%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP137^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n007.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 08:55:51 AM UTC 2026
% Result : Theorem 2.77s 1.84s
% Output : Refutation 2.77s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP137^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n007.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:26:38 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.47 ---- Original DTF file ---
% 0.19/0.47 thf(spec,logic,$$dhol).
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 % File : ITP137^2 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.47 % Domain : Interactive Theorem Proving
% 0.19/0.47 % Problem : Sledgehammer Paraconsistency problem prob_407__3265932_1
% 0.19/0.47 % Version : Especial.
% 0.19/0.47 % English :
% 0.19/0.47
% 0.19/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.47 % Source : [Des21]
% 0.19/0.47 % Names : Paraconsistency/prob_407__3265932_1 [Des21]
% 0.19/0.47
% 0.19/0.47 % Status : Theorem
% 0.19/0.47 % Rating : 0.33 v8.1.0, 0.25 v7.5.0
% 0.19/0.47 % Syntax : Number of formulae : 316 ( 180 unt; 56 typ; 0 def)
% 0.19/0.47 % Number of atoms : 531 ( 386 equ; 0 cnn)
% 0.19/0.47 % Maximal formula atoms : 16 ( 2 avg)
% 0.19/0.47 % Number of connectives : 4410 ( 99 ~; 8 |; 53 &;4048 @)
% 0.19/0.47 % ( 0 <=>; 202 =>; 0 <=; 0 <~>)
% 0.19/0.47 % Maximal formula depth : 28 ( 8 avg)
% 0.19/0.47 % Number of types : 6 ( 5 usr)
% 0.19/0.47 % Number of type conns : 471 ( 471 >; 0 *; 0 +; 0 <<)
% 0.19/0.47 % Number of symbols : 54 ( 51 usr; 4 con; 0-8 aty)
% 0.19/0.47 % Number of variables : 1493 ( 197 ^;1218 !; 13 ?;1493 :)
% 0.19/0.47 % ( 65 !>; 0 ?*; 0 @-; 0 @+)
% 0.19/0.47 % SPC : TH1_THM_EQU_NAR
% 0.19/0.47
% 0.19/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:21:07.434
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 % Could-be-implicit typings (8)
% 0.19/0.47 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv,type,
% 0.19/0.47 paraco415392788lle_tv: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm,type,
% 0.19/0.47 paraco414474393lle_fm: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Product__Type_Ounit,type,
% 0.19/0.47 product_unit: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.19/0.47 product_prod: $tType > $tType > $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_String_Ochar,type,
% 0.19/0.47 char: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_List_Olist,type,
% 0.19/0.47 list: $tType > $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Set_Oset,type,
% 0.19/0.47 set: $tType > $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Nat_Onat,type,
% 0.19/0.47 nat: $tType ).
% 0.19/0.47
% 0.19/0.47 % Explicit typings (48)
% 0.19/0.47 thf(sy_c_Fun__Def_Oreduction__pair,type,
% 0.19/0.47 fun_reduction_pair:
% 0.19/0.47 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 0.19/0.47 fun_rp_inv_image:
% 0.19/0.47 !>[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 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_HOL_OThe,type,
% 0.19/0.47 the:
% 0.19/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_If,type,
% 0.19/0.47 if:
% 0.19/0.47 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Order__Relation_Oabove,type,
% 0.19/0.47 order_above:
% 0.19/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Order__Relation_OaboveS,type,
% 0.19/0.47 order_aboveS:
% 0.19/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Order__Relation_Orelation__of,type,
% 0.19/0.47 order_relation_of:
% 0.19/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.19/0.47 bot_bot:
% 0.19/0.47 !>[A: $tType] : A ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.19/0.47 ord_less_eq:
% 0.19/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.19/0.47 top_top:
% 0.19/0.47 !>[A: $tType] : A ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval,type,
% 0.19/0.47 paraco876059933e_eval: ( ( list @ char ) > paraco415392788lle_tv ) > paraco414474393lle_fm > paraco415392788lle_tv ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H,type,
% 0.19/0.47 paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql,type,
% 0.19/0.47 paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H,type,
% 0.19/0.47 paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H,type,
% 0.19/0.47 paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro,type,
% 0.19/0.47 paraco27778325le_Pro: ( list @ char ) > paraco414474393lle_fm ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth,type,
% 0.19/0.47 paraco251304083_Truth: paraco414474393lle_fm ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Ocase__fm,type,
% 0.19/0.47 paraco1246693743ase_fm:
% 0.19/0.47 !>[A: $tType] : ( ( ( list @ char ) > A ) > A > ( paraco414474393lle_fm > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A ) > paraco414474393lle_fm > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet,type,
% 0.19/0.47 paraco2040174112le_Det: $o > paraco415392788lle_tv ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet,type,
% 0.19/0.47 paraco676387099_Indet: nat > paraco415392788lle_tv ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Ocase__tv,type,
% 0.19/0.47 paraco490622181ase_tv:
% 0.19/0.47 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv,type,
% 0.19/0.47 paraco152590079rec_tv:
% 0.19/0.47 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid,type,
% 0.19/0.47 paraco769098683_valid: paraco414474393lle_fm > $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.19/0.47 product_Pair:
% 0.19/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_OSigma,type,
% 0.19/0.47 product_Sigma:
% 0.19/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_Obool_Ocase__bool,type,
% 0.19/0.47 product_case_bool:
% 0.19/0.47 !>[A: $tType] : ( A > A > $o > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_Ocurry,type,
% 0.19/0.47 product_curry:
% 0.19/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.19/0.47 produc2004651681e_prod:
% 0.19/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_Omap__prod,type,
% 0.19/0.47 product_map_prod:
% 0.19/0.47 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.19/0.47 product_rec_prod:
% 0.19/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 0.19/0.47 product_rec_set_prod:
% 0.19/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 0.19/0.47 product_rec_set_unit:
% 0.19/0.47 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 0.19/0.47 product_rec_unit:
% 0.30/0.47 !>[T: $tType] : ( T > product_unit > T ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.30/0.47 product_case_prod:
% 0.30/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.30/0.47 product_fst:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.30/0.47 product_snd:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.30/0.47 product_swap:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oproduct,type,
% 0.30/0.47 product_product:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oscomp,type,
% 0.30/0.47 product_scomp:
% 0.30/0.47 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Relation_Oinv__image,type,
% 0.30/0.47 inv_image:
% 0.30/0.47 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Relation_Oinv__imagep,type,
% 0.30/0.47 inv_imagep:
% 0.30/0.47 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_OCollect,type,
% 0.30/0.47 collect:
% 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_Oimage,type,
% 0.30/0.47 image:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_Othe__elem,type,
% 0.30/0.47 the_elem:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Wellfounded_Olex__prod,type,
% 0.30/0.47 lex_prod:
% 0.30/0.47 !>[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 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Wfrec_Osame__fst,type,
% 0.30/0.47 same_fst:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_member,type,
% 0.30/0.47 member:
% 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_p,type,
% 0.30/0.47 p: paraco414474393lle_fm ).
