%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT222^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n012.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 08:45:12 AM UTC 2026
% Result : Theorem 8.05s 2.05s
% Output : Refutation 8.05s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : DAT222^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.07 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.07/0.26 % Computer : n012.cluster.edu
% 0.07/0.26 % Model : x86_64 x86_64
% 0.07/0.26 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.26 % Memory : 8042.1875MB
% 0.07/0.26 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.26 % CPULimit : 300
% 0.07/0.26 % WCLimit : 300
% 0.07/0.26 % DateTime : Tue Feb 24 06:11:00 EST 2026
% 0.07/0.26 % CPUTime :
% 0.07/0.26 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.07/0.32 ---- Original DTF file ---
% 0.16/0.33 thf(spec,logic,$$dhol).
% 0.16/0.33 %------------------------------------------------------------------------------
% 0.16/0.33 % File : DAT222^1 : TPTP v9.2.1. Released v7.0.0.
% 0.16/0.33 % Domain : Data Structures
% 0.16/0.33 % Problem : Tllist 370
% 0.16/0.33 % Version : [Bla16] axioms : Especial.
% 0.16/0.33 % English :
% 0.16/0.33
% 0.16/0.33 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.16/0.33 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.16/0.33 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.16/0.33 % Source : [Bla16]
% 0.16/0.33 % Names : tllist__370.p [Bla16]
% 0.16/0.33
% 0.16/0.33 % Status : Theorem
% 0.16/0.33 % Rating : 0.67 v8.1.0, 0.75 v7.5.0, 1.00 v7.1.0
% 0.16/0.33 % Syntax : Number of formulae : 321 ( 156 unt; 58 typ; 0 def)
% 0.16/0.33 % Number of atoms : 565 ( 315 equ; 0 cnn)
% 0.16/0.33 % Maximal formula atoms : 7 ( 2 avg)
% 0.16/0.33 % Number of connectives : 4819 ( 42 ~; 3 |; 68 &;4466 @)
% 0.16/0.33 % ( 0 <=>; 240 =>; 0 <=; 0 <~>)
% 0.16/0.33 % Maximal formula depth : 28 ( 9 avg)
% 0.16/0.33 % Number of types : 4 ( 3 usr)
% 0.16/0.33 % Number of type conns : 562 ( 562 >; 0 *; 0 +; 0 <<)
% 0.16/0.33 % Number of symbols : 58 ( 55 usr; 5 con; 0-7 aty)
% 0.16/0.33 % Number of variables : 1657 ( 255 ^;1304 !; 17 ?;1657 :)
% 0.16/0.33 % ( 81 !>; 0 ?*; 0 @-; 0 @+)
% 0.16/0.33 % SPC : TH1_THM_EQU_NAR
% 0.16/0.33
% 0.16/0.33 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.16/0.33 % 2016-07-13 14:51:42.139
% 0.16/0.33 %------------------------------------------------------------------------------
% 0.16/0.33 %----Could-be-implicit typings (8)
% 0.16/0.33 thf(ty_t_TLList__Mirabelle__qhjoikztpd_Otllist,type,
% 0.16/0.33 tLList446370796tllist: $tType > $tType > $tType ).
% 0.16/0.33
% 0.16/0.33 thf(ty_t_Coinductive__List_Ollist,type,
% 0.16/0.33 coinductive_llist: $tType > $tType ).
% 0.16/0.33
% 0.16/0.33 thf(ty_t_Product__Type_Ounit,type,
% 0.16/0.33 product_unit: $tType ).
% 0.16/0.33
% 0.16/0.33 thf(ty_t_Product__Type_Oprod,type,
% 0.16/0.33 product_prod: $tType > $tType > $tType ).
% 0.16/0.33
% 0.16/0.33 thf(ty_t_Set_Oset,type,
% 0.16/0.33 set: $tType > $tType ).
% 0.16/0.33
% 0.16/0.33 thf(ty_t_itself,type,
% 0.16/0.33 itself: $tType > $tType ).
% 0.16/0.33
% 0.16/0.33 thf(ty_tf_b,type,
% 0.16/0.33 b: $tType ).
% 0.16/0.33
% 0.16/0.33 thf(ty_tf_a,type,
% 0.16/0.33 a: $tType ).
% 0.16/0.33
% 0.16/0.33 %----Explicit typings (50)
% 0.16/0.33 thf(sy_cl_Orderings_Olinorder,type,
% 0.16/0.33 linorder:
% 0.16/0.33 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_BNF__Def_OGr,type,
% 0.16/0.33 bNF_Gr:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Coinductive__List_Olfilter,type,
% 0.16/0.33 coinductive_lfilter:
% 0.16/0.33 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.16/0.33 coinductive_lfinite:
% 0.16/0.33 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Fun__Def_Oreduction__pair,type,
% 0.16/0.33 fun_reduction_pair:
% 0.16/0.33 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 0.16/0.33 fun_rp_inv_image:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_HOL_OThe,type,
% 0.16/0.33 the:
% 0.16/0.33 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_If,type,
% 0.16/0.33 if:
% 0.16/0.33 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Lifting_OQuotient,type,
% 0.16/0.33 quotient:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Order__Relation_Oabove,type,
% 0.16/0.33 order_above:
% 0.16/0.33 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Order__Relation_OaboveS,type,
% 0.16/0.33 order_aboveS:
% 0.16/0.33 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.16/0.33 ord_less_eq:
% 0.16/0.33 !>[A: $tType] : ( A > A > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.16/0.33 top_top:
% 0.16/0.33 !>[A: $tType] : A ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_OPair,type,
% 0.16/0.33 product_Pair:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_OSigma,type,
% 0.16/0.33 product_Sigma:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.16/0.33 produc2004651681e_prod:
% 0.16/0.33 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Omap__prod,type,
% 0.16/0.33 product_map_prod:
% 0.16/0.33 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 0.16/0.33 product_rec_bool:
% 0.16/0.33 !>[T: $tType] : ( T > T > $o > T ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
% 0.16/0.33 product_rec_set_bool:
% 0.16/0.33 !>[T: $tType] : ( T > T > $o > T > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.16/0.33 product_rec_prod:
% 0.16/0.33 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 0.16/0.33 product_rec_set_prod:
% 0.16/0.33 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 0.16/0.33 product_rec_set_unit:
% 0.16/0.33 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 0.16/0.33 product_rec_unit:
% 0.16/0.33 !>[T: $tType] : ( T > product_unit > T ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.16/0.33 product_case_prod:
% 0.16/0.33 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.16/0.33 product_fst:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.16/0.33 product_snd:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.16/0.33 product_swap:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Oproduct,type,
% 0.16/0.33 product_product:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Product__Type_Oscomp,type,
% 0.16/0.33 product_scomp:
% 0.16/0.33 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Pure_Otype,type,
% 0.16/0.33 type:
% 0.16/0.33 !>[A: $tType] : ( itself @ A ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Relation_Oconversep,type,
% 0.16/0.33 conversep:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > A > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Relation_Oinv__image,type,
% 0.16/0.33 inv_image:
% 0.16/0.33 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Relation_Oinv__imagep,type,
% 0.16/0.33 inv_imagep:
% 0.16/0.33 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Relation_Orefl__on,type,
% 0.16/0.33 refl_on:
% 0.16/0.33 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Relation_Oreflp,type,
% 0.16/0.33 reflp:
% 0.16/0.33 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Set_OCollect,type,
% 0.16/0.33 collect:
% 0.16/0.33 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Set_Oimage,type,
% 0.16/0.33 image:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Ocr__tllist,type,
% 0.16/0.33 tLList47617868tllist:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( product_prod @ ( coinductive_llist @ A ) @ B ) > ( tLList446370796tllist @ A @ B ) > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Ollist__of__tllist,type,
% 0.16/0.33 tLList798109904tllist:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > ( coinductive_llist @ A ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Oterminal0,type,
% 0.16/0.33 tLList1825092077minal0:
% 0.16/0.33 !>[A: $tType] : A ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Ocase__tllist,type,
% 0.16/0.33 tLList200813139tllist:
% 0.16/0.33 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > ( tLList446370796tllist @ A @ B ) > C ) > ( tLList446370796tllist @ A @ B ) > C ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Oterminal,type,
% 0.16/0.33 tLList2110128105rminal:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > B ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist__of__llist,type,
% 0.16/0.33 tLList1672613558_llist:
% 0.16/0.33 !>[B: $tType,A: $tType] : ( B > ( coinductive_llist @ A ) > ( tLList446370796tllist @ A @ B ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Wellfounded_Olex__prod,type,
% 0.16/0.33 lex_prod:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_Wfrec_Osame__fst,type,
% 0.16/0.33 same_fst:
% 0.16/0.33 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_c_member,type,
% 0.16/0.33 member:
% 0.16/0.33 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(sy_v_b,type,
% 0.16/0.33 b2: b ).
% 0.16/0.33
% 0.16/0.33 thf(sy_v_fun,type,
% 0.16/0.33 fun: a > $o ).
% 0.16/0.33
% 0.16/0.33 thf(sy_v_prod1,type,
% 0.16/0.33 prod1: product_prod @ ( coinductive_llist @ a ) @ b ).
% 0.16/0.33
% 0.16/0.33 thf(sy_v_prod2,type,
% 0.16/0.33 prod2: product_prod @ ( coinductive_llist @ a ) @ b ).
% 0.16/0.33
% 0.16/0.33 %----Relevant facts (256)
% 0.16/0.33 thf(fact_0_case__prodI2_H,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,C: $tType,P: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
% 0.16/0.33 ( ! [A2: A,B2: B] :
% 0.16/0.33 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.16/0.33 = P )
% 0.16/0.33 => ( C2 @ A2 @ B2 @ X ) )
% 0.16/0.33 => ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prodI2'
% 0.16/0.33 thf(fact_1_case__prodI,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
% 0.16/0.33 ( ( F @ A3 @ B3 )
% 0.16/0.33 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prodI
% 0.16/0.33 thf(fact_2_case__prodI2,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P: product_prod @ A @ B,C2: A > B > $o] :
% 0.16/0.33 ( ! [A2: A,B2: B] :
% 0.16/0.33 ( ( P
% 0.16/0.33 = ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.16/0.33 => ( C2 @ A2 @ B2 ) )
% 0.16/0.33 => ( product_case_prod @ A @ B @ $o @ C2 @ P ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prodI2
% 0.16/0.33 thf(fact_3_case__prod__conv,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B3: C] :
% 0.16/0.33 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 0.16/0.33 = ( F @ A3 @ B3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prod_conv
% 0.16/0.33 thf(fact_4_lfilter__K__True,axiom,
% 0.16/0.33 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.16/0.33 ( ( coinductive_lfilter @ A
% 0.16/0.33 @ ^ [Uu: A] : $true
% 0.16/0.33 @ Xs )
% 0.16/0.33 = Xs ) ).
% 0.16/0.33
% 0.16/0.33 % lfilter_K_True
% 0.16/0.33 thf(fact_5_split__part,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P2: $o,Q: A > B > $o] :
% 0.16/0.33 ( ( product_case_prod @ A @ B @ $o
% 0.16/0.33 @ ^ [A4: A,B4: B] :
% 0.16/0.33 ( P2
% 0.16/0.33 & ( Q @ A4 @ B4 ) ) )
% 0.16/0.33 = ( ^ [Ab: product_prod @ A @ B] :
% 0.16/0.33 ( P2
% 0.16/0.33 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % split_part
% 0.16/0.33 thf(fact_6_lfilter__idem,axiom,
% 0.16/0.33 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 0.16/0.33 ( ( coinductive_lfilter @ A @ P2 @ ( coinductive_lfilter @ A @ P2 @ Xs ) )
% 0.16/0.33 = ( coinductive_lfilter @ A @ P2 @ Xs ) ) ).
% 0.16/0.33
% 0.16/0.33 % lfilter_idem
% 0.16/0.33 thf(fact_7_case__prodD_H,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A3: A,B3: B,C2: C] :
% 0.16/0.33 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A3 @ B3 ) @ C2 )
% 0.16/0.33 => ( R @ A3 @ B3 @ C2 ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prodD'
% 0.16/0.33 thf(fact_8_case__prodE_H,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P: product_prod @ A @ B,Z: C] :
% 0.16/0.33 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P @ Z )
% 0.16/0.33 => ~ ! [X2: A,Y: B] :
% 0.16/0.33 ( ( P
% 0.16/0.33 = ( product_Pair @ A @ B @ X2 @ Y ) )
% 0.16/0.33 => ~ ( C2 @ X2 @ Y @ Z ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prodE'
% 0.16/0.33 thf(fact_9_case__prod__Pair__iden,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 0.16/0.33 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P )
% 0.16/0.33 = P ) ).
% 0.16/0.33
% 0.16/0.33 % case_prod_Pair_iden
% 0.16/0.33 thf(fact_10_prod_Oinject,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.16/0.33 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.16/0.33 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.16/0.33 = ( ( X1 = Y1 )
% 0.16/0.33 & ( X22 = Y2 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod.inject
% 0.16/0.33 thf(fact_11_old_Oprod_Oinject,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B5: B] :
% 0.16/0.33 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.16/0.33 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.16/0.33 = ( ( A3 = A5 )
% 0.16/0.33 & ( B3 = B5 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % old.prod.inject
% 0.16/0.33 thf(fact_12_mem__case__prodI2,axiom,
% 0.16/0.33 ! [C: $tType,B: $tType,A: $tType,P: product_prod @ A @ B,Z: C,C2: A > B > ( set @ C )] :
% 0.16/0.33 ( ! [A2: A,B2: B] :
% 0.16/0.33 ( ( P
% 0.16/0.33 = ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.16/0.33 => ( member @ C @ Z @ ( C2 @ A2 @ B2 ) ) )
% 0.16/0.33 => ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % mem_case_prodI2
% 0.16/0.33 thf(fact_13_mem__case__prodI,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),A3: B,B3: C] :
% 0.16/0.33 ( ( member @ A @ Z @ ( C2 @ A3 @ B3 ) )
% 0.16/0.33 => ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % mem_case_prodI
% 0.16/0.33 thf(fact_14_mem__case__prodE,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),P: product_prod @ B @ C] :
% 0.16/0.33 ( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P ) )
% 0.16/0.33 => ~ ! [X2: B,Y: C] :
% 0.16/0.33 ( ( P
% 0.16/0.33 = ( product_Pair @ B @ C @ X2 @ Y ) )
% 0.16/0.33 => ~ ( member @ A @ Z @ ( C2 @ X2 @ Y ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % mem_case_prodE
% 0.16/0.33 thf(fact_15_old_Oprod_Oinducts,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.16/0.33 ( ! [A2: A,B2: B] : ( P2 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.16/0.33 => ( P2 @ Prod ) ) ).
% 0.16/0.33
% 0.16/0.33 % old.prod.inducts
% 0.16/0.33 thf(fact_16_old_Oprod_Oexhaust,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 0.16/0.33 ~ ! [A2: A,B2: B] :
% 0.16/0.33 ( Y3
% 0.16/0.33 != ( product_Pair @ A @ B @ A2 @ B2 ) ) ).
% 0.16/0.33
% 0.16/0.33 % old.prod.exhaust
% 0.16/0.33 thf(fact_17_prod__induct7,axiom,
% 0.16/0.33 ! [G: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) )] :
% 0.16/0.33 ( ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G ) @ E2 @ ( product_Pair @ F2 @ G @ F3 @ G2 ) ) ) ) ) ) )
% 0.16/0.33 => ( P2 @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_induct7
% 0.16/0.33 thf(fact_18_prod__induct6,axiom,
% 0.16/0.33 ! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.16/0.33 ( ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 0.16/0.33 => ( P2 @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_induct6
% 0.16/0.33 thf(fact_19_prod__induct5,axiom,
% 0.16/0.33 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( 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 ) ) )] :
% 0.16/0.33 ( ! [A2: A,B2: B,C3: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.16/0.33 => ( P2 @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_induct5
% 0.16/0.33 thf(fact_20_prod__induct4,axiom,
% 0.16/0.33 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.16/0.33 ( ! [A2: A,B2: B,C3: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 0.16/0.33 => ( P2 @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_induct4
% 0.16/0.33 thf(fact_21_prod__induct3,axiom,
% 0.16/0.33 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.16/0.33 ( ! [A2: A,B2: B,C3: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C3 ) ) )
% 0.16/0.33 => ( P2 @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_induct3
% 0.16/0.33 thf(fact_22_prod__cases7,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) )] :
% 0.16/0.33 ~ ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2,G2: G] :
% 0.16/0.33 ( Y3
% 0.16/0.33 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G ) @ E2 @ ( product_Pair @ F2 @ G @ F3 @ G2 ) ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_cases7
% 0.16/0.33 thf(fact_23_prod__cases6,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.16/0.33 ~ ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2] :
% 0.16/0.33 ( Y3
% 0.16/0.33 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_cases6
% 0.16/0.33 thf(fact_24_prod__cases5,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.16/0.33 ~ ! [A2: A,B2: B,C3: C,D2: D,E2: E] :
% 0.16/0.33 ( Y3
% 0.16/0.33 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_cases5
% 0.16/0.33 thf(fact_25_prod__cases4,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.16/0.33 ~ ! [A2: A,B2: B,C3: C,D2: D] :
% 0.16/0.33 ( Y3
% 0.16/0.33 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_cases4
% 0.16/0.33 thf(fact_26_prod__cases3,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.16/0.33 ~ ! [A2: A,B2: B,C3: C] :
% 0.16/0.33 ( Y3
% 0.16/0.33 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C3 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_cases3
% 0.16/0.33 thf(fact_27_Pair__inject,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B5: B] :
% 0.16/0.33 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.16/0.33 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.16/0.33 => ~ ( ( A3 = A5 )
% 0.16/0.33 => ( B3 != B5 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Pair_inject
% 0.16/0.33 thf(fact_28_prod__cases,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 0.16/0.33 ( ! [A2: A,B2: B] : ( P2 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.16/0.33 => ( P2 @ P ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_cases
% 0.16/0.33 thf(fact_29_surj__pair,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 0.16/0.33 ? [X2: A,Y: B] :
% 0.16/0.33 ( P
% 0.16/0.33 = ( product_Pair @ A @ B @ X2 @ Y ) ) ).
% 0.16/0.33
% 0.16/0.33 % surj_pair
% 0.16/0.33 thf(fact_30_prod_Ocase__distrib,axiom,
% 0.16/0.33 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.16/0.33 ( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.16/0.33 = ( product_case_prod @ A @ B @ D
% 0.16/0.33 @ ^ [X12: A,X23: B] : ( H @ ( F @ X12 @ X23 ) )
% 0.16/0.33 @ Prod ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod.case_distrib
% 0.16/0.33 thf(fact_31_prod_Odisc__eq__case,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.16/0.33 ( product_case_prod @ A @ B @ $o
% 0.16/0.33 @ ^ [Uu: A,Uv: B] : $true
% 0.16/0.33 @ Prod ) ).
% 0.16/0.33
% 0.16/0.33 % prod.disc_eq_case
% 0.16/0.33 thf(fact_32_lfilter__lfilter,axiom,
% 0.16/0.33 ! [A: $tType,P2: A > $o,Q: A > $o,Xs: coinductive_llist @ A] :
% 0.16/0.33 ( ( coinductive_lfilter @ A @ P2 @ ( coinductive_lfilter @ A @ Q @ Xs ) )
% 0.16/0.33 = ( coinductive_lfilter @ A
% 0.16/0.33 @ ^ [X3: A] :
% 0.16/0.33 ( ( P2 @ X3 )
% 0.16/0.33 & ( Q @ X3 ) )
% 0.16/0.33 @ Xs ) ) ).
% 0.16/0.33
% 0.16/0.33 % lfilter_lfilter
% 0.16/0.33 thf(fact_33_old_Oprod_Ocase,axiom,
% 0.16/0.33 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 0.16/0.33 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.16/0.33 = ( F @ X1 @ X22 ) ) ).
% 0.16/0.33
% 0.16/0.33 % old.prod.case
% 0.16/0.33 thf(fact_34_lfinite__lfilterI,axiom,
% 0.16/0.33 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 0.16/0.33 ( ( coinductive_lfinite @ A @ Xs )
% 0.16/0.33 => ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P2 @ Xs ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % lfinite_lfilterI
% 0.16/0.33 thf(fact_35_cond__case__prod__eta,axiom,
% 0.16/0.33 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G3: ( product_prod @ A @ B ) > C] :
% 0.16/0.33 ( ! [X2: A,Y: B] :
% 0.16/0.33 ( ( F @ X2 @ Y )
% 0.16/0.33 = ( G3 @ ( product_Pair @ A @ B @ X2 @ Y ) ) )
% 0.16/0.33 => ( ( product_case_prod @ A @ B @ C @ F )
% 0.16/0.33 = G3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % cond_case_prod_eta
% 0.16/0.33 thf(fact_36_case__prod__eta,axiom,
% 0.16/0.33 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.16/0.33 ( ( product_case_prod @ A @ B @ C
% 0.16/0.33 @ ^ [X3: A,Y4: B] : ( F @ ( product_Pair @ A @ B @ X3 @ Y4 ) ) )
% 0.16/0.33 = F ) ).
% 0.16/0.33
% 0.16/0.33 % case_prod_eta
% 0.16/0.33 thf(fact_37_case__prodE2,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P2: B > C > A,Z: product_prod @ B @ C] :
% 0.16/0.33 ( ( Q @ ( product_case_prod @ B @ C @ A @ P2 @ Z ) )
% 0.16/0.33 => ~ ! [X2: B,Y: C] :
% 0.16/0.33 ( ( Z
% 0.16/0.33 = ( product_Pair @ B @ C @ X2 @ Y ) )
% 0.16/0.33 => ~ ( Q @ ( P2 @ X2 @ Y ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prodE2
% 0.16/0.33 thf(fact_38_case__prod__app,axiom,
% 0.16/0.33 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.16/0.33 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 0.16/0.33 = ( ^ [F4: B > C > D > A,X3: product_prod @ B @ C,Y4: D] :
% 0.16/0.33 ( product_case_prod @ B @ C @ A
% 0.16/0.33 @ ^ [L: B,R2: C] : ( F4 @ L @ R2 @ Y4 )
% 0.16/0.33 @ X3 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prod_app
% 0.16/0.33 thf(fact_39_case__prodE,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,C2: A > B > $o,P: product_prod @ A @ B] :
% 0.16/0.33 ( ( product_case_prod @ A @ B @ $o @ C2 @ P )
% 0.16/0.33 => ~ ! [X2: A,Y: B] :
% 0.16/0.33 ( ( P
% 0.16/0.33 = ( product_Pair @ A @ B @ X2 @ Y ) )
% 0.16/0.33 => ~ ( C2 @ X2 @ Y ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prodE
% 0.16/0.33 thf(fact_40_case__prodD,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
% 0.16/0.33 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.16/0.33 => ( F @ A3 @ B3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prodD
% 0.16/0.33 thf(fact_41_old_Oprod_Orec,axiom,
% 0.16/0.33 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 0.16/0.33 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.16/0.33 = ( F1 @ A3 @ B3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % old.prod.rec
% 0.16/0.33 thf(fact_42_split__cong,axiom,
% 0.16/0.33 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G3: A > B > C,P: product_prod @ A @ B] :
% 0.16/0.33 ( ! [X2: A,Y: B] :
% 0.16/0.33 ( ( ( product_Pair @ A @ B @ X2 @ Y )
% 0.16/0.33 = Q2 )
% 0.16/0.33 => ( ( F @ X2 @ Y )
% 0.16/0.33 = ( G3 @ X2 @ Y ) ) )
% 0.16/0.33 => ( ( P = Q2 )
% 0.16/0.33 => ( ( product_case_prod @ A @ B @ C @ F @ P )
% 0.16/0.33 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % split_cong
% 0.16/0.33 thf(fact_43_same__fst__def,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( same_fst @ A @ B )
% 0.16/0.33 = ( ^ [P3: A > $o,R3: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.16/0.33 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 0.16/0.33 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 0.16/0.33 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 0.16/0.33 @ ^ [X4: A,Y5: B] :
% 0.16/0.33 ( product_case_prod @ A @ B @ $o
% 0.16/0.33 @ ^ [X3: A,Y4: B] :
% 0.16/0.33 ( ( X4 = X3 )
% 0.16/0.33 & ( P3 @ X3 )
% 0.16/0.33 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y5 @ Y4 ) @ ( R3 @ X3 ) ) ) ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % same_fst_def
% 0.16/0.33 thf(fact_44_lex__prod__def,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( lex_prod @ A @ B )
% 0.16/0.33 = ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
% 0.16/0.33 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 0.16/0.33 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 0.16/0.33 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 0.16/0.33 @ ^ [A4: A,B4: B] :
% 0.16/0.33 ( product_case_prod @ A @ B @ $o
% 0.16/0.33 @ ^ [A6: A,B6: B] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A6 ) @ Ra )
% 0.16/0.33 | ( ( A4 = A6 )
% 0.16/0.33 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % lex_prod_def
% 0.16/0.33 thf(fact_45_mem__Collect__eq,axiom,
% 0.16/0.33 ! [A: $tType,A3: A,P2: A > $o] :
% 0.16/0.33 ( ( member @ A @ A3 @ ( collect @ A @ P2 ) )
% 0.16/0.33 = ( P2 @ A3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % mem_Collect_eq
% 0.16/0.33 thf(fact_46_Collect__mem__eq,axiom,
% 0.16/0.33 ! [A: $tType,A7: set @ A] :
% 0.16/0.33 ( ( collect @ A
% 0.16/0.33 @ ^ [X3: A] : ( member @ A @ X3 @ A7 ) )
% 0.16/0.33 = A7 ) ).
% 0.16/0.33
% 0.16/0.33 % Collect_mem_eq
% 0.16/0.33 thf(fact_47_Collect__cong,axiom,
% 0.16/0.33 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.16/0.33 ( ! [X2: A] :
% 0.16/0.33 ( ( P2 @ X2 )
% 0.16/0.33 = ( Q @ X2 ) )
% 0.16/0.33 => ( ( collect @ A @ P2 )
% 0.16/0.33 = ( collect @ A @ Q ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Collect_cong
% 0.16/0.33 thf(fact_48_ext,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,F: A > B,G3: A > B] :
% 0.16/0.33 ( ! [X2: A] :
% 0.16/0.33 ( ( F @ X2 )
% 0.16/0.33 = ( G3 @ X2 ) )
% 0.16/0.33 => ( F = G3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % ext
% 0.16/0.33 thf(fact_49_internal__case__prod__def,axiom,
% 0.16/0.33 ! [C: $tType,B: $tType,A: $tType] :
% 0.16/0.33 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.16/0.33 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.16/0.33
% 0.16/0.33 % internal_case_prod_def
% 0.16/0.33 thf(fact_50_tllist_Oabs__eq__iff,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,X: product_prod @ ( coinductive_llist @ A ) @ B,Y3: product_prod @ ( coinductive_llist @ A ) @ B] :
% 0.16/0.33 ( ( ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( tLList446370796tllist @ A @ B )
% 0.16/0.33 @ ^ [Xs2: coinductive_llist @ A,A4: B] : ( tLList1672613558_llist @ B @ A @ A4 @ Xs2 )
% 0.16/0.33 @ X )
% 0.16/0.33 = ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( tLList446370796tllist @ A @ B )
% 0.16/0.33 @ ^ [Xs2: coinductive_llist @ A,A4: B] : ( tLList1672613558_llist @ B @ A @ A4 @ Xs2 )
% 0.16/0.33 @ Y3 ) )
% 0.16/0.33 = ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( ( product_prod @ ( coinductive_llist @ A ) @ B ) > $o )
% 0.16/0.33 @ ^ [Xs2: coinductive_llist @ A,A4: B] :
% 0.16/0.33 ( product_case_prod @ ( coinductive_llist @ A ) @ B @ $o
% 0.16/0.33 @ ^ [Ys: coinductive_llist @ A,B4: B] :
% 0.16/0.33 ( ( Xs2 = Ys )
% 0.16/0.33 & ( ( coinductive_lfinite @ A @ Ys )
% 0.16/0.33 => ( A4 = B4 ) ) ) )
% 0.16/0.33 @ X
% 0.16/0.33 @ Y3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % tllist.abs_eq_iff
% 0.16/0.33 thf(fact_51_pred__equals__eq2,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.16/0.33 ( ( ( ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R ) )
% 0.16/0.33 = ( ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S ) ) )
% 0.16/0.33 = ( R = S ) ) ).
% 0.16/0.33
% 0.16/0.33 % pred_equals_eq2
% 0.16/0.33 thf(fact_52_internal__case__prod__conv,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B3: C] :
% 0.16/0.33 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 0.16/0.33 = ( C2 @ A3 @ B3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % internal_case_prod_conv
% 0.16/0.33 thf(fact_53_reflp__tllist,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( reflp @ ( product_prod @ ( coinductive_llist @ A ) @ B )
% 0.16/0.33 @ ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( ( product_prod @ ( coinductive_llist @ A ) @ B ) > $o )
% 0.16/0.33 @ ^ [Xs2: coinductive_llist @ A,A4: B] :
% 0.16/0.33 ( product_case_prod @ ( coinductive_llist @ A ) @ B @ $o
% 0.16/0.33 @ ^ [Ys: coinductive_llist @ A,B4: B] :
% 0.16/0.33 ( ( Xs2 = Ys )
% 0.16/0.33 & ( ( coinductive_lfinite @ A @ Ys )
% 0.16/0.33 => ( A4 = B4 ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % reflp_tllist
% 0.16/0.33 thf(fact_54_ssubst__Pair__rhs,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R4: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R4 @ S2 ) @ R )
% 0.16/0.33 => ( ( S3 = S2 )
% 0.16/0.33 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R4 @ S3 ) @ R ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % ssubst_Pair_rhs
% 0.16/0.33 thf(fact_55_The__split__eq,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,X: A,Y3: B] :
% 0.16/0.33 ( ( the @ ( product_prod @ A @ B )
% 0.16/0.33 @ ( product_case_prod @ A @ B @ $o
% 0.16/0.33 @ ^ [X4: A,Y5: B] :
% 0.16/0.33 ( ( X = X4 )
% 0.16/0.33 & ( Y3 = Y5 ) ) ) )
% 0.16/0.33 = ( product_Pair @ A @ B @ X @ Y3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % The_split_eq
% 0.16/0.33 thf(fact_56_tllist__of__llist__inject,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,B3: B,Xs: coinductive_llist @ A,C2: B,Ys2: coinductive_llist @ A] :
% 0.16/0.33 ( ( ( tLList1672613558_llist @ B @ A @ B3 @ Xs )
% 0.16/0.33 = ( tLList1672613558_llist @ B @ A @ C2 @ Ys2 ) )
% 0.16/0.33 = ( ( Xs = Ys2 )
% 0.16/0.33 & ( ( coinductive_lfinite @ A @ Ys2 )
% 0.16/0.33 => ( B3 = C2 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % tllist_of_llist_inject
% 0.16/0.33 thf(fact_57_in__lex__prod,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B5: B,R4: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
% 0.16/0.33 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Pair @ A @ B @ A5 @ B5 ) ) @ ( lex_prod @ A @ B @ R4 @ S2 ) )
% 0.16/0.33 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A5 ) @ R4 )
% 0.16/0.33 | ( ( A3 = A5 )
% 0.16/0.33 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B5 ) @ S2 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % in_lex_prod
% 0.16/0.33 thf(fact_58_same__fstI,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P2: A > $o,X: A,Y6: B,Y3: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.16/0.33 ( ( P2 @ X )
% 0.16/0.33 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y3 ) @ ( R @ X ) )
% 0.16/0.33 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y6 ) @ ( product_Pair @ A @ B @ X @ Y3 ) ) @ ( same_fst @ A @ B @ P2 @ R ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % same_fstI
% 0.16/0.33 thf(fact_59_reflp__mono,axiom,
% 0.16/0.33 ! [A: $tType,R: A > A > $o,Q: A > A > $o] :
% 0.16/0.33 ( ( reflp @ A @ R )
% 0.16/0.33 => ( ! [X2: A,Y: A] :
% 0.16/0.33 ( ( R @ X2 @ Y )
% 0.16/0.33 => ( Q @ X2 @ Y ) )
% 0.16/0.33 => ( reflp @ A @ Q ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % reflp_mono
% 0.16/0.33 thf(fact_60_reflp__def,axiom,
% 0.16/0.33 ! [A: $tType] :
% 0.16/0.33 ( ( reflp @ A )
% 0.16/0.33 = ( ^ [R2: A > A > $o] :
% 0.16/0.33 ! [X3: A] : ( R2 @ X3 @ X3 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % reflp_def
% 0.16/0.33 thf(fact_61_theI__unique,axiom,
% 0.16/0.33 ! [A: $tType,P2: A > $o,X: A] :
% 0.16/0.33 ( ? [X5: A] :
% 0.16/0.33 ( ( P2 @ X5 )
% 0.16/0.33 & ! [Y: A] :
% 0.16/0.33 ( ( P2 @ Y )
% 0.16/0.33 => ( Y = X5 ) ) )
% 0.16/0.33 => ( ( P2 @ X )
% 0.16/0.33 = ( X
% 0.16/0.33 = ( the @ A @ P2 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % theI_unique
% 0.16/0.33 thf(fact_62_reflpI,axiom,
% 0.16/0.33 ! [A: $tType,R4: A > A > $o] :
% 0.16/0.33 ( ! [X2: A] : ( R4 @ X2 @ X2 )
% 0.16/0.33 => ( reflp @ A @ R4 ) ) ).
% 0.16/0.33
% 0.16/0.33 % reflpI
% 0.16/0.33 thf(fact_63_reflpE,axiom,
% 0.16/0.33 ! [A: $tType,R4: A > A > $o,X: A] :
% 0.16/0.33 ( ( reflp @ A @ R4 )
% 0.16/0.33 => ( R4 @ X @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % reflpE
% 0.16/0.33 thf(fact_64_reflpD,axiom,
% 0.16/0.33 ! [A: $tType,R4: A > A > $o,X: A] :
% 0.16/0.33 ( ( reflp @ A @ R4 )
% 0.16/0.33 => ( R4 @ X @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % reflpD
% 0.16/0.33 thf(fact_65_tllist_Oabs__induct,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P2: ( tLList446370796tllist @ A @ B ) > $o,X: tLList446370796tllist @ A @ B] :
% 0.16/0.33 ( ! [Y: product_prod @ ( coinductive_llist @ A ) @ B] :
% 0.16/0.33 ( P2
% 0.16/0.33 @ ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( tLList446370796tllist @ A @ B )
% 0.16/0.33 @ ^ [Xs2: coinductive_llist @ A,A4: B] : ( tLList1672613558_llist @ B @ A @ A4 @ Xs2 )
% 0.16/0.33 @ Y ) )
% 0.16/0.33 => ( P2 @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % tllist.abs_induct
% 0.16/0.33 thf(fact_66_tllist__of__llist__cong,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,Xs: coinductive_llist @ A,Xs3: coinductive_llist @ A,B3: B,B5: B] :
% 0.16/0.33 ( ( Xs = Xs3 )
% 0.16/0.33 => ( ( ( coinductive_lfinite @ A @ Xs3 )
% 0.16/0.33 => ( B3 = B5 ) )
% 0.16/0.33 => ( ( tLList1672613558_llist @ B @ A @ B3 @ Xs )
% 0.16/0.33 = ( tLList1672613558_llist @ B @ A @ B5 @ Xs3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % tllist_of_llist_cong
% 0.16/0.33 thf(fact_67_old_Orec__prod__def,axiom,
% 0.16/0.33 ! [T: $tType,B: $tType,A: $tType] :
% 0.16/0.33 ( ( product_rec_prod @ A @ B @ T )
% 0.16/0.33 = ( ^ [F12: A > B > T,X3: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % old.rec_prod_def
% 0.16/0.33 thf(fact_68_the__sym__eq__trivial,axiom,
% 0.16/0.33 ! [A: $tType,X: A] :
% 0.16/0.33 ( ( the @ A
% 0.16/0.33 @ ( ^ [Y7: A,Z2: A] : ( Y7 = Z2 )
% 0.16/0.33 @ X ) )
% 0.16/0.33 = X ) ).
% 0.16/0.33
% 0.16/0.33 % the_sym_eq_trivial
% 0.16/0.33 thf(fact_69_the__eq__trivial,axiom,
% 0.16/0.33 ! [A: $tType,A3: A] :
% 0.16/0.33 ( ( the @ A
% 0.16/0.33 @ ^ [X3: A] : ( X3 = A3 ) )
% 0.16/0.33 = A3 ) ).
% 0.16/0.33
% 0.16/0.33 % the_eq_trivial
% 0.16/0.33 thf(fact_70_the__equality,axiom,
% 0.16/0.33 ! [A: $tType,P2: A > $o,A3: A] :
% 0.16/0.33 ( ( P2 @ A3 )
% 0.16/0.33 => ( ! [X2: A] :
% 0.16/0.33 ( ( P2 @ X2 )
% 0.16/0.33 => ( X2 = A3 ) )
% 0.16/0.33 => ( ( the @ A @ P2 )
% 0.16/0.33 = A3 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % the_equality
% 0.16/0.33 thf(fact_71_the1__equality,axiom,
% 0.16/0.33 ! [A: $tType,P2: A > $o,A3: A] :
% 0.16/0.33 ( ? [X5: A] :
% 0.16/0.33 ( ( P2 @ X5 )
% 0.16/0.33 & ! [Y: A] :
% 0.16/0.33 ( ( P2 @ Y )
% 0.16/0.33 => ( Y = X5 ) ) )
% 0.16/0.33 => ( ( P2 @ A3 )
% 0.16/0.33 => ( ( the @ A @ P2 )
% 0.16/0.33 = A3 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % the1_equality
% 0.16/0.33 thf(fact_72_the1I2,axiom,
% 0.16/0.33 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.16/0.33 ( ? [X5: A] :
% 0.16/0.33 ( ( P2 @ X5 )
% 0.16/0.33 & ! [Y: A] :
% 0.16/0.33 ( ( P2 @ Y )
% 0.16/0.33 => ( Y = X5 ) ) )
% 0.16/0.33 => ( ! [X2: A] :
% 0.16/0.33 ( ( P2 @ X2 )
% 0.16/0.33 => ( Q @ X2 ) )
% 0.16/0.33 => ( Q @ ( the @ A @ P2 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % the1I2
% 0.16/0.33 thf(fact_73_If__def,axiom,
% 0.16/0.33 ! [A: $tType] :
% 0.16/0.33 ( ( if @ A )
% 0.16/0.33 = ( ^ [P3: $o,X3: A,Y4: A] :
% 0.16/0.33 ( the @ A
% 0.16/0.33 @ ^ [Z3: A] :
% 0.16/0.33 ( ( P3
% 0.16/0.33 => ( Z3 = X3 ) )
% 0.16/0.33 & ( ~ P3
% 0.16/0.33 => ( Z3 = Y4 ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % If_def
% 0.16/0.33 thf(fact_74_theI2,axiom,
% 0.16/0.33 ! [A: $tType,P2: A > $o,A3: A,Q: A > $o] :
% 0.16/0.33 ( ( P2 @ A3 )
% 0.16/0.33 => ( ! [X2: A] :
% 0.16/0.33 ( ( P2 @ X2 )
% 0.16/0.33 => ( X2 = A3 ) )
% 0.16/0.33 => ( ! [X2: A] :
% 0.16/0.33 ( ( P2 @ X2 )
% 0.16/0.33 => ( Q @ X2 ) )
% 0.16/0.33 => ( Q @ ( the @ A @ P2 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % theI2
% 0.16/0.33 thf(fact_75_theI,axiom,
% 0.16/0.33 ! [A: $tType,P2: A > $o,A3: A] :
% 0.16/0.33 ( ( P2 @ A3 )
% 0.16/0.33 => ( ! [X2: A] :
% 0.16/0.33 ( ( P2 @ X2 )
% 0.16/0.33 => ( X2 = A3 ) )
% 0.16/0.33 => ( P2 @ ( the @ A @ P2 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % theI
% 0.16/0.33 thf(fact_76_theI_H,axiom,
% 0.16/0.33 ! [A: $tType,P2: A > $o] :
% 0.16/0.33 ( ? [X5: A] :
% 0.16/0.33 ( ( P2 @ X5 )
% 0.16/0.33 & ! [Y: A] :
% 0.16/0.33 ( ( P2 @ Y )
% 0.16/0.33 => ( Y = X5 ) ) )
% 0.16/0.33 => ( P2 @ ( the @ A @ P2 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % theI'
% 0.16/0.33 thf(fact_77_cr__tllist__def,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( tLList47617868tllist @ A @ B )
% 0.16/0.33 = ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( ( tLList446370796tllist @ A @ B ) > $o )
% 0.16/0.33 @ ^ [Xs2: coinductive_llist @ A,B4: B] :
% 0.16/0.33 ( ^ [Y7: tLList446370796tllist @ A @ B,Z2: tLList446370796tllist @ A @ B] : ( Y7 = Z2 )
% 0.16/0.33 @ ( tLList1672613558_llist @ B @ A @ B4 @ Xs2 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % cr_tllist_def
% 0.16/0.33 thf(fact_78_inv__image__def,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType] :
% 0.16/0.33 ( ( inv_image @ B @ A )
% 0.16/0.33 = ( ^ [R2: set @ ( product_prod @ B @ B ),F4: A > B] :
% 0.16/0.33 ( collect @ ( product_prod @ A @ A )
% 0.16/0.33 @ ( product_case_prod @ A @ A @ $o
% 0.16/0.33 @ ^ [X3: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X3 ) @ ( F4 @ Y4 ) ) @ R2 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % inv_image_def
% 0.16/0.33 thf(fact_79_old_Orec__unit__def,axiom,
% 0.16/0.33 ! [T: $tType] :
% 0.16/0.33 ( ( product_rec_unit @ T )
% 0.16/0.33 = ( ^ [F12: T,X3: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % old.rec_unit_def
% 0.16/0.33 thf(fact_80_DEADID_Orel__reflp,axiom,
% 0.16/0.33 ! [A: $tType] :
% 0.16/0.33 ( reflp @ A
% 0.16/0.33 @ ^ [Y7: A,Z2: A] : ( Y7 = Z2 ) ) ).
% 0.16/0.33
% 0.16/0.33 % DEADID.rel_reflp
% 0.16/0.33 thf(fact_81_Nitpick_OThe__psimp,axiom,
% 0.16/0.33 ! [A: $tType,P2: A > $o,X: A] :
% 0.16/0.33 ( ( P2
% 0.16/0.33 = ( ^ [Y7: A,Z2: A] : ( Y7 = Z2 )
% 0.16/0.33 @ X ) )
% 0.16/0.33 => ( ( the @ A @ P2 )
% 0.16/0.33 = X ) ) ).
% 0.16/0.33
% 0.16/0.33 % Nitpick.The_psimp
% 0.16/0.33 thf(fact_82_old_Orec__bool__def,axiom,
% 0.16/0.33 ! [T: $tType] :
% 0.16/0.33 ( ( product_rec_bool @ T )
% 0.16/0.33 = ( ^ [F12: T,F22: T,X3: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F22 @ X3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % old.rec_bool_def
% 0.16/0.33 thf(fact_83_old_Obool_Osimps_I5_J,axiom,
% 0.16/0.33 ! [T: $tType,F1: T,F23: T] :
% 0.16/0.33 ( ( product_rec_bool @ T @ F1 @ F23 @ $true )
% 0.16/0.33 = F1 ) ).
% 0.16/0.33
% 0.16/0.33 % old.bool.simps(5)
% 0.16/0.33 thf(fact_84_old_Obool_Osimps_I6_J,axiom,
% 0.16/0.33 ! [T: $tType,F1: T,F23: T] :
% 0.16/0.33 ( ( product_rec_bool @ T @ F1 @ F23 @ $false )
% 0.16/0.33 = F23 ) ).
% 0.16/0.33
% 0.16/0.33 % old.bool.simps(6)
% 0.16/0.33 thf(fact_85_in__inv__image,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,X: A,Y3: A,R4: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( inv_image @ B @ A @ R4 @ F ) )
% 0.16/0.33 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y3 ) ) @ R4 ) ) ).
% 0.16/0.33
% 0.16/0.33 % in_inv_image
% 0.16/0.33 thf(fact_86_rp__inv__image__def,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( fun_rp_inv_image @ A @ B )
% 0.16/0.33 = ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
% 0.16/0.33 @ ^ [R3: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A ),F4: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R3 @ F4 ) @ ( inv_image @ A @ B @ S4 @ F4 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % rp_inv_image_def
% 0.16/0.33 thf(fact_87_Quotient__tllist,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( quotient @ ( product_prod @ ( coinductive_llist @ A ) @ B ) @ ( tLList446370796tllist @ A @ B )
% 0.16/0.33 @ ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( ( product_prod @ ( coinductive_llist @ A ) @ B ) > $o )
% 0.16/0.33 @ ^ [Xs2: coinductive_llist @ A,A4: B] :
% 0.16/0.33 ( product_case_prod @ ( coinductive_llist @ A ) @ B @ $o
% 0.16/0.33 @ ^ [Ys: coinductive_llist @ A,B4: B] :
% 0.16/0.33 ( ( Xs2 = Ys )
% 0.16/0.33 & ( ( coinductive_lfinite @ A @ Ys )
% 0.16/0.33 => ( A4 = B4 ) ) ) ) )
% 0.16/0.33 @ ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( tLList446370796tllist @ A @ B )
% 0.16/0.33 @ ^ [Xs2: coinductive_llist @ A,A4: B] : ( tLList1672613558_llist @ B @ A @ A4 @ Xs2 ) )
% 0.16/0.33 @ ^ [Ys: tLList446370796tllist @ A @ B] : ( product_Pair @ ( coinductive_llist @ A ) @ B @ ( tLList798109904tllist @ A @ B @ Ys ) @ ( tLList2110128105rminal @ A @ B @ Ys ) )
% 0.16/0.33 @ ( tLList47617868tllist @ A @ B ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_tllist
% 0.16/0.33 thf(fact_88_terminal__tllist__of__llist__lfinite,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,B3: B] :
% 0.16/0.33 ( ( coinductive_lfinite @ A @ Xs )
% 0.16/0.33 => ( ( tLList2110128105rminal @ A @ B @ ( tLList1672613558_llist @ B @ A @ B3 @ Xs ) )
% 0.16/0.33 = B3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % terminal_tllist_of_llist_lfinite
% 0.16/0.33 thf(fact_89_in__inv__imagep,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( inv_imagep @ A @ B )
% 0.16/0.33 = ( ^ [R2: A > A > $o,F4: B > A,X3: B,Y4: B] : ( R2 @ ( F4 @ X3 ) @ ( F4 @ Y4 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % in_inv_imagep
% 0.16/0.33 thf(fact_90_case__swap,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,C: $tType,F: C > B > A,P: product_prod @ C @ B] :
% 0.16/0.33 ( ( product_case_prod @ B @ C @ A
% 0.16/0.33 @ ^ [Y4: B,X3: C] : ( F @ X3 @ Y4 )
% 0.16/0.33 @ ( product_swap @ C @ B @ P ) )
% 0.16/0.33 = ( product_case_prod @ C @ B @ A @ F @ P ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_swap
% 0.16/0.33 thf(fact_91_pair__imageI,axiom,
% 0.16/0.33 ! [C: $tType,B: $tType,A: $tType,A3: A,B3: B,A7: set @ ( product_prod @ A @ B ),F: A > B > C] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ A7 )
% 0.16/0.33 => ( member @ C @ ( F @ A3 @ B3 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A7 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % pair_imageI
% 0.16/0.33 thf(fact_92_swap__swap,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 0.16/0.33 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 0.16/0.33 = P ) ).
% 0.16/0.33
% 0.16/0.33 % swap_swap
% 0.16/0.33 thf(fact_93_pair__in__swap__image,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,Y3: A,X: B,A7: set @ ( product_prod @ B @ A )] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A7 ) )
% 0.16/0.33 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) @ A7 ) ) ).
% 0.16/0.33
% 0.16/0.33 % pair_in_swap_image
% 0.16/0.33 thf(fact_94_swap__simp,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,X: B,Y3: A] :
% 0.16/0.33 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
% 0.16/0.33 = ( product_Pair @ A @ B @ Y3 @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % swap_simp
% 0.16/0.33 thf(fact_95_tllist__of__llist__inverse,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,B3: B,Xs: coinductive_llist @ A] :
% 0.16/0.33 ( ( tLList798109904tllist @ A @ B @ ( tLList1672613558_llist @ B @ A @ B3 @ Xs ) )
% 0.16/0.33 = Xs ) ).
% 0.16/0.33
% 0.16/0.33 % tllist_of_llist_inverse
% 0.16/0.33 thf(fact_96_llist__of__tllist__inverse,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,B3: tLList446370796tllist @ A @ B] :
% 0.16/0.33 ( ( tLList1672613558_llist @ B @ A @ ( tLList2110128105rminal @ A @ B @ B3 ) @ ( tLList798109904tllist @ A @ B @ B3 ) )
% 0.16/0.33 = B3 ) ).
% 0.16/0.33
% 0.16/0.33 % llist_of_tllist_inverse
% 0.16/0.33 thf(fact_97_inv__imagep__def,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType] :
% 0.16/0.33 ( ( inv_imagep @ B @ A )
% 0.16/0.33 = ( ^ [R2: B > B > $o,F4: A > B,X3: A,Y4: A] : ( R2 @ ( F4 @ X3 ) @ ( F4 @ Y4 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % inv_imagep_def
% 0.16/0.33 thf(fact_98_image__ident,axiom,
% 0.16/0.33 ! [A: $tType,Y8: set @ A] :
% 0.16/0.33 ( ( image @ A @ A
% 0.16/0.33 @ ^ [X3: A] : X3
% 0.16/0.33 @ Y8 )
% 0.16/0.33 = Y8 ) ).
