @ (refine1665802226e_nres @ d)) @ (set @ ((product_prod @ (a > (refine1665802226e_nres @ b))) @ (c > (refine1665802226e_nres @ d))))) @ (((((relAPP @ a) @ c) @ ((set @ ((product_prod @ (refine1665802226e_nres @ b)) @ (refine1665802226e_nres @ d))) > (set @ ((product_prod @ (a > (refine1665802226e_nres @ b))) @ (c > (refine1665802226e_nres @ d)))))) @ ((((fun_rel @ a) @ c) @ (refine1665802226e_nres @ b)) @ (refine1665802226e_nres @ d))) @ r1)) @ (((((relAPP @ b) @ d) @ (set @ ((product_prod @ (refine1665802226e_nres @ b)) @ (refine1665802226e_nres @ d)))) @ ((refine476890328es_rel @ b) @ d)) @ r2)))) @ (((((relAPP @ b) @ d) @ (set @ ((product_prod @ (refine1665802226e_nres @ b)) @ (refine1665802226e_nres @ d)))) @ ((refine476890328es_rel @ b) @ d)) @ r2))))).
% 3.09/1.54 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.LLLxuS3oP5/DTF2DTF2166.p
% 3.09/1.54 ------------------------
% 3.09/1.59 ---- Cleaned THF ---
% 3.09/1.60 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
% 3.09/1.60 refine1665802226e_nres: $tType > $tType ).
% 3.09/1.60
% 3.09/1.60 thf(ty_t_Autoref__Tagging_Oannot,type,
% 3.09/1.60 autoref_annot: $tType ).
% 3.09/1.60
% 3.09/1.60 thf(ty_t_Product__Type_Oprod,type,
% 3.09/1.60 product_prod: $tType > $tType > $tType ).
% 3.09/1.60
% 3.09/1.60 thf(ty_t_Set_Oset,type,
% 3.09/1.60 set: $tType > $tType ).
% 3.09/1.60
% 3.09/1.60 thf(ty_t_Nat_Onat,type,
% 3.09/1.60 nat: $tType ).
% 3.09/1.60
% 3.09/1.60 thf(ty_t_Int_Oint,type,
% 3.09/1.60 int: $tType ).
% 3.09/1.60
% 3.09/1.60 thf(ty_tf_d,type,
% 3.09/1.60 d: $tType ).
% 3.09/1.60
% 3.09/1.60 thf(ty_tf_c,type,
% 3.09/1.60 c: $tType ).
% 3.09/1.60
% 3.09/1.60 thf(ty_tf_b,type,
% 3.09/1.60 b: $tType ).
% 3.09/1.60
% 3.09/1.60 thf(ty_tf_a,type,
% 3.09/1.60 a: $tType ).
% 3.09/1.60
% 3.09/1.60 thf(sy_cl_HOL_Otype,type,
% 3.09/1.60 type:
% 3.09/1.60 !>[A: $tType] : $o ).
% 3.09/1.60
% 3.09/1.60 thf(sy_cl_Orderings_Oord,type,
% 3.09/1.60 ord:
% 3.09/1.60 !>[A: $tType] : $o ).
% 3.09/1.60
% 3.09/1.60 thf(sy_cl_Orderings_Oorder,type,
% 3.09/1.60 order:
% 3.09/1.60 !>[A: $tType] : $o ).
% 3.09/1.60
% 3.09/1.60 thf(sy_cl_Orderings_Olinorder,type,
% 3.09/1.60 linorder:
% 3.09/1.60 !>[A: $tType] : $o ).
% 3.09/1.60
% 3.09/1.60 thf(sy_cl_Orderings_Opreorder,type,
% 3.09/1.60 preorder:
% 3.09/1.60 !>[A: $tType] : $o ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Autoref__Tagging_OANNOT,type,
% 3.09/1.60 autoref_ANNOT:
% 3.09/1.60 !>[A: $tType] : ( A > autoref_annot > A ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Autoref__Tagging_OAPP,type,
% 3.09/1.60 autoref_APP:
% 3.09/1.60 !>[B: $tType,A: $tType] : ( ( B > A ) > B > A ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Autoref__Tagging_OOP,type,
% 3.09/1.60 autoref_OP:
% 3.09/1.60 !>[A: $tType] : ( A > A ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Autoref__Tagging_Orel__annot,type,
% 3.09/1.60 autoref_rel_annot:
% 3.09/1.60 !>[C: $tType,A: $tType] : ( ( set @ ( product_prod @ C @ A ) ) > autoref_annot ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Autoref__Translate_OREMOVE__INTERNAL,type,
% 3.09/1.60 autore2013370551TERNAL:
% 3.09/1.60 !>[A: $tType] : ( A > A ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Autoref__Translate_OREMOVE__INTERNAL__EQ,type,
% 3.09/1.60 autore813025987NAL_EQ:
% 3.09/1.60 !>[A: $tType] : ( A > A > $o ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Fun_Ocomp,type,
% 3.09/1.60 comp:
% 3.09/1.60 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Fun_Ofun__upd,type,
% 3.09/1.60 fun_upd:
% 3.09/1.60 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Fun_Oid,type,
% 3.09/1.60 id:
% 3.09/1.60 !>[A: $tType] : ( A > A ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_If,type,
% 3.09/1.60 if:
% 3.09/1.60 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_List_Oall__interval__nat,type,
% 3.09/1.60 all_interval_nat: ( nat > $o ) > nat > nat > $o ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Misc_Ouncurry,type,
% 3.09/1.60 uncurry:
% 3.09/1.60 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Nat_Onat_Ocase__nat,type,
% 3.09/1.60 case_nat:
% 3.09/1.60 !>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
% 3.09/1.60 rec_nat:
% 3.09/1.60 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Orderings_Oord__class_Oless,type,
% 3.09/1.60 ord_less:
% 3.09/1.60 !>[A: $tType] : ( A > A > $o ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.09/1.60 ord_less_eq:
% 3.09/1.60 !>[A: $tType] : ( A > A > $o ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Product__Type_OPair,type,
% 3.09/1.60 product_Pair:
% 3.09/1.60 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Product__Type_Oapfst,type,
% 3.09/1.60 product_apfst:
% 3.09/1.60 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Product__Type_Oapsnd,type,
% 3.09/1.60 product_apsnd:
% 3.09/1.60 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Product__Type_Obool_Ocase__bool,type,
% 3.09/1.60 product_case_bool:
% 3.09/1.60 !>[A: $tType] : ( A > A > $o > A ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Product__Type_Ocurry,type,
% 3.09/1.60 product_curry:
% 3.09/1.60 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 3.09/1.60 produc2004651681e_prod:
% 3.09/1.60 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Product__Type_Omap__prod,type,
% 3.09/1.60 product_map_prod:
% 3.09/1.60 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 3.09/1.60 product_rec_bool:
% 3.09/1.60 !>[T: $tType] : ( T > T > $o > T ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 3.09/1.60 product_rec_prod:
% 3.09/1.60 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 3.09/1.60 product_case_prod:
% 3.09/1.60 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 3.09/1.60 product_fst:
% 3.09/1.60 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 3.09/1.60 product_snd:
% 3.09/1.60 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 3.09/1.60 product_swap:
% 3.09/1.60 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
% 3.09/1.60 refine1687780735RETURN:
% 3.09/1.60 !>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind,type,
% 3.09/1.60 refine463715084e_bind:
% 3.09/1.60 !>[B: $tType,A: $tType] : ( ( refine1665802226e_nres @ B ) > ( B > ( refine1665802226e_nres @ A ) ) > ( refine1665802226e_nres @ A ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Olift__assn,type,
% 3.09/1.60 refine1580981607t_assn:
% 3.09/1.60 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( B > $o ) > A > $o ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres__rel,type,
% 3.09/1.60 refine476890328es_rel:
% 3.09/1.60 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oop__nres__ASSERT__bnd,type,
% 3.09/1.60 refine171938416RT_bnd:
% 3.09/1.60 !>[A: $tType] : ( $o > ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Refine__Misc_Oless__than__bool,type,
% 3.09/1.60 refine1361002633n_bool: set @ ( product_prod @ $o @ $o ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Relation_OId,type,
% 3.09/1.60 id2:
% 3.09/1.60 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Relators_Ofun__rel,type,
% 3.09/1.60 fun_rel:
% 3.09/1.60 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Relators_Oprod__rel,type,
% 3.09/1.60 prod_rel:
% 3.09/1.60 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Relators_OrelAPP,type,
% 3.09/1.60 relAPP:
% 3.09/1.60 !>[C1: $tType,A1: $tType,A: $tType] : ( ( ( set @ ( product_prod @ C1 @ A1 ) ) > A ) > ( set @ ( product_prod @ C1 @ A1 ) ) > A ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_Set_OCollect,type,
% 3.09/1.60 collect:
% 3.09/1.60 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_c_member,type,
% 3.09/1.60 member:
% 3.09/1.60 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_v_R1,type,
% 3.09/1.60 r1: set @ ( product_prod @ a @ c ) ).
% 3.09/1.60
% 3.09/1.60 thf(sy_v_R2,type,
% 3.09/1.60 r2: set @ ( product_prod @ b @ d ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_0_param__bind,axiom,
% 3.09/1.60 ! [C: $tType,A: $tType,D: $tType,B: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( ( refine1665802226e_nres @ A ) > ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( refine1665802226e_nres @ C ) > ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) @ ( product_Pair @ ( ( refine1665802226e_nres @ A ) > ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( refine1665802226e_nres @ C ) > ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) @ ( refine463715084e_bind @ A @ B ) @ ( refine463715084e_bind @ C @ D ) ) @ ( relAPP @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) @ ( set @ ( product_prod @ ( ( refine1665802226e_nres @ A ) > ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( refine1665802226e_nres @ C ) > ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) @ ( relAPP @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ C ) @ ( ( set @ ( product_prod @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) > ( set @ ( product_prod @ ( ( refine1665802226e_nres @ A ) > ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( refine1665802226e_nres @ C ) > ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) ) @ ( fun_rel @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ C ) @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) @ ( relAPP @ A @ C @ ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ C ) ) ) @ ( refine476890328es_rel @ A @ C ) @ Ra ) ) @ ( relAPP @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) @ ( set @ ( product_prod @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) @ ( relAPP @ ( A > ( refine1665802226e_nres @ B ) ) @ ( C > ( refine1665802226e_nres @ D ) ) @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) ) > ( set @ ( product_prod @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) ) @ ( fun_rel @ ( A > ( refine1665802226e_nres @ B ) ) @ ( C > ( refine1665802226e_nres @ D ) ) @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) @ ( relAPP @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) @ ( set @ ( product_prod @ ( A > ( refine1665802226e_nres @ B ) ) @ ( C > ( refine1665802226e_nres @ D ) ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) ) > ( set @ ( product_prod @ ( A > ( refine1665802226e_nres @ B ) ) @ ( C > ( refine1665802226e_nres @ D ) ) ) ) ) @ ( fun_rel @ A @ C @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) @ Ra ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) ) @ ( refine476890328es_rel @ B @ D ) @ Rb ) ) ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) ) @ ( refine476890328es_rel @ B @ D ) @ Rb ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_1_fun__relI,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: set @ ( product_prod @ A @ B ),F: A > C,F2: B > D,B2: set @ ( product_prod @ C @ D )] :
% 3.09/1.60 ( ! [A3: A,A4: B] :
% 3.09/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ A4 ) @ A2 )
% 3.09/1.60 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( F2 @ A4 ) ) @ B2 ) )
% 3.09/1.60 => ( member @ ( product_prod @ ( A > C ) @ ( B > D ) ) @ ( product_Pair @ ( A > C ) @ ( B > D ) @ F @ F2 ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ A2 ) @ B2 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_2_autoref__Let,axiom,
% 3.09/1.60 ! [C: $tType,A: $tType,D: $tType,B: $tType,Ra: set @ ( product_prod @ A @ C ),Rr: set @ ( product_prod @ B @ D )] :
% 3.09/1.60 ( member @ ( product_prod @ ( A > ( A > B ) > B ) @ ( C > ( C > D ) > D ) )
% 3.09/1.60 @ ( product_Pair @ ( A > ( A > B ) > B ) @ ( C > ( C > D ) > D )
% 3.09/1.60 @ ^ [S: A,F3: A > B] : ( F3 @ S )
% 3.09/1.60 @ ^ [S: C,F3: C > D] : ( F3 @ S ) )
% 3.09/1.60 @ ( relAPP @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) @ ( set @ ( product_prod @ ( A > ( A > B ) > B ) @ ( C > ( C > D ) > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) ) ) > ( set @ ( product_prod @ ( A > ( A > B ) > B ) @ ( C > ( C > D ) > D ) ) ) ) @ ( fun_rel @ A @ C @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) ) @ Ra ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) ) ) @ ( relAPP @ ( A > B ) @ ( C > D ) @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( C > D ) @ B @ D ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ Ra ) @ Rr ) ) @ Rr ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_3_fun__relD1,axiom,
% 3.09/1.60 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,Ra: set @ ( product_prod @ A @ C ),Rr: set @ ( product_prod @ B @ D ),X: A,R: B] :
% 3.09/1.60 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ Ra ) @ Rr ) )
% 3.09/1.60 => ( ( ( F @ X )
% 3.09/1.60 = R )
% 3.09/1.60 => ! [X2: C] :
% 3.09/1.60 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X2 ) @ Ra )
% 3.09/1.60 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ R @ ( F2 @ X2 ) ) @ Rr ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_4_fun__relD2,axiom,
% 3.09/1.60 ! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,Ra: set @ ( product_prod @ A @ C ),Rr: set @ ( product_prod @ B @ D ),X3: C,R2: D] :
% 3.09/1.60 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ Ra ) @ Rr ) )
% 3.09/1.60 => ( ( ( F2 @ X3 )
% 3.09/1.60 = R2 )
% 3.09/1.60 => ! [X4: A] :
% 3.09/1.60 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ X3 ) @ Ra )
% 3.09/1.60 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X4 ) @ R2 ) @ Rr ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_5_tagged__fun__relD__lhs,axiom,
% 3.09/1.60 ! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D ),X: A,X3: C] :
% 3.09/1.60 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) )
% 3.09/1.60 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ A2 )
% 3.09/1.60 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X ) @ ( F2 @ X3 ) ) @ B2 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_6_tagged__fun__relD__rhs,axiom,
% 3.09/1.60 ! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D ),X: A,X3: C] :
% 3.09/1.60 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) )
% 3.09/1.60 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ A2 )
% 3.09/1.60 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X ) @ ( F2 @ X3 ) ) @ B2 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_7_tagged__fun__relD__both,axiom,
% 3.09/1.60 ! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D ),X: A,X3: C] :
% 3.09/1.60 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) )
% 3.09/1.60 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ A2 )
% 3.09/1.60 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X ) @ ( F2 @ X3 ) ) @ B2 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_8_tagged__fun__relD__none,axiom,
% 3.09/1.60 ! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D ),X: A,X3: C] :
% 3.09/1.60 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) )
% 3.09/1.60 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ A2 )
% 3.09/1.60 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X ) @ ( F2 @ X3 ) ) @ B2 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_9_prod_Oinject,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 3.09/1.60 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 3.09/1.60 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 3.09/1.60 = ( ( X1 = Y1 )
% 3.09/1.60 & ( X22 = Y2 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_10_old_Oprod_Oinject,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,A5: A,B3: B,A6: A,B4: B] :
% 3.09/1.60 ( ( ( product_Pair @ A @ B @ A5 @ B3 )
% 3.09/1.60 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 3.09/1.60 = ( ( A5 = A6 )
% 3.09/1.60 & ( B3 = B4 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_11_autoref__APP,axiom,
% 3.09/1.60 ! [C: $tType,A: $tType,D: $tType,B: $tType,C2: A > B,A5: C > D,Ra: set @ ( product_prod @ A @ C ),Rr: set @ ( product_prod @ B @ D ),X: A,X3: C] :
% 3.09/1.60 ( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ C2 @ A5 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ Ra ) @ Rr ) )
% 3.09/1.60 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ Ra )
% 3.09/1.60 => ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( autoref_APP @ A @ B @ C2 @ X ) @ ( autoref_APP @ C @ D @ A5 @ X3 ) ) @ Rr ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_12_lift__assnI,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,S2: A,S3: B,R3: set @ ( product_prod @ A @ B ),Phi: B > $o] :
% 3.09/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S2 @ S3 ) @ R3 )
% 3.09/1.60 => ( ( Phi @ S3 )
% 3.09/1.60 => ( refine1580981607t_assn @ A @ B @ R3 @ Phi @ S2 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_13_old_Oprod_Oinducts,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 3.09/1.60 ( ! [A3: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A3 @ B5 ) )
% 3.09/1.60 => ( P @ Prod ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_14_old_Oprod_Oexhaust,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 3.09/1.60 ~ ! [A3: A,B5: B] :
% 3.09/1.60 ( Y
% 3.09/1.60 != ( product_Pair @ A @ B @ A3 @ B5 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_15_prod__induct7,axiom,
% 3.09/1.60 ! [G: $tType,F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) )] :
% 3.09/1.60 ( ! [A3: A,B5: B,C3: C,D2: D,E2: E,F5: F4,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G ) @ E2 @ ( product_Pair @ F4 @ G @ F5 @ G2 ) ) ) ) ) ) )
% 3.09/1.60 => ( P @ X ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_16_prod__induct6,axiom,
% 3.09/1.60 ! [F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
% 3.09/1.60 ( ! [A3: A,B5: B,C3: C,D2: D,E2: E,F5: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) )
% 3.09/1.60 => ( P @ X ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_17_prod__induct5,axiom,
% 3.09/1.60 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 3.09/1.60 ( ! [A3: A,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 3.09/1.60 => ( P @ X ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_18_prod__induct4,axiom,
% 3.09/1.60 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.09/1.60 ( ! [A3: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 3.09/1.60 => ( P @ X ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_19_prod__induct3,axiom,
% 3.09/1.60 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.09/1.60 ( ! [A3: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
% 3.09/1.60 => ( P @ X ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_20_prod__cases7,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) )] :
% 3.09/1.60 ~ ! [A3: A,B5: B,C3: C,D2: D,E2: E,F5: F4,G2: G] :
% 3.09/1.60 ( Y
% 3.09/1.60 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G ) @ E2 @ ( product_Pair @ F4 @ G @ F5 @ G2 ) ) ) ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_21_prod__cases6,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
% 3.09/1.60 ~ ! [A3: A,B5: B,C3: C,D2: D,E2: E,F5: F4] :
% 3.09/1.60 ( Y
% 3.09/1.60 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_22_prod__cases5,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 3.09/1.60 ~ ! [A3: A,B5: B,C3: C,D2: D,E2: E] :
% 3.09/1.60 ( Y
% 3.09/1.60 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_23_prod__cases4,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.09/1.60 ~ ! [A3: A,B5: B,C3: C,D2: D] :
% 3.09/1.60 ( Y
% 3.09/1.60 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_24_prod__cases3,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.09/1.60 ~ ! [A3: A,B5: B,C3: C] :
% 3.09/1.60 ( Y
% 3.09/1.60 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_25_Pair__inject,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,A5: A,B3: B,A6: A,B4: B] :
% 3.09/1.60 ( ( ( product_Pair @ A @ B @ A5 @ B3 )
% 3.09/1.60 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 3.09/1.60 => ~ ( ( A5 = A6 )
% 3.09/1.60 => ( B3 != B4 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_26_prod__cases,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 3.09/1.60 ( ! [A3: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A3 @ B5 ) )
% 3.09/1.60 => ( P @ P2 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_27_surj__pair,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 3.09/1.60 ? [X5: A,Y3: B] :
% 3.09/1.60 ( P2
% 3.09/1.60 = ( product_Pair @ A @ B @ X5 @ Y3 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_28_relAPP__def,axiom,
% 3.09/1.60 ! [A: $tType,A1: $tType,C1: $tType] :
% 3.09/1.60 ( ( relAPP @ C1 @ A1 @ A )
% 3.09/1.60 = ( ^ [F3: ( set @ ( product_prod @ C1 @ A1 ) ) > A] : F3 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_29_lift__assn__def,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType] :