% 0.30/0.47
% 0.30/0.47 % Relevant facts (256)
% 0.30/0.47 thf(fact_0_tv_Oinject_I1_J,axiom,
% 0.30/0.47 ! [X1: $o,Y1: $o] :
% 0.30/0.47 ( ( ( paraco2040174112le_Det @ X1 )
% 0.30/0.47 = ( paraco2040174112le_Det @ Y1 ) )
% 0.30/0.47 = ( X1 = Y1 ) ) ).
% 0.30/0.47
% 0.30/0.47 % tv.inject(1)
% 0.30/0.47 thf(fact_1_assms,axiom,
% 0.30/0.47 paraco769098683_valid @ ( paraco329115265le_Neg @ p ) ).
% 0.30/0.47
% 0.30/0.47 % assms
% 0.30/0.47 thf(fact_2_valid__def,axiom,
% 0.30/0.47 ( paraco769098683_valid
% 0.30/0.47 = ( ^ [P: paraco414474393lle_fm] :
% 0.30/0.47 ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 0.30/0.47 ( ( paraco876059933e_eval @ I @ P )
% 0.30/0.47 = ( paraco2040174112le_Det @ $true ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % valid_def
% 0.30/0.47 thf(fact_3_eval_Osimps_I2_J,axiom,
% 0.30/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv] :
% 0.30/0.47 ( ( paraco876059933e_eval @ I2 @ paraco251304083_Truth )
% 0.30/0.47 = ( paraco2040174112le_Det @ $true ) ) ).
% 0.30/0.47
% 0.30/0.47 % eval.simps(2)
% 0.30/0.47 thf(fact_4_eval_Osimps_I4_J,axiom,
% 0.30/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.30/0.47 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 = ( paraco876059933e_eval @ I2 @ Q ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 0.30/0.47 = ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 0.30/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 != ( paraco876059933e_eval @ I2 @ Q ) )
% 0.30/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 = ( paraco2040174112le_Det @ $true ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 0.30/0.47 = ( paraco876059933e_eval @ I2 @ Q ) ) )
% 0.30/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.30/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ Q )
% 0.30/0.47 = ( paraco2040174112le_Det @ $true ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 0.30/0.47 = ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 0.30/0.47 & ( ( ( paraco876059933e_eval @ I2 @ Q )
% 0.30/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 0.30/0.47 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eval.simps(4)
% 0.30/0.47 thf(fact_5_eval_Osimps_I5_J,axiom,
% 0.30/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.30/0.47 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 = ( paraco876059933e_eval @ I2 @ Q ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
% 0.30/0.47 = ( paraco2040174112le_Det @ $true ) ) )
% 0.30/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 != ( paraco876059933e_eval @ I2 @ Q ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
% 0.30/0.47 = ( paraco2040174112le_Det @ $false ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eval.simps(5)
% 0.30/0.47 thf(fact_6_eval__equality__simplify,axiom,
% 0.30/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
% 0.30/0.47 = ( paraco2040174112le_Det
% 0.30/0.47 @ ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 = ( paraco876059933e_eval @ I2 @ Q ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eval_equality_simplify
% 0.30/0.47 thf(fact_7_tv_Osimps_I7_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
% 0.30/0.47 ( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
% 0.30/0.47 = ( F1 @ X1 ) ) ).
% 0.30/0.47
% 0.30/0.47 % tv.simps(7)
% 0.30/0.47 thf(fact_8_eval__negation,axiom,
% 0.30/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm] :
% 0.30/0.47 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 = ( paraco2040174112le_Det @ $false ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 0.30/0.47 = ( paraco2040174112le_Det @ $true ) ) )
% 0.30/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 != ( paraco2040174112le_Det @ $false ) )
% 0.30/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 = ( paraco2040174112le_Det @ $true ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 0.30/0.47 = ( paraco2040174112le_Det @ $false ) ) )
% 0.30/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 0.30/0.47 = ( paraco876059933e_eval @ I2 @ P2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eval_negation
% 0.30/0.47 thf(fact_9_tv_Osimps_I5_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
% 0.30/0.47 ( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
% 0.30/0.47 = ( F1 @ X1 ) ) ).
% 0.30/0.47
% 0.30/0.47 % tv.simps(5)
% 0.30/0.47 thf(fact_10_eval_Osimps_I1_J,axiom,
% 0.30/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,S: list @ char] :
% 0.30/0.47 ( ( paraco876059933e_eval @ I2 @ ( paraco27778325le_Pro @ S ) )
% 0.30/0.47 = ( I2 @ S ) ) ).