% 0.16/0.33
% 0.16/0.33 % image_ident
% 0.16/0.33 thf(fact_99_image__eqI,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,B3: A,F: B > A,X: B,A7: set @ B] :
% 0.16/0.33 ( ( B3
% 0.16/0.33 = ( F @ X ) )
% 0.16/0.33 => ( ( member @ B @ X @ A7 )
% 0.16/0.33 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A7 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % image_eqI
% 0.16/0.33 thf(fact_100_Quotient__total__abs__eq__iff,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,X: A,Y3: A] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( reflp @ A @ R )
% 0.16/0.33 => ( ( ( Abs @ X )
% 0.16/0.33 = ( Abs @ Y3 ) )
% 0.16/0.33 = ( R @ X @ Y3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_total_abs_eq_iff
% 0.16/0.33 thf(fact_101_Quotient__total__abs__induct,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,P2: B > $o,X: B] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( reflp @ A @ R )
% 0.16/0.33 => ( ! [Y: A] : ( P2 @ ( Abs @ Y ) )
% 0.16/0.33 => ( P2 @ X ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_total_abs_induct
% 0.16/0.33 thf(fact_102_rp__inv__image__rp,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,P2: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
% 0.16/0.33 ( ( fun_reduction_pair @ A @ P2 )
% 0.16/0.33 => ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P2 @ F ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % rp_inv_image_rp
% 0.16/0.33 thf(fact_103_rev__image__eqI,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,X: A,A7: set @ A,B3: B,F: A > B] :
% 0.16/0.33 ( ( member @ A @ X @ A7 )
% 0.16/0.33 => ( ( B3
% 0.16/0.33 = ( F @ X ) )
% 0.16/0.33 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A7 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % rev_image_eqI
% 0.16/0.33 thf(fact_104_ball__imageD,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,F: B > A,A7: set @ B,P2: A > $o] :
% 0.16/0.33 ( ! [X2: A] :
% 0.16/0.33 ( ( member @ A @ X2 @ ( image @ B @ A @ F @ A7 ) )
% 0.16/0.33 => ( P2 @ X2 ) )
% 0.16/0.33 => ! [X5: B] :
% 0.16/0.33 ( ( member @ B @ X5 @ A7 )
% 0.16/0.33 => ( P2 @ ( F @ X5 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % ball_imageD
% 0.16/0.33 thf(fact_105_image__cong,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G3: A > B] :
% 0.16/0.33 ( ( M = N )
% 0.16/0.33 => ( ! [X2: A] :
% 0.16/0.33 ( ( member @ A @ X2 @ N )
% 0.16/0.33 => ( ( F @ X2 )
% 0.16/0.33 = ( G3 @ X2 ) ) )
% 0.16/0.33 => ( ( image @ A @ B @ F @ M )
% 0.16/0.33 = ( image @ A @ B @ G3 @ N ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % image_cong
% 0.16/0.33 thf(fact_106_bex__imageD,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,F: B > A,A7: set @ B,P2: A > $o] :
% 0.16/0.33 ( ? [X5: A] :
% 0.16/0.33 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A7 ) )
% 0.16/0.33 & ( P2 @ X5 ) )
% 0.16/0.33 => ? [X2: B] :
% 0.16/0.33 ( ( member @ B @ X2 @ A7 )
% 0.16/0.33 & ( P2 @ ( F @ X2 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % bex_imageD
% 0.16/0.33 thf(fact_107_image__iff,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,Z: A,F: B > A,A7: set @ B] :
% 0.16/0.33 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A7 ) )
% 0.16/0.33 = ( ? [X3: B] :
% 0.16/0.33 ( ( member @ B @ X3 @ A7 )
% 0.16/0.33 & ( Z
% 0.16/0.33 = ( F @ X3 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % image_iff
% 0.16/0.33 thf(fact_108_imageI,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,X: A,A7: set @ A,F: A > B] :
% 0.16/0.33 ( ( member @ A @ X @ A7 )
% 0.16/0.33 => ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A7 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % imageI
% 0.16/0.33 thf(fact_109_Quotient__rep__abs__fold__unmap,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,X6: B,X: A,Rep2: B > A] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( X6
% 0.16/0.33 = ( Abs @ X ) )
% 0.16/0.33 => ( ( R @ X @ X )
% 0.16/0.33 => ( ( ( Rep @ X6 )
% 0.16/0.33 = ( Rep2 @ X6 ) )
% 0.16/0.33 => ( R @ ( Rep2 @ X6 ) @ X ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_rep_abs_fold_unmap
% 0.16/0.33 thf(fact_110_Quotient__to__transfer,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,C2: A,C4: B] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( R @ C2 @ C2 )
% 0.16/0.33 => ( ( C4
% 0.16/0.33 = ( Abs @ C2 ) )
% 0.16/0.33 => ( T2 @ C2 @ C4 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_to_transfer
% 0.16/0.33 thf(fact_111_Quotient__abs__induct,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,P2: B > $o,X: B] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ! [Y: A] :
% 0.16/0.33 ( ( R @ Y @ Y )
% 0.16/0.33 => ( P2 @ ( Abs @ Y ) ) )
% 0.16/0.33 => ( P2 @ X ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_abs_induct
% 0.16/0.33 thf(fact_112_Quotient__rep__reflp,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,A3: B] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_rep_reflp
% 0.16/0.33 thf(fact_113_Quotient__rel__abs2,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,X: B,Y3: A] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( R @ ( Rep @ X ) @ Y3 )
% 0.16/0.33 => ( X
% 0.16/0.33 = ( Abs @ Y3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_rel_abs2
% 0.16/0.33 thf(fact_114_Quotient__alt__def3,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( quotient @ A @ B )
% 0.16/0.33 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep3: B > A,T3: A > B > $o] :
% 0.16/0.33 ( ! [A4: A,B4: B] :
% 0.16/0.33 ( ( T3 @ A4 @ B4 )
% 0.16/0.33 => ( ( Abs2 @ A4 )
% 0.16/0.33 = B4 ) )
% 0.16/0.33 & ! [B4: B] : ( T3 @ ( Rep3 @ B4 ) @ B4 )
% 0.16/0.33 & ! [X3: A,Y4: A] :
% 0.16/0.33 ( ( R3 @ X3 @ Y4 )
% 0.16/0.33 = ( ? [Z3: B] :
% 0.16/0.33 ( ( T3 @ X3 @ Z3 )
% 0.16/0.33 & ( T3 @ Y4 @ Z3 ) ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_alt_def3
% 0.16/0.33 thf(fact_115_Quotient__alt__def2,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( quotient @ A @ B )
% 0.16/0.33 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep3: B > A,T3: A > B > $o] :
% 0.16/0.33 ( ! [A4: A,B4: B] :
% 0.16/0.33 ( ( T3 @ A4 @ B4 )
% 0.16/0.33 => ( ( Abs2 @ A4 )
% 0.16/0.33 = B4 ) )
% 0.16/0.33 & ! [B4: B] : ( T3 @ ( Rep3 @ B4 ) @ B4 )
% 0.16/0.33 & ! [X3: A,Y4: A] :
% 0.16/0.33 ( ( R3 @ X3 @ Y4 )
% 0.16/0.33 = ( ( T3 @ X3 @ ( Abs2 @ Y4 ) )
% 0.16/0.33 & ( T3 @ Y4 @ ( Abs2 @ X3 ) ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_alt_def2
% 0.16/0.33 thf(fact_116_Quotient__rep__abs,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( R @ R4 @ R4 )
% 0.16/0.33 => ( R @ ( Rep @ ( Abs @ R4 ) ) @ R4 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_rep_abs
% 0.16/0.33 thf(fact_117_Quotient__rel__rep,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,A3: B,B3: B] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( R @ ( Rep @ A3 ) @ ( Rep @ B3 ) )
% 0.16/0.33 = ( A3 = B3 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_rel_rep
% 0.16/0.33 thf(fact_118_Quotient__rel__abs,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A,S2: A] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( R @ R4 @ S2 )
% 0.16/0.33 => ( ( Abs @ R4 )
% 0.16/0.33 = ( Abs @ S2 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_rel_abs
% 0.16/0.33 thf(fact_119_Quotient__alt__def,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( quotient @ A @ B )
% 0.16/0.33 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep3: B > A,T3: A > B > $o] :
% 0.16/0.33 ( ! [A4: A,B4: B] :
% 0.16/0.33 ( ( T3 @ A4 @ B4 )
% 0.16/0.33 => ( ( Abs2 @ A4 )
% 0.16/0.33 = B4 ) )
% 0.16/0.33 & ! [B4: B] : ( T3 @ ( Rep3 @ B4 ) @ B4 )
% 0.16/0.33 & ! [X3: A,Y4: A] :
% 0.16/0.33 ( ( R3 @ X3 @ Y4 )
% 0.16/0.33 = ( ( T3 @ X3 @ ( Abs2 @ X3 ) )
% 0.16/0.33 & ( T3 @ Y4 @ ( Abs2 @ Y4 ) )
% 0.16/0.33 & ( ( Abs2 @ X3 )
% 0.16/0.33 = ( Abs2 @ Y4 ) ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_alt_def
% 0.16/0.33 thf(fact_120_Quotient__abs__rep,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,A3: B] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( Abs @ ( Rep @ A3 ) )
% 0.16/0.33 = A3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_abs_rep
% 0.16/0.33 thf(fact_121_Quotient__Rep__eq,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,X6: B,X: A] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( X6
% 0.16/0.33 = ( Abs @ X ) )
% 0.16/0.33 => ( ( Rep @ X6 )
% 0.16/0.33 = ( Rep @ X6 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_Rep_eq
% 0.16/0.33 thf(fact_122_Quotient__refl2,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A,S2: A] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( R @ R4 @ S2 )
% 0.16/0.33 => ( R @ S2 @ S2 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_refl2
% 0.16/0.33 thf(fact_123_Quotient__refl1,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A,S2: A] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( R @ R4 @ S2 )
% 0.16/0.33 => ( R @ R4 @ R4 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_refl1
% 0.16/0.33 thf(fact_124_Quotient__rel,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A,S2: A] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( ( R @ R4 @ R4 )
% 0.16/0.33 & ( R @ S2 @ S2 )
% 0.16/0.33 & ( ( Abs @ R4 )
% 0.16/0.33 = ( Abs @ S2 ) ) )
% 0.16/0.33 = ( R @ R4 @ S2 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_rel
% 0.16/0.33 thf(fact_125_Compr__image__eq,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,F: B > A,A7: set @ B,P2: A > $o] :
% 0.16/0.33 ( ( collect @ A
% 0.16/0.33 @ ^ [X3: A] :
% 0.16/0.33 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A7 ) )
% 0.16/0.33 & ( P2 @ X3 ) ) )
% 0.16/0.33 = ( image @ B @ A @ F
% 0.16/0.33 @ ( collect @ B
% 0.16/0.33 @ ^ [X3: B] :
% 0.16/0.33 ( ( member @ B @ X3 @ A7 )
% 0.16/0.33 & ( P2 @ ( F @ X3 ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Compr_image_eq
% 0.16/0.33 thf(fact_126_image__image,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,C: $tType,F: B > A,G3: C > B,A7: set @ C] :
% 0.16/0.33 ( ( image @ B @ A @ F @ ( image @ C @ B @ G3 @ A7 ) )
% 0.16/0.33 = ( image @ C @ A
% 0.16/0.33 @ ^ [X3: C] : ( F @ ( G3 @ X3 ) )
% 0.16/0.33 @ A7 ) ) ).
% 0.16/0.33
% 0.16/0.33 % image_image
% 0.16/0.33 thf(fact_127_imageE,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,B3: A,F: B > A,A7: set @ B] :
% 0.16/0.33 ( ( member @ A @ B3 @ ( image @ B @ A @ F @ A7 ) )
% 0.16/0.33 => ~ ! [X2: B] :
% 0.16/0.33 ( ( B3
% 0.16/0.33 = ( F @ X2 ) )
% 0.16/0.33 => ~ ( member @ B @ X2 @ A7 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % imageE
% 0.16/0.33 thf(fact_128_Quotient__cr__rel,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( T2
% 0.16/0.33 = ( ^ [X3: A,Y4: B] :
% 0.16/0.33 ( ( R @ X3 @ X3 )
% 0.16/0.33 & ( ( Abs @ X3 )
% 0.16/0.33 = Y4 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_cr_rel
% 0.16/0.33 thf(fact_129_Quotient__def,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( quotient @ A @ B )
% 0.16/0.33 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep3: B > A,T3: A > B > $o] :
% 0.16/0.33 ( ! [A4: B] :
% 0.16/0.33 ( ( Abs2 @ ( Rep3 @ A4 ) )
% 0.16/0.33 = A4 )
% 0.16/0.33 & ! [A4: B] : ( R3 @ ( Rep3 @ A4 ) @ ( Rep3 @ A4 ) )
% 0.16/0.33 & ! [R2: A,S5: A] :
% 0.16/0.33 ( ( R3 @ R2 @ S5 )
% 0.16/0.33 = ( ( R3 @ R2 @ R2 )
% 0.16/0.33 & ( R3 @ S5 @ S5 )
% 0.16/0.33 & ( ( Abs2 @ R2 )
% 0.16/0.33 = ( Abs2 @ S5 ) ) ) )
% 0.16/0.33 & ( T3
% 0.16/0.33 = ( ^ [X3: A,Y4: B] :
% 0.16/0.33 ( ( R3 @ X3 @ X3 )
% 0.16/0.33 & ( ( Abs2 @ X3 )
% 0.16/0.33 = Y4 ) ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_def
% 0.16/0.33 thf(fact_130_QuotientI,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,Abs: B > A,Rep: A > B,R: B > B > $o,T2: B > A > $o] :
% 0.16/0.33 ( ! [A2: A] :
% 0.16/0.33 ( ( Abs @ ( Rep @ A2 ) )
% 0.16/0.33 = A2 )
% 0.16/0.33 => ( ! [A2: A] : ( R @ ( Rep @ A2 ) @ ( Rep @ A2 ) )
% 0.16/0.33 => ( ! [R5: B,S6: B] :
% 0.16/0.33 ( ( R @ R5 @ S6 )
% 0.16/0.33 = ( ( R @ R5 @ R5 )
% 0.16/0.33 & ( R @ S6 @ S6 )
% 0.16/0.33 & ( ( Abs @ R5 )
% 0.16/0.33 = ( Abs @ S6 ) ) ) )
% 0.16/0.33 => ( ( T2
% 0.16/0.33 = ( ^ [X3: B,Y4: A] :
% 0.16/0.33 ( ( R @ X3 @ X3 )
% 0.16/0.33 & ( ( Abs @ X3 )
% 0.16/0.33 = Y4 ) ) ) )
% 0.16/0.33 => ( quotient @ B @ A @ R @ Abs @ Rep @ T2 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % QuotientI
% 0.16/0.33 thf(fact_131_Sup_OSUP__identity__eq,axiom,
% 0.16/0.33 ! [A: $tType,Sup: ( set @ A ) > A,A7: set @ A] :
% 0.16/0.33 ( ( Sup
% 0.16/0.33 @ ( image @ A @ A
% 0.16/0.33 @ ^ [X3: A] : X3
% 0.16/0.33 @ A7 ) )
% 0.16/0.33 = ( Sup @ A7 ) ) ).
% 0.16/0.33
% 0.16/0.33 % Sup.SUP_identity_eq
% 0.16/0.33 thf(fact_132_Inf_OINF__identity__eq,axiom,
% 0.16/0.33 ! [A: $tType,Inf: ( set @ A ) > A,A7: set @ A] :
% 0.16/0.33 ( ( Inf
% 0.16/0.33 @ ( image @ A @ A
% 0.16/0.33 @ ^ [X3: A] : X3
% 0.16/0.33 @ A7 ) )
% 0.16/0.33 = ( Inf @ A7 ) ) ).
% 0.16/0.33
% 0.16/0.33 % Inf.INF_identity_eq
% 0.16/0.33 thf(fact_133_terminal0__terminal,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( tLList1825092077minal0 @ ( ( tLList446370796tllist @ A @ B ) > B ) )
% 0.16/0.33 = ( tLList2110128105rminal @ A @ B ) ) ).
% 0.16/0.33
% 0.16/0.33 % terminal0_terminal
% 0.16/0.33 thf(fact_134_surj__swap,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.16/0.33 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % surj_swap
% 0.16/0.33 thf(fact_135_swap__product,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A7: set @ B,B7: set @ A] :
% 0.16/0.33 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
% 0.16/0.33 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 0.16/0.33 @ ^ [I: B,J: A] : ( product_Pair @ A @ B @ J @ I ) )
% 0.16/0.33 @ ( product_Sigma @ B @ A @ A7
% 0.16/0.33 @ ^ [Uu: B] : B7 ) )
% 0.16/0.33 = ( product_Sigma @ A @ B @ B7
% 0.16/0.33 @ ^ [Uu: A] : A7 ) ) ).
% 0.16/0.33
% 0.16/0.33 % swap_product
% 0.16/0.33 thf(fact_136_mem__Sigma__iff,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A3: A,B3: B,A7: set @ A,B7: A > ( set @ B )] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
% 0.16/0.33 = ( ( member @ A @ A3 @ A7 )
% 0.16/0.33 & ( member @ B @ B3 @ ( B7 @ A3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % mem_Sigma_iff
% 0.16/0.33 thf(fact_137_SigmaI,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A3: A,A7: set @ A,B3: B,B7: A > ( set @ B )] :
% 0.16/0.33 ( ( member @ A @ A3 @ A7 )
% 0.16/0.33 => ( ( member @ B @ B3 @ ( B7 @ A3 ) )
% 0.16/0.33 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % SigmaI
% 0.16/0.33 thf(fact_138_Collect__case__prod,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P2: A > $o,Q: B > $o] :
% 0.16/0.33 ( ( collect @ ( product_prod @ A @ B )
% 0.16/0.33 @ ( product_case_prod @ A @ B @ $o
% 0.16/0.33 @ ^ [A4: A,B4: B] :
% 0.16/0.33 ( ( P2 @ A4 )
% 0.16/0.33 & ( Q @ B4 ) ) ) )
% 0.16/0.33 = ( product_Sigma @ A @ B @ ( collect @ A @ P2 )
% 0.16/0.33 @ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Collect_case_prod
% 0.16/0.33 thf(fact_139_UNIV__Times__UNIV,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 0.16/0.33 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
% 0.16/0.33 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % UNIV_Times_UNIV
% 0.16/0.33 thf(fact_140_UNIV__def,axiom,
% 0.16/0.33 ! [A: $tType] :
% 0.16/0.33 ( ( top_top @ ( set @ A ) )
% 0.16/0.33 = ( collect @ A
% 0.16/0.33 @ ^ [X3: A] : $true ) ) ).
% 0.16/0.33
% 0.16/0.33 % UNIV_def
% 0.16/0.33 thf(fact_141_SigmaE,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,C2: product_prod @ A @ B,A7: set @ A,B7: A > ( set @ B )] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
% 0.16/0.33 => ~ ! [X2: A] :
% 0.16/0.33 ( ( member @ A @ X2 @ A7 )
% 0.16/0.33 => ! [Y: B] :
% 0.16/0.33 ( ( member @ B @ Y @ ( B7 @ X2 ) )
% 0.16/0.33 => ( C2
% 0.16/0.33 != ( product_Pair @ A @ B @ X2 @ Y ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % SigmaE
% 0.16/0.33 thf(fact_142_SigmaD1,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A3: A,B3: B,A7: set @ A,B7: A > ( set @ B )] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
% 0.16/0.33 => ( member @ A @ A3 @ A7 ) ) ).
% 0.16/0.33
% 0.16/0.33 % SigmaD1
% 0.16/0.33 thf(fact_143_SigmaD2,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A3: A,B3: B,A7: set @ A,B7: A > ( set @ B )] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
% 0.16/0.33 => ( member @ B @ B3 @ ( B7 @ A3 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % SigmaD2
% 0.16/0.33 thf(fact_144_SigmaE2,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A3: A,B3: B,A7: set @ A,B7: A > ( set @ B )] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
% 0.16/0.33 => ~ ( ( member @ A @ A3 @ A7 )
% 0.16/0.33 => ~ ( member @ B @ B3 @ ( B7 @ A3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % SigmaE2
% 0.16/0.33 thf(fact_145_Sigma__cong,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A7: set @ A,B7: set @ A,C5: A > ( set @ B ),D3: A > ( set @ B )] :
% 0.16/0.33 ( ( A7 = B7 )
% 0.16/0.33 => ( ! [X2: A] :
% 0.16/0.33 ( ( member @ A @ X2 @ B7 )
% 0.16/0.33 => ( ( C5 @ X2 )
% 0.16/0.33 = ( D3 @ X2 ) ) )
% 0.16/0.33 => ( ( product_Sigma @ A @ B @ A7 @ C5 )
% 0.16/0.33 = ( product_Sigma @ A @ B @ B7 @ D3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Sigma_cong
% 0.16/0.33 thf(fact_146_Times__eq__cancel2,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,X: A,C5: set @ A,A7: set @ B,B7: set @ B] :
% 0.16/0.33 ( ( member @ A @ X @ C5 )
% 0.16/0.33 => ( ( ( product_Sigma @ B @ A @ A7
% 0.16/0.33 @ ^ [Uu: B] : C5 )
% 0.16/0.33 = ( product_Sigma @ B @ A @ B7
% 0.16/0.33 @ ^ [Uu: B] : C5 ) )
% 0.16/0.33 = ( A7 = B7 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Times_eq_cancel2
% 0.16/0.33 thf(fact_147_range__eqI,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,B3: A,F: B > A,X: B] :
% 0.16/0.33 ( ( B3
% 0.16/0.33 = ( F @ X ) )
% 0.16/0.33 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % range_eqI
% 0.16/0.33 thf(fact_148_rangeI,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % rangeI
% 0.16/0.33 thf(fact_149_Collect__case__prod__Sigma,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P2: A > $o,Q: A > B > $o] :
% 0.16/0.33 ( ( collect @ ( product_prod @ A @ B )
% 0.16/0.33 @ ( product_case_prod @ A @ B @ $o
% 0.16/0.33 @ ^ [X3: A,Y4: B] :
% 0.16/0.33 ( ( P2 @ X3 )
% 0.16/0.33 & ( Q @ X3 @ Y4 ) ) ) )
% 0.16/0.33 = ( product_Sigma @ A @ B @ ( collect @ A @ P2 )
% 0.16/0.33 @ ^ [X3: A] : ( collect @ B @ ( Q @ X3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Collect_case_prod_Sigma
% 0.16/0.33 thf(fact_150_range__composition,axiom,
% 0.16/0.33 ! [A: $tType,C: $tType,B: $tType,F: C > A,G3: B > C] :
% 0.16/0.33 ( ( image @ B @ A
% 0.16/0.33 @ ^ [X3: B] : ( F @ ( G3 @ X3 ) )
% 0.16/0.33 @ ( top_top @ ( set @ B ) ) )
% 0.16/0.33 = ( image @ C @ A @ F @ ( image @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % range_composition
% 0.16/0.33 thf(fact_151_rangeE,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,B3: A,F: B > A] :
% 0.16/0.33 ( ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
% 0.16/0.33 => ~ ! [X2: B] :
% 0.16/0.33 ( B3
% 0.16/0.33 != ( F @ X2 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % rangeE
% 0.16/0.33 thf(fact_152_product__swap,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A7: set @ B,B7: set @ A] :
% 0.16/0.33 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
% 0.16/0.33 @ ( product_Sigma @ B @ A @ A7
% 0.16/0.33 @ ^ [Uu: B] : B7 ) )
% 0.16/0.33 = ( product_Sigma @ A @ B @ B7
% 0.16/0.33 @ ^ [Uu: A] : A7 ) ) ).
% 0.16/0.33
% 0.16/0.33 % product_swap
% 0.16/0.33 thf(fact_153_Inf_OINF__cong,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,A7: set @ B,B7: set @ B,C5: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 0.16/0.33 ( ( A7 = B7 )
% 0.16/0.33 => ( ! [X2: B] :
% 0.16/0.33 ( ( member @ B @ X2 @ B7 )
% 0.16/0.33 => ( ( C5 @ X2 )
% 0.16/0.33 = ( D3 @ X2 ) ) )
% 0.16/0.33 => ( ( Inf @ ( image @ B @ A @ C5 @ A7 ) )
% 0.16/0.33 = ( Inf @ ( image @ B @ A @ D3 @ B7 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Inf.INF_cong
% 0.16/0.33 thf(fact_154_Sup_OSUP__cong,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,A7: set @ B,B7: set @ B,C5: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 0.16/0.33 ( ( A7 = B7 )
% 0.16/0.33 => ( ! [X2: B] :
% 0.16/0.33 ( ( member @ B @ X2 @ B7 )
% 0.16/0.33 => ( ( C5 @ X2 )
% 0.16/0.33 = ( D3 @ X2 ) ) )
% 0.16/0.33 => ( ( Sup @ ( image @ B @ A @ C5 @ A7 ) )
% 0.16/0.33 = ( Sup @ ( image @ B @ A @ D3 @ B7 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Sup.SUP_cong
% 0.16/0.33 thf(fact_155_surjD,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,F: B > A,Y3: A] :
% 0.16/0.33 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.16/0.33 = ( top_top @ ( set @ A ) ) )
% 0.16/0.33 => ? [X2: B] :
% 0.16/0.33 ( Y3
% 0.16/0.33 = ( F @ X2 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % surjD
% 0.16/0.33 thf(fact_156_surjE,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,F: B > A,Y3: A] :
% 0.16/0.33 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.16/0.33 = ( top_top @ ( set @ A ) ) )
% 0.16/0.33 => ~ ! [X2: B] :
% 0.16/0.33 ( Y3
% 0.16/0.33 != ( F @ X2 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % surjE
% 0.16/0.33 thf(fact_157_surjI,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,G3: B > A,F: A > B] :
% 0.16/0.33 ( ! [X2: A] :
% 0.16/0.33 ( ( G3 @ ( F @ X2 ) )
% 0.16/0.33 = X2 )
% 0.16/0.33 => ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
% 0.16/0.33 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % surjI
% 0.16/0.33 thf(fact_158_surj__def,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,F: B > A] :
% 0.16/0.33 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.16/0.33 = ( top_top @ ( set @ A ) ) )
% 0.16/0.33 = ( ! [Y4: A] :
% 0.16/0.33 ? [X3: B] :
% 0.16/0.33 ( Y4
% 0.16/0.33 = ( F @ X3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % surj_def
% 0.16/0.33 thf(fact_159_member__product,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A7: set @ A,B7: set @ B] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A7 @ B7 ) )
% 0.16/0.33 = ( member @ ( product_prod @ A @ B ) @ X
% 0.16/0.33 @ ( product_Sigma @ A @ B @ A7
% 0.16/0.33 @ ^ [Uu: A] : B7 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % member_product
% 0.16/0.33 thf(fact_160_Product__Type_Oproduct__def,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( product_product @ A @ B )
% 0.16/0.33 = ( ^ [A8: set @ A,B8: set @ B] :
% 0.16/0.33 ( product_Sigma @ A @ B @ A8
% 0.16/0.33 @ ^ [Uu: A] : B8 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Product_Type.product_def
% 0.16/0.33 thf(fact_161_map__prod__surj,axiom,
% 0.16/0.33 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G3: C > D] :
% 0.16/0.33 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.16/0.33 = ( top_top @ ( set @ B ) ) )
% 0.16/0.33 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 0.16/0.33 = ( top_top @ ( set @ D ) ) )
% 0.16/0.33 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.16/0.33 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % map_prod_surj
% 0.16/0.33 thf(fact_162_map__prod__ident,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( product_map_prod @ A @ A @ B @ B
% 0.16/0.33 @ ^ [X3: A] : X3
% 0.16/0.33 @ ^ [Y4: B] : Y4 )
% 0.16/0.33 = ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % map_prod_ident
% 0.16/0.33 thf(fact_163_map__prod__simp,axiom,
% 0.16/0.33 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G3: D > B,A3: C,B3: D] :
% 0.16/0.33 ( ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
% 0.16/0.33 = ( product_Pair @ A @ B @ ( F @ A3 ) @ ( G3 @ B3 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % map_prod_simp
% 0.16/0.33 thf(fact_164_map__prod__imageI,axiom,
% 0.16/0.33 ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R: set @ ( product_prod @ A @ B ),F: A > C,G3: B > D] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R )
% 0.16/0.33 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( G3 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G3 ) @ R ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % map_prod_imageI
% 0.16/0.33 thf(fact_165_case__prod__map__prod,axiom,
% 0.16/0.33 ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G3: E > C,X: product_prod @ D @ E] :
% 0.16/0.33 ( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G3 @ X ) )
% 0.16/0.33 = ( product_case_prod @ D @ E @ A
% 0.16/0.33 @ ^ [L: D,R2: E] : ( H @ ( F @ L ) @ ( G3 @ R2 ) )
% 0.16/0.33 @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prod_map_prod
% 0.16/0.33 thf(fact_166_prod__fun__imageE,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F: C > A,G3: D > B,R: set @ ( product_prod @ C @ D )] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G3 ) @ R ) )
% 0.16/0.33 => ~ ! [X2: C,Y: D] :
% 0.16/0.33 ( ( C2
% 0.16/0.33 = ( product_Pair @ A @ B @ ( F @ X2 ) @ ( G3 @ Y ) ) )
% 0.16/0.33 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X2 @ Y ) @ R ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_fun_imageE
% 0.16/0.33 thf(fact_167_map__prod__def,axiom,
% 0.16/0.33 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 0.16/0.33 ( ( product_map_prod @ A @ C @ B @ D )
% 0.16/0.33 = ( ^ [F4: A > C,G4: B > D] :
% 0.16/0.33 ( product_case_prod @ A @ B @ ( product_prod @ C @ D )
% 0.16/0.33 @ ^ [X3: A,Y4: B] : ( product_Pair @ C @ D @ ( F4 @ X3 ) @ ( G4 @ Y4 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % map_prod_def
% 0.16/0.33 thf(fact_168_map__prod__surj__on,axiom,
% 0.16/0.33 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A7: set @ B,A9: set @ A,G3: D > C,B7: set @ D,B9: set @ C] :
% 0.16/0.33 ( ( ( image @ B @ A @ F @ A7 )
% 0.16/0.33 = A9 )
% 0.16/0.33 => ( ( ( image @ D @ C @ G3 @ B7 )
% 0.16/0.33 = B9 )
% 0.16/0.33 => ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G3 )
% 0.16/0.33 @ ( product_Sigma @ B @ D @ A7
% 0.16/0.33 @ ^ [Uu: B] : B7 ) )
% 0.16/0.33 = ( product_Sigma @ A @ C @ A9
% 0.16/0.33 @ ^ [Uu: A] : B9 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % map_prod_surj_on
% 0.16/0.33 thf(fact_169_prod_Omap__ident,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,T4: product_prod @ A @ B] :
% 0.16/0.33 ( ( product_map_prod @ A @ A @ B @ B
% 0.16/0.33 @ ^ [X3: A] : X3
% 0.16/0.33 @ ^ [X3: B] : X3
% 0.16/0.33 @ T4 )
% 0.16/0.33 = T4 ) ).
% 0.16/0.33
% 0.16/0.33 % prod.map_ident
% 0.16/0.33 thf(fact_170_terminal__def,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( tLList2110128105rminal @ A @ B )
% 0.16/0.33 = ( tLList200813139tllist @ B @ B @ A
% 0.16/0.33 @ ^ [X12: B] : X12
% 0.16/0.33 @ ^ [X_dummy: A] : ( tLList1825092077minal0 @ ( ( tLList446370796tllist @ A @ B ) > B ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % terminal_def
% 0.16/0.33 thf(fact_171_reflp__refl__eq,axiom,
% 0.16/0.33 ! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
% 0.16/0.33 ( ( reflp @ A
% 0.16/0.33 @ ^ [X3: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R4 ) )
% 0.16/0.33 = ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R4 ) ) ).
% 0.16/0.33
% 0.16/0.33 % reflp_refl_eq
% 0.16/0.33 thf(fact_172_refl__onD2,axiom,
% 0.16/0.33 ! [A: $tType,A7: set @ A,R4: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
% 0.16/0.33 ( ( refl_on @ A @ A7 @ R4 )
% 0.16/0.33 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R4 )
% 0.16/0.33 => ( member @ A @ Y3 @ A7 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % refl_onD2
% 0.16/0.33 thf(fact_173_refl__onD1,axiom,
% 0.16/0.33 ! [A: $tType,A7: set @ A,R4: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
% 0.16/0.33 ( ( refl_on @ A @ A7 @ R4 )
% 0.16/0.33 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R4 )
% 0.16/0.33 => ( member @ A @ X @ A7 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % refl_onD1
% 0.16/0.33 thf(fact_174_refl__onD,axiom,
% 0.16/0.33 ! [A: $tType,A7: set @ A,R4: set @ ( product_prod @ A @ A ),A3: A] :
% 0.16/0.33 ( ( refl_on @ A @ A7 @ R4 )
% 0.16/0.33 => ( ( member @ A @ A3 @ A7 )
% 0.16/0.33 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R4 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % refl_onD
% 0.16/0.33 thf(fact_175_tllist_Ocase__distrib,axiom,
% 0.16/0.33 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F1: B > C,F23: A > ( tLList446370796tllist @ A @ B ) > C,Tllist: tLList446370796tllist @ A @ B] :
% 0.16/0.33 ( ( H @ ( tLList200813139tllist @ B @ C @ A @ F1 @ F23 @ Tllist ) )
% 0.16/0.33 = ( tLList200813139tllist @ B @ D @ A
% 0.16/0.33 @ ^ [X3: B] : ( H @ ( F1 @ X3 ) )
% 0.16/0.33 @ ^ [X12: A,X23: tLList446370796tllist @ A @ B] : ( H @ ( F23 @ X12 @ X23 ) )
% 0.16/0.33 @ Tllist ) ) ).
% 0.16/0.33
% 0.16/0.33 % tllist.case_distrib
% 0.16/0.33 thf(fact_176_refl__on__domain,axiom,
% 0.16/0.33 ! [A: $tType,A7: set @ A,R4: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
% 0.16/0.33 ( ( refl_on @ A @ A7 @ R4 )
% 0.16/0.33 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R4 )
% 0.16/0.33 => ( ( member @ A @ A3 @ A7 )
% 0.16/0.33 & ( member @ A @ B3 @ A7 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % refl_on_domain
% 0.16/0.33 thf(fact_177_range__fst,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.16/0.33 = ( top_top @ ( set @ A ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % range_fst
% 0.16/0.33 thf(fact_178_range__snd,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.16/0.33 = ( top_top @ ( set @ A ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % range_snd
% 0.16/0.33 thf(fact_179_fst__map__prod,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G3: D > B,X: product_prod @ C @ D] :
% 0.16/0.33 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ X ) )
% 0.16/0.33 = ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % fst_map_prod
% 0.16/0.33 thf(fact_180_snd__map__prod,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G3: D > A,X: product_prod @ C @ D] :
% 0.16/0.33 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G3 @ X ) )
% 0.16/0.33 = ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % snd_map_prod
% 0.16/0.33 thf(fact_181_prod_Ocollapse,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.16/0.33 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.16/0.33 = Prod ) ).
% 0.16/0.33
% 0.16/0.33 % prod.collapse
% 0.16/0.33 thf(fact_182_fst__swap,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 0.16/0.33 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 0.16/0.33 = ( product_snd @ B @ A @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % fst_swap
% 0.16/0.33 thf(fact_183_snd__swap,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 0.16/0.33 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 0.16/0.33 = ( product_fst @ A @ B @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % snd_swap
% 0.16/0.33 thf(fact_184_prod__eq__iff,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( ^ [Y7: product_prod @ A @ B,Z2: product_prod @ A @ B] : ( Y7 = Z2 ) )
% 0.16/0.33 = ( ^ [S5: product_prod @ A @ B,T5: product_prod @ A @ B] :
% 0.16/0.33 ( ( ( product_fst @ A @ B @ S5 )
% 0.16/0.33 = ( product_fst @ A @ B @ T5 ) )
% 0.16/0.33 & ( ( product_snd @ A @ B @ S5 )
% 0.16/0.33 = ( product_snd @ A @ B @ T5 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_eq_iff
% 0.16/0.33 thf(fact_185_prod_Oexpand,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.16/0.33 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.16/0.33 = ( product_fst @ A @ B @ Prod2 ) )
% 0.16/0.33 & ( ( product_snd @ A @ B @ Prod )
% 0.16/0.33 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.16/0.33 => ( Prod = Prod2 ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod.expand
% 0.16/0.33 thf(fact_186_prod__eqI,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.16/0.33 ( ( ( product_fst @ A @ B @ P )
% 0.16/0.33 = ( product_fst @ A @ B @ Q2 ) )
% 0.16/0.33 => ( ( ( product_snd @ A @ B @ P )
% 0.16/0.33 = ( product_snd @ A @ B @ Q2 ) )
% 0.16/0.33 => ( P = Q2 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod_eqI
% 0.16/0.33 thf(fact_187_split__fst,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: A > $o,P: product_prod @ A @ B] :
% 0.16/0.33 ( ( R @ ( product_fst @ A @ B @ P ) )
% 0.16/0.33 = ( ! [X3: A,Y4: B] :
% 0.16/0.33 ( ( P
% 0.16/0.33 = ( product_Pair @ A @ B @ X3 @ Y4 ) )
% 0.16/0.33 => ( R @ X3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % split_fst
% 0.16/0.33 thf(fact_188_split__fst__asm,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: A > $o,P: product_prod @ A @ B] :
% 0.16/0.33 ( ( R @ ( product_fst @ A @ B @ P ) )
% 0.16/0.33 = ( ~ ? [X3: A,Y4: B] :
% 0.16/0.33 ( ( P
% 0.16/0.33 = ( product_Pair @ A @ B @ X3 @ Y4 ) )
% 0.16/0.33 & ~ ( R @ X3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % split_fst_asm
% 0.16/0.33 thf(fact_189_fst__eqD,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,X: A,Y3: B,A3: A] :
% 0.16/0.33 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y3 ) )
% 0.16/0.33 = A3 )
% 0.16/0.33 => ( X = A3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % fst_eqD
% 0.16/0.33 thf(fact_190_fst__conv,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 0.16/0.33 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.16/0.33 = X1 ) ).
% 0.16/0.33
% 0.16/0.33 % fst_conv
% 0.16/0.33 thf(fact_191_snd__eqD,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,X: B,Y3: A,A3: A] :
% 0.16/0.33 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
% 0.16/0.33 = A3 )
% 0.16/0.33 => ( Y3 = A3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % snd_eqD
% 0.16/0.33 thf(fact_192_snd__conv,axiom,
% 0.16/0.33 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 0.16/0.33 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 0.16/0.33 = X22 ) ).
% 0.16/0.33
% 0.16/0.33 % snd_conv
% 0.16/0.33 thf(fact_193_prod_Oswap__def,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( product_swap @ A @ B )
% 0.16/0.33 = ( ^ [P4: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P4 ) @ ( product_fst @ A @ B @ P4 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod.swap_def
% 0.16/0.33 thf(fact_194_prod_Oexhaust__sel,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.16/0.33 ( Prod
% 0.16/0.33 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod.exhaust_sel
% 0.16/0.33 thf(fact_195_surjective__pairing,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,T4: product_prod @ A @ B] :
% 0.16/0.33 ( T4
% 0.16/0.33 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T4 ) @ ( product_snd @ A @ B @ T4 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % surjective_pairing
% 0.16/0.33 thf(fact_196_case__prod__beta,axiom,
% 0.16/0.33 ! [A: $tType,C: $tType,B: $tType] :
% 0.16/0.33 ( ( product_case_prod @ B @ C @ A )
% 0.16/0.33 = ( ^ [F4: B > C > A,P4: product_prod @ B @ C] : ( F4 @ ( product_fst @ B @ C @ P4 ) @ ( product_snd @ B @ C @ P4 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prod_beta
% 0.16/0.33 thf(fact_197_prod_Ocase__eq__if,axiom,
% 0.16/0.33 ! [C: $tType,B: $tType,A: $tType] :
% 0.16/0.33 ( ( product_case_prod @ A @ B @ C )
% 0.16/0.33 = ( ^ [F4: A > B > C,Prod3: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod.case_eq_if
% 0.16/0.33 thf(fact_198_Product__Type_OCollect__case__prodD,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A7: A > B > $o] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A7 ) ) )
% 0.16/0.33 => ( A7 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Product_Type.Collect_case_prodD
% 0.16/0.33 thf(fact_199_split__comp__eq,axiom,
% 0.16/0.33 ! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G3: D > A] :
% 0.16/0.33 ( ( ^ [U: product_prod @ D @ B] : ( F @ ( G3 @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
% 0.16/0.33 = ( product_case_prod @ D @ B @ C
% 0.16/0.33 @ ^ [X3: D] : ( F @ ( G3 @ X3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % split_comp_eq
% 0.16/0.33 thf(fact_200_case__prod__beta_H,axiom,
% 0.16/0.33 ! [C: $tType,B: $tType,A: $tType] :
% 0.16/0.33 ( ( product_case_prod @ A @ B @ C )
% 0.16/0.33 = ( ^ [F4: A > B > C,X3: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prod_beta'
% 0.16/0.33 thf(fact_201_case__prod__unfold,axiom,
% 0.16/0.33 ! [C: $tType,B: $tType,A: $tType] :
% 0.16/0.33 ( ( product_case_prod @ A @ B @ C )
% 0.16/0.33 = ( ^ [C6: A > B > C,P4: product_prod @ A @ B] : ( C6 @ ( product_fst @ A @ B @ P4 ) @ ( product_snd @ A @ B @ P4 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % case_prod_unfold
% 0.16/0.33 thf(fact_202_prod_Osplit__sel,axiom,
% 0.16/0.33 ! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.16/0.33 ( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.16/0.33 = ( ( Prod
% 0.16/0.33 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.16/0.33 => ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod.split_sel
% 0.16/0.33 thf(fact_203_prod_Osplit__sel__asm,axiom,
% 0.16/0.33 ! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.16/0.33 ( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.16/0.33 = ( ~ ( ( Prod
% 0.16/0.33 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.16/0.33 & ~ ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prod.split_sel_asm
% 0.16/0.33 thf(fact_204_fst__def,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( product_fst @ A @ B )
% 0.16/0.33 = ( product_case_prod @ A @ B @ A
% 0.16/0.33 @ ^ [X12: A,X23: B] : X12 ) ) ).
% 0.16/0.33
% 0.16/0.33 % fst_def
% 0.16/0.33 thf(fact_205_snd__def,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( product_snd @ A @ B )
% 0.16/0.33 = ( product_case_prod @ A @ B @ B
% 0.16/0.33 @ ^ [X12: A,X23: B] : X23 ) ) ).
% 0.16/0.33
% 0.16/0.33 % snd_def
% 0.16/0.33 thf(fact_206_The__case__prod,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P2: A > B > $o] :
% 0.16/0.33 ( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) )
% 0.16/0.33 = ( the @ ( product_prod @ A @ B )
% 0.16/0.33 @ ^ [Xy: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % The_case_prod
% 0.16/0.33 thf(fact_207_mem__Times__iff,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,X: product_prod @ A @ B,A7: set @ A,B7: set @ B] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ X
% 0.16/0.33 @ ( product_Sigma @ A @ B @ A7
% 0.16/0.33 @ ^ [Uu: A] : B7 ) )
% 0.16/0.33 = ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A7 )
% 0.16/0.33 & ( member @ B @ ( product_snd @ A @ B @ X ) @ B7 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % mem_Times_iff
% 0.16/0.33 thf(fact_208_exE__realizer,axiom,
% 0.16/0.33 ! [C: $tType,A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A,Q: C > $o,F: B > A > C] :
% 0.16/0.33 ( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
% 0.16/0.33 => ( ! [X2: B,Y: A] :
% 0.16/0.33 ( ( P2 @ Y @ X2 )
% 0.16/0.33 => ( Q @ ( F @ X2 @ Y ) ) )
% 0.16/0.33 => ( Q @ ( product_case_prod @ B @ A @ C @ F @ P ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % exE_realizer
% 0.16/0.33 thf(fact_209_exI__realizer,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P2: A > B > $o,Y3: A,X: B] :
% 0.16/0.33 ( ( P2 @ Y3 @ X )
% 0.16/0.33 => ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % exI_realizer
% 0.16/0.33 thf(fact_210_conjI__realizer,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
% 0.16/0.33 ( ( P2 @ P )
% 0.16/0.33 => ( ( Q @ Q2 )
% 0.16/0.33 => ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
% 0.16/0.33 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % conjI_realizer
% 0.16/0.33 thf(fact_211_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,P2: A > B > $o,X: A,Y3: B,A3: product_prod @ A @ B] :
% 0.16/0.33 ( ( P2 @ X @ Y3 )
% 0.16/0.33 => ( ( A3
% 0.16/0.33 = ( product_Pair @ A @ B @ X @ Y3 ) )
% 0.16/0.33 => ( P2 @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % BNF_Greatest_Fixpoint.subst_Pair
% 0.16/0.33 thf(fact_212_above__def,axiom,
% 0.16/0.33 ! [A: $tType] :
% 0.16/0.33 ( ( order_above @ A )
% 0.16/0.33 = ( ^ [R2: set @ ( product_prod @ A @ A ),A4: A] :
% 0.16/0.33 ( collect @ A
% 0.16/0.33 @ ^ [B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % above_def
% 0.16/0.33 thf(fact_213_aboveS__def,axiom,
% 0.16/0.33 ! [A: $tType] :
% 0.16/0.33 ( ( order_aboveS @ A )
% 0.16/0.33 = ( ^ [R2: set @ ( product_prod @ A @ A ),A4: A] :
% 0.16/0.33 ( collect @ A
% 0.16/0.33 @ ^ [B4: A] :
% 0.16/0.33 ( ( B4 != A4 )
% 0.16/0.33 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % aboveS_def
% 0.16/0.33 thf(fact_214_sndI,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y3: A,Z: B] :
% 0.16/0.33 ( ( X
% 0.16/0.33 = ( product_Pair @ A @ B @ Y3 @ Z ) )
% 0.16/0.33 => ( ( product_snd @ A @ B @ X )
% 0.16/0.33 = Z ) ) ).
% 0.16/0.33
% 0.16/0.33 % sndI
% 0.16/0.33 thf(fact_215_fstI,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y3: A,Z: B] :
% 0.16/0.33 ( ( X
% 0.16/0.33 = ( product_Pair @ A @ B @ Y3 @ Z ) )
% 0.16/0.33 => ( ( product_fst @ A @ B @ X )
% 0.16/0.33 = Y3 ) ) ).
% 0.16/0.33
% 0.16/0.33 % fstI
% 0.16/0.33 thf(fact_216_eq__snd__iff,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,B3: A,P: product_prod @ B @ A] :
% 0.16/0.33 ( ( B3
% 0.16/0.33 = ( product_snd @ B @ A @ P ) )
% 0.16/0.33 = ( ? [A4: B] :
% 0.16/0.33 ( P
% 0.16/0.33 = ( product_Pair @ B @ A @ A4 @ B3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % eq_snd_iff
% 0.16/0.33 thf(fact_217_eq__fst__iff,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,A3: A,P: product_prod @ A @ B] :
% 0.16/0.33 ( ( A3
% 0.16/0.33 = ( product_fst @ A @ B @ P ) )
% 0.16/0.33 = ( ? [B4: B] :
% 0.16/0.33 ( P
% 0.16/0.33 = ( product_Pair @ A @ B @ A3 @ B4 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % eq_fst_iff
% 0.16/0.33 thf(fact_218_subset__fst__snd,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A7: set @ ( product_prod @ A @ B )] :
% 0.16/0.33 ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A7
% 0.16/0.33 @ ( product_Sigma @ A @ B @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A7 )
% 0.16/0.33 @ ^ [Uu: A] : ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A7 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % subset_fst_snd
% 0.16/0.33 thf(fact_219_scomp__unfold,axiom,
% 0.16/0.33 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.16/0.33 ( ( product_scomp @ A @ B @ C @ D )
% 0.16/0.33 = ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X3: A] : ( G4 @ ( product_fst @ B @ C @ ( F4 @ X3 ) ) @ ( product_snd @ B @ C @ ( F4 @ X3 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % scomp_unfold
% 0.16/0.33 thf(fact_220_scomp__apply,axiom,
% 0.16/0.33 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.16/0.33 ( ( product_scomp @ B @ C @ D @ A )
% 0.16/0.33 = ( ^ [F4: B > ( product_prod @ C @ D ),G4: C > D > A,X3: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F4 @ X3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % scomp_apply
% 0.16/0.33 thf(fact_221_subset__Collect__iff,axiom,
% 0.16/0.33 ! [A: $tType,B7: set @ A,A7: set @ A,P2: A > $o] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ A ) @ B7 @ A7 )
% 0.16/0.33 => ( ( ord_less_eq @ ( set @ A ) @ B7
% 0.16/0.33 @ ( collect @ A
% 0.16/0.33 @ ^ [X3: A] :
% 0.16/0.33 ( ( member @ A @ X3 @ A7 )
% 0.16/0.33 & ( P2 @ X3 ) ) ) )
% 0.16/0.33 = ( ! [X3: A] :
% 0.16/0.33 ( ( member @ A @ X3 @ B7 )
% 0.16/0.33 => ( P2 @ X3 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % subset_Collect_iff
% 0.16/0.33 thf(fact_222_subset__CollectI,axiom,
% 0.16/0.33 ! [A: $tType,B7: set @ A,A7: set @ A,Q: A > $o,P2: A > $o] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ A ) @ B7 @ A7 )
% 0.16/0.33 => ( ! [X2: A] :
% 0.16/0.33 ( ( member @ A @ X2 @ B7 )
% 0.16/0.33 => ( ( Q @ X2 )
% 0.16/0.33 => ( P2 @ X2 ) ) )
% 0.16/0.33 => ( ord_less_eq @ ( set @ A )
% 0.16/0.33 @ ( collect @ A
% 0.16/0.33 @ ^ [X3: A] :
% 0.16/0.33 ( ( member @ A @ X3 @ B7 )
% 0.16/0.33 & ( Q @ X3 ) ) )
% 0.16/0.33 @ ( collect @ A
% 0.16/0.33 @ ^ [X3: A] :
% 0.16/0.33 ( ( member @ A @ X3 @ A7 )
% 0.16/0.33 & ( P2 @ X3 ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % subset_CollectI
% 0.16/0.33 thf(fact_223_Sigma__mono,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A7: set @ A,C5: set @ A,B7: A > ( set @ B ),D3: A > ( set @ B )] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ A ) @ A7 @ C5 )
% 0.16/0.33 => ( ! [X2: A] :
% 0.16/0.33 ( ( member @ A @ X2 @ A7 )
% 0.16/0.33 => ( ord_less_eq @ ( set @ B ) @ ( B7 @ X2 ) @ ( D3 @ X2 ) ) )
% 0.16/0.33 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) @ ( product_Sigma @ A @ B @ C5 @ D3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Sigma_mono
% 0.16/0.33 thf(fact_224_Times__subset__cancel2,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,X: A,C5: set @ A,A7: set @ B,B7: set @ B] :
% 0.16/0.33 ( ( member @ A @ X @ C5 )
% 0.16/0.33 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 0.16/0.33 @ ( product_Sigma @ B @ A @ A7
% 0.16/0.33 @ ^ [Uu: B] : C5 )
% 0.16/0.33 @ ( product_Sigma @ B @ A @ B7
% 0.16/0.33 @ ^ [Uu: B] : C5 ) )
% 0.16/0.33 = ( ord_less_eq @ ( set @ B ) @ A7 @ B7 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Times_subset_cancel2
% 0.16/0.33 thf(fact_225_subrelI,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.16/0.33 ( ! [X2: A,Y: B] :
% 0.16/0.33 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R4 )
% 0.16/0.33 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S2 ) )
% 0.16/0.33 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S2 ) ) ).
% 0.16/0.33
% 0.16/0.33 % subrelI
% 0.16/0.33 thf(fact_226_subset__image__iff,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,B7: set @ A,F: B > A,A7: set @ B] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F @ A7 ) )
% 0.16/0.33 = ( ? [AA: set @ B] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ B ) @ AA @ A7 )
% 0.16/0.33 & ( B7
% 0.16/0.33 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % subset_image_iff
% 0.16/0.33 thf(fact_227_image__subset__iff,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,F: B > A,A7: set @ B,B7: set @ A] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A7 ) @ B7 )
% 0.16/0.33 = ( ! [X3: B] :
% 0.16/0.33 ( ( member @ B @ X3 @ A7 )
% 0.16/0.33 => ( member @ A @ ( F @ X3 ) @ B7 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % image_subset_iff
% 0.16/0.33 thf(fact_228_subset__imageE,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,B7: set @ A,F: B > A,A7: set @ B] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F @ A7 ) )
% 0.16/0.33 => ~ ! [C7: set @ B] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ B ) @ C7 @ A7 )
% 0.16/0.33 => ( B7
% 0.16/0.33 != ( image @ B @ A @ F @ C7 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % subset_imageE
% 0.16/0.33 thf(fact_229_image__subsetI,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,A7: set @ A,F: A > B,B7: set @ B] :
% 0.16/0.33 ( ! [X2: A] :
% 0.16/0.33 ( ( member @ A @ X2 @ A7 )
% 0.16/0.33 => ( member @ B @ ( F @ X2 ) @ B7 ) )
% 0.16/0.33 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A7 ) @ B7 ) ) ).
% 0.16/0.33
% 0.16/0.33 % image_subsetI
% 0.16/0.33 thf(fact_230_image__mono,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A7: set @ A,B7: set @ A,F: A > B] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.16/0.33 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A7 ) @ ( image @ A @ B @ F @ B7 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % image_mono
% 0.16/0.33 thf(fact_231_conj__subset__def,axiom,
% 0.16/0.33 ! [A: $tType,A7: set @ A,P2: A > $o,Q: A > $o] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ A ) @ A7
% 0.16/0.33 @ ( collect @ A
% 0.16/0.33 @ ^ [X3: A] :
% 0.16/0.33 ( ( P2 @ X3 )
% 0.16/0.33 & ( Q @ X3 ) ) ) )
% 0.16/0.33 = ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( collect @ A @ P2 ) )
% 0.16/0.33 & ( ord_less_eq @ ( set @ A ) @ A7 @ ( collect @ A @ Q ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % conj_subset_def
% 0.16/0.33 thf(fact_232_Collect__restrict,axiom,
% 0.16/0.33 ! [A: $tType,X7: set @ A,P2: A > $o] :
% 0.16/0.33 ( ord_less_eq @ ( set @ A )
% 0.16/0.33 @ ( collect @ A
% 0.16/0.33 @ ^ [X3: A] :
% 0.16/0.33 ( ( member @ A @ X3 @ X7 )
% 0.16/0.33 & ( P2 @ X3 ) ) )
% 0.16/0.33 @ X7 ) ).
% 0.16/0.33
% 0.16/0.33 % Collect_restrict
% 0.16/0.33 thf(fact_233_prop__restrict,axiom,
% 0.16/0.33 ! [A: $tType,X: A,Z4: set @ A,X7: set @ A,P2: A > $o] :
% 0.16/0.33 ( ( member @ A @ X @ Z4 )
% 0.16/0.33 => ( ( ord_less_eq @ ( set @ A ) @ Z4
% 0.16/0.33 @ ( collect @ A
% 0.16/0.33 @ ^ [X3: A] :
% 0.16/0.33 ( ( member @ A @ X3 @ X7 )
% 0.16/0.33 & ( P2 @ X3 ) ) ) )
% 0.16/0.33 => ( P2 @ X ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % prop_restrict
% 0.16/0.33 thf(fact_234_scomp__scomp,axiom,
% 0.16/0.33 ! [A: $tType,C: $tType,D: $tType,B: $tType,F2: $tType,E: $tType,F: A > ( product_prod @ E @ F2 ),G3: E > F2 > ( product_prod @ C @ D ),H: C > D > B] :
% 0.16/0.33 ( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F2 @ ( product_prod @ C @ D ) @ F @ G3 ) @ H )
% 0.16/0.33 = ( product_scomp @ A @ E @ F2 @ B @ F
% 0.16/0.33 @ ^ [X3: E] : ( product_scomp @ F2 @ C @ D @ B @ ( G3 @ X3 ) @ H ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % scomp_scomp
% 0.16/0.33 thf(fact_235_wlog__linorder__le,axiom,
% 0.16/0.33 ! [A: $tType] :
% 0.16/0.33 ( ( linorder @ A @ ( type @ A ) )
% 0.16/0.33 => ! [P2: A > A > $o,B3: A,A3: A] :
% 0.16/0.33 ( ! [A2: A,B2: A] :
% 0.16/0.33 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.33 => ( P2 @ A2 @ B2 ) )
% 0.16/0.33 => ( ( ( P2 @ B3 @ A3 )
% 0.16/0.33 => ( P2 @ A3 @ B3 ) )
% 0.16/0.33 => ( P2 @ A3 @ B3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % wlog_linorder_le
% 0.16/0.33 thf(fact_236_scomp__Pair,axiom,
% 0.16/0.33 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 0.16/0.33 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 0.16/0.33 = X ) ).