% 3.09/1.60 ( ( refine1580981607t_assn @ A @ B )
% 3.09/1.60 = ( ^ [R4: set @ ( product_prod @ A @ B ),Phi2: B > $o,S: A] :
% 3.09/1.60 ? [S4: B] :
% 3.09/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S @ S4 ) @ R4 )
% 3.09/1.60 & ( Phi2 @ S4 ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_30_autoref__RETURN,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > ( refine1665802226e_nres @ A ) ) @ ( B > ( refine1665802226e_nres @ B ) ) ) @ ( product_Pair @ ( A > ( refine1665802226e_nres @ A ) ) @ ( B > ( refine1665802226e_nres @ B ) ) @ ( refine1687780735RETURN @ A ) @ ( refine1687780735RETURN @ B ) ) @ ( relAPP @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) @ ( set @ ( product_prod @ ( A > ( refine1665802226e_nres @ A ) ) @ ( B > ( refine1665802226e_nres @ B ) ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) > ( set @ ( product_prod @ ( A > ( refine1665802226e_nres @ A ) ) @ ( B > ( refine1665802226e_nres @ B ) ) ) ) ) @ ( fun_rel @ A @ B @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) @ ( refine476890328es_rel @ A @ B ) @ R3 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_31_param__op__nres__ASSERT__bnd,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,Phi3: $o,Phi: $o,M: refine1665802226e_nres @ A,M2: refine1665802226e_nres @ B,R3: set @ ( product_prod @ A @ B )] :
% 3.09/1.60 ( ( Phi3
% 3.09/1.60 => Phi )
% 3.09/1.60 => ( ( Phi3
% 3.09/1.60 => ( Phi
% 3.09/1.60 => ( member @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) @ M @ M2 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) @ ( refine476890328es_rel @ A @ B ) @ R3 ) ) ) )
% 3.09/1.60 => ( member @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) @ ( refine171938416RT_bnd @ A @ Phi @ M ) @ ( refine171938416RT_bnd @ B @ Phi3 @ M2 ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) @ ( refine476890328es_rel @ A @ B ) @ R3 ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_32_autoref__If,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) @ ( product_Pair @ ( $o > A > A > A ) @ ( $o > B > B > B ) @ ( if @ A ) @ ( if @ B ) ) @ ( relAPP @ ( A > A > A ) @ ( B > B > B ) @ ( set @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) > ( set @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) ) ) @ ( fun_rel @ $o @ $o @ ( A > A > A ) @ ( B > B > B ) ) @ ( id2 @ $o ) ) @ ( relAPP @ ( A > A ) @ ( B > B ) @ ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) > ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) ) @ ( fun_rel @ A @ B @ ( A > A ) @ ( B > B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) ) @ ( fun_rel @ A @ B @ A @ B ) @ R3 ) @ R3 ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_33_fun__relE2,axiom,
% 3.09/1.60 ! [A: $tType,C: $tType,B: $tType,F: A > B,F2: A > C,Rv: set @ ( product_prod @ B @ C ),T2: B,X: A] :
% 3.09/1.60 ( ( member @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_Pair @ ( A > B ) @ ( A > C ) @ F @ F2 ) @ ( relAPP @ B @ C @ ( set @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( fun_rel @ A @ A @ B @ C ) @ ( id2 @ A ) ) @ Rv ) )
% 3.09/1.60 => ( ( T2
% 3.09/1.60 = ( F @ X ) )
% 3.09/1.60 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ T2 @ ( F2 @ X ) ) @ Rv ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_34_fun__relE1,axiom,
% 3.09/1.60 ! [A: $tType,C: $tType,B: $tType,F: A > B,F2: A > C,Rv: set @ ( product_prod @ B @ C ),T3: C,X: A] :
% 3.09/1.60 ( ( member @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_Pair @ ( A > B ) @ ( A > C ) @ F @ F2 ) @ ( relAPP @ B @ C @ ( set @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( fun_rel @ A @ A @ B @ C ) @ ( id2 @ A ) ) @ Rv ) )
% 3.09/1.60 => ( ( T3
% 3.09/1.60 = ( F2 @ X ) )
% 3.09/1.60 => ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ ( F @ X ) @ T3 ) @ Rv ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_35_autoref__Let__cong,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,D: $tType,C: $tType,X3: A,X: B,Ra: set @ ( product_prod @ A @ B ),F2: A > C,F: B > D,Rr: set @ ( product_prod @ C @ D )] :
% 3.09/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ X ) @ Ra )
% 3.09/1.60 => ( ! [Y3: B] :
% 3.09/1.60 ( ( autore2013370551TERNAL @ $o @ ( X = Y3 ) )
% 3.09/1.60 => ! [Y4: A] :
% 3.09/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y4 @ Y3 ) @ Ra )
% 3.09/1.60 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F2 @ Y4 ) @ ( autoref_APP @ B @ D @ F @ Y3 ) ) @ Rr ) ) )
% 3.09/1.60 => ( member @ ( product_prod @ C @ D )
% 3.09/1.60 @ ( product_Pair @ C @ D @ ( F2 @ X3 )
% 3.09/1.60 @ ( autoref_APP @ ( B > D ) @ D
% 3.09/1.60 @ ( autoref_APP @ B @ ( ( B > D ) > D )
% 3.09/1.60 @ ( autoref_ANNOT @ ( B > ( B > D ) > D )
% 3.09/1.60 @ ( autoref_OP @ ( B > ( B > D ) > D )
% 3.09/1.60 @ ^ [S: B,F3: B > D] : ( F3 @ S ) )
% 3.09/1.60 @ ( autoref_rel_annot @ ( A > ( A > C ) > C ) @ ( B > ( B > D ) > D ) @ ( relAPP @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ ( set @ ( product_prod @ ( A > ( A > C ) > C ) @ ( B > ( B > D ) > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) ) ) > ( set @ ( product_prod @ ( A > ( A > C ) > C ) @ ( B > ( B > D ) > D ) ) ) ) @ ( fun_rel @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) ) @ Ra ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) ) ) @ ( relAPP @ ( A > C ) @ ( B > D ) @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) ) ) ) @ ( fun_rel @ ( A > C ) @ ( B > D ) @ C @ D ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ Ra ) @ Rr ) ) @ Rr ) ) ) )
% 3.09/1.60 @ X )
% 3.09/1.60 @ F ) )
% 3.09/1.60 @ Rr ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_36_internal__case__prod__conv,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A5: B,B3: C] :
% 3.09/1.60 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A5 @ B3 ) )
% 3.09/1.60 = ( C2 @ A5 @ B3 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_37_old_Oprod_Orec,axiom,
% 3.09/1.60 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A5: A,B3: B] :
% 3.09/1.60 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 3.09/1.60 = ( F1 @ A5 @ B3 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_38_APP__def,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType] :
% 3.09/1.60 ( ( autoref_APP @ B @ A )
% 3.09/1.60 = ( ^ [F3: B > A] : F3 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_39_autoref__id,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > A ) @ ( B > B ) ) @ ( product_Pair @ ( A > A ) @ ( B > B ) @ ( id @ A ) @ ( id @ B ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) ) @ ( fun_rel @ A @ B @ A @ B ) @ R3 ) @ R3 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_40_autoref__REMOVE__INTERNAL__EQ,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,C2: A,A5: B,R3: set @ ( product_prod @ A @ B ),C4: A] :
% 3.09/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ C2 @ A5 ) @ R3 )
% 3.09/1.60 => ( ( autore813025987NAL_EQ @ A @ C2 @ C4 )
% 3.09/1.60 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ C4 @ A5 ) @ R3 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_41_nres__more__simps_I6_J,axiom,
% 3.09/1.60 ! [A: $tType,X: A,Y: A] :
% 3.09/1.60 ( ( ( refine1687780735RETURN @ A @ X )
% 3.09/1.60 = ( refine1687780735RETURN @ A @ Y ) )
% 3.09/1.60 = ( X = Y ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_42_mem__Collect__eq,axiom,
% 3.09/1.60 ! [A: $tType,A5: A,P: A > $o] :
% 3.09/1.60 ( ( member @ A @ A5 @ ( collect @ A @ P ) )
% 3.09/1.60 = ( P @ A5 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_43_Collect__mem__eq,axiom,
% 3.09/1.60 ! [A: $tType,A2: set @ A] :
% 3.09/1.60 ( ( collect @ A
% 3.09/1.60 @ ^ [X6: A] : ( member @ A @ X6 @ A2 ) )
% 3.09/1.60 = A2 ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_44_Collect__cong,axiom,
% 3.09/1.60 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.09/1.60 ( ! [X5: A] :
% 3.09/1.60 ( ( P @ X5 )
% 3.09/1.60 = ( Q @ X5 ) )
% 3.09/1.60 => ( ( collect @ A @ P )
% 3.09/1.60 = ( collect @ A @ Q ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_45_ext,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,F: A > B,G3: A > B] :
% 3.09/1.60 ( ! [X5: A] :
% 3.09/1.60 ( ( F @ X5 )
% 3.09/1.60 = ( G3 @ X5 ) )
% 3.09/1.60 => ( F = G3 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_46_nres__monad1,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,X: B,F: B > ( refine1665802226e_nres @ A )] :
% 3.09/1.60 ( ( refine463715084e_bind @ B @ A @ ( refine1687780735RETURN @ B @ X ) @ F )
% 3.09/1.60 = ( F @ X ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_47_nres__monad2,axiom,
% 3.09/1.60 ! [A: $tType,M3: refine1665802226e_nres @ A] :
% 3.09/1.60 ( ( refine463715084e_bind @ A @ A @ M3 @ ( refine1687780735RETURN @ A ) )
% 3.09/1.60 = M3 ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_48_fun__rel__id__simp,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType] :
% 3.09/1.60 ( ( relAPP @ B @ B @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( fun_rel @ A @ A @ B @ B ) @ ( id2 @ A ) ) @ ( id2 @ B ) )
% 3.09/1.60 = ( id2 @ ( A > B ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_49_relprop__triggers_I3_J,axiom,
% 3.09/1.60 ! [D: $tType,R3: set @ ( product_prod @ D @ D )] :
% 3.09/1.60 ( ( R3
% 3.09/1.60 = ( id2 @ D ) )
% 3.09/1.60 => ( ( id2 @ D )
% 3.09/1.60 = R3 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_50_relprop__triggers_I2_J,axiom,
% 3.09/1.60 ! [C: $tType,R3: set @ ( product_prod @ C @ C )] :
% 3.09/1.60 ( ( R3
% 3.09/1.60 = ( id2 @ C ) )
% 3.09/1.60 => ( R3
% 3.09/1.60 = ( id2 @ C ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_51_autoref__int_I6_J,axiom,
% 3.09/1.60 ( member @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) )
% 3.09/1.60 @ ( product_Pair @ ( int > int > $o ) @ ( int > int > $o )
% 3.09/1.60 @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 3.09/1.60 @ ^ [Y5: int,Z: int] : ( Y5 = Z ) )
% 3.09/1.60 @ ( relAPP @ ( int > $o ) @ ( int > $o ) @ ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) > ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) ) @ ( fun_rel @ int @ int @ ( int > $o ) @ ( int > $o ) ) @ ( id2 @ int ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) ) @ ( fun_rel @ int @ int @ $o @ $o ) @ ( id2 @ int ) ) @ ( id2 @ $o ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_52_autoref__nat_I7_J,axiom,
% 3.09/1.60 ( member @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) )
% 3.09/1.60 @ ( product_Pair @ ( nat > nat > $o ) @ ( nat > nat > $o )
% 3.09/1.60 @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 3.09/1.60 @ ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 3.09/1.60 @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) > ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ ( nat > $o ) @ ( nat > $o ) ) @ ( id2 @ nat ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_53_autoref__bool_I8_J,axiom,
% 3.09/1.60 member @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) @ ( product_Pair @ ( $o > $o > $o ) @ ( $o > $o > $o ) @ (=>) @ (=>) ) @ ( relAPP @ ( $o > $o ) @ ( $o > $o ) @ ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) > ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ ( $o > $o ) @ ( $o > $o ) ) @ ( id2 @ $o ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ $o @ $o ) @ ( id2 @ $o ) ) @ ( id2 @ $o ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_54_autoref__bool_I4_J,axiom,
% 3.09/1.60 member @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) @ ( product_Pair @ ( $o > $o ) @ ( $o > $o ) @ (~) @ (~) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ $o @ $o ) @ ( id2 @ $o ) ) @ ( id2 @ $o ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_55_autoref__bool_I3_J,axiom,
% 3.09/1.60 member @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) @ ( product_Pair @ ( $o > $o > $o ) @ ( $o > $o > $o ) @ (|) @ (|) ) @ ( relAPP @ ( $o > $o ) @ ( $o > $o ) @ ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) > ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ ( $o > $o ) @ ( $o > $o ) ) @ ( id2 @ $o ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ $o @ $o ) @ ( id2 @ $o ) ) @ ( id2 @ $o ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_56_autoref__bool_I2_J,axiom,
% 3.09/1.60 member @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) @ ( product_Pair @ ( $o > $o > $o ) @ ( $o > $o > $o ) @ (&) @ (&) ) @ ( relAPP @ ( $o > $o ) @ ( $o > $o ) @ ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) > ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ ( $o > $o ) @ ( $o > $o ) ) @ ( id2 @ $o ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ $o @ $o ) @ ( id2 @ $o ) ) @ ( id2 @ $o ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_57_autoref__bool_I1_J,axiom,
% 3.09/1.60 ! [X: $o] : ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ X @ X ) @ ( id2 @ $o ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_58_relprop__id__orient,axiom,
% 3.09/1.60 ! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
% 3.09/1.60 ( ( R3
% 3.09/1.60 = ( id2 @ A ) )
% 3.09/1.60 => ( ( id2 @ A )
% 3.09/1.60 = R3 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_59_rel__ANNOT__eq,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,T2: A,R3: set @ ( product_prod @ B @ A )] :
% 3.09/1.60 ( T2
% 3.09/1.60 = ( autoref_ANNOT @ A @ T2 @ ( autoref_rel_annot @ B @ A @ R3 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_60_ANNOT__def,axiom,
% 3.09/1.60 ! [A: $tType] :
% 3.09/1.60 ( ( autoref_ANNOT @ A )
% 3.09/1.60 = ( ^ [X6: A,A7: autoref_annot] : X6 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_61_OP__def,axiom,
% 3.09/1.60 ! [A: $tType] :
% 3.09/1.60 ( ( autoref_OP @ A )
% 3.09/1.60 = ( ^ [X6: A] : X6 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_62_rel__cong,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,F: A > B,G3: A > B,X: A,Y: A] :
% 3.09/1.60 ( ( member @ ( product_prod @ ( A > B ) @ ( A > B ) ) @ ( product_Pair @ ( A > B ) @ ( A > B ) @ F @ G3 ) @ ( id2 @ ( A > B ) ) )
% 3.09/1.60 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id2 @ A ) )
% 3.09/1.60 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( G3 @ Y ) ) @ ( id2 @ B ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_63_rel__fun__cong,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,F: A > B,G3: A > B,X: A] :
% 3.09/1.60 ( ( member @ ( product_prod @ ( A > B ) @ ( A > B ) ) @ ( product_Pair @ ( A > B ) @ ( A > B ) @ F @ G3 ) @ ( id2 @ ( A > B ) ) )
% 3.09/1.60 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( G3 @ X ) ) @ ( id2 @ B ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_64_REMOVE__INTERNAL__EQ__def,axiom,
% 3.09/1.60 ! [A: $tType] :
% 3.09/1.60 ( ( autore813025987NAL_EQ @ A )
% 3.09/1.60 = ( ^ [Y5: A,Z: A] : ( Y5 = Z ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_65_REMOVE__INTERNAL__def,axiom,
% 3.09/1.60 ! [A: $tType] :
% 3.09/1.60 ( ( autore2013370551TERNAL @ A )
% 3.09/1.60 = ( ^ [X6: A] : X6 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_66_REMOVE__INTERNAL__EQI,axiom,
% 3.09/1.60 ! [A: $tType,A5: A] : ( autore813025987NAL_EQ @ A @ A5 @ A5 ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_67_rel__arg__cong,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,X: A,Y: A,F: A > B] :
% 3.09/1.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id2 @ A ) )
% 3.09/1.60 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y ) ) @ ( id2 @ B ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_68_fun__rel__id,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 3.09/1.60 ( ( R1
% 3.09/1.60 = ( id2 @ A ) )
% 3.09/1.60 => ( ( R22
% 3.09/1.60 = ( id2 @ B ) )
% 3.09/1.60 => ( ( relAPP @ B @ B @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( fun_rel @ A @ A @ B @ B ) @ R1 ) @ R22 )
% 3.09/1.60 = ( id2 @ ( A > B ) ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_69_IdI,axiom,
% 3.09/1.60 ! [A: $tType,A5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( id2 @ A ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_70_pair__in__Id__conv,axiom,
% 3.09/1.60 ! [A: $tType,A5: A,B3: A] :
% 3.09/1.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ ( id2 @ A ) )
% 3.09/1.60 = ( A5 = B3 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_71_autoref__bool_I6_J,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) @ ( product_Pair @ ( A > A > $o > A ) @ ( B > B > $o > B ) @ ( product_rec_bool @ A ) @ ( product_rec_bool @ B ) ) @ ( relAPP @ ( A > $o > A ) @ ( B > $o > B ) @ ( set @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) > ( set @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) ) ) @ ( fun_rel @ A @ B @ ( A > $o > A ) @ ( B > $o > B ) ) @ R3 ) @ ( relAPP @ ( $o > A ) @ ( $o > B ) @ ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) > ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) ) @ ( fun_rel @ A @ B @ ( $o > A ) @ ( $o > B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) ) @ ( fun_rel @ $o @ $o @ A @ B ) @ ( id2 @ $o ) ) @ R3 ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_72_autoref__If__cong,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,C4: $o,C2: $o,T3: A,T2: B,R3: set @ ( product_prod @ A @ B ),E3: A,E4: B] :
% 3.09/1.60 ( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ C4 @ C2 ) @ ( id2 @ $o ) )
% 3.09/1.60 => ( ( ( autore2013370551TERNAL @ $o @ C2 )
% 3.09/1.60 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ T3 @ T2 ) @ R3 ) )
% 3.09/1.60 => ( ( ~ ( autore2013370551TERNAL @ $o @ C2 )
% 3.09/1.60 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ E3 @ E4 ) @ R3 ) )
% 3.09/1.60 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( if @ A @ C4 @ T3 @ E3 ) @ ( autoref_APP @ B @ B @ ( autoref_APP @ B @ ( B > B ) @ ( autoref_APP @ $o @ ( B > B > B ) @ ( autoref_ANNOT @ ( $o > B > B > B ) @ ( autoref_OP @ ( $o > B > B > B ) @ ( if @ B ) ) @ ( autoref_rel_annot @ ( $o > A > A > A ) @ ( $o > B > B > B ) @ ( relAPP @ ( A > A > A ) @ ( B > B > B ) @ ( set @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) > ( set @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) ) ) @ ( fun_rel @ $o @ $o @ ( A > A > A ) @ ( B > B > B ) ) @ ( id2 @ $o ) ) @ ( relAPP @ ( A > A ) @ ( B > B ) @ ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) > ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) ) @ ( fun_rel @ A @ B @ ( A > A ) @ ( B > B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) ) @ ( fun_rel @ A @ B @ A @ B ) @ R3 ) @ R3 ) ) ) ) ) @ C2 ) @ T2 ) @ E4 ) ) @ R3 ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_73_param__fun__upd,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,D: $tType,C: $tType,Ra: set @ ( product_prod @ A @ B ),Rb: set @ ( product_prod @ C @ D )] :
% 3.09/1.60 ( ( member @ ( product_prod @ ( A > A > $o ) @ ( B > B > $o ) )
% 3.09/1.60 @ ( product_Pair @ ( A > A > $o ) @ ( B > B > $o )
% 3.09/1.60 @ ^ [Y5: A,Z: A] : ( Y5 = Z )
% 3.09/1.60 @ ^ [Y5: B,Z: B] : ( Y5 = Z ) )
% 3.09/1.60 @ ( relAPP @ ( A > $o ) @ ( B > $o ) @ ( set @ ( product_prod @ ( A > A > $o ) @ ( B > B > $o ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) ) > ( set @ ( product_prod @ ( A > A > $o ) @ ( B > B > $o ) ) ) ) @ ( fun_rel @ A @ B @ ( A > $o ) @ ( B > $o ) ) @ Ra ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) ) ) @ ( fun_rel @ A @ B @ $o @ $o ) @ Ra ) @ ( id2 @ $o ) ) ) )
% 3.09/1.60 => ( member @ ( product_prod @ ( ( A > C ) > A > C > A > C ) @ ( ( B > D ) > B > D > B > D ) ) @ ( product_Pair @ ( ( A > C ) > A > C > A > C ) @ ( ( B > D ) > B > D > B > D ) @ ( fun_upd @ A @ C ) @ ( fun_upd @ B @ D ) ) @ ( relAPP @ ( A > C > A > C ) @ ( B > D > B > D ) @ ( set @ ( product_prod @ ( ( A > C ) > A > C > A > C ) @ ( ( B > D ) > B > D > B > D ) ) ) @ ( relAPP @ ( A > C ) @ ( B > D ) @ ( ( set @ ( product_prod @ ( A > C > A > C ) @ ( B > D > B > D ) ) ) > ( set @ ( product_prod @ ( ( A > C ) > A > C > A > C ) @ ( ( B > D ) > B > D > B > D ) ) ) ) @ ( fun_rel @ ( A > C ) @ ( B > D ) @ ( A > C > A > C ) @ ( B > D > B > D ) ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ Ra ) @ Rb ) ) @ ( relAPP @ ( C > A > C ) @ ( D > B > D ) @ ( set @ ( product_prod @ ( A > C > A > C ) @ ( B > D > B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( C > A > C ) @ ( D > B > D ) ) ) > ( set @ ( product_prod @ ( A > C > A > C ) @ ( B > D > B > D ) ) ) ) @ ( fun_rel @ A @ B @ ( C > A > C ) @ ( D > B > D ) ) @ Ra ) @ ( relAPP @ ( A > C ) @ ( B > D ) @ ( set @ ( product_prod @ ( C > A > C ) @ ( D > B > D ) ) ) @ ( relAPP @ C @ D @ ( ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) > ( set @ ( product_prod @ ( C > A > C ) @ ( D > B > D ) ) ) ) @ ( fun_rel @ C @ D @ ( A > C ) @ ( B > D ) ) @ Rb ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ Ra ) @ Rb ) ) ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_74_id__apply,axiom,