% 0.30/0.47
% 0.30/0.47 % eval.simps(1)
% 0.30/0.47 thf(fact_11_tv_Odistinct_I1_J,axiom,
% 0.30/0.47 ! [X1: $o,X2: nat] :
% 0.30/0.47 ( ( paraco2040174112le_Det @ X1 )
% 0.30/0.47 != ( paraco676387099_Indet @ X2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % tv.distinct(1)
% 0.30/0.47 thf(fact_12_tv_Oinduct,axiom,
% 0.30/0.47 ! [P3: paraco415392788lle_tv > $o,Tv: paraco415392788lle_tv] :
% 0.30/0.47 ( ! [X: $o] : ( P3 @ ( paraco2040174112le_Det @ X ) )
% 0.30/0.47 => ( ! [X: nat] : ( P3 @ ( paraco676387099_Indet @ X ) )
% 0.30/0.47 => ( P3 @ Tv ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % tv.induct
% 0.30/0.47 thf(fact_13_tv_Oexhaust,axiom,
% 0.30/0.47 ! [Y: paraco415392788lle_tv] :
% 0.30/0.47 ( ! [X12: $o] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( paraco2040174112le_Det @ X12 ) )
% 0.30/0.47 => ~ ! [X22: nat] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( paraco676387099_Indet @ X22 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % tv.exhaust
% 0.30/0.47 thf(fact_14_fm_Oinject_I2_J,axiom,
% 0.30/0.47 ! [X3: paraco414474393lle_fm,Y3: paraco414474393lle_fm] :
% 0.30/0.47 ( ( ( paraco329115265le_Neg @ X3 )
% 0.30/0.47 = ( paraco329115265le_Neg @ Y3 ) )
% 0.30/0.47 = ( X3 = Y3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.inject(2)
% 0.30/0.47 thf(fact_15_tv_Oinject_I2_J,axiom,
% 0.30/0.47 ! [X2: nat,Y2: nat] :
% 0.30/0.47 ( ( ( paraco676387099_Indet @ X2 )
% 0.30/0.47 = ( paraco676387099_Indet @ Y2 ) )
% 0.30/0.47 = ( X2 = Y2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % tv.inject(2)
% 0.30/0.47 thf(fact_16_fm_Oinject_I4_J,axiom,
% 0.30/0.47 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,Y51: paraco414474393lle_fm,Y52: paraco414474393lle_fm] :
% 0.30/0.47 ( ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 0.30/0.47 = ( paraco2084319816le_Eql @ Y51 @ Y52 ) )
% 0.30/0.47 = ( ( X51 = Y51 )
% 0.30/0.47 & ( X52 = Y52 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.inject(4)
% 0.30/0.47 thf(fact_17_fm_Oinject_I1_J,axiom,
% 0.30/0.47 ! [X1: list @ char,Y1: list @ char] :
% 0.30/0.47 ( ( ( paraco27778325le_Pro @ X1 )
% 0.30/0.47 = ( paraco27778325le_Pro @ Y1 ) )
% 0.30/0.47 = ( X1 = Y1 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.inject(1)
% 0.30/0.47 thf(fact_18_fm_Oinject_I3_J,axiom,
% 0.30/0.47 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,Y41: paraco414474393lle_fm,Y42: paraco414474393lle_fm] :
% 0.30/0.47 ( ( ( paraco2100061555le_Con @ X41 @ X42 )
% 0.30/0.47 = ( paraco2100061555le_Con @ Y41 @ Y42 ) )
% 0.30/0.47 = ( ( X41 = Y41 )
% 0.30/0.47 & ( X42 = Y42 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.inject(3)
% 0.30/0.47 thf(fact_19_tv_Osimps_I8_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: $o > A,F2: nat > A,X2: nat] :
% 0.30/0.47 ( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X2 ) )
% 0.30/0.47 = ( F2 @ X2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % tv.simps(8)
% 0.30/0.47 thf(fact_20_tv_Osimps_I6_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: $o > A,F2: nat > A,X2: nat] :
% 0.30/0.47 ( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X2 ) )
% 0.30/0.47 = ( F2 @ X2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % tv.simps(6)
% 0.30/0.47 thf(fact_21_fm_Odistinct_I25_J,axiom,
% 0.30/0.47 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 0.30/0.47 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(25)
% 0.30/0.47 thf(fact_22_fm_Odistinct_I21_J,axiom,
% 0.30/0.47 ! [X3: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco329115265le_Neg @ X3 )
% 0.30/0.47 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(21)
% 0.30/0.47 thf(fact_23_fm_Odistinct_I19_J,axiom,
% 0.30/0.47 ! [X3: paraco414474393lle_fm,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco329115265le_Neg @ X3 )
% 0.30/0.47 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(19)
% 0.30/0.47 thf(fact_24_fm_Odistinct_I15_J,axiom,
% 0.30/0.47 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.30/0.47 ( paraco251304083_Truth
% 0.30/0.47 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(15)
% 0.30/0.47 thf(fact_25_fm_Odistinct_I13_J,axiom,
% 0.30/0.47 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.30/0.47 ( paraco251304083_Truth
% 0.30/0.47 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(13)
% 0.30/0.47 thf(fact_26_fm_Odistinct_I11_J,axiom,
% 0.30/0.47 ! [X3: paraco414474393lle_fm] :
% 0.30/0.47 ( paraco251304083_Truth
% 0.30/0.47 != ( paraco329115265le_Neg @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(11)
% 0.30/0.47 thf(fact_27_fm_Odistinct_I7_J,axiom,
% 0.30/0.47 ! [X1: list @ char,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.30/0.47 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(7)
% 0.30/0.47 thf(fact_28_fm_Odistinct_I5_J,axiom,
% 0.30/0.47 ! [X1: list @ char,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.30/0.47 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(5)
% 0.30/0.47 thf(fact_29_fm_Odistinct_I3_J,axiom,
% 0.30/0.47 ! [X1: list @ char,X3: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.30/0.47 != ( paraco329115265le_Neg @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(3)
% 0.30/0.47 thf(fact_30_fm_Odistinct_I1_J,axiom,
% 0.30/0.47 ! [X1: list @ char] :
% 0.30/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.30/0.47 != paraco251304083_Truth ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(1)
% 0.30/0.47 thf(fact_31_double__negation,axiom,
% 0.30/0.47 ( paraco876059933e_eval
% 0.30/0.47 = ( ^ [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] : ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ ( paraco329115265le_Neg @ P ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % double_negation
% 0.30/0.47 thf(fact_32_string__tv_Oinduct,axiom,
% 0.30/0.47 ! [P3: paraco415392788lle_tv > $o,A0: paraco415392788lle_tv] :
% 0.30/0.47 ( ( P3 @ ( paraco2040174112le_Det @ $true ) )
% 0.30/0.47 => ( ( P3 @ ( paraco2040174112le_Det @ $false ) )