% 0.16/0.33
% 0.16/0.33 % scomp_Pair
% 0.16/0.33 thf(fact_237_image__Collect__subsetI,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,P2: A > $o,F: A > B,B7: set @ B] :
% 0.16/0.33 ( ! [X2: A] :
% 0.16/0.33 ( ( P2 @ X2 )
% 0.16/0.33 => ( member @ B @ ( F @ X2 ) @ B7 ) )
% 0.16/0.33 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( collect @ A @ P2 ) ) @ B7 ) ) ).
% 0.16/0.33
% 0.16/0.33 % image_Collect_subsetI
% 0.16/0.33 thf(fact_238_scomp__def,axiom,
% 0.16/0.33 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.16/0.33 ( ( product_scomp @ A @ B @ C @ D )
% 0.16/0.33 = ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X3: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F4 @ X3 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % scomp_def
% 0.16/0.33 thf(fact_239_Pair__scomp,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,C: $tType,X: C,F: C > A > B] :
% 0.16/0.33 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F )
% 0.16/0.33 = ( F @ X ) ) ).
% 0.16/0.33
% 0.16/0.33 % Pair_scomp
% 0.16/0.33 thf(fact_240_refl__onI,axiom,
% 0.16/0.33 ! [A: $tType,R4: set @ ( product_prod @ A @ A ),A7: set @ A] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4
% 0.16/0.33 @ ( product_Sigma @ A @ A @ A7
% 0.16/0.33 @ ^ [Uu: A] : A7 ) )
% 0.16/0.33 => ( ! [X2: A] :
% 0.16/0.33 ( ( member @ A @ X2 @ A7 )
% 0.16/0.33 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R4 ) )
% 0.16/0.33 => ( refl_on @ A @ A7 @ R4 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % refl_onI
% 0.16/0.33 thf(fact_241_refl__on__def,axiom,
% 0.16/0.33 ! [A: $tType] :
% 0.16/0.33 ( ( refl_on @ A )
% 0.16/0.33 = ( ^ [A8: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
% 0.16/0.33 @ ( product_Sigma @ A @ A @ A8
% 0.16/0.33 @ ^ [Uu: A] : A8 ) )
% 0.16/0.33 & ! [X3: A] :
% 0.16/0.33 ( ( member @ A @ X3 @ A8 )
% 0.16/0.33 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R2 ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % refl_on_def
% 0.16/0.33 thf(fact_242_Collect__split__mono__strong,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,X7: set @ A,A7: set @ ( product_prod @ A @ B ),Y8: set @ B,P2: A > B > $o,Q: A > B > $o] :
% 0.16/0.33 ( ( X7
% 0.16/0.33 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A7 ) )
% 0.16/0.33 => ( ( Y8
% 0.16/0.33 = ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A7 ) )
% 0.16/0.33 => ( ! [X2: A] :
% 0.16/0.33 ( ( member @ A @ X2 @ X7 )
% 0.16/0.33 => ! [Xa: B] :
% 0.16/0.33 ( ( member @ B @ Xa @ Y8 )
% 0.16/0.33 => ( ( P2 @ X2 @ Xa )
% 0.16/0.33 => ( Q @ X2 @ Xa ) ) ) )
% 0.16/0.33 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A7 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) ) )
% 0.16/0.33 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A7 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Collect_split_mono_strong
% 0.16/0.33 thf(fact_243_subset__snd__imageI,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A7: set @ A,B7: set @ B,S: set @ ( product_prod @ A @ B ),X: A] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
% 0.16/0.33 @ ( product_Sigma @ A @ B @ A7
% 0.16/0.33 @ ^ [Uu: A] : B7 )
% 0.16/0.33 @ S )
% 0.16/0.33 => ( ( member @ A @ X @ A7 )
% 0.16/0.33 => ( ord_less_eq @ ( set @ B ) @ B7 @ ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % subset_snd_imageI
% 0.16/0.33 thf(fact_244_subset__fst__imageI,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A7: set @ A,B7: set @ B,S: set @ ( product_prod @ A @ B ),Y3: B] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
% 0.16/0.33 @ ( product_Sigma @ A @ B @ A7
% 0.16/0.33 @ ^ [Uu: A] : B7 )
% 0.16/0.33 @ S )
% 0.16/0.33 => ( ( member @ B @ Y3 @ B7 )
% 0.16/0.33 => ( ord_less_eq @ ( set @ A ) @ A7 @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % subset_fst_imageI
% 0.16/0.33 thf(fact_245_Quotient__rep__abs__eq,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,T4: A] :
% 0.16/0.33 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 0.16/0.33 => ( ( R @ T4 @ T4 )
% 0.16/0.33 => ( ( ord_less_eq @ ( A > A > $o ) @ R
% 0.16/0.33 @ ^ [Y7: A,Z2: A] : ( Y7 = Z2 ) )
% 0.16/0.33 => ( ( Rep @ ( Abs @ T4 ) )
% 0.16/0.33 = T4 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Quotient_rep_abs_eq
% 0.16/0.33 thf(fact_246_reflp__ge__eq,axiom,
% 0.16/0.33 ! [A: $tType,R: A > A > $o] :
% 0.16/0.33 ( ( reflp @ A @ R )
% 0.16/0.33 => ( ord_less_eq @ ( A > A > $o )
% 0.16/0.33 @ ^ [Y7: A,Z2: A] : ( Y7 = Z2 )
% 0.16/0.33 @ R ) ) ).
% 0.16/0.33
% 0.16/0.33 % reflp_ge_eq
% 0.16/0.33 thf(fact_247_Collect__case__prod__mono,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,A7: A > B > $o,B7: A > B > $o] :
% 0.16/0.33 ( ( ord_less_eq @ ( A > B > $o ) @ A7 @ B7 )
% 0.16/0.33 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A7 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B7 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % Collect_case_prod_mono
% 0.16/0.33 thf(fact_248_pred__subset__eq,axiom,
% 0.16/0.33 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.16/0.33 ( ( ord_less_eq @ ( A > $o )
% 0.16/0.33 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 0.16/0.33 @ ^ [X3: A] : ( member @ A @ X3 @ S ) )
% 0.16/0.33 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 0.16/0.33
% 0.16/0.33 % pred_subset_eq
% 0.16/0.33 thf(fact_249_pred__subset__eq2,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.16/0.33 ( ( ord_less_eq @ ( A > B > $o )
% 0.16/0.33 @ ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R )
% 0.16/0.33 @ ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S ) )
% 0.16/0.33 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.16/0.33
% 0.16/0.33 % pred_subset_eq2
% 0.16/0.33 thf(fact_250_less__eq__set__def,axiom,
% 0.16/0.33 ! [A: $tType] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ A ) )
% 0.16/0.33 = ( ^ [A8: set @ A,B8: set @ A] :
% 0.16/0.33 ( ord_less_eq @ ( A > $o )
% 0.16/0.33 @ ^ [X3: A] : ( member @ A @ X3 @ A8 )
% 0.16/0.33 @ ^ [X3: A] : ( member @ A @ X3 @ B8 ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % less_eq_set_def
% 0.16/0.33 thf(fact_251_reflp__eq,axiom,
% 0.16/0.33 ! [A: $tType] :
% 0.16/0.33 ( ( reflp @ A )
% 0.16/0.33 = ( ord_less_eq @ ( A > A > $o )
% 0.16/0.33 @ ^ [Y7: A,Z2: A] : ( Y7 = Z2 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % reflp_eq
% 0.16/0.33 thf(fact_252_Gr__incl,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,A7: set @ A,F: A > B,B7: set @ B] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A7 @ F )
% 0.16/0.33 @ ( product_Sigma @ A @ B @ A7
% 0.16/0.33 @ ^ [Uu: A] : B7 ) )
% 0.16/0.33 = ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A7 ) @ B7 ) ) ).
% 0.16/0.33
% 0.16/0.33 % Gr_incl
% 0.16/0.33 thf(fact_253_flip__pred,axiom,
% 0.16/0.33 ! [A: $tType,B: $tType,A7: set @ ( product_prod @ A @ B ),R: B > A > $o] :
% 0.16/0.33 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A7 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( conversep @ B @ A @ R ) ) ) )
% 0.16/0.33 => ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 0.16/0.33 @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
% 0.16/0.33 @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
% 0.16/0.33 @ ^ [X3: A,Y4: B] : ( product_Pair @ B @ A @ Y4 @ X3 ) )
% 0.16/0.33 @ A7 )
% 0.16/0.33 @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R ) ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % flip_pred
% 0.16/0.33 thf(fact_254_conversep__eq,axiom,
% 0.16/0.33 ! [A: $tType] :
% 0.16/0.33 ( ( conversep @ A @ A
% 0.16/0.33 @ ^ [Y7: A,Z2: A] : ( Y7 = Z2 ) )
% 0.16/0.33 = ( ^ [Y7: A,Z2: A] : ( Y7 = Z2 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % conversep_eq
% 0.16/0.33 thf(fact_255_conversep__iff,axiom,
% 0.16/0.33 ! [B: $tType,A: $tType] :
% 0.16/0.33 ( ( conversep @ A @ B )
% 0.16/0.33 = ( ^ [R2: A > B > $o,A4: B,B4: A] : ( R2 @ B4 @ A4 ) ) ) ).
% 0.16/0.33
% 0.16/0.33 % conversep_iff
% 0.16/0.33
% 0.16/0.33 %----Type constructors (2)
% 0.16/0.33 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.16/0.33 linorder @ $o @ ( type @ $o ) ).
% 0.16/0.33
% 0.16/0.33 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_1,axiom,
% 0.16/0.33 linorder @ product_unit @ ( type @ product_unit ) ).
% 0.16/0.33
% 0.16/0.33 %----Helper facts (3)
% 0.16/0.33 thf(help_If_3_1_T,axiom,
% 0.16/0.33 ! [P2: $o] :
% 0.16/0.33 ( ( P2 = $true )
% 0.16/0.33 | ( P2 = $false ) ) ).
% 0.16/0.33
% 0.16/0.33 thf(help_If_2_1_T,axiom,
% 0.16/0.33 ! [A: $tType,X: A,Y3: A] :
% 0.16/0.33 ( ( if @ A @ $false @ X @ Y3 )
% 0.16/0.33 = Y3 ) ).
% 0.16/0.33
% 0.16/0.33 thf(help_If_1_1_T,axiom,
% 0.16/0.33 ! [A: $tType,X: A,Y3: A] :
% 0.16/0.33 ( ( if @ A @ $true @ X @ Y3 )
% 0.16/0.33 = X ) ).
% 0.16/0.33
% 0.16/0.33 %----Conjectures (2)
% 0.16/0.33 thf(conj_0,hypothesis,
% 0.16/0.33 ( product_case_prod @ ( coinductive_llist @ a ) @ b @ ( ( product_prod @ ( coinductive_llist @ a ) @ b ) > $o )
% 0.16/0.33 @ ^ [Xs2: coinductive_llist @ a,A4: b] :
% 0.16/0.33 ( product_case_prod @ ( coinductive_llist @ a ) @ b @ $o
% 0.16/0.33 @ ^ [Ys: coinductive_llist @ a,B4: b] :
% 0.16/0.33 ( ( Xs2 = Ys )
% 0.16/0.33 & ( ( coinductive_lfinite @ a @ Ys )
% 0.16/0.33 => ( A4 = B4 ) ) ) )
% 0.16/0.33 @ prod1
% 0.16/0.33 @ prod2 ) ).
% 0.16/0.33
% 0.16/0.33 thf(conj_1,conjecture,
% 0.16/0.33 ( product_case_prod @ ( coinductive_llist @ a ) @ b @ ( ( product_prod @ ( coinductive_llist @ a ) @ b ) > $o )
% 0.16/0.33 @ ^ [Xs2: coinductive_llist @ a,A4: b] :
% 0.16/0.33 ( product_case_prod @ ( coinductive_llist @ a ) @ b @ $o
% 0.16/0.33 @ ^ [Ys: coinductive_llist @ a,B4: b] :
% 0.16/0.33 ( ( Xs2 = Ys )
% 0.16/0.33 & ( ( coinductive_lfinite @ a @ Ys )
% 0.16/0.33 => ( A4 = B4 ) ) ) )
% 0.16/0.33 @ ( product_case_prod @ ( coinductive_llist @ a ) @ b @ ( product_prod @ ( coinductive_llist @ a ) @ b )
% 0.16/0.33 @ ^ [Xs2: coinductive_llist @ a,B6: b] : ( product_Pair @ ( coinductive_llist @ a ) @ b @ ( coinductive_lfilter @ a @ fun @ Xs2 ) @ ( if @ b @ ( coinductive_lfinite @ a @ Xs2 ) @ B6 @ b2 ) )
% 0.16/0.33 @ prod1 )
% 0.16/0.33 @ ( product_case_prod @ ( coinductive_llist @ a ) @ b @ ( product_prod @ ( coinductive_llist @ a ) @ b )
% 0.16/0.33 @ ^ [Xs2: coinductive_llist @ a,B6: b] : ( product_Pair @ ( coinductive_llist @ a ) @ b @ ( coinductive_lfilter @ a @ fun @ Xs2 ) @ ( if @ b @ ( coinductive_lfinite @ a @ Xs2 ) @ B6 @ b2 ) )
% 0.16/0.33 @ prod2 ) ) ).
% 0.16/0.33
% 0.16/0.33 %------------------------------------------------------------------------------
% 0.16/0.33 ------------------------
% 1.75/1.11 ---- Embedded in THF ---
% 1.75/1.11 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.75/1.11 %%% Generated on Tue Feb 24 06:11:01 EST 2026
% 1.75/1.11 %%% using '$$dhol' embedding, version 1.3.0.
% 1.75/1.11 %%% Logic specification used:
% 1.75/1.11 %%% thf(spec, logic, $$dhol).
% 1.75/1.11
% 1.75/1.11 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.ihDMkoi04K/DTF2DTF29522.p
% 1.75/1.11 thf(ty_t_TLList__Mirabelle__qhjoikztpd_Otllist, type, tLList446370796tllist: ($tType > ($tType > $tType))).
% 1.75/1.11 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 1.75/1.11 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 1.75/1.11 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 1.75/1.11 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 1.75/1.11 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 1.75/1.11 thf(ty_tf_b, type, b: $tType).
% 1.75/1.11 thf(ty_tf_a, type, a: $tType).
% 1.75/1.11 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.75/1.11 thf(sy_c_BNF__Def_OGr, type, bNF_Gr: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (set @ ((product_prod @ A) @ B))))))).
% 1.75/1.11 thf(sy_c_Coinductive__List_Olfilter, type, coinductive_lfilter: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 1.75/1.11 thf(sy_c_Coinductive__List_Olfinite, type, coinductive_lfinite: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 1.75/1.11 thf(sy_c_Fun__Def_Oreduction__pair, type, fun_reduction_pair: (!> [A:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > $o)))).
% 1.75/1.11 thf(sy_c_Fun__Def_Orp__inv__image, type, fun_rp_inv_image: (!> [A:$tType,B:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))))))).
% 1.75/1.11 thf(sy_c_HOL_OThe, type, the: (!> [A:$tType]: (((A > $o) > A)))).
% 1.75/1.11 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 1.75/1.11 thf(sy_c_Lifting_OQuotient, type, quotient: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > ((B > A) > ((A > (B > $o)) > $o))))))).
% 1.75/1.11 thf(sy_c_Order__Relation_Oabove, type, order_above: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 1.75/1.11 thf(sy_c_Order__Relation_OaboveS, type, order_aboveS: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 1.75/1.11 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 1.75/1.11 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 1.75/1.11 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 1.75/1.11 thf(sy_c_Product__Type_OSigma, type, product_Sigma: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ ((product_prod @ A) @ B))))))).
% 1.75/1.11 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 1.75/1.11 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 1.75/1.11 thf(sy_c_Product__Type_Oold_Obool_Orec__bool, type, product_rec_bool: (!> [T:$tType]: ((T > (T > ($o > T)))))).
% 1.75/1.11 thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool, type, product_rec_set_bool: (!> [T:$tType]: ((T > (T > ($o > (T > $o))))))).
% 1.75/1.11 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 1.75/1.11 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod, type, product_rec_set_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > (T > $o)))))).
% 1.75/1.11 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit, type, product_rec_set_unit: (!> [T:$tType]: ((T > (product_unit > (T > $o)))))).
% 1.75/1.11 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit, type, product_rec_unit: (!> [T:$tType]: ((T > (product_unit > T))))).
% 1.75/1.11 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 1.75/1.11 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 1.75/1.11 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 1.75/1.11 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 1.75/1.11 thf(sy_c_Product__Type_Oproduct, type, product_product: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((product_prod @ A) @ B))))))).
% 1.75/1.11 thf(sy_c_Product__Type_Oscomp, type, product_scomp: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: (((A > ((product_prod @ B) @ C)) > ((B > (C > D)) > (A > D)))))).
% 1.75/1.11 thf(sy_c_Pure_Otype, type, type: (!> [A:$tType]: ((itself @ A)))).
% 1.75/1.11 thf(sy_c_Relation_Oconversep, type, conversep: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > (B > (A > $o)))))).
% 1.75/1.11 thf(sy_c_Relation_Oinv__image, type, inv_image: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((A > B) > (set @ ((product_prod @ A) @ A))))))).
% 1.75/1.11 thf(sy_c_Relation_Oinv__imagep, type, inv_imagep: (!> [B:$tType,A:$tType]: (((B > (B > $o)) > ((A > B) > (A > (A > $o))))))).
% 1.75/1.11 thf(sy_c_Relation_Orefl__on, type, refl_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 1.75/1.11 thf(sy_c_Relation_Oreflp, type, reflp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 1.75/1.11 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 1.75/1.11 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 1.75/1.11 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Ocr__tllist, type, tLList47617868tllist: (!> [A:$tType,B:$tType]: ((((product_prod @ (coinductive_llist @ A)) @ B) > (((tLList446370796tllist @ A) @ B) > $o))))).
% 1.75/1.11 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Ollist__of__tllist, type, tLList798109904tllist: (!> [A:$tType,B:$tType]: ((((tLList446370796tllist @ A) @ B) > (coinductive_llist @ A))))).
% 1.75/1.11 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Oterminal0, type, tLList1825092077minal0: (!> [A:$tType]: (A))).
% 1.75/1.11 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Ocase__tllist, type, tLList200813139tllist: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > (((tLList446370796tllist @ A) @ B) > C)) > (((tLList446370796tllist @ A) @ B) > C)))))).
% 1.75/1.11 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Oterminal, type, tLList2110128105rminal: (!> [A:$tType,B:$tType]: ((((tLList446370796tllist @ A) @ B) > B)))).
% 1.75/1.11 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist__of__llist, type, tLList1672613558_llist: (!> [B:$tType,A:$tType]: ((B > ((coinductive_llist @ A) > ((tLList446370796tllist @ A) @ B)))))).
% 1.75/1.11 thf(sy_c_Wellfounded_Olex__prod, type, lex_prod: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 1.75/1.11 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 1.75/1.11 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 1.75/1.11 thf(sy_v_b, type, b2: b).
% 1.75/1.11 thf(sy_v_fun, type, fun: (a > $o)).
% 1.75/1.11 thf(sy_v_prod1, type, prod1: ((product_prod @ (coinductive_llist @ a)) @ b)).
% 1.75/1.11 thf(sy_v_prod2, type, prod2: ((product_prod @ (coinductive_llist @ a)) @ b)).
% 1.75/1.11 thf(fact_0_case__prodI2_H, axiom, (! [A:$tType,B:$tType,C:$tType,P:((product_prod @ A) @ B),C2:(A > (B > (C > $o))),X:C]: (((! [A2:A,B2:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = P) => (((C2 @ A2) @ B2) @ X)))) => ((((((product_case_prod @ A) @ B) @ (C > $o)) @ C2) @ P) @ X))))).
% 1.75/1.11 thf(fact_1_case__prodI, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A3:A,B3:B]: ((((F @ A3) @ B3) => (((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A3) @ B3)))))).
% 1.75/1.11 thf(fact_2_case__prodI2, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B),C2:(A > (B > $o))]: (((! [A2:A,B2:B]: (((P = ((((product_Pair @ A) @ B) @ A2) @ B2)) => ((C2 @ A2) @ B2)))) => (((((product_case_prod @ A) @ B) @ $o) @ C2) @ P))))).
% 1.75/1.11 thf(fact_3_case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),A3:B,B3:C]: (((((((product_case_prod @ B) @ C) @ A) @ F) @ ((((product_Pair @ B) @ C) @ A3) @ B3)) = ((F @ A3) @ B3))))).
% 1.75/1.11 thf(fact_4_lfilter__K__True, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_lfilter @ A) @ (^ [Uu:A]: ($true))) @ Xs) = Xs)))).
% 1.75/1.11 thf(fact_5_split__part, axiom, (! [B:$tType,A:$tType,P2:$o,Q:(A > (B > $o))]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [A4:A,B4:B]: ((P2 & ((Q @ A4) @ B4))))) = (^ [Ab:((product_prod @ A) @ B)]: ((P2 & (((((product_case_prod @ A) @ B) @ $o) @ Q) @ Ab)))))))).
% 1.75/1.11 thf(fact_6_lfilter__idem, axiom, (! [A:$tType,P2:(A > $o),Xs:(coinductive_llist @ A)]: (((((coinductive_lfilter @ A) @ P2) @ (((coinductive_lfilter @ A) @ P2) @ Xs)) = (((coinductive_lfilter @ A) @ P2) @ Xs))))).
% 1.75/1.11 thf(fact_7_case__prodD_H, axiom, (! [B:$tType,A:$tType,C:$tType,R:(A > (B > (C > $o))),A3:A,B3:B,C2:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ R) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ C2) => (((R @ A3) @ B3) @ C2))))).
% 1.75/1.11 thf(fact_8_case__prodE_H, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(A > (B > (C > $o))),P:((product_prod @ A) @ B),Z:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ C2) @ P) @ Z) => (~ ((! [X2:A,Y:B]: (((P = ((((product_Pair @ A) @ B) @ X2) @ Y)) => (~ ((((C2 @ X2) @ Y) @ Z)))))))))))).
% 1.75/1.11 thf(fact_9_case__prod__Pair__iden, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) @ P) = P)))).
% 1.75/1.11 thf(fact_10_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 1.75/1.11 thf(fact_11_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A5:A,B5:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A5) @ B5)) = ((A3 = A5) & (B3 = B5)))))).
% 1.75/1.11 thf(fact_12_mem__case__prodI2, axiom, (! [C:$tType,B:$tType,A:$tType,P:((product_prod @ A) @ B),Z:C,C2:(A > (B > (set @ C)))]: (((! [A2:A,B2:B]: (((P = ((((product_Pair @ A) @ B) @ A2) @ B2)) => (((member @ C) @ Z) @ ((C2 @ A2) @ B2))))) => (((member @ C) @ Z) @ (((((product_case_prod @ A) @ B) @ (set @ C)) @ C2) @ P)))))).
% 1.75/1.11 thf(fact_13_mem__case__prodI, axiom, (! [A:$tType,B:$tType,C:$tType,Z:A,C2:(B > (C > (set @ A))),A3:B,B3:C]: (((((member @ A) @ Z) @ ((C2 @ A3) @ B3)) => (((member @ A) @ Z) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ ((((product_Pair @ B) @ C) @ A3) @ B3))))))).
% 1.75/1.11 thf(fact_14_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z:A,C2:(B > (C > (set @ A))),P:((product_prod @ B) @ C)]: (((((member @ A) @ Z) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ P)) => (~ ((! [X2:B,Y:C]: (((P = ((((product_Pair @ B) @ C) @ X2) @ Y)) => (~ ((((member @ A) @ Z) @ ((C2 @ X2) @ Y))))))))))))).
% 1.75/1.11 thf(fact_15_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A2:A,B2:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A2) @ B2)))) => (P2 @ Prod))))).
% 1.75/1.11 thf(fact_16_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y3:((product_prod @ A) @ B)]: ((~ ((! [A2:A,B2:B]: ((Y3 != ((((product_Pair @ A) @ B) @ A2) @ B2))))))))).
% 1.75/1.11 thf(fact_17_prod__induct7, axiom, (! [G:$tType,F2:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G))))))]: (((! [A2:A,B2:B,C3:C,D2:D,E2:E,F3:F2,G2:G]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G)))))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G))))) @ B2) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G)) @ E2) @ ((((product_Pair @ F2) @ G) @ F3) @ G2))))))))) => (P2 @ X))))).
% 1.75/1.11 thf(fact_18_prod__induct6, axiom, (! [F2:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))))]: (((! [A2:A,B2:B,C3:C,D2:D,E2:E,F3:F2]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))) @ B2) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F2)) @ D2) @ ((((product_Pair @ E) @ F2) @ E2) @ F3)))))))) => (P2 @ X))))).
% 1.75/1.11 thf(fact_19_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A2:A,B2:B,C3:C,D2:D,E2:E]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B2) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P2 @ X))))).
% 1.75/1.11 thf(fact_20_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A2:A,B2:B,C3:C,D2:D]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B2) @ ((((product_Pair @ C) @ D) @ C3) @ D2)))))) => (P2 @ X))))).
% 1.75/1.11 thf(fact_21_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A2:A,B2:B,C3:C]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A2) @ ((((product_Pair @ B) @ C) @ B2) @ C3))))) => (P2 @ X))))).
% 1.75/1.11 thf(fact_22_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F2:$tType,G:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G))))))]: ((~ ((! [A2:A,B2:B,C3:C,D2:D,E2:E,F3:F2,G2:G]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G)))))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G))))) @ B2) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G)) @ E2) @ ((((product_Pair @ F2) @ G) @ F3) @ G2)))))))))))))).
% 1.75/1.11 thf(fact_23_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F2:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))))]: ((~ ((! [A2:A,B2:B,C3:C,D2:D,E2:E,F3:F2]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))) @ B2) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F2)) @ D2) @ ((((product_Pair @ E) @ F2) @ E2) @ F3))))))))))))).
% 1.75/1.11 thf(fact_24_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A2:A,B2:B,C3:C,D2:D,E2:E]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B2) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 1.75/1.11 thf(fact_25_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A2:A,B2:B,C3:C,D2:D]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A2) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B2) @ ((((product_Pair @ C) @ D) @ C3) @ D2))))))))))).
% 1.75/1.11 thf(fact_26_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A2:A,B2:B,C3:C]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A2) @ ((((product_Pair @ B) @ C) @ B2) @ C3)))))))))).
% 1.75/1.11 thf(fact_27_Pair__inject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A5:A,B5:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A5) @ B5)) => (~ (((A3 = A5) => (B3 != B5)))))))).
% 1.75/1.11 thf(fact_28_prod__cases, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),P:((product_prod @ A) @ B)]: (((! [A2:A,B2:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A2) @ B2)))) => (P2 @ P))))).
% 1.75/1.11 thf(fact_29_surj__pair, axiom, (! [A:$tType,B:$tType,P:((product_prod @ A) @ B)]: ((? [X2:A,Y:B]: ((P = ((((product_Pair @ A) @ B) @ X2) @ Y))))))).
% 1.75/1.11 thf(fact_30_prod_Ocase__distrib, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,H:(C > D),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((H @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = (((((product_case_prod @ A) @ B) @ D) @ (^ [X12:A,X23:B]: ((H @ ((F @ X12) @ X23))))) @ Prod))))).
% 1.75/1.11 thf(fact_31_prod_Odisc__eq__case, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [Uu:A,Uv:B]: ($true))) @ Prod)))).
% 1.75/1.11 thf(fact_32_lfilter__lfilter, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o),Xs:(coinductive_llist @ A)]: (((((coinductive_lfilter @ A) @ P2) @ (((coinductive_lfilter @ A) @ Q) @ Xs)) = (((coinductive_lfilter @ A) @ (^ [X3:A]: (((P2 @ X3) & (Q @ X3))))) @ Xs))))).
% 1.75/1.11 thf(fact_33_old_Oprod_Ocase, axiom, (! [A:$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))))).
% 1.75/1.11 thf(fact_34_lfinite__lfilterI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P2:(A > $o)]: ((((coinductive_lfinite @ A) @ Xs) => ((coinductive_lfinite @ A) @ (((coinductive_lfilter @ A) @ P2) @ Xs)))))).
% 1.75/1.11 thf(fact_35_cond__case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C)),G3:(((product_prod @ A) @ B) > C)]: (((! [X2:A,Y:B]: ((((F @ X2) @ Y) = (G3 @ ((((product_Pair @ A) @ B) @ X2) @ Y))))) => (((((product_case_prod @ A) @ B) @ C) @ F) = G3))))).
% 1.75/1.11 thf(fact_36_case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ (^ [X3:A,Y4:B]: ((F @ ((((product_Pair @ A) @ B) @ X3) @ Y4))))) = F)))).
% 1.75/1.11 thf(fact_37_case__prodE2, axiom, (! [B:$tType,A:$tType,C:$tType,Q:(A > $o),P2:(B > (C > A)),Z:((product_prod @ B) @ C)]: (((Q @ (((((product_case_prod @ B) @ C) @ A) @ P2) @ Z)) => (~ ((! [X2:B,Y:C]: (((Z = ((((product_Pair @ B) @ C) @ X2) @ Y)) => (~ ((Q @ ((P2 @ X2) @ Y))))))))))))).
% 1.75/1.11 thf(fact_38_case__prod__app, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ (D > A)) = (^ [F4:(B > (C > (D > A))),X3:((product_prod @ B) @ C),Y4:D]: ((((((product_case_prod @ B) @ C) @ A) @ (^ [L:B,R2:C]: ((((F4 @ L) @ R2) @ Y4)))) @ X3))))))).
% 1.75/1.11 thf(fact_39_case__prodE, axiom, (! [A:$tType,B:$tType,C2:(A > (B > $o)),P:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ $o) @ C2) @ P) => (~ ((! [X2:A,Y:B]: (((P = ((((product_Pair @ A) @ B) @ X2) @ Y)) => (~ (((C2 @ X2) @ Y)))))))))))).
% 1.75/1.11 thf(fact_40_case__prodD, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A3:A,B3:B]: (((((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) => ((F @ A3) @ B3))))).
% 1.75/1.11 thf(fact_41_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A3:A,B3:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((F1 @ A3) @ B3))))).
% 1.75/1.11 thf(fact_42_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q2:((product_prod @ A) @ B),F:(A > (B > C)),G3:(A > (B > C)),P:((product_prod @ A) @ B)]: (((! [X2:A,Y:B]: (((((((product_Pair @ A) @ B) @ X2) @ Y) = Q2) => (((F @ X2) @ Y) = ((G3 @ X2) @ Y))))) => ((P = Q2) => ((((((product_case_prod @ A) @ B) @ C) @ F) @ P) = (((((product_case_prod @ A) @ B) @ C) @ G3) @ Q2))))))).
% 1.75/1.11 thf(fact_43_same__fst__def, axiom, (! [B:$tType,A:$tType]: ((((same_fst @ A) @ B) = (^ [P3:(A > $o),R3:(A > (set @ ((product_prod @ B) @ B)))]: (((collect @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_case_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ $o) @ ((((product_case_prod @ A) @ B) @ (((product_prod @ A) @ B) > $o)) @ (^ [X4:A,Y5:B]: (((((product_case_prod @ A) @ B) @ $o) @ (^ [X3:A,Y4:B]: (((X4 = X3) & ((P3 @ X3) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y5) @ Y4)) @ (R3 @ X3)))))))))))))))))).
% 1.75/1.11 thf(fact_44_lex__prod__def, axiom, (! [B:$tType,A:$tType]: ((((lex_prod @ A) @ B) = (^ [Ra:(set @ ((product_prod @ A) @ A)),Rb:(set @ ((product_prod @ B) @ B))]: (((collect @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_case_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ $o) @ ((((product_case_prod @ A) @ B) @ (((product_prod @ A) @ B) > $o)) @ (^ [A4:A,B4:B]: (((((product_case_prod @ A) @ B) @ $o) @ (^ [A6:A,B6:B]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ A6)) @ Ra) | ((A4 = A6) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B4) @ B6)) @ Rb))))))))))))))))).
% 1.75/1.11 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P2:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P2)) = (P2 @ A3))))).
% 1.75/1.11 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A7:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A7)))) = A7)))).
% 1.75/1.11 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P2 @ X2) = (Q @ X2)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 1.75/1.11 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G3:(A > B)]: (((! [X2:A]: (((F @ X2) = (G3 @ X2)))) => (F = G3))))).
% 1.75/1.11 thf(fact_49_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc2004651681e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 1.75/1.11 thf(fact_50_tllist_Oabs__eq__iff, axiom, (! [B:$tType,A:$tType,X:((product_prod @ (coinductive_llist @ A)) @ B),Y3:((product_prod @ (coinductive_llist @ A)) @ B)]: ((((((((product_case_prod @ (coinductive_llist @ A)) @ B) @ ((tLList446370796tllist @ A) @ B)) @ (^ [Xs2:(coinductive_llist @ A),A4:B]: (((((tLList1672613558_llist @ B) @ A) @ A4) @ Xs2)))) @ X) = (((((product_case_prod @ (coinductive_llist @ A)) @ B) @ ((tLList446370796tllist @ A) @ B)) @ (^ [Xs2:(coinductive_llist @ A),A4:B]: (((((tLList1672613558_llist @ B) @ A) @ A4) @ Xs2)))) @ Y3)) = ((((((product_case_prod @ (coinductive_llist @ A)) @ B) @ (((product_prod @ (coinductive_llist @ A)) @ B) > $o)) @ (^ [Xs2:(coinductive_llist @ A),A4:B]: (((((product_case_prod @ (coinductive_llist @ A)) @ B) @ $o) @ (^ [Ys:(coinductive_llist @ A),B4:B]: (((Xs2 = Ys) & (((coinductive_lfinite @ A) @ Ys) => (A4 = B4))))))))) @ X) @ Y3))))).
% 1.75/1.11 thf(fact_51_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: ((((^ [X3:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y4)) @ R))) = (^ [X3:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y4)) @ S)))) = (R = S))))).
% 1.75/1.11 thf(fact_52_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A3:B,B3:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A3) @ B3)) = ((C2 @ A3) @ B3))))).
% 1.75/1.11 thf(fact_53_reflp__tllist, axiom, (! [B:$tType,A:$tType]: (((reflp @ ((product_prod @ (coinductive_llist @ A)) @ B)) @ ((((product_case_prod @ (coinductive_llist @ A)) @ B) @ (((product_prod @ (coinductive_llist @ A)) @ B) > $o)) @ (^ [Xs2:(coinductive_llist @ A),A4:B]: (((((product_case_prod @ (coinductive_llist @ A)) @ B) @ $o) @ (^ [Ys:(coinductive_llist @ A),B4:B]: (((Xs2 = Ys) & (((coinductive_lfinite @ A) @ Ys) => (A4 = B4))))))))))))).
% 1.75/1.11 thf(fact_54_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R4:A,S2:B,R:(set @ ((product_prod @ A) @ B)),S3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R4) @ S2)) @ R) => ((S3 = S2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R4) @ S3)) @ R)))))).
% 1.75/1.11 thf(fact_55_The__split__eq, axiom, (! [A:$tType,B:$tType,X:A,Y3:B]: ((((the @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X4:A,Y5:B]: (((X = X4) & (Y3 = Y5)))))) = ((((product_Pair @ A) @ B) @ X) @ Y3))))).
% 1.75/1.11 thf(fact_56_tllist__of__llist__inject, axiom, (! [A:$tType,B:$tType,B3:B,Xs:(coinductive_llist @ A),C2:B,Ys2:(coinductive_llist @ A)]: (((((((tLList1672613558_llist @ B) @ A) @ B3) @ Xs) = ((((tLList1672613558_llist @ B) @ A) @ C2) @ Ys2)) = ((Xs = Ys2) & (((coinductive_lfinite @ A) @ Ys2) => (B3 = C2))))))).
% 1.75/1.11 thf(fact_57_in__lex__prod, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A5:A,B5:B,R4:(set @ ((product_prod @ A) @ A)),S2:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ ((((product_Pair @ A) @ B) @ A5) @ B5))) @ ((((lex_prod @ A) @ B) @ R4) @ S2)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A5)) @ R4) | ((A3 = A5) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B3) @ B5)) @ S2))))))).
% 1.75/1.11 thf(fact_58_same__fstI, axiom, (! [B:$tType,A:$tType,P2:(A > $o),X:A,Y6:B,Y3:B,R:(A > (set @ ((product_prod @ B) @ B)))]: (((P2 @ X) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y6) @ Y3)) @ (R @ X)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y6)) @ ((((product_Pair @ A) @ B) @ X) @ Y3))) @ ((((same_fst @ A) @ B) @ P2) @ R))))))).
% 1.75/1.11 thf(fact_59_reflp__mono, axiom, (! [A:$tType,R:(A > (A > $o)),Q:(A > (A > $o))]: ((((reflp @ A) @ R) => ((! [X2:A,Y:A]: ((((R @ X2) @ Y) => ((Q @ X2) @ Y)))) => ((reflp @ A) @ Q)))))).
% 1.75/1.11 thf(fact_60_reflp__def, axiom, (! [A:$tType]: (((reflp @ A) = (^ [R2:(A > (A > $o))]: ((! [X3:A]: (((R2 @ X3) @ X3))))))))).
% 1.75/1.11 thf(fact_61_theI__unique, axiom, (! [A:$tType,P2:(A > $o),X:A]: (((? [X5:A]: (((P2 @ X5) & (! [Y:A]: (((P2 @ Y) => (Y = X5))))))) => ((P2 @ X) = (X = ((the @ A) @ P2))))))).
% 1.75/1.11 thf(fact_62_reflpI, axiom, (! [A:$tType,R4:(A > (A > $o))]: (((! [X2:A]: (((R4 @ X2) @ X2))) => ((reflp @ A) @ R4))))).
% 1.75/1.11 thf(fact_63_reflpE, axiom, (! [A:$tType,R4:(A > (A > $o)),X:A]: ((((reflp @ A) @ R4) => ((R4 @ X) @ X))))).
% 1.75/1.11 thf(fact_64_reflpD, axiom, (! [A:$tType,R4:(A > (A > $o)),X:A]: ((((reflp @ A) @ R4) => ((R4 @ X) @ X))))).
% 1.75/1.11 thf(fact_65_tllist_Oabs__induct, axiom, (! [B:$tType,A:$tType,P2:(((tLList446370796tllist @ A) @ B) > $o),X:((tLList446370796tllist @ A) @ B)]: (((! [Y:((product_prod @ (coinductive_llist @ A)) @ B)]: ((P2 @ (((((product_case_prod @ (coinductive_llist @ A)) @ B) @ ((tLList446370796tllist @ A) @ B)) @ (^ [Xs2:(coinductive_llist @ A),A4:B]: (((((tLList1672613558_llist @ B) @ A) @ A4) @ Xs2)))) @ Y)))) => (P2 @ X))))).
% 1.75/1.11 thf(fact_66_tllist__of__llist__cong, axiom, (! [B:$tType,A:$tType,Xs:(coinductive_llist @ A),Xs3:(coinductive_llist @ A),B3:B,B5:B]: (((Xs = Xs3) => ((((coinductive_lfinite @ A) @ Xs3) => (B3 = B5)) => (((((tLList1672613558_llist @ B) @ A) @ B3) @ Xs) = ((((tLList1672613558_llist @ B) @ A) @ B5) @ Xs3))))))).
% 1.75/1.11 thf(fact_67_old_Orec__prod__def, axiom, (! [T:$tType,B:$tType,A:$tType]: (((((product_rec_prod @ A) @ B) @ T) = (^ [F12:(A > (B > T)),X3:((product_prod @ A) @ B)]: (((the @ T) @ (((((product_rec_set_prod @ A) @ B) @ T) @ F12) @ X3)))))))).
% 1.75/1.11 thf(fact_68_the__sym__eq__trivial, axiom, (! [A:$tType,X:A]: ((((the @ A) @ ((^ [Y7:A,Z2:A]: ((Y7 = Z2))) @ X)) = X)))).
% 1.75/1.11 thf(fact_69_the__eq__trivial, axiom, (! [A:$tType,A3:A]: ((((the @ A) @ (^ [X3:A]: ((X3 = A3)))) = A3)))).
% 1.75/1.11 thf(fact_70_the__equality, axiom, (! [A:$tType,P2:(A > $o),A3:A]: (((P2 @ A3) => ((! [X2:A]: (((P2 @ X2) => (X2 = A3)))) => (((the @ A) @ P2) = A3)))))).
% 1.75/1.11 thf(fact_71_the1__equality, axiom, (! [A:$tType,P2:(A > $o),A3:A]: (((? [X5:A]: (((P2 @ X5) & (! [Y:A]: (((P2 @ Y) => (Y = X5))))))) => ((P2 @ A3) => (((the @ A) @ P2) = A3)))))).
% 1.75/1.11 thf(fact_72_the1I2, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((? [X5:A]: (((P2 @ X5) & (! [Y:A]: (((P2 @ Y) => (Y = X5))))))) => ((! [X2:A]: (((P2 @ X2) => (Q @ X2)))) => (Q @ ((the @ A) @ P2))))))).
% 1.75/1.11 thf(fact_73_If__def, axiom, (! [A:$tType]: (((if @ A) = (^ [P3:$o,X3:A,Y4:A]: (((the @ A) @ (^ [Z3:A]: (((P3 => (Z3 = X3)) & ((~ (P3)) => (Z3 = Y4)))))))))))).
% 1.75/1.11 thf(fact_74_theI2, axiom, (! [A:$tType,P2:(A > $o),A3:A,Q:(A > $o)]: (((P2 @ A3) => ((! [X2:A]: (((P2 @ X2) => (X2 = A3)))) => ((! [X2:A]: (((P2 @ X2) => (Q @ X2)))) => (Q @ ((the @ A) @ P2)))))))).
% 1.75/1.11 thf(fact_75_theI, axiom, (! [A:$tType,P2:(A > $o),A3:A]: (((P2 @ A3) => ((! [X2:A]: (((P2 @ X2) => (X2 = A3)))) => (P2 @ ((the @ A) @ P2))))))).
% 1.75/1.11 thf(fact_76_theI_H, axiom, (! [A:$tType,P2:(A > $o)]: (((? [X5:A]: (((P2 @ X5) & (! [Y:A]: (((P2 @ Y) => (Y = X5))))))) => (P2 @ ((the @ A) @ P2)))))).
% 1.75/1.11 thf(fact_77_cr__tllist__def, axiom, (! [B:$tType,A:$tType]: ((((tLList47617868tllist @ A) @ B) = ((((product_case_prod @ (coinductive_llist @ A)) @ B) @ (((tLList446370796tllist @ A) @ B) > $o)) @ (^ [Xs2:(coinductive_llist @ A),B4:B]: (((^ [Y7:((tLList446370796tllist @ A) @ B),Z2:((tLList446370796tllist @ A) @ B)]: ((Y7 = Z2))) @ ((((tLList1672613558_llist @ B) @ A) @ B4) @ Xs2))))))))).
% 1.75/1.11 thf(fact_78_inv__image__def, axiom, (! [A:$tType,B:$tType]: ((((inv_image @ B) @ A) = (^ [R2:(set @ ((product_prod @ B) @ B)),F4:(A > B)]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X3:A,Y4:A]: ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F4 @ X3)) @ (F4 @ Y4))) @ R2))))))))))).
% 1.75/1.11 thf(fact_79_old_Orec__unit__def, axiom, (! [T:$tType]: (((product_rec_unit @ T) = (^ [F12:T,X3:product_unit]: (((the @ T) @ (((product_rec_set_unit @ T) @ F12) @ X3)))))))).
% 1.75/1.11 thf(fact_80_DEADID_Orel__reflp, axiom, (! [A:$tType]: (((reflp @ A) @ (^ [Y7:A,Z2:A]: ((Y7 = Z2))))))).
% 1.75/1.11 thf(fact_81_Nitpick_OThe__psimp, axiom, (! [A:$tType,P2:(A > $o),X:A]: (((P2 = ((^ [Y7:A,Z2:A]: ((Y7 = Z2))) @ X)) => (((the @ A) @ P2) = X))))).
% 1.75/1.11 thf(fact_82_old_Orec__bool__def, axiom, (! [T:$tType]: (((product_rec_bool @ T) = (^ [F12:T,F22:T,X3:$o]: (((the @ T) @ ((((product_rec_set_bool @ T) @ F12) @ F22) @ X3)))))))).
% 1.75/1.11 thf(fact_83_old_Obool_Osimps_I5_J, axiom, (! [T:$tType,F1:T,F23:T]: ((((((product_rec_bool @ T) @ F1) @ F23) @ $true) = F1)))).
% 1.75/1.11 thf(fact_84_old_Obool_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F23:T]: ((((((product_rec_bool @ T) @ F1) @ F23) @ $false) = F23)))).
% 1.75/1.11 thf(fact_85_in__inv__image, axiom, (! [A:$tType,B:$tType,X:A,Y3:A,R4:(set @ ((product_prod @ B) @ B)),F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ ((((inv_image @ B) @ A) @ R4) @ F)) = (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X)) @ (F @ Y3))) @ R4))))).
% 1.75/1.11 thf(fact_86_rp__inv__image__def, axiom, (! [B:$tType,A:$tType]: ((((fun_rp_inv_image @ A) @ B) = ((((product_case_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))))) @ (^ [R3:(set @ ((product_prod @ A) @ A)),S4:(set @ ((product_prod @ A) @ A)),F4:(B > A)]: (((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ ((((inv_image @ A) @ B) @ R3) @ F4)) @ ((((inv_image @ A) @ B) @ S4) @ F4))))))))).
% 1.75/1.11 thf(fact_87_Quotient__tllist, axiom, (! [B:$tType,A:$tType]: (((((((quotient @ ((product_prod @ (coinductive_llist @ A)) @ B)) @ ((tLList446370796tllist @ A) @ B)) @ ((((product_case_prod @ (coinductive_llist @ A)) @ B) @ (((product_prod @ (coinductive_llist @ A)) @ B) > $o)) @ (^ [Xs2:(coinductive_llist @ A),A4:B]: (((((product_case_prod @ (coinductive_llist @ A)) @ B) @ $o) @ (^ [Ys:(coinductive_llist @ A),B4:B]: (((Xs2 = Ys) & (((coinductive_lfinite @ A) @ Ys) => (A4 = B4)))))))))) @ ((((product_case_prod @ (coinductive_llist @ A)) @ B) @ ((tLList446370796tllist @ A) @ B)) @ (^ [Xs2:(coinductive_llist @ A),A4:B]: (((((tLList1672613558_llist @ B) @ A) @ A4) @ Xs2))))) @ (^ [Ys:((tLList446370796tllist @ A) @ B)]: (((((product_Pair @ (coinductive_llist @ A)) @ B) @ (((tLList798109904tllist @ A) @ B) @ Ys)) @ (((tLList2110128105rminal @ A) @ B) @ Ys))))) @ ((tLList47617868tllist @ A) @ B))))).
% 1.75/1.11 thf(fact_88_terminal__tllist__of__llist__lfinite, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),B3:B]: ((((coinductive_lfinite @ A) @ Xs) => ((((tLList2110128105rminal @ A) @ B) @ ((((tLList1672613558_llist @ B) @ A) @ B3) @ Xs)) = B3))))).
% 1.75/1.11 thf(fact_89_in__inv__imagep, axiom, (! [B:$tType,A:$tType]: ((((inv_imagep @ A) @ B) = (^ [R2:(A > (A > $o)),F4:(B > A),X3:B,Y4:B]: (((R2 @ (F4 @ X3)) @ (F4 @ Y4)))))))).
% 1.75/1.11 thf(fact_90_case__swap, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > (B > A)),P:((product_prod @ C) @ B)]: (((((((product_case_prod @ B) @ C) @ A) @ (^ [Y4:B,X3:C]: (((F @ X3) @ Y4)))) @ (((product_swap @ C) @ B) @ P)) = (((((product_case_prod @ C) @ B) @ A) @ F) @ P))))).
% 1.75/1.11 thf(fact_91_pair__imageI, axiom, (! [C:$tType,B:$tType,A:$tType,A3:A,B3:B,A7:(set @ ((product_prod @ A) @ B)),F:(A > (B > C))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ A7) => (((member @ C) @ ((F @ A3) @ B3)) @ ((((image @ ((product_prod @ A) @ B)) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F)) @ A7)))))).
% 1.75/1.11 thf(fact_92_swap__swap, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P)) = P)))).
% 1.75/1.11 thf(fact_93_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y3:A,X:B,A7:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y3) @ X)) @ ((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A7)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y3)) @ A7))))).
% 1.75/1.11 thf(fact_94_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y3:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y3)) = ((((product_Pair @ A) @ B) @ Y3) @ X))))).
% 1.75/1.11 thf(fact_95_tllist__of__llist__inverse, axiom, (! [B:$tType,A:$tType,B3:B,Xs:(coinductive_llist @ A)]: (((((tLList798109904tllist @ A) @ B) @ ((((tLList1672613558_llist @ B) @ A) @ B3) @ Xs)) = Xs)))).
% 1.75/1.11 thf(fact_96_llist__of__tllist__inverse, axiom, (! [B:$tType,A:$tType,B3:((tLList446370796tllist @ A) @ B)]: ((((((tLList1672613558_llist @ B) @ A) @ (((tLList2110128105rminal @ A) @ B) @ B3)) @ (((tLList798109904tllist @ A) @ B) @ B3)) = B3)))).
% 1.75/1.11 thf(fact_97_inv__imagep__def, axiom, (! [A:$tType,B:$tType]: ((((inv_imagep @ B) @ A) = (^ [R2:(B > (B > $o)),F4:(A > B),X3:A,Y4:A]: (((R2 @ (F4 @ X3)) @ (F4 @ Y4)))))))).
% 1.75/1.11 thf(fact_98_image__ident, axiom, (! [A:$tType,Y8:(set @ A)]: ((((((image @ A) @ A) @ (^ [X3:A]: (X3))) @ Y8) = Y8)))).
% 1.75/1.11 thf(fact_99_image__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X:B,A7:(set @ B)]: (((B3 = (F @ X)) => ((((member @ B) @ X) @ A7) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ A7))))))).
% 1.75/1.11 thf(fact_100_Quotient__total__abs__eq__iff, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),X:A,Y3:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => (((reflp @ A) @ R) => (((Abs @ X) = (Abs @ Y3)) = ((R @ X) @ Y3))))))).
% 1.75/1.11 thf(fact_101_Quotient__total__abs__induct, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),P2:(B > $o),X:B]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => (((reflp @ A) @ R) => ((! [Y:A]: ((P2 @ (Abs @ Y)))) => (P2 @ X))))))).
% 1.75/1.11 thf(fact_102_rp__inv__image__rp, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))),F:(B > A)]: ((((fun_reduction_pair @ A) @ P2) => ((fun_reduction_pair @ B) @ ((((fun_rp_inv_image @ A) @ B) @ P2) @ F)))))).
% 1.75/1.11 thf(fact_103_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X:A,A7:(set @ A),B3:B,F:(A > B)]: (((((member @ A) @ X) @ A7) => ((B3 = (F @ X)) => (((member @ B) @ B3) @ ((((image @ A) @ B) @ F) @ A7))))))).
% 1.75/1.11 thf(fact_104_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A7:(set @ B),P2:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ ((((image @ B) @ A) @ F) @ A7)) => (P2 @ X2)))) => (! [X5:B]: (((((member @ B) @ X5) @ A7) => (P2 @ (F @ X5))))))))).
% 1.75/1.11 thf(fact_105_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N:(set @ A),F:(A > B),G3:(A > B)]: (((M = N) => ((! [X2:A]: (((((member @ A) @ X2) @ N) => ((F @ X2) = (G3 @ X2))))) => (((((image @ A) @ B) @ F) @ M) = ((((image @ A) @ B) @ G3) @ N))))))).
% 1.75/1.11 thf(fact_106_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A7:(set @ B),P2:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F) @ A7)) & (P2 @ X5)))) => (? [X2:B]: (((((member @ B) @ X2) @ A7) & (P2 @ (F @ X2))))))))).
% 1.75/1.11 thf(fact_107_image__iff, axiom, (! [A:$tType,B:$tType,Z:A,F:(B > A),A7:(set @ B)]: (((((member @ A) @ Z) @ ((((image @ B) @ A) @ F) @ A7)) = (? [X3:B]: (((((member @ B) @ X3) @ A7) & (Z = (F @ X3))))))))).
% 1.75/1.11 thf(fact_108_imageI, axiom, (! [B:$tType,A:$tType,X:A,A7:(set @ A),F:(A > B)]: (((((member @ A) @ X) @ A7) => (((member @ B) @ (F @ X)) @ ((((image @ A) @ B) @ F) @ A7)))))).