% 3.09/1.60 ! [A: $tType] :
% 3.09/1.60 ( ( id @ A )
% 3.09/1.60 = ( ^ [X6: A] : X6 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_75_prod__refine_I3_J,axiom,
% 3.09/1.60 ! [B: $tType,D: $tType,C: $tType,A: $tType,F4: $tType,E: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D ),Rr: set @ ( product_prod @ E @ F4 )] : ( member @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) @ ( product_Pair @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) @ ( product_rec_prod @ A @ B @ E ) @ ( product_rec_prod @ C @ D @ F4 ) ) @ ( relAPP @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) @ ( set @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) ) @ ( relAPP @ ( A > B > E ) @ ( C > D > F4 ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) > ( set @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) ) ) @ ( fun_rel @ ( A > B > E ) @ ( C > D > F4 ) @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) @ ( relAPP @ ( B > E ) @ ( D > F4 ) @ ( set @ ( product_prod @ ( A > B > E ) @ ( C > D > F4 ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) > ( set @ ( product_prod @ ( A > B > E ) @ ( C > D > F4 ) ) ) ) @ ( fun_rel @ A @ C @ ( B > E ) @ ( D > F4 ) ) @ Ra ) @ ( relAPP @ E @ F4 @ ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) @ ( relAPP @ B @ D @ ( ( set @ ( product_prod @ E @ F4 ) ) > ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) ) @ ( fun_rel @ B @ D @ E @ F4 ) @ Rb ) @ Rr ) ) ) @ ( relAPP @ E @ F4 @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ E @ F4 ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ E @ F4 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) @ Rr ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_76_autoref__bool_I5_J,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) @ ( product_Pair @ ( A > A > $o > A ) @ ( B > B > $o > B ) @ ( product_case_bool @ A ) @ ( product_case_bool @ B ) ) @ ( relAPP @ ( A > $o > A ) @ ( B > $o > B ) @ ( set @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) > ( set @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) ) ) @ ( fun_rel @ A @ B @ ( A > $o > A ) @ ( B > $o > B ) ) @ R3 ) @ ( relAPP @ ( $o > A ) @ ( $o > B ) @ ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) > ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) ) @ ( fun_rel @ A @ B @ ( $o > A ) @ ( $o > B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) ) @ ( fun_rel @ $o @ $o @ A @ B ) @ ( id2 @ $o ) ) @ R3 ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_77_fun__upd__upd,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z2: B] :
% 3.09/1.60 ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z2 )
% 3.09/1.60 = ( fun_upd @ A @ B @ F @ X @ Z2 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_78_fun__upd__triv,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 3.09/1.60 ( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
% 3.09/1.60 = F ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_79_fun__upd__apply,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType] :
% 3.09/1.60 ( ( fun_upd @ B @ A )
% 3.09/1.60 = ( ^ [F3: B > A,X6: B,Y6: A,Z3: B] : ( if @ A @ ( Z3 = X6 ) @ Y6 @ ( F3 @ Z3 ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_80_old_Obool_Osimps_I6_J,axiom,
% 3.09/1.60 ! [T: $tType,F1: T,F22: T] :
% 3.09/1.60 ( ( product_rec_bool @ T @ F1 @ F22 @ $false )
% 3.09/1.60 = F22 ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_81_old_Obool_Osimps_I5_J,axiom,
% 3.09/1.60 ! [T: $tType,F1: T,F22: T] :
% 3.09/1.60 ( ( product_rec_bool @ T @ F1 @ F22 @ $true )
% 3.09/1.60 = F1 ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_82_prod__rel__simp,axiom,
% 3.09/1.60 ! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A,B3: B,A6: C,B4: D,R1: set @ ( product_prod @ A @ C ),R22: set @ ( product_prod @ B @ D )] :
% 3.09/1.60 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ C @ D @ A6 @ B4 ) ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ R1 ) @ R22 ) )
% 3.09/1.60 = ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A5 @ A6 ) @ R1 )
% 3.09/1.60 & ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ B3 @ B4 ) @ R22 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_83_prod__rel__id__simp,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType] :
% 3.09/1.60 ( ( relAPP @ B @ B @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) @ ( prod_rel @ A @ A @ B @ B ) @ ( id2 @ A ) ) @ ( id2 @ B ) )
% 3.09/1.60 = ( id2 @ ( product_prod @ A @ B ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_84_fun__upd__def,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType] :
% 3.09/1.60 ( ( fun_upd @ A @ B )
% 3.09/1.60 = ( ^ [F3: A > B,A7: A,B6: B,X6: A] : ( if @ B @ ( X6 = A7 ) @ B6 @ ( F3 @ X6 ) ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_85_fun__upd__eqD,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G3: A > B,Z2: B] :
% 3.09/1.60 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 3.09/1.60 = ( fun_upd @ A @ B @ G3 @ X @ Z2 ) )
% 3.09/1.60 => ( Y = Z2 ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_86_fun__upd__idem,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
% 3.09/1.60 ( ( ( F @ X )
% 3.09/1.60 = Y )
% 3.09/1.60 => ( ( fun_upd @ B @ A @ F @ X @ Y )
% 3.09/1.60 = F ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_87_fun__upd__same,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
% 3.09/1.60 ( ( fun_upd @ B @ A @ F @ X @ Y @ X )
% 3.09/1.60 = Y ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_88_fun__upd__other,axiom,
% 3.09/1.60 ! [B: $tType,A: $tType,Z2: A,X: A,F: A > B,Y: B] :
% 3.09/1.60 ( ( Z2 != X )
% 3.09/1.60 => ( ( fun_upd @ A @ B @ F @ X @ Y @ Z2 )
% 3.09/1.60 = ( F @ Z2 ) ) ) ).
% 3.09/1.60
% 3.09/1.60 thf(fact_89_fun__upd__twist,axiom,
% 3.09/1.60 ! [A: $tType,B: $tType,A5: A,C2: A,M: A > B,B3: B,D3: B] :
% 3.09/1.60 ( ( A5 != C2 )
% 3.09/1.60 => ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A5 @ B3 ) @ C2 @ D3 )
% 3.09/1.60 = ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C2 @ D3 ) @ A5 @ B3 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_90_fun__upd__idem__iff,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
% 3.24/1.60 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 3.24/1.60 = F )
% 3.24/1.60 = ( ( F @ X )
% 3.24/1.60 = Y ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_91_rec__bool__is__case,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( product_rec_bool @ A )
% 3.24/1.60 = ( product_case_bool @ A ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_92_bool_Osplit__sel__asm,axiom,
% 3.24/1.60 ! [A: $tType,P: A > $o,F1: A,F22: A,Bool: $o] :
% 3.24/1.60 ( ( P @ ( product_case_bool @ A @ F1 @ F22 @ Bool ) )
% 3.24/1.60 = ( ~ ( ( Bool
% 3.24/1.60 & ~ ( P @ F1 ) )
% 3.24/1.60 | ( ~ Bool
% 3.24/1.60 & ~ ( P @ F22 ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_93_bool_Ocase__distrib,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,H: A > B,F1: A,F22: A,Bool: $o] :
% 3.24/1.60 ( ( H @ ( product_case_bool @ A @ F1 @ F22 @ Bool ) )
% 3.24/1.60 = ( product_case_bool @ B @ ( H @ F1 ) @ ( H @ F22 ) @ Bool ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_94_bool_Ocase__eq__if,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( product_case_bool @ A )
% 3.24/1.60 = ( ^ [F12: A,F23: A,Bool2: $o] : ( if @ A @ Bool2 @ F12 @ F23 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_95_bool_Osplit__sel,axiom,
% 3.24/1.60 ! [A: $tType,P: A > $o,F1: A,F22: A,Bool: $o] :
% 3.24/1.60 ( ( P @ ( product_case_bool @ A @ F1 @ F22 @ Bool ) )
% 3.24/1.60 = ( ( Bool
% 3.24/1.60 => ( P @ F1 ) )
% 3.24/1.60 & ( ~ Bool
% 3.24/1.60 => ( P @ F22 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_96_old_Obool_Osimps_I3_J,axiom,
% 3.24/1.60 ! [A: $tType,F1: A,F22: A] :
% 3.24/1.60 ( ( product_case_bool @ A @ F1 @ F22 @ $true )
% 3.24/1.60 = F1 ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_97_old_Obool_Osimps_I4_J,axiom,
% 3.24/1.60 ! [A: $tType,F1: A,F22: A] :
% 3.24/1.60 ( ( product_case_bool @ A @ F1 @ F22 @ $false )
% 3.24/1.60 = F22 ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_98_prod__relI,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,D: $tType,C: $tType,A5: A,A6: B,R1: set @ ( product_prod @ A @ B ),B3: C,B4: D,R22: set @ ( product_prod @ C @ D )] :
% 3.24/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ A6 ) @ R1 )
% 3.24/1.60 => ( ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ B3 @ B4 ) @ R22 )
% 3.24/1.60 => ( member @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_Pair @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_Pair @ A @ C @ A5 @ B3 ) @ ( product_Pair @ B @ D @ A6 @ B4 ) ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) @ ( prod_rel @ A @ B @ C @ D ) @ R1 ) @ R22 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_99_prod__relE,axiom,
% 3.24/1.60 ! [A: $tType,C: $tType,D: $tType,B: $tType,P2: product_prod @ A @ B,P3: product_prod @ C @ D,R1: set @ ( product_prod @ A @ C ),R22: set @ ( product_prod @ B @ D )] :
% 3.24/1.60 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ P2 @ P3 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ R1 ) @ R22 ) )
% 3.24/1.60 => ~ ! [A3: A,B5: B] :
% 3.24/1.60 ( ( P2
% 3.24/1.60 = ( product_Pair @ A @ B @ A3 @ B5 ) )
% 3.24/1.60 => ! [A4: C,B7: D] :
% 3.24/1.60 ( ( P3
% 3.24/1.60 = ( product_Pair @ C @ D @ A4 @ B7 ) )
% 3.24/1.60 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ A4 ) @ R1 )
% 3.24/1.60 => ~ ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ B5 @ B7 ) @ R22 ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_100_prod__refine_I1_J,axiom,
% 3.24/1.60 ! [A: $tType,C: $tType,D: $tType,B: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( A > B > ( product_prod @ A @ B ) ) @ ( C > D > ( product_prod @ C @ D ) ) ) @ ( product_Pair @ ( A > B > ( product_prod @ A @ B ) ) @ ( C > D > ( product_prod @ C @ D ) ) @ ( product_Pair @ A @ B ) @ ( product_Pair @ C @ D ) ) @ ( relAPP @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) @ ( set @ ( product_prod @ ( A > B > ( product_prod @ A @ B ) ) @ ( C > D > ( product_prod @ C @ D ) ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) ) ) > ( set @ ( product_prod @ ( A > B > ( product_prod @ A @ B ) ) @ ( C > D > ( product_prod @ C @ D ) ) ) ) ) @ ( fun_rel @ A @ C @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) ) @ Ra ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) ) ) @ ( relAPP @ B @ D @ ( ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) > ( set @ ( product_prod @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) ) ) ) @ ( fun_rel @ B @ D @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) @ Rb ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_101_prod__rel__id,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 3.24/1.60 ( ( R1
% 3.24/1.60 = ( id2 @ A ) )
% 3.24/1.60 => ( ( R22
% 3.24/1.60 = ( id2 @ B ) )
% 3.24/1.60 => ( ( relAPP @ B @ B @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) @ ( prod_rel @ A @ A @ B @ B ) @ R1 ) @ R22 )
% 3.24/1.60 = ( id2 @ ( product_prod @ A @ B ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_102_id__def,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( id @ A )
% 3.24/1.60 = ( ^ [X6: A] : X6 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_103_eq__id__iff,axiom,
% 3.24/1.60 ! [A: $tType,F: A > A] :
% 3.24/1.60 ( ( ! [X6: A] :
% 3.24/1.60 ( ( F @ X6 )
% 3.24/1.60 = X6 ) )
% 3.24/1.60 = ( F
% 3.24/1.60 = ( id @ A ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_104_param__and__cong2,axiom,
% 3.24/1.60 ! [A5: $o,A6: $o,B3: $o,B4: $o] :
% 3.24/1.60 ( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ A5 @ A6 ) @ ( id2 @ $o ) )
% 3.24/1.60 => ( ( A5
% 3.24/1.60 => ( A6
% 3.24/1.60 => ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ B3 @ B4 ) @ ( id2 @ $o ) ) ) )
% 3.24/1.60 => ( member @ ( product_prod @ $o @ $o )
% 3.24/1.60 @ ( product_Pair @ $o @ $o
% 3.24/1.60 @ ( B3
% 3.24/1.60 & A5 )
% 3.24/1.60 @ ( B4
% 3.24/1.60 & A6 ) )
% 3.24/1.60 @ ( id2 @ $o ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_105_param__and__cong1,axiom,
% 3.24/1.60 ! [A5: $o,A6: $o,B3: $o,B4: $o] :
% 3.24/1.60 ( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ A5 @ A6 ) @ ( id2 @ $o ) )
% 3.24/1.60 => ( ( A5
% 3.24/1.60 => ( A6
% 3.24/1.60 => ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ B3 @ B4 ) @ ( id2 @ $o ) ) ) )
% 3.24/1.60 => ( member @ ( product_prod @ $o @ $o )
% 3.24/1.60 @ ( product_Pair @ $o @ $o
% 3.24/1.60 @ ( A5
% 3.24/1.60 & B3 )
% 3.24/1.60 @ ( A6
% 3.24/1.60 & B4 ) )
% 3.24/1.60 @ ( id2 @ $o ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_106_param__bool_I1_J,axiom,
% 3.24/1.60 member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ $true @ $true ) @ ( id2 @ $o ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_107_param__bool_I2_J,axiom,
% 3.24/1.60 member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ $false @ $false ) @ ( id2 @ $o ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_108_IdE,axiom,
% 3.24/1.60 ! [A: $tType,P2: product_prod @ A @ A] :
% 3.24/1.60 ( ( member @ ( product_prod @ A @ A ) @ P2 @ ( id2 @ A ) )
% 3.24/1.60 => ~ ! [X5: A] :
% 3.24/1.60 ( P2
% 3.24/1.60 != ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_109_param__if,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,C2: $o,C4: $o,T2: A,T3: B,R3: set @ ( product_prod @ A @ B ),E4: A,E3: B] :
% 3.24/1.60 ( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ C2 @ C4 ) @ ( id2 @ $o ) )
% 3.24/1.60 => ( ( C2
% 3.24/1.60 => ( C4
% 3.24/1.60 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ T2 @ T3 ) @ R3 ) ) )
% 3.24/1.60 => ( ( ~ C2
% 3.24/1.60 => ( ~ C4
% 3.24/1.60 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ E4 @ E3 ) @ R3 ) ) )
% 3.24/1.60 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( if @ A @ C2 @ T2 @ E4 ) @ ( if @ B @ C4 @ T3 @ E3 ) ) @ R3 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_110_param__uncurry,axiom,
% 3.24/1.60 ! [B: $tType,E: $tType,D: $tType,A: $tType,F4: $tType,C: $tType,A2: set @ ( product_prod @ A @ D ),B2: set @ ( product_prod @ B @ E ),C5: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) @ ( ( D > E > F4 ) > ( product_prod @ D @ E ) > F4 ) ) @ ( product_Pair @ ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) @ ( ( D > E > F4 ) > ( product_prod @ D @ E ) > F4 ) @ ( uncurry @ A @ B @ C ) @ ( uncurry @ D @ E @ F4 ) ) @ ( relAPP @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) @ ( set @ ( product_prod @ ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) @ ( ( D > E > F4 ) > ( product_prod @ D @ E ) > F4 ) ) ) @ ( relAPP @ ( A > B > C ) @ ( D > E > F4 ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) > ( set @ ( product_prod @ ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) @ ( ( D > E > F4 ) > ( product_prod @ D @ E ) > F4 ) ) ) ) @ ( fun_rel @ ( A > B > C ) @ ( D > E > F4 ) @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) @ ( relAPP @ ( B > C ) @ ( E > F4 ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) > ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) ) @ ( fun_rel @ A @ D @ ( B > C ) @ ( E > F4 ) ) @ A2 ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) @ ( relAPP @ B @ E @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) ) @ ( fun_rel @ B @ E @ C @ F4 ) @ B2 ) @ C5 ) ) ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ C @ F4 ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) ) ) ) @ ( prod_rel @ A @ D @ B @ E ) @ A2 ) @ B2 ) ) @ C5 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_111_param__all__interval__nat,axiom,
% 3.24/1.60 member @ ( product_prod @ ( ( nat > $o ) > nat > nat > $o ) @ ( ( nat > $o ) > nat > nat > $o ) ) @ ( product_Pair @ ( ( nat > $o ) > nat > nat > $o ) @ ( ( nat > $o ) > nat > nat > $o ) @ all_interval_nat @ all_interval_nat ) @ ( relAPP @ ( nat > nat > $o ) @ ( nat > nat > $o ) @ ( set @ ( product_prod @ ( ( nat > $o ) > nat > nat > $o ) @ ( ( nat > $o ) > nat > nat > $o ) ) ) @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) > ( set @ ( product_prod @ ( ( nat > $o ) > nat > nat > $o ) @ ( ( nat > $o ) > nat > nat > $o ) ) ) ) @ ( fun_rel @ ( nat > $o ) @ ( nat > $o ) @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) > ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ ( nat > $o ) @ ( nat > $o ) ) @ ( id2 @ nat ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_112_param__curry,axiom,
% 3.24/1.60 ! [A: $tType,D: $tType,B: $tType,E: $tType,F4: $tType,C: $tType,Ra: set @ ( product_prod @ A @ D ),Rb: set @ ( product_prod @ B @ E ),Rc: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) @ ( ( ( product_prod @ D @ E ) > F4 ) > D > E > F4 ) ) @ ( product_Pair @ ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) @ ( ( ( product_prod @ D @ E ) > F4 ) > D > E > F4 ) @ ( product_curry @ A @ B @ C ) @ ( product_curry @ D @ E @ F4 ) ) @ ( relAPP @ ( A > B > C ) @ ( D > E > F4 ) @ ( set @ ( product_prod @ ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) @ ( ( ( product_prod @ D @ E ) > F4 ) > D > E > F4 ) ) ) @ ( relAPP @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) @ ( ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) > ( set @ ( product_prod @ ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) @ ( ( ( product_prod @ D @ E ) > F4 ) > D > E > F4 ) ) ) ) @ ( fun_rel @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) @ ( A > B > C ) @ ( D > E > F4 ) ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ C @ F4 ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) ) ) ) @ ( prod_rel @ A @ D @ B @ E ) @ Ra ) @ Rb ) ) @ Rc ) ) @ ( relAPP @ ( B > C ) @ ( E > F4 ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) > ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) ) @ ( fun_rel @ A @ D @ ( B > C ) @ ( E > F4 ) ) @ Ra ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) @ ( relAPP @ B @ E @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) ) @ ( fun_rel @ B @ E @ C @ F4 ) @ Rb ) @ Rc ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_113_prod__refine_I2_J,axiom,
% 3.24/1.60 ! [B: $tType,D: $tType,C: $tType,A: $tType,F4: $tType,E: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D ),Rr: set @ ( product_prod @ E @ F4 )] : ( member @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) @ ( product_Pair @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) @ ( product_case_prod @ A @ B @ E ) @ ( product_case_prod @ C @ D @ F4 ) ) @ ( relAPP @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) @ ( set @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) ) @ ( relAPP @ ( A > B > E ) @ ( C > D > F4 ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) > ( set @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) ) ) @ ( fun_rel @ ( A > B > E ) @ ( C > D > F4 ) @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) @ ( relAPP @ ( B > E ) @ ( D > F4 ) @ ( set @ ( product_prod @ ( A > B > E ) @ ( C > D > F4 ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) > ( set @ ( product_prod @ ( A > B > E ) @ ( C > D > F4 ) ) ) ) @ ( fun_rel @ A @ C @ ( B > E ) @ ( D > F4 ) ) @ Ra ) @ ( relAPP @ E @ F4 @ ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) @ ( relAPP @ B @ D @ ( ( set @ ( product_prod @ E @ F4 ) ) > ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) ) @ ( fun_rel @ B @ D @ E @ F4 ) @ Rb ) @ Rr ) ) ) @ ( relAPP @ E @ F4 @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ E @ F4 ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ E @ F4 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) @ Rr ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_114_autoref__rec__nat,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > ( nat > A > A ) > nat > A ) @ ( B > ( nat > B > B ) > nat > B ) ) @ ( product_Pair @ ( A > ( nat > A > A ) > nat > A ) @ ( B > ( nat > B > B ) > nat > B ) @ ( rec_nat @ A ) @ ( rec_nat @ B ) ) @ ( relAPP @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) @ ( set @ ( product_prod @ ( A > ( nat > A > A ) > nat > A ) @ ( B > ( nat > B > B ) > nat > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) ) ) > ( set @ ( product_prod @ ( A > ( nat > A > A ) > nat > A ) @ ( B > ( nat > B > B ) > nat > B ) ) ) ) @ ( fun_rel @ A @ B @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) ) @ R3 ) @ ( relAPP @ ( nat > A ) @ ( nat > B ) @ ( set @ ( product_prod @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) ) ) @ ( relAPP @ ( nat > A > A ) @ ( nat > B > B ) @ ( ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) > ( set @ ( product_prod @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) ) ) ) @ ( fun_rel @ ( nat > A > A ) @ ( nat > B > B ) @ ( nat > A ) @ ( nat > B ) ) @ ( relAPP @ ( A > A ) @ ( B > B ) @ ( set @ ( product_prod @ ( nat > A > A ) @ ( nat > B > B ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) > ( set @ ( product_prod @ ( nat > A > A ) @ ( nat > B > B ) ) ) ) @ ( fun_rel @ nat @ nat @ ( A > A ) @ ( B > B ) ) @ ( id2 @ nat ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) ) @ ( fun_rel @ A @ B @ A @ B ) @ R3 ) @ R3 ) ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) ) @ ( fun_rel @ nat @ nat @ A @ B ) @ ( id2 @ nat ) ) @ R3 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_115_autoref__int_I5_J,axiom,