% 0.30/0.47 => ( ! [N: nat] : ( P3 @ ( paraco676387099_Indet @ N ) )
% 0.30/0.47 => ( P3 @ A0 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % string_tv.induct
% 0.30/0.47 thf(fact_33_string__tv_Ocases,axiom,
% 0.30/0.47 ! [X4: paraco415392788lle_tv] :
% 0.30/0.47 ( ( X4
% 0.30/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.30/0.47 => ( ( X4
% 0.30/0.47 != ( paraco2040174112le_Det @ $false ) )
% 0.30/0.47 => ~ ! [N: nat] :
% 0.30/0.47 ( X4
% 0.30/0.47 != ( paraco676387099_Indet @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % string_tv.cases
% 0.30/0.47 thf(fact_34_fm_Oinduct,axiom,
% 0.30/0.47 ! [P3: paraco414474393lle_fm > $o,Fm: paraco414474393lle_fm] :
% 0.30/0.47 ( ! [X: list @ char] : ( P3 @ ( paraco27778325le_Pro @ X ) )
% 0.30/0.47 => ( ( P3 @ paraco251304083_Truth )
% 0.30/0.47 => ( ! [X: paraco414474393lle_fm] :
% 0.30/0.47 ( ( P3 @ X )
% 0.30/0.47 => ( P3 @ ( paraco329115265le_Neg @ X ) ) )
% 0.30/0.47 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 0.30/0.47 ( ( P3 @ X1a )
% 0.30/0.47 => ( ( P3 @ X22 )
% 0.30/0.47 => ( P3 @ ( paraco2100061555le_Con @ X1a @ X22 ) ) ) )
% 0.30/0.47 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 0.30/0.47 ( ( P3 @ X1a )
% 0.30/0.47 => ( ( P3 @ X22 )
% 0.30/0.47 => ( P3 @ ( paraco2084319816le_Eql @ X1a @ X22 ) ) ) )
% 0.30/0.47 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 0.30/0.47 ( ( P3 @ X1a )
% 0.30/0.47 => ( ( P3 @ X22 )
% 0.30/0.47 => ( P3 @ ( paraco1628874225le_Eql @ X1a @ X22 ) ) ) )
% 0.30/0.47 => ( P3 @ Fm ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.induct
% 0.30/0.47 thf(fact_35_fm_Oexhaust,axiom,
% 0.30/0.47 ! [Y: paraco414474393lle_fm] :
% 0.30/0.47 ( ! [X12: list @ char] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( paraco27778325le_Pro @ X12 ) )
% 0.30/0.47 => ( ( Y != paraco251304083_Truth )
% 0.30/0.47 => ( ! [X32: paraco414474393lle_fm] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( paraco329115265le_Neg @ X32 ) )
% 0.30/0.47 => ( ! [X412: paraco414474393lle_fm,X422: paraco414474393lle_fm] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( paraco2100061555le_Con @ X412 @ X422 ) )
% 0.30/0.47 => ( ! [X512: paraco414474393lle_fm,X522: paraco414474393lle_fm] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( paraco2084319816le_Eql @ X512 @ X522 ) )
% 0.30/0.47 => ~ ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.exhaust
% 0.30/0.47 thf(fact_36_eval_Osimps_I3_J,axiom,
% 0.30/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 0.30/0.47 = ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ P2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eval.simps(3)
% 0.30/0.47 thf(fact_37_eval__equality,axiom,
% 0.30/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.30/0.47 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 = ( paraco876059933e_eval @ I2 @ Q ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.30/0.47 = ( paraco2040174112le_Det @ $true ) ) )
% 0.30/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 != ( paraco876059933e_eval @ I2 @ Q ) )
% 0.30/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 = ( paraco2040174112le_Det @ $true ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.30/0.47 = ( paraco876059933e_eval @ I2 @ Q ) ) )
% 0.30/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.30/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ Q )
% 0.30/0.47 = ( paraco2040174112le_Det @ $true ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.30/0.47 = ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 0.30/0.47 & ( ( ( paraco876059933e_eval @ I2 @ Q )
% 0.30/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.30/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 = ( paraco2040174112le_Det @ $false ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.30/0.47 = ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ Q ) ) ) )
% 0.30/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 != ( paraco2040174112le_Det @ $false ) )
% 0.30/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ Q )
% 0.30/0.47 = ( paraco2040174112le_Det @ $false ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.30/0.47 = ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) ) ) )
% 0.30/0.47 & ( ( ( paraco876059933e_eval @ I2 @ Q )
% 0.30/0.47 != ( paraco2040174112le_Det @ $false ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.30/0.47 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eval_equality
% 0.30/0.47 thf(fact_38_eval_Ocases,axiom,
% 0.30/0.47 ! [X4: product_prod @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm] :
% 0.30/0.47 ( ! [I3: ( list @ char ) > paraco415392788lle_tv,S2: list @ char] :
% 0.30/0.47 ( X4
% 0.30/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco27778325le_Pro @ S2 ) ) )
% 0.30/0.47 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv] :
% 0.30/0.47 ( X4
% 0.30/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ paraco251304083_Truth ) )
% 0.30/0.47 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm] :
% 0.30/0.47 ( X4
% 0.30/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco329115265le_Neg @ P4 ) ) )
% 0.30/0.47 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 0.30/0.47 ( X4
% 0.30/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2100061555le_Con @ P4 @ Q2 ) ) )
% 0.30/0.47 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 0.30/0.47 ( X4
% 0.30/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2084319816le_Eql @ P4 @ Q2 ) ) )
% 0.30/0.47 => ~ ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 0.30/0.47 ( X4
% 0.30/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco1628874225le_Eql @ P4 @ Q2 ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eval.cases
% 0.30/0.47 thf(fact_39_fm_Osimps_I37_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A] :
% 0.30/0.47 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ paraco251304083_Truth )
% 0.30/0.47 = F2 ) ).