% 1.75/1.11 thf(fact_109_Quotient__rep__abs__fold__unmap, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),X6:B,X:A,Rep2:(B > A)]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => ((X6 = (Abs @ X)) => (((R @ X) @ X) => (((Rep @ X6) = (Rep2 @ X6)) => ((R @ (Rep2 @ X6)) @ X)))))))).
% 1.75/1.11 thf(fact_110_Quotient__to__transfer, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),C2:A,C4:B]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => (((R @ C2) @ C2) => ((C4 = (Abs @ C2)) => ((T2 @ C2) @ C4))))))).
% 1.75/1.11 thf(fact_111_Quotient__abs__induct, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),P2:(B > $o),X:B]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => ((! [Y:A]: ((((R @ Y) @ Y) => (P2 @ (Abs @ Y))))) => (P2 @ X)))))).
% 1.75/1.11 thf(fact_112_Quotient__rep__reflp, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),A3:B]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => ((R @ (Rep @ A3)) @ (Rep @ A3)))))).
% 1.75/1.11 thf(fact_113_Quotient__rel__abs2, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),X:B,Y3:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => (((R @ (Rep @ X)) @ Y3) => (X = (Abs @ Y3))))))).
% 1.75/1.11 thf(fact_114_Quotient__alt__def3, axiom, (! [B:$tType,A:$tType]: ((((quotient @ A) @ B) = (^ [R3:(A > (A > $o)),Abs2:(A > B),Rep3:(B > A),T3:(A > (B > $o))]: (((! [A4:A,B4:B]: ((((T3 @ A4) @ B4) => ((Abs2 @ A4) = B4)))) & ((! [B4:B]: (((T3 @ (Rep3 @ B4)) @ B4))) & (! [X3:A,Y4:A]: ((((R3 @ X3) @ Y4) = (? [Z3:B]: ((((T3 @ X3) @ Z3) & ((T3 @ Y4) @ Z3))))))))))))))).
% 1.75/1.11 thf(fact_115_Quotient__alt__def2, axiom, (! [B:$tType,A:$tType]: ((((quotient @ A) @ B) = (^ [R3:(A > (A > $o)),Abs2:(A > B),Rep3:(B > A),T3:(A > (B > $o))]: (((! [A4:A,B4:B]: ((((T3 @ A4) @ B4) => ((Abs2 @ A4) = B4)))) & ((! [B4:B]: (((T3 @ (Rep3 @ B4)) @ B4))) & (! [X3:A,Y4:A]: ((((R3 @ X3) @ Y4) = (((T3 @ X3) @ (Abs2 @ Y4)) & ((T3 @ Y4) @ (Abs2 @ X3)))))))))))))).
% 1.75/1.11 thf(fact_116_Quotient__rep__abs, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),R4:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => (((R @ R4) @ R4) => ((R @ (Rep @ (Abs @ R4))) @ R4)))))).
% 1.75/1.11 thf(fact_117_Quotient__rel__rep, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),A3:B,B3:B]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => (((R @ (Rep @ A3)) @ (Rep @ B3)) = (A3 = B3)))))).
% 1.75/1.11 thf(fact_118_Quotient__rel__abs, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),R4:A,S2:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => (((R @ R4) @ S2) => ((Abs @ R4) = (Abs @ S2))))))).
% 1.75/1.11 thf(fact_119_Quotient__alt__def, axiom, (! [B:$tType,A:$tType]: ((((quotient @ A) @ B) = (^ [R3:(A > (A > $o)),Abs2:(A > B),Rep3:(B > A),T3:(A > (B > $o))]: (((! [A4:A,B4:B]: ((((T3 @ A4) @ B4) => ((Abs2 @ A4) = B4)))) & ((! [B4:B]: (((T3 @ (Rep3 @ B4)) @ B4))) & (! [X3:A,Y4:A]: ((((R3 @ X3) @ Y4) = (((T3 @ X3) @ (Abs2 @ X3)) & (((T3 @ Y4) @ (Abs2 @ Y4)) & ((Abs2 @ X3) = (Abs2 @ Y4))))))))))))))).
% 1.75/1.11 thf(fact_120_Quotient__abs__rep, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),A3:B]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => ((Abs @ (Rep @ A3)) = A3))))).
% 1.75/1.11 thf(fact_121_Quotient__Rep__eq, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),X6:B,X:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => ((X6 = (Abs @ X)) => ((Rep @ X6) = (Rep @ X6))))))).
% 1.75/1.11 thf(fact_122_Quotient__refl2, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),R4:A,S2:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => (((R @ R4) @ S2) => ((R @ S2) @ S2)))))).
% 1.75/1.11 thf(fact_123_Quotient__refl1, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),R4:A,S2:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => (((R @ R4) @ S2) => ((R @ R4) @ R4)))))).
% 1.75/1.11 thf(fact_124_Quotient__rel, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),R4:A,S2:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => ((((R @ R4) @ R4) & (((R @ S2) @ S2) & ((Abs @ R4) = (Abs @ S2)))) = ((R @ R4) @ S2)))))).
% 1.75/1.11 thf(fact_125_Compr__image__eq, axiom, (! [A:$tType,B:$tType,F:(B > A),A7:(set @ B),P2:(A > $o)]: ((((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F) @ A7)) & (P2 @ X3))))) = ((((image @ B) @ A) @ F) @ ((collect @ B) @ (^ [X3:B]: (((((member @ B) @ X3) @ A7) & (P2 @ (F @ X3))))))))))).
% 1.75/1.11 thf(fact_126_image__image, axiom, (! [A:$tType,B:$tType,C:$tType,F:(B > A),G3:(C > B),A7:(set @ C)]: ((((((image @ B) @ A) @ F) @ ((((image @ C) @ B) @ G3) @ A7)) = ((((image @ C) @ A) @ (^ [X3:C]: ((F @ (G3 @ X3))))) @ A7))))).
% 1.75/1.11 thf(fact_127_imageE, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),A7:(set @ B)]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ A7)) => (~ ((! [X2:B]: (((B3 = (F @ X2)) => (~ ((((member @ B) @ X2) @ A7)))))))))))).
% 1.75/1.11 thf(fact_128_Quotient__cr__rel, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o))]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => (T2 = (^ [X3:A,Y4:B]: ((((R @ X3) @ X3) & ((Abs @ X3) = Y4))))))))).
% 1.75/1.11 thf(fact_129_Quotient__def, axiom, (! [B:$tType,A:$tType]: ((((quotient @ A) @ B) = (^ [R3:(A > (A > $o)),Abs2:(A > B),Rep3:(B > A),T3:(A > (B > $o))]: (((! [A4:B]: (((Abs2 @ (Rep3 @ A4)) = A4))) & ((! [A4:B]: (((R3 @ (Rep3 @ A4)) @ (Rep3 @ A4)))) & ((! [R2:A,S5:A]: ((((R3 @ R2) @ S5) = (((R3 @ R2) @ R2) & (((R3 @ S5) @ S5) & ((Abs2 @ R2) = (Abs2 @ S5))))))) & (T3 = (^ [X3:A,Y4:B]: ((((R3 @ X3) @ X3) & ((Abs2 @ X3) = Y4)))))))))))))).
% 1.75/1.11 thf(fact_130_QuotientI, axiom, (! [A:$tType,B:$tType,Abs:(B > A),Rep:(A > B),R:(B > (B > $o)),T2:(B > (A > $o))]: (((! [A2:A]: (((Abs @ (Rep @ A2)) = A2))) => ((! [A2:A]: (((R @ (Rep @ A2)) @ (Rep @ A2)))) => ((! [R5:B,S6:B]: ((((R @ R5) @ S6) = (((R @ R5) @ R5) & (((R @ S6) @ S6) & ((Abs @ R5) = (Abs @ S6))))))) => ((T2 = (^ [X3:B,Y4:A]: ((((R @ X3) @ X3) & ((Abs @ X3) = Y4))))) => ((((((quotient @ B) @ A) @ R) @ Abs) @ Rep) @ T2)))))))).
% 1.75/1.11 thf(fact_131_Sup_OSUP__identity__eq, axiom, (! [A:$tType,Sup:((set @ A) > A),A7:(set @ A)]: (((Sup @ ((((image @ A) @ A) @ (^ [X3:A]: (X3))) @ A7)) = (Sup @ A7))))).
% 1.75/1.11 thf(fact_132_Inf_OINF__identity__eq, axiom, (! [A:$tType,Inf:((set @ A) > A),A7:(set @ A)]: (((Inf @ ((((image @ A) @ A) @ (^ [X3:A]: (X3))) @ A7)) = (Inf @ A7))))).
% 1.75/1.11 thf(fact_133_terminal0__terminal, axiom, (! [B:$tType,A:$tType]: (((tLList1825092077minal0 @ (((tLList446370796tllist @ A) @ B) > B)) = ((tLList2110128105rminal @ A) @ B))))).
% 1.75/1.11 thf(fact_134_surj__swap, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 1.75/1.11 thf(fact_135_swap__product, axiom, (! [B:$tType,A:$tType,A7:(set @ B),B7:(set @ A)]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ B) @ A) @ ((product_prod @ A) @ B)) @ (^ [I:B,J:A]: (((((product_Pair @ A) @ B) @ J) @ I))))) @ ((((product_Sigma @ B) @ A) @ A7) @ (^ [Uu:B]: (B7)))) = ((((product_Sigma @ A) @ B) @ B7) @ (^ [Uu:A]: (A7))))))).
% 1.75/1.11 thf(fact_136_mem__Sigma__iff, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,A7:(set @ A),B7:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ ((((product_Sigma @ A) @ B) @ A7) @ B7)) = ((((member @ A) @ A3) @ A7) & (((member @ B) @ B3) @ (B7 @ A3))))))).
% 1.75/1.11 thf(fact_137_SigmaI, axiom, (! [B:$tType,A:$tType,A3:A,A7:(set @ A),B3:B,B7:(A > (set @ B))]: (((((member @ A) @ A3) @ A7) => ((((member @ B) @ B3) @ (B7 @ A3)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ ((((product_Sigma @ A) @ B) @ A7) @ B7))))))).
% 1.75/1.11 thf(fact_138_Collect__case__prod, axiom, (! [B:$tType,A:$tType,P2:(A > $o),Q:(B > $o)]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A4:A,B4:B]: (((P2 @ A4) & (Q @ B4)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P2)) @ (^ [Uu:A]: (((collect @ B) @ Q)))))))).
% 1.75/1.11 thf(fact_139_UNIV__Times__UNIV, axiom, (! [B:$tType,A:$tType]: ((((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu:A]: ((top_top @ (set @ B))))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 1.75/1.11 thf(fact_140_UNIV__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (^ [X3:A]: ($true))))))).
% 1.75/1.11 thf(fact_141_SigmaE, axiom, (! [A:$tType,B:$tType,C2:((product_prod @ A) @ B),A7:(set @ A),B7:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ C2) @ ((((product_Sigma @ A) @ B) @ A7) @ B7)) => (~ ((! [X2:A]: (((((member @ A) @ X2) @ A7) => (! [Y:B]: (((((member @ B) @ Y) @ (B7 @ X2)) => (C2 != ((((product_Pair @ A) @ B) @ X2) @ Y)))))))))))))).
% 1.75/1.11 thf(fact_142_SigmaD1, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,A7:(set @ A),B7:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ ((((product_Sigma @ A) @ B) @ A7) @ B7)) => (((member @ A) @ A3) @ A7))))).
% 1.75/1.11 thf(fact_143_SigmaD2, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,A7:(set @ A),B7:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ ((((product_Sigma @ A) @ B) @ A7) @ B7)) => (((member @ B) @ B3) @ (B7 @ A3)))))).
% 1.75/1.11 thf(fact_144_SigmaE2, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,A7:(set @ A),B7:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ ((((product_Sigma @ A) @ B) @ A7) @ B7)) => (~ (((((member @ A) @ A3) @ A7) => (~ ((((member @ B) @ B3) @ (B7 @ A3))))))))))).
% 1.75/1.11 thf(fact_145_Sigma__cong, axiom, (! [B:$tType,A:$tType,A7:(set @ A),B7:(set @ A),C5:(A > (set @ B)),D3:(A > (set @ B))]: (((A7 = B7) => ((! [X2:A]: (((((member @ A) @ X2) @ B7) => ((C5 @ X2) = (D3 @ X2))))) => (((((product_Sigma @ A) @ B) @ A7) @ C5) = ((((product_Sigma @ A) @ B) @ B7) @ D3))))))).
% 1.75/1.11 thf(fact_146_Times__eq__cancel2, axiom, (! [A:$tType,B:$tType,X:A,C5:(set @ A),A7:(set @ B),B7:(set @ B)]: (((((member @ A) @ X) @ C5) => ((((((product_Sigma @ B) @ A) @ A7) @ (^ [Uu:B]: (C5))) = ((((product_Sigma @ B) @ A) @ B7) @ (^ [Uu:B]: (C5)))) = (A7 = B7)))))).
% 1.75/1.11 thf(fact_147_range__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X:B]: (((B3 = (F @ X)) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 1.75/1.11 thf(fact_148_rangeI, axiom, (! [A:$tType,B:$tType,F:(B > A),X:B]: ((((member @ A) @ (F @ X)) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B))))))).
% 1.75/1.11 thf(fact_149_Collect__case__prod__Sigma, axiom, (! [B:$tType,A:$tType,P2:(A > $o),Q:(A > (B > $o))]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X3:A,Y4:B]: (((P2 @ X3) & ((Q @ X3) @ Y4)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P2)) @ (^ [X3:A]: (((collect @ B) @ (Q @ X3))))))))).
% 1.75/1.11 thf(fact_150_range__composition, axiom, (! [A:$tType,C:$tType,B:$tType,F:(C > A),G3:(B > C)]: ((((((image @ B) @ A) @ (^ [X3:B]: ((F @ (G3 @ X3))))) @ (top_top @ (set @ B))) = ((((image @ C) @ A) @ F) @ ((((image @ B) @ C) @ G3) @ (top_top @ (set @ B)))))))).
% 1.75/1.11 thf(fact_151_rangeE, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A)]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))) => (~ ((! [X2:B]: ((B3 != (F @ X2)))))))))).
% 1.75/1.11 thf(fact_152_product__swap, axiom, (! [B:$tType,A:$tType,A7:(set @ B),B7:(set @ A)]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A7) @ (^ [Uu:B]: (B7)))) = ((((product_Sigma @ A) @ B) @ B7) @ (^ [Uu:A]: (A7))))))).
% 1.75/1.11 thf(fact_153_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A7:(set @ B),B7:(set @ B),C5:(B > A),D3:(B > A),Inf:((set @ A) > A)]: (((A7 = B7) => ((! [X2:B]: (((((member @ B) @ X2) @ B7) => ((C5 @ X2) = (D3 @ X2))))) => ((Inf @ ((((image @ B) @ A) @ C5) @ A7)) = (Inf @ ((((image @ B) @ A) @ D3) @ B7)))))))).
% 1.75/1.11 thf(fact_154_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A7:(set @ B),B7:(set @ B),C5:(B > A),D3:(B > A),Sup:((set @ A) > A)]: (((A7 = B7) => ((! [X2:B]: (((((member @ B) @ X2) @ B7) => ((C5 @ X2) = (D3 @ X2))))) => ((Sup @ ((((image @ B) @ A) @ C5) @ A7)) = (Sup @ ((((image @ B) @ A) @ D3) @ B7)))))))).
% 1.75/1.11 thf(fact_155_surjD, axiom, (! [A:$tType,B:$tType,F:(B > A),Y3:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X2:B]: ((Y3 = (F @ X2)))))))).
% 1.75/1.11 thf(fact_156_surjE, axiom, (! [A:$tType,B:$tType,F:(B > A),Y3:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X2:B]: ((Y3 != (F @ X2)))))))))).
% 1.75/1.11 thf(fact_157_surjI, axiom, (! [B:$tType,A:$tType,G3:(B > A),F:(A > B)]: (((! [X2:A]: (((G3 @ (F @ X2)) = X2))) => (((((image @ B) @ A) @ G3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 1.75/1.11 thf(fact_158_surj__def, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y4:A]: ((? [X3:B]: ((Y4 = (F @ X3)))))))))).
% 1.75/1.11 thf(fact_159_member__product, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A7:(set @ A),B7:(set @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_product @ A) @ B) @ A7) @ B7)) = (((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_Sigma @ A) @ B) @ A7) @ (^ [Uu:A]: (B7)))))))).
% 1.75/1.11 thf(fact_160_Product__Type_Oproduct__def, axiom, (! [B:$tType,A:$tType]: ((((product_product @ A) @ B) = (^ [A8:(set @ A),B8:(set @ B)]: (((((product_Sigma @ A) @ B) @ A8) @ (^ [Uu:A]: (B8))))))))).
% 1.75/1.11 thf(fact_161_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F:(A > B),G3:(C > D)]: (((((((image @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image @ C) @ D) @ G3) @ (top_top @ (set @ C))) = (top_top @ (set @ D))) => (((((image @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F) @ G3)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 1.75/1.11 thf(fact_162_map__prod__ident, axiom, (! [B:$tType,A:$tType]: ((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X3:A]: (X3))) @ (^ [Y4:B]: (Y4))) = (^ [Z3:((product_prod @ A) @ B)]: (Z3)))))).
% 1.75/1.11 thf(fact_163_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G3:(D > B),A3:C,B3:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G3) @ ((((product_Pair @ C) @ D) @ A3) @ B3)) = ((((product_Pair @ A) @ B) @ (F @ A3)) @ (G3 @ B3)))))).
% 1.75/1.11 thf(fact_164_map__prod__imageI, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,A3:A,B3:B,R:(set @ ((product_prod @ A) @ B)),F:(A > C),G3:(B > D)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F @ A3)) @ (G3 @ B3))) @ ((((image @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F) @ G3)) @ R)))))).
% 1.75/1.11 thf(fact_165_case__prod__map__prod, axiom, (! [C:$tType,A:$tType,B:$tType,E:$tType,D:$tType,H:(B > (C > A)),F:(D > B),G3:(E > C),X:((product_prod @ D) @ E)]: (((((((product_case_prod @ B) @ C) @ A) @ H) @ (((((((product_map_prod @ D) @ B) @ E) @ C) @ F) @ G3) @ X)) = (((((product_case_prod @ D) @ E) @ A) @ (^ [L:D,R2:E]: (((H @ (F @ L)) @ (G3 @ R2))))) @ X))))).
% 1.75/1.11 thf(fact_166_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C2:((product_prod @ A) @ B),F:(C > A),G3:(D > B),R:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C2) @ ((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G3)) @ R)) => (~ ((! [X2:C,Y:D]: (((C2 = ((((product_Pair @ A) @ B) @ (F @ X2)) @ (G3 @ Y))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X2) @ Y)) @ R)))))))))))).
% 1.75/1.11 thf(fact_167_map__prod__def, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType]: ((((((product_map_prod @ A) @ C) @ B) @ D) = (^ [F4:(A > C),G4:(B > D)]: (((((product_case_prod @ A) @ B) @ ((product_prod @ C) @ D)) @ (^ [X3:A,Y4:B]: (((((product_Pair @ C) @ D) @ (F4 @ X3)) @ (G4 @ Y4))))))))))).
% 1.75/1.11 thf(fact_168_map__prod__surj__on, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F:(B > A),A7:(set @ B),A9:(set @ A),G3:(D > C),B7:(set @ D),B9:(set @ C)]: (((((((image @ B) @ A) @ F) @ A7) = A9) => ((((((image @ D) @ C) @ G3) @ B7) = B9) => (((((image @ ((product_prod @ B) @ D)) @ ((product_prod @ A) @ C)) @ ((((((product_map_prod @ B) @ A) @ D) @ C) @ F) @ G3)) @ ((((product_Sigma @ B) @ D) @ A7) @ (^ [Uu:B]: (B7)))) = ((((product_Sigma @ A) @ C) @ A9) @ (^ [Uu:A]: (B9))))))))).
% 1.75/1.11 thf(fact_169_prod_Omap__ident, axiom, (! [B:$tType,A:$tType,T4:((product_prod @ A) @ B)]: (((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X3:A]: (X3))) @ (^ [X3:B]: (X3))) @ T4) = T4)))).
% 1.75/1.11 thf(fact_170_terminal__def, axiom, (! [B:$tType,A:$tType]: ((((tLList2110128105rminal @ A) @ B) = (((((tLList200813139tllist @ B) @ B) @ A) @ (^ [X12:B]: (X12))) @ (^ [X_dummy:A]: ((tLList1825092077minal0 @ (((tLList446370796tllist @ A) @ B) > B))))))))).
% 1.75/1.11 thf(fact_171_reflp__refl__eq, axiom, (! [A:$tType,R4:(set @ ((product_prod @ A) @ A))]: ((((reflp @ A) @ (^ [X3:A,Y4:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ R4)))) = (((refl_on @ A) @ (top_top @ (set @ A))) @ R4))))).
% 1.75/1.11 thf(fact_172_refl__onD2, axiom, (! [A:$tType,A7:(set @ A),R4:(set @ ((product_prod @ A) @ A)),X:A,Y3:A]: (((((refl_on @ A) @ A7) @ R4) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ R4) => (((member @ A) @ Y3) @ A7)))))).
% 1.75/1.11 thf(fact_173_refl__onD1, axiom, (! [A:$tType,A7:(set @ A),R4:(set @ ((product_prod @ A) @ A)),X:A,Y3:A]: (((((refl_on @ A) @ A7) @ R4) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ R4) => (((member @ A) @ X) @ A7)))))).
% 1.75/1.11 thf(fact_174_refl__onD, axiom, (! [A:$tType,A7:(set @ A),R4:(set @ ((product_prod @ A) @ A)),A3:A]: (((((refl_on @ A) @ A7) @ R4) => ((((member @ A) @ A3) @ A7) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A3)) @ R4)))))).
% 1.75/1.11 thf(fact_175_tllist_Ocase__distrib, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,H:(C > D),F1:(B > C),F23:(A > (((tLList446370796tllist @ A) @ B) > C)),Tllist:((tLList446370796tllist @ A) @ B)]: (((H @ ((((((tLList200813139tllist @ B) @ C) @ A) @ F1) @ F23) @ Tllist)) = ((((((tLList200813139tllist @ B) @ D) @ A) @ (^ [X3:B]: ((H @ (F1 @ X3))))) @ (^ [X12:A,X23:((tLList446370796tllist @ A) @ B)]: ((H @ ((F23 @ X12) @ X23))))) @ Tllist))))).
% 1.75/1.11 thf(fact_176_refl__on__domain, axiom, (! [A:$tType,A7:(set @ A),R4:(set @ ((product_prod @ A) @ A)),A3:A,B3:A]: (((((refl_on @ A) @ A7) @ R4) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B3)) @ R4) => ((((member @ A) @ A3) @ A7) & (((member @ A) @ B3) @ A7))))))).
% 1.75/1.11 thf(fact_177_range__fst, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = (top_top @ (set @ A)))))).
% 1.75/1.11 thf(fact_178_range__snd, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ A)))))).
% 1.75/1.11 thf(fact_179_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)))))).
% 1.75/1.11 thf(fact_180_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)))))).
% 1.75/1.11 thf(fact_181_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)))).
% 1.75/1.11 thf(fact_182_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))))).
% 1.75/1.11 thf(fact_183_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))))).
% 1.75/1.11 thf(fact_184_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y7:((product_prod @ A) @ B),Z2:((product_prod @ A) @ B)]: ((Y7 = Z2))) = (^ [S5:((product_prod @ A) @ B),T5:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S5) = (((product_fst @ A) @ B) @ T5)) & ((((product_snd @ A) @ B) @ S5) = (((product_snd @ A) @ B) @ T5))))))))).
% 1.75/1.11 thf(fact_185_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))))).
% 1.75/1.11 thf(fact_186_prod__eqI, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B),Q2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P) = (((product_fst @ A) @ B) @ Q2)) => (((((product_snd @ A) @ B) @ P) = (((product_snd @ A) @ B) @ Q2)) => (P = Q2)))))).
% 1.75/1.11 thf(fact_187_split__fst, axiom, (! [B:$tType,A:$tType,R:(A > $o),P:((product_prod @ A) @ B)]: (((R @ (((product_fst @ A) @ B) @ P)) = (! [X3:A,Y4:B]: (((P = ((((product_Pair @ A) @ B) @ X3) @ Y4)) => (R @ X3)))))))).
% 1.75/1.11 thf(fact_188_split__fst__asm, axiom, (! [B:$tType,A:$tType,R:(A > $o),P:((product_prod @ A) @ B)]: (((R @ (((product_fst @ A) @ B) @ P)) = (~ ((? [X3:A,Y4:B]: (((P = ((((product_Pair @ A) @ B) @ X3) @ Y4)) & (~ ((R @ X3)))))))))))).
% 1.75/1.11 thf(fact_189_fst__eqD, axiom, (! [B:$tType,A:$tType,X:A,Y3:B,A3:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X) @ Y3)) = A3) => (X = A3))))).
% 1.75/1.11 thf(fact_190_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X22:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = X1)))).
% 1.75/1.11 thf(fact_191_snd__eqD, axiom, (! [B:$tType,A:$tType,X:B,Y3:A,A3:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y3)) = A3) => (Y3 = A3))))).
% 1.75/1.11 thf(fact_192_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X22:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X22)) = X22)))).
% 1.75/1.11 thf(fact_193_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P4:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P4)) @ (((product_fst @ A) @ B) @ P4)))))))).
% 1.75/1.11 thf(fact_194_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 1.75/1.11 thf(fact_195_surjective__pairing, axiom, (! [B:$tType,A:$tType,T4:((product_prod @ A) @ B)]: ((T4 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T4)) @ (((product_snd @ A) @ B) @ T4)))))).
% 1.75/1.11 thf(fact_196_case__prod__beta, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ A) = (^ [F4:(B > (C > A)),P4:((product_prod @ B) @ C)]: (((F4 @ (((product_fst @ B) @ C) @ P4)) @ (((product_snd @ B) @ C) @ P4)))))))).
% 1.75/1.11 thf(fact_197_prod_Ocase__eq__if, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F4:(A > (B > C)),Prod3:((product_prod @ A) @ B)]: (((F4 @ (((product_fst @ A) @ B) @ Prod3)) @ (((product_snd @ A) @ B) @ Prod3)))))))).
% 1.75/1.11 thf(fact_198_Product__Type_OCollect__case__prodD, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A7:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A7))) => ((A7 @ (((product_fst @ A) @ B) @ X)) @ (((product_snd @ A) @ B) @ X)))))).
% 1.75/1.11 thf(fact_199_split__comp__eq, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,F:(A > (B > C)),G3:(D > A)]: (((^ [U:((product_prod @ D) @ B)]: (((F @ (G3 @ (((product_fst @ D) @ B) @ U))) @ (((product_snd @ D) @ B) @ U)))) = ((((product_case_prod @ D) @ B) @ C) @ (^ [X3:D]: ((F @ (G3 @ X3))))))))).
% 1.75/1.11 thf(fact_200_case__prod__beta_H, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F4:(A > (B > C)),X3:((product_prod @ A) @ B)]: (((F4 @ (((product_fst @ A) @ B) @ X3)) @ (((product_snd @ A) @ B) @ X3)))))))).
% 1.75/1.11 thf(fact_201_case__prod__unfold, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [C6:(A > (B > C)),P4:((product_prod @ A) @ B)]: (((C6 @ (((product_fst @ A) @ B) @ P4)) @ (((product_snd @ A) @ B) @ P4)))))))).
% 1.75/1.11 thf(fact_202_prod_Osplit__sel, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(C > $o),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P2 @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) => (P2 @ ((F @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))).
% 1.75/1.11 thf(fact_203_prod_Osplit__sel__asm, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(C > $o),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P2 @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = (~ (((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) & (~ ((P2 @ ((F @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))))))).
% 1.75/1.11 thf(fact_204_fst__def, axiom, (! [B:$tType,A:$tType]: ((((product_fst @ A) @ B) = ((((product_case_prod @ A) @ B) @ A) @ (^ [X12:A,X23:B]: (X12))))))).
% 1.75/1.11 thf(fact_205_snd__def, axiom, (! [B:$tType,A:$tType]: ((((product_snd @ A) @ B) = ((((product_case_prod @ A) @ B) @ B) @ (^ [X12:A,X23:B]: (X23))))))).
% 1.75/1.11 thf(fact_206_The__case__prod, axiom, (! [B:$tType,A:$tType,P2:(A > (B > $o))]: ((((the @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P2)) = ((the @ ((product_prod @ A) @ B)) @ (^ [Xy:((product_prod @ A) @ B)]: (((P2 @ (((product_fst @ A) @ B) @ Xy)) @ (((product_snd @ A) @ B) @ Xy))))))))).
% 1.75/1.11 thf(fact_207_mem__Times__iff, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),A7:(set @ A),B7:(set @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_Sigma @ A) @ B) @ A7) @ (^ [Uu:A]: (B7)))) = ((((member @ A) @ (((product_fst @ A) @ B) @ X)) @ A7) & (((member @ B) @ (((product_snd @ A) @ B) @ X)) @ B7)))))).
% 1.75/1.11 thf(fact_208_exE__realizer, axiom, (! [C:$tType,A:$tType,B:$tType,P2:(A > (B > $o)),P:((product_prod @ B) @ A),Q:(C > $o),F:(B > (A > C))]: ((((P2 @ (((product_snd @ B) @ A) @ P)) @ (((product_fst @ B) @ A) @ P)) => ((! [X2:B,Y:A]: ((((P2 @ Y) @ X2) => (Q @ ((F @ X2) @ Y))))) => (Q @ (((((product_case_prod @ B) @ A) @ C) @ F) @ P))))))).
% 1.75/1.11 thf(fact_209_exI__realizer, axiom, (! [B:$tType,A:$tType,P2:(A > (B > $o)),Y3:A,X:B]: ((((P2 @ Y3) @ X) => ((P2 @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y3))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y3))))))).
% 1.75/1.11 thf(fact_210_conjI__realizer, axiom, (! [A:$tType,B:$tType,P2:(A > $o),P:A,Q:(B > $o),Q2:B]: (((P2 @ P) => ((Q @ Q2) => ((P2 @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P) @ Q2))) & (Q @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P) @ Q2))))))))).
% 1.75/1.11 thf(fact_211_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P2:(A > (B > $o)),X:A,Y3:B,A3:((product_prod @ A) @ B)]: ((((P2 @ X) @ Y3) => ((A3 = ((((product_Pair @ A) @ B) @ X) @ Y3)) => ((P2 @ (((product_fst @ A) @ B) @ A3)) @ (((product_snd @ A) @ B) @ A3))))))).
% 1.75/1.11 thf(fact_212_above__def, axiom, (! [A:$tType]: (((order_above @ A) = (^ [R2:(set @ ((product_prod @ A) @ A)),A4:A]: (((collect @ A) @ (^ [B4:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ B4)) @ R2)))))))))).
% 1.75/1.11 thf(fact_213_aboveS__def, axiom, (! [A:$tType]: (((order_aboveS @ A) = (^ [R2:(set @ ((product_prod @ A) @ A)),A4:A]: (((collect @ A) @ (^ [B4:A]: (((B4 != A4) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ B4)) @ R2))))))))))).
% 1.75/1.11 thf(fact_214_sndI, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),Y3:A,Z:B]: (((X = ((((product_Pair @ A) @ B) @ Y3) @ Z)) => ((((product_snd @ A) @ B) @ X) = Z))))).
% 1.75/1.11 thf(fact_215_fstI, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),Y3:A,Z:B]: (((X = ((((product_Pair @ A) @ B) @ Y3) @ Z)) => ((((product_fst @ A) @ B) @ X) = Y3))))).
% 1.75/1.11 thf(fact_216_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B3:A,P:((product_prod @ B) @ A)]: (((B3 = (((product_snd @ B) @ A) @ P)) = (? [A4:B]: ((P = ((((product_Pair @ B) @ A) @ A4) @ B3)))))))).
% 1.75/1.11 thf(fact_217_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A3:A,P:((product_prod @ A) @ B)]: (((A3 = (((product_fst @ A) @ B) @ P)) = (? [B4:B]: ((P = ((((product_Pair @ A) @ B) @ A3) @ B4)))))))).
% 1.75/1.11 thf(fact_218_subset__fst__snd, axiom, (! [B:$tType,A:$tType,A7:(set @ ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A7) @ ((((product_Sigma @ A) @ B) @ ((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A7)) @ (^ [Uu:A]: (((((image @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A7)))))))).
% 1.75/1.11 thf(fact_219_scomp__unfold, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((((product_scomp @ A) @ B) @ C) @ D) = (^ [F4:(A > ((product_prod @ B) @ C)),G4:(B > (C > D)),X3:A]: (((G4 @ (((product_fst @ B) @ C) @ (F4 @ X3))) @ (((product_snd @ B) @ C) @ (F4 @ X3))))))))).
% 1.75/1.11 thf(fact_220_scomp__apply, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: ((((((product_scomp @ B) @ C) @ D) @ A) = (^ [F4:(B > ((product_prod @ C) @ D)),G4:(C > (D > A)),X3:B]: ((((((product_case_prod @ C) @ D) @ A) @ G4) @ (F4 @ X3)))))))).
% 1.75/1.11 thf(fact_221_subset__Collect__iff, axiom, (! [A:$tType,B7:(set @ A),A7:(set @ A),P2:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B7) @ A7) => ((((ord_less_eq @ (set @ A)) @ B7) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A7) & (P2 @ X3)))))) = (! [X3:A]: (((((member @ A) @ X3) @ B7) => (P2 @ X3))))))))).
% 1.75/1.11 thf(fact_222_subset__CollectI, axiom, (! [A:$tType,B7:(set @ A),A7:(set @ A),Q:(A > $o),P2:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B7) @ A7) => ((! [X2:A]: (((((member @ A) @ X2) @ B7) => ((Q @ X2) => (P2 @ X2))))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ B7) & (Q @ X3)))))) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A7) & (P2 @ X3))))))))))).
% 1.75/1.11 thf(fact_223_Sigma__mono, axiom, (! [B:$tType,A:$tType,A7:(set @ A),C5:(set @ A),B7:(A > (set @ B)),D3:(A > (set @ B))]: (((((ord_less_eq @ (set @ A)) @ A7) @ C5) => ((! [X2:A]: (((((member @ A) @ X2) @ A7) => (((ord_less_eq @ (set @ B)) @ (B7 @ X2)) @ (D3 @ X2))))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A7) @ B7)) @ ((((product_Sigma @ A) @ B) @ C5) @ D3))))))).
% 1.75/1.11 thf(fact_224_Times__subset__cancel2, axiom, (! [A:$tType,B:$tType,X:A,C5:(set @ A),A7:(set @ B),B7:(set @ B)]: (((((member @ A) @ X) @ C5) => ((((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ ((((product_Sigma @ B) @ A) @ A7) @ (^ [Uu:B]: (C5)))) @ ((((product_Sigma @ B) @ A) @ B7) @ (^ [Uu:B]: (C5)))) = (((ord_less_eq @ (set @ B)) @ A7) @ B7)))))).
% 1.75/1.11 thf(fact_225_subrelI, axiom, (! [B:$tType,A:$tType,R4:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: (((! [X2:A,Y:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ R4) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ S2)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R4) @ S2))))).
% 1.75/1.11 thf(fact_226_subset__image__iff, axiom, (! [A:$tType,B:$tType,B7:(set @ A),F:(B > A),A7:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B7) @ ((((image @ B) @ A) @ F) @ A7)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A7) & (B7 = ((((image @ B) @ A) @ F) @ AA))))))))).
% 1.75/1.11 thf(fact_227_image__subset__iff, axiom, (! [A:$tType,B:$tType,F:(B > A),A7:(set @ B),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A7)) @ B7) = (! [X3:B]: (((((member @ B) @ X3) @ A7) => (((member @ A) @ (F @ X3)) @ B7)))))))).
% 1.75/1.11 thf(fact_228_subset__imageE, axiom, (! [A:$tType,B:$tType,B7:(set @ A),F:(B > A),A7:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B7) @ ((((image @ B) @ A) @ F) @ A7)) => (~ ((! [C7:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C7) @ A7) => (B7 != ((((image @ B) @ A) @ F) @ C7))))))))))).
% 1.75/1.11 thf(fact_229_image__subsetI, axiom, (! [A:$tType,B:$tType,A7:(set @ A),F:(A > B),B7:(set @ B)]: (((! [X2:A]: (((((member @ A) @ X2) @ A7) => (((member @ B) @ (F @ X2)) @ B7)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A7)) @ B7))))).
% 1.75/1.11 thf(fact_230_image__mono, axiom, (! [B:$tType,A:$tType,A7:(set @ A),B7:(set @ A),F:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A7)) @ ((((image @ A) @ B) @ F) @ B7)))))).
% 1.75/1.11 thf(fact_231_conj__subset__def, axiom, (! [A:$tType,A7:(set @ A),P2:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A7) @ ((collect @ A) @ (^ [X3:A]: (((P2 @ X3) & (Q @ X3)))))) = ((((ord_less_eq @ (set @ A)) @ A7) @ ((collect @ A) @ P2)) & (((ord_less_eq @ (set @ A)) @ A7) @ ((collect @ A) @ Q))))))).
% 1.75/1.11 thf(fact_232_Collect__restrict, axiom, (! [A:$tType,X7:(set @ A),P2:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ X7) & (P2 @ X3)))))) @ X7)))).
% 1.75/1.11 thf(fact_233_prop__restrict, axiom, (! [A:$tType,X:A,Z4:(set @ A),X7:(set @ A),P2:(A > $o)]: (((((member @ A) @ X) @ Z4) => ((((ord_less_eq @ (set @ A)) @ Z4) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ X7) & (P2 @ X3)))))) => (P2 @ X)))))).
% 1.75/1.11 thf(fact_234_scomp__scomp, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F2:$tType,E:$tType,F:(A > ((product_prod @ E) @ F2)),G3:(E > (F2 > ((product_prod @ C) @ D))),H:(C > (D > B))]: ((((((((product_scomp @ A) @ C) @ D) @ B) @ ((((((product_scomp @ A) @ E) @ F2) @ ((product_prod @ C) @ D)) @ F) @ G3)) @ H) = ((((((product_scomp @ A) @ E) @ F2) @ B) @ F) @ (^ [X3:E]: (((((((product_scomp @ F2) @ C) @ D) @ B) @ (G3 @ X3)) @ H)))))))).
% 1.75/1.11 thf(fact_235_wlog__linorder__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type @ A)) => (! [P2:(A > (A > $o)),B3:A,A3:A]: (((! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((P2 @ A2) @ B2)))) => ((((P2 @ B3) @ A3) => ((P2 @ A3) @ B3)) => ((P2 @ A3) @ B3))))))))).
% 1.75/1.11 thf(fact_236_scomp__Pair, axiom, (! [C:$tType,B:$tType,A:$tType,X:(A > ((product_prod @ B) @ C))]: ((((((((product_scomp @ A) @ B) @ C) @ ((product_prod @ B) @ C)) @ X) @ ((product_Pair @ B) @ C)) = X)))).
% 1.75/1.11 thf(fact_237_image__Collect__subsetI, axiom, (! [A:$tType,B:$tType,P2:(A > $o),F:(A > B),B7:(set @ B)]: (((! [X2:A]: (((P2 @ X2) => (((member @ B) @ (F @ X2)) @ B7)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ ((collect @ A) @ P2))) @ B7))))).
% 1.75/1.11 thf(fact_238_scomp__def, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((((product_scomp @ A) @ B) @ C) @ D) = (^ [F4:(A > ((product_prod @ B) @ C)),G4:(B > (C > D)),X3:A]: ((((((product_case_prod @ B) @ C) @ D) @ G4) @ (F4 @ X3)))))))).
% 1.75/1.11 thf(fact_239_Pair__scomp, axiom, (! [A:$tType,B:$tType,C:$tType,X:C,F:(C > (A > B))]: ((((((((product_scomp @ A) @ C) @ A) @ B) @ (((product_Pair @ C) @ A) @ X)) @ F) = (F @ X))))).
% 1.75/1.11 thf(fact_240_refl__onI, axiom, (! [A:$tType,R4:(set @ ((product_prod @ A) @ A)),A7:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R4) @ ((((product_Sigma @ A) @ A) @ A7) @ (^ [Uu:A]: (A7)))) => ((! [X2:A]: (((((member @ A) @ X2) @ A7) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ R4)))) => (((refl_on @ A) @ A7) @ R4)))))).
% 1.75/1.11 thf(fact_241_refl__on__def, axiom, (! [A:$tType]: (((refl_on @ A) = (^ [A8:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R2) @ ((((product_Sigma @ A) @ A) @ A8) @ (^ [Uu:A]: (A8)))) & (! [X3:A]: (((((member @ A) @ X3) @ A8) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ R2))))))))))).
% 1.75/1.11 thf(fact_242_Collect__split__mono__strong, axiom, (! [B:$tType,A:$tType,X7:(set @ A),A7:(set @ ((product_prod @ A) @ B)),Y8:(set @ B),P2:(A > (B > $o)),Q:(A > (B > $o))]: (((X7 = ((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A7)) => ((Y8 = ((((image @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A7)) => ((! [X2:A]: (((((member @ A) @ X2) @ X7) => (! [Xa:B]: (((((member @ B) @ Xa) @ Y8) => (((P2 @ X2) @ Xa) => ((Q @ X2) @ Xa)))))))) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A7) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P2))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A7) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ Q)))))))))).
% 1.75/1.11 thf(fact_243_subset__snd__imageI, axiom, (! [B:$tType,A:$tType,A7:(set @ A),B7:(set @ B),S:(set @ ((product_prod @ A) @ B)),X:A]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A7) @ (^ [Uu:A]: (B7)))) @ S) => ((((member @ A) @ X) @ A7) => (((ord_less_eq @ (set @ B)) @ B7) @ ((((image @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ S))))))).
% 1.75/1.11 thf(fact_244_subset__fst__imageI, axiom, (! [B:$tType,A:$tType,A7:(set @ A),B7:(set @ B),S:(set @ ((product_prod @ A) @ B)),Y3:B]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A7) @ (^ [Uu:A]: (B7)))) @ S) => ((((member @ B) @ Y3) @ B7) => (((ord_less_eq @ (set @ A)) @ A7) @ ((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ S))))))).
% 1.75/1.11 thf(fact_245_Quotient__rep__abs__eq, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T2:(A > (B > $o)),T4:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T2) => (((R @ T4) @ T4) => ((((ord_less_eq @ (A > (A > $o))) @ R) @ (^ [Y7:A,Z2:A]: ((Y7 = Z2)))) => ((Rep @ (Abs @ T4)) = T4))))))).
% 1.75/1.11 thf(fact_246_reflp__ge__eq, axiom, (! [A:$tType,R:(A > (A > $o))]: ((((reflp @ A) @ R) => (((ord_less_eq @ (A > (A > $o))) @ (^ [Y7:A,Z2:A]: ((Y7 = Z2)))) @ R))))).
% 1.75/1.11 thf(fact_247_Collect__case__prod__mono, axiom, (! [B:$tType,A:$tType,A7:(A > (B > $o)),B7:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ A7) @ B7) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A7))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ B7))))))).
% 1.75/1.11 thf(fact_248_pred__subset__eq, axiom, (! [A:$tType,R:(set @ A),S:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ R)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ S)))) = (((ord_less_eq @ (set @ A)) @ R) @ S))))).
% 1.75/1.11 thf(fact_249_pred__subset__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (A > (B > $o))) @ (^ [X3:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y4)) @ R)))) @ (^ [X3:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y4)) @ S)))) = (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S))))).
% 1.75/1.11 thf(fact_250_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A8:(set @ A),B8:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A8)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B8)))))))))).
% 1.75/1.11 thf(fact_251_reflp__eq, axiom, (! [A:$tType]: (((reflp @ A) = ((ord_less_eq @ (A > (A > $o))) @ (^ [Y7:A,Z2:A]: ((Y7 = Z2)))))))).
% 1.75/1.11 thf(fact_252_Gr__incl, axiom, (! [A:$tType,B:$tType,A7:(set @ A),F:(A > B),B7:(set @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((bNF_Gr @ A) @ B) @ A7) @ F)) @ ((((product_Sigma @ A) @ B) @ A7) @ (^ [Uu:A]: (B7)))) = (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A7)) @ B7))))).
% 1.75/1.11 thf(fact_253_flip__pred, axiom, (! [A:$tType,B:$tType,A7:(set @ ((product_prod @ A) @ B)),R:(B > (A > $o))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A7) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (((conversep @ B) @ A) @ R)))) => (((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ ((((image @ ((product_prod @ A) @ B)) @ ((product_prod @ B) @ A)) @ ((((product_case_prod @ A) @ B) @ ((product_prod @ B) @ A)) @ (^ [X3:A,Y4:B]: (((((product_Pair @ B) @ A) @ Y4) @ X3))))) @ A7)) @ ((collect @ ((product_prod @ B) @ A)) @ ((((product_case_prod @ B) @ A) @ $o) @ R))))))).
% 1.75/1.11 thf(fact_254_conversep__eq, axiom, (! [A:$tType]: (((((conversep @ A) @ A) @ (^ [Y7:A,Z2:A]: ((Y7 = Z2)))) = (^ [Y7:A,Z2:A]: ((Y7 = Z2))))))).
% 1.75/1.11 thf(fact_255_conversep__iff, axiom, (! [B:$tType,A:$tType]: ((((conversep @ A) @ B) = (^ [R2:(A > (B > $o)),A4:B,B4:A]: (((R2 @ B4) @ A4))))))).
% 1.75/1.11 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type @ $o))).
% 1.75/1.11 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_1, axiom, ((linorder @ product_unit) @ (type @ product_unit))).
% 1.75/1.11 thf(help_If_3_1_T, axiom, (! [P2:$o]: (((P2 = $true) | (P2 = $false))))).
% 1.75/1.11 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y3:A]: ((((((if @ A) @ $false) @ X) @ Y3) = Y3)))).
% 1.75/1.11 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y3:A]: ((((((if @ A) @ $true) @ X) @ Y3) = X)))).
% 1.75/1.11 thf(conj_0, hypothesis, ((((((product_case_prod @ (coinductive_llist @ a)) @ b) @ (((product_prod @ (coinductive_llist @ a)) @ b) > $o)) @ (^ [Xs2:(coinductive_llist @ a),A4:b]: (((((product_case_prod @ (coinductive_llist @ a)) @ b) @ $o) @ (^ [Ys:(coinductive_llist @ a),B4:b]: (((Xs2 = Ys) & (((coinductive_lfinite @ a) @ Ys) => (A4 = B4))))))))) @ prod1) @ prod2)).
% 1.75/1.11 thf(conj_1, conjecture, ((((((product_case_prod @ (coinductive_llist @ a)) @ b) @ (((product_prod @ (coinductive_llist @ a)) @ b) > $o)) @ (^ [Xs2:(coinductive_llist @ a),A4:b]: (((((product_case_prod @ (coinductive_llist @ a)) @ b) @ $o) @ (^ [Ys:(coinductive_llist @ a),B4:b]: (((Xs2 = Ys) & (((coinductive_lfinite @ a) @ Ys) => (A4 = B4))))))))) @ (((((product_case_prod @ (coinductive_llist @ a)) @ b) @ ((product_prod @ (coinductive_llist @ a)) @ b)) @ (^ [Xs2:(coinductive_llist @ a),B6:b]: (((((product_Pair @ (coinductive_llist @ a)) @ b) @ (((coinductive_lfilter @ a) @ fun) @ Xs2)) @ ((((if @ b) @ ((coinductive_lfinite @ a) @ Xs2)) @ B6) @ b2))))) @ prod1)) @ (((((product_case_prod @ (coinductive_llist @ a)) @ b) @ ((product_prod @ (coinductive_llist @ a)) @ b)) @ (^ [Xs2:(coinductive_llist @ a),B6:b]: (((((product_Pair @ (coinductive_llist @ a)) @ b) @ (((coinductive_lfilter @ a) @ fun) @ Xs2)) @ ((((if @ b) @ ((coinductive_lfinite @ a) @ Xs2)) @ B6) @ b2))))) @ prod2))).
% 1.75/1.11 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.ihDMkoi04K/DTF2DTF29522.p
% 1.75/1.11 ------------------------
% 2.08/1.14 ---- Cleaned THF ---
% 2.08/1.14 thf(ty_t_TLList__Mirabelle__qhjoikztpd_Otllist,type,
% 2.08/1.14 tLList446370796tllist: $tType > $tType > $tType ).
% 2.08/1.14
% 2.08/1.14 thf(ty_t_Coinductive__List_Ollist,type,
% 2.08/1.14 coinductive_llist: $tType > $tType ).
% 2.08/1.14
% 2.08/1.14 thf(ty_t_Product__Type_Ounit,type,
% 2.08/1.14 product_unit: $tType ).
% 2.08/1.14
% 2.08/1.14 thf(ty_t_Product__Type_Oprod,type,
% 2.08/1.14 product_prod: $tType > $tType > $tType ).
% 2.08/1.14
% 2.08/1.14 thf(ty_t_Set_Oset,type,
% 2.08/1.14 set: $tType > $tType ).
% 2.08/1.14
% 2.08/1.14 thf(ty_t_itself,type,
% 2.08/1.14 itself: $tType > $tType ).
% 2.08/1.14
% 2.08/1.14 thf(ty_tf_b,type,
% 2.08/1.14 b: $tType ).
% 2.08/1.14
% 2.08/1.14 thf(ty_tf_a,type,
% 2.08/1.14 a: $tType ).
% 2.08/1.14
% 2.08/1.14 thf(sy_cl_Orderings_Olinorder,type,
% 2.08/1.14 linorder:
% 2.08/1.14 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_BNF__Def_OGr,type,
% 2.08/1.14 bNF_Gr:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Coinductive__List_Olfilter,type,
% 2.08/1.14 coinductive_lfilter:
% 2.08/1.14 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Coinductive__List_Olfinite,type,
% 2.08/1.14 coinductive_lfinite:
% 2.08/1.14 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Fun__Def_Oreduction__pair,type,
% 2.08/1.14 fun_reduction_pair:
% 2.08/1.14 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 2.08/1.14 fun_rp_inv_image:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_HOL_OThe,type,
% 2.08/1.14 the:
% 2.08/1.14 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_If,type,
% 2.08/1.14 if:
% 2.08/1.14 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Lifting_OQuotient,type,
% 2.08/1.14 quotient:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Order__Relation_Oabove,type,
% 2.08/1.14 order_above:
% 2.08/1.14 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Order__Relation_OaboveS,type,
% 2.08/1.14 order_aboveS:
% 2.08/1.14 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.08/1.14 ord_less_eq:
% 2.08/1.14 !>[A: $tType] : ( A > A > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.08/1.14 top_top:
% 2.08/1.14 !>[A: $tType] : A ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_OPair,type,
% 2.08/1.14 product_Pair:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_OSigma,type,
% 2.08/1.14 product_Sigma:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.08/1.14 produc2004651681e_prod:
% 2.08/1.14 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Omap__prod,type,
% 2.08/1.14 product_map_prod:
% 2.08/1.14 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 2.08/1.14 product_rec_bool:
% 2.08/1.14 !>[T: $tType] : ( T > T > $o > T ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
% 2.08/1.14 product_rec_set_bool:
% 2.08/1.14 !>[T: $tType] : ( T > T > $o > T > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.08/1.14 product_rec_prod:
% 2.08/1.14 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 2.08/1.14 product_rec_set_prod:
% 2.08/1.14 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 2.08/1.14 product_rec_set_unit:
% 2.08/1.14 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 2.08/1.14 product_rec_unit:
% 2.08/1.14 !>[T: $tType] : ( T > product_unit > T ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 2.08/1.14 product_case_prod:
% 2.08/1.14 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 2.08/1.14 product_fst:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 2.08/1.14 product_snd:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.08/1.14 product_swap:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Oproduct,type,
% 2.08/1.14 product_product:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Product__Type_Oscomp,type,
% 2.08/1.14 product_scomp:
% 2.08/1.14 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Pure_Otype,type,
% 2.08/1.14 type:
% 2.08/1.14 !>[A: $tType] : ( itself @ A ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Relation_Oconversep,type,
% 2.08/1.14 conversep:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > A > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Relation_Oinv__image,type,
% 2.08/1.14 inv_image:
% 2.08/1.14 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Relation_Oinv__imagep,type,
% 2.08/1.14 inv_imagep:
% 2.08/1.14 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Relation_Orefl__on,type,
% 2.08/1.14 refl_on:
% 2.08/1.14 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Relation_Oreflp,type,
% 2.08/1.14 reflp:
% 2.08/1.14 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Set_OCollect,type,
% 2.08/1.14 collect:
% 2.08/1.14 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Set_Oimage,type,
% 2.08/1.14 image:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Ocr__tllist,type,
% 2.08/1.14 tLList47617868tllist:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( product_prod @ ( coinductive_llist @ A ) @ B ) > ( tLList446370796tllist @ A @ B ) > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Ollist__of__tllist,type,
% 2.08/1.14 tLList798109904tllist:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > ( coinductive_llist @ A ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Oterminal0,type,
% 2.08/1.14 tLList1825092077minal0:
% 2.08/1.14 !>[A: $tType] : A ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Ocase__tllist,type,
% 2.08/1.14 tLList200813139tllist:
% 2.08/1.14 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > ( tLList446370796tllist @ A @ B ) > C ) > ( tLList446370796tllist @ A @ B ) > C ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Oterminal,type,
% 2.08/1.14 tLList2110128105rminal:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > B ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist__of__llist,type,
% 2.08/1.14 tLList1672613558_llist:
% 2.08/1.14 !>[B: $tType,A: $tType] : ( B > ( coinductive_llist @ A ) > ( tLList446370796tllist @ A @ B ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Wellfounded_Olex__prod,type,
% 2.08/1.14 lex_prod:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_Wfrec_Osame__fst,type,
% 2.08/1.14 same_fst:
% 2.08/1.14 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_c_member,type,
% 2.08/1.14 member:
% 2.08/1.14 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(sy_v_b,type,
% 2.08/1.14 b2: b ).