% 3.24/1.60 member @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) @ ( product_Pair @ ( int > int > $o ) @ ( int > int > $o ) @ ( ord_less_eq @ int ) @ ( ord_less_eq @ int ) ) @ ( relAPP @ ( int > $o ) @ ( int > $o ) @ ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) > ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) ) @ ( fun_rel @ int @ int @ ( int > $o ) @ ( int > $o ) ) @ ( id2 @ int ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) ) @ ( fun_rel @ int @ int @ $o @ $o ) @ ( id2 @ int ) ) @ ( id2 @ $o ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_116_case__prod__Pair,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType] :
% 3.24/1.60 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
% 3.24/1.60 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_117_curryI,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A5: A,B3: B] :
% 3.24/1.60 ( ( F @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 3.24/1.60 => ( product_curry @ A @ B @ $o @ F @ A5 @ B3 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_118_curry__conv,axiom,
% 3.24/1.60 ! [A: $tType,C: $tType,B: $tType] :
% 3.24/1.60 ( ( product_curry @ B @ C @ A )
% 3.24/1.60 = ( ^ [F3: ( product_prod @ B @ C ) > A,A7: B,B6: C] : ( F3 @ ( product_Pair @ B @ C @ A7 @ B6 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_119_case__prod__curry,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 3.24/1.60 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 3.24/1.60 = F ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_120_curry__case__prod,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 3.24/1.60 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
% 3.24/1.60 = F ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_121_lhs__step__If,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( ord @ A )
% 3.24/1.60 => ! [B3: $o,T2: A,M: A,E4: A] :
% 3.24/1.60 ( ( B3
% 3.24/1.60 => ( ord_less_eq @ A @ T2 @ M ) )
% 3.24/1.60 => ( ( ~ B3
% 3.24/1.60 => ( ord_less_eq @ A @ E4 @ M ) )
% 3.24/1.60 => ( ord_less_eq @ A @ ( if @ A @ B3 @ T2 @ E4 ) @ M ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_122_mem__case__prodE,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 3.24/1.60 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P2 ) )
% 3.24/1.60 => ~ ! [X5: B,Y3: C] :
% 3.24/1.60 ( ( P2
% 3.24/1.60 = ( product_Pair @ B @ C @ X5 @ Y3 ) )
% 3.24/1.60 => ~ ( member @ A @ Z2 @ ( C2 @ X5 @ Y3 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_123_old_Oprod_Ocase,axiom,
% 3.24/1.60 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 3.24/1.60 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 3.24/1.60 = ( F @ X1 @ X22 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_124_param__case__prod_H_H,axiom,
% 3.24/1.60 ! [C: $tType,D: $tType,B: $tType,A: $tType,F4: $tType,E: $tType,P2: product_prod @ A @ B,P3: product_prod @ C @ D,F: A > B > E,F2: C > D > F4,R3: set @ ( product_prod @ E @ F4 )] :
% 3.24/1.60 ( ! [A3: A,B5: B,A4: C,B7: D] :
% 3.24/1.60 ( ( P2
% 3.24/1.60 = ( product_Pair @ A @ B @ A3 @ B5 ) )
% 3.24/1.60 => ( ( P3
% 3.24/1.60 = ( product_Pair @ C @ D @ A4 @ B7 ) )
% 3.24/1.60 => ( member @ ( product_prod @ E @ F4 ) @ ( product_Pair @ E @ F4 @ ( F @ A3 @ B5 ) @ ( F2 @ A4 @ B7 ) ) @ R3 ) ) )
% 3.24/1.60 => ( member @ ( product_prod @ E @ F4 ) @ ( product_Pair @ E @ F4 @ ( product_case_prod @ A @ B @ E @ F @ P2 ) @ ( product_case_prod @ C @ D @ F4 @ F2 @ P3 ) ) @ R3 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_125_rec__prod__is__case,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType] :
% 3.24/1.60 ( ( product_rec_prod @ A @ B @ C )
% 3.24/1.60 = ( product_case_prod @ A @ B @ C ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_126_curryD,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A5: A,B3: B] :
% 3.24/1.60 ( ( product_curry @ A @ B @ $o @ F @ A5 @ B3 )
% 3.24/1.60 => ( F @ ( product_Pair @ A @ B @ A5 @ B3 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_127_curryE,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A5: A,B3: B] :
% 3.24/1.60 ( ( product_curry @ A @ B @ $o @ F @ A5 @ B3 )
% 3.24/1.60 => ( F @ ( product_Pair @ A @ B @ A5 @ B3 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_128_internal__case__prod__def,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType] :
% 3.24/1.60 ( ( produc2004651681e_prod @ A @ B @ C )
% 3.24/1.60 = ( product_case_prod @ A @ B @ C ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_129_param__case__prod_H,axiom,
% 3.24/1.60 ! [C: $tType,D: $tType,B: $tType,A: $tType,F4: $tType,E: $tType,P2: product_prod @ A @ B,P3: product_prod @ C @ D,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D ),F: A > B > E,F2: C > D > F4,R3: set @ ( product_prod @ E @ F4 )] :
% 3.24/1.60 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ P2 @ P3 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) )
% 3.24/1.60 => ( ! [A3: A,B5: B,A4: C,B7: D] :
% 3.24/1.60 ( ( P2
% 3.24/1.60 = ( product_Pair @ A @ B @ A3 @ B5 ) )
% 3.24/1.60 => ( ( P3
% 3.24/1.60 = ( product_Pair @ C @ D @ A4 @ B7 ) )
% 3.24/1.60 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ A4 ) @ Ra )
% 3.24/1.60 => ( ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ B5 @ B7 ) @ Rb )
% 3.24/1.60 => ( member @ ( product_prod @ E @ F4 ) @ ( product_Pair @ E @ F4 @ ( F @ A3 @ B5 ) @ ( F2 @ A4 @ B7 ) ) @ R3 ) ) ) ) )
% 3.24/1.60 => ( member @ ( product_prod @ E @ F4 ) @ ( product_Pair @ E @ F4 @ ( product_case_prod @ A @ B @ E @ F @ P2 ) @ ( product_case_prod @ C @ D @ F4 @ F2 @ P3 ) ) @ R3 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_130_autoref__case__nat,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,Ra: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > ( nat > A ) > nat > A ) @ ( B > ( nat > B ) > nat > B ) ) @ ( product_Pair @ ( A > ( nat > A ) > nat > A ) @ ( B > ( nat > B ) > nat > B ) @ ( case_nat @ A ) @ ( case_nat @ B ) ) @ ( relAPP @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) @ ( set @ ( product_prod @ ( A > ( nat > A ) > nat > A ) @ ( B > ( nat > B ) > nat > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) ) ) > ( set @ ( product_prod @ ( A > ( nat > A ) > nat > A ) @ ( B > ( nat > B ) > nat > B ) ) ) ) @ ( fun_rel @ A @ B @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) ) @ Ra ) @ ( relAPP @ ( nat > A ) @ ( nat > B ) @ ( set @ ( product_prod @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) ) ) @ ( relAPP @ ( nat > A ) @ ( nat > B ) @ ( ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) > ( set @ ( product_prod @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) ) ) ) @ ( fun_rel @ ( nat > A ) @ ( nat > B ) @ ( nat > A ) @ ( nat > B ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) ) @ ( fun_rel @ nat @ nat @ A @ B ) @ ( id2 @ nat ) ) @ Ra ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) ) @ ( fun_rel @ nat @ nat @ A @ B ) @ ( id2 @ nat ) ) @ Ra ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_131_uncurry__curry__id,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 3.24/1.60 ( ( uncurry @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 3.24/1.60 = F ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_132_curry__uncurry__id,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 3.24/1.60 ( ( product_curry @ A @ B @ C @ ( uncurry @ A @ B @ C @ F ) )
% 3.24/1.60 = F ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_133_uncurry__apply,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A5: B,B3: C] :
% 3.24/1.60 ( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A5 @ B3 ) )
% 3.24/1.60 = ( F @ A5 @ B3 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_134_nres__order__simps_I20_J,axiom,
% 3.24/1.60 ! [W: $tType,X: W,Y: W] :
% 3.24/1.60 ( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X ) @ ( refine1687780735RETURN @ W @ Y ) )
% 3.24/1.60 = ( X = Y ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_135_subrelI,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 3.24/1.60 ( ! [X5: A,Y3: B] :
% 3.24/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ R )
% 3.24/1.60 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ S2 ) )
% 3.24/1.60 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_136_fun__rel__mono,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,D: $tType,C: $tType,R1: set @ ( product_prod @ A @ B ),R22: set @ ( product_prod @ A @ B ),R32: set @ ( product_prod @ C @ D ),R42: set @ ( product_prod @ C @ D )] :
% 3.24/1.60 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R1 @ R22 )
% 3.24/1.60 => ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ R32 @ R42 )
% 3.24/1.60 => ( ord_less_eq @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ R22 ) @ R32 ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ R1 ) @ R42 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_137_prod__rel__mono,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,D: $tType,C: $tType,R22: set @ ( product_prod @ A @ B ),R1: set @ ( product_prod @ A @ B ),R32: set @ ( product_prod @ C @ D ),R42: set @ ( product_prod @ C @ D )] :
% 3.24/1.60 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R22 @ R1 )
% 3.24/1.60 => ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ R32 @ R42 )
% 3.24/1.60 => ( ord_less_eq @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) @ ( prod_rel @ A @ B @ C @ D ) @ R22 ) @ R32 ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) @ ( prod_rel @ A @ B @ C @ D ) @ R1 ) @ R42 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_138_bind__cong,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M2: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F2: A > ( refine1665802226e_nres @ B )] :
% 3.24/1.60 ( ( M = M2 )
% 3.24/1.60 => ( ! [X5: A] :
% 3.24/1.60 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X5 ) @ M2 )
% 3.24/1.60 => ( ( F @ X5 )
% 3.24/1.60 = ( F2 @ X5 ) ) )
% 3.24/1.60 => ( ( refine463715084e_bind @ A @ B @ M @ F )
% 3.24/1.60 = ( refine463715084e_bind @ A @ B @ M2 @ F2 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_139_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,M3: refine1665802226e_nres @ A,M4: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F2: A > ( refine1665802226e_nres @ B )] :
% 3.24/1.60 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ M4 )
% 3.24/1.60 => ( ! [X5: A] :
% 3.24/1.60 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X5 ) @ M3 )
% 3.24/1.60 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X5 ) @ ( F2 @ X5 ) ) )
% 3.24/1.60 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M3 @ F ) @ ( refine463715084e_bind @ A @ B @ M4 @ F2 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_140_ord__eq__le__eq__trans,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( ord @ A )
% 3.24/1.60 => ! [A5: A,B3: A,C2: A,D3: A] :
% 3.24/1.60 ( ( A5 = B3 )
% 3.24/1.60 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 3.24/1.60 => ( ( C2 = D3 )
% 3.24/1.60 => ( ord_less_eq @ A @ A5 @ D3 ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_141_pairself_Oinduct,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,P: ( A > B ) > ( product_prod @ A @ A ) > $o,A0: A > B,A12: product_prod @ A @ A] :
% 3.24/1.60 ( ! [F5: A > B,A3: A,B5: A] : ( P @ F5 @ ( product_Pair @ A @ A @ A3 @ B5 ) )
% 3.24/1.60 => ( P @ A0 @ A12 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_142_pairself_Ocases,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
% 3.24/1.60 ~ ! [F5: A > B,A3: A,B5: A] :
% 3.24/1.60 ( X
% 3.24/1.60 != ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F5 @ ( product_Pair @ A @ A @ A3 @ B5 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_143_bex2I,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,A5: A,B3: B,S5: set @ ( product_prod @ A @ B ),P: A > B > $o] :
% 3.24/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ S5 )
% 3.24/1.60 => ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ S5 )
% 3.24/1.60 => ( P @ A5 @ B3 ) )
% 3.24/1.60 => ? [A3: A,B5: B] :
% 3.24/1.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ S5 )
% 3.24/1.60 & ( P @ A3 @ B5 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_144_autoref__nat_I6_J,axiom,
% 3.24/1.60 member @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) @ ( product_Pair @ ( nat > nat > $o ) @ ( nat > nat > $o ) @ ( ord_less_eq @ nat ) @ ( ord_less_eq @ nat ) ) @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) > ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ ( nat > $o ) @ ( nat > $o ) ) @ ( id2 @ nat ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_145_autoref__int_I4_J,axiom,
% 3.24/1.60 member @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) @ ( product_Pair @ ( int > int > $o ) @ ( int > int > $o ) @ ( ord_less @ int ) @ ( ord_less @ int ) ) @ ( relAPP @ ( int > $o ) @ ( int > $o ) @ ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) > ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) ) @ ( fun_rel @ int @ int @ ( int > $o ) @ ( int > $o ) ) @ ( id2 @ int ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) ) @ ( fun_rel @ int @ int @ $o @ $o ) @ ( id2 @ int ) ) @ ( id2 @ $o ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_146_autoref__nat_I5_J,axiom,
% 3.24/1.60 member @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) @ ( product_Pair @ ( nat > nat > $o ) @ ( nat > nat > $o ) @ ( ord_less @ nat ) @ ( ord_less @ nat ) ) @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) > ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ ( nat > $o ) @ ( nat > $o ) ) @ ( id2 @ nat ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_147_case__prod__mono,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType] :
% 3.24/1.60 ( ( ord @ C )
% 3.24/1.60 => ! [P2: product_prod @ A @ B,F: A > B > C,F2: A > B > C] :
% 3.24/1.60 ( ! [A3: A,B5: B] :
% 3.24/1.60 ( ( P2
% 3.24/1.60 = ( product_Pair @ A @ B @ A3 @ B5 ) )
% 3.24/1.60 => ( ord_less_eq @ C @ ( F @ A3 @ B5 ) @ ( F2 @ A3 @ B5 ) ) )
% 3.24/1.60 => ( ord_less_eq @ C @ ( product_case_prod @ A @ B @ C @ F @ P2 ) @ ( product_case_prod @ A @ B @ C @ F2 @ P2 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_148_param__apsnd,axiom,
% 3.24/1.60 ! [A: $tType,D: $tType,C: $tType,F4: $tType,E: $tType,B: $tType,Rb: set @ ( product_prod @ A @ D ),Rc: set @ ( product_prod @ B @ E ),Ra: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( A > B ) > ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( D > E ) > ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) @ ( product_Pair @ ( ( A > B ) > ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( D > E ) > ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) @ ( product_apsnd @ A @ B @ C ) @ ( product_apsnd @ D @ E @ F4 ) ) @ ( relAPP @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) @ ( set @ ( product_prod @ ( ( A > B ) > ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( D > E ) > ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) @ ( relAPP @ ( A > B ) @ ( D > E ) @ ( ( set @ ( product_prod @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) > ( set @ ( product_prod @ ( ( A > B ) > ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( D > E ) > ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( D > E ) @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) ) @ ( fun_rel @ A @ D @ B @ E ) @ Rb ) @ Rc ) ) @ ( relAPP @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) @ ( relAPP @ ( product_prod @ C @ A ) @ ( product_prod @ F4 @ D ) @ ( ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) ) ) > ( set @ ( product_prod @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) ) @ ( fun_rel @ ( product_prod @ C @ A ) @ ( product_prod @ F4 @ D ) @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) ) @ ( relAPP @ A @ D @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ F4 @ D ) ) ) @ ( relAPP @ C @ F4 @ ( ( set @ ( product_prod @ A @ D ) ) > ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ F4 @ D ) ) ) ) @ ( prod_rel @ C @ F4 @ A @ D ) @ Ra ) @ Rb ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) ) ) @ ( relAPP @ C @ F4 @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) ) ) ) @ ( prod_rel @ C @ F4 @ B @ E ) @ Ra ) @ Rc ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_149_apsnd__conv,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y: C] :
% 3.24/1.60 ( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y ) )
% 3.24/1.60 = ( product_Pair @ A @ B @ X @ ( F @ Y ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_150_apsnd__id,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType] :
% 3.24/1.60 ( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
% 3.24/1.60 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_151_relprop__triggers_I6_J,axiom,
% 3.24/1.60 ! [I: $tType,R3: set @ I,R5: set @ I] :
% 3.24/1.60 ( ( ord_less_eq @ ( set @ I ) @ R3 @ R5 )
% 3.24/1.60 => ( ord_less_eq @ ( set @ I ) @ R3 @ R5 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_152_Collect__case__prod__mono,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,A2: A > B > $o,B2: A > B > $o] :
% 3.24/1.60 ( ( ord_less_eq @ ( A > B > $o ) @ A2 @ B2 )
% 3.24/1.60 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A2 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B2 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_153_subset__Collect__conv,axiom,
% 3.24/1.60 ! [A: $tType,S5: set @ A,P: A > $o] :
% 3.24/1.60 ( ( ord_less_eq @ ( set @ A ) @ S5 @ ( collect @ A @ P ) )
% 3.24/1.60 = ( ! [X6: A] :
% 3.24/1.60 ( ( member @ A @ X6 @ S5 )
% 3.24/1.60 => ( P @ X6 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_154_exists__leI,axiom,
% 3.24/1.60 ! [N: nat,P: nat > $o] :
% 3.24/1.60 ( ( ! [N2: nat] :
% 3.24/1.60 ( ( ord_less @ nat @ N2 @ N )
% 3.24/1.60 => ~ ( P @ N2 ) )
% 3.24/1.60 => ( P @ N ) )
% 3.24/1.60 => ? [N3: nat] :
% 3.24/1.60 ( ( ord_less_eq @ nat @ N3 @ N )
% 3.24/1.60 & ( P @ N3 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_155_Refine__Misc_Oif__mono,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( ord @ A )
% 3.24/1.60 => ! [B3: $o,M1: A,M12: A,M22: A,M23: A] :
% 3.24/1.60 ( ( B3
% 3.24/1.60 => ( ord_less_eq @ A @ M1 @ M12 ) )
% 3.24/1.60 => ( ( ~ B3
% 3.24/1.60 => ( ord_less_eq @ A @ M22 @ M23 ) )
% 3.24/1.60 => ( ord_less_eq @ A @ ( if @ A @ B3 @ M1 @ M22 ) @ ( if @ A @ B3 @ M12 @ M23 ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_156_param__apfst,axiom,
% 3.24/1.60 ! [D: $tType,A: $tType,B: $tType,E: $tType,F4: $tType,C: $tType,Ra: set @ ( product_prod @ A @ D ),Rb: set @ ( product_prod @ B @ E ),Rc: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( A > B ) > ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( D > E ) > ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) @ ( product_Pair @ ( ( A > B ) > ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( D > E ) > ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) @ ( product_apfst @ A @ B @ C ) @ ( product_apfst @ D @ E @ F4 ) ) @ ( relAPP @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) @ ( set @ ( product_prod @ ( ( A > B ) > ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( D > E ) > ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) @ ( relAPP @ ( A > B ) @ ( D > E ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) > ( set @ ( product_prod @ ( ( A > B ) > ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( D > E ) > ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( D > E ) @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) ) @ ( fun_rel @ A @ D @ B @ E ) @ Ra ) @ Rb ) ) @ ( relAPP @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) @ ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) @ ( relAPP @ ( product_prod @ A @ C ) @ ( product_prod @ D @ F4 ) @ ( ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ C ) @ ( product_prod @ D @ F4 ) @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ D @ F4 ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ D @ F4 ) ) ) ) @ ( prod_rel @ A @ D @ C @ F4 ) @ Ra ) @ Rc ) ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) ) ) @ ( relAPP @ B @ E @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) ) ) ) @ ( prod_rel @ B @ E @ C @ F4 ) @ Rb ) @ Rc ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_157_param__map__prod,axiom,