% 0.30/0.47
% 0.30/0.47 % fm.simps(37)
% 0.30/0.47 thf(fact_40_fm_Osimps_I39_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2100061555le_Con @ X41 @ X42 ) )
% 0.30/0.47 = ( F4 @ X41 @ X42 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.simps(39)
% 0.30/0.47 thf(fact_41_fm_Osimps_I36_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X1: list @ char] :
% 0.30/0.47 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco27778325le_Pro @ X1 ) )
% 0.30/0.47 = ( F1 @ X1 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.simps(36)
% 0.30/0.47 thf(fact_42_fm_Osimps_I40_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2084319816le_Eql @ X51 @ X52 ) )
% 0.30/0.47 = ( F5 @ X51 @ X52 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.simps(40)
% 0.30/0.47 thf(fact_43_fm_Osimps_I38_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X3: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco329115265le_Neg @ X3 ) )
% 0.30/0.47 = ( F3 @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.simps(38)
% 0.30/0.47 thf(fact_44_fm_Odistinct_I17_J,axiom,
% 0.30/0.47 ! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 0.30/0.47 ( paraco251304083_Truth
% 0.30/0.47 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(17)
% 0.30/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,P3: A > $o] :
% 0.30/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 0.30/0.47 = ( P3 @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_Collect_eq
% 0.30/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A] :
% 0.30/0.47 ( ( collect @ A
% 0.30/0.47 @ ^ [X5: A] : ( member @ A @ X5 @ A3 ) )
% 0.30/0.47 = A3 ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_mem_eq
% 0.30/0.47 thf(fact_47_Collect__cong,axiom,
% 0.30/0.47 ! [A: $tType,P3: A > $o,Q3: A > $o] :
% 0.30/0.47 ( ! [X: A] :
% 0.30/0.47 ( ( P3 @ X )
% 0.30/0.47 = ( Q3 @ X ) )
% 0.30/0.47 => ( ( collect @ A @ P3 )
% 0.30/0.47 = ( collect @ A @ Q3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_cong
% 0.30/0.47 thf(fact_48_ext,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.47 ( ! [X: A] :
% 0.30/0.47 ( ( F @ X )
% 0.30/0.47 = ( G @ X ) )
% 0.30/0.47 => ( F = G ) ) ).
% 0.30/0.47
% 0.30/0.47 % ext
% 0.30/0.47 thf(fact_49_fm_Oinject_I5_J,axiom,
% 0.30/0.47 ! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm,Y61: paraco414474393lle_fm,Y62: paraco414474393lle_fm] :
% 0.30/0.47 ( ( ( paraco1628874225le_Eql @ X612 @ X622 )
% 0.30/0.47 = ( paraco1628874225le_Eql @ Y61 @ Y62 ) )
% 0.30/0.47 = ( ( X612 = Y61 )
% 0.30/0.47 & ( X622 = Y62 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.inject(5)
% 0.30/0.47 thf(fact_50_fm_Osimps_I41_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco1628874225le_Eql @ X612 @ X622 ) )
% 0.30/0.47 = ( F6 @ X612 @ X622 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.simps(41)
% 0.30/0.47 thf(fact_51_fm_Odistinct_I23_J,axiom,
% 0.30/0.47 ! [X3: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco329115265le_Neg @ X3 )
% 0.30/0.47 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(23)
% 0.30/0.47 thf(fact_52_fm_Odistinct_I29_J,axiom,
% 0.30/0.47 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 0.30/0.47 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(29)
% 0.30/0.47 thf(fact_53_fm_Odistinct_I9_J,axiom,
% 0.30/0.47 ! [X1: list @ char,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.30/0.47 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(9)
% 0.30/0.47 thf(fact_54_fm_Odistinct_I27_J,axiom,
% 0.30/0.47 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 0.30/0.47 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 0.30/0.47 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.distinct(27)
% 0.30/0.47 thf(fact_55_old_Oprod_Oinject,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.30/0.47 = ( product_Pair @ A @ B @ A4 @ B3 ) )
% 0.30/0.47 = ( ( A2 = A4 )
% 0.30/0.47 & ( B2 = B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.inject
% 0.30/0.47 thf(fact_56_prod_Oinject,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 0.30/0.47 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.30/0.47 = ( ( X1 = Y1 )
% 0.30/0.47 & ( X2 = Y2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod.inject
% 0.30/0.47 thf(fact_57_bool_Osplit__sel__asm,axiom,
% 0.30/0.47 ! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
% 0.30/0.47 ( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 0.30/0.47 = ( ~ ( ( Bool
% 0.30/0.47 & ~ ( P3 @ F1 ) )
% 0.30/0.47 | ( ~ Bool
% 0.30/0.47 & ~ ( P3 @ F2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bool.split_sel_asm
% 0.30/0.47 thf(fact_58_bool_Ocase__distrib,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,H: A > B,F1: A,F2: A,Bool: $o] :
% 0.30/0.47 ( ( H @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 0.30/0.47 = ( product_case_bool @ B @ ( H @ F1 ) @ ( H @ F2 ) @ Bool ) ) ).
% 0.30/0.47
% 0.30/0.47 % bool.case_distrib
% 0.30/0.47 thf(fact_59_bool_Ocase__eq__if,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( product_case_bool @ A )
% 0.30/0.47 = ( ^ [F12: A,F22: A,Bool2: $o] : ( if @ A @ Bool2 @ F12 @ F22 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bool.case_eq_if
% 0.30/0.47 thf(fact_60_bool_Osplit__sel,axiom,
% 0.30/0.47 ! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
% 0.30/0.47 ( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 0.30/0.47 = ( ( Bool
% 0.30/0.47 => ( P3 @ F1 ) )
% 0.30/0.47 & ( ~ Bool
% 0.30/0.47 => ( P3 @ F2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bool.split_sel
% 0.30/0.47 thf(fact_61_old_Obool_Osimps_I3_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: A,F2: A] :
% 0.30/0.47 ( ( product_case_bool @ A @ F1 @ F2 @ $true )
% 0.30/0.47 = F1 ) ).