% 2.08/1.14
% 2.08/1.14 thf(sy_v_fun,type,
% 2.08/1.14 fun: a > $o ).
% 2.08/1.14
% 2.08/1.14 thf(sy_v_prod1,type,
% 2.08/1.14 prod1: product_prod @ ( coinductive_llist @ a ) @ b ).
% 2.08/1.14
% 2.08/1.14 thf(sy_v_prod2,type,
% 2.08/1.14 prod2: product_prod @ ( coinductive_llist @ a ) @ b ).
% 2.08/1.14
% 2.08/1.14 thf(fact_0_case__prodI2_H,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,C: $tType,P: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
% 2.08/1.14 ( ! [A2: A,B2: B] :
% 2.08/1.14 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.08/1.14 = P )
% 2.08/1.14 => ( C2 @ A2 @ B2 @ X ) )
% 2.08/1.14 => ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_1_case__prodI,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
% 2.08/1.14 ( ( F @ A3 @ B3 )
% 2.08/1.14 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_2_case__prodI2,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P: product_prod @ A @ B,C2: A > B > $o] :
% 2.08/1.14 ( ! [A2: A,B2: B] :
% 2.08/1.14 ( ( P
% 2.08/1.14 = ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.08/1.14 => ( C2 @ A2 @ B2 ) )
% 2.08/1.14 => ( product_case_prod @ A @ B @ $o @ C2 @ P ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_3_case__prod__conv,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B3: C] :
% 2.08/1.14 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 2.08/1.14 = ( F @ A3 @ B3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_4_lfilter__K__True,axiom,
% 2.08/1.14 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.08/1.14 ( ( coinductive_lfilter @ A
% 2.08/1.14 @ ^ [Uu: A] : $true
% 2.08/1.14 @ Xs )
% 2.08/1.14 = Xs ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_5_split__part,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P2: $o,Q: A > B > $o] :
% 2.08/1.14 ( ( product_case_prod @ A @ B @ $o
% 2.08/1.14 @ ^ [A4: A,B4: B] :
% 2.08/1.14 ( P2
% 2.08/1.14 & ( Q @ A4 @ B4 ) ) )
% 2.08/1.14 = ( ^ [Ab: product_prod @ A @ B] :
% 2.08/1.14 ( P2
% 2.08/1.14 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_6_lfilter__idem,axiom,
% 2.08/1.14 ! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
% 2.08/1.14 ( ( coinductive_lfilter @ A @ P2 @ ( coinductive_lfilter @ A @ P2 @ Xs ) )
% 2.08/1.14 = ( coinductive_lfilter @ A @ P2 @ Xs ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_7_case__prodD_H,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A3: A,B3: B,C2: C] :
% 2.08/1.14 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A3 @ B3 ) @ C2 )
% 2.08/1.14 => ( R @ A3 @ B3 @ C2 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_8_case__prodE_H,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P: product_prod @ A @ B,Z: C] :
% 2.08/1.14 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P @ Z )
% 2.08/1.14 => ~ ! [X2: A,Y: B] :
% 2.08/1.14 ( ( P
% 2.08/1.14 = ( product_Pair @ A @ B @ X2 @ Y ) )
% 2.08/1.14 => ~ ( C2 @ X2 @ Y @ Z ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_9_case__prod__Pair__iden,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 2.08/1.14 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P )
% 2.08/1.14 = P ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_10_prod_Oinject,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 2.08/1.14 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.08/1.14 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.08/1.14 = ( ( X1 = Y1 )
% 2.08/1.14 & ( X22 = Y2 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_11_old_Oprod_Oinject,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B5: B] :
% 2.08/1.14 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 2.08/1.14 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.08/1.14 = ( ( A3 = A5 )
% 2.08/1.14 & ( B3 = B5 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_12_mem__case__prodI2,axiom,
% 2.08/1.14 ! [C: $tType,B: $tType,A: $tType,P: product_prod @ A @ B,Z: C,C2: A > B > ( set @ C )] :
% 2.08/1.14 ( ! [A2: A,B2: B] :
% 2.08/1.14 ( ( P
% 2.08/1.14 = ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.08/1.14 => ( member @ C @ Z @ ( C2 @ A2 @ B2 ) ) )
% 2.08/1.14 => ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_13_mem__case__prodI,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),A3: B,B3: C] :
% 2.08/1.14 ( ( member @ A @ Z @ ( C2 @ A3 @ B3 ) )
% 2.08/1.14 => ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_14_mem__case__prodE,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),P: product_prod @ B @ C] :
% 2.08/1.14 ( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P ) )
% 2.08/1.14 => ~ ! [X2: B,Y: C] :
% 2.08/1.14 ( ( P
% 2.08/1.14 = ( product_Pair @ B @ C @ X2 @ Y ) )
% 2.08/1.14 => ~ ( member @ A @ Z @ ( C2 @ X2 @ Y ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_15_old_Oprod_Oinducts,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.08/1.14 ( ! [A2: A,B2: B] : ( P2 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.08/1.14 => ( P2 @ Prod ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_16_old_Oprod_Oexhaust,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 2.08/1.14 ~ ! [A2: A,B2: B] :
% 2.08/1.14 ( Y3
% 2.08/1.14 != ( product_Pair @ A @ B @ A2 @ B2 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_17_prod__induct7,axiom,
% 2.08/1.14 ! [G: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) )] :
% 2.08/1.14 ( ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G ) @ E2 @ ( product_Pair @ F2 @ G @ F3 @ G2 ) ) ) ) ) ) )
% 2.08/1.14 => ( P2 @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_18_prod__induct6,axiom,
% 2.08/1.14 ! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 2.08/1.14 ( ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 2.08/1.14 => ( P2 @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_19_prod__induct5,axiom,
% 2.08/1.14 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( 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 ) ) )] :
% 2.08/1.14 ( ! [A2: A,B2: B,C3: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 2.08/1.14 => ( P2 @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_20_prod__induct4,axiom,
% 2.08/1.14 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.08/1.14 ( ! [A2: A,B2: B,C3: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 2.08/1.14 => ( P2 @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_21_prod__induct3,axiom,
% 2.08/1.14 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.08/1.14 ( ! [A2: A,B2: B,C3: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C3 ) ) )
% 2.08/1.14 => ( P2 @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_22_prod__cases7,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) )] :
% 2.08/1.14 ~ ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2,G2: G] :
% 2.08/1.14 ( Y3
% 2.08/1.14 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G ) @ E2 @ ( product_Pair @ F2 @ G @ F3 @ G2 ) ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_23_prod__cases6,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 2.08/1.14 ~ ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2] :
% 2.08/1.14 ( Y3
% 2.08/1.14 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_24_prod__cases5,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.08/1.14 ~ ! [A2: A,B2: B,C3: C,D2: D,E2: E] :
% 2.08/1.14 ( Y3
% 2.08/1.14 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_25_prod__cases4,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.08/1.14 ~ ! [A2: A,B2: B,C3: C,D2: D] :
% 2.08/1.14 ( Y3
% 2.08/1.14 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_26_prod__cases3,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.08/1.14 ~ ! [A2: A,B2: B,C3: C] :
% 2.08/1.14 ( Y3
% 2.08/1.14 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C3 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_27_Pair__inject,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B5: B] :
% 2.08/1.14 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 2.08/1.14 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.08/1.14 => ~ ( ( A3 = A5 )
% 2.08/1.14 => ( B3 != B5 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_28_prod__cases,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 2.08/1.14 ( ! [A2: A,B2: B] : ( P2 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.08/1.14 => ( P2 @ P ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_29_surj__pair,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 2.08/1.14 ? [X2: A,Y: B] :
% 2.08/1.14 ( P
% 2.08/1.14 = ( product_Pair @ A @ B @ X2 @ Y ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_30_prod_Ocase__distrib,axiom,
% 2.08/1.14 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
% 2.08/1.14 ( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 2.08/1.14 = ( product_case_prod @ A @ B @ D
% 2.08/1.14 @ ^ [X12: A,X23: B] : ( H @ ( F @ X12 @ X23 ) )
% 2.08/1.14 @ Prod ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_31_prod_Odisc__eq__case,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.08/1.14 ( product_case_prod @ A @ B @ $o
% 2.08/1.14 @ ^ [Uu: A,Uv: B] : $true
% 2.08/1.14 @ Prod ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_32_lfilter__lfilter,axiom,
% 2.08/1.14 ! [A: $tType,P2: A > $o,Q: A > $o,Xs: coinductive_llist @ A] :
% 2.08/1.14 ( ( coinductive_lfilter @ A @ P2 @ ( coinductive_lfilter @ A @ Q @ Xs ) )
% 2.08/1.14 = ( coinductive_lfilter @ A
% 2.08/1.14 @ ^ [X3: A] :
% 2.08/1.14 ( ( P2 @ X3 )
% 2.08/1.14 & ( Q @ X3 ) )
% 2.08/1.14 @ Xs ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_33_old_Oprod_Ocase,axiom,
% 2.08/1.14 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 2.08/1.14 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 2.08/1.14 = ( F @ X1 @ X22 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_34_lfinite__lfilterI,axiom,
% 2.08/1.14 ! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
% 2.08/1.14 ( ( coinductive_lfinite @ A @ Xs )
% 2.08/1.14 => ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P2 @ Xs ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_35_cond__case__prod__eta,axiom,
% 2.08/1.14 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G3: ( product_prod @ A @ B ) > C] :
% 2.08/1.14 ( ! [X2: A,Y: B] :
% 2.08/1.14 ( ( F @ X2 @ Y )
% 2.08/1.14 = ( G3 @ ( product_Pair @ A @ B @ X2 @ Y ) ) )
% 2.08/1.14 => ( ( product_case_prod @ A @ B @ C @ F )
% 2.08/1.14 = G3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_36_case__prod__eta,axiom,
% 2.08/1.14 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 2.08/1.14 ( ( product_case_prod @ A @ B @ C
% 2.08/1.14 @ ^ [X3: A,Y4: B] : ( F @ ( product_Pair @ A @ B @ X3 @ Y4 ) ) )
% 2.08/1.14 = F ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_37_case__prodE2,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P2: B > C > A,Z: product_prod @ B @ C] :
% 2.08/1.14 ( ( Q @ ( product_case_prod @ B @ C @ A @ P2 @ Z ) )
% 2.08/1.14 => ~ ! [X2: B,Y: C] :
% 2.08/1.14 ( ( Z
% 2.08/1.14 = ( product_Pair @ B @ C @ X2 @ Y ) )
% 2.08/1.14 => ~ ( Q @ ( P2 @ X2 @ Y ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_38_case__prod__app,axiom,
% 2.08/1.14 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 2.08/1.14 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 2.08/1.14 = ( ^ [F4: B > C > D > A,X3: product_prod @ B @ C,Y4: D] :
% 2.08/1.14 ( product_case_prod @ B @ C @ A
% 2.08/1.14 @ ^ [L: B,R2: C] : ( F4 @ L @ R2 @ Y4 )
% 2.08/1.14 @ X3 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_39_case__prodE,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,C2: A > B > $o,P: product_prod @ A @ B] :
% 2.08/1.14 ( ( product_case_prod @ A @ B @ $o @ C2 @ P )
% 2.08/1.14 => ~ ! [X2: A,Y: B] :
% 2.08/1.14 ( ( P
% 2.08/1.14 = ( product_Pair @ A @ B @ X2 @ Y ) )
% 2.08/1.14 => ~ ( C2 @ X2 @ Y ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_40_case__prodD,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
% 2.08/1.14 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.08/1.14 => ( F @ A3 @ B3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_41_old_Oprod_Orec,axiom,
% 2.08/1.14 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 2.08/1.14 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.08/1.14 = ( F1 @ A3 @ B3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_42_split__cong,axiom,
% 2.08/1.14 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G3: A > B > C,P: product_prod @ A @ B] :
% 2.08/1.14 ( ! [X2: A,Y: B] :
% 2.08/1.14 ( ( ( product_Pair @ A @ B @ X2 @ Y )
% 2.08/1.14 = Q2 )
% 2.08/1.14 => ( ( F @ X2 @ Y )
% 2.08/1.14 = ( G3 @ X2 @ Y ) ) )
% 2.08/1.14 => ( ( P = Q2 )
% 2.08/1.14 => ( ( product_case_prod @ A @ B @ C @ F @ P )
% 2.08/1.14 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_43_same__fst__def,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( same_fst @ A @ B )
% 2.08/1.14 = ( ^ [P3: A > $o,R3: A > ( set @ ( product_prod @ B @ B ) )] :
% 2.08/1.14 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 2.08/1.14 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 2.08/1.14 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 2.08/1.14 @ ^ [X4: A,Y5: B] :
% 2.08/1.14 ( product_case_prod @ A @ B @ $o
% 2.08/1.14 @ ^ [X3: A,Y4: B] :
% 2.08/1.14 ( ( X4 = X3 )
% 2.08/1.14 & ( P3 @ X3 )
% 2.08/1.14 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y5 @ Y4 ) @ ( R3 @ X3 ) ) ) ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_44_lex__prod__def,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( lex_prod @ A @ B )
% 2.08/1.14 = ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
% 2.08/1.14 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 2.08/1.14 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 2.08/1.14 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 2.08/1.14 @ ^ [A4: A,B4: B] :
% 2.08/1.14 ( product_case_prod @ A @ B @ $o
% 2.08/1.14 @ ^ [A6: A,B6: B] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A6 ) @ Ra )
% 2.08/1.14 | ( ( A4 = A6 )
% 2.08/1.14 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_45_mem__Collect__eq,axiom,
% 2.08/1.14 ! [A: $tType,A3: A,P2: A > $o] :
% 2.08/1.14 ( ( member @ A @ A3 @ ( collect @ A @ P2 ) )
% 2.08/1.14 = ( P2 @ A3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_46_Collect__mem__eq,axiom,
% 2.08/1.14 ! [A: $tType,A7: set @ A] :
% 2.08/1.14 ( ( collect @ A
% 2.08/1.14 @ ^ [X3: A] : ( member @ A @ X3 @ A7 ) )
% 2.08/1.14 = A7 ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_47_Collect__cong,axiom,
% 2.08/1.14 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.08/1.14 ( ! [X2: A] :
% 2.08/1.14 ( ( P2 @ X2 )
% 2.08/1.14 = ( Q @ X2 ) )
% 2.08/1.14 => ( ( collect @ A @ P2 )
% 2.08/1.14 = ( collect @ A @ Q ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_48_ext,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,F: A > B,G3: A > B] :
% 2.08/1.14 ( ! [X2: A] :
% 2.08/1.14 ( ( F @ X2 )
% 2.08/1.14 = ( G3 @ X2 ) )
% 2.08/1.14 => ( F = G3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_49_internal__case__prod__def,axiom,
% 2.08/1.14 ! [C: $tType,B: $tType,A: $tType] :
% 2.08/1.14 ( ( produc2004651681e_prod @ A @ B @ C )
% 2.08/1.14 = ( product_case_prod @ A @ B @ C ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_50_tllist_Oabs__eq__iff,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,X: product_prod @ ( coinductive_llist @ A ) @ B,Y3: product_prod @ ( coinductive_llist @ A ) @ B] :
% 2.08/1.14 ( ( ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( tLList446370796tllist @ A @ B )
% 2.08/1.14 @ ^ [Xs2: coinductive_llist @ A,A4: B] : ( tLList1672613558_llist @ B @ A @ A4 @ Xs2 )
% 2.08/1.14 @ X )
% 2.08/1.14 = ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( tLList446370796tllist @ A @ B )
% 2.08/1.14 @ ^ [Xs2: coinductive_llist @ A,A4: B] : ( tLList1672613558_llist @ B @ A @ A4 @ Xs2 )
% 2.08/1.14 @ Y3 ) )
% 2.08/1.14 = ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( ( product_prod @ ( coinductive_llist @ A ) @ B ) > $o )
% 2.08/1.14 @ ^ [Xs2: coinductive_llist @ A,A4: B] :
% 2.08/1.14 ( product_case_prod @ ( coinductive_llist @ A ) @ B @ $o
% 2.08/1.14 @ ^ [Ys: coinductive_llist @ A,B4: B] :
% 2.08/1.14 ( ( Xs2 = Ys )
% 2.08/1.14 & ( ( coinductive_lfinite @ A @ Ys )
% 2.08/1.14 => ( A4 = B4 ) ) ) )
% 2.08/1.14 @ X
% 2.08/1.14 @ Y3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_51_pred__equals__eq2,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.08/1.14 ( ( ( ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R ) )
% 2.08/1.14 = ( ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S ) ) )
% 2.08/1.14 = ( R = S ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_52_internal__case__prod__conv,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B3: C] :
% 2.08/1.14 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 2.08/1.14 = ( C2 @ A3 @ B3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_53_reflp__tllist,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( reflp @ ( product_prod @ ( coinductive_llist @ A ) @ B )
% 2.08/1.14 @ ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( ( product_prod @ ( coinductive_llist @ A ) @ B ) > $o )
% 2.08/1.14 @ ^ [Xs2: coinductive_llist @ A,A4: B] :
% 2.08/1.14 ( product_case_prod @ ( coinductive_llist @ A ) @ B @ $o
% 2.08/1.14 @ ^ [Ys: coinductive_llist @ A,B4: B] :
% 2.08/1.14 ( ( Xs2 = Ys )
% 2.08/1.14 & ( ( coinductive_lfinite @ A @ Ys )
% 2.08/1.14 => ( A4 = B4 ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_54_ssubst__Pair__rhs,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R4: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R4 @ S2 ) @ R )
% 2.08/1.14 => ( ( S3 = S2 )
% 2.08/1.14 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R4 @ S3 ) @ R ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_55_The__split__eq,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,X: A,Y3: B] :
% 2.08/1.14 ( ( the @ ( product_prod @ A @ B )
% 2.08/1.14 @ ( product_case_prod @ A @ B @ $o
% 2.08/1.14 @ ^ [X4: A,Y5: B] :
% 2.08/1.14 ( ( X = X4 )
% 2.08/1.14 & ( Y3 = Y5 ) ) ) )
% 2.08/1.14 = ( product_Pair @ A @ B @ X @ Y3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_56_tllist__of__llist__inject,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,B3: B,Xs: coinductive_llist @ A,C2: B,Ys2: coinductive_llist @ A] :
% 2.08/1.14 ( ( ( tLList1672613558_llist @ B @ A @ B3 @ Xs )
% 2.08/1.14 = ( tLList1672613558_llist @ B @ A @ C2 @ Ys2 ) )
% 2.08/1.14 = ( ( Xs = Ys2 )
% 2.08/1.14 & ( ( coinductive_lfinite @ A @ Ys2 )
% 2.08/1.14 => ( B3 = C2 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_57_in__lex__prod,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B5: B,R4: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
% 2.08/1.14 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Pair @ A @ B @ A5 @ B5 ) ) @ ( lex_prod @ A @ B @ R4 @ S2 ) )
% 2.08/1.14 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A5 ) @ R4 )
% 2.08/1.14 | ( ( A3 = A5 )
% 2.08/1.14 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B5 ) @ S2 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_58_same__fstI,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P2: A > $o,X: A,Y6: B,Y3: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
% 2.08/1.14 ( ( P2 @ X )
% 2.08/1.14 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y3 ) @ ( R @ X ) )
% 2.08/1.14 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y6 ) @ ( product_Pair @ A @ B @ X @ Y3 ) ) @ ( same_fst @ A @ B @ P2 @ R ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_59_reflp__mono,axiom,
% 2.08/1.14 ! [A: $tType,R: A > A > $o,Q: A > A > $o] :
% 2.08/1.14 ( ( reflp @ A @ R )
% 2.08/1.14 => ( ! [X2: A,Y: A] :
% 2.08/1.14 ( ( R @ X2 @ Y )
% 2.08/1.14 => ( Q @ X2 @ Y ) )
% 2.08/1.14 => ( reflp @ A @ Q ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_60_reflp__def,axiom,
% 2.08/1.14 ! [A: $tType] :
% 2.08/1.14 ( ( reflp @ A )
% 2.08/1.14 = ( ^ [R2: A > A > $o] :
% 2.08/1.14 ! [X3: A] : ( R2 @ X3 @ X3 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_61_theI__unique,axiom,
% 2.08/1.14 ! [A: $tType,P2: A > $o,X: A] :
% 2.08/1.14 ( ? [X5: A] :
% 2.08/1.14 ( ( P2 @ X5 )
% 2.08/1.14 & ! [Y: A] :
% 2.08/1.14 ( ( P2 @ Y )
% 2.08/1.14 => ( Y = X5 ) ) )
% 2.08/1.14 => ( ( P2 @ X )
% 2.08/1.14 = ( X
% 2.08/1.14 = ( the @ A @ P2 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_62_reflpI,axiom,
% 2.08/1.14 ! [A: $tType,R4: A > A > $o] :
% 2.08/1.14 ( ! [X2: A] : ( R4 @ X2 @ X2 )
% 2.08/1.14 => ( reflp @ A @ R4 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_63_reflpE,axiom,
% 2.08/1.14 ! [A: $tType,R4: A > A > $o,X: A] :
% 2.08/1.14 ( ( reflp @ A @ R4 )
% 2.08/1.14 => ( R4 @ X @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_64_reflpD,axiom,
% 2.08/1.14 ! [A: $tType,R4: A > A > $o,X: A] :
% 2.08/1.14 ( ( reflp @ A @ R4 )
% 2.08/1.14 => ( R4 @ X @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_65_tllist_Oabs__induct,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P2: ( tLList446370796tllist @ A @ B ) > $o,X: tLList446370796tllist @ A @ B] :
% 2.08/1.14 ( ! [Y: product_prod @ ( coinductive_llist @ A ) @ B] :
% 2.08/1.14 ( P2
% 2.08/1.14 @ ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( tLList446370796tllist @ A @ B )
% 2.08/1.14 @ ^ [Xs2: coinductive_llist @ A,A4: B] : ( tLList1672613558_llist @ B @ A @ A4 @ Xs2 )
% 2.08/1.14 @ Y ) )
% 2.08/1.14 => ( P2 @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_66_tllist__of__llist__cong,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,Xs: coinductive_llist @ A,Xs3: coinductive_llist @ A,B3: B,B5: B] :
% 2.08/1.14 ( ( Xs = Xs3 )
% 2.08/1.14 => ( ( ( coinductive_lfinite @ A @ Xs3 )
% 2.08/1.14 => ( B3 = B5 ) )
% 2.08/1.14 => ( ( tLList1672613558_llist @ B @ A @ B3 @ Xs )
% 2.08/1.14 = ( tLList1672613558_llist @ B @ A @ B5 @ Xs3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_67_old_Orec__prod__def,axiom,
% 2.08/1.14 ! [T: $tType,B: $tType,A: $tType] :
% 2.08/1.14 ( ( product_rec_prod @ A @ B @ T )
% 2.08/1.14 = ( ^ [F12: A > B > T,X3: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_68_the__sym__eq__trivial,axiom,
% 2.08/1.14 ! [A: $tType,X: A] :
% 2.08/1.14 ( ( the @ A
% 2.08/1.14 @ ( ^ [Y7: A,Z2: A] : ( Y7 = Z2 )
% 2.08/1.14 @ X ) )
% 2.08/1.14 = X ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_69_the__eq__trivial,axiom,
% 2.08/1.14 ! [A: $tType,A3: A] :
% 2.08/1.14 ( ( the @ A
% 2.08/1.14 @ ^ [X3: A] : ( X3 = A3 ) )
% 2.08/1.14 = A3 ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_70_the__equality,axiom,
% 2.08/1.14 ! [A: $tType,P2: A > $o,A3: A] :
% 2.08/1.14 ( ( P2 @ A3 )
% 2.08/1.14 => ( ! [X2: A] :
% 2.08/1.14 ( ( P2 @ X2 )
% 2.08/1.14 => ( X2 = A3 ) )
% 2.08/1.14 => ( ( the @ A @ P2 )
% 2.08/1.14 = A3 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_71_the1__equality,axiom,
% 2.08/1.14 ! [A: $tType,P2: A > $o,A3: A] :
% 2.08/1.14 ( ? [X5: A] :
% 2.08/1.14 ( ( P2 @ X5 )
% 2.08/1.14 & ! [Y: A] :
% 2.08/1.14 ( ( P2 @ Y )
% 2.08/1.14 => ( Y = X5 ) ) )
% 2.08/1.14 => ( ( P2 @ A3 )
% 2.08/1.14 => ( ( the @ A @ P2 )
% 2.08/1.14 = A3 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_72_the1I2,axiom,
% 2.08/1.14 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.08/1.14 ( ? [X5: A] :
% 2.08/1.14 ( ( P2 @ X5 )
% 2.08/1.14 & ! [Y: A] :
% 2.08/1.14 ( ( P2 @ Y )
% 2.08/1.14 => ( Y = X5 ) ) )
% 2.08/1.14 => ( ! [X2: A] :
% 2.08/1.14 ( ( P2 @ X2 )
% 2.08/1.14 => ( Q @ X2 ) )
% 2.08/1.14 => ( Q @ ( the @ A @ P2 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_73_If__def,axiom,
% 2.08/1.14 ! [A: $tType] :
% 2.08/1.14 ( ( if @ A )
% 2.08/1.14 = ( ^ [P3: $o,X3: A,Y4: A] :
% 2.08/1.14 ( the @ A
% 2.08/1.14 @ ^ [Z3: A] :
% 2.08/1.14 ( ( P3
% 2.08/1.14 => ( Z3 = X3 ) )
% 2.08/1.14 & ( ~ P3
% 2.08/1.14 => ( Z3 = Y4 ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_74_theI2,axiom,
% 2.08/1.14 ! [A: $tType,P2: A > $o,A3: A,Q: A > $o] :
% 2.08/1.14 ( ( P2 @ A3 )
% 2.08/1.14 => ( ! [X2: A] :
% 2.08/1.14 ( ( P2 @ X2 )
% 2.08/1.14 => ( X2 = A3 ) )
% 2.08/1.14 => ( ! [X2: A] :
% 2.08/1.14 ( ( P2 @ X2 )
% 2.08/1.14 => ( Q @ X2 ) )
% 2.08/1.14 => ( Q @ ( the @ A @ P2 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_75_theI,axiom,
% 2.08/1.14 ! [A: $tType,P2: A > $o,A3: A] :
% 2.08/1.14 ( ( P2 @ A3 )
% 2.08/1.14 => ( ! [X2: A] :
% 2.08/1.14 ( ( P2 @ X2 )
% 2.08/1.14 => ( X2 = A3 ) )
% 2.08/1.14 => ( P2 @ ( the @ A @ P2 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_76_theI_H,axiom,
% 2.08/1.14 ! [A: $tType,P2: A > $o] :
% 2.08/1.14 ( ? [X5: A] :
% 2.08/1.14 ( ( P2 @ X5 )
% 2.08/1.14 & ! [Y: A] :
% 2.08/1.14 ( ( P2 @ Y )
% 2.08/1.14 => ( Y = X5 ) ) )
% 2.08/1.14 => ( P2 @ ( the @ A @ P2 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_77_cr__tllist__def,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( tLList47617868tllist @ A @ B )
% 2.08/1.14 = ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( ( tLList446370796tllist @ A @ B ) > $o )
% 2.08/1.14 @ ^ [Xs2: coinductive_llist @ A,B4: B] :
% 2.08/1.14 ( ^ [Y7: tLList446370796tllist @ A @ B,Z2: tLList446370796tllist @ A @ B] : ( Y7 = Z2 )
% 2.08/1.14 @ ( tLList1672613558_llist @ B @ A @ B4 @ Xs2 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_78_inv__image__def,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType] :
% 2.08/1.14 ( ( inv_image @ B @ A )
% 2.08/1.14 = ( ^ [R2: set @ ( product_prod @ B @ B ),F4: A > B] :
% 2.08/1.14 ( collect @ ( product_prod @ A @ A )
% 2.08/1.14 @ ( product_case_prod @ A @ A @ $o
% 2.08/1.14 @ ^ [X3: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X3 ) @ ( F4 @ Y4 ) ) @ R2 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_79_old_Orec__unit__def,axiom,
% 2.08/1.14 ! [T: $tType] :
% 2.08/1.14 ( ( product_rec_unit @ T )
% 2.08/1.14 = ( ^ [F12: T,X3: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_80_DEADID_Orel__reflp,axiom,
% 2.08/1.14 ! [A: $tType] :
% 2.08/1.14 ( reflp @ A
% 2.08/1.14 @ ^ [Y7: A,Z2: A] : ( Y7 = Z2 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_81_Nitpick_OThe__psimp,axiom,
% 2.08/1.14 ! [A: $tType,P2: A > $o,X: A] :
% 2.08/1.14 ( ( P2
% 2.08/1.14 = ( ^ [Y7: A,Z2: A] : ( Y7 = Z2 )
% 2.08/1.14 @ X ) )
% 2.08/1.14 => ( ( the @ A @ P2 )
% 2.08/1.14 = X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_82_old_Orec__bool__def,axiom,
% 2.08/1.14 ! [T: $tType] :
% 2.08/1.14 ( ( product_rec_bool @ T )
% 2.08/1.14 = ( ^ [F12: T,F22: T,X3: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F22 @ X3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_83_old_Obool_Osimps_I5_J,axiom,
% 2.08/1.14 ! [T: $tType,F1: T,F23: T] :
% 2.08/1.14 ( ( product_rec_bool @ T @ F1 @ F23 @ $true )
% 2.08/1.14 = F1 ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_84_old_Obool_Osimps_I6_J,axiom,
% 2.08/1.14 ! [T: $tType,F1: T,F23: T] :
% 2.08/1.14 ( ( product_rec_bool @ T @ F1 @ F23 @ $false )
% 2.08/1.14 = F23 ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_85_in__inv__image,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,X: A,Y3: A,R4: set @ ( product_prod @ B @ B ),F: A > B] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( inv_image @ B @ A @ R4 @ F ) )
% 2.08/1.14 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y3 ) ) @ R4 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_86_rp__inv__image__def,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( fun_rp_inv_image @ A @ B )
% 2.08/1.14 = ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
% 2.08/1.14 @ ^ [R3: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A ),F4: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R3 @ F4 ) @ ( inv_image @ A @ B @ S4 @ F4 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_87_Quotient__tllist,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( quotient @ ( product_prod @ ( coinductive_llist @ A ) @ B ) @ ( tLList446370796tllist @ A @ B )
% 2.08/1.14 @ ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( ( product_prod @ ( coinductive_llist @ A ) @ B ) > $o )
% 2.08/1.14 @ ^ [Xs2: coinductive_llist @ A,A4: B] :
% 2.08/1.14 ( product_case_prod @ ( coinductive_llist @ A ) @ B @ $o
% 2.08/1.14 @ ^ [Ys: coinductive_llist @ A,B4: B] :
% 2.08/1.14 ( ( Xs2 = Ys )
% 2.08/1.14 & ( ( coinductive_lfinite @ A @ Ys )
% 2.08/1.14 => ( A4 = B4 ) ) ) ) )
% 2.08/1.14 @ ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( tLList446370796tllist @ A @ B )
% 2.08/1.14 @ ^ [Xs2: coinductive_llist @ A,A4: B] : ( tLList1672613558_llist @ B @ A @ A4 @ Xs2 ) )
% 2.08/1.14 @ ^ [Ys: tLList446370796tllist @ A @ B] : ( product_Pair @ ( coinductive_llist @ A ) @ B @ ( tLList798109904tllist @ A @ B @ Ys ) @ ( tLList2110128105rminal @ A @ B @ Ys ) )
% 2.08/1.14 @ ( tLList47617868tllist @ A @ B ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_88_terminal__tllist__of__llist__lfinite,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,B3: B] :
% 2.08/1.14 ( ( coinductive_lfinite @ A @ Xs )
% 2.08/1.14 => ( ( tLList2110128105rminal @ A @ B @ ( tLList1672613558_llist @ B @ A @ B3 @ Xs ) )
% 2.08/1.14 = B3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_89_in__inv__imagep,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( inv_imagep @ A @ B )
% 2.08/1.14 = ( ^ [R2: A > A > $o,F4: B > A,X3: B,Y4: B] : ( R2 @ ( F4 @ X3 ) @ ( F4 @ Y4 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_90_case__swap,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,C: $tType,F: C > B > A,P: product_prod @ C @ B] :
% 2.08/1.14 ( ( product_case_prod @ B @ C @ A
% 2.08/1.14 @ ^ [Y4: B,X3: C] : ( F @ X3 @ Y4 )
% 2.08/1.14 @ ( product_swap @ C @ B @ P ) )
% 2.08/1.14 = ( product_case_prod @ C @ B @ A @ F @ P ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_91_pair__imageI,axiom,
% 2.08/1.14 ! [C: $tType,B: $tType,A: $tType,A3: A,B3: B,A7: set @ ( product_prod @ A @ B ),F: A > B > C] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ A7 )
% 2.08/1.14 => ( member @ C @ ( F @ A3 @ B3 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A7 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_92_swap__swap,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 2.08/1.14 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 2.08/1.14 = P ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_93_pair__in__swap__image,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,Y3: A,X: B,A7: set @ ( product_prod @ B @ A )] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A7 ) )
% 2.08/1.14 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) @ A7 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_94_swap__simp,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,X: B,Y3: A] :
% 2.08/1.14 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
% 2.08/1.14 = ( product_Pair @ A @ B @ Y3 @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_95_tllist__of__llist__inverse,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,B3: B,Xs: coinductive_llist @ A] :
% 2.08/1.14 ( ( tLList798109904tllist @ A @ B @ ( tLList1672613558_llist @ B @ A @ B3 @ Xs ) )
% 2.08/1.14 = Xs ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_96_llist__of__tllist__inverse,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,B3: tLList446370796tllist @ A @ B] :
% 2.08/1.14 ( ( tLList1672613558_llist @ B @ A @ ( tLList2110128105rminal @ A @ B @ B3 ) @ ( tLList798109904tllist @ A @ B @ B3 ) )
% 2.08/1.14 = B3 ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_97_inv__imagep__def,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType] :
% 2.08/1.14 ( ( inv_imagep @ B @ A )
% 2.08/1.14 = ( ^ [R2: B > B > $o,F4: A > B,X3: A,Y4: A] : ( R2 @ ( F4 @ X3 ) @ ( F4 @ Y4 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_98_image__ident,axiom,
% 2.08/1.14 ! [A: $tType,Y8: set @ A] :
% 2.08/1.14 ( ( image @ A @ A
% 2.08/1.14 @ ^ [X3: A] : X3
% 2.08/1.14 @ Y8 )
% 2.08/1.14 = Y8 ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_99_image__eqI,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,B3: A,F: B > A,X: B,A7: set @ B] :
% 2.08/1.14 ( ( B3
% 2.08/1.14 = ( F @ X ) )
% 2.08/1.14 => ( ( member @ B @ X @ A7 )
% 2.08/1.14 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A7 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_100_Quotient__total__abs__eq__iff,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,X: A,Y3: A] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( reflp @ A @ R )
% 2.08/1.14 => ( ( ( Abs @ X )
% 2.08/1.14 = ( Abs @ Y3 ) )
% 2.08/1.14 = ( R @ X @ Y3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_101_Quotient__total__abs__induct,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,P2: B > $o,X: B] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( reflp @ A @ R )
% 2.08/1.14 => ( ! [Y: A] : ( P2 @ ( Abs @ Y ) )
% 2.08/1.14 => ( P2 @ X ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_102_rp__inv__image__rp,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,P2: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
% 2.08/1.14 ( ( fun_reduction_pair @ A @ P2 )
% 2.08/1.14 => ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P2 @ F ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_103_rev__image__eqI,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,X: A,A7: set @ A,B3: B,F: A > B] :
% 2.08/1.14 ( ( member @ A @ X @ A7 )
% 2.08/1.14 => ( ( B3
% 2.08/1.14 = ( F @ X ) )
% 2.08/1.14 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A7 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_104_ball__imageD,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,F: B > A,A7: set @ B,P2: A > $o] :
% 2.08/1.14 ( ! [X2: A] :
% 2.08/1.14 ( ( member @ A @ X2 @ ( image @ B @ A @ F @ A7 ) )
% 2.08/1.14 => ( P2 @ X2 ) )
% 2.08/1.14 => ! [X5: B] :
% 2.08/1.14 ( ( member @ B @ X5 @ A7 )
% 2.08/1.14 => ( P2 @ ( F @ X5 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_105_image__cong,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G3: A > B] :
% 2.08/1.14 ( ( M = N )
% 2.08/1.14 => ( ! [X2: A] :
% 2.08/1.14 ( ( member @ A @ X2 @ N )
% 2.08/1.14 => ( ( F @ X2 )
% 2.08/1.14 = ( G3 @ X2 ) ) )
% 2.08/1.14 => ( ( image @ A @ B @ F @ M )
% 2.08/1.14 = ( image @ A @ B @ G3 @ N ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_106_bex__imageD,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,F: B > A,A7: set @ B,P2: A > $o] :
% 2.08/1.14 ( ? [X5: A] :
% 2.08/1.14 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A7 ) )
% 2.08/1.14 & ( P2 @ X5 ) )
% 2.08/1.14 => ? [X2: B] :
% 2.08/1.14 ( ( member @ B @ X2 @ A7 )
% 2.08/1.14 & ( P2 @ ( F @ X2 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_107_image__iff,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,Z: A,F: B > A,A7: set @ B] :
% 2.08/1.14 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A7 ) )
% 2.08/1.14 = ( ? [X3: B] :
% 2.08/1.14 ( ( member @ B @ X3 @ A7 )
% 2.08/1.14 & ( Z
% 2.08/1.14 = ( F @ X3 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_108_imageI,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,X: A,A7: set @ A,F: A > B] :
% 2.08/1.14 ( ( member @ A @ X @ A7 )
% 2.08/1.14 => ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A7 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_109_Quotient__rep__abs__fold__unmap,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,X6: B,X: A,Rep2: B > A] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( X6
% 2.08/1.14 = ( Abs @ X ) )
% 2.08/1.14 => ( ( R @ X @ X )
% 2.08/1.14 => ( ( ( Rep @ X6 )
% 2.08/1.14 = ( Rep2 @ X6 ) )
% 2.08/1.14 => ( R @ ( Rep2 @ X6 ) @ X ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_110_Quotient__to__transfer,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,C2: A,C4: B] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( R @ C2 @ C2 )
% 2.08/1.14 => ( ( C4
% 2.08/1.14 = ( Abs @ C2 ) )
% 2.08/1.14 => ( T2 @ C2 @ C4 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_111_Quotient__abs__induct,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,P2: B > $o,X: B] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ! [Y: A] :
% 2.08/1.14 ( ( R @ Y @ Y )
% 2.08/1.14 => ( P2 @ ( Abs @ Y ) ) )
% 2.08/1.14 => ( P2 @ X ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_112_Quotient__rep__reflp,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,A3: B] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_113_Quotient__rel__abs2,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,X: B,Y3: A] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( R @ ( Rep @ X ) @ Y3 )
% 2.08/1.14 => ( X
% 2.08/1.14 = ( Abs @ Y3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_114_Quotient__alt__def3,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( quotient @ A @ B )
% 2.08/1.14 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep3: B > A,T3: A > B > $o] :
% 2.08/1.14 ( ! [A4: A,B4: B] :
% 2.08/1.14 ( ( T3 @ A4 @ B4 )
% 2.08/1.14 => ( ( Abs2 @ A4 )
% 2.08/1.14 = B4 ) )
% 2.08/1.14 & ! [B4: B] : ( T3 @ ( Rep3 @ B4 ) @ B4 )
% 2.08/1.14 & ! [X3: A,Y4: A] :
% 2.08/1.14 ( ( R3 @ X3 @ Y4 )
% 2.08/1.14 = ( ? [Z3: B] :
% 2.08/1.14 ( ( T3 @ X3 @ Z3 )
% 2.08/1.14 & ( T3 @ Y4 @ Z3 ) ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_115_Quotient__alt__def2,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( quotient @ A @ B )
% 2.08/1.14 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep3: B > A,T3: A > B > $o] :
% 2.08/1.14 ( ! [A4: A,B4: B] :
% 2.08/1.14 ( ( T3 @ A4 @ B4 )
% 2.08/1.14 => ( ( Abs2 @ A4 )
% 2.08/1.14 = B4 ) )
% 2.08/1.14 & ! [B4: B] : ( T3 @ ( Rep3 @ B4 ) @ B4 )
% 2.08/1.14 & ! [X3: A,Y4: A] :
% 2.08/1.14 ( ( R3 @ X3 @ Y4 )
% 2.08/1.14 = ( ( T3 @ X3 @ ( Abs2 @ Y4 ) )
% 2.08/1.14 & ( T3 @ Y4 @ ( Abs2 @ X3 ) ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_116_Quotient__rep__abs,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( R @ R4 @ R4 )
% 2.08/1.14 => ( R @ ( Rep @ ( Abs @ R4 ) ) @ R4 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_117_Quotient__rel__rep,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,A3: B,B3: B] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( R @ ( Rep @ A3 ) @ ( Rep @ B3 ) )
% 2.08/1.14 = ( A3 = B3 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_118_Quotient__rel__abs,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A,S2: A] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( R @ R4 @ S2 )
% 2.08/1.14 => ( ( Abs @ R4 )
% 2.08/1.14 = ( Abs @ S2 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_119_Quotient__alt__def,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( quotient @ A @ B )
% 2.08/1.14 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep3: B > A,T3: A > B > $o] :
% 2.08/1.14 ( ! [A4: A,B4: B] :
% 2.08/1.14 ( ( T3 @ A4 @ B4 )
% 2.08/1.14 => ( ( Abs2 @ A4 )
% 2.08/1.14 = B4 ) )
% 2.08/1.14 & ! [B4: B] : ( T3 @ ( Rep3 @ B4 ) @ B4 )
% 2.08/1.14 & ! [X3: A,Y4: A] :
% 2.08/1.14 ( ( R3 @ X3 @ Y4 )
% 2.08/1.14 = ( ( T3 @ X3 @ ( Abs2 @ X3 ) )
% 2.08/1.14 & ( T3 @ Y4 @ ( Abs2 @ Y4 ) )
% 2.08/1.14 & ( ( Abs2 @ X3 )
% 2.08/1.14 = ( Abs2 @ Y4 ) ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_120_Quotient__abs__rep,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,A3: B] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( Abs @ ( Rep @ A3 ) )
% 2.08/1.14 = A3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_121_Quotient__Rep__eq,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,X6: B,X: A] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( X6
% 2.08/1.14 = ( Abs @ X ) )
% 2.08/1.14 => ( ( Rep @ X6 )
% 2.08/1.14 = ( Rep @ X6 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_122_Quotient__refl2,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A,S2: A] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( R @ R4 @ S2 )
% 2.08/1.14 => ( R @ S2 @ S2 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_123_Quotient__refl1,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A,S2: A] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( R @ R4 @ S2 )
% 2.08/1.14 => ( R @ R4 @ R4 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_124_Quotient__rel,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A,S2: A] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( ( R @ R4 @ R4 )
% 2.08/1.14 & ( R @ S2 @ S2 )
% 2.08/1.14 & ( ( Abs @ R4 )
% 2.08/1.14 = ( Abs @ S2 ) ) )
% 2.08/1.14 = ( R @ R4 @ S2 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_125_Compr__image__eq,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,F: B > A,A7: set @ B,P2: A > $o] :
% 2.08/1.14 ( ( collect @ A
% 2.08/1.14 @ ^ [X3: A] :
% 2.08/1.14 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A7 ) )
% 2.08/1.14 & ( P2 @ X3 ) ) )
% 2.08/1.14 = ( image @ B @ A @ F
% 2.08/1.14 @ ( collect @ B
% 2.08/1.14 @ ^ [X3: B] :
% 2.08/1.14 ( ( member @ B @ X3 @ A7 )
% 2.08/1.14 & ( P2 @ ( F @ X3 ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_126_image__image,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,C: $tType,F: B > A,G3: C > B,A7: set @ C] :
% 2.08/1.14 ( ( image @ B @ A @ F @ ( image @ C @ B @ G3 @ A7 ) )
% 2.08/1.14 = ( image @ C @ A
% 2.08/1.14 @ ^ [X3: C] : ( F @ ( G3 @ X3 ) )
% 2.08/1.14 @ A7 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_127_imageE,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,B3: A,F: B > A,A7: set @ B] :
% 2.08/1.14 ( ( member @ A @ B3 @ ( image @ B @ A @ F @ A7 ) )
% 2.08/1.14 => ~ ! [X2: B] :
% 2.08/1.14 ( ( B3
% 2.08/1.14 = ( F @ X2 ) )
% 2.08/1.14 => ~ ( member @ B @ X2 @ A7 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_128_Quotient__cr__rel,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( T2
% 2.08/1.14 = ( ^ [X3: A,Y4: B] :
% 2.08/1.14 ( ( R @ X3 @ X3 )
% 2.08/1.14 & ( ( Abs @ X3 )
% 2.08/1.14 = Y4 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_129_Quotient__def,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( quotient @ A @ B )
% 2.08/1.14 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep3: B > A,T3: A > B > $o] :
% 2.08/1.14 ( ! [A4: B] :
% 2.08/1.14 ( ( Abs2 @ ( Rep3 @ A4 ) )
% 2.08/1.14 = A4 )
% 2.08/1.14 & ! [A4: B] : ( R3 @ ( Rep3 @ A4 ) @ ( Rep3 @ A4 ) )
% 2.08/1.14 & ! [R2: A,S5: A] :
% 2.08/1.14 ( ( R3 @ R2 @ S5 )
% 2.08/1.14 = ( ( R3 @ R2 @ R2 )
% 2.08/1.14 & ( R3 @ S5 @ S5 )
% 2.08/1.14 & ( ( Abs2 @ R2 )
% 2.08/1.14 = ( Abs2 @ S5 ) ) ) )
% 2.08/1.14 & ( T3
% 2.08/1.14 = ( ^ [X3: A,Y4: B] :
% 2.08/1.14 ( ( R3 @ X3 @ X3 )
% 2.08/1.14 & ( ( Abs2 @ X3 )
% 2.08/1.14 = Y4 ) ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_130_QuotientI,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,Abs: B > A,Rep: A > B,R: B > B > $o,T2: B > A > $o] :
% 2.08/1.14 ( ! [A2: A] :
% 2.08/1.14 ( ( Abs @ ( Rep @ A2 ) )
% 2.08/1.14 = A2 )
% 2.08/1.14 => ( ! [A2: A] : ( R @ ( Rep @ A2 ) @ ( Rep @ A2 ) )
% 2.08/1.14 => ( ! [R5: B,S6: B] :
% 2.08/1.14 ( ( R @ R5 @ S6 )
% 2.08/1.14 = ( ( R @ R5 @ R5 )
% 2.08/1.14 & ( R @ S6 @ S6 )
% 2.08/1.14 & ( ( Abs @ R5 )
% 2.08/1.14 = ( Abs @ S6 ) ) ) )
% 2.08/1.14 => ( ( T2
% 2.08/1.14 = ( ^ [X3: B,Y4: A] :
% 2.08/1.14 ( ( R @ X3 @ X3 )
% 2.08/1.14 & ( ( Abs @ X3 )
% 2.08/1.14 = Y4 ) ) ) )
% 2.08/1.14 => ( quotient @ B @ A @ R @ Abs @ Rep @ T2 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_131_Sup_OSUP__identity__eq,axiom,
% 2.08/1.14 ! [A: $tType,Sup: ( set @ A ) > A,A7: set @ A] :
% 2.08/1.14 ( ( Sup
% 2.08/1.14 @ ( image @ A @ A
% 2.08/1.14 @ ^ [X3: A] : X3
% 2.08/1.14 @ A7 ) )
% 2.08/1.14 = ( Sup @ A7 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_132_Inf_OINF__identity__eq,axiom,
% 2.08/1.14 ! [A: $tType,Inf: ( set @ A ) > A,A7: set @ A] :
% 2.08/1.14 ( ( Inf
% 2.08/1.14 @ ( image @ A @ A
% 2.08/1.14 @ ^ [X3: A] : X3
% 2.08/1.14 @ A7 ) )
% 2.08/1.14 = ( Inf @ A7 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_133_terminal0__terminal,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( tLList1825092077minal0 @ ( ( tLList446370796tllist @ A @ B ) > B ) )
% 2.08/1.14 = ( tLList2110128105rminal @ A @ B ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_134_surj__swap,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.08/1.14 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_135_swap__product,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A7: set @ B,B7: set @ A] :
% 2.08/1.14 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
% 2.08/1.14 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 2.08/1.14 @ ^ [I: B,J: A] : ( product_Pair @ A @ B @ J @ I ) )
% 2.08/1.14 @ ( product_Sigma @ B @ A @ A7
% 2.08/1.14 @ ^ [Uu: B] : B7 ) )
% 2.08/1.14 = ( product_Sigma @ A @ B @ B7
% 2.08/1.14 @ ^ [Uu: A] : A7 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_136_mem__Sigma__iff,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A3: A,B3: B,A7: set @ A,B7: A > ( set @ B )] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
% 2.08/1.14 = ( ( member @ A @ A3 @ A7 )
% 2.08/1.14 & ( member @ B @ B3 @ ( B7 @ A3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_137_SigmaI,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A3: A,A7: set @ A,B3: B,B7: A > ( set @ B )] :
% 2.08/1.14 ( ( member @ A @ A3 @ A7 )
% 2.08/1.14 => ( ( member @ B @ B3 @ ( B7 @ A3 ) )
% 2.08/1.14 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_138_Collect__case__prod,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P2: A > $o,Q: B > $o] :
% 2.08/1.14 ( ( collect @ ( product_prod @ A @ B )
% 2.08/1.14 @ ( product_case_prod @ A @ B @ $o
% 2.08/1.14 @ ^ [A4: A,B4: B] :
% 2.08/1.14 ( ( P2 @ A4 )
% 2.08/1.14 & ( Q @ B4 ) ) ) )
% 2.08/1.14 = ( product_Sigma @ A @ B @ ( collect @ A @ P2 )
% 2.08/1.14 @ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_139_UNIV__Times__UNIV,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 2.08/1.14 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
% 2.08/1.14 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_140_UNIV__def,axiom,
% 2.08/1.14 ! [A: $tType] :
% 2.08/1.14 ( ( top_top @ ( set @ A ) )
% 2.08/1.14 = ( collect @ A
% 2.08/1.14 @ ^ [X3: A] : $true ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_141_SigmaE,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,C2: product_prod @ A @ B,A7: set @ A,B7: A > ( set @ B )] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
% 2.08/1.14 => ~ ! [X2: A] :
% 2.08/1.14 ( ( member @ A @ X2 @ A7 )
% 2.08/1.14 => ! [Y: B] :
% 2.08/1.14 ( ( member @ B @ Y @ ( B7 @ X2 ) )
% 2.08/1.14 => ( C2
% 2.08/1.14 != ( product_Pair @ A @ B @ X2 @ Y ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_142_SigmaD1,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A3: A,B3: B,A7: set @ A,B7: A > ( set @ B )] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
% 2.08/1.14 => ( member @ A @ A3 @ A7 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_143_SigmaD2,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A3: A,B3: B,A7: set @ A,B7: A > ( set @ B )] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
% 2.08/1.14 => ( member @ B @ B3 @ ( B7 @ A3 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_144_SigmaE2,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A3: A,B3: B,A7: set @ A,B7: A > ( set @ B )] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
% 2.08/1.14 => ~ ( ( member @ A @ A3 @ A7 )
% 2.08/1.14 => ~ ( member @ B @ B3 @ ( B7 @ A3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_145_Sigma__cong,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A7: set @ A,B7: set @ A,C5: A > ( set @ B ),D3: A > ( set @ B )] :
% 2.08/1.14 ( ( A7 = B7 )
% 2.08/1.14 => ( ! [X2: A] :
% 2.08/1.14 ( ( member @ A @ X2 @ B7 )
% 2.08/1.14 => ( ( C5 @ X2 )
% 2.08/1.14 = ( D3 @ X2 ) ) )
% 2.08/1.14 => ( ( product_Sigma @ A @ B @ A7 @ C5 )
% 2.08/1.14 = ( product_Sigma @ A @ B @ B7 @ D3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_146_Times__eq__cancel2,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,X: A,C5: set @ A,A7: set @ B,B7: set @ B] :
% 2.08/1.14 ( ( member @ A @ X @ C5 )
% 2.08/1.14 => ( ( ( product_Sigma @ B @ A @ A7
% 2.08/1.14 @ ^ [Uu: B] : C5 )
% 2.08/1.14 = ( product_Sigma @ B @ A @ B7
% 2.08/1.14 @ ^ [Uu: B] : C5 ) )
% 2.08/1.14 = ( A7 = B7 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_147_range__eqI,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,B3: A,F: B > A,X: B] :
% 2.08/1.14 ( ( B3
% 2.08/1.14 = ( F @ X ) )
% 2.08/1.14 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_148_rangeI,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_149_Collect__case__prod__Sigma,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P2: A > $o,Q: A > B > $o] :
% 2.08/1.14 ( ( collect @ ( product_prod @ A @ B )
% 2.08/1.14 @ ( product_case_prod @ A @ B @ $o
% 2.08/1.14 @ ^ [X3: A,Y4: B] :
% 2.08/1.14 ( ( P2 @ X3 )
% 2.08/1.14 & ( Q @ X3 @ Y4 ) ) ) )
% 2.08/1.14 = ( product_Sigma @ A @ B @ ( collect @ A @ P2 )