% 3.24/1.60 ! [C: $tType,G: $tType,E: $tType,A: $tType,B: $tType,F4: $tType,H2: $tType,D: $tType,Ra: set @ ( product_prod @ A @ E ),Rb: set @ ( product_prod @ B @ F4 ),Rc: set @ ( product_prod @ C @ G ),Rd: set @ ( product_prod @ D @ H2 )] : ( member @ ( product_prod @ ( ( A > B ) > ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( E > F4 ) > ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) @ ( product_Pair @ ( ( A > B ) > ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( E > F4 ) > ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) @ ( product_map_prod @ A @ B @ C @ D ) @ ( product_map_prod @ E @ F4 @ G @ H2 ) ) @ ( relAPP @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) @ ( set @ ( product_prod @ ( ( A > B ) > ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( E > F4 ) > ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) @ ( relAPP @ ( A > B ) @ ( E > F4 ) @ ( ( set @ ( product_prod @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) > ( set @ ( product_prod @ ( ( A > B ) > ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( E > F4 ) > ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( E > F4 ) @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) @ ( relAPP @ B @ F4 @ ( set @ ( product_prod @ ( A > B ) @ ( E > F4 ) ) ) @ ( relAPP @ A @ E @ ( ( set @ ( product_prod @ B @ F4 ) ) > ( set @ ( product_prod @ ( A > B ) @ ( E > F4 ) ) ) ) @ ( fun_rel @ A @ E @ B @ F4 ) @ Ra ) @ Rb ) ) @ ( relAPP @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) @ ( set @ ( product_prod @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) @ ( relAPP @ ( C > D ) @ ( G > H2 ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) > ( set @ ( product_prod @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) ) @ ( fun_rel @ ( C > D ) @ ( G > H2 ) @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) @ ( relAPP @ D @ H2 @ ( set @ ( product_prod @ ( C > D ) @ ( G > H2 ) ) ) @ ( relAPP @ C @ G @ ( ( set @ ( product_prod @ D @ H2 ) ) > ( set @ ( product_prod @ ( C > D ) @ ( G > H2 ) ) ) ) @ ( fun_rel @ C @ G @ D @ H2 ) @ Rc ) @ Rd ) ) @ ( relAPP @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) @ ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) @ ( relAPP @ ( product_prod @ A @ C ) @ ( product_prod @ E @ G ) @ ( ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ C ) @ ( product_prod @ E @ G ) @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) ) @ ( relAPP @ C @ G @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ E @ G ) ) ) @ ( relAPP @ A @ E @ ( ( set @ ( product_prod @ C @ G ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ E @ G ) ) ) ) @ ( prod_rel @ A @ E @ C @ G ) @ Ra ) @ Rc ) ) @ ( relAPP @ D @ H2 @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) ) ) @ ( relAPP @ B @ F4 @ ( ( set @ ( product_prod @ D @ H2 ) ) > ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) ) ) ) @ ( prod_rel @ B @ F4 @ D @ H2 ) @ Rb ) @ Rd ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_158_param__prod__swap,axiom,
% 3.24/1.60 ! [B: $tType,D: $tType,C: $tType,A: $tType,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) @ ( ( product_prod @ C @ D ) > ( product_prod @ D @ C ) ) ) @ ( product_Pair @ ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) @ ( ( product_prod @ C @ D ) > ( product_prod @ D @ C ) ) @ ( product_swap @ A @ B ) @ ( product_swap @ C @ D ) ) @ ( relAPP @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) @ ( ( product_prod @ C @ D ) > ( product_prod @ D @ C ) ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) @ ( ( product_prod @ C @ D ) > ( product_prod @ D @ C ) ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) ) @ ( relAPP @ A @ C @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) ) ) @ ( relAPP @ B @ D @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) ) ) ) @ ( prod_rel @ B @ D @ A @ C ) @ B2 ) @ A2 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_159_swap__swap,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 3.24/1.60 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 3.24/1.60 = P2 ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_160_map__prod__simp,axiom,
% 3.24/1.60 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G3: D > B,A5: C,B3: D] :
% 3.24/1.60 ( ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ ( product_Pair @ C @ D @ A5 @ B3 ) )
% 3.24/1.60 = ( product_Pair @ A @ B @ ( F @ A5 ) @ ( G3 @ B3 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_161_apfst__conv,axiom,
% 3.24/1.60 ! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y: B] :
% 3.24/1.60 ( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y ) )
% 3.24/1.60 = ( product_Pair @ A @ B @ ( F @ X ) @ Y ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_162_swap__simp,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,X: B,Y: A] :
% 3.24/1.60 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 3.24/1.60 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_163_apfst__id,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType] :
% 3.24/1.60 ( ( product_apfst @ A @ A @ B @ ( id @ A ) )
% 3.24/1.60 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_164_apfst__def,axiom,
% 3.24/1.60 ! [B: $tType,C: $tType,A: $tType] :
% 3.24/1.60 ( ( product_apfst @ A @ C @ B )
% 3.24/1.60 = ( ^ [F3: A > C] : ( product_map_prod @ A @ C @ B @ B @ F3 @ ( id @ B ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_165_apsnd__apfst__commute,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G3: D > A,P2: product_prod @ D @ C] :
% 3.24/1.60 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G3 @ P2 ) )
% 3.24/1.60 = ( product_apfst @ D @ A @ B @ G3 @ ( product_apsnd @ C @ B @ D @ F @ P2 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_166_apsnd__def,axiom,
% 3.24/1.60 ! [B: $tType,C: $tType,A: $tType] :
% 3.24/1.60 ( ( product_apsnd @ B @ C @ A )
% 3.24/1.60 = ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_167_prod_Omap__id0,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType] :
% 3.24/1.60 ( ( product_map_prod @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) )
% 3.24/1.60 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_168_autoref__comp,axiom,
% 3.24/1.60 ! [A: $tType,D: $tType,C: $tType,F4: $tType,E: $tType,B: $tType,Ra: set @ ( product_prod @ A @ D ),Rb: set @ ( product_prod @ B @ E ),Rc: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( A > B ) > ( C > A ) > C > B ) @ ( ( D > E ) > ( F4 > D ) > F4 > E ) ) @ ( product_Pair @ ( ( A > B ) > ( C > A ) > C > B ) @ ( ( D > E ) > ( F4 > D ) > F4 > E ) @ ( comp @ A @ B @ C ) @ ( comp @ D @ E @ F4 ) ) @ ( relAPP @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) @ ( set @ ( product_prod @ ( ( A > B ) > ( C > A ) > C > B ) @ ( ( D > E ) > ( F4 > D ) > F4 > E ) ) ) @ ( relAPP @ ( A > B ) @ ( D > E ) @ ( ( set @ ( product_prod @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) ) ) > ( set @ ( product_prod @ ( ( A > B ) > ( C > A ) > C > B ) @ ( ( D > E ) > ( F4 > D ) > F4 > E ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( D > E ) @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) ) @ ( fun_rel @ A @ D @ B @ E ) @ Ra ) @ Rb ) ) @ ( relAPP @ ( C > B ) @ ( F4 > E ) @ ( set @ ( product_prod @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) ) ) @ ( relAPP @ ( C > A ) @ ( F4 > D ) @ ( ( set @ ( product_prod @ ( C > B ) @ ( F4 > E ) ) ) > ( set @ ( product_prod @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) ) ) ) @ ( fun_rel @ ( C > A ) @ ( F4 > D ) @ ( C > B ) @ ( F4 > E ) ) @ ( relAPP @ A @ D @ ( set @ ( product_prod @ ( C > A ) @ ( F4 > D ) ) ) @ ( relAPP @ C @ F4 @ ( ( set @ ( product_prod @ A @ D ) ) > ( set @ ( product_prod @ ( C > A ) @ ( F4 > D ) ) ) ) @ ( fun_rel @ C @ F4 @ A @ D ) @ Rc ) @ Ra ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( C > B ) @ ( F4 > E ) ) ) @ ( relAPP @ C @ F4 @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( C > B ) @ ( F4 > E ) ) ) ) @ ( fun_rel @ C @ F4 @ B @ E ) @ Rc ) @ Rb ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_169_less__than__bool__iff,axiom,
% 3.24/1.60 ! [X: $o,Y: $o] :
% 3.24/1.60 ( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ X @ Y ) @ refine1361002633n_bool )
% 3.24/1.60 = ( ~ X
% 3.24/1.60 & Y ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_170_comp__apply,axiom,
% 3.24/1.60 ! [C: $tType,A: $tType,B: $tType] :
% 3.24/1.60 ( ( comp @ B @ A @ C )
% 3.24/1.60 = ( ^ [F3: B > A,G4: C > B,X6: C] : ( F3 @ ( G4 @ X6 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_171_comp__id,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,F: A > B] :
% 3.24/1.60 ( ( comp @ A @ B @ A @ F @ ( id @ A ) )
% 3.24/1.60 = F ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_172_id__comp,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,G3: A > B] :
% 3.24/1.60 ( ( comp @ B @ B @ A @ ( id @ B ) @ G3 )
% 3.24/1.60 = G3 ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_173_fun_Omap__id,axiom,
% 3.24/1.60 ! [A: $tType,D: $tType,T2: D > A] :
% 3.24/1.60 ( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
% 3.24/1.60 = T2 ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_174_swap__comp__swap,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType] :
% 3.24/1.60 ( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
% 3.24/1.60 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_175_map__prod_Ocompositionality,axiom,
% 3.24/1.60 ! [D: $tType,F4: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G3: D > F4,H: A > C,I2: B > D,Prod: product_prod @ A @ B] :
% 3.24/1.60 ( ( product_map_prod @ C @ E @ D @ F4 @ F @ G3 @ ( product_map_prod @ A @ C @ B @ D @ H @ I2 @ Prod ) )
% 3.24/1.60 = ( product_map_prod @ A @ E @ B @ F4 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F4 @ B @ G3 @ I2 ) @ Prod ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_176_map__prod__compose,axiom,
% 3.24/1.60 ! [D: $tType,C: $tType,A: $tType,E: $tType,F4: $tType,B: $tType,F1: E > C,F22: A > E,G1: F4 > D,G22: B > F4] :
% 3.24/1.60 ( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F4 @ D @ B @ G1 @ G22 ) )
% 3.24/1.60 = ( comp @ ( product_prod @ E @ F4 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F4 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F4 @ F22 @ G22 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_177_map__prod_Ocomp,axiom,
% 3.24/1.60 ! [A: $tType,C: $tType,E: $tType,F4: $tType,D: $tType,B: $tType,F: C > E,G3: D > F4,H: A > C,I2: B > D] :
% 3.24/1.60 ( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F4 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F4 @ F @ G3 ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I2 ) )
% 3.24/1.60 = ( product_map_prod @ A @ E @ B @ F4 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F4 @ B @ G3 @ I2 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_178_apfst__compose,axiom,
% 3.24/1.60 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G3: D > C,X: product_prod @ D @ B] :
% 3.24/1.60 ( ( product_apfst @ C @ A @ B @ F @ ( product_apfst @ D @ C @ B @ G3 @ X ) )
% 3.24/1.60 = ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F @ G3 ) @ X ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_179_comp__eq__id__dest,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType,A5: C > B,B3: A > C,C2: A > B,V: A] :
% 3.24/1.60 ( ( ( comp @ C @ B @ A @ A5 @ B3 )
% 3.24/1.60 = ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
% 3.24/1.60 => ( ( A5 @ ( B3 @ V ) )
% 3.24/1.60 = ( C2 @ V ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_180_fun_Omap__id0,axiom,
% 3.24/1.60 ! [A: $tType,D: $tType] :
% 3.24/1.60 ( ( comp @ A @ A @ D @ ( id @ A ) )
% 3.24/1.60 = ( id @ ( D > A ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_181_comp__cong__left,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,C: $tType,X: A > B,Y: A > B,F: C > A] :
% 3.24/1.60 ( ( X = Y )
% 3.24/1.60 => ( ( comp @ A @ B @ C @ X @ F )
% 3.24/1.60 = ( comp @ A @ B @ C @ Y @ F ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_182_comp__cong__right,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType,X: A > B,Y: A > B,F: B > C] :
% 3.24/1.60 ( ( X = Y )
% 3.24/1.60 => ( ( comp @ B @ C @ A @ F @ X )
% 3.24/1.60 = ( comp @ B @ C @ A @ F @ Y ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_183_fun__comp__eq__conv,axiom,
% 3.24/1.60 ! [B: $tType,C: $tType,A: $tType,F: C > B,G3: A > C,Fg: A > B] :
% 3.24/1.60 ( ( ( comp @ C @ B @ A @ F @ G3 )
% 3.24/1.60 = Fg )
% 3.24/1.60 = ( ! [X6: A] :
% 3.24/1.60 ( ( F @ ( G3 @ X6 ) )
% 3.24/1.60 = ( Fg @ X6 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_184_comp__def,axiom,
% 3.24/1.60 ! [A: $tType,C: $tType,B: $tType] :
% 3.24/1.60 ( ( comp @ B @ C @ A )
% 3.24/1.60 = ( ^ [F3: B > C,G4: A > B,X6: A] : ( F3 @ ( G4 @ X6 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_185_comp__assoc,axiom,
% 3.24/1.60 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G3: C > D,H: A > C] :
% 3.24/1.60 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G3 ) @ H )
% 3.24/1.60 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G3 @ H ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_186_comp__eq__dest,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,D: $tType,A: $tType,A5: C > B,B3: A > C,C2: D > B,D3: A > D,V: A] :
% 3.24/1.60 ( ( ( comp @ C @ B @ A @ A5 @ B3 )
% 3.24/1.60 = ( comp @ D @ B @ A @ C2 @ D3 ) )
% 3.24/1.60 => ( ( A5 @ ( B3 @ V ) )
% 3.24/1.60 = ( C2 @ ( D3 @ V ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_187_comp__eq__elim,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,D: $tType,A: $tType,A5: C > B,B3: A > C,C2: D > B,D3: A > D] :
% 3.24/1.60 ( ( ( comp @ C @ B @ A @ A5 @ B3 )
% 3.24/1.60 = ( comp @ D @ B @ A @ C2 @ D3 ) )
% 3.24/1.60 => ! [V2: A] :
% 3.24/1.60 ( ( A5 @ ( B3 @ V2 ) )
% 3.24/1.60 = ( C2 @ ( D3 @ V2 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_188_comp__eq__dest__lhs,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType,A5: C > B,B3: A > C,C2: A > B,V: A] :
% 3.24/1.60 ( ( ( comp @ C @ B @ A @ A5 @ B3 )
% 3.24/1.60 = C2 )
% 3.24/1.60 => ( ( A5 @ ( B3 @ V ) )
% 3.24/1.60 = ( C2 @ V ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_189_fun__upd__comp,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,C: $tType,F: C > B,G3: A > C,X: A,Y: C] :
% 3.24/1.60 ( ( comp @ C @ B @ A @ F @ ( fun_upd @ A @ C @ G3 @ X @ Y ) )
% 3.24/1.60 = ( fun_upd @ A @ B @ ( comp @ C @ B @ A @ F @ G3 ) @ X @ ( F @ Y ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_190_apsnd__compose,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G3: D > C,X: product_prod @ A @ D] :
% 3.24/1.60 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G3 @ X ) )
% 3.24/1.60 = ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G3 ) @ X ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_191_prod_Omap__id,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 3.24/1.60 ( ( product_map_prod @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) @ T2 )
% 3.24/1.60 = T2 ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_192_predicate2I,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 3.24/1.60 ( ! [X5: A,Y3: B] :
% 3.24/1.60 ( ( P @ X5 @ Y3 )
% 3.24/1.60 => ( Q @ X5 @ Y3 ) )
% 3.24/1.60 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_193_prod__refine_I4_J,axiom,
% 3.24/1.60 ! [B: $tType,D: $tType,C: $tType,A: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( ( product_prod @ A @ B ) > A ) @ ( ( product_prod @ C @ D ) > C ) ) @ ( product_Pair @ ( ( product_prod @ A @ B ) > A ) @ ( ( product_prod @ C @ D ) > C ) @ ( product_fst @ A @ B ) @ ( product_fst @ C @ D ) ) @ ( relAPP @ A @ C @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > A ) @ ( ( product_prod @ C @ D ) > C ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > A ) @ ( ( product_prod @ C @ D ) > C ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ A @ C ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) @ Ra ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_194_prod__refine_I5_J,axiom,
% 3.24/1.60 ! [C: $tType,A: $tType,D: $tType,B: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( ( product_prod @ A @ B ) > B ) @ ( ( product_prod @ C @ D ) > D ) ) @ ( product_Pair @ ( ( product_prod @ A @ B ) > B ) @ ( ( product_prod @ C @ D ) > D ) @ ( product_snd @ A @ B ) @ ( product_snd @ C @ D ) ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > B ) @ ( ( product_prod @ C @ D ) > D ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > B ) @ ( ( product_prod @ C @ D ) > D ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ B @ D ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) @ Rb ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_195_fst__map__prod,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G3: D > B,X: product_prod @ C @ D] :
% 3.24/1.60 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ X ) )
% 3.24/1.60 = ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_196_snd__map__prod,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G3: D > A,X: product_prod @ C @ D] :
% 3.24/1.60 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G3 @ X ) )
% 3.24/1.60 = ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_197_fst__apfst,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
% 3.24/1.60 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
% 3.24/1.60 = ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_198_apfst__eq__conv,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G3: C > A] :
% 3.24/1.60 ( ( ( product_apfst @ C @ A @ B @ F @ X )
% 3.24/1.60 = ( product_apfst @ C @ A @ B @ G3 @ X ) )
% 3.24/1.60 = ( ( F @ ( product_fst @ C @ B @ X ) )
% 3.24/1.60 = ( G3 @ ( product_fst @ C @ B @ X ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_199_fst__apsnd,axiom,
% 3.24/1.60 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
% 3.24/1.60 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
% 3.24/1.60 = ( product_fst @ A @ C @ X ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_200_snd__apfst,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
% 3.24/1.60 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
% 3.24/1.60 = ( product_snd @ C @ A @ X ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_201_snd__apsnd,axiom,
% 3.24/1.60 ! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
% 3.24/1.60 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
% 3.24/1.60 = ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_202_apsnd__eq__conv,axiom,
% 3.24/1.60 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G3: C > B] :
% 3.24/1.60 ( ( ( product_apsnd @ C @ B @ A @ F @ X )
% 3.24/1.60 = ( product_apsnd @ C @ B @ A @ G3 @ X ) )
% 3.24/1.60 = ( ( F @ ( product_snd @ A @ C @ X ) )
% 3.24/1.60 = ( G3 @ ( product_snd @ A @ C @ X ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_203_prod_Ocollapse,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 3.24/1.60 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 3.24/1.60 = Prod ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_204_fst__swap,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 3.24/1.60 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 3.24/1.60 = ( product_snd @ B @ A @ X ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_205_snd__swap,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 3.24/1.60 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 3.24/1.60 = ( product_fst @ A @ B @ X ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_206_fst__comp__apsnd,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType,F: B > C] :