% 0.30/0.47
% 0.30/0.47 % old.bool.simps(3)
% 0.30/0.47 thf(fact_62_old_Obool_Osimps_I4_J,axiom,
% 0.30/0.47 ! [A: $tType,F1: A,F2: A] :
% 0.30/0.47 ( ( product_case_bool @ A @ F1 @ F2 @ $false )
% 0.30/0.47 = F2 ) ).
% 0.30/0.47
% 0.30/0.47 % old.bool.simps(4)
% 0.30/0.47 thf(fact_63_surj__pair,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.30/0.47 ? [X: A,Y4: B] :
% 0.30/0.47 ( P2
% 0.30/0.47 = ( product_Pair @ A @ B @ X @ Y4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_pair
% 0.30/0.47 thf(fact_64_prod__cases,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.30/0.47 ( ! [A5: A,B4: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.30/0.47 => ( P3 @ P2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases
% 0.30/0.47 thf(fact_65_Pair__inject,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.30/0.47 = ( product_Pair @ A @ B @ A4 @ B3 ) )
% 0.30/0.47 => ~ ( ( A2 = A4 )
% 0.30/0.47 => ( B2 != B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pair_inject
% 0.30/0.47 thf(fact_66_prod__cases3,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.47 ~ ! [A5: A,B4: B,C2: C] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases3
% 0.30/0.47 thf(fact_67_prod__cases4,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.30/0.47 ~ ! [A5: A,B4: B,C2: C,D2: D] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases4
% 0.30/0.47 thf(fact_68_prod__cases5,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.30/0.47 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases5
% 0.30/0.47 thf(fact_69_prod__cases6,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F7: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) )] :
% 0.30/0.47 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F8: F7] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F7 ) @ D2 @ ( product_Pair @ E @ F7 @ E2 @ F8 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases6
% 0.30/0.47 thf(fact_70_prod__cases7,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F7: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) )] :
% 0.30/0.47 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F8: F7,G3: G2] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F7 @ G2 ) @ E2 @ ( product_Pair @ F7 @ G2 @ F8 @ G3 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases7
% 0.30/0.47 thf(fact_71_prod__induct3,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.47 ( ! [A5: A,B4: B,C2: C] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 0.30/0.47 => ( P3 @ X4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct3
% 0.30/0.47 thf(fact_72_prod__induct4,axiom,
% 0.30/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.30/0.47 ( ! [A5: A,B4: B,C2: C,D2: D] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.30/0.47 => ( P3 @ X4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct4
% 0.30/0.47 thf(fact_73_prod__induct5,axiom,
% 0.30/0.47 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.30/0.47 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.30/0.47 => ( P3 @ X4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct5
% 0.30/0.47 thf(fact_74_prod__induct6,axiom,
% 0.30/0.47 ! [F7: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) )] :
% 0.30/0.47 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F8: F7] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F7 ) @ D2 @ ( product_Pair @ E @ F7 @ E2 @ F8 ) ) ) ) ) )
% 0.30/0.47 => ( P3 @ X4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct6
% 0.30/0.47 thf(fact_75_prod__induct7,axiom,
% 0.30/0.47 ! [G2: $tType,F7: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) )] :
% 0.30/0.47 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F8: F7,G3: G2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F7 @ G2 ) @ E2 @ ( product_Pair @ F7 @ G2 @ F8 @ G3 ) ) ) ) ) ) )
% 0.30/0.47 => ( P3 @ X4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct7
% 0.30/0.47 thf(fact_76_old_Oprod_Oexhaust,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.30/0.47 ~ ! [A5: A,B4: B] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.exhaust
% 0.30/0.47 thf(fact_77_old_Oprod_Oinducts,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.30/0.47 ( ! [A5: A,B4: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.30/0.47 => ( P3 @ Prod ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.inducts
% 0.30/0.47 thf(fact_78_old_Oprod_Orec,axiom,
% 0.30/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.30/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.30/0.47 = ( F1 @ A2 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.rec
% 0.30/0.47 thf(fact_79_internal__case__prod__conv,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 0.30/0.47 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.30/0.47 = ( C3 @ A2 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % internal_case_prod_conv
% 0.30/0.47 thf(fact_80_ssubst__Pair__rhs,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S3: B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
% 0.30/0.47 => ( ( S3 = S )
% 0.30/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ssubst_Pair_rhs
% 0.30/0.47 thf(fact_81_curry__conv,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.47 ( ( product_curry @ B @ C @ A )
% 0.30/0.47 = ( ^ [F9: ( product_prod @ B @ C ) > A,A6: B,B5: C] : ( F9 @ ( product_Pair @ B @ C @ A6 @ B5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % curry_conv
% 0.30/0.47 thf(fact_82_curryI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.30/0.47 ( ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.30/0.47 => ( product_curry @ A @ B @ $o @ F @ A2 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % curryI
% 0.30/0.47 thf(fact_83_swap__simp,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X4: B,Y: A] :
% 0.30/0.47 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y ) )
% 0.30/0.47 = ( product_Pair @ A @ B @ Y @ X4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % swap_simp
% 0.30/0.47 thf(fact_84_eval_Osimps_I6_J,axiom,
% 0.30/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.30/0.47 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 = ( paraco876059933e_eval @ I2 @ Q ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.30/0.47 = ( paraco2040174112le_Det @ $true ) ) )
% 0.30/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.30/0.47 != ( paraco876059933e_eval @ I2 @ Q ) )
% 0.30/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.30/0.47 = ( product_case_prod @ paraco415392788lle_tv @ paraco415392788lle_tv @ paraco415392788lle_tv
% 0.30/0.47 @ ^ [X5: paraco415392788lle_tv,Y5: paraco415392788lle_tv] :
% 0.30/0.47 ( paraco490622181ase_tv @ paraco415392788lle_tv
% 0.30/0.47 @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco876059933e_eval @ I2 @ Q )
% 0.30/0.47 @ ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco876059933e_eval @ I2 @ P2 ) @ ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ Q ) ) )
% 0.30/0.47 @ ^ [Nat: nat] : ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ Q ) )
% 0.30/0.47 @ Y5 ) )
% 0.30/0.47 @ ^ [Nat: nat] :
% 0.30/0.47 ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco876059933e_eval @ I2 @ P2 ) @ ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 0.30/0.47 @ ^ [Nata: nat] : ( paraco2040174112le_Det @ $false )
% 0.30/0.47 @ Y5 )
% 0.30/0.47 @ X5 )
% 0.30/0.47 @ ( product_Pair @ paraco415392788lle_tv @ paraco415392788lle_tv @ ( paraco876059933e_eval @ I2 @ P2 ) @ ( paraco876059933e_eval @ I2 @ Q ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eval.simps(6)
% 0.30/0.47 thf(fact_85_swap__swap,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.30/0.47 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 0.30/0.47 = P2 ) ).