% 2.08/1.14 @ ^ [X3: A] : ( collect @ B @ ( Q @ X3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_150_range__composition,axiom,
% 2.08/1.14 ! [A: $tType,C: $tType,B: $tType,F: C > A,G3: B > C] :
% 2.08/1.14 ( ( image @ B @ A
% 2.08/1.14 @ ^ [X3: B] : ( F @ ( G3 @ X3 ) )
% 2.08/1.14 @ ( top_top @ ( set @ B ) ) )
% 2.08/1.14 = ( image @ C @ A @ F @ ( image @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_151_rangeE,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,B3: A,F: B > A] :
% 2.08/1.14 ( ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
% 2.08/1.14 => ~ ! [X2: B] :
% 2.08/1.14 ( B3
% 2.08/1.14 != ( F @ X2 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_152_product__swap,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A7: set @ B,B7: set @ A] :
% 2.08/1.14 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
% 2.08/1.14 @ ( product_Sigma @ B @ A @ A7
% 2.08/1.14 @ ^ [Uu: B] : B7 ) )
% 2.08/1.14 = ( product_Sigma @ A @ B @ B7
% 2.08/1.14 @ ^ [Uu: A] : A7 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_153_Inf_OINF__cong,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,A7: set @ B,B7: set @ B,C5: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 2.08/1.14 ( ( A7 = B7 )
% 2.08/1.14 => ( ! [X2: B] :
% 2.08/1.14 ( ( member @ B @ X2 @ B7 )
% 2.08/1.14 => ( ( C5 @ X2 )
% 2.08/1.14 = ( D3 @ X2 ) ) )
% 2.08/1.14 => ( ( Inf @ ( image @ B @ A @ C5 @ A7 ) )
% 2.08/1.14 = ( Inf @ ( image @ B @ A @ D3 @ B7 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_154_Sup_OSUP__cong,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,A7: set @ B,B7: set @ B,C5: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 2.08/1.14 ( ( A7 = B7 )
% 2.08/1.14 => ( ! [X2: B] :
% 2.08/1.14 ( ( member @ B @ X2 @ B7 )
% 2.08/1.14 => ( ( C5 @ X2 )
% 2.08/1.14 = ( D3 @ X2 ) ) )
% 2.08/1.14 => ( ( Sup @ ( image @ B @ A @ C5 @ A7 ) )
% 2.08/1.14 = ( Sup @ ( image @ B @ A @ D3 @ B7 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_155_surjD,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,F: B > A,Y3: A] :
% 2.08/1.14 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.08/1.14 = ( top_top @ ( set @ A ) ) )
% 2.08/1.14 => ? [X2: B] :
% 2.08/1.14 ( Y3
% 2.08/1.14 = ( F @ X2 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_156_surjE,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,F: B > A,Y3: A] :
% 2.08/1.14 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.08/1.14 = ( top_top @ ( set @ A ) ) )
% 2.08/1.14 => ~ ! [X2: B] :
% 2.08/1.14 ( Y3
% 2.08/1.14 != ( F @ X2 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_157_surjI,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,G3: B > A,F: A > B] :
% 2.08/1.14 ( ! [X2: A] :
% 2.08/1.14 ( ( G3 @ ( F @ X2 ) )
% 2.08/1.14 = X2 )
% 2.08/1.14 => ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
% 2.08/1.14 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_158_surj__def,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,F: B > A] :
% 2.08/1.14 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.08/1.14 = ( top_top @ ( set @ A ) ) )
% 2.08/1.14 = ( ! [Y4: A] :
% 2.08/1.14 ? [X3: B] :
% 2.08/1.14 ( Y4
% 2.08/1.14 = ( F @ X3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_159_member__product,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A7: set @ A,B7: set @ B] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A7 @ B7 ) )
% 2.08/1.14 = ( member @ ( product_prod @ A @ B ) @ X
% 2.08/1.14 @ ( product_Sigma @ A @ B @ A7
% 2.08/1.14 @ ^ [Uu: A] : B7 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_160_Product__Type_Oproduct__def,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( product_product @ A @ B )
% 2.08/1.14 = ( ^ [A8: set @ A,B8: set @ B] :
% 2.08/1.14 ( product_Sigma @ A @ B @ A8
% 2.08/1.14 @ ^ [Uu: A] : B8 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_161_map__prod__surj,axiom,
% 2.08/1.14 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G3: C > D] :
% 2.08/1.14 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.08/1.14 = ( top_top @ ( set @ B ) ) )
% 2.08/1.14 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 2.08/1.14 = ( top_top @ ( set @ D ) ) )
% 2.08/1.14 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 2.08/1.14 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_162_map__prod__ident,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( product_map_prod @ A @ A @ B @ B
% 2.08/1.14 @ ^ [X3: A] : X3
% 2.08/1.14 @ ^ [Y4: B] : Y4 )
% 2.08/1.14 = ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_163_map__prod__simp,axiom,
% 2.08/1.14 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G3: D > B,A3: C,B3: D] :
% 2.08/1.14 ( ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
% 2.08/1.14 = ( product_Pair @ A @ B @ ( F @ A3 ) @ ( G3 @ B3 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_164_map__prod__imageI,axiom,
% 2.08/1.14 ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R: set @ ( product_prod @ A @ B ),F: A > C,G3: B > D] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R )
% 2.08/1.14 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( G3 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G3 ) @ R ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_165_case__prod__map__prod,axiom,
% 2.08/1.14 ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G3: E > C,X: product_prod @ D @ E] :
% 2.08/1.14 ( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G3 @ X ) )
% 2.08/1.14 = ( product_case_prod @ D @ E @ A
% 2.08/1.14 @ ^ [L: D,R2: E] : ( H @ ( F @ L ) @ ( G3 @ R2 ) )
% 2.08/1.14 @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_166_prod__fun__imageE,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F: C > A,G3: D > B,R: set @ ( product_prod @ C @ D )] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G3 ) @ R ) )
% 2.08/1.14 => ~ ! [X2: C,Y: D] :
% 2.08/1.14 ( ( C2
% 2.08/1.14 = ( product_Pair @ A @ B @ ( F @ X2 ) @ ( G3 @ Y ) ) )
% 2.08/1.14 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X2 @ Y ) @ R ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_167_map__prod__def,axiom,
% 2.08/1.14 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 2.08/1.14 ( ( product_map_prod @ A @ C @ B @ D )
% 2.08/1.14 = ( ^ [F4: A > C,G4: B > D] :
% 2.08/1.14 ( product_case_prod @ A @ B @ ( product_prod @ C @ D )
% 2.08/1.14 @ ^ [X3: A,Y4: B] : ( product_Pair @ C @ D @ ( F4 @ X3 ) @ ( G4 @ Y4 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_168_map__prod__surj__on,axiom,
% 2.08/1.14 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A7: set @ B,A9: set @ A,G3: D > C,B7: set @ D,B9: set @ C] :
% 2.08/1.14 ( ( ( image @ B @ A @ F @ A7 )
% 2.08/1.14 = A9 )
% 2.08/1.14 => ( ( ( image @ D @ C @ G3 @ B7 )
% 2.08/1.14 = B9 )
% 2.08/1.14 => ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G3 )
% 2.08/1.14 @ ( product_Sigma @ B @ D @ A7
% 2.08/1.14 @ ^ [Uu: B] : B7 ) )
% 2.08/1.14 = ( product_Sigma @ A @ C @ A9
% 2.08/1.14 @ ^ [Uu: A] : B9 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_169_prod_Omap__ident,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,T4: product_prod @ A @ B] :
% 2.08/1.14 ( ( product_map_prod @ A @ A @ B @ B
% 2.08/1.14 @ ^ [X3: A] : X3
% 2.08/1.14 @ ^ [X3: B] : X3
% 2.08/1.14 @ T4 )
% 2.08/1.14 = T4 ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_170_terminal__def,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( tLList2110128105rminal @ A @ B )
% 2.08/1.14 = ( tLList200813139tllist @ B @ B @ A
% 2.08/1.14 @ ^ [X12: B] : X12
% 2.08/1.14 @ ^ [X_dummy: A] : ( tLList1825092077minal0 @ ( ( tLList446370796tllist @ A @ B ) > B ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_171_reflp__refl__eq,axiom,
% 2.08/1.14 ! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
% 2.08/1.14 ( ( reflp @ A
% 2.08/1.14 @ ^ [X3: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R4 ) )
% 2.08/1.14 = ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R4 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_172_refl__onD2,axiom,
% 2.08/1.14 ! [A: $tType,A7: set @ A,R4: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
% 2.08/1.14 ( ( refl_on @ A @ A7 @ R4 )
% 2.08/1.14 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R4 )
% 2.08/1.14 => ( member @ A @ Y3 @ A7 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_173_refl__onD1,axiom,
% 2.08/1.14 ! [A: $tType,A7: set @ A,R4: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
% 2.08/1.14 ( ( refl_on @ A @ A7 @ R4 )
% 2.08/1.14 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R4 )
% 2.08/1.14 => ( member @ A @ X @ A7 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_174_refl__onD,axiom,
% 2.08/1.14 ! [A: $tType,A7: set @ A,R4: set @ ( product_prod @ A @ A ),A3: A] :
% 2.08/1.14 ( ( refl_on @ A @ A7 @ R4 )
% 2.08/1.14 => ( ( member @ A @ A3 @ A7 )
% 2.08/1.14 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R4 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_175_tllist_Ocase__distrib,axiom,
% 2.08/1.14 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F1: B > C,F23: A > ( tLList446370796tllist @ A @ B ) > C,Tllist: tLList446370796tllist @ A @ B] :
% 2.08/1.14 ( ( H @ ( tLList200813139tllist @ B @ C @ A @ F1 @ F23 @ Tllist ) )
% 2.08/1.14 = ( tLList200813139tllist @ B @ D @ A
% 2.08/1.14 @ ^ [X3: B] : ( H @ ( F1 @ X3 ) )
% 2.08/1.14 @ ^ [X12: A,X23: tLList446370796tllist @ A @ B] : ( H @ ( F23 @ X12 @ X23 ) )
% 2.08/1.14 @ Tllist ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_176_refl__on__domain,axiom,
% 2.08/1.14 ! [A: $tType,A7: set @ A,R4: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
% 2.08/1.14 ( ( refl_on @ A @ A7 @ R4 )
% 2.08/1.14 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R4 )
% 2.08/1.14 => ( ( member @ A @ A3 @ A7 )
% 2.08/1.14 & ( member @ A @ B3 @ A7 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_177_range__fst,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.08/1.14 = ( top_top @ ( set @ A ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_178_range__snd,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.08/1.14 = ( top_top @ ( set @ A ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_179_fst__map__prod,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G3: D > B,X: product_prod @ C @ D] :
% 2.08/1.14 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ X ) )
% 2.08/1.14 = ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_180_snd__map__prod,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G3: D > A,X: product_prod @ C @ D] :
% 2.08/1.14 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G3 @ X ) )
% 2.08/1.14 = ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_181_prod_Ocollapse,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.08/1.14 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 2.08/1.14 = Prod ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_182_fst__swap,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 2.08/1.14 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 2.08/1.14 = ( product_snd @ B @ A @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_183_snd__swap,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 2.08/1.14 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 2.08/1.14 = ( product_fst @ A @ B @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_184_prod__eq__iff,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( ^ [Y7: product_prod @ A @ B,Z2: product_prod @ A @ B] : ( Y7 = Z2 ) )
% 2.08/1.14 = ( ^ [S5: product_prod @ A @ B,T5: product_prod @ A @ B] :
% 2.08/1.14 ( ( ( product_fst @ A @ B @ S5 )
% 2.08/1.14 = ( product_fst @ A @ B @ T5 ) )
% 2.08/1.14 & ( ( product_snd @ A @ B @ S5 )
% 2.08/1.14 = ( product_snd @ A @ B @ T5 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_185_prod_Oexpand,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 2.08/1.14 ( ( ( ( product_fst @ A @ B @ Prod )
% 2.08/1.14 = ( product_fst @ A @ B @ Prod2 ) )
% 2.08/1.14 & ( ( product_snd @ A @ B @ Prod )
% 2.08/1.14 = ( product_snd @ A @ B @ Prod2 ) ) )
% 2.08/1.14 => ( Prod = Prod2 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_186_prod__eqI,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 2.08/1.14 ( ( ( product_fst @ A @ B @ P )
% 2.08/1.14 = ( product_fst @ A @ B @ Q2 ) )
% 2.08/1.14 => ( ( ( product_snd @ A @ B @ P )
% 2.08/1.14 = ( product_snd @ A @ B @ Q2 ) )
% 2.08/1.14 => ( P = Q2 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_187_split__fst,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: A > $o,P: product_prod @ A @ B] :
% 2.08/1.14 ( ( R @ ( product_fst @ A @ B @ P ) )
% 2.08/1.14 = ( ! [X3: A,Y4: B] :
% 2.08/1.14 ( ( P
% 2.08/1.14 = ( product_Pair @ A @ B @ X3 @ Y4 ) )
% 2.08/1.14 => ( R @ X3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_188_split__fst__asm,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: A > $o,P: product_prod @ A @ B] :
% 2.08/1.14 ( ( R @ ( product_fst @ A @ B @ P ) )
% 2.08/1.14 = ( ~ ? [X3: A,Y4: B] :
% 2.08/1.14 ( ( P
% 2.08/1.14 = ( product_Pair @ A @ B @ X3 @ Y4 ) )
% 2.08/1.14 & ~ ( R @ X3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_189_fst__eqD,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,X: A,Y3: B,A3: A] :
% 2.08/1.14 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y3 ) )
% 2.08/1.14 = A3 )
% 2.08/1.14 => ( X = A3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_190_fst__conv,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 2.08/1.14 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 2.08/1.14 = X1 ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_191_snd__eqD,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,X: B,Y3: A,A3: A] :
% 2.08/1.14 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
% 2.08/1.14 = A3 )
% 2.08/1.14 => ( Y3 = A3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_192_snd__conv,axiom,
% 2.08/1.14 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 2.08/1.14 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 2.08/1.14 = X22 ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_193_prod_Oswap__def,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( product_swap @ A @ B )
% 2.08/1.14 = ( ^ [P4: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P4 ) @ ( product_fst @ A @ B @ P4 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_194_prod_Oexhaust__sel,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.08/1.14 ( Prod
% 2.08/1.14 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_195_surjective__pairing,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,T4: product_prod @ A @ B] :
% 2.08/1.14 ( T4
% 2.08/1.14 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T4 ) @ ( product_snd @ A @ B @ T4 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_196_case__prod__beta,axiom,
% 2.08/1.14 ! [A: $tType,C: $tType,B: $tType] :
% 2.08/1.14 ( ( product_case_prod @ B @ C @ A )
% 2.08/1.14 = ( ^ [F4: B > C > A,P4: product_prod @ B @ C] : ( F4 @ ( product_fst @ B @ C @ P4 ) @ ( product_snd @ B @ C @ P4 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_197_prod_Ocase__eq__if,axiom,
% 2.08/1.14 ! [C: $tType,B: $tType,A: $tType] :
% 2.08/1.14 ( ( product_case_prod @ A @ B @ C )
% 2.08/1.14 = ( ^ [F4: A > B > C,Prod3: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_198_Product__Type_OCollect__case__prodD,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A7: A > B > $o] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A7 ) ) )
% 2.08/1.14 => ( A7 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_199_split__comp__eq,axiom,
% 2.08/1.14 ! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G3: D > A] :
% 2.08/1.14 ( ( ^ [U: product_prod @ D @ B] : ( F @ ( G3 @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
% 2.08/1.14 = ( product_case_prod @ D @ B @ C
% 2.08/1.14 @ ^ [X3: D] : ( F @ ( G3 @ X3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_200_case__prod__beta_H,axiom,
% 2.08/1.14 ! [C: $tType,B: $tType,A: $tType] :
% 2.08/1.14 ( ( product_case_prod @ A @ B @ C )
% 2.08/1.14 = ( ^ [F4: A > B > C,X3: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_201_case__prod__unfold,axiom,
% 2.08/1.14 ! [C: $tType,B: $tType,A: $tType] :
% 2.08/1.14 ( ( product_case_prod @ A @ B @ C )
% 2.08/1.14 = ( ^ [C6: A > B > C,P4: product_prod @ A @ B] : ( C6 @ ( product_fst @ A @ B @ P4 ) @ ( product_snd @ A @ B @ P4 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_202_prod_Osplit__sel,axiom,
% 2.08/1.14 ! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 2.08/1.14 ( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 2.08/1.14 = ( ( Prod
% 2.08/1.14 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 2.08/1.14 => ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_203_prod_Osplit__sel__asm,axiom,
% 2.08/1.14 ! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 2.08/1.14 ( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 2.08/1.14 = ( ~ ( ( Prod
% 2.08/1.14 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 2.08/1.14 & ~ ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_204_fst__def,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( product_fst @ A @ B )
% 2.08/1.14 = ( product_case_prod @ A @ B @ A
% 2.08/1.14 @ ^ [X12: A,X23: B] : X12 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_205_snd__def,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( product_snd @ A @ B )
% 2.08/1.14 = ( product_case_prod @ A @ B @ B
% 2.08/1.14 @ ^ [X12: A,X23: B] : X23 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_206_The__case__prod,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P2: A > B > $o] :
% 2.08/1.14 ( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) )
% 2.08/1.14 = ( the @ ( product_prod @ A @ B )
% 2.08/1.14 @ ^ [Xy: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_207_mem__Times__iff,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,X: product_prod @ A @ B,A7: set @ A,B7: set @ B] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ X
% 2.08/1.14 @ ( product_Sigma @ A @ B @ A7
% 2.08/1.14 @ ^ [Uu: A] : B7 ) )
% 2.08/1.14 = ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A7 )
% 2.08/1.14 & ( member @ B @ ( product_snd @ A @ B @ X ) @ B7 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_208_exE__realizer,axiom,
% 2.08/1.14 ! [C: $tType,A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A,Q: C > $o,F: B > A > C] :
% 2.08/1.14 ( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
% 2.08/1.14 => ( ! [X2: B,Y: A] :
% 2.08/1.14 ( ( P2 @ Y @ X2 )
% 2.08/1.14 => ( Q @ ( F @ X2 @ Y ) ) )
% 2.08/1.14 => ( Q @ ( product_case_prod @ B @ A @ C @ F @ P ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_209_exI__realizer,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P2: A > B > $o,Y3: A,X: B] :
% 2.08/1.14 ( ( P2 @ Y3 @ X )
% 2.08/1.14 => ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_210_conjI__realizer,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
% 2.08/1.14 ( ( P2 @ P )
% 2.08/1.14 => ( ( Q @ Q2 )
% 2.08/1.14 => ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
% 2.08/1.14 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_211_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,P2: A > B > $o,X: A,Y3: B,A3: product_prod @ A @ B] :
% 2.08/1.14 ( ( P2 @ X @ Y3 )
% 2.08/1.14 => ( ( A3
% 2.08/1.14 = ( product_Pair @ A @ B @ X @ Y3 ) )
% 2.08/1.14 => ( P2 @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_212_above__def,axiom,
% 2.08/1.14 ! [A: $tType] :
% 2.08/1.14 ( ( order_above @ A )
% 2.08/1.14 = ( ^ [R2: set @ ( product_prod @ A @ A ),A4: A] :
% 2.08/1.14 ( collect @ A
% 2.08/1.14 @ ^ [B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_213_aboveS__def,axiom,
% 2.08/1.14 ! [A: $tType] :
% 2.08/1.14 ( ( order_aboveS @ A )
% 2.08/1.14 = ( ^ [R2: set @ ( product_prod @ A @ A ),A4: A] :
% 2.08/1.14 ( collect @ A
% 2.08/1.14 @ ^ [B4: A] :
% 2.08/1.14 ( ( B4 != A4 )
% 2.08/1.14 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_214_sndI,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y3: A,Z: B] :
% 2.08/1.14 ( ( X
% 2.08/1.14 = ( product_Pair @ A @ B @ Y3 @ Z ) )
% 2.08/1.14 => ( ( product_snd @ A @ B @ X )
% 2.08/1.14 = Z ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_215_fstI,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y3: A,Z: B] :
% 2.08/1.14 ( ( X
% 2.08/1.14 = ( product_Pair @ A @ B @ Y3 @ Z ) )
% 2.08/1.14 => ( ( product_fst @ A @ B @ X )
% 2.08/1.14 = Y3 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_216_eq__snd__iff,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,B3: A,P: product_prod @ B @ A] :
% 2.08/1.14 ( ( B3
% 2.08/1.14 = ( product_snd @ B @ A @ P ) )
% 2.08/1.14 = ( ? [A4: B] :
% 2.08/1.14 ( P
% 2.08/1.14 = ( product_Pair @ B @ A @ A4 @ B3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_217_eq__fst__iff,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,A3: A,P: product_prod @ A @ B] :
% 2.08/1.14 ( ( A3
% 2.08/1.14 = ( product_fst @ A @ B @ P ) )
% 2.08/1.14 = ( ? [B4: B] :
% 2.08/1.14 ( P
% 2.08/1.14 = ( product_Pair @ A @ B @ A3 @ B4 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_218_subset__fst__snd,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A7: set @ ( product_prod @ A @ B )] :
% 2.08/1.14 ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A7
% 2.08/1.14 @ ( product_Sigma @ A @ B @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A7 )
% 2.08/1.14 @ ^ [Uu: A] : ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A7 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_219_scomp__unfold,axiom,
% 2.08/1.14 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 2.08/1.14 ( ( product_scomp @ A @ B @ C @ D )
% 2.08/1.14 = ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X3: A] : ( G4 @ ( product_fst @ B @ C @ ( F4 @ X3 ) ) @ ( product_snd @ B @ C @ ( F4 @ X3 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_220_scomp__apply,axiom,
% 2.08/1.14 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 2.08/1.14 ( ( product_scomp @ B @ C @ D @ A )
% 2.08/1.14 = ( ^ [F4: B > ( product_prod @ C @ D ),G4: C > D > A,X3: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F4 @ X3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_221_subset__Collect__iff,axiom,
% 2.08/1.14 ! [A: $tType,B7: set @ A,A7: set @ A,P2: A > $o] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ A ) @ B7 @ A7 )
% 2.08/1.14 => ( ( ord_less_eq @ ( set @ A ) @ B7
% 2.08/1.14 @ ( collect @ A
% 2.08/1.14 @ ^ [X3: A] :
% 2.08/1.14 ( ( member @ A @ X3 @ A7 )
% 2.08/1.14 & ( P2 @ X3 ) ) ) )
% 2.08/1.14 = ( ! [X3: A] :
% 2.08/1.14 ( ( member @ A @ X3 @ B7 )
% 2.08/1.14 => ( P2 @ X3 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_222_subset__CollectI,axiom,
% 2.08/1.14 ! [A: $tType,B7: set @ A,A7: set @ A,Q: A > $o,P2: A > $o] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ A ) @ B7 @ A7 )
% 2.08/1.14 => ( ! [X2: A] :
% 2.08/1.14 ( ( member @ A @ X2 @ B7 )
% 2.08/1.14 => ( ( Q @ X2 )
% 2.08/1.14 => ( P2 @ X2 ) ) )
% 2.08/1.14 => ( ord_less_eq @ ( set @ A )
% 2.08/1.14 @ ( collect @ A
% 2.08/1.14 @ ^ [X3: A] :
% 2.08/1.14 ( ( member @ A @ X3 @ B7 )
% 2.08/1.14 & ( Q @ X3 ) ) )
% 2.08/1.14 @ ( collect @ A
% 2.08/1.14 @ ^ [X3: A] :
% 2.08/1.14 ( ( member @ A @ X3 @ A7 )
% 2.08/1.14 & ( P2 @ X3 ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_223_Sigma__mono,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A7: set @ A,C5: set @ A,B7: A > ( set @ B ),D3: A > ( set @ B )] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ A ) @ A7 @ C5 )
% 2.08/1.14 => ( ! [X2: A] :
% 2.08/1.14 ( ( member @ A @ X2 @ A7 )
% 2.08/1.14 => ( ord_less_eq @ ( set @ B ) @ ( B7 @ X2 ) @ ( D3 @ X2 ) ) )
% 2.08/1.14 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) @ ( product_Sigma @ A @ B @ C5 @ D3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_224_Times__subset__cancel2,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,X: A,C5: set @ A,A7: set @ B,B7: set @ B] :
% 2.08/1.14 ( ( member @ A @ X @ C5 )
% 2.08/1.14 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 2.08/1.14 @ ( product_Sigma @ B @ A @ A7
% 2.08/1.14 @ ^ [Uu: B] : C5 )
% 2.08/1.14 @ ( product_Sigma @ B @ A @ B7
% 2.08/1.14 @ ^ [Uu: B] : C5 ) )
% 2.08/1.14 = ( ord_less_eq @ ( set @ B ) @ A7 @ B7 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_225_subrelI,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 2.08/1.14 ( ! [X2: A,Y: B] :
% 2.08/1.14 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R4 )
% 2.08/1.14 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S2 ) )
% 2.08/1.14 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S2 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_226_subset__image__iff,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,B7: set @ A,F: B > A,A7: set @ B] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F @ A7 ) )
% 2.08/1.14 = ( ? [AA: set @ B] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ B ) @ AA @ A7 )
% 2.08/1.14 & ( B7
% 2.08/1.14 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_227_image__subset__iff,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,F: B > A,A7: set @ B,B7: set @ A] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A7 ) @ B7 )
% 2.08/1.14 = ( ! [X3: B] :
% 2.08/1.14 ( ( member @ B @ X3 @ A7 )
% 2.08/1.14 => ( member @ A @ ( F @ X3 ) @ B7 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_228_subset__imageE,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,B7: set @ A,F: B > A,A7: set @ B] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F @ A7 ) )
% 2.08/1.14 => ~ ! [C7: set @ B] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ B ) @ C7 @ A7 )
% 2.08/1.14 => ( B7
% 2.08/1.14 != ( image @ B @ A @ F @ C7 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_229_image__subsetI,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,A7: set @ A,F: A > B,B7: set @ B] :
% 2.08/1.14 ( ! [X2: A] :
% 2.08/1.14 ( ( member @ A @ X2 @ A7 )
% 2.08/1.14 => ( member @ B @ ( F @ X2 ) @ B7 ) )
% 2.08/1.14 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A7 ) @ B7 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_230_image__mono,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A7: set @ A,B7: set @ A,F: A > B] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.08/1.14 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A7 ) @ ( image @ A @ B @ F @ B7 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_231_conj__subset__def,axiom,
% 2.08/1.14 ! [A: $tType,A7: set @ A,P2: A > $o,Q: A > $o] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ A ) @ A7
% 2.08/1.14 @ ( collect @ A
% 2.08/1.14 @ ^ [X3: A] :
% 2.08/1.14 ( ( P2 @ X3 )
% 2.08/1.14 & ( Q @ X3 ) ) ) )
% 2.08/1.14 = ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( collect @ A @ P2 ) )
% 2.08/1.14 & ( ord_less_eq @ ( set @ A ) @ A7 @ ( collect @ A @ Q ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_232_Collect__restrict,axiom,
% 2.08/1.14 ! [A: $tType,X7: set @ A,P2: A > $o] :
% 2.08/1.14 ( ord_less_eq @ ( set @ A )
% 2.08/1.14 @ ( collect @ A
% 2.08/1.14 @ ^ [X3: A] :
% 2.08/1.14 ( ( member @ A @ X3 @ X7 )
% 2.08/1.14 & ( P2 @ X3 ) ) )
% 2.08/1.14 @ X7 ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_233_prop__restrict,axiom,
% 2.08/1.14 ! [A: $tType,X: A,Z4: set @ A,X7: set @ A,P2: A > $o] :
% 2.08/1.14 ( ( member @ A @ X @ Z4 )
% 2.08/1.14 => ( ( ord_less_eq @ ( set @ A ) @ Z4
% 2.08/1.14 @ ( collect @ A
% 2.08/1.14 @ ^ [X3: A] :
% 2.08/1.14 ( ( member @ A @ X3 @ X7 )
% 2.08/1.14 & ( P2 @ X3 ) ) ) )
% 2.08/1.14 => ( P2 @ X ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_234_scomp__scomp,axiom,
% 2.08/1.14 ! [A: $tType,C: $tType,D: $tType,B: $tType,F2: $tType,E: $tType,F: A > ( product_prod @ E @ F2 ),G3: E > F2 > ( product_prod @ C @ D ),H: C > D > B] :
% 2.08/1.14 ( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F2 @ ( product_prod @ C @ D ) @ F @ G3 ) @ H )
% 2.08/1.14 = ( product_scomp @ A @ E @ F2 @ B @ F
% 2.08/1.14 @ ^ [X3: E] : ( product_scomp @ F2 @ C @ D @ B @ ( G3 @ X3 ) @ H ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_235_wlog__linorder__le,axiom,
% 2.08/1.14 ! [A: $tType] :
% 2.08/1.14 ( ( linorder @ A @ ( type @ A ) )
% 2.08/1.14 => ! [P2: A > A > $o,B3: A,A3: A] :
% 2.08/1.14 ( ! [A2: A,B2: A] :
% 2.08/1.14 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.08/1.14 => ( P2 @ A2 @ B2 ) )
% 2.08/1.14 => ( ( ( P2 @ B3 @ A3 )
% 2.08/1.14 => ( P2 @ A3 @ B3 ) )
% 2.08/1.14 => ( P2 @ A3 @ B3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_236_scomp__Pair,axiom,
% 2.08/1.14 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 2.08/1.14 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 2.08/1.14 = X ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_237_image__Collect__subsetI,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,P2: A > $o,F: A > B,B7: set @ B] :
% 2.08/1.14 ( ! [X2: A] :
% 2.08/1.14 ( ( P2 @ X2 )
% 2.08/1.14 => ( member @ B @ ( F @ X2 ) @ B7 ) )
% 2.08/1.14 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( collect @ A @ P2 ) ) @ B7 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_238_scomp__def,axiom,
% 2.08/1.14 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 2.08/1.14 ( ( product_scomp @ A @ B @ C @ D )
% 2.08/1.14 = ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X3: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F4 @ X3 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_239_Pair__scomp,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,C: $tType,X: C,F: C > A > B] :
% 2.08/1.14 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F )
% 2.08/1.14 = ( F @ X ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_240_refl__onI,axiom,
% 2.08/1.14 ! [A: $tType,R4: set @ ( product_prod @ A @ A ),A7: set @ A] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4
% 2.08/1.14 @ ( product_Sigma @ A @ A @ A7
% 2.08/1.14 @ ^ [Uu: A] : A7 ) )
% 2.08/1.14 => ( ! [X2: A] :
% 2.08/1.14 ( ( member @ A @ X2 @ A7 )
% 2.08/1.14 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R4 ) )
% 2.08/1.14 => ( refl_on @ A @ A7 @ R4 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_241_refl__on__def,axiom,
% 2.08/1.14 ! [A: $tType] :
% 2.08/1.14 ( ( refl_on @ A )
% 2.08/1.14 = ( ^ [A8: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
% 2.08/1.14 @ ( product_Sigma @ A @ A @ A8
% 2.08/1.14 @ ^ [Uu: A] : A8 ) )
% 2.08/1.14 & ! [X3: A] :
% 2.08/1.14 ( ( member @ A @ X3 @ A8 )
% 2.08/1.14 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R2 ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_242_Collect__split__mono__strong,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,X7: set @ A,A7: set @ ( product_prod @ A @ B ),Y8: set @ B,P2: A > B > $o,Q: A > B > $o] :
% 2.08/1.14 ( ( X7
% 2.08/1.14 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A7 ) )
% 2.08/1.14 => ( ( Y8
% 2.08/1.14 = ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A7 ) )
% 2.08/1.14 => ( ! [X2: A] :
% 2.08/1.14 ( ( member @ A @ X2 @ X7 )
% 2.08/1.14 => ! [Xa: B] :
% 2.08/1.14 ( ( member @ B @ Xa @ Y8 )
% 2.08/1.14 => ( ( P2 @ X2 @ Xa )
% 2.08/1.14 => ( Q @ X2 @ Xa ) ) ) )
% 2.08/1.14 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A7 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) ) )
% 2.08/1.14 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A7 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_243_subset__snd__imageI,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A7: set @ A,B7: set @ B,S: set @ ( product_prod @ A @ B ),X: A] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
% 2.08/1.14 @ ( product_Sigma @ A @ B @ A7
% 2.08/1.14 @ ^ [Uu: A] : B7 )
% 2.08/1.14 @ S )
% 2.08/1.14 => ( ( member @ A @ X @ A7 )
% 2.08/1.14 => ( ord_less_eq @ ( set @ B ) @ B7 @ ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_244_subset__fst__imageI,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A7: set @ A,B7: set @ B,S: set @ ( product_prod @ A @ B ),Y3: B] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
% 2.08/1.14 @ ( product_Sigma @ A @ B @ A7
% 2.08/1.14 @ ^ [Uu: A] : B7 )
% 2.08/1.14 @ S )
% 2.08/1.14 => ( ( member @ B @ Y3 @ B7 )
% 2.08/1.14 => ( ord_less_eq @ ( set @ A ) @ A7 @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_245_Quotient__rep__abs__eq,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,T4: A] :
% 2.08/1.14 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
% 2.08/1.14 => ( ( R @ T4 @ T4 )
% 2.08/1.14 => ( ( ord_less_eq @ ( A > A > $o ) @ R
% 2.08/1.14 @ ^ [Y7: A,Z2: A] : ( Y7 = Z2 ) )
% 2.08/1.14 => ( ( Rep @ ( Abs @ T4 ) )
% 2.08/1.14 = T4 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_246_reflp__ge__eq,axiom,
% 2.08/1.14 ! [A: $tType,R: A > A > $o] :
% 2.08/1.14 ( ( reflp @ A @ R )
% 2.08/1.14 => ( ord_less_eq @ ( A > A > $o )
% 2.08/1.14 @ ^ [Y7: A,Z2: A] : ( Y7 = Z2 )
% 2.08/1.14 @ R ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_247_Collect__case__prod__mono,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,A7: A > B > $o,B7: A > B > $o] :
% 2.08/1.14 ( ( ord_less_eq @ ( A > B > $o ) @ A7 @ B7 )
% 2.08/1.14 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A7 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B7 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_248_pred__subset__eq,axiom,
% 2.08/1.14 ! [A: $tType,R: set @ A,S: set @ A] :
% 2.08/1.14 ( ( ord_less_eq @ ( A > $o )
% 2.08/1.14 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 2.08/1.14 @ ^ [X3: A] : ( member @ A @ X3 @ S ) )
% 2.08/1.14 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_249_pred__subset__eq2,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.08/1.14 ( ( ord_less_eq @ ( A > B > $o )
% 2.08/1.14 @ ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R )
% 2.08/1.14 @ ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S ) )
% 2.08/1.14 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_250_less__eq__set__def,axiom,
% 2.08/1.14 ! [A: $tType] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ A ) )
% 2.08/1.14 = ( ^ [A8: set @ A,B8: set @ A] :
% 2.08/1.14 ( ord_less_eq @ ( A > $o )
% 2.08/1.14 @ ^ [X3: A] : ( member @ A @ X3 @ A8 )
% 2.08/1.14 @ ^ [X3: A] : ( member @ A @ X3 @ B8 ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_251_reflp__eq,axiom,
% 2.08/1.14 ! [A: $tType] :
% 2.08/1.14 ( ( reflp @ A )
% 2.08/1.14 = ( ord_less_eq @ ( A > A > $o )
% 2.08/1.14 @ ^ [Y7: A,Z2: A] : ( Y7 = Z2 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_252_Gr__incl,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,A7: set @ A,F: A > B,B7: set @ B] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A7 @ F )
% 2.08/1.14 @ ( product_Sigma @ A @ B @ A7
% 2.08/1.14 @ ^ [Uu: A] : B7 ) )
% 2.08/1.14 = ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A7 ) @ B7 ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_253_flip__pred,axiom,
% 2.08/1.14 ! [A: $tType,B: $tType,A7: set @ ( product_prod @ A @ B ),R: B > A > $o] :
% 2.08/1.14 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A7 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( conversep @ B @ A @ R ) ) ) )
% 2.08/1.14 => ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 2.08/1.14 @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
% 2.08/1.14 @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
% 2.08/1.14 @ ^ [X3: A,Y4: B] : ( product_Pair @ B @ A @ Y4 @ X3 ) )
% 2.08/1.14 @ A7 )
% 2.08/1.14 @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R ) ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_254_conversep__eq,axiom,
% 2.08/1.14 ! [A: $tType] :
% 2.08/1.14 ( ( conversep @ A @ A
% 2.08/1.14 @ ^ [Y7: A,Z2: A] : ( Y7 = Z2 ) )
% 2.08/1.14 = ( ^ [Y7: A,Z2: A] : ( Y7 = Z2 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(fact_255_conversep__iff,axiom,
% 2.08/1.14 ! [B: $tType,A: $tType] :
% 2.08/1.14 ( ( conversep @ A @ B )
% 2.08/1.14 = ( ^ [R2: A > B > $o,A4: B,B4: A] : ( R2 @ B4 @ A4 ) ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.08/1.14 linorder @ $o @ ( type @ $o ) ).
% 2.08/1.14
% 2.08/1.14 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_1,axiom,
% 2.08/1.14 linorder @ product_unit @ ( type @ product_unit ) ).
% 2.08/1.14
% 2.08/1.14 thf(help_If_3_1_T,axiom,
% 2.08/1.14 ! [P2: $o] :
% 2.08/1.14 ( ( P2 = $true )
% 2.08/1.14 | ( P2 = $false ) ) ).
% 2.08/1.14
% 2.08/1.14 thf(help_If_2_1_T,axiom,
% 2.08/1.14 ! [A: $tType,X: A,Y3: A] :
% 2.08/1.14 ( ( if @ A @ $false @ X @ Y3 )
% 2.08/1.14 = Y3 ) ).
% 2.08/1.14
% 2.08/1.14 thf(help_If_1_1_T,axiom,
% 2.08/1.14 ! [A: $tType,X: A,Y3: A] :
% 2.08/1.14 ( ( if @ A @ $true @ X @ Y3 )
% 2.08/1.14 = X ) ).
% 2.08/1.14
% 2.08/1.14 thf(conj_0,hypothesis,
% 2.08/1.14 ( product_case_prod @ ( coinductive_llist @ a ) @ b @ ( ( product_prod @ ( coinductive_llist @ a ) @ b ) > $o )
% 2.08/1.14 @ ^ [Xs2: coinductive_llist @ a,A4: b] :
% 2.08/1.14 ( product_case_prod @ ( coinductive_llist @ a ) @ b @ $o
% 2.08/1.14 @ ^ [Ys: coinductive_llist @ a,B4: b] :
% 2.08/1.14 ( ( Xs2 = Ys )
% 2.08/1.14 & ( ( coinductive_lfinite @ a @ Ys )
% 2.08/1.14 => ( A4 = B4 ) ) ) )
% 2.08/1.14 @ prod1
% 2.08/1.14 @ prod2 ) ).
% 2.08/1.14
% 2.08/1.14 thf(conj_1,conjecture,
% 2.08/1.14 ( product_case_prod @ ( coinductive_llist @ a ) @ b @ ( ( product_prod @ ( coinductive_llist @ a ) @ b ) > $o )
% 2.08/1.14 @ ^ [Xs2: coinductive_llist @ a,A4: b] :
% 2.08/1.14 ( product_case_prod @ ( coinductive_llist @ a ) @ b @ $o
% 2.08/1.14 @ ^ [Ys: coinductive_llist @ a,B4: b] :
% 2.08/1.14 ( ( Xs2 = Ys )
% 2.08/1.14 & ( ( coinductive_lfinite @ a @ Ys )
% 2.08/1.14 => ( A4 = B4 ) ) ) )
% 2.08/1.14 @ ( product_case_prod @ ( coinductive_llist @ a ) @ b @ ( product_prod @ ( coinductive_llist @ a ) @ b )
% 2.08/1.14 @ ^ [Xs2: coinductive_llist @ a,B6: b] : ( product_Pair @ ( coinductive_llist @ a ) @ b @ ( coinductive_lfilter @ a @ fun @ Xs2 ) @ ( if @ b @ ( coinductive_lfinite @ a @ Xs2 ) @ B6 @ b2 ) )
% 2.08/1.14 @ prod1 )
% 2.08/1.14 @ ( product_case_prod @ ( coinductive_llist @ a ) @ b @ ( product_prod @ ( coinductive_llist @ a ) @ b )
% 2.08/1.14 @ ^ [Xs2: coinductive_llist @ a,B6: b] : ( product_Pair @ ( coinductive_llist @ a ) @ b @ ( coinductive_lfilter @ a @ fun @ Xs2 ) @ ( if @ b @ ( coinductive_lfinite @ a @ Xs2 ) @ B6 @ b2 ) )
% 2.08/1.14 @ prod2 ) ) ).
% 2.08/1.14 ------------------------
% 2.08/1.16 % (29766)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_29522 for (2999ds/27Mi)
% 2.08/1.16 % (29769)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_29522 for (2999ds/275Mi)
% 2.08/1.16 % (29764)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/183Mi)
% 2.08/1.16 % (29768)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_29522 for (2999ds/2Mi)
% 2.08/1.16 % (29765)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_29522 for (2999ds/4Mi)
% 2.08/1.16 % (29767)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/2Mi)
% 2.08/1.16 % (29770)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/18Mi)
% 2.08/1.17 % (29765)Instruction limit reached!
% 2.08/1.17 % (29765)------------------------------
% 2.08/1.17 % (29765)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.08/1.17 % (29765)Termination reason: Unknown
% 2.08/1.17 % (29765)Termination phase: shuffling
% 2.08/1.17
% 2.08/1.17 % (29765)Memory used [KB]: 1535
% 2.08/1.17 % (29765)Time elapsed: 0.003 s
% 2.08/1.17 % (29765)Instructions burned: 4 (million)
% 2.08/1.17 % (29765)------------------------------
% 2.08/1.17 % (29765)------------------------------
% 2.08/1.17 % (29767)Instruction limit reached!
% 2.08/1.17 % (29767)------------------------------
% 2.08/1.17 % (29767)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.08/1.17 % (29767)Termination reason: Unknown
% 2.08/1.17 % (29767)Termination phase: shuffling
% 2.08/1.17
% 2.08/1.17 % (29767)Memory used [KB]: 1535
% 2.08/1.17 % (29767)Time elapsed: 0.003 s
% 2.08/1.17 % (29767)Instructions burned: 4 (million)
% 2.08/1.17 % (29767)------------------------------
% 2.08/1.17 % (29767)------------------------------
% 2.08/1.17 % (29768)Instruction limit reached!
% 2.08/1.17 % (29768)------------------------------
% 2.08/1.17 % (29768)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.08/1.17 % (29768)Termination reason: Unknown
% 2.08/1.17 % (29768)Termination phase: shuffling
% 2.08/1.17
% 2.08/1.17 % (29768)Memory used [KB]: 1535
% 2.08/1.17 % (29768)Time elapsed: 0.003 s
% 2.08/1.17 % (29768)Instructions burned: 5 (million)
% 2.08/1.17 % (29768)------------------------------
% 2.08/1.17 % (29768)------------------------------
% 2.08/1.17 % (29770)Instruction limit reached!
% 2.08/1.17 % (29770)------------------------------
% 2.08/1.17 % (29770)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.08/1.17 % (29770)Termination reason: Unknown
% 2.08/1.17 % (29770)Termination phase: shuffling
% 2.08/1.17
% 2.08/1.17 % (29770)Memory used [KB]: 1918
% 2.08/1.17 % (29770)Time elapsed: 0.008 s
% 2.08/1.17 % (29770)Instructions burned: 19 (million)
% 2.08/1.17 % (29770)------------------------------
% 2.08/1.17 % (29770)------------------------------
% 2.08/1.17 % (29766)Instruction limit reached!
% 2.08/1.17 % (29766)------------------------------
% 2.08/1.17 % (29766)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.08/1.17 % (29766)Termination reason: Unknown
% 2.08/1.17 % (29766)Termination phase: shuffling
% 2.08/1.17
% 2.08/1.17 % (29766)Memory used [KB]: 2046
% 2.08/1.17 % (29766)Time elapsed: 0.011 s
% 2.08/1.17 % (29766)Instructions burned: 28 (million)
% 2.08/1.17 % (29766)------------------------------
% 2.08/1.17 % (29766)------------------------------
% 2.08/1.17 % (29771)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/3Mi)
% 2.08/1.17 % (29772)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_29522 for (2999ds/37Mi)
% 2.08/1.17 % (29773)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_29522 for (2999ds/15Mi)
% 2.08/1.18 % (29771)Instruction limit reached!
% 2.08/1.18 % (29771)------------------------------
% 2.08/1.18 % (29771)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.08/1.18 % (29771)Termination reason: Unknown
% 2.08/1.18 % (29771)Termination phase: shuffling
% 2.08/1.18
% 2.08/1.18 % (29771)Memory used [KB]: 1535
% 2.08/1.18 % (29771)Time elapsed: 0.002 s
% 2.08/1.18 % (29771)Instructions burned: 3 (million)
% 2.08/1.18 % (29771)------------------------------
% 2.08/1.18 % (29771)------------------------------
% 2.08/1.18 % (29773)Instruction limit reached!
% 2.08/1.18 % (29773)------------------------------
% 2.08/1.18 % (29773)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.08/1.18 % (29773)Termination reason: Unknown
% 2.08/1.18 % (29773)Termination phase: shuffling
% 2.08/1.18
% 2.08/1.18 % (29773)Memory used [KB]: 1791
% 2.08/1.18 % (29773)Time elapsed: 0.006 s
% 2.08/1.18 % (29773)Instructions burned: 15 (million)
% 2.08/1.18 % (29773)------------------------------
% 2.08/1.18 % (29773)------------------------------
% 2.08/1.18 % (29774)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_29522 for (2999ds/3Mi)
% 2.08/1.18 % (29774)Instruction limit reached!
% 2.08/1.18 % (29774)------------------------------
% 2.08/1.18 % (29774)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.08/1.18 % (29774)Termination reason: Unknown
% 2.08/1.18 % (29774)Termination phase: shuffling
% 2.08/1.18
% 2.08/1.18 % (29774)Memory used [KB]: 1535
% 2.08/1.18 % (29774)Time elapsed: 0.002 s
% 2.08/1.18 % (29774)Instructions burned: 3 (million)
% 2.08/1.18 % (29774)------------------------------
% 2.08/1.18 % (29774)------------------------------
% 2.08/1.18 % (29775)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_29522 for (2999ds/1041Mi)
% 2.08/1.18 % (29776)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/7Mi)
% 2.08/1.19 % (29772)Instruction limit reached!
% 2.08/1.19 % (29772)------------------------------
% 2.08/1.19 % (29772)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.08/1.19 % (29772)Termination reason: Unknown
% 2.08/1.19 % (29772)Termination phase: shuffling
% 2.08/1.19
% 2.08/1.19 % (29772)Memory used [KB]: 2174
% 2.08/1.19 % (29772)Time elapsed: 0.012 s
% 2.08/1.19 % (29772)Instructions burned: 38 (million)
% 2.08/1.19 % (29772)------------------------------
% 2.08/1.19 % (29772)------------------------------
% 2.08/1.19 % (29776)Instruction limit reached!
% 2.08/1.19 % (29776)------------------------------
% 2.08/1.19 % (29776)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.08/1.19 % (29776)Termination reason: Unknown
% 2.08/1.19 % (29776)Termination phase: shuffling
% 2.08/1.19
% 2.08/1.19 % (29776)Memory used [KB]: 1663
% 2.08/1.19 % (29776)Time elapsed: 0.004 s
% 2.08/1.19 % (29776)Instructions burned: 9 (million)
% 2.08/1.19 % (29776)------------------------------
% 2.08/1.19 % (29776)------------------------------
% 2.08/1.19 % (29777)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/16Mi)
% 2.08/1.19 % (29778)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_29522 for (2999ds/3Mi)
% 2.08/1.19 % (29778)Instruction limit reached!
% 2.08/1.19 % (29778)------------------------------
% 2.08/1.19 % (29778)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.08/1.19 % (29778)Termination reason: Unknown
% 2.08/1.19 % (29778)Termination phase: shuffling
% 2.08/1.19
% 2.08/1.19 % (29778)Memory used [KB]: 1535
% 2.08/1.19 % (29778)Time elapsed: 0.002 s
% 2.08/1.19 % (29778)Instructions burned: 3 (million)
% 2.08/1.19 % (29778)------------------------------
% 2.08/1.19 % (29778)------------------------------
% 2.08/1.20 % (29777)Instruction limit reached!
% 2.08/1.20 % (29777)------------------------------
% 2.08/1.20 % (29777)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.08/1.20 % (29777)Termination reason: Unknown
% 2.08/1.20 % (29777)Termination phase: shuffling
% 2.08/1.20
% 2.08/1.20 % (29777)Memory used [KB]: 1918
% 2.08/1.20 % (29777)Time elapsed: 0.007 s
% 2.08/1.20 % (29777)Instructions burned: 18 (million)
% 2.08/1.20 % (29777)------------------------------
% 2.08/1.20 % (29777)------------------------------
% 2.08/1.20 % (29779)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_29522 for (2999ds/3Mi)
% 2.10/1.20 % (29779)Instruction limit reached!
% 2.10/1.20 % (29779)------------------------------
% 2.10/1.20 % (29779)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.20 % (29779)Termination reason: Unknown
% 2.10/1.20 % (29779)Termination phase: shuffling
% 2.10/1.20
% 2.10/1.20 % (29779)Memory used [KB]: 1535
% 2.10/1.20 % (29779)Time elapsed: 0.002 s
% 2.10/1.20 % (29779)Instructions burned: 3 (million)
% 2.10/1.20 % (29779)------------------------------
% 2.10/1.20 % (29779)------------------------------
% 2.10/1.20 % (29780)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/7Mi)
% 2.10/1.20 % (29780)Instruction limit reached!
% 2.10/1.20 % (29780)------------------------------
% 2.10/1.20 % (29780)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.20 % (29780)Termination reason: Unknown
% 2.10/1.20 % (29780)Termination phase: shuffling
% 2.10/1.20
% 2.10/1.20 % (29780)Memory used [KB]: 1663
% 2.10/1.20 % (29780)Time elapsed: 0.004 s
% 2.10/1.20 % (29780)Instructions burned: 9 (million)
% 2.10/1.20 % (29780)------------------------------
% 2.10/1.20 % (29780)------------------------------
% 2.10/1.20 % (29781)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/3Mi)
% 2.10/1.20 % (29781)Instruction limit reached!
% 2.10/1.20 % (29781)------------------------------
% 2.10/1.20 % (29781)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.20 % (29781)Termination reason: Unknown
% 2.10/1.20 % (29781)Termination phase: shuffling
% 2.10/1.20
% 2.10/1.20 % (29781)Memory used [KB]: 1535
% 2.10/1.20 % (29781)Time elapsed: 0.002 s
% 2.10/1.20 % (29781)Instructions burned: 3 (million)
% 2.10/1.20 % (29781)------------------------------
% 2.10/1.20 % (29781)------------------------------
% 2.10/1.20 % (29782)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/4Mi)
% 2.10/1.21 % (29783)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/18Mi)
% 2.10/1.21 % (29782)Instruction limit reached!
% 2.10/1.21 % (29782)------------------------------
% 2.10/1.21 % (29782)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.21 % (29782)Termination reason: Unknown
% 2.10/1.21 % (29782)Termination phase: shuffling
% 2.10/1.21
% 2.10/1.21 % (29782)Memory used [KB]: 1535
% 2.10/1.21 % (29782)Time elapsed: 0.003 s
% 2.10/1.21 % (29782)Instructions burned: 6 (million)
% 2.10/1.21 % (29782)------------------------------
% 2.10/1.21 % (29782)------------------------------
% 2.10/1.21 % (29784)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/710Mi)
% 2.10/1.21 % (29785)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/6Mi)
% 2.10/1.21 % (29783)Instruction limit reached!