% 3.24/1.60 ( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
% 3.24/1.60 = ( product_fst @ A @ B ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_207_snd__comp__apfst,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType,F: A > C] :
% 3.24/1.60 ( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
% 3.24/1.60 = ( product_snd @ A @ B ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_208_fst__comp__map__prod,axiom,
% 3.24/1.60 ! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > C,G3: B > D] :
% 3.24/1.60 ( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G3 ) )
% 3.24/1.60 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_209_snd__comp__map__prod,axiom,
% 3.24/1.60 ! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > D,G3: B > C] :
% 3.24/1.60 ( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F @ G3 ) )
% 3.24/1.60 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ G3 @ ( product_snd @ A @ B ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_210_fst__comp__apfst,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType,F: A > C] :
% 3.24/1.60 ( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
% 3.24/1.60 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_211_snd__comp__apsnd,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType,F: B > C] :
% 3.24/1.60 ( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
% 3.24/1.60 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_212_apfst__apsnd,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G3: D > B,X: product_prod @ C @ D] :
% 3.24/1.60 ( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G3 @ X ) )
% 3.24/1.60 = ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G3 @ ( product_snd @ C @ D @ X ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_213_apsnd__apfst,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G3: D > A,X: product_prod @ D @ C] :
% 3.24/1.60 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G3 @ X ) )
% 3.24/1.60 = ( product_Pair @ A @ B @ ( G3 @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_214_prod_Oswap__def,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType] :
% 3.24/1.60 ( ( product_swap @ A @ B )
% 3.24/1.60 = ( ^ [P4: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P4 ) @ ( product_fst @ A @ B @ P4 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_215_prod_Osplit__sel,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 3.24/1.60 ( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 3.24/1.60 = ( ( Prod
% 3.24/1.60 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 3.24/1.60 => ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_216_prod_Oexhaust__sel,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 3.24/1.60 ( Prod
% 3.24/1.60 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_217_prod_Osplit__sel__asm,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 3.24/1.60 ( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 3.24/1.60 = ( ~ ( ( Prod
% 3.24/1.60 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 3.24/1.60 & ~ ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_218_surjective__pairing,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 3.24/1.60 ( T2
% 3.24/1.60 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_219_sndE,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType,X: product_prod @ A @ B,A5: A,B3: B,P: B > $o] :
% 3.24/1.60 ( ( X
% 3.24/1.60 = ( product_Pair @ A @ B @ A5 @ B3 ) )
% 3.24/1.60 => ( ( P @ ( product_snd @ A @ B @ X ) )
% 3.24/1.60 => ( P @ B3 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_220_fstE,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A5: A,B3: B,P: A > $o] :
% 3.24/1.60 ( ( X
% 3.24/1.60 = ( product_Pair @ A @ B @ A5 @ B3 ) )
% 3.24/1.60 => ( ( P @ ( product_fst @ A @ B @ X ) )
% 3.24/1.60 => ( P @ A5 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_221_snd__eqD,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,X: B,Y: A,A5: A] :
% 3.24/1.60 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 3.24/1.60 = A5 )
% 3.24/1.60 => ( Y = A5 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_222_snd__conv,axiom,
% 3.24/1.60 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 3.24/1.60 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 3.24/1.60 = X22 ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_223_fst__eqD,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,X: A,Y: B,A5: A] :
% 3.24/1.60 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 3.24/1.60 = A5 )
% 3.24/1.60 => ( X = A5 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_224_fst__conv,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 3.24/1.60 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 3.24/1.60 = X1 ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_225_prod__eq__iff,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType] :
% 3.24/1.60 ( ( ^ [Y5: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y5 = Z ) )
% 3.24/1.60 = ( ^ [S: product_prod @ A @ B,T4: product_prod @ A @ B] :
% 3.24/1.60 ( ( ( product_fst @ A @ B @ S )
% 3.24/1.60 = ( product_fst @ A @ B @ T4 ) )
% 3.24/1.60 & ( ( product_snd @ A @ B @ S )
% 3.24/1.60 = ( product_snd @ A @ B @ T4 ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_226_prod_Oexpand,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 3.24/1.60 ( ( ( ( product_fst @ A @ B @ Prod )
% 3.24/1.60 = ( product_fst @ A @ B @ Prod2 ) )
% 3.24/1.60 & ( ( product_snd @ A @ B @ Prod )
% 3.24/1.60 = ( product_snd @ A @ B @ Prod2 ) ) )
% 3.24/1.60 => ( Prod = Prod2 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_227_prod__eqI,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 3.24/1.60 ( ( ( product_fst @ A @ B @ P2 )
% 3.24/1.60 = ( product_fst @ A @ B @ Q2 ) )
% 3.24/1.60 => ( ( ( product_snd @ A @ B @ P2 )
% 3.24/1.60 = ( product_snd @ A @ B @ Q2 ) )
% 3.24/1.60 => ( P2 = Q2 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_228_All__prod__contract,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,P: A > B > $o] :
% 3.24/1.60 ( ( ! [A7: A,X7: B] : ( P @ A7 @ X7 ) )
% 3.24/1.60 = ( ! [Z3: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z3 ) @ ( product_snd @ A @ B @ Z3 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_229_Ex__prod__contract,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,P: A > B > $o] :
% 3.24/1.60 ( ( ? [A7: A,X7: B] : ( P @ A7 @ X7 ) )
% 3.24/1.60 = ( ? [Z3: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z3 ) @ ( product_snd @ A @ B @ Z3 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_230_case__prod__beta,axiom,
% 3.24/1.60 ! [A: $tType,C: $tType,B: $tType] :
% 3.24/1.60 ( ( product_case_prod @ B @ C @ A )
% 3.24/1.60 = ( ^ [F3: B > C > A,P4: product_prod @ B @ C] : ( F3 @ ( product_fst @ B @ C @ P4 ) @ ( product_snd @ B @ C @ P4 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_231_prod_Ocase__eq__if,axiom,
% 3.24/1.60 ! [C: $tType,B: $tType,A: $tType] :
% 3.24/1.60 ( ( product_case_prod @ A @ B @ C )
% 3.24/1.60 = ( ^ [F3: A > B > C,Prod3: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_232_Product__Type_OCollect__case__prodD,axiom,
% 3.24/1.60 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A2: A > B > $o] :
% 3.24/1.60 ( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A2 ) ) )
% 3.24/1.60 => ( A2 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_233_order__subst1,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType] :
% 3.24/1.60 ( ( ( order @ B )
% 3.24/1.60 & ( order @ A ) )
% 3.24/1.60 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 3.24/1.60 ( ( ord_less_eq @ A @ A5 @ ( F @ B3 ) )
% 3.24/1.60 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 3.24/1.60 => ( ! [X5: B,Y3: B] :
% 3.24/1.60 ( ( ord_less_eq @ B @ X5 @ Y3 )
% 3.24/1.60 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 3.24/1.60 => ( ord_less_eq @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_234_order__subst2,axiom,
% 3.24/1.60 ! [A: $tType,C: $tType] :
% 3.24/1.60 ( ( ( order @ C )
% 3.24/1.60 & ( order @ A ) )
% 3.24/1.60 => ! [A5: A,B3: A,F: A > C,C2: C] :
% 3.24/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.24/1.60 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 3.24/1.60 => ( ! [X5: A,Y3: A] :
% 3.24/1.60 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 3.24/1.60 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 3.24/1.60 => ( ord_less_eq @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_235_ord__eq__le__subst,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType] :
% 3.24/1.60 ( ( ( ord @ B )
% 3.24/1.60 & ( ord @ A ) )
% 3.24/1.60 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 3.24/1.60 ( ( A5
% 3.24/1.60 = ( F @ B3 ) )
% 3.24/1.60 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 3.24/1.60 => ( ! [X5: B,Y3: B] :
% 3.24/1.60 ( ( ord_less_eq @ B @ X5 @ Y3 )
% 3.24/1.60 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 3.24/1.60 => ( ord_less_eq @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_236_ord__le__eq__subst,axiom,
% 3.24/1.60 ! [A: $tType,B: $tType] :
% 3.24/1.60 ( ( ( ord @ B )
% 3.24/1.60 & ( ord @ A ) )
% 3.24/1.60 => ! [A5: A,B3: A,F: A > B,C2: B] :
% 3.24/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.24/1.60 => ( ( ( F @ B3 )
% 3.24/1.60 = C2 )
% 3.24/1.60 => ( ! [X5: A,Y3: A] :
% 3.24/1.60 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 3.24/1.60 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 3.24/1.60 => ( ord_less_eq @ B @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_237_eq__iff,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( order @ A )
% 3.24/1.60 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 3.24/1.60 = ( ^ [X6: A,Y6: A] :
% 3.24/1.60 ( ( ord_less_eq @ A @ X6 @ Y6 )
% 3.24/1.60 & ( ord_less_eq @ A @ Y6 @ X6 ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_238_antisym,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( order @ A )
% 3.24/1.60 => ! [X: A,Y: A] :
% 3.24/1.60 ( ( ord_less_eq @ A @ X @ Y )
% 3.24/1.60 => ( ( ord_less_eq @ A @ Y @ X )
% 3.24/1.60 => ( X = Y ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_239_linear,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( linorder @ A )
% 3.24/1.60 => ! [X: A,Y: A] :
% 3.24/1.60 ( ( ord_less_eq @ A @ X @ Y )
% 3.24/1.60 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_240_eq__refl,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( preorder @ A )
% 3.24/1.60 => ! [X: A,Y: A] :
% 3.24/1.60 ( ( X = Y )
% 3.24/1.60 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_241_le__cases,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( linorder @ A )
% 3.24/1.60 => ! [X: A,Y: A] :
% 3.24/1.60 ( ~ ( ord_less_eq @ A @ X @ Y )
% 3.24/1.60 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_242_order_Otrans,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( order @ A )
% 3.24/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.24/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.24/1.60 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 3.24/1.60 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_243_le__cases3,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( linorder @ A )
% 3.24/1.60 => ! [X: A,Y: A,Z2: A] :
% 3.24/1.60 ( ( ( ord_less_eq @ A @ X @ Y )
% 3.24/1.60 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 3.24/1.60 => ( ( ( ord_less_eq @ A @ Y @ X )
% 3.24/1.60 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 3.24/1.60 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 3.24/1.60 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 3.24/1.60 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 3.24/1.60 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 3.24/1.60 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 3.24/1.60 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 3.24/1.60 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 3.24/1.60 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_244_antisym__conv,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( order @ A )
% 3.24/1.60 => ! [Y: A,X: A] :
% 3.24/1.60 ( ( ord_less_eq @ A @ Y @ X )
% 3.24/1.60 => ( ( ord_less_eq @ A @ X @ Y )
% 3.24/1.60 = ( X = Y ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_245_order__class_Oorder_Oeq__iff,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( order @ A )
% 3.24/1.60 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 3.24/1.60 = ( ^ [A7: A,B6: A] :
% 3.24/1.60 ( ( ord_less_eq @ A @ A7 @ B6 )
% 3.24/1.60 & ( ord_less_eq @ A @ B6 @ A7 ) ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_246_ord__eq__le__trans,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( ord @ A )
% 3.24/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.24/1.60 ( ( A5 = B3 )
% 3.24/1.60 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 3.24/1.60 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_247_ord__le__eq__trans,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( ord @ A )
% 3.24/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.24/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.24/1.60 => ( ( B3 = C2 )
% 3.24/1.60 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_248_order__class_Oorder_Oantisym,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( order @ A )
% 3.24/1.60 => ! [A5: A,B3: A] :
% 3.24/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.24/1.60 => ( ( ord_less_eq @ A @ B3 @ A5 )
% 3.24/1.60 => ( A5 = B3 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_249_order__trans,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( preorder @ A )
% 3.24/1.60 => ! [X: A,Y: A,Z2: A] :
% 3.24/1.60 ( ( ord_less_eq @ A @ X @ Y )
% 3.24/1.60 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 3.24/1.60 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_250_dual__order_Orefl,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( order @ A )
% 3.24/1.60 => ! [A5: A] : ( ord_less_eq @ A @ A5 @ A5 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(fact_251_linorder__wlog,axiom,
% 3.24/1.60 ! [A: $tType] :
% 3.24/1.60 ( ( linorder @ A )
% 3.24/1.60 => ! [P: A > A > $o,A5: A,B3: A] :
% 3.24/1.60 ( ! [A3: A,B5: A] :
% 3.24/1.60 ( ( ord_less_eq @ A @ A3 @ B5 )
% 3.24/1.60 => ( P @ A3 @ B5 ) )
% 3.24/1.60 => ( ! [A3: A,B5: A] :
% 3.24/1.60 ( ( P @ B5 @ A3 )
% 3.24/1.60 => ( P @ A3 @ B5 ) )
% 3.24/1.60 => ( P @ A5 @ B3 ) ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.24/1.60 ! [A8: $tType,A9: $tType] :
% 3.24/1.60 ( ( preorder @ A9 )
% 3.24/1.60 => ( preorder @ ( A8 > A9 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.24/1.60 ! [A8: $tType,A9: $tType] :
% 3.24/1.60 ( ( order @ A9 )
% 3.24/1.60 => ( order @ ( A8 > A9 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_fun___Orderings_Oord,axiom,
% 3.24/1.60 ! [A8: $tType,A9: $tType] :
% 3.24/1.60 ( ( ord @ A9 )
% 3.24/1.60 => ( ord @ ( A8 > A9 ) ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 3.24/1.60 preorder @ int ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 3.24/1.60 linorder @ int ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 3.24/1.60 order @ int ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 3.24/1.60 ord @ int ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
% 3.24/1.60 preorder @ nat ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Nat_Onat___Orderings_Olinorder_5,axiom,
% 3.24/1.60 linorder @ nat ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Nat_Onat___Orderings_Oorder_6,axiom,
% 3.24/1.60 order @ nat ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Nat_Onat___Orderings_Oord_7,axiom,
% 3.24/1.60 ord @ nat ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Set_Oset___Orderings_Opreorder_8,axiom,
% 3.24/1.60 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 3.24/1.60 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
% 3.24/1.60 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_HOL_Obool___Orderings_Opreorder_11,axiom,
% 3.24/1.60 preorder @ $o ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_HOL_Obool___Orderings_Olinorder_12,axiom,
% 3.24/1.60 linorder @ $o ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
% 3.24/1.60 order @ $o ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
% 3.24/1.60 ord @ $o ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_15,axiom,
% 3.24/1.60 ! [A8: $tType] : ( preorder @ ( refine1665802226e_nres @ A8 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_16,axiom,
% 3.24/1.60 ! [A8: $tType] : ( order @ ( refine1665802226e_nres @ A8 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_17,axiom,
% 3.24/1.60 ! [A8: $tType] : ( ord @ ( refine1665802226e_nres @ A8 ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(help_If_3_1_T,axiom,
% 3.24/1.60 ! [P: $o] :
% 3.24/1.60 ( ( P = $true )
% 3.24/1.60 | ( P = $false ) ) ).
% 3.24/1.60
% 3.24/1.60 thf(help_If_2_1_T,axiom,
% 3.24/1.60 ! [A: $tType,X: A,Y: A] :
% 3.24/1.60 ( ( if @ A @ $false @ X @ Y )
% 3.24/1.60 = Y ) ).
% 3.24/1.60
% 3.24/1.60 thf(help_If_1_1_T,axiom,
% 3.24/1.60 ! [A: $tType,X: A,Y: A] :
% 3.24/1.60 ( ( if @ A @ $true @ X @ Y )
% 3.24/1.60 = X ) ).
% 3.24/1.60
% 3.24/1.60 thf(conj_0,conjecture,
% 3.24/1.60 member @ ( product_prod @ ( ( refine1665802226e_nres @ a ) > ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( refine1665802226e_nres @ c ) > ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) @ ( product_Pair @ ( ( refine1665802226e_nres @ a ) > ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( refine1665802226e_nres @ c ) > ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) @ ( refine463715084e_bind @ a @ b ) @ ( refine463715084e_bind @ c @ d ) ) @ ( relAPP @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) @ ( set @ ( product_prod @ ( ( refine1665802226e_nres @ a ) > ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( refine1665802226e_nres @ c ) > ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) @ ( relAPP @ ( refine1665802226e_nres @ a ) @ ( refine1665802226e_nres @ c ) @ ( ( set @ ( product_prod @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) > ( set @ ( product_prod @ ( ( refine1665802226e_nres @ a ) > ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( refine1665802226e_nres @ c ) > ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) ) @ ( fun_rel @ ( refine1665802226e_nres @ a ) @ ( refine1665802226e_nres @ c ) @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) @ ( relAPP @ a @ c @ ( set @ ( product_prod @ ( refine1665802226e_nres @ a ) @ ( refine1665802226e_nres @ c ) ) ) @ ( refine476890328es_rel @ a @ c ) @ r1 ) ) @ ( relAPP @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) @ ( set @ ( product_prod @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) @ ( relAPP @ ( a > ( refine1665802226e_nres @ b ) ) @ ( c > ( refine1665802226e_nres @ d ) ) @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) ) > ( set @ ( product_prod @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) ) @ ( fun_rel @ ( a > ( refine1665802226e_nres @ b ) ) @ ( c > ( refine1665802226e_nres @ d ) ) @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) @ ( relAPP @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) @ ( set @ ( product_prod @ ( a > ( refine1665802226e_nres @ b ) ) @ ( c > ( refine1665802226e_nres @ d ) ) ) ) @ ( relAPP @ a @ c @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) ) > ( set @ ( product_prod @ ( a > ( refine1665802226e_nres @ b ) ) @ ( c > ( refine1665802226e_nres @ d ) ) ) ) ) @ ( fun_rel @ a @ c @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) @ r1 ) @ ( relAPP @ b @ d @ ( set @ ( product_prod @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) ) @ ( refine476890328es_rel @ b @ d ) @ r2 ) ) ) @ ( relAPP @ b @ d @ ( set @ ( product_prod @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) ) @ ( refine476890328es_rel @ b @ d ) @ r2 ) ) ) ).
% 3.24/1.60 ------------------------
% 3.24/1.63 % (2365)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_2166 for (2999ds/275Mi)
% 3.24/1.63 % (2360)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_2166 for (2999ds/4Mi)
% 3.24/1.63 % (2359)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_2166 for (2999ds/183Mi)
% 3.24/1.63 % (2362)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_2166 for (2999ds/2Mi)
% 3.24/1.63 % (2366)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_2166 for (2999ds/18Mi)
% 3.24/1.63 % (2364)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_2166 for (2999ds/2Mi)
% 3.24/1.63 % (2362)Instruction limit reached!