% 0.30/0.47
% 0.30/0.47 % swap_swap
% 0.30/0.47 thf(fact_86_case__prod__conv,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
% 0.30/0.47 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.30/0.47 = ( F @ A2 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % case_prod_conv
% 0.30/0.47 thf(fact_87_case__prod__curry,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.30/0.47 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 0.30/0.47 = F ) ).
% 0.30/0.47
% 0.30/0.47 % case_prod_curry
% 0.30/0.47 thf(fact_88_curry__case__prod,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 0.30/0.47 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
% 0.30/0.47 = F ) ).
% 0.30/0.47
% 0.30/0.47 % curry_case_prod
% 0.30/0.47 thf(fact_89_case__swap,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,F: C > B > A,P2: product_prod @ C @ B] :
% 0.30/0.47 ( ( product_case_prod @ B @ C @ A
% 0.30/0.47 @ ^ [Y5: B,X5: C] : ( F @ X5 @ Y5 )
% 0.30/0.47 @ ( product_swap @ C @ B @ P2 ) )
% 0.30/0.47 = ( product_case_prod @ C @ B @ A @ F @ P2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % case_swap
% 0.30/0.47 thf(fact_90_cond__case__prod__eta,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
% 0.30/0.47 ( ! [X: A,Y4: B] :
% 0.30/0.47 ( ( F @ X @ Y4 )
% 0.30/0.47 = ( G @ ( product_Pair @ A @ B @ X @ Y4 ) ) )
% 0.30/0.47 => ( ( product_case_prod @ A @ B @ C @ F )
% 0.30/0.47 = G ) ) ).
% 0.30/0.47
% 0.30/0.47 % cond_case_prod_eta
% 0.30/0.47 thf(fact_91_case__prod__eta,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.30/0.47 ( ( product_case_prod @ A @ B @ C
% 0.30/0.47 @ ^ [X5: A,Y5: B] : ( F @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) )
% 0.30/0.47 = F ) ).
% 0.30/0.47
% 0.30/0.47 % case_prod_eta
% 0.30/0.47 thf(fact_92_case__prodE2,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,C: $tType,Q3: A > $o,P3: B > C > A,Z: product_prod @ B @ C] :
% 0.30/0.47 ( ( Q3 @ ( product_case_prod @ B @ C @ A @ P3 @ Z ) )
% 0.30/0.47 => ~ ! [X: B,Y4: C] :
% 0.30/0.47 ( ( Z
% 0.30/0.47 = ( product_Pair @ B @ C @ X @ Y4 ) )
% 0.30/0.47 => ~ ( Q3 @ ( P3 @ X @ Y4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % case_prodE2
% 0.30/0.47 thf(fact_93_curry__def,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.47 ( ( product_curry @ A @ B @ C )
% 0.30/0.47 = ( ^ [C4: ( product_prod @ A @ B ) > C,X5: A,Y5: B] : ( C4 @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % curry_def
% 0.30/0.47 thf(fact_94_curry__K,axiom,
% 0.30/0.47 ! [B: $tType,C: $tType,A: $tType,C3: C] :
% 0.30/0.47 ( ( product_curry @ A @ B @ C
% 0.30/0.47 @ ^ [X5: product_prod @ A @ B] : C3 )
% 0.30/0.47 = ( ^ [X5: A,Y5: B] : C3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % curry_K
% 0.30/0.47 thf(fact_95_prod_Ocase__distrib,axiom,
% 0.30/0.47 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.30/0.47 ( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.30/0.47 = ( product_case_prod @ A @ B @ D
% 0.30/0.47 @ ^ [X13: A,X23: B] : ( H @ ( F @ X13 @ X23 ) )
% 0.30/0.47 @ Prod ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod.case_distrib
% 0.30/0.47 thf(fact_96_internal__case__prod__def,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.47 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.30/0.47 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.30/0.47
% 0.30/0.47 % internal_case_prod_def
% 0.30/0.47 thf(fact_97_tv_Ocase__distrib,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,H: A > B,F1: $o > A,F2: nat > A,Tv: paraco415392788lle_tv] :
% 0.30/0.47 ( ( H @ ( paraco490622181ase_tv @ A @ F1 @ F2 @ Tv ) )
% 0.30/0.47 = ( paraco490622181ase_tv @ B
% 0.30/0.47 @ ^ [X5: $o] : ( H @ ( F1 @ X5 ) )
% 0.30/0.47 @ ^ [X5: nat] : ( H @ ( F2 @ X5 ) )
% 0.30/0.47 @ Tv ) ) ).
% 0.30/0.47
% 0.30/0.47 % tv.case_distrib
% 0.30/0.47 thf(fact_98_fm_Ocase__distrib,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,H: A > B,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,Fm: paraco414474393lle_fm] :
% 0.30/0.47 ( ( H @ ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ Fm ) )
% 0.30/0.47 = ( paraco1246693743ase_fm @ B
% 0.30/0.47 @ ^ [X5: list @ char] : ( H @ ( F1 @ X5 ) )
% 0.30/0.47 @ ( H @ F2 )
% 0.30/0.47 @ ^ [X5: paraco414474393lle_fm] : ( H @ ( F3 @ X5 ) )
% 0.30/0.47 @ ^ [X13: paraco414474393lle_fm,X23: paraco414474393lle_fm] : ( H @ ( F4 @ X13 @ X23 ) )
% 0.30/0.47 @ ^ [X13: paraco414474393lle_fm,X23: paraco414474393lle_fm] : ( H @ ( F5 @ X13 @ X23 ) )
% 0.30/0.47 @ ^ [X13: paraco414474393lle_fm,X23: paraco414474393lle_fm] : ( H @ ( F6 @ X13 @ X23 ) )
% 0.30/0.47 @ Fm ) ) ).