% 2.10/1.21 % (29783)------------------------------
% 2.10/1.21 % (29783)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.21 % (29783)Termination reason: Unknown
% 2.10/1.21 % (29783)Termination phase: shuffling
% 2.10/1.21
% 2.10/1.21 % (29783)Memory used [KB]: 1918
% 2.10/1.21 % (29783)Time elapsed: 0.008 s
% 2.10/1.21 % (29783)Instructions burned: 20 (million)
% 2.10/1.21 % (29783)------------------------------
% 2.10/1.21 % (29783)------------------------------
% 2.10/1.21 % (29785)Instruction limit reached!
% 2.10/1.21 % (29785)------------------------------
% 2.10/1.21 % (29785)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.21 % (29785)Termination reason: Unknown
% 2.10/1.21 % (29785)Termination phase: shuffling
% 2.10/1.21
% 2.10/1.21 % (29785)Memory used [KB]: 1663
% 2.10/1.21 % (29785)Time elapsed: 0.004 s
% 2.10/1.21 % (29785)Instructions burned: 8 (million)
% 2.10/1.21 % (29785)------------------------------
% 2.10/1.21 % (29785)------------------------------
% 2.10/1.22 % (29786)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/902Mi)
% 2.10/1.22 % (29764)Instruction limit reached!
% 2.10/1.22 % (29764)------------------------------
% 2.10/1.22 % (29764)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.22 % (29764)Termination reason: Unknown
% 2.10/1.22 % (29764)Termination phase: Saturation
% 2.10/1.22
% 2.10/1.22 % (29764)Memory used [KB]: 7675
% 2.10/1.22 % (29764)Time elapsed: 0.056 s
% 2.10/1.22 % (29764)Instructions burned: 183 (million)
% 2.10/1.22 % (29764)------------------------------
% 2.10/1.22 % (29764)------------------------------
% 2.10/1.22 % (29787)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/21Mi)
% 2.10/1.22 % (29788)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/5Mi)
% 2.10/1.22 % (29788)Instruction limit reached!
% 2.10/1.22 % (29788)------------------------------
% 2.10/1.22 % (29788)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.22 % (29788)Termination reason: Unknown
% 2.10/1.22 % (29788)Termination phase: shuffling
% 2.10/1.22
% 2.10/1.22 % (29788)Memory used [KB]: 1663
% 2.10/1.22 % (29788)Time elapsed: 0.003 s
% 2.10/1.22 % (29788)Instructions burned: 7 (million)
% 2.10/1.22 % (29788)------------------------------
% 2.10/1.22 % (29788)------------------------------
% 2.10/1.23 % (29789)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/6Mi)
% 2.10/1.23 % (29787)Instruction limit reached!
% 2.10/1.23 % (29787)------------------------------
% 2.10/1.23 % (29787)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.23 % (29787)Termination reason: Unknown
% 2.10/1.23 % (29787)Termination phase: shuffling
% 2.10/1.23
% 2.10/1.23 % (29787)Memory used [KB]: 1918
% 2.10/1.23 % (29787)Time elapsed: 0.008 s
% 2.10/1.23 % (29787)Instructions burned: 21 (million)
% 2.10/1.23 % (29787)------------------------------
% 2.10/1.23 % (29787)------------------------------
% 2.10/1.23 % (29789)Instruction limit reached!
% 2.10/1.23 % (29789)------------------------------
% 2.10/1.23 % (29789)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.23 % (29789)Termination reason: Unknown
% 2.10/1.23 % (29789)Termination phase: shuffling
% 2.10/1.23
% 2.10/1.23 % (29789)Memory used [KB]: 1663
% 2.10/1.23 % (29789)Time elapsed: 0.003 s
% 2.10/1.23 % (29789)Instructions burned: 6 (million)
% 2.10/1.23 % (29789)------------------------------
% 2.10/1.23 % (29789)------------------------------
% 2.10/1.23 % (29790)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/377Mi)
% 2.10/1.24 % (29791)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/779Mi)
% 2.10/1.24 % (29792)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/19Mi)
% 2.10/1.24 % (29792)Instruction limit reached!
% 2.10/1.24 % (29792)------------------------------
% 2.10/1.24 % (29792)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.24 % (29792)Termination reason: Unknown
% 2.10/1.25 % (29792)Termination phase: shuffling
% 2.10/1.25
% 2.10/1.25 % (29792)Memory used [KB]: 1918
% 2.10/1.25 % (29792)Time elapsed: 0.007 s
% 2.10/1.25 % (29792)Instructions burned: 21 (million)
% 2.10/1.25 % (29792)------------------------------
% 2.10/1.25 % (29792)------------------------------
% 2.10/1.25 % (29769)Instruction limit reached!
% 2.10/1.25 % (29769)------------------------------
% 2.10/1.25 % (29769)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.25 % (29769)Termination reason: Unknown
% 2.10/1.25 % (29769)Termination phase: Saturation
% 2.10/1.25
% 2.10/1.25 % (29769)Memory used [KB]: 7931
% 2.10/1.25 % (29769)Time elapsed: 0.087 s
% 2.10/1.25 % (29769)Instructions burned: 276 (million)
% 2.10/1.25 % (29769)------------------------------
% 2.10/1.25 % (29769)------------------------------
% 2.10/1.25 % (29793)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/879Mi)
% 2.10/1.26 % (29794)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_29522 for (2999ds/17Mi)
% 2.10/1.26 % (29794)Instruction limit reached!
% 2.10/1.26 % (29794)------------------------------
% 2.10/1.26 % (29794)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.27 % (29794)Termination reason: Unknown
% 2.10/1.27 % (29794)Termination phase: shuffling
% 2.10/1.27
% 2.10/1.27 % (29794)Memory used [KB]: 1791
% 2.10/1.27 % (29794)Time elapsed: 0.007 s
% 2.10/1.27 % (29794)Instructions burned: 18 (million)
% 2.10/1.27 % (29794)------------------------------
% 2.10/1.27 % (29794)------------------------------
% 2.10/1.27 % (29795)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_29522 for (2998ds/3Mi)
% 2.10/1.28 % (29795)Instruction limit reached!
% 2.10/1.28 % (29795)------------------------------
% 2.10/1.28 % (29795)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.28 % (29795)Termination reason: Unknown
% 2.10/1.28 % (29795)Termination phase: shuffling
% 2.10/1.28
% 2.10/1.28 % (29795)Memory used [KB]: 1535
% 2.10/1.28 % (29795)Time elapsed: 0.003 s
% 2.10/1.28 % (29795)Instructions burned: 5 (million)
% 2.10/1.28 % (29795)------------------------------
% 2.10/1.28 % (29795)------------------------------
% 2.10/1.29 % (29796)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_29522 for (2998ds/30Mi)
% 2.10/1.30 % (29796)Instruction limit reached!
% 2.10/1.30 % (29796)------------------------------
% 2.10/1.30 % (29796)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.30 % (29796)Termination reason: Unknown
% 2.10/1.30 % (29796)Termination phase: shuffling
% 2.10/1.30
% 2.10/1.30 % (29796)Memory used [KB]: 2046
% 2.10/1.30 % (29796)Time elapsed: 0.012 s
% 2.10/1.30 % (29796)Instructions burned: 31 (million)
% 2.10/1.30 % (29796)------------------------------
% 2.10/1.30 % (29796)------------------------------
% 2.10/1.31 % (29797)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_29522 for (2998ds/127Mi)
% 2.10/1.34 % (29797)Instruction limit reached!
% 2.10/1.34 % (29797)------------------------------
% 2.10/1.34 % (29797)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.34 % (29797)Termination reason: Unknown
% 2.10/1.34 % (29797)Termination phase: Saturation
% 2.10/1.34
% 2.10/1.34 % (29797)Memory used [KB]: 7164
% 2.10/1.34 % (29797)Time elapsed: 0.039 s
% 2.10/1.34 % (29797)Instructions burned: 129 (million)
% 2.10/1.34 % (29797)------------------------------
% 2.10/1.34 % (29797)------------------------------
% 2.10/1.35 % (29790)Instruction limit reached!
% 2.10/1.35 % (29790)------------------------------
% 2.10/1.35 % (29790)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.35 % (29790)Termination reason: Unknown
% 2.10/1.35 % (29790)Termination phase: Saturation
% 2.10/1.35
% 2.10/1.35 % (29790)Memory used [KB]: 8699
% 2.10/1.35 % (29790)Time elapsed: 0.113 s
% 2.10/1.35 % (29790)Instructions burned: 377 (million)
% 2.10/1.35 % (29790)------------------------------
% 2.10/1.35 % (29790)------------------------------
% 2.10/1.35 % (29798)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_29522 for (2997ds/100Mi)
% 2.10/1.36 % (29799)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_29522 for (2997ds/3Mi)
% 2.10/1.36 % (29799)Instruction limit reached!
% 2.10/1.36 % (29799)------------------------------
% 2.10/1.36 % (29799)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.36 % (29799)Termination reason: Unknown
% 2.10/1.36 % (29799)Termination phase: shuffling
% 2.10/1.36
% 2.10/1.36 % (29799)Memory used [KB]: 1535
% 2.10/1.36 % (29799)Time elapsed: 0.002 s
% 2.10/1.36 % (29799)Instructions burned: 3 (million)
% 2.10/1.36 % (29799)------------------------------
% 2.10/1.36 % (29799)------------------------------
% 2.10/1.37 % (29800)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_29522 for (2997ds/20Mi)
% 2.10/1.37 % (29800)Instruction limit reached!
% 2.10/1.37 % (29800)------------------------------
% 2.10/1.37 % (29800)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.37 % (29800)Termination reason: Unknown
% 2.10/1.37 % (29800)Termination phase: shuffling
% 2.10/1.37
% 2.10/1.37 % (29800)Memory used [KB]: 1918
% 2.10/1.37 % (29800)Time elapsed: 0.008 s
% 2.10/1.37 % (29800)Instructions burned: 21 (million)
% 2.10/1.37 % (29800)------------------------------
% 2.10/1.37 % (29800)------------------------------
% 2.10/1.38 % (29798)Instruction limit reached!
% 2.10/1.38 % (29798)------------------------------
% 2.10/1.38 % (29798)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.38 % (29798)Termination reason: Unknown
% 2.10/1.38 % (29798)Termination phase: Preprocessing 3
% 2.10/1.38
% 2.10/1.38 % (29798)Memory used [KB]: 2814
% 2.10/1.38 % (29798)Time elapsed: 0.030 s
% 2.10/1.38 % (29798)Instructions burned: 102 (million)
% 2.10/1.38 % (29798)------------------------------
% 2.10/1.38 % (29798)------------------------------
% 2.10/1.38 % (29801)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_29522 for (2997ds/86Mi)
% 2.10/1.39 % (29802)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_29522 for (2997ds/107Mi)
% 2.10/1.41 % (29801)Instruction limit reached!
% 2.10/1.41 % (29801)------------------------------
% 2.10/1.41 % (29801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.41 % (29801)Termination reason: Unknown
% 2.10/1.41 % (29801)Termination phase: Function definition elimination
% 2.10/1.41
% 2.10/1.41 % (29801)Memory used [KB]: 2558
% 2.10/1.41 % (29801)Time elapsed: 0.028 s
% 2.10/1.41 % (29801)Instructions burned: 89 (million)
% 2.10/1.41 % (29801)------------------------------
% 2.10/1.41 % (29801)------------------------------
% 2.10/1.42 % (29803)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_29522 for (2997ds/39Mi)
% 2.10/1.42 % (29802)Instruction limit reached!
% 2.10/1.42 % (29802)------------------------------
% 2.10/1.42 % (29802)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.42 % (29802)Termination reason: Unknown
% 2.10/1.42 % (29802)Termination phase: Property scanning
% 2.10/1.42
% 2.10/1.42 % (29802)Memory used [KB]: 2686
% 2.10/1.42 % (29802)Time elapsed: 0.033 s
% 2.10/1.42 % (29802)Instructions burned: 107 (million)
% 2.10/1.42 % (29802)------------------------------
% 2.10/1.42 % (29802)------------------------------
% 2.10/1.43 % (29803)Instruction limit reached!
% 2.10/1.43 % (29803)------------------------------
% 2.10/1.43 % (29803)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.43 % (29803)Termination reason: Unknown
% 2.10/1.43 % (29803)Termination phase: shuffling
% 2.10/1.43
% 2.10/1.43 % (29803)Memory used [KB]: 2302
% 2.10/1.43 % (29803)Time elapsed: 0.012 s
% 2.10/1.43 % (29803)Instructions burned: 39 (million)
% 2.10/1.43 % (29803)------------------------------
% 2.10/1.43 % (29803)------------------------------
% 2.10/1.43 % (29804)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_29522 for (2997ds/448Mi)
% 2.10/1.43 % (29784)Instruction limit reached!
% 2.10/1.43 % (29784)------------------------------
% 2.10/1.43 % (29784)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.43 % (29784)Termination reason: Unknown
% 2.10/1.43 % (29784)Termination phase: Saturation
% 2.10/1.43
% 2.10/1.43 % (29784)Memory used [KB]: 9338
% 2.10/1.43 % (29784)Time elapsed: 0.225 s
% 2.10/1.43 % (29784)Instructions burned: 710 (million)
% 2.10/1.43 % (29784)------------------------------
% 2.10/1.43 % (29784)------------------------------
% 2.10/1.44 % (29805)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_29522 for (2997ds/46Mi)
% 2.10/1.44 % (29806)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_29522 for (2997ds/98Mi)
% 2.10/1.45 % (29805)Instruction limit reached!
% 2.10/1.45 % (29805)------------------------------
% 2.10/1.45 % (29805)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.45 % (29805)Termination reason: Unknown
% 2.10/1.45 % (29805)Termination phase: SInE selection
% 2.10/1.45
% 2.10/1.45 % (29805)Memory used [KB]: 2046
% 2.10/1.45 % (29805)Time elapsed: 0.016 s
% 2.10/1.45 % (29805)Instructions burned: 48 (million)
% 2.10/1.45 % (29805)------------------------------
% 2.10/1.45 % (29805)------------------------------
% 2.10/1.46 % (29807)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_29522 for (2996ds/507Mi)
% 2.10/1.47 % (29806)Instruction limit reached!
% 2.10/1.47 % (29806)------------------------------
% 2.10/1.47 % (29806)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.47 % (29806)Termination reason: Unknown
% 2.10/1.47 % (29806)Termination phase: Preprocessing 3
% 2.10/1.47
% 2.10/1.47 % (29806)Memory used [KB]: 2814
% 2.10/1.47 % (29806)Time elapsed: 0.029 s
% 2.10/1.47 % (29806)Instructions burned: 98 (million)
% 2.10/1.47 % (29806)------------------------------
% 2.10/1.47 % (29806)------------------------------
% 2.10/1.47 % (29786)Instruction limit reached!
% 2.10/1.47 % (29786)------------------------------
% 2.10/1.47 % (29786)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.47 % (29786)Termination reason: Unknown
% 2.10/1.47 % (29786)Termination phase: Saturation
% 2.10/1.47
% 2.10/1.47 % (29786)Memory used [KB]: 8827
% 2.10/1.47 % (29786)Time elapsed: 0.259 s
% 2.10/1.47 % (29786)Instructions burned: 903 (million)
% 2.10/1.47 % (29786)------------------------------
% 2.10/1.47 % (29786)------------------------------
% 2.10/1.48 % (29791)Instruction limit reached!
% 2.10/1.48 % (29791)------------------------------
% 2.10/1.48 % (29791)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.48 % (29791)Termination reason: Unknown
% 2.10/1.48 % (29791)Termination phase: Saturation
% 2.10/1.48
% 2.10/1.48 % (29791)Memory used [KB]: 9850
% 2.10/1.48 % (29791)Time elapsed: 0.263 s
% 2.10/1.48 % (29791)Instructions burned: 781 (million)
% 2.10/1.48 % (29791)------------------------------
% 2.10/1.48 % (29791)------------------------------
% 2.10/1.48 % (29808)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_29522 for (2996ds/149Mi)
% 2.10/1.48 % (29809)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_29522 for (2996ds/39Mi)
% 2.10/1.49 % (29810)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_29522 for (2996ds/6Mi)
% 2.10/1.49 % (29810)Instruction limit reached!
% 2.10/1.49 % (29810)------------------------------
% 2.10/1.49 % (29810)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.49 % (29810)Termination reason: Unknown
% 2.10/1.49 % (29810)Termination phase: shuffling
% 2.10/1.49
% 2.10/1.49 % (29810)Memory used [KB]: 1663
% 2.10/1.49 % (29810)Time elapsed: 0.003 s
% 2.10/1.49 % (29810)Instructions burned: 6 (million)
% 2.10/1.49 % (29810)------------------------------
% 2.10/1.49 % (29810)------------------------------
% 2.10/1.50 % (29809)Instruction limit reached!
% 2.10/1.50 % (29809)------------------------------
% 2.10/1.50 % (29809)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.50 % (29809)Termination reason: Unknown
% 2.10/1.50 % (29809)Termination phase: Property scanning
% 2.10/1.50
% 2.10/1.50 % (29809)Memory used [KB]: 2174
% 2.10/1.50 % (29809)Time elapsed: 0.014 s
% 2.10/1.50 % (29809)Instructions burned: 41 (million)
% 2.10/1.50 % (29809)------------------------------
% 2.10/1.50 % (29809)------------------------------
% 2.10/1.50 % (29811)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_29522 for (2996ds/1087Mi)
% 2.10/1.50 % (29775)Instruction limit reached!
% 2.10/1.50 % (29775)------------------------------
% 2.10/1.50 % (29775)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.50 % (29775)Termination reason: Unknown
% 2.10/1.50 % (29775)Termination phase: Saturation
% 2.10/1.50
% 2.10/1.50 % (29775)Memory used [KB]: 10106
% 2.10/1.50 % (29775)Time elapsed: 0.321 s
% 2.10/1.50 % (29775)Instructions burned: 1042 (million)
% 2.10/1.50 % (29775)------------------------------
% 2.10/1.50 % (29775)------------------------------
% 2.10/1.51 % (29812)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_29522 for (2996ds/47Mi)
% 2.10/1.51 % (29813)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_29522 for (2996ds/44Mi)
% 2.10/1.52 % (29812)Instruction limit reached!
% 2.10/1.52 % (29812)------------------------------
% 2.10/1.52 % (29812)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.52 % (29812)Termination reason: Unknown
% 2.10/1.52 % (29812)Termination phase: Preprocessing 1
% 2.10/1.52
% 2.10/1.52 % (29812)Memory used [KB]: 2174
% 2.10/1.52 % (29812)Time elapsed: 0.016 s
% 2.10/1.52 % (29812)Instructions burned: 47 (million)
% 2.10/1.52 % (29812)------------------------------
% 2.10/1.52 % (29812)------------------------------
% 2.10/1.52 % (29793)Instruction limit reached!
% 2.10/1.52 % (29793)------------------------------
% 2.10/1.52 % (29793)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.52 % (29793)Termination reason: Unknown
% 2.10/1.52 % (29793)Termination phase: Saturation
% 2.10/1.52
% 2.10/1.52 % (29793)Memory used [KB]: 8827
% 2.10/1.52 % (29793)Time elapsed: 0.295 s
% 2.10/1.52 % (29793)Instructions burned: 882 (million)
% 2.10/1.52 % (29793)------------------------------
% 2.10/1.52 % (29793)------------------------------
% 2.10/1.53 % (29813)Instruction limit reached!
% 2.10/1.53 % (29813)------------------------------
% 2.10/1.53 % (29813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.53 % (29813)Termination reason: Unknown
% 2.10/1.53 % (29813)Termination phase: shuffling
% 2.10/1.53
% 2.10/1.53 % (29813)Memory used [KB]: 2302
% 2.10/1.53 % (29813)Time elapsed: 0.014 s
% 2.10/1.53 % (29813)Instructions burned: 45 (million)
% 2.10/1.53 % (29813)------------------------------
% 2.10/1.53 % (29813)------------------------------
% 2.10/1.53 % (29808)Instruction limit reached!
% 2.10/1.53 % (29808)------------------------------
% 2.10/1.53 % (29808)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.53 % (29808)Termination reason: Unknown
% 2.10/1.53 % (29808)Termination phase: Saturation
% 2.10/1.53
% 2.10/1.53 % (29808)Memory used [KB]: 7419
% 2.10/1.53 % (29808)Time elapsed: 0.046 s
% 2.10/1.53 % (29808)Instructions burned: 151 (million)
% 2.10/1.53 % (29808)------------------------------
% 2.10/1.53 % (29808)------------------------------
% 2.10/1.53 % (29814)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_29522 for (2996ds/432Mi)
% 2.10/1.53 % (29815)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_29522 for (2996ds/62Mi)
% 2.10/1.53 % (29816)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_29522 for (2996ds/31Mi)
% 2.10/1.53 % (29817)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_29522 for (2996ds/96Mi)
% 2.10/1.54 % (29816)Instruction limit reached!
% 2.10/1.54 % (29816)------------------------------
% 2.10/1.54 % (29816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.54 % (29816)Termination reason: Unknown
% 2.10/1.54 % (29816)Termination phase: shuffling
% 2.10/1.54
% 2.10/1.54 % (29816)Memory used [KB]: 2046
% 2.10/1.54 % (29816)Time elapsed: 0.011 s
% 2.10/1.54 % (29816)Instructions burned: 31 (million)
% 2.10/1.54 % (29816)------------------------------
% 2.10/1.54 % (29816)------------------------------
% 2.10/1.55 % (29815)Instruction limit reached!
% 2.10/1.55 % (29815)------------------------------
% 2.10/1.55 % (29815)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.55 % (29815)Termination reason: Unknown
% 2.10/1.55 % (29815)Termination phase: Property scanning
% 2.10/1.55
% 2.10/1.55 % (29815)Memory used [KB]: 2430
% 2.10/1.55 % (29815)Time elapsed: 0.019 s
% 2.10/1.55 % (29815)Instructions burned: 65 (million)
% 2.10/1.55 % (29815)------------------------------
% 2.10/1.55 % (29815)------------------------------
% 2.10/1.55 % (29818)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_29522 for (2996ds/453Mi)
% 2.10/1.56 % (29819)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_29522 for (2995ds/71Mi)
% 2.10/1.56 % (29817)Instruction limit reached!
% 2.10/1.56 % (29817)------------------------------
% 2.10/1.56 % (29817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.56 % (29817)Termination reason: Unknown
% 2.10/1.56 % (29817)Termination phase: Property scanning
% 2.10/1.56
% 2.10/1.56 % (29817)Memory used [KB]: 2686
% 2.10/1.56 % (29817)Time elapsed: 0.030 s
% 2.10/1.56 % (29817)Instructions burned: 97 (million)
% 2.10/1.56 % (29817)------------------------------
% 2.10/1.56 % (29817)------------------------------
% 2.10/1.57 % (29804)Instruction limit reached!
% 2.10/1.57 % (29804)------------------------------
% 2.10/1.57 % (29804)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.57 % (29804)Termination reason: Unknown
% 2.10/1.57 % (29804)Termination phase: Saturation
% 2.10/1.57
% 2.10/1.57 % (29804)Memory used [KB]: 9466
% 2.10/1.57 % (29804)Time elapsed: 0.134 s
% 2.10/1.57 % (29804)Instructions burned: 450 (million)
% 2.10/1.57 % (29804)------------------------------
% 2.10/1.57 % (29804)------------------------------
% 2.10/1.57 % (29820)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_29522 for (2995ds/83Mi)
% 2.10/1.58 % (29821)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_29522 for (2995ds/11Mi)
% 2.10/1.58 % (29821)Instruction limit reached!
% 2.10/1.58 % (29821)------------------------------
% 2.10/1.58 % (29821)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.58 % (29821)Termination reason: Unknown
% 2.10/1.58 % (29821)Termination phase: shuffling
% 2.10/1.58
% 2.10/1.58 % (29821)Memory used [KB]: 1663
% 2.10/1.58 % (29821)Time elapsed: 0.004 s
% 2.10/1.58 % (29821)Instructions burned: 11 (million)
% 2.10/1.58 % (29821)------------------------------
% 2.10/1.58 % (29821)------------------------------
% 2.10/1.58 % (29819)Instruction limit reached!
% 2.10/1.58 % (29819)------------------------------
% 2.10/1.58 % (29819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.58 % (29819)Termination reason: Unknown
% 2.10/1.58 % (29819)Termination phase: Property scanning
% 2.10/1.58
% 2.10/1.58 % (29819)Memory used [KB]: 2558
% 2.10/1.58 % (29819)Time elapsed: 0.023 s
% 2.10/1.58 % (29819)Instructions burned: 71 (million)
% 2.10/1.58 % (29819)------------------------------
% 2.10/1.58 % (29819)------------------------------
% 2.10/1.59 % (29822)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_29522 for (2995ds/41Mi)
% 2.10/1.59 % (29823)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_29522 for (2995ds/6Mi)
% 2.10/1.59 % (29823)Instruction limit reached!
% 2.10/1.59 % (29823)------------------------------
% 2.10/1.59 % (29823)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.59 % (29823)Termination reason: Unknown
% 2.10/1.59 % (29823)Termination phase: shuffling
% 2.10/1.59
% 2.10/1.59 % (29823)Memory used [KB]: 1535
% 2.10/1.59 % (29823)Time elapsed: 0.003 s
% 2.10/1.59 % (29823)Instructions burned: 6 (million)
% 2.10/1.59 % (29823)------------------------------
% 2.10/1.59 % (29823)------------------------------
% 2.10/1.60 % (29820)Instruction limit reached!
% 2.10/1.60 % (29820)------------------------------
% 2.10/1.60 % (29820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.60 % (29820)Termination reason: Unknown
% 2.10/1.60 % (29820)Termination phase: Function definition elimination
% 2.10/1.60
% 2.10/1.60 % (29820)Memory used [KB]: 2430
% 2.10/1.60 % (29820)Time elapsed: 0.027 s
% 2.10/1.60 % (29820)Instructions burned: 86 (million)
% 2.10/1.60 % (29820)------------------------------
% 2.10/1.60 % (29820)------------------------------
% 2.10/1.60 % (29822)Instruction limit reached!
% 2.10/1.60 % (29822)------------------------------
% 2.10/1.60 % (29822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.60 % (29822)Termination reason: Unknown
% 2.10/1.60 % (29822)Termination phase: shuffling
% 2.10/1.60
% 2.10/1.60 % (29822)Memory used [KB]: 2302
% 2.10/1.60 % (29822)Time elapsed: 0.013 s
% 2.10/1.60 % (29822)Instructions burned: 43 (million)
% 2.10/1.60 % (29822)------------------------------
% 2.10/1.60 % (29822)------------------------------
% 2.10/1.60 % (29824)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_29522 for (2995ds/150Mi)
% 2.10/1.61 % (29825)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_29522 for (2995ds/343Mi)
% 2.10/1.61 % (29826)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_29522 for (2995ds/109Mi)
% 2.10/1.62 % (29807)Instruction limit reached!
% 2.10/1.62 % (29807)------------------------------
% 2.10/1.62 % (29807)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.62 % (29807)Termination reason: Unknown
% 2.10/1.62 % (29807)Termination phase: Saturation
% 2.10/1.62
% 2.10/1.62 % (29807)Memory used [KB]: 8955
% 2.10/1.62 % (29807)Time elapsed: 0.153 s
% 2.10/1.62 % (29807)Instructions burned: 507 (million)
% 2.10/1.62 % (29807)------------------------------
% 2.10/1.62 % (29807)------------------------------
% 2.10/1.63 % (29827)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_29522 for (2995ds/253Mi)
% 2.10/1.64 % (29826)Instruction limit reached!
% 2.10/1.64 % (29826)------------------------------
% 2.10/1.64 % (29826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.64 % (29826)Termination reason: Unknown
% 2.10/1.64 % (29826)Termination phase: Property scanning
% 2.10/1.64
% 2.10/1.64 % (29826)Memory used [KB]: 2686
% 2.10/1.64 % (29826)Time elapsed: 0.034 s
% 2.10/1.64 % (29826)Instructions burned: 112 (million)
% 2.10/1.64 % (29826)------------------------------
% 2.10/1.64 % (29826)------------------------------
% 2.10/1.65 % (29824)Instruction limit reached!
% 2.10/1.65 % (29824)------------------------------
% 2.10/1.65 % (29824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.65 % (29824)Termination reason: Unknown
% 2.10/1.65 % (29824)Termination phase: Saturation
% 2.10/1.65
% 2.10/1.65 % (29824)Memory used [KB]: 7419
% 2.10/1.65 % (29824)Time elapsed: 0.045 s
% 2.10/1.65 % (29824)Instructions burned: 152 (million)
% 2.10/1.65 % (29824)------------------------------
% 2.10/1.65 % (29824)------------------------------
% 2.10/1.65 % (29828)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_29522 for (2994ds/140Mi)
% 2.10/1.66 % (29829)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_29522 for (2994ds/614Mi)
% 2.10/1.66 % (29814)Instruction limit reached!
% 2.10/1.66 % (29814)------------------------------
% 2.10/1.66 % (29814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.66 % (29814)Termination reason: Unknown
% 2.10/1.66 % (29814)Termination phase: Saturation
% 2.10/1.66
% 2.10/1.66 % (29814)Memory used [KB]: 4733
% 2.10/1.66 % (29814)Time elapsed: 0.133 s
% 2.10/1.66 % (29814)Instructions burned: 435 (million)
% 2.10/1.66 % (29814)------------------------------
% 2.10/1.66 % (29814)------------------------------
% 2.10/1.67 % (29830)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_29522 for (2994ds/202Mi)
% 2.10/1.68 % (29818)Instruction limit reached!
% 2.10/1.68 % (29818)------------------------------
% 2.10/1.68 % (29818)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.68 % (29818)Termination reason: Unknown
% 2.10/1.68 % (29818)Termination phase: Saturation
% 2.10/1.68
% 2.10/1.68 % (29818)Memory used [KB]: 9083
% 2.10/1.68 % (29818)Time elapsed: 0.129 s
% 2.10/1.68 % (29818)Instructions burned: 453 (million)
% 2.10/1.68 % (29818)------------------------------
% 2.10/1.68 % (29818)------------------------------
% 2.10/1.69 % (29831)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_29522 for (2994ds/339Mi)
% 2.10/1.69 % (29828)Instruction limit reached!
% 2.10/1.69 % (29828)------------------------------
% 2.10/1.69 % (29828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.69 % (29828)Termination reason: Unknown
% 2.10/1.69 % (29828)Termination phase: Saturation
% 2.10/1.69
% 2.10/1.69 % (29828)Memory used [KB]: 7547
% 2.10/1.69 % (29828)Time elapsed: 0.043 s
% 2.10/1.69 % (29828)Instructions burned: 140 (million)
% 2.10/1.69 % (29828)------------------------------
% 2.10/1.69 % (29828)------------------------------
% 2.10/1.70 % (29827)Instruction limit reached!
% 2.10/1.70 % (29827)------------------------------
% 2.10/1.70 % (29827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.70 % (29827)Termination reason: Unknown
% 2.10/1.70 % (29827)Termination phase: Saturation
% 2.10/1.70
% 2.10/1.70 % (29827)Memory used [KB]: 8187
% 2.10/1.70 % (29827)Time elapsed: 0.078 s
% 2.10/1.70 % (29827)Instructions burned: 253 (million)
% 2.10/1.70 % (29827)------------------------------
% 2.10/1.70 % (29827)------------------------------
% 2.10/1.70 % (29832)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_29522 for (2994ds/283Mi)
% 2.10/1.71 % (29825)Instruction limit reached!
% 2.10/1.71 % (29825)------------------------------
% 2.10/1.71 % (29825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.71 % (29825)Termination reason: Unknown
% 2.10/1.71 % (29825)Termination phase: Saturation
% 2.10/1.71
% 2.10/1.71 % (29825)Memory used [KB]: 8187
% 2.10/1.71 % (29825)Time elapsed: 0.102 s
% 2.10/1.71 % (29825)Instructions burned: 345 (million)
% 2.10/1.71 % (29825)------------------------------
% 2.10/1.71 % (29825)------------------------------
% 2.10/1.71 % (29833)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_29522 for (2994ds/134Mi)
% 2.10/1.72 % (29834)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_29522 for (2994ds/298Mi)
% 2.10/1.73 % (29830)Instruction limit reached!
% 2.10/1.73 % (29830)------------------------------
% 2.10/1.73 % (29830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.73 % (29830)Termination reason: Unknown
% 2.10/1.73 % (29830)Termination phase: Saturation
% 2.10/1.73
% 2.10/1.73 % (29830)Memory used [KB]: 8187
% 2.10/1.73 % (29830)Time elapsed: 0.061 s
% 2.10/1.73 % (29830)Instructions burned: 203 (million)
% 2.10/1.73 % (29830)------------------------------
% 2.10/1.73 % (29830)------------------------------
% 2.10/1.74 % (29835)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_29522 for (2994ds/192Mi)
% 2.10/1.75 % (29833)Instruction limit reached!
% 2.10/1.75 % (29833)------------------------------
% 2.10/1.75 % (29833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.75 % (29833)Termination reason: Unknown
% 2.10/1.75 % (29833)Termination phase: Saturation
% 2.10/1.75
% 2.10/1.75 % (29833)Memory used [KB]: 7291
% 2.10/1.75 % (29833)Time elapsed: 0.040 s
% 2.10/1.75 % (29833)Instructions burned: 135 (million)
% 2.10/1.75 % (29833)------------------------------
% 2.10/1.75 % (29833)------------------------------
% 2.10/1.76 % (29836)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_29522 for (2993ds/258Mi)
% 2.10/1.79 % (29832)Instruction limit reached!
% 2.10/1.79 % (29832)------------------------------
% 2.10/1.79 % (29832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.79 % (29832)Termination reason: Unknown
% 2.10/1.79 % (29832)Termination phase: Saturation
% 2.10/1.79
% 2.10/1.79 % (29832)Memory used [KB]: 7931
% 2.10/1.79 % (29832)Time elapsed: 0.085 s
% 2.10/1.79 % (29832)Instructions burned: 283 (million)
% 2.10/1.79 % (29832)------------------------------
% 2.10/1.79 % (29832)------------------------------
% 2.10/1.79 % (29831)Instruction limit reached!
% 2.10/1.79 % (29831)------------------------------
% 2.10/1.79 % (29831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.79 % (29831)Termination reason: Unknown
% 2.10/1.79 % (29831)Termination phase: Saturation
% 2.10/1.79
% 2.10/1.79 % (29831)Memory used [KB]: 7803
% 2.10/1.79 % (29831)Time elapsed: 0.098 s
% 2.10/1.79 % (29831)Instructions burned: 341 (million)
% 2.10/1.79 % (29831)------------------------------
% 2.10/1.79 % (29831)------------------------------
% 2.10/1.80 % (29837)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_29522 for (2993ds/90Mi)
% 2.10/1.80 % (29835)Instruction limit reached!
% 2.10/1.80 % (29835)------------------------------
% 2.10/1.80 % (29835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.80 % (29835)Termination reason: Unknown
% 2.10/1.80 % (29835)Termination phase: Saturation
% 2.10/1.80
% 2.10/1.80 % (29835)Memory used [KB]: 3837
% 2.10/1.80 % (29835)Time elapsed: 0.058 s
% 2.10/1.80 % (29835)Instructions burned: 193 (million)
% 2.10/1.80 % (29835)------------------------------
% 2.10/1.80 % (29835)------------------------------
% 2.10/1.80 % (29838)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_29522 for (2993ds/231Mi)
% 2.10/1.80 % (29834)Instruction limit reached!
% 2.10/1.80 % (29834)------------------------------
% 2.10/1.80 % (29834)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.80 % (29834)Termination reason: Unknown
% 2.10/1.80 % (29834)Termination phase: Saturation
% 2.10/1.80
% 2.10/1.80 % (29834)Memory used [KB]: 9083
% 2.10/1.80 % (29834)Time elapsed: 0.085 s
% 2.10/1.80 % (29834)Instructions burned: 301 (million)
% 2.10/1.80 % (29834)------------------------------
% 2.10/1.80 % (29834)------------------------------
% 2.10/1.81 % (29839)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_29522 for (2993ds/458Mi)
% 2.10/1.81 % (29840)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_29522 for (2993ds/31Mi)
% 2.10/1.82 % (29840)Instruction limit reached!
% 2.10/1.82 % (29840)------------------------------
% 2.10/1.82 % (29840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.82 % (29840)Termination reason: Unknown
% 2.10/1.82 % (29840)Termination phase: shuffling
% 2.10/1.82
% 2.10/1.82 % (29840)Memory used [KB]: 2046
% 2.10/1.82 % (29840)Time elapsed: 0.011 s
% 2.10/1.82 % (29840)Instructions burned: 31 (million)
% 2.10/1.82 % (29840)------------------------------
% 2.10/1.82 % (29840)------------------------------
% 2.10/1.82 % (29837)Instruction limit reached!
% 2.10/1.82 % (29837)------------------------------
% 2.10/1.82 % (29837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.82 % (29837)Termination reason: Unknown
% 2.10/1.82 % (29837)Termination phase: Function definition elimination
% 2.10/1.82
% 2.10/1.82 % (29837)Memory used [KB]: 2686
% 2.10/1.82 % (29837)Time elapsed: 0.028 s
% 2.10/1.82 % (29837)Instructions burned: 90 (million)
% 2.10/1.82 % (29837)------------------------------
% 2.10/1.82 % (29837)------------------------------
% 2.10/1.83 % (29811)Instruction limit reached!
% 2.10/1.83 % (29811)------------------------------
% 2.10/1.83 % (29811)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.83 % (29811)Termination reason: Unknown
% 2.10/1.83 % (29811)Termination phase: Saturation
% 2.10/1.83
% 2.10/1.83 % (29811)Memory used [KB]: 8699
% 2.10/1.83 % (29811)Time elapsed: 0.332 s
% 2.10/1.83 % (29811)Instructions burned: 1087 (million)
% 2.10/1.83 % (29811)------------------------------
% 2.10/1.83 % (29811)------------------------------
% 2.10/1.83 % (29841)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_29522 for (2993ds/297Mi)
% 2.10/1.83 % (29842)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_29522 for (2993ds/987Mi)
% 2.10/1.84 % (29836)Instruction limit reached!
% 2.10/1.84 % (29836)------------------------------
% 2.10/1.84 % (29836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.84 % (29836)Termination reason: Unknown
% 2.10/1.84 % (29836)Termination phase: Saturation
% 2.10/1.84
% 2.10/1.84 % (29836)Memory used [KB]: 8571
% 2.10/1.84 % (29836)Time elapsed: 0.078 s
% 2.10/1.84 % (29836)Instructions burned: 260 (million)
% 2.10/1.84 % (29836)------------------------------
% 2.10/1.84 % (29836)------------------------------
% 2.10/1.84 % (29843)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_29522 for (2993ds/182Mi)
% 2.10/1.84 % (29829)Instruction limit reached!
% 2.10/1.84 % (29829)------------------------------
% 2.10/1.84 % (29829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.84 % (29829)Termination reason: Unknown
% 2.10/1.84 % (29829)Termination phase: Saturation
% 2.10/1.84
% 2.10/1.84 % (29829)Memory used [KB]: 9083
% 2.10/1.84 % (29829)Time elapsed: 0.185 s
% 2.10/1.84 % (29829)Instructions burned: 616 (million)
% 2.10/1.84 % (29829)------------------------------
% 2.10/1.84 % (29829)------------------------------
% 2.10/1.85 % (29844)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_29522 for (2993ds/182Mi)
% 2.10/1.85 % (29845)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_29522 for (2992ds/154Mi)
% 2.10/1.86 % (29838)Instruction limit reached!
% 2.10/1.86 % (29838)------------------------------
% 2.10/1.86 % (29838)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.86 % (29838)Termination reason: Unknown
% 2.10/1.86 % (29838)Termination phase: Saturation
% 2.10/1.86
% 2.10/1.86 % (29838)Memory used [KB]: 7675
% 2.10/1.86 % (29838)Time elapsed: 0.065 s
% 2.10/1.86 % (29838)Instructions burned: 232 (million)
% 2.10/1.86 % (29838)------------------------------
% 2.10/1.86 % (29838)------------------------------
% 2.10/1.87 % (29846)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_29522 for (2992ds/79Mi)
% 2.10/1.89 % (29843)Instruction limit reached!
% 2.10/1.89 % (29843)------------------------------
% 2.10/1.89 % (29843)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.89 % (29843)Termination reason: Unknown
% 2.10/1.89 % (29843)Termination phase: Saturation
% 2.10/1.89
% 2.10/1.89 % (29843)Memory used [KB]: 7419
% 2.10/1.89 % (29843)Time elapsed: 0.053 s
% 2.10/1.89 % (29843)Instructions burned: 182 (million)
% 2.10/1.89 % (29843)------------------------------
% 2.10/1.89 % (29843)------------------------------
% 2.10/1.89 % (29845)Instruction limit reached!
% 2.10/1.89 % (29845)------------------------------
% 2.10/1.89 % (29845)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.89 % (29845)Termination reason: Unknown
% 2.10/1.89 % (29845)Termination phase: Saturation
% 2.10/1.89
% 2.10/1.89 % (29845)Memory used [KB]: 3070
% 2.10/1.89 % (29845)Time elapsed: 0.044 s
% 2.10/1.89 % (29845)Instructions burned: 156 (million)
% 2.10/1.89 % (29845)------------------------------
% 2.10/1.89 % (29845)------------------------------
% 2.10/1.90 % (29846)Instruction limit reached!
% 2.10/1.90 % (29846)------------------------------
% 2.10/1.90 % (29846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.90 % (29846)Termination reason: Unknown
% 2.10/1.90 % (29846)Termination phase: Property scanning
% 2.10/1.90
% 2.10/1.90 % (29846)Memory used [KB]: 2558
% 2.10/1.90 % (29846)Time elapsed: 0.025 s
% 2.10/1.90 % (29846)Instructions burned: 79 (million)
% 2.10/1.90 % (29846)------------------------------
% 2.10/1.90 % (29846)------------------------------
% 2.10/1.90 % (29847)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_29522 for (2992ds/216Mi)
% 2.10/1.90 % (29844)Instruction limit reached!
% 2.10/1.90 % (29844)------------------------------
% 2.10/1.90 % (29844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.90 % (29844)Termination reason: Unknown
% 2.10/1.90 % (29844)Termination phase: Saturation
% 2.10/1.90
% 2.10/1.90 % (29844)Memory used [KB]: 7419
% 2.10/1.90 % (29844)Time elapsed: 0.057 s
% 2.10/1.90 % (29844)Instructions burned: 183 (million)
% 2.10/1.90 % (29844)------------------------------
% 2.10/1.90 % (29844)------------------------------
% 2.10/1.90 % (29848)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_29522 for (2992ds/114Mi)
% 2.10/1.91 % (29849)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_29522 for (2992ds/922Mi)
% 2.10/1.91 % (29850)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_29522 for (2992ds/182Mi)
% 2.10/1.92 % (29841)Instruction limit reached!
% 2.10/1.92 % (29841)------------------------------
% 2.10/1.92 % (29841)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.92 % (29841)Termination reason: Unknown
% 2.10/1.92 % (29841)Termination phase: Saturation
% 2.10/1.92
% 2.10/1.92 % (29841)Memory used [KB]: 8571
% 2.10/1.92 % (29841)Time elapsed: 0.086 s
% 2.10/1.92 % (29841)Instructions burned: 298 (million)
% 2.10/1.92 % (29841)------------------------------
% 2.10/1.92 % (29841)------------------------------
% 2.10/1.93 % (29851)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_29522 for (2992ds/164Mi)
% 2.10/1.94 % (29848)Instruction limit reached!
% 2.10/1.94 % (29848)------------------------------
% 2.10/1.94 % (29848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.94 % (29848)Termination reason: Unknown
% 2.10/1.94 % (29848)Termination phase: Property scanning
% 2.10/1.94
% 2.10/1.94 % (29848)Memory used [KB]: 2686
% 2.10/1.94 % (29848)Time elapsed: 0.035 s
% 2.10/1.94 % (29848)Instructions burned: 114 (million)
% 2.10/1.94 % (29848)------------------------------
% 2.10/1.94 % (29848)------------------------------
% 2.10/1.94 % (29839)Instruction limit reached!
% 2.10/1.94 % (29839)------------------------------
% 2.10/1.94 % (29839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.94 % (29839)Termination reason: Unknown
% 2.10/1.94 % (29839)Termination phase: Saturation
% 2.10/1.94
% 2.10/1.94 % (29839)Memory used [KB]: 8827
% 2.10/1.94 % (29839)Time elapsed: 0.137 s
% 2.10/1.94 % (29839)Instructions burned: 460 (million)
% 2.10/1.94 % (29839)------------------------------
% 2.10/1.94 % (29839)------------------------------
% 2.10/1.95 % (29852)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_29522 for (2992ds/50Mi)
% 2.10/1.95 % (29853)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_29522 for (2991ds/368Mi)
% 2.10/1.96 % (29852)Instruction limit reached!
% 2.10/1.96 % (29852)------------------------------
% 2.10/1.96 % (29852)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.96 % (29852)Termination reason: Unknown
% 2.10/1.96 % (29852)Termination phase: Property scanning
% 2.10/1.96
% 2.10/1.96 % (29852)Memory used [KB]: 2430
% 2.10/1.96 % (29852)Time elapsed: 0.016 s
% 2.10/1.96 % (29852)Instructions burned: 52 (million)
% 2.10/1.96 % (29852)------------------------------
% 2.10/1.96 % (29852)------------------------------
% 2.10/1.96 % (29847)Instruction limit reached!
% 2.10/1.96 % (29847)------------------------------
% 2.10/1.96 % (29847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.96 % (29847)Termination reason: Unknown
% 2.10/1.96 % (29847)Termination phase: Saturation
% 2.10/1.96
% 2.10/1.96 % (29847)Memory used [KB]: 7675
% 2.10/1.96 % (29847)Time elapsed: 0.062 s
% 2.10/1.96 % (29847)Instructions burned: 216 (million)
% 2.10/1.96 % (29847)------------------------------
% 2.10/1.96 % (29847)------------------------------
% 2.10/1.96 % (29850)Instruction limit reached!
% 2.10/1.96 % (29850)------------------------------
% 2.10/1.96 % (29850)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.96 % (29850)Termination reason: Unknown
% 2.10/1.96 % (29850)Termination phase: Saturation
% 2.10/1.96
% 2.10/1.96 % (29850)Memory used [KB]: 7675
% 2.10/1.96 % (29850)Time elapsed: 0.052 s
% 2.10/1.96 % (29850)Instructions burned: 182 (million)
% 2.10/1.96 % (29850)------------------------------
% 2.10/1.96 % (29850)------------------------------
% 2.10/1.97 % (29854)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_29522 for (2991ds/1021Mi)
% 2.10/1.97 % (29855)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_29522 for (2991ds/267Mi)
% 2.10/1.97 % (29856)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_29522 for (2991ds/204Mi)
% 2.10/1.97 % (29851)Instruction limit reached!
% 2.10/1.97 % (29851)------------------------------
% 2.10/1.97 % (29851)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.97 % (29851)Termination reason: Unknown
% 2.10/1.97 % (29851)Termination phase: Saturation
% 2.10/1.97
% 2.10/1.97 % (29851)Memory used [KB]: 2942
% 2.10/1.97 % (29851)Time elapsed: 0.048 s
% 2.10/1.97 % (29851)Instructions burned: 167 (million)
% 2.10/1.97 % (29851)------------------------------
% 2.10/1.97 % (29851)------------------------------
% 2.10/1.98 % (29857)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_29522 for (2991ds/26Mi)
% 2.10/1.99 % (29857)Instruction limit reached!
% 2.10/1.99 % (29857)------------------------------
% 2.10/1.99 % (29857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.99 % (29857)Termination reason: Unknown
% 2.10/1.99 % (29857)Termination phase: shuffling
% 2.10/1.99
% 2.10/1.99 % (29857)Memory used [KB]: 2046
% 2.10/1.99 % (29857)Time elapsed: 0.009 s
% 2.10/1.99 % (29857)Instructions burned: 28 (million)
% 2.10/1.99 % (29857)------------------------------
% 2.10/1.99 % (29857)------------------------------
% 2.10/2.00 % (29858)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_29522 for (2991ds/53Mi)
% 8.05/2.02 % (29858)Instruction limit reached!
% 8.05/2.02 % (29858)------------------------------
% 8.05/2.02 % (29858)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.05/2.02 % (29858)Termination reason: Unknown
% 8.05/2.02 % (29858)Termination phase: Property scanning
% 8.05/2.02
% 8.05/2.02 % (29858)Memory used [KB]: 2430
% 8.05/2.02 % (29858)Time elapsed: 0.016 s
% 8.05/2.02 % (29858)Instructions burned: 53 (million)
% 8.05/2.02 % (29858)------------------------------
% 8.05/2.02 % (29858)------------------------------
% 8.05/2.02 % (29859)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_29522 for (2991ds/4436Mi)
% 8.05/2.03 % (29856)Instruction limit reached!
% 8.05/2.03 % (29856)------------------------------
% 8.05/2.03 % (29856)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.05/2.03 % (29856)Termination reason: Unknown
% 8.05/2.03 % (29856)Termination phase: Saturation
% 8.05/2.03
% 8.05/2.03 % (29856)Memory used [KB]: 7675
% 8.05/2.03 % (29856)Time elapsed: 0.060 s
% 8.05/2.03 % (29856)Instructions burned: 204 (million)
% 8.05/2.03 % (29856)------------------------------
% 8.05/2.03 % (29856)------------------------------
% 8.05/2.04 % (29860)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_29522 for (2991ds/1539Mi)
% 8.05/2.05 % (29853)First to succeed.
% 8.05/2.05 % (29855)Instruction limit reached!
% 8.05/2.05 % (29855)------------------------------
% 8.05/2.05 % (29855)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.05/2.05 % (29855)Termination reason: Unknown
% 8.05/2.05 % (29855)Termination phase: Saturation
% 8.05/2.05
% 8.05/2.05 % (29855)Memory used [KB]: 7931
% 8.05/2.05 % (29855)Time elapsed: 0.081 s
% 8.05/2.05 % (29855)Instructions burned: 267 (million)
% 8.05/2.05 % (29855)------------------------------
% 8.05/2.05 % (29855)------------------------------
% 8.05/2.05 % (29853)Refutation found. Thanks to Tanya!
% 8.05/2.05 % SZS status Theorem for DTF2THF_29522
% 8.05/2.05 % SZS output start Proof for DTF2THF_29522
% 8.05/2.05 thf(type_def_5, type, itself: ($tType) > $tType).
% 8.05/2.05 thf(type_def_7, type, set: ($tType) > $tType).
% 8.05/2.05 thf(type_def_8, type, product_prod: ($tType * $tType) > $tType).
% 8.05/2.05 thf(type_def_9, type, coinductive_llist: ($tType) > $tType).
% 8.05/2.05 thf(type_def_10, type, product_unit: $tType).
% 8.05/2.05 thf(type_def_11, type, tLList446370796tllist: ($tType * $tType) > $tType).
% 8.05/2.05 thf(type_def_12, type, b: $tType).
% 8.05/2.05 thf(type_def_13, type, a: $tType).
% 8.05/2.05 thf(func_def_0, type, tLList446370796tllist: ($tType * $tType) > $tType).
% 8.05/2.05 thf(func_def_1, type, coinductive_llist: ($tType) > $tType).
% 8.05/2.05 thf(func_def_2, type, product_unit: $tType).
% 8.05/2.05 thf(func_def_3, type, product_prod: ($tType * $tType) > $tType).
% 8.05/2.05 thf(func_def_4, type, set: ($tType) > $tType).
% 8.05/2.05 thf(func_def_5, type, itself: ($tType) > $tType).
% 8.05/2.05 thf(func_def_6, type, b: $tType).
% 8.05/2.05 thf(func_def_7, type, a: $tType).
% 8.05/2.05 thf(func_def_8, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 8.05/2.05 thf(func_def_9, type, bNF_Gr: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > set @ product_prod @ X0 @ X1)).
% 8.05/2.05 thf(func_def_10, type, coinductive_lfilter: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 8.05/2.05 thf(func_def_11, type, coinductive_lfinite: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 8.05/2.05 thf(func_def_12, type, fun_reduction_pair: !>[X0: $tType]:(product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X0 @ X0 > $o)).
% 8.05/2.05 thf(func_def_13, type, fun_rp_inv_image: !>[X0: $tType, X1: $tType]:(product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X0 @ X0 > (X1 > X0) > product_prod @ set @ product_prod @ X1 @ X1 @ set @ product_prod @ X1 @ X1)).
% 8.05/2.05 thf(func_def_14, type, the: !>[X0: $tType]:((X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_15, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 8.05/2.05 thf(func_def_16, type, quotient: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > (X1 > X0) > (X0 > X1 > $o) > $o)).
% 8.05/2.05 thf(func_def_17, type, order_above: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 8.05/2.05 thf(func_def_18, type, order_aboveS: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 8.05/2.05 thf(func_def_19, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 8.05/2.05 thf(func_def_20, type, top_top: !>[X0: $tType]:(X0)).
% 8.05/2.05 thf(func_def_21, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 8.05/2.05 thf(func_def_22, type, product_Sigma: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ product_prod @ X0 @ X1)).