% 3.24/1.63 % (2362)------------------------------
% 3.24/1.63 % (2362)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.63 % (2362)Termination reason: Unknown
% 3.24/1.63 % (2362)Termination phase: shuffling
% 3.24/1.63
% 3.24/1.63 % (2362)Memory used [KB]: 1663
% 3.24/1.63 % (2362)Time elapsed: 0.002 s
% 3.24/1.63 % (2362)Instructions burned: 3 (million)
% 3.24/1.63 % (2362)------------------------------
% 3.24/1.63 % (2362)------------------------------
% 3.24/1.63 % (2364)Instruction limit reached!
% 3.24/1.63 % (2364)------------------------------
% 3.24/1.63 % (2364)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.63 % (2364)Termination reason: Unknown
% 3.24/1.63 % (2364)Termination phase: shuffling
% 3.24/1.63
% 3.24/1.63 % (2364)Memory used [KB]: 1663
% 3.24/1.63 % (2364)Time elapsed: 0.002 s
% 3.24/1.63 % (2364)Instructions burned: 2 (million)
% 3.24/1.63 % (2364)------------------------------
% 3.24/1.63 % (2364)------------------------------
% 3.24/1.63 % (2360)Instruction limit reached!
% 3.24/1.63 % (2360)------------------------------
% 3.24/1.63 % (2360)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.63 % (2360)Termination reason: Unknown
% 3.24/1.63 % (2360)Termination phase: shuffling
% 3.24/1.63
% 3.24/1.63 % (2360)Memory used [KB]: 1663
% 3.24/1.63 % (2360)Time elapsed: 0.003 s
% 3.24/1.63 % (2360)Instructions burned: 7 (million)
% 3.24/1.63 % (2360)------------------------------
% 3.24/1.63 % (2360)------------------------------
% 3.24/1.63 % (2366)Instruction limit reached!
% 3.24/1.63 % (2366)------------------------------
% 3.24/1.63 % (2366)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.63 % (2366)Termination reason: Unknown
% 3.24/1.63 % (2366)Termination phase: shuffling
% 3.24/1.63
% 3.24/1.63 % (2366)Memory used [KB]: 1918
% 3.24/1.63 % (2366)Time elapsed: 0.007 s
% 3.24/1.63 % (2366)Instructions burned: 18 (million)
% 3.24/1.63 % (2366)------------------------------
% 3.24/1.63 % (2366)------------------------------
% 3.24/1.64 % (2361)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_2166 for (2999ds/27Mi)
% 3.24/1.64 % (2367)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2166 for (2999ds/3Mi)
% 3.24/1.64 % (2368)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_2166 for (2999ds/37Mi)
% 3.24/1.64 % (2369)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_2166 for (2999ds/15Mi)
% 3.24/1.64 % (2367)Instruction limit reached!
% 3.24/1.64 % (2367)------------------------------
% 3.24/1.64 % (2367)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.64 % (2367)Termination reason: Unknown
% 3.24/1.64 % (2367)Termination phase: shuffling
% 3.24/1.64
% 3.24/1.64 % (2367)Memory used [KB]: 1663
% 3.24/1.64 % (2367)Time elapsed: 0.002 s
% 3.24/1.64 % (2367)Instructions burned: 3 (million)
% 3.24/1.64 % (2367)------------------------------
% 3.24/1.64 % (2367)------------------------------
% 3.24/1.64 % (2370)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_2166 for (2999ds/3Mi)
% 3.24/1.64 % (2369)Instruction limit reached!
% 3.24/1.64 % (2369)------------------------------
% 3.24/1.64 % (2369)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.64 % (2369)Termination reason: Unknown
% 3.24/1.64 % (2369)Termination phase: shuffling
% 3.24/1.64
% 3.24/1.64 % (2369)Memory used [KB]: 1791
% 3.24/1.64 % (2369)Time elapsed: 0.006 s
% 3.24/1.64 % (2369)Instructions burned: 16 (million)
% 3.24/1.64 % (2369)------------------------------
% 3.24/1.64 % (2369)------------------------------
% 3.24/1.64 % (2370)Instruction limit reached!
% 3.24/1.64 % (2370)------------------------------
% 3.24/1.64 % (2370)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.64 % (2370)Termination reason: Unknown
% 3.24/1.64 % (2370)Termination phase: shuffling
% 3.24/1.64
% 3.24/1.64 % (2370)Memory used [KB]: 1663
% 3.24/1.64 % (2370)Time elapsed: 0.002 s
% 3.24/1.64 % (2370)Instructions burned: 3 (million)
% 3.24/1.64 % (2370)------------------------------
% 3.24/1.64 % (2370)------------------------------
% 3.24/1.65 % (2361)Instruction limit reached!
% 3.24/1.65 % (2361)------------------------------
% 3.24/1.65 % (2361)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.65 % (2361)Termination reason: Unknown
% 3.24/1.65 % (2361)Termination phase: shuffling
% 3.24/1.65
% 3.24/1.65 % (2361)Memory used [KB]: 2046
% 3.24/1.65 % (2361)Time elapsed: 0.012 s
% 3.24/1.65 % (2361)Instructions burned: 27 (million)
% 3.24/1.65 % (2361)------------------------------
% 3.24/1.65 % (2361)------------------------------
% 3.24/1.65 % (2371)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_2166 for (2999ds/1041Mi)
% 3.24/1.65 % (2368)Instruction limit reached!
% 3.24/1.65 % (2368)------------------------------
% 3.24/1.65 % (2368)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.65 % (2368)Termination reason: Unknown
% 3.24/1.65 % (2368)Termination phase: shuffling
% 3.24/1.65
% 3.24/1.65 % (2368)Memory used [KB]: 2302
% 3.24/1.65 % (2368)Time elapsed: 0.012 s
% 3.24/1.65 % (2368)Instructions burned: 38 (million)
% 3.24/1.65 % (2368)------------------------------
% 3.24/1.65 % (2368)------------------------------
% 3.24/1.65 % (2373)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_2166 for (2999ds/16Mi)
% 3.24/1.66 % (2372)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_2166 for (2999ds/7Mi)
% 3.24/1.66 % (2374)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_2166 for (2999ds/3Mi)
% 3.24/1.66 % (2374)Instruction limit reached!
% 3.24/1.66 % (2374)------------------------------
% 3.24/1.66 % (2374)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.66 % (2374)Termination reason: Unknown
% 3.24/1.66 % (2374)Termination phase: shuffling
% 3.24/1.66
% 3.24/1.66 % (2374)Memory used [KB]: 1663
% 3.24/1.66 % (2374)Time elapsed: 0.003 s
% 3.24/1.66 % (2374)Instructions burned: 3 (million)
% 3.24/1.66 % (2374)------------------------------
% 3.24/1.66 % (2374)------------------------------
% 3.24/1.66 % (2372)Instruction limit reached!
% 3.24/1.66 % (2372)------------------------------
% 3.24/1.66 % (2372)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.66 % (2372)Termination reason: Unknown
% 3.24/1.66 % (2372)Termination phase: shuffling
% 3.24/1.66
% 3.24/1.66 % (2372)Memory used [KB]: 1791
% 3.24/1.66 % (2372)Time elapsed: 0.005 s
% 3.24/1.66 % (2372)Instructions burned: 8 (million)
% 3.24/1.66 % (2372)------------------------------
% 3.24/1.66 % (2372)------------------------------
% 3.24/1.66 % (2378)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_2166 for (2999ds/3Mi)
% 3.24/1.66 % (2373)Instruction limit reached!
% 3.24/1.66 % (2373)------------------------------
% 3.24/1.66 % (2373)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.66 % (2373)Termination reason: Unknown
% 3.24/1.66 % (2373)Termination phase: shuffling
% 3.24/1.66
% 3.24/1.66 % (2373)Memory used [KB]: 1791
% 3.24/1.66 % (2373)Time elapsed: 0.007 s
% 3.24/1.66 % (2373)Instructions burned: 17 (million)
% 3.24/1.66 % (2373)------------------------------
% 3.24/1.66 % (2373)------------------------------
% 3.24/1.66 % (2378)Instruction limit reached!
% 3.24/1.66 % (2378)------------------------------
% 3.24/1.66 % (2378)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.66 % (2378)Termination reason: Unknown
% 3.24/1.66 % (2378)Termination phase: shuffling
% 3.24/1.66
% 3.24/1.66 % (2378)Memory used [KB]: 1663
% 3.24/1.66 % (2378)Time elapsed: 0.002 s
% 3.24/1.66 % (2378)Instructions burned: 3 (million)
% 3.24/1.66 % (2378)------------------------------
% 3.24/1.66 % (2378)------------------------------
% 3.24/1.67 % (2381)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_2166 for (2999ds/7Mi)
% 3.24/1.67 % (2382)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_2166 for (2999ds/3Mi)
% 3.24/1.67 % (2383)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_2166 for (2999ds/4Mi)
% 3.24/1.67 % (2384)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_2166 for (2999ds/18Mi)
% 3.24/1.67 % (2382)Instruction limit reached!
% 3.24/1.67 % (2382)------------------------------
% 3.24/1.67 % (2382)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.67 % (2382)Termination reason: Unknown
% 3.24/1.67 % (2382)Termination phase: shuffling
% 3.24/1.67
% 3.24/1.67 % (2382)Memory used [KB]: 1663
% 3.24/1.67 % (2382)Time elapsed: 0.003 s
% 3.24/1.67 % (2382)Instructions burned: 4 (million)
% 3.24/1.67 % (2382)------------------------------
% 3.24/1.67 % (2382)------------------------------
% 3.24/1.67 % (2381)Instruction limit reached!
% 3.24/1.67 % (2381)------------------------------
% 3.24/1.67 % (2381)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.67 % (2381)Termination reason: Unknown
% 3.24/1.67 % (2381)Termination phase: shuffling
% 3.24/1.67
% 3.24/1.67 % (2381)Memory used [KB]: 1791
% 3.24/1.67 % (2381)Time elapsed: 0.004 s
% 3.24/1.67 % (2381)Instructions burned: 9 (million)
% 3.24/1.67 % (2381)------------------------------
% 3.24/1.67 % (2381)------------------------------
% 3.24/1.67 % (2365)First to succeed.
% 3.24/1.67 % (2383)Instruction limit reached!
% 3.24/1.67 % (2383)------------------------------
% 3.24/1.67 % (2383)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.67 % (2383)Termination reason: Unknown
% 3.24/1.67 % (2383)Termination phase: shuffling
% 3.24/1.67
% 3.24/1.67 % (2383)Memory used [KB]: 1663
% 3.24/1.67 % (2383)Time elapsed: 0.003 s
% 3.24/1.67 % (2383)Instructions burned: 7 (million)
% 3.24/1.67 % (2383)------------------------------
% 3.24/1.67 % (2383)------------------------------
% 3.24/1.67 % (2365)Refutation found. Thanks to Tanya!
% 3.24/1.67 % SZS status Theorem for DTF2THF_2166
% 3.24/1.67 % SZS output start Proof for DTF2THF_2166
% 3.24/1.67 thf(type_def_5, type, autoref_annot: $tType).
% 3.24/1.67 thf(type_def_7, type, product_prod: ($tType * $tType) > $tType).
% 3.24/1.67 thf(type_def_8, type, set: ($tType) > $tType).
% 3.24/1.67 thf(type_def_9, type, nat: $tType).
% 3.24/1.67 thf(type_def_10, type, refine1665802226e_nres: ($tType) > $tType).
% 3.24/1.67 thf(type_def_11, type, a: $tType).
% 3.24/1.67 thf(type_def_12, type, c: $tType).
% 3.24/1.67 thf(type_def_13, type, b: $tType).
% 3.24/1.67 thf(type_def_14, type, d: $tType).
% 3.24/1.67 thf(type_def_15, type, int: $tType).
% 3.24/1.67 thf(func_def_0, type, refine1665802226e_nres: ($tType) > $tType).
% 3.24/1.67 thf(func_def_1, type, autoref_annot: $tType).
% 3.24/1.67 thf(func_def_2, type, product_prod: ($tType * $tType) > $tType).
% 3.24/1.67 thf(func_def_3, type, set: ($tType) > $tType).
% 3.24/1.67 thf(func_def_4, type, nat: $tType).
% 3.24/1.67 thf(func_def_5, type, int: $tType).
% 3.24/1.67 thf(func_def_6, type, d: $tType).
% 3.24/1.67 thf(func_def_7, type, c: $tType).
% 3.24/1.67 thf(func_def_8, type, b: $tType).
% 3.24/1.67 thf(func_def_9, type, a: $tType).
% 3.24/1.67 thf(func_def_10, type, type: !>[X0: $tType]:($o)).
% 3.24/1.67 thf(func_def_11, type, ord: !>[X0: $tType]:($o)).
% 3.24/1.67 thf(func_def_12, type, order: !>[X0: $tType]:($o)).
% 3.24/1.67 thf(func_def_13, type, linorder: !>[X0: $tType]:($o)).
% 3.24/1.67 thf(func_def_14, type, preorder: !>[X0: $tType]:($o)).
% 3.24/1.67 thf(func_def_15, type, autoref_ANNOT: !>[X0: $tType]:(X0 > autoref_annot > X0)).
% 3.24/1.67 thf(func_def_16, type, autoref_APP: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0 > X1)).
% 3.24/1.67 thf(func_def_17, type, autoref_OP: !>[X0: $tType]:(X0 > X0)).
% 3.24/1.67 thf(func_def_18, type, autoref_rel_annot: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > autoref_annot)).
% 3.24/1.67 thf(func_def_19, type, autore2013370551TERNAL: !>[X0: $tType]:(X0 > X0)).
% 3.24/1.67 thf(func_def_20, type, autore813025987NAL_EQ: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.24/1.67 thf(func_def_21, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 3.24/1.67 thf(func_def_22, type, fun_upd: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0 > X1 > X0 > X1)).
% 3.24/1.67 thf(func_def_23, type, id: !>[X0: $tType]:(X0 > X0)).
% 3.24/1.67 thf(func_def_24, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.24/1.67 thf(func_def_25, type, all_interval_nat: (nat > $o) > nat > nat > $o).
% 3.24/1.67 thf(func_def_26, type, uncurry: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.24/1.67 thf(func_def_27, type, case_nat: !>[X0: $tType]:(X0 > (nat > X0) > nat > X0)).
% 3.24/1.67 thf(func_def_28, type, rec_nat: !>[X0: $tType]:(X0 > (nat > X0 > X0) > nat > X0)).
% 3.24/1.67 thf(func_def_29, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.24/1.67 thf(func_def_30, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.24/1.67 thf(func_def_31, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.24/1.67 thf(func_def_32, type, product_apfst: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 3.24/1.67 thf(func_def_33, type, product_apsnd: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X2 @ X0 > product_prod @ X2 @ X1)).
% 3.24/1.67 thf(func_def_34, type, product_case_bool: !>[X0: $tType]:(X0 > X0 > $o > X0)).
% 3.24/1.67 thf(func_def_35, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 3.24/1.67 thf(func_def_36, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.24/1.67 thf(func_def_37, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 3.24/1.67 thf(func_def_38, type, product_rec_bool: !>[X0: $tType]:(X0 > X0 > $o > X0)).
% 3.24/1.67 thf(func_def_39, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.24/1.67 thf(func_def_40, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.24/1.67 thf(func_def_41, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.24/1.67 thf(func_def_42, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.24/1.67 thf(func_def_43, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 3.24/1.67 thf(func_def_44, type, refine1687780735RETURN: !>[X0: $tType]:(X0 > refine1665802226e_nres @ X0)).
% 3.24/1.67 thf(func_def_45, type, refine463715084e_bind: !>[X0: $tType, X1: $tType]:(refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X1)).
% 3.24/1.67 thf(func_def_46, type, refine1580981607t_assn: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > (X1 > $o) > X0 > $o)).
% 3.24/1.67 thf(func_def_47, type, refine476890328es_rel: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ refine1665802226e_nres @ X0 @ refine1665802226e_nres @ X1)).
% 3.24/1.67 thf(func_def_48, type, refine171938416RT_bnd: !>[X0: $tType]:($o > refine1665802226e_nres @ X0 > refine1665802226e_nres @ X0)).
% 3.24/1.67 thf(func_def_49, type, refine1361002633n_bool: set @ product_prod @ $o @ $o).
% 3.24/1.67 thf(func_def_50, type, id2: !>[X0: $tType]:(set @ product_prod @ X0 @ X0)).
% 3.24/1.67 thf(func_def_51, type, fun_rel: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ X2 @ X3 > set @ product_prod @ (X0 > X2) @ (X1 > X3))).
% 3.24/1.67 thf(func_def_52, type, prod_rel: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ X2 @ X3 > set @ product_prod @ product_prod @ X0 @ X2 @ product_prod @ X1 @ X3)).
% 3.24/1.67 thf(func_def_53, type, relAPP: !>[X0: $tType, X1: $tType, X2: $tType]:((set @ product_prod @ X0 @ X1 > X2) > set @ product_prod @ X0 @ X1 > X2)).
% 3.24/1.67 thf(func_def_54, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.24/1.67 thf(func_def_55, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.24/1.67 thf(func_def_56, type, r1: set @ product_prod @ a @ c).
% 3.24/1.67 thf(func_def_57, type, r2: set @ product_prod @ b @ d).
% 3.24/1.67 thf(func_def_72, type, sK0: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.24/1.67 thf(func_def_73, type, sK1: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.24/1.67 thf(func_def_74, type, sK2: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.24/1.67 thf(func_def_75, type, sK3: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.24/1.67 thf(func_def_76, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 3.24/1.67 thf(func_def_77, type, sK5: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 3.24/1.67 thf(func_def_78, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 3.24/1.67 thf(func_def_79, type, sK7: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > set @ product_prod @ X0 @ X1 > X0)).
% 3.24/1.67 thf(func_def_80, type, sK8: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > set @ product_prod @ X0 @ X1 > X1)).
% 3.24/1.67 thf(func_def_81, type, sK9: !>[X0: $tType, X1: $tType]:(((X1 > X0) > product_prod @ X1 @ X1 > $o) > X1 > X0)).
% 3.24/1.67 thf(func_def_82, type, sK10: !>[X0: $tType, X1: $tType]:(((X1 > X0) > product_prod @ X1 @ X1 > $o) > X1)).
% 3.24/1.67 thf(func_def_83, type, sK11: !>[X0: $tType, X1: $tType]:(((X1 > X0) > product_prod @ X1 @ X1 > $o) > X1)).
% 3.24/1.67 thf(func_def_84, type, sK12: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.24/1.67 thf(func_def_85, type, sK13: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.24/1.67 thf(func_def_86, type, sK14: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 3.24/1.67 thf(func_def_87, type, sK15: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 3.24/1.67 thf(func_def_88, type, sK16: !>[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)).
% 3.24/1.67 thf(func_def_89, type, sK17: !>[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)).
% 3.24/1.67 thf(func_def_90, type, sK18: !>[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)).
% 3.24/1.67 thf(func_def_91, type, sK19: !>[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)).
% 3.24/1.67 thf(func_def_92, type, sK20: !>[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)).
% 3.24/1.67 thf(func_def_93, type, sK21: !>[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)).
% 3.24/1.67 thf(func_def_94, type, sK22: !>[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)).
% 3.24/1.67 thf(func_def_95, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X1 > X0) > (X2 > X1 > X0) > product_prod @ X2 @ X1 > X2)).
% 3.24/1.67 thf(func_def_96, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X1 > X0) > (X2 > X1 > X0) > product_prod @ X2 @ X1 > X1)).
% 3.24/1.67 thf(func_def_97, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.24/1.67 thf(func_def_98, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.24/1.67 thf(func_def_99, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.24/1.67 thf(func_def_100, type, sK28: !>[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)).
% 3.24/1.67 thf(func_def_101, type, sK29: !>[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)).
% 3.24/1.67 thf(func_def_102, type, sK30: !>[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)).
% 3.24/1.67 thf(func_def_103, type, sK31: !>[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)).
% 3.24/1.67 thf(func_def_104, type, sK32: !>[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)).
% 3.24/1.67 thf(func_def_105, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 3.24/1.67 thf(func_def_106, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 3.24/1.67 thf(func_def_107, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 3.24/1.67 thf(func_def_108, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 3.24/1.67 thf(func_def_109, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 3.24/1.67 thf(func_def_110, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(set @ product_prod @ X3 @ X2 > (X1 > X2) > (X0 > X3) > set @ product_prod @ X0 @ X1 > X0)).