% 0.30/0.47
% 0.30/0.47 % fm.case_distrib
% 0.30/0.47 thf(fact_99_old_Oprod_Ocase,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X2: B] :
% 0.30/0.47 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 0.30/0.47 = ( F @ X1 @ X2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.case
% 0.30/0.47 thf(fact_100_curryE,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.30/0.47 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 0.30/0.47 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % curryE
% 0.30/0.47 thf(fact_101_curryD,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.30/0.47 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 0.30/0.47 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % curryD
% 0.30/0.47 thf(fact_102_split__cong,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,Q: product_prod @ A @ B,F: A > B > C,G: A > B > C,P2: product_prod @ A @ B] :
% 0.30/0.47 ( ! [X: A,Y4: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ X @ Y4 )
% 0.30/0.47 = Q )
% 0.30/0.47 => ( ( F @ X @ Y4 )
% 0.30/0.47 = ( G @ X @ Y4 ) ) )
% 0.30/0.47 => ( ( P2 = Q )
% 0.30/0.47 => ( ( product_case_prod @ A @ B @ C @ F @ P2 )
% 0.30/0.47 = ( product_case_prod @ A @ B @ C @ G @ Q ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % split_cong
% 0.30/0.47 thf(fact_103_case__bool__if,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( product_case_bool @ A )
% 0.30/0.47 = ( ^ [X5: A,Y5: A,P5: $o] : ( if @ A @ P5 @ X5 @ Y5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % case_bool_if
% 0.30/0.47 thf(fact_104_case__prod__app,axiom,
% 0.30/0.47 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.30/0.47 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 0.30/0.47 = ( ^ [F9: B > C > D > A,X5: product_prod @ B @ C,Y5: D] :
% 0.30/0.47 ( product_case_prod @ B @ C @ A
% 0.30/0.47 @ ^ [L: B,R3: C] : ( F9 @ L @ R3 @ Y5 )
% 0.30/0.47 @ X5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % case_prod_app
% 0.30/0.47 thf(fact_105_pred__equals__eq2,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
% 0.30/0.47 ( ( ( ^ [X5: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) @ R2 ) )
% 0.30/0.47 = ( ^ [X5: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) @ S4 ) ) )
% 0.30/0.47 = ( R2 = S4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % pred_equals_eq2
% 0.30/0.47 thf(fact_106_pair__imageI,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set @ ( product_prod @ A @ B ),F: A > B > C] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A3 )
% 0.30/0.47 => ( member @ C @ ( F @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % pair_imageI
% 0.30/0.47 thf(fact_107_scomp__apply,axiom,
% 0.30/0.47 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.30/0.47 ( ( product_scomp @ B @ C @ D @ A )
% 0.30/0.47 = ( ^ [F9: B > ( product_prod @ C @ D ),G4: C > D > A,X5: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F9 @ X5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % scomp_apply
% 0.30/0.47 thf(fact_108_case__prodI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
% 0.30/0.47 ( ( F @ A2 @ B2 )
% 0.30/0.47 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % case_prodI
% 0.30/0.47 thf(fact_109_case__prodI2,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,C3: A > B > $o] :
% 0.30/0.47 ( ! [A5: A,B4: B] :
% 0.30/0.47 ( ( P2
% 0.30/0.47 = ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.30/0.47 => ( C3 @ A5 @ B4 ) )
% 0.30/0.47 => ( product_case_prod @ A @ B @ $o @ C3 @ P2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % case_prodI2
% 0.30/0.47 thf(fact_110_mem__case__prodI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),A2: B,B2: C] :
% 0.30/0.47 ( ( member @ A @ Z @ ( C3 @ A2 @ B2 ) )
% 0.30/0.47 => ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_case_prodI
% 0.30/0.47 thf(fact_111_mem__case__prodI2,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z: C,C3: A > B > ( set @ C )] :
% 0.30/0.47 ( ! [A5: A,B4: B] :
% 0.30/0.47 ( ( P2
% 0.30/0.47 = ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.30/0.47 => ( member @ C @ Z @ ( C3 @ A5 @ B4 ) ) )
% 0.30/0.47 => ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_case_prodI2
% 0.30/0.47 thf(fact_112_case__prodI2_H,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C3: A > B > C > $o,X4: C] :
% 0.30/0.47 ( ! [A5: A,B4: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ A5 @ B4 )
% 0.30/0.47 = P2 )
% 0.30/0.47 => ( C3 @ A5 @ B4 @ X4 ) )
% 0.30/0.47 => ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ X4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % case_prodI2'
% 0.30/0.47 thf(fact_113_pair__in__swap__image,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,Y: A,X4: B,A3: set @ ( product_prod @ B @ A )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X4 ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A3 ) )
% 0.30/0.47 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y ) @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % pair_in_swap_image
% 0.30/0.47 thf(fact_114_case__prod__Pair__iden,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.30/0.47 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
% 0.30/0.47 = P2 ) ).
% 0.30/0.47
% 0.30/0.47 % case_prod_Pair_iden
% 0.30/0.47 thf(fact_115_scomp__scomp,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,D: $tType,B: $tType,F7: $tType,E: $tType,F: A > ( product_prod @ E @ F7 ),G: E > F7 > ( product_prod @ C @ D ),H: C > D > B] :
% 0.30/0.47 ( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F7 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
% 0.30/0.47 = ( product_scomp @ A @ E @ F7 @ B @ F
% 0.30/0.47 @ ^ [X5: E] : ( product_scomp @ F7 @ C @ D @