% 8.05/2.05 thf(func_def_23, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 8.05/2.05 thf(func_def_24, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 8.05/2.05 thf(func_def_25, type, product_rec_bool: !>[X0: $tType]:(X0 > X0 > $o > X0)).
% 8.05/2.05 thf(func_def_26, type, product_rec_set_bool: !>[X0: $tType]:(X0 > X0 > $o > X0 > $o)).
% 8.05/2.05 thf(func_def_27, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 8.05/2.05 thf(func_def_28, type, product_rec_set_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2 > $o)).
% 8.05/2.05 thf(func_def_29, type, product_rec_set_unit: !>[X0: $tType]:(X0 > product_unit > X0 > $o)).
% 8.05/2.05 thf(func_def_30, type, product_rec_unit: !>[X0: $tType]:(X0 > product_unit > X0)).
% 8.05/2.05 thf(func_def_31, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 8.05/2.05 thf(func_def_32, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 8.05/2.05 thf(func_def_33, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 8.05/2.05 thf(func_def_34, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 8.05/2.05 thf(func_def_35, type, product_product: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ product_prod @ X0 @ X1)).
% 8.05/2.05 thf(func_def_36, type, product_scomp: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > product_prod @ X1 @ X2) > (X1 > X2 > X3) > X0 > X3)).
% 8.05/2.05 thf(func_def_37, type, type: !>[X0: $tType]:(itself @ X0)).
% 8.05/2.05 thf(func_def_38, type, conversep: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X1 > X0 > $o)).
% 8.05/2.05 thf(func_def_39, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 8.05/2.05 thf(func_def_40, type, inv_imagep: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > X1 > X1 > $o)).
% 8.05/2.05 thf(func_def_41, type, refl_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 8.05/2.05 thf(func_def_42, type, reflp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 8.05/2.05 thf(func_def_43, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 8.05/2.05 thf(func_def_44, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 8.05/2.05 thf(func_def_45, type, tLList47617868tllist: !>[X0: $tType, X1: $tType]:(product_prod @ coinductive_llist @ X0 @ X1 > tLList446370796tllist @ X0 @ X1 > $o)).
% 8.05/2.05 thf(func_def_46, type, tLList798109904tllist: !>[X0: $tType, X1: $tType]:(tLList446370796tllist @ X0 @ X1 > coinductive_llist @ X0)).
% 8.05/2.05 thf(func_def_47, type, tLList1825092077minal0: !>[X0: $tType]:(X0)).
% 8.05/2.05 thf(func_def_48, type, tLList200813139tllist: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > tLList446370796tllist @ X2 @ X0 > X1) > tLList446370796tllist @ X2 @ X0 > X1)).
% 8.05/2.05 thf(func_def_49, type, tLList2110128105rminal: !>[X0: $tType, X1: $tType]:(tLList446370796tllist @ X0 @ X1 > X1)).
% 8.05/2.05 thf(func_def_50, type, tLList1672613558_llist: !>[X0: $tType, X1: $tType]:(X0 > coinductive_llist @ X1 > tLList446370796tllist @ X1 @ X0)).
% 8.05/2.05 thf(func_def_51, type, lex_prod: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 8.05/2.05 thf(func_def_52, type, same_fst: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > set @ product_prod @ X1 @ X1) > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 8.05/2.05 thf(func_def_53, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 8.05/2.05 thf(func_def_54, type, b2: b).
% 8.05/2.05 thf(func_def_55, type, fun: a > $o).
% 8.05/2.05 thf(func_def_56, type, prod1: product_prod @ coinductive_llist @ a @ b).
% 8.05/2.05 thf(func_def_57, type, prod2: product_prod @ coinductive_llist @ a @ b).
% 8.05/2.05 thf(func_def_75, type, sK0: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > set @ X1)).
% 8.05/2.05 thf(func_def_76, type, sK1: !>[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)).
% 8.05/2.05 thf(func_def_77, type, sK2: !>[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)).
% 8.05/2.05 thf(func_def_78, type, sK3: !>[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)).
% 8.05/2.05 thf(func_def_79, type, sK4: !>[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)).
% 8.05/2.05 thf(func_def_80, type, sK5: !>[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)).
% 8.05/2.05 thf(func_def_81, type, sK6: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > X1) > set @ X0 > X0)).
% 8.05/2.05 thf(func_def_82, type, sK7: !>[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)).
% 8.05/2.05 thf(func_def_83, type, sK8: !>[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)).
% 8.05/2.05 thf(func_def_84, type, sK9: !>[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)).
% 8.05/2.05 thf(func_def_85, type, sK10: !>[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)).
% 8.05/2.05 thf(func_def_86, type, sK11: !>[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)).
% 8.05/2.05 thf(func_def_87, type, sK12: !>[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)).
% 8.05/2.05 thf(func_def_88, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ X2 > (X0 > X2 > X1) > (X1 > $o) > X0)).
% 8.05/2.05 thf(func_def_89, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ X2 > (X0 > X2 > X1) > (X1 > $o) > X2)).
% 8.05/2.05 thf(func_def_90, type, sK15: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 8.05/2.05 thf(func_def_91, type, sK16: !>[X0: $tType, X1: $tType]:((X1 > X0) > X0 > X1)).
% 8.05/2.05 thf(func_def_92, type, sK17: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_93, type, sK18: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_94, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 8.05/2.05 thf(func_def_95, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 8.05/2.05 thf(func_def_96, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 8.05/2.05 thf(func_def_97, type, sK22: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 8.05/2.05 thf(func_def_98, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 8.05/2.05 thf(func_def_99, type, sK24: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_100, type, sK25: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > product_prod @ X1 @ X0 > X1)).
% 8.05/2.05 thf(func_def_101, type, sK26: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > product_prod @ X1 @ X0 > X0)).
% 8.05/2.05 thf(func_def_102, type, sK27: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 8.05/2.05 thf(func_def_103, type, sK28: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 8.05/2.05 thf(func_def_104, type, sK29: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_105, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X1 > set @ X0) > X0 > product_prod @ X2 @ X1 > X2)).
% 8.05/2.05 thf(func_def_106, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X1 > set @ X0) > X0 > product_prod @ X2 @ X1 > X1)).
% 8.05/2.05 thf(func_def_107, type, sK32: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_108, type, sK33: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_109, type, sK34: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_110, type, sK35: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 8.05/2.05 thf(func_def_111, type, sK36: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 8.05/2.05 thf(func_def_112, type, sK37: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > X0)).
% 8.05/2.05 thf(func_def_113, type, sK38: !>[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)).
% 8.05/2.05 thf(func_def_114, type, sK39: !>[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)).
% 8.05/2.05 thf(func_def_115, type, sK40: !>[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)).
% 8.05/2.05 thf(func_def_116, type, sK41: !>[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)).
% 8.05/2.05 thf(func_def_117, type, sK42: !>[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)).
% 8.05/2.05 thf(func_def_118, type, sK43: !>[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)).
% 8.05/2.05 thf(func_def_119, type, sK44: !>[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)).
% 8.05/2.05 thf(func_def_120, type, sK45: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > set @ X0 > set @ X1 > X1)).
% 8.05/2.05 thf(func_def_121, type, sK46: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > set @ X0 > set @ X1 > X0)).
% 8.05/2.05 thf(func_def_122, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ X2 > (X0 > X2 > set @ X1) > X1 > X0)).
% 8.05/2.05 thf(func_def_123, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ X2 > (X0 > X2 > set @ X1) > X1 > X2)).
% 8.05/2.05 thf(func_def_124, type, sK49: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > set @ X0 > X0)).
% 8.05/2.05 thf(func_def_125, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X1 > X0) > (X0 > $o) > (X1 > X2 > $o) > X2)).
% 8.05/2.05 thf(func_def_126, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X1 > X0) > (X0 > $o) > (X1 > X2 > $o) > X1)).
% 8.05/2.05 thf(func_def_127, type, sK52: !>[X0: $tType, X1: $tType]:((X0 > set @ X1) > set @ X0 > product_prod @ X0 @ X1 > X0)).
% 8.05/2.05 thf(func_def_128, type, sK53: !>[X0: $tType, X1: $tType]:((X0 > set @ X1) > set @ X0 > product_prod @ X0 @ X1 > X1)).
% 8.05/2.05 thf(func_def_129, type, sK54: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 8.05/2.05 thf(func_def_130, type, sK55: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 8.05/2.05 thf(func_def_131, type, sK56: !>[X0: $tType, X1: $tType]:((X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_132, type, sK57: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > (X0 > X1 > $o) > X0)).
% 8.05/2.05 thf(func_def_133, type, sK58: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > (X0 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_134, type, sK59: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_135, type, sK60: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_136, type, sK61: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 8.05/2.05 thf(func_def_137, type, sK62: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_138, type, sK63: !>[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)).
% 8.05/2.05 thf(func_def_139, type, sK64: !>[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)).
% 8.05/2.05 thf(func_def_140, type, sK65: !>[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)).
% 8.05/2.05 thf(func_def_141, type, sK66: !>[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)).
% 8.05/2.05 thf(func_def_142, type, sK67: !>[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)).
% 8.05/2.05 thf(func_def_143, type, sK68: !>[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)).
% 8.05/2.05 thf(func_def_144, type, sK69: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 8.05/2.05 thf(func_def_145, type, sK70: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 8.05/2.05 thf(func_def_146, type, sK71: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 8.05/2.05 thf(func_def_147, type, sK72: !>[X0: $tType, X1: $tType]:((X1 > X1 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_148, type, sK73: !>[X0: $tType, X1: $tType]:((X1 > X1 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_149, type, sK74: !>[X0: $tType, X1: $tType]:((X1 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_150, type, sK75: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_151, type, sK76: !>[X0: $tType, X1: $tType]:((X1 > $o) > (X0 > X1) > (X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_152, type, sK77: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(set @ product_prod @ X3 @ X2 > (X2 > X0) > (X3 > X1) > product_prod @ X1 @ X0 > X3)).
% 8.05/2.05 thf(func_def_153, type, sK78: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(set @ product_prod @ X3 @ X2 > (X2 > X0) > (X3 > X1) > product_prod @ X1 @ X0 > X2)).
% 8.05/2.05 thf(func_def_154, type, sK79: !>[X0: $tType, X1: $tType]:((X1 > set @ X0) > (X1 > set @ X0) > set @ X1 > X1)).
% 8.05/2.05 thf(func_def_155, type, sK80: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X1 > X0) > (X2 > X1 > X0) > product_prod @ X2 @ X1 > X2)).
% 8.05/2.05 thf(func_def_156, type, sK81: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X1 > X0) > (X2 > X1 > X0) > product_prod @ X2 @ X1 > X1)).
% 8.05/2.05 thf(func_def_157, type, sK82: !>[X0: $tType, X1: $tType, X2: $tType]:(X2 > (X0 > X1 > X2 > $o) > product_prod @ X0 @ X1 > X0)).
% 8.05/2.05 thf(func_def_158, type, sK83: !>[X0: $tType, X1: $tType, X2: $tType]:(X2 > (X0 > X1 > X2 > $o) > product_prod @ X0 @ X1 > X1)).
% 8.05/2.05 thf(func_def_159, type, sK84: !>[X0: $tType, X1: $tType]:((tLList446370796tllist @ X1 @ X0 > $o) > product_prod @ coinductive_llist @ X1 @ X0)).
% 8.05/2.05 thf(func_def_160, type, sK85: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 8.05/2.05 thf(func_def_161, type, sK86: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 8.05/2.05 thf(func_def_162, type, sK87: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 8.05/2.05 thf(func_def_163, type, sK88: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 8.05/2.05 thf(func_def_164, type, sK89: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 8.05/2.05 thf(func_def_165, type, sK90: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_166, type, sK91: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ X1 > X0) > (X2 > X1 > X0) > X2)).
% 8.05/2.05 thf(func_def_167, type, sK92: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ X1 > X0) > (X2 > X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_168, type, sK93: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 8.05/2.05 thf(func_def_169, type, sK94: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 8.05/2.05 thf(func_def_170, type, sK95: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_171, type, sK96: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_172, type, sK97: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 8.05/2.05 thf(func_def_173, type, sK98: !>[X0: $tType, X1: $tType]:((X1 > set @ X0) > (X1 > set @ X0) > set @ X1 > X1)).
% 8.05/2.05 thf(func_def_174, type, sK99: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > X1)).
% 8.05/2.05 thf(func_def_175, type, sK100: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 8.05/2.05 thf(func_def_176, type, sK101: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_177, type, sK102: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_178, type, sK103: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 8.05/2.05 thf(func_def_179, type, sK104: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 8.05/2.05 thf(func_def_180, type, sK105: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 8.05/2.05 thf(func_def_181, type, sK106: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 8.05/2.05 thf(func_def_182, type, sK107: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_183, type, sK108: !>[X0: $tType, X1: $tType, X2: $tType]:(X2 > product_prod @ X1 @ X0 > (X1 > X0 > X2 > $o) > X1)).
% 8.05/2.05 thf(func_def_184, type, sK109: !>[X0: $tType, X1: $tType, X2: $tType]:(X2 > product_prod @ X1 @ X0 > (X1 > X0 > X2 > $o) > X0)).
% 8.05/2.05 thf(func_def_185, type, sK110: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > X1) > (X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_186, type, sK111: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_187, type, sK112: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_188, type, sK113: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_189, type, sK114: !>[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)).
% 8.05/2.05 thf(func_def_190, type, sK115: !>[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)).
% 8.05/2.05 thf(func_def_191, type, sK116: !>[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)).
% 8.05/2.05 thf(func_def_192, type, sK117: !>[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)).
% 8.05/2.05 thf(func_def_193, type, sK118: !>[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)).
% 8.05/2.05 thf(func_def_194, type, sK119: !>[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)).
% 8.05/2.05 thf(func_def_195, type, sK120: !>[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)).
% 8.05/2.05 thf(func_def_196, type, sK121: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_197, type, sK122: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_198, type, sK123: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_199, type, sK124: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_200, type, sK125: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_202, type, sK127: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X1)).
% 8.05/2.05 thf(func_def_203, type, sK128: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_204, type, sK129: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_205, type, sK130: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_206, type, sK131: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_207, type, sK132: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_208, type, sK133: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X1)).
% 8.05/2.05 thf(func_def_209, type, sK134: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_210, type, sK135: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_211, type, sK136: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_212, type, sK137: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_213, type, sK138: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_214, type, sK139: !>[X0: $tType, X1: $tType]:(product_prod @ X1 @ X0 > (X1 > $o) > X0)).
% 8.05/2.05 thf(func_def_215, type, sK140: !>[X0: $tType, X1: $tType]:(product_prod @ X1 @ X0 > (X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_216, type, sK141: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_217, type, sK142: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_218, type, sK143: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_219, type, sK144: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_220, type, sK145: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_221, type, sK146: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_222, type, sK147: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_223, type, sK148: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_224, type, sK149: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_225, type, sK150: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_226, type, sK151: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_227, type, sK152: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_228, type, sK153: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_229, type, sK154: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_230, type, sK155: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_231, type, sK156: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_232, type, sK157: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_233, type, sK158: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_234, type, sK159: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_235, type, sK160: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_236, type, sK161: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_237, type, sK162: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_238, type, sK163: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_239, type, sK164: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_240, type, sK165: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > X0)).
% 8.05/2.05 thf(func_def_241, type, sK166: !>[X0: $tType, X1: $tType]:(set @ X1 > set @ X0 > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_242, type, sK167: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_243, type, sK168: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X1)).
% 8.05/2.05 thf(func_def_244, type, sK169: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_245, type, sK170: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_246, type, sK171: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_247, type, sK172: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_248, type, sK173: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_249, type, sK174: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_250, type, sK175: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_251, type, sK176: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_252, type, sK177: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_253, type, sK178: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_254, type, sK179: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_255, type, sK180: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_256, type, sK181: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X1)).
% 8.05/2.05 thf(func_def_257, type, sK182: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_258, type, sK183: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_259, type, sK184: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_260, type, sK185: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_261, type, sK186: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_262, type, sK187: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_263, type, sK188: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_264, type, sK189: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X1)).
% 8.05/2.05 thf(func_def_265, type, sK190: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_266, type, sK191: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_267, type, sK192: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_268, type, sK193: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_269, type, sK194: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_270, type, sK195: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_271, type, sK196: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_272, type, sK197: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_273, type, sK198: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_274, type, sK199: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_275, type, sK200: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_276, type, sK201: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_277, type, sK202: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_278, type, sK203: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_279, type, sK204: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_280, type, sK205: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_281, type, sK206: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_282, type, sK207: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_283, type, sK208: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_284, type, sK209: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_285, type, sK210: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_286, type, sK211: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_287, type, sK212: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_288, type, sK213: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_289, type, sK214: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_290, type, sK215: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_291, type, sK216: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_292, type, sK217: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_293, type, sK218: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_294, type, sK219: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_295, type, sK220: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_296, type, sK221: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > X0)).
% 8.05/2.05 thf(func_def_297, type, sK222: !>[X0: $tType, X1: $tType]:(product_prod @ X1 @ X0 > X0 > X1)).
% 8.05/2.05 thf(func_def_298, type, sK223: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0 > X1)).
% 8.05/2.05 thf(func_def_299, type, sK224: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_300, type, sK225: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X1)).
% 8.05/2.05 thf(func_def_301, type, sK226: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_302, type, sK227: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_303, type, sK228: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_304, type, sK229: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_305, type, sK230: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_306, type, sK231: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_307, type, sK232: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_308, type, sK233: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > X0)).
% 8.05/2.05 thf(func_def_309, type, sK234: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > set @ X0 > set @ X1)).
% 8.05/2.05 thf(func_def_310, type, sK235: !>[X0: $tType]:(set @ X0 > (X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_311, type, sK236: !>[X0: $tType, X1: $tType]:(product_prod @ X1 @ X0 > (X1 > $o) > X0)).
% 8.05/2.05 thf(func_def_312, type, sK237: !>[X0: $tType, X1: $tType]:(product_prod @ X1 @ X0 > (X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_313, type, sK238: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_314, type, sK239: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_315, type, sK240: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_316, type, sK241: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_317, type, sK242: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_318, type, sK243: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_319, type, sK244: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_320, type, sK245: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_321, type, sK246: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_322, type, sK247: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X0 > X1)).
% 8.05/2.05 thf(func_def_323, type, sK248: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_324, type, sK249: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 8.05/2.05 thf(func_def_325, type, sK250: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > X0)).
% 8.05/2.05 thf(func_def_326, type, sK251: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X1)).
% 8.05/2.05 thf(func_def_327, type, sK252: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_328, type, sK253: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_329, type, sK254: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_330, type, sK255: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_331, type, sK256: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_332, type, sK257: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_333, type, sK258: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_334, type, sK259: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_335, type, sK260: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_336, type, sK261: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_337, type, sK262: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_338, type, sK263: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_339, type, sK264: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > X0)).
% 8.05/2.05 thf(func_def_340, type, sK265: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_341, type, sK266: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_342, type, sK267: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_343, type, sK268: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_344, type, sK269: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_345, type, sK270: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_346, type, sK271: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_347, type, sK272: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0)).
% 8.05/2.05 thf(func_def_348, type, sK273: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0)).
% 8.05/2.05 thf(func_def_349, type, sK274: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1)).
% 8.05/2.05 thf(func_def_350, type, sK275: !>[X0: $tType, X1: $tType]:(X1 > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_351, type, sK276: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_352, type, sK277: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_353, type, sK278: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_354, type, sK279: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_355, type, sK280: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_356, type, sK281: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_357, type, sK282: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_358, type, sK283: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_359, type, sK284: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_360, type, sK285: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_361, type, sK286: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_362, type, sK287: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_363, type, sK288: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_364, type, sK289: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_365, type, sK290: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_366, type, sK291: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X0)).
% 8.05/2.05 thf(func_def_367, type, sK292: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_368, type, sK293: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X0 > X1) > X0)).
% 8.05/2.05 thf(func_def_369, type, sK294: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > (X1 > X1 > $o) > X1)).
% 8.05/2.05 thf(func_def_370, type, sK295: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0) > X1)).
% 8.05/2.05 thf(func_def_371, type, ph296: !>[X0: $tType]:(X0)).
% 8.05/2.05 thf(func_def_372, type, sK297: b > b).
% 8.05/2.05 thf(func_def_373, type, sK298: b > b).
% 8.05/2.05 thf(func_def_374, type, sK299: b).
% 8.05/2.05 thf(f3257,plain,(
% 8.05/2.05 $false),
% 8.05/2.05 inference(avatar_sat_refutation,[status(thm)],[f2913,f2922,f2941,f2946,f2947,f3030,f3119,f3130,f3233,f3252,f3255,f3256])).
% 8.05/2.05 thf(f3256,plain,(
% 8.05/2.05 ((product_fst @ coinductive_llist @ a @ b @ prod2) != (product_fst @ coinductive_llist @ a @ b @ prod1)) | ((^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0)))) != (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))) | ((product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))) = (product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0))))))))),
% 8.05/2.05 introduced(definition,[],[theory_tautology_sat_conflict])).
% 8.05/2.05 thf(f3255,plain,(
% 8.05/2.05 ((product_fst @ coinductive_llist @ a @ b @ prod2) != (product_fst @ coinductive_llist @ a @ b @ prod1)) | ($false != (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | ($true != (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) | ($true = $false)),
% 8.05/2.05 introduced(definition,[],[theory_tautology_sat_conflict])).
% 8.05/2.05 thf(f3252,plain,(
% 8.05/2.05 ~spl126_22 | spl126_10 | spl126_15 | spl126_21),
% 8.05/2.05 inference(avatar_split_clause,[status(thm)],[f3164,f3127,f3096,f3010,f3230])).
% 8.05/2.05 thf(f3230,definition,(
% 8.05/2.05 spl126_22 <=> (b2 = sK299)),
% 8.05/2.05 introduced(definition,[new_symbols(naming,[spl126_22])],[avatar_definition])).
% 8.05/2.05 thf(f3010,definition,(
% 8.05/2.05 spl126_10 <=> ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)))),
% 8.05/2.05 introduced(definition,[new_symbols(naming,[spl126_10])],[avatar_definition])).
% 8.05/2.05 thf(f3096,definition,(
% 8.05/2.05 spl126_15 <=> ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)))),
% 8.05/2.05 introduced(definition,[new_symbols(naming,[spl126_15])],[avatar_definition])).
% 8.05/2.05 thf(f3127,definition,(
% 8.05/2.05 spl126_21 <=> ((^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0)))) = (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0)))))),
% 8.05/2.05 introduced(definition,[new_symbols(naming,[spl126_21])],[avatar_definition])).
% 8.05/2.05 thf(f3164,plain,(
% 8.05/2.05 ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) | (b2 != sK299) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | spl126_21),
% 8.05/2.05 inference(duplicate_literal_removal,[status(thm)],[f3163])).
% 8.05/2.05 thf(f3163,plain,(
% 8.05/2.05 ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | (b2 != sK299) | (b2 != sK299) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) | spl126_21),
% 8.05/2.05 inference(equality_proxy_clausification,[status(thm)],[f3160])).
% 8.05/2.05 thf(f3160,plain,(
% 8.05/2.05 (b2 != sK299) | ((b2 = sK299) = $false) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) | spl126_21),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f3159])).
% 8.05/2.05 thf(f3159,plain,(
% 8.05/2.05 (b2 != sK299) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | ($false = ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = sK299))) | spl126_21),
% 8.05/2.05 inference(equality_proxy_clausification,[status(thm)],[f3158])).
% 8.05/2.05 thf(f3158,plain,(
% 8.05/2.05 ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | ((b2 = sK299) = $false) | ($false = ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = sK299))) | spl126_21),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f3140])).
% 8.05/2.05 thf(f3140,plain,(
% 8.05/2.05 (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = sK299)) = $false) | ((b2 = sK299) = $false) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | ($false = ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = sK299))) | spl126_21),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f3139])).
% 8.05/2.05 thf(f3139,plain,(
% 8.05/2.05 (((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = sK299)) = $false) | (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = sK299)) = $false) | ($false = ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = sK299))) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | spl126_21),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f3137])).
% 8.05/2.05 thf(f3137,plain,(
% 8.05/2.05 ($false = ((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = sK299))) | (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = sK299)) = $false) | (((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = sK299)) = $false) | ($false = ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = sK299))) | spl126_21),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f3136])).
% 8.05/2.05 thf(f3136,plain,(
% 8.05/2.05 ($false = (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = sK299)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = sK299)))) | ($false = ((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = sK299))) | ($false = ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = sK299))) | spl126_21),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f3135])).
% 8.05/2.05 thf(f3135,plain,(
% 8.05/2.05 ($false = (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = sK299)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = sK299)))) | ($false = (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = sK299)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = sK299)))) | spl126_21),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f3133])).
% 8.05/2.05 thf(f3133,plain,(
% 8.05/2.05 ((((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = sK299)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = sK299))) != (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = sK299)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = sK299)))) | spl126_21),
% 8.05/2.05 inference(beta_eta_normalization,[status(thm)],[f3131])).
% 8.05/2.05 thf(f3131,plain,(
% 8.05/2.05 (((^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0)))) @ sK299) != ((^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0)))) @ sK299)) | spl126_21),
% 8.05/2.05 inference(negative_extensionality,[status(thm)],[f3129])).
% 8.05/2.05 thf(f3129,plain,(
% 8.05/2.05 ((^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0)))) != (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))) | spl126_21),
% 8.05/2.05 inference(avatar_component_clause,[status(thm)],[f3127])).
% 8.05/2.05 thf(f3233,plain,(
% 8.05/2.05 spl126_15 | spl126_10 | spl126_22 | spl126_21),
% 8.05/2.05 inference(avatar_split_clause,[status(thm)],[f3228,f3127,f3230,f3010,f3096])).
% 8.05/2.05 thf(f3228,plain,(
% 8.05/2.05 ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) | (b2 = sK299) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | spl126_21),
% 8.05/2.05 inference(duplicate_literal_removal,[status(thm)],[f3227])).
% 8.05/2.05 thf(f3227,plain,(
% 8.05/2.05 (b2 = sK299) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) | (b2 = sK299) | spl126_21),
% 8.05/2.05 inference(equality_proxy_clausification,[status(thm)],[f3226])).
% 8.05/2.05 thf(f3226,plain,(
% 8.05/2.05 ($true = (b2 = sK299)) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) | (b2 = sK299) | spl126_21),
% 8.05/2.05 inference(not_proxy_clausification,[status(thm)],[f3225])).
% 8.05/2.05 thf(f3225,plain,(
% 8.05/2.05 ($false = (~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)))) | ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | ($true = (b2 = sK299)) | (b2 = sK299) | spl126_21),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f3224])).
% 8.05/2.05 thf(f3224,plain,(
% 8.05/2.05 ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | (b2 = sK299) | ($true = ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = sK299))) | spl126_21),
% 8.05/2.05 inference(equality_proxy_clausification,[status(thm)],[f3223])).
% 8.05/2.05 thf(f3223,plain,(
% 8.05/2.05 ($true = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | ($true = (b2 = sK299)) | ($true = ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = sK299))) | spl126_21),
% 8.05/2.05 inference(not_proxy_clausification,[status(thm)],[f3222])).
% 8.05/2.05 thf(f3222,plain,(
% 8.05/2.05 ($false = (~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)))) | ($true = (b2 = sK299)) | ($true = ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = sK299))) | spl126_21),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f3215])).
% 8.05/2.05 thf(f3215,plain,(
% 8.05/2.05 ($true = ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = sK299))) | ($true = ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = sK299))) | spl126_21),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f3202])).
% 8.05/2.05 thf(f3202,plain,(
% 8.05/2.05 ($true = (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = sK299)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = sK299)))) | ($true = ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = sK299))) | spl126_21),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f3134])).
% 8.05/2.05 thf(f3134,plain,(
% 8.05/2.05 ($true = (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = sK299)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = sK299)))) | ($true = (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = sK299)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = sK299)))) | spl126_21),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f3133])).
% 8.05/2.05 thf(f3130,plain,(
% 8.05/2.05 ~spl126_21 | ~spl126_9 | spl126_4),
% 8.05/2.05 inference(avatar_split_clause,[status(thm)],[f3123,f2919,f2943,f3127])).
% 8.05/2.05 thf(f2943,definition,(
% 8.05/2.05 spl126_9 <=> ((product_fst @ coinductive_llist @ a @ b @ prod2) = (product_fst @ coinductive_llist @ a @ b @ prod1))),
% 8.05/2.05 introduced(definition,[new_symbols(naming,[spl126_9])],[avatar_definition])).
% 8.05/2.05 thf(f2919,definition,(
% 8.05/2.05 spl126_4 <=> ((product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0))))))) = (product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))))),
% 8.05/2.05 introduced(definition,[new_symbols(naming,[spl126_4])],[avatar_definition])).
% 8.05/2.05 thf(f3123,plain,(
% 8.05/2.05 ((^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0)))) != (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))) | ((product_fst @ coinductive_llist @ a @ b @ prod2) != (product_fst @ coinductive_llist @ a @ b @ prod1)) | spl126_4),
% 8.05/2.05 inference(equality_resolution,[status(thm)],[f2921])).
% 8.05/2.05 thf(f2921,plain,(
% 8.05/2.05 ((product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0))))))) != (product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0)))))))) | spl126_4),
% 8.05/2.05 inference(avatar_component_clause,[status(thm)],[f2919])).
% 8.05/2.05 thf(f3119,plain,(
% 8.05/2.05 ((product_snd @ coinductive_llist @ a @ b @ prod1) != (product_snd @ coinductive_llist @ a @ b @ prod2)) | ((product_fst @ coinductive_llist @ a @ b @ prod2) != (product_fst @ coinductive_llist @ a @ b @ prod1)) | ((product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0))))))) = (product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))))),
% 8.05/2.05 introduced(definition,[],[theory_tautology_sat_conflict])).
% 8.05/2.05 thf(f3030,plain,(
% 8.05/2.05 ($true != (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | ($false != (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | ($true = $false)),
% 8.05/2.05 introduced(definition,[],[theory_tautology_sat_conflict])).
% 8.05/2.05 thf(f2947,plain,(
% 8.05/2.05 ((product_snd @ coinductive_llist @ a @ b @ prod1) != (product_snd @ coinductive_llist @ a @ b @ prod2)) | ((product_fst @ coinductive_llist @ a @ b @ prod2) != (product_fst @ coinductive_llist @ a @ b @ prod1)) | ((product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))) = (product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0))))))))),
% 8.05/2.05 introduced(definition,[],[theory_tautology_sat_conflict])).
% 8.05/2.05 thf(f2946,plain,(
% 8.05/2.05 spl126_9),
% 8.05/2.05 inference(avatar_split_clause,[status(thm)],[f1672,f2943])).
% 8.05/2.05 thf(f1672,plain,(
% 8.05/2.05 ((product_fst @ coinductive_llist @ a @ b @ prod2) = (product_fst @ coinductive_llist @ a @ b @ prod1))),
% 8.05/2.05 inference(equality_proxy_clausification,[status(thm)],[f1671])).
% 8.05/2.05 thf(f1671,plain,(
% 8.05/2.05 ($true = ((product_fst @ coinductive_llist @ a @ b @ prod1) = (product_fst @ coinductive_llist @ a @ b @ prod2)))),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f1669])).
% 8.05/2.05 thf(f1669,plain,(
% 8.05/2.05 ((((product_fst @ coinductive_llist @ a @ b @ prod1) = (product_fst @ coinductive_llist @ a @ b @ prod2)) & ((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = (product_snd @ coinductive_llist @ a @ b @ prod2)))) = $true)),
% 8.05/2.05 inference(beta_eta_normalization,[status(thm)],[f1397])).
% 8.05/2.05 thf(f1397,plain,(
% 8.05/2.05 ($true = ((^[Y0 : coinductive_llist @ a > b > product_prod @ coinductive_llist @ a @ b > $o]: ((^[Y1 : product_prod @ coinductive_llist @ a @ b]: (Y0 @ (product_fst @ coinductive_llist @ a @ b @ Y1) @ (product_snd @ coinductive_llist @ a @ b @ Y1))))) @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: ((^[Y2 : coinductive_llist @ a > b > $o]: ((^[Y3 : product_prod @ coinductive_llist @ a @ b]: (Y2 @ (product_fst @ coinductive_llist @ a @ b @ Y3) @ (product_snd @ coinductive_llist @ a @ b @ Y3))))) @ (^[Y2 : coinductive_llist @ a]: ((^[Y3 : b]: ((Y0 = Y2) & ((coinductive_lfinite @ a @ Y2) => (Y1 = Y3)))))))))) @ prod1 @ prod2))),
% 8.05/2.05 inference(definition_unfolding,[status(thm)],[f1124,f1103,f1103])).
% 8.05/2.05 thf(f1103,plain,(
% 8.05/2.05 ( ! [X1 : $tType,X0 : $tType,X2 : $tType] : ((product_case_prod @ X2 @ X1 @ X0 = (^[Y0 : X2 > X1 > X0]: ((^[Y1 : product_prod @ X2 @ X1]: (Y0 @ (product_fst @ X2 @ X1 @ Y1) @ (product_snd @ X2 @ X1 @ Y1))))))) )),
% 8.05/2.05 inference(cnf_transformation,[status(thm)],[f475])).
% 8.05/2.05 thf(f475,plain,(
% 8.05/2.05 ! [X0 : $tType,X1 : $tType,X2 : $tType] : (product_case_prod @ X2 @ X1 @ X0 = (^[Y0 : X2 > X1 > X0]: ((^[Y1 : product_prod @ X2 @ X1]: (Y0 @ (product_fst @ X2 @ X1 @ Y1) @ (product_snd @ X2 @ X1 @ Y1))))))),
% 8.05/2.05 inference(fool_elimination,[status(thm)],[f474])).
% 8.05/2.05 thf(f474,plain,(
% 8.05/2.05 ! [X0 : $tType,X1 : $tType,X2 : $tType] : (product_case_prod @ X2 @ X1 @ X0 = (^[X3 : X2 > X1 > X0, X4 : product_prod @ X2 @ X1] : (X3 @ (product_fst @ X2 @ X1 @ X4) @ (product_snd @ X2 @ X1 @ X4))))),
% 8.05/2.05 inference(rectify,[status(thm)],[f202])).
% 8.05/2.05 thf(f202,axiom,(
% 8.05/2.05 ! [X2 : $tType,X1 : $tType,X0 : $tType] : (product_case_prod @ X0 @ X1 @ X2 = (^[X114 : X0 > X1 > X2, X111 : product_prod @ X0 @ X1] : (X114 @ (product_fst @ X0 @ X1 @ X111) @ (product_snd @ X0 @ X1 @ X111))))),
% 8.05/2.05 file('/export/starexec/sandbox2/tmp/tmp.ihDMkoi04K/DTF2THF_29522.p',fact_201_case__prod__unfold)).
% 8.05/2.05 thf(f1124,plain,(
% 8.05/2.05 ($true = (product_case_prod @ coinductive_llist @ a @ b @ (product_prod @ coinductive_llist @ a @ b > $o) @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: (product_case_prod @ coinductive_llist @ a @ b @ $o @ (^[Y2 : coinductive_llist @ a]: ((^[Y3 : b]: ((Y0 = Y2) & ((coinductive_lfinite @ a @ Y2) => (Y1 = Y3)))))))))) @ prod1 @ prod2))),
% 8.05/2.05 inference(cnf_transformation,[status(thm)],[f557])).
% 8.05/2.05 thf(f557,plain,(
% 8.05/2.05 ($true = (product_case_prod @ coinductive_llist @ a @ b @ (product_prod @ coinductive_llist @ a @ b > $o) @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: (product_case_prod @ coinductive_llist @ a @ b @ $o @ (^[Y2 : coinductive_llist @ a]: ((^[Y3 : b]: ((Y0 = Y2) & ((coinductive_lfinite @ a @ Y2) => (Y1 = Y3)))))))))) @ prod1 @ prod2))),
% 8.05/2.05 inference(fool_elimination,[status(thm)],[f556])).
% 8.05/2.05 thf(f556,plain,(
% 8.05/2.05 (product_case_prod @ coinductive_llist @ a @ b @ (product_prod @ coinductive_llist @ a @ b > $o) @ (^[X0 : coinductive_llist @ a, X1 : b] : (product_case_prod @ coinductive_llist @ a @ b @ $o @ (^[X2 : coinductive_llist @ a, X3 : b] : (((coinductive_lfinite @ a @ X2) => (X1 = X3)) & (X0 = X2))))) @ prod1 @ prod2)),
% 8.05/2.05 inference(rectify,[status(thm)],[f262])).
% 8.05/2.05 thf(f262,axiom,(
% 8.05/2.05 (product_case_prod @ coinductive_llist @ a @ b @ (product_prod @ coinductive_llist @ a @ b > $o) @ (^[X61 : coinductive_llist @ a, X17 : b] : (product_case_prod @ coinductive_llist @ a @ b @ $o @ (^[X62 : coinductive_llist @ a, X18 : b] : (((coinductive_lfinite @ a @ X62) => (X17 = X18)) & (X61 = X62))))) @ prod1 @ prod2)),
% 8.05/2.05 file('/export/starexec/sandbox2/tmp/tmp.ihDMkoi04K/DTF2THF_29522.p',conj_0)).
% 8.05/2.05 thf(f2941,plain,(
% 8.05/2.05 spl126_7 | spl126_8),
% 8.05/2.05 inference(avatar_split_clause,[status(thm)],[f1674,f2938,f2934])).
% 8.05/2.05 thf(f2934,definition,(
% 8.05/2.05 spl126_7 <=> ((product_snd @ coinductive_llist @ a @ b @ prod1) = (product_snd @ coinductive_llist @ a @ b @ prod2))),
% 8.05/2.05 introduced(definition,[new_symbols(naming,[spl126_7])],[avatar_definition])).
% 8.05/2.05 thf(f2938,definition,(
% 8.05/2.05 spl126_8 <=> ($false = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)))),
% 8.05/2.05 introduced(definition,[new_symbols(naming,[spl126_8])],[avatar_definition])).
% 8.05/2.05 thf(f1674,plain,(
% 8.05/2.05 ($false = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) | ((product_snd @ coinductive_llist @ a @ b @ prod1) = (product_snd @ coinductive_llist @ a @ b @ prod2))),
% 8.05/2.05 inference(equality_proxy_clausification,[status(thm)],[f1673])).
% 8.05/2.05 thf(f1673,plain,(
% 8.05/2.05 ($true = ((product_snd @ coinductive_llist @ a @ b @ prod1) = (product_snd @ coinductive_llist @ a @ b @ prod2))) | ($false = (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)))),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f1670])).
% 8.05/2.05 thf(f1670,plain,(
% 8.05/2.05 ($true = ((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = (product_snd @ coinductive_llist @ a @ b @ prod2))))),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f1669])).
% 8.05/2.05 thf(f2922,plain,(
% 8.05/2.05 ~spl126_3 | ~spl126_4),
% 8.05/2.05 inference(avatar_split_clause,[status(thm)],[f2309,f2919,f2915])).
% 8.05/2.05 thf(f2915,definition,(
% 8.05/2.05 spl126_3 <=> ((product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))) = (product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0))))))))),
% 8.05/2.05 introduced(definition,[new_symbols(naming,[spl126_3])],[avatar_definition])).
% 8.05/2.05 thf(f2309,plain,(
% 8.05/2.05 ((product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))) != (product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0)))))))) | ((product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0))))))) != (product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))))),
% 8.05/2.05 inference(equality_proxy_clausification,[status(thm)],[f2307])).
% 8.05/2.05 thf(f2307,plain,(
% 8.05/2.05 ((product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))) != (product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0)))))))) | ($false = ((product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0))))))) = (product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0)))))))))),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f2306])).
% 8.05/2.05 thf(f2306,plain,(
% 8.05/2.05 ((product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))) != (product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0)))))))) | (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0)))))))) => ((product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0))))))) = (product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))))) = $false)),
% 8.05/2.05 inference(equality_proxy_clausification,[status(thm)],[f2305])).
% 8.05/2.05 thf(f2305,plain,(
% 8.05/2.05 ($false = ((product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0))))))) = (product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))))) | (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0)))))))) => ((product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0))))))) = (product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))))) = $false)),
% 8.05/2.05 inference(binary_proxy_clausification,[status(thm)],[f2304])).
% 8.05/2.05 thf(f2304,plain,(
% 8.05/2.05 ($true != (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0)))))))) => ((product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0))))))) = (product_snd @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))))) & ((product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod1)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1)) => ((product_snd @ coinductive_llist @ a @ b @ prod1) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod1))) => (b2 = Y0))))))) = (product_fst @ coinductive_llist @ a @ b @ (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ (product_fst @ coinductive_llist @ a @ b @ prod2)) @ (the @ b @ (^[Y0 : b]: (((coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2)) => ((product_snd @ coinductive_llist @ a @ b @ prod2) = Y0)) & ((~ (coinductive_lfinite @ a @ (product_fst @ coinductive_llist @ a @ b @ prod2))) => (b2 = Y0))))))))))),
% 8.05/2.05 inference(beta_eta_normalization,[status(thm)],[f1483])).
% 8.05/2.05 thf(f1483,plain,(
% 8.05/2.05 ($true != ((^[Y0 : coinductive_llist @ a > b > product_prod @ coinductive_llist @ a @ b > $o]: ((^[Y1 : product_prod @ coinductive_llist @ a @ b]: (Y0 @ (product_fst @ coinductive_llist @ a @ b @ Y1) @ (product_snd @ coinductive_llist @ a @ b @ Y1))))) @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: ((^[Y2 : coinductive_llist @ a > b > $o]: ((^[Y3 : product_prod @ coinductive_llist @ a @ b]: (Y2 @ (product_fst @ coinductive_llist @ a @ b @ Y3) @ (product_snd @ coinductive_llist @ a @ b @ Y3))))) @ (^[Y2 : coinductive_llist @ a]: ((^[Y3 : b]: (((coinductive_lfinite @ a @ Y2) => (Y1 = Y3)) & (Y0 = Y2))))))))) @ ((^[Y0 : coinductive_llist @ a > b > product_prod @ coinductive_llist @ a @ b]: ((^[Y1 : product_prod @ coinductive_llist @ a @ b]: (Y0 @ (product_fst @ coinductive_llist @ a @ b @ Y1) @ (product_snd @ coinductive_llist @ a @ b @ Y1))))) @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ Y0) @ ((^[Y2 : $o]: ((^[Y3 : b]: ((^[Y4 : b]: (the @ b @ (^[Y5 : b]: ((Y2 => (Y3 = Y5)) & ((~ Y2) => (Y4 = Y5)))))))))) @ (coinductive_lfinite @ a @ Y0) @ Y1 @ b2))))) @ prod1) @ ((^[Y0 : coinductive_llist @ a > b > product_prod @ coinductive_llist @ a @ b]: ((^[Y1 : product_prod @ coinductive_llist @ a @ b]: (Y0 @ (product_fst @ coinductive_llist @ a @ b @ Y1) @ (product_snd @ coinductive_llist @ a @ b @ Y1))))) @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ Y0) @ ((^[Y2 : $o]: ((^[Y3 : b]: ((^[Y4 : b]: (the @ b @ (^[Y5 : b]: ((Y2 => (Y3 = Y5)) & ((~ Y2) => (Y4 = Y5)))))))))) @ (coinductive_lfinite @ a @ Y0) @ Y1 @ b2))))) @ prod2)))),
% 8.05/2.05 inference(definition_unfolding,[status(thm)],[f1311,f1103,f1103,f1103,f1074,f1103,f1074])).
% 8.05/2.05 thf(f1074,plain,(
% 8.05/2.05 ( ! [X0 : $tType] : ((if @ X0 = (^[Y0 : $o]: ((^[Y1 : X0]: ((^[Y2 : X0]: (the @ X0 @ (^[Y3 : X0]: ((Y0 => (Y1 = Y3)) & ((~ Y0) => (Y2 = Y3)))))))))))) )),
% 8.05/2.05 inference(cnf_transformation,[status(thm)],[f387])).
% 8.05/2.05 thf(f387,plain,(
% 8.05/2.05 ! [X0 : $tType] : (if @ X0 = (^[Y0 : $o]: ((^[Y1 : X0]: ((^[Y2 : X0]: (the @ X0 @ (^[Y3 : X0]: ((Y0 => (Y1 = Y3)) & ((~ Y0) => (Y2 = Y3)))))))))))),
% 8.05/2.05 inference(fool_elimination,[status(thm)],[f386])).
% 8.05/2.05 thf(f386,plain,(
% 8.05/2.05 ! [X0 : $tType] : (if @ X0 = (^[X1 : $o, X2 : X0, X3 : X0] : (the @ X0 @ (^[X4 : X0] : ((~X1 => (X3 = X4)) & (X1 => (X2 = X4)))))))),
% 8.05/2.05 inference(rectify,[status(thm)],[f74])).
% 8.05/2.05 thf(f74,axiom,(
% 8.05/2.05 ! [X0 : $tType] : (if @ X0 = (^[X52 : $o, X44 : X0, X46 : X0] : (the @ X0 @ (^[X74 : X0] : ((~X52 => (X46 = X74)) & (X52 => (X44 = X74)))))))),
% 8.05/2.05 file('/export/starexec/sandbox2/tmp/tmp.ihDMkoi04K/DTF2THF_29522.p',fact_73_If__def)).
% 8.05/2.05 thf(f1311,plain,(
% 8.05/2.05 ($true != (product_case_prod @ coinductive_llist @ a @ b @ (product_prod @ coinductive_llist @ a @ b > $o) @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: (product_case_prod @ coinductive_llist @ a @ b @ $o @ (^[Y2 : coinductive_llist @ a]: ((^[Y3 : b]: (((coinductive_lfinite @ a @ Y2) => (Y1 = Y3)) & (Y0 = Y2))))))))) @ (product_case_prod @ coinductive_llist @ a @ b @ product_prod @ coinductive_llist @ a @ b @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ Y0) @ (if @ b @ (coinductive_lfinite @ a @ Y0) @ Y1 @ b2))))) @ prod1) @ (product_case_prod @ coinductive_llist @ a @ b @ product_prod @ coinductive_llist @ a @ b @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ Y0) @ (if @ b @ (coinductive_lfinite @ a @ Y0) @ Y1 @ b2))))) @ prod2)))),
% 8.05/2.05 inference(cnf_transformation,[status(thm)],[f733])).
% 8.05/2.05 thf(f733,plain,(
% 8.05/2.05 ($true != (product_case_prod @ coinductive_llist @ a @ b @ (product_prod @ coinductive_llist @ a @ b > $o) @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: (product_case_prod @ coinductive_llist @ a @ b @ $o @ (^[Y2 : coinductive_llist @ a]: ((^[Y3 : b]: (((coinductive_lfinite @ a @ Y2) => (Y1 = Y3)) & (Y0 = Y2))))))))) @ (product_case_prod @ coinductive_llist @ a @ b @ product_prod @ coinductive_llist @ a @ b @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ Y0) @ (if @ b @ (coinductive_lfinite @ a @ Y0) @ Y1 @ b2))))) @ prod1) @ (product_case_prod @ coinductive_llist @ a @ b @ product_prod @ coinductive_llist @ a @ b @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ Y0) @ (if @ b @ (coinductive_lfinite @ a @ Y0) @ Y1 @ b2))))) @ prod2)))),
% 8.05/2.05 inference(flattening,[status(thm)],[f521])).
% 8.05/2.05 thf(f521,plain,(
% 8.05/2.05 ~($true = (product_case_prod @ coinductive_llist @ a @ b @ (product_prod @ coinductive_llist @ a @ b > $o) @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: (product_case_prod @ coinductive_llist @ a @ b @ $o @ (^[Y2 : coinductive_llist @ a]: ((^[Y3 : b]: (((coinductive_lfinite @ a @ Y2) => (Y1 = Y3)) & (Y0 = Y2))))))))) @ (product_case_prod @ coinductive_llist @ a @ b @ product_prod @ coinductive_llist @ a @ b @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ Y0) @ (if @ b @ (coinductive_lfinite @ a @ Y0) @ Y1 @ b2))))) @ prod1) @ (product_case_prod @ coinductive_llist @ a @ b @ product_prod @ coinductive_llist @ a @ b @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : b]: (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ Y0) @ (if @ b @ (coinductive_lfinite @ a @ Y0) @ Y1 @ b2))))) @ prod2)))),
% 8.05/2.05 inference(fool_elimination,[status(thm)],[f520])).
% 8.05/2.05 thf(f520,plain,(
% 8.05/2.05 ~(product_case_prod @ coinductive_llist @ a @ b @ (product_prod @ coinductive_llist @ a @ b > $o) @ (^[X0 : coinductive_llist @ a, X1 : b] : (product_case_prod @ coinductive_llist @ a @ b @ $o @ (^[X2 : coinductive_llist @ a, X3 : b] : ((X0 = X2) & ((coinductive_lfinite @ a @ X2) => (X1 = X3)))))) @ (product_case_prod @ coinductive_llist @ a @ b @ product_prod @ coinductive_llist @ a @ b @ (^[X4 : coinductive_llist @ a, X5 : b] : (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ X4) @ (if @ b @ (coinductive_lfinite @ a @ X4) @ X5 @ b2))) @ prod1) @ (product_case_prod @ coinductive_llist @ a @ b @ product_prod @ coinductive_llist @ a @ b @ (^[X6 : coinductive_llist @ a, X7 : b] : (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ X6) @ (if @ b @ (coinductive_lfinite @ a @ X6) @ X7 @ b2))) @ prod2))),
% 8.05/2.05 inference(rectify,[status(thm)],[f264])).
% 8.05/2.05 thf(f264,negated_conjecture,(
% 8.05/2.05 ~(product_case_prod @ coinductive_llist @ a @ b @ (product_prod @ coinductive_llist @ a @ b > $o) @ (^[X61 : coinductive_llist @ a, X17 : b] : (product_case_prod @ coinductive_llist @ a @ b @ $o @ (^[X62 : coinductive_llist @ a, X18 : b] : ((X61 = X62) & ((coinductive_lfinite @ a @ X62) => (X17 = X18)))))) @ (product_case_prod @ coinductive_llist @ a @ b @ product_prod @ coinductive_llist @ a @ b @ (^[X61 : coinductive_llist @ a, X59 : b] : (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ X61) @ (if @ b @ (coinductive_lfinite @ a @ X61) @ X59 @ b2))) @ prod1) @ (product_case_prod @ coinductive_llist @ a @ b @ product_prod @ coinductive_llist @ a @ b @ (^[X61 : coinductive_llist @ a, X59 : b] : (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ X61) @ (if @ b @ (coinductive_lfinite @ a @ X61) @ X59 @ b2))) @ prod2))),
% 8.05/2.05 inference(negated_conjecture,[status(cth)],[f263])).
% 8.05/2.05 thf(f263,conjecture,(
% 8.05/2.05 (product_case_prod @ coinductive_llist @ a @ b @ (product_prod @ coinductive_llist @ a @ b > $o) @ (^[X61 : coinductive_llist @ a, X17 : b] : (product_case_prod @ coinductive_llist @ a @ b @ $o @ (^[X62 : coinductive_llist @ a, X18 : b] : ((X61 = X62) & ((coinductive_lfinite @ a @ X62) => (X17 = X18)))))) @ (product_case_prod @ coinductive_llist @ a @ b @ product_prod @ coinductive_llist @ a @ b @ (^[X61 : coinductive_llist @ a, X59 : b] : (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ X61) @ (if @ b @ (coinductive_lfinite @ a @ X61) @ X59 @ b2))) @ prod1) @ (product_case_prod @ coinductive_llist @ a @ b @ product_prod @ coinductive_llist @ a @ b @ (^[X61 : coinductive_llist @ a, X59 : b] : (product_Pair @ coinductive_llist @ a @ b @ (coinductive_lfilter @ a @ fun @ X61) @ (if @ b @ (coinductive_lfinite @ a @ X61) @ X59 @ b2))) @ prod2))),
% 8.05/2.05 file('/export/starexec/sandbox2/tmp/tmp.ihDMkoi04K/DTF2THF_29522.p',conj_1)).
% 8.05/2.05 thf(f2913,plain,(
% 8.05/2.05 ~spl126_2),
% 8.05/2.05 inference(avatar_split_clause,[status(thm)],[f265,f2910])).
% 8.05/2.05 thf(f2910,definition,(
% 8.05/2.05 spl126_2 <=> ($true = $false)),
% 8.05/2.05 introduced(definition,[new_symbols(naming,[spl126_2])],[avatar_definition])).
% 8.05/2.05 thf(f265,plain,(
% 8.05/2.05 ($true != $false)),
% 8.05/2.05 introduced(definition,[],[fool_axiom])).
% 8.05/2.05 % SZS output end Proof for DTF2THF_29522
% 8.05/2.05 % (29853)------------------------------
% 8.05/2.05 % (29853)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.05/2.05 % (29853)Termination reason: Refutation
% 8.05/2.05
% 8.05/2.05 % (29853)Memory used [KB]: 8443
% 8.05/2.05 % (29853)Time elapsed: 0.099 s
% 8.05/2.05 % (29853)Instructions burned: 320 (million)
% 8.05/2.05 % (29853)------------------------------
% 8.05/2.05 % (29853)------------------------------
% 8.05/2.05 % (29763)Success in time 0.915 s
%------------------------------------------------------------------------------