% 3.24/1.67 thf(func_def_111, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(set @ product_prod @ X3 @ X2 > (X1 > X2) > (X0 > X3) > set @ product_prod @ X0 @ X1 > X1)).
% 3.24/1.67 thf(func_def_112, type, sK40: !>[X0: $tType, X1: $tType]:((X1 > refine1665802226e_nres @ X0) > (X1 > refine1665802226e_nres @ X0) > refine1665802226e_nres @ X1 > X1)).
% 3.24/1.67 thf(func_def_113, type, sK41: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.24/1.67 thf(func_def_114, type, sK42: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.24/1.67 thf(func_def_115, type, sK43: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.24/1.67 thf(func_def_116, type, sK44: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.24/1.67 thf(func_def_117, type, sK45: !>[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)).
% 3.24/1.67 thf(func_def_118, type, sK46: !>[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)).
% 3.24/1.67 thf(func_def_119, type, sK47: !>[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)).
% 3.24/1.67 thf(func_def_120, type, sK48: !>[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)).
% 3.24/1.67 thf(func_def_121, type, sK49: !>[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)).
% 3.24/1.67 thf(func_def_122, type, sK50: !>[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)).
% 3.24/1.67 thf(func_def_123, type, sK51: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.24/1.67 thf(func_def_124, type, sK52: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.24/1.67 thf(func_def_125, type, sK53: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.24/1.67 thf(func_def_126, type, sK54: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.24/1.67 thf(func_def_127, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.24/1.67 thf(func_def_128, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.24/1.67 thf(func_def_129, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.24/1.67 thf(func_def_130, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.24/1.67 thf(func_def_131, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 3.24/1.67 thf(func_def_132, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 3.24/1.67 thf(func_def_133, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 3.24/1.67 thf(func_def_134, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 3.24/1.67 thf(func_def_135, type, sK63: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.24/1.67 thf(func_def_136, type, sK64: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.24/1.67 thf(func_def_137, type, sK65: !>[X0: $tType, X1: $tType]:((X1 > refine1665802226e_nres @ X0) > (X1 > refine1665802226e_nres @ X0) > refine1665802226e_nres @ X1 > X1)).
% 3.24/1.67 thf(func_def_138, type, sK66: !>[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)).
% 3.24/1.67 thf(func_def_139, type, sK67: !>[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)).
% 3.24/1.67 thf(func_def_140, type, sK68: !>[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)).
% 3.24/1.67 thf(func_def_141, type, sK69: !>[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)).
% 3.24/1.67 thf(func_def_142, type, sK70: !>[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)).
% 3.24/1.67 thf(func_def_143, type, sK71: !>[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)).
% 3.24/1.67 thf(func_def_144, type, sK72: !>[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)).
% 3.24/1.67 thf(func_def_145, type, sK73: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X1)).
% 3.24/1.67 thf(func_def_146, type, sK74: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X0)).
% 3.24/1.67 thf(func_def_147, type, sK75: !>[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)).
% 3.24/1.67 thf(func_def_148, type, sK76: !>[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)).
% 3.24/1.67 thf(func_def_149, type, sK77: !>[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)).
% 3.24/1.67 thf(func_def_150, type, sK78: !>[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)).
% 3.24/1.67 thf(func_def_151, type, sK79: !>[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)).
% 3.24/1.67 thf(func_def_152, type, sK80: !>[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)).
% 3.24/1.67 thf(func_def_153, type, sK81: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.24/1.67 thf(func_def_154, type, sK82: !>[X0: $tType, X1: $tType]:(product_prod @ (X1 > X0) @ product_prod @ X1 @ X1 > X1 > X0)).
% 3.24/1.67 thf(func_def_155, type, sK83: !>[X0: $tType, X1: $tType]:(product_prod @ (X1 > X0) @ product_prod @ X1 @ X1 > X1)).
% 3.24/1.67 thf(func_def_156, type, sK84: !>[X0: $tType, X1: $tType]:(product_prod @ (X1 > X0) @ product_prod @ X1 @ X1 > X1)).
% 3.24/1.67 thf(func_def_158, type, ph86: !>[X0: $tType]:(X0)).
% 3.24/1.67 thf(f890,plain,(
% 3.24/1.67 $false),
% 3.24/1.67 inference(subsumption_resolution,[status(thm)],[f830,f860])).
% 3.24/1.67 thf(f860,plain,(
% 3.24/1.67 ( ! [X1 : $tType,X0 : $tType,X3 : $tType,X2 : $tType,X4 : set @ product_prod @ X1 @ X0,X5 : set @ product_prod @ X3 @ X2] : (($true = (member @ product_prod @ (refine1665802226e_nres @ X1 > (X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ (product_Pair @ (refine1665802226e_nres @ X1 > (X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ refine463715084e_bind @ X1 @ X3 @ refine463715084e_bind @ X0 @ X2) @ (relAPP @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ set @ product_prod @ (refine1665802226e_nres @ X1 > (X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ (relAPP @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X0 @ (set @ product_prod @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) > set @ product_prod @ (refine1665802226e_nres @ X1 > (X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2)) @ fun_rel @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X0 @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ (relAPP @ X1 @ X0 @ set @ product_prod @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X0 @ refine476890328es_rel @ X1 @ X0 @ X4)) @ (relAPP @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ set @ product_prod @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ (relAPP @ (X1 > refine1665802226e_nres @ X3) @ (X0 > refine1665802226e_nres @ X2) @ (set @ product_prod @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 > set @ product_prod @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2)) @ fun_rel @ (X1 > refine1665802226e_nres @ X3) @ (X0 > refine1665802226e_nres @ X2) @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ (relAPP @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ set @ product_prod @ (X1 > refine1665802226e_nres @ X3) @ (X0 > refine1665802226e_nres @ X2) @ (relAPP @ X1 @ X0 @ (set @ product_prod @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 > set @ product_prod @ (X1 > refine1665802226e_nres @ X3) @ (X0 > refine1665802226e_nres @ X2)) @ fun_rel @ X1 @ X0 @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ X4) @ (relAPP @ X3 @ X2 @ set @ product_prod @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ refine476890328es_rel @ X3 @ X2 @ X5))) @ (relAPP @ X3 @ X2 @ set @ product_prod @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ refine476890328es_rel @ X3 @ X2 @ X5)))))) )),
% 3.24/1.67 inference(cnf_transformation,[status(thm)],[f339])).
% 3.24/1.67 thf(f339,plain,(
% 3.24/1.67 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : $tType,X4 : set @ product_prod @ X1 @ X0,X5 : set @ product_prod @ X3 @ X2] : ($true = (member @ product_prod @ (refine1665802226e_nres @ X1 > (X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ (product_Pair @ (refine1665802226e_nres @ X1 > (X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ refine463715084e_bind @ X1 @ X3 @ refine463715084e_bind @ X0 @ X2) @ (relAPP @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ set @ product_prod @ (refine1665802226e_nres @ X1 > (X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ (relAPP @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X0 @ (set @ product_prod @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) > set @ product_prod @ (refine1665802226e_nres @ X1 > (X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2)) @ fun_rel @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X0 @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ (relAPP @ X1 @ X0 @ set @ product_prod @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X0 @ refine476890328es_rel @ X1 @ X0 @ X4)) @ (relAPP @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ set @ product_prod @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ (relAPP @ (X1 > refine1665802226e_nres @ X3) @ (X0 > refine1665802226e_nres @ X2) @ (set @ product_prod @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 > set @ product_prod @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2)) @ fun_rel @ (X1 > refine1665802226e_nres @ X3) @ (X0 > refine1665802226e_nres @ X2) @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ (relAPP @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ set @ product_prod @ (X1 > refine1665802226e_nres @ X3) @ (X0 > refine1665802226e_nres @ X2) @ (relAPP @ X1 @ X0 @ (set @ product_prod @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 > set @ product_prod @ (X1 > refine1665802226e_nres @ X3) @ (X0 > refine1665802226e_nres @ X2)) @ fun_rel @ X1 @ X0 @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ X4) @ (relAPP @ X3 @ X2 @ set @ product_prod @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ refine476890328es_rel @ X3 @ X2 @ X5))) @ (relAPP @ X3 @ X2 @ set @ product_prod @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ refine476890328es_rel @ X3 @ X2 @ X5)))))),
% 3.24/1.67 inference(fool_elimination,[status(thm)],[f338])).
% 3.24/1.67 thf(f338,plain,(
% 3.24/1.67 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : $tType,X4 : set @ product_prod @ X1 @ X0,X5 : set @ product_prod @ X3 @ X2] : (member @ product_prod @ (refine1665802226e_nres @ X1 > (X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ (product_Pair @ (refine1665802226e_nres @ X1 > (X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ refine463715084e_bind @ X1 @ X3 @ refine463715084e_bind @ X0 @ X2) @ (relAPP @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ set @ product_prod @ (refine1665802226e_nres @ X1 > (X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ (relAPP @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X0 @ (set @ product_prod @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) > set @ product_prod @ (refine1665802226e_nres @ X1 > (X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2)) @ fun_rel @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X0 @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ (relAPP @ X1 @ X0 @ set @ product_prod @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X0 @ refine476890328es_rel @ X1 @ X0 @ X4)) @ (relAPP @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ set @ product_prod @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2) @ (relAPP @ (X1 > refine1665802226e_nres @ X3) @ (X0 > refine1665802226e_nres @ X2) @ (set @ product_prod @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 > set @ product_prod @ ((X1 > refine1665802226e_nres @ X3) > refine1665802226e_nres @ X3) @ ((X0 > refine1665802226e_nres @ X2) > refine1665802226e_nres @ X2)) @ fun_rel @ (X1 > refine1665802226e_nres @ X3) @ (X0 > refine1665802226e_nres @ X2) @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ (relAPP @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ set @ product_prod @ (X1 > refine1665802226e_nres @ X3) @ (X0 > refine1665802226e_nres @ X2) @ (relAPP @ X1 @ X0 @ (set @ product_prod @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 > set @ product_prod @ (X1 > refine1665802226e_nres @ X3) @ (X0 > refine1665802226e_nres @ X2)) @ fun_rel @ X1 @ X0 @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ X4) @ (relAPP @ X3 @ X2 @ set @ product_prod @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ refine476890328es_rel @ X3 @ X2 @ X5))) @ (relAPP @ X3 @ X2 @ set @ product_prod @ refine1665802226e_nres @ X3 @ refine1665802226e_nres @ X2 @ refine476890328es_rel @ X3 @ X2 @ X5))))),
% 3.24/1.67 inference(rectify,[status(thm)],[f1])).
% 3.24/1.67 thf(f1,axiom,(
% 3.24/1.67 ! [X2 : $tType,X0 : $tType,X4 : $tType,X1 : $tType,X7 : set @ product_prod @ X0 @ X2,X8 : set @ product_prod @ X1 @ X4] : (member @ product_prod @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X1) @ (refine1665802226e_nres @ X2 > (X2 > refine1665802226e_nres @ X4) > refine1665802226e_nres @ X4) @ (product_Pair @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X1) @ (refine1665802226e_nres @ X2 > (X2 > refine1665802226e_nres @ X4) > refine1665802226e_nres @ X4) @ refine463715084e_bind @ X0 @ X1 @ refine463715084e_bind @ X2 @ X4) @ (relAPP @ ((X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X1) @ ((X2 > refine1665802226e_nres @ X4) > refine1665802226e_nres @ X4) @ set @ product_prod @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X1) @ (refine1665802226e_nres @ X2 > (X2 > refine1665802226e_nres @ X4) > refine1665802226e_nres @ X4) @ (relAPP @ refine1665802226e_nres @ X0 @ refine1665802226e_nres @ X2 @ (set @ product_prod @ ((X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X1) @ ((X2 > refine1665802226e_nres @ X4) > refine1665802226e_nres @ X4) > set @ product_prod @ (refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X1) @ (refine1665802226e_nres @ X2 > (X2 > refine1665802226e_nres @ X4) > refine1665802226e_nres @ X4)) @ fun_rel @ refine1665802226e_nres @ X0 @ refine1665802226e_nres @ X2 @ ((X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X1) @ ((X2 > refine1665802226e_nres @ X4) > refine1665802226e_nres @ X4) @ (relAPP @ X0 @ X2 @ set @ product_prod @ refine1665802226e_nres @ X0 @ refine1665802226e_nres @ X2 @ refine476890328es_rel @ X0 @ X2 @ X7)) @ (relAPP @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X4 @ set @ product_prod @ ((X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X1) @ ((X2 > refine1665802226e_nres @ X4) > refine1665802226e_nres @ X4) @ (relAPP @ (X0 > refine1665802226e_nres @ X1) @ (X2 > refine1665802226e_nres @ X4) @ (set @ product_prod @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X4 > set @ product_prod @ ((X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X1) @ ((X2 > refine1665802226e_nres @ X4) > refine1665802226e_nres @ X4)) @ fun_rel @ (X0 > refine1665802226e_nres @ X1) @ (X2 > refine1665802226e_nres @ X4) @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X4 @ (relAPP @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X4 @ set @ product_prod @ (X0 > refine1665802226e_nres @ X1) @ (X2 > refine1665802226e_nres @ X4) @ (relAPP @ X0 @ X2 @ (set @ product_prod @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X4 > set @ product_prod @ (X0 > refine1665802226e_nres @ X1) @ (X2 > refine1665802226e_nres @ X4)) @ fun_rel @ X0 @ X2 @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X4 @ X7) @ (relAPP @ X1 @ X4 @ set @ product_prod @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X4 @ refine476890328es_rel @ X1 @ X4 @ X8))) @ (relAPP @ X1 @ X4 @ set @ product_prod @ refine1665802226e_nres @ X1 @ refine1665802226e_nres @ X4 @ refine476890328es_rel @ X1 @ X4 @ X8))))),
% 3.24/1.67 file('/export/starexec/sandbox/tmp/tmp.LLLxuS3oP5/DTF2THF_2166.p',fact_0_param__bind)).
% 3.24/1.67 thf(f830,plain,(
% 3.24/1.67 ((member @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (product_Pair @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ refine463715084e_bind @ a @ b @ refine463715084e_bind @ c @ d) @ (relAPP @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ set @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ (set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) > set @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d)) @ fun_rel @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ a @ c @ set @ product_prod @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ refine476890328es_rel @ a @ c @ r1)) @ (relAPP @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ (set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d > set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d)) @ fun_rel @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ (relAPP @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ set @ product_prod @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ (relAPP @ a @ c @ (set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d > set @ product_prod @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d)) @ fun_rel @ a @ c @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ r1) @ (relAPP @ b @ d @ set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ refine476890328es_rel @ b @ d @ r2))) @ (relAPP @ b @ d @ set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ refine476890328es_rel @ b @ d @ r2)))) != $true)),
% 3.24/1.67 inference(cnf_transformation,[status(thm)],[f642])).
% 3.24/1.67 thf(f642,plain,(
% 3.24/1.67 ((member @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (product_Pair @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ refine463715084e_bind @ a @ b @ refine463715084e_bind @ c @ d) @ (relAPP @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ set @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ (set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) > set @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d)) @ fun_rel @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ a @ c @ set @ product_prod @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ refine476890328es_rel @ a @ c @ r1)) @ (relAPP @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ (set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d > set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d)) @ fun_rel @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ (relAPP @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ set @ product_prod @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ (relAPP @ a @ c @ (set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d > set @ product_prod @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d)) @ fun_rel @ a @ c @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ r1) @ (relAPP @ b @ d @ set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ refine476890328es_rel @ b @ d @ r2))) @ (relAPP @ b @ d @ set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ refine476890328es_rel @ b @ d @ r2)))) != $true)),
% 3.24/1.67 inference(flattening,[status(thm)],[f415])).
% 3.24/1.67 thf(f415,plain,(
% 3.24/1.67 ~((member @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (product_Pair @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ refine463715084e_bind @ a @ b @ refine463715084e_bind @ c @ d) @ (relAPP @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ set @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ (set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) > set @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d)) @ fun_rel @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ a @ c @ set @ product_prod @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ refine476890328es_rel @ a @ c @ r1)) @ (relAPP @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ (set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d > set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d)) @ fun_rel @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ (relAPP @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ set @ product_prod @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ (relAPP @ a @ c @ (set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d > set @ product_prod @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d)) @ fun_rel @ a @ c @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ r1) @ (relAPP @ b @ d @ set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ refine476890328es_rel @ b @ d @ r2))) @ (relAPP @ b @ d @ set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ refine476890328es_rel @ b @ d @ r2)))) = $true)),
% 3.24/1.67 inference(fool_elimination,[status(thm)],[f414])).
% 3.24/1.67 thf(f414,plain,(
% 3.24/1.67 ~(member @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (product_Pair @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ refine463715084e_bind @ a @ b @ refine463715084e_bind @ c @ d) @ (relAPP @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ set @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ (set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) > set @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d)) @ fun_rel @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ a @ c @ set @ product_prod @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ refine476890328es_rel @ a @ c @ r1)) @ (relAPP @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ (set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d > set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d)) @ fun_rel @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ (relAPP @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ set @ product_prod @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ (relAPP @ a @ c @ (set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d > set @ product_prod @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d)) @ fun_rel @ a @ c @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ r1) @ (relAPP @ b @ d @ set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ refine476890328es_rel @ b @ d @ r2))) @ (relAPP @ b @ d @ set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ refine476890328es_rel @ b @ d @ r2))))),
% 3.24/1.67 inference(rectify,[status(thm)],[f278])).
% 3.24/1.67 thf(f278,negated_conjecture,(
% 3.24/1.67 ~(member @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (product_Pair @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ refine463715084e_bind @ a @ b @ refine463715084e_bind @ c @ d) @ (relAPP @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ set @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ (set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) > set @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d)) @ fun_rel @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ a @ c @ set @ product_prod @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ refine476890328es_rel @ a @ c @ r1)) @ (relAPP @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ (set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d > set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d)) @ fun_rel @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ (relAPP @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ set @ product_prod @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ (relAPP @ a @ c @ (set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d > set @ product_prod @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d)) @ fun_rel @ a @ c @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ r1) @ (relAPP @ b @ d @ set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ refine476890328es_rel @ b @ d @ r2))) @ (relAPP @ b @ d @ set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ refine476890328es_rel @ b @ d @ r2))))),
% 3.24/1.67 inference(negated_conjecture,[status(cth)],[f277])).
% 3.24/1.67 thf(f277,conjecture,(
% 3.24/1.67 (member @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (product_Pair @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ refine463715084e_bind @ a @ b @ refine463715084e_bind @ c @ d) @ (relAPP @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ set @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ (set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) > set @ product_prod @ (refine1665802226e_nres @ a > (a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ (refine1665802226e_nres @ c > (c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d)) @ fun_rel @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ a @ c @ set @ product_prod @ refine1665802226e_nres @ a @ refine1665802226e_nres @ c @ refine476890328es_rel @ a @ c @ r1)) @ (relAPP @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d) @ (relAPP @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ (set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d > set @ product_prod @ ((a > refine1665802226e_nres @ b) > refine1665802226e_nres @ b) @ ((c > refine1665802226e_nres @ d) > refine1665802226e_nres @ d)) @ fun_rel @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ (relAPP @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ set @ product_prod @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d) @ (relAPP @ a @ c @ (set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d > set @ product_prod @ (a > refine1665802226e_nres @ b) @ (c > refine1665802226e_nres @ d)) @ fun_rel @ a @ c @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ r1) @ (relAPP @ b @ d @ set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ refine476890328es_rel @ b @ d @ r2))) @ (relAPP @ b @ d @ set @ product_prod @ refine1665802226e_nres @ b @ refine1665802226e_nres @ d @ refine476890328es_rel @ b @ d @ r2))))),
% 3.24/1.67 file('/export/starexec/sandbox/tmp/tmp.LLLxuS3oP5/DTF2THF_2166.p',conj_0)).
% 3.24/1.67 % SZS output end Proof for DTF2THF_2166
% 3.24/1.67 % (2365)------------------------------
% 3.24/1.67 % (2365)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.67 % (2365)Termination reason: Refutation
% 3.24/1.67
% 3.24/1.67 % (2365)Memory used [KB]: 8059
% 3.24/1.67 % (2365)Time elapsed: 0.047 s
% 3.24/1.67 % (2365)Instructions burned: 140 (million)
% 3.24/1.67 % (2365)------------------------------
% 3.24/1.67 % (2365)------------------------------
% 3.24/1.67 % (2358)Success in time 0.067 s
%------------------------------------------------------------------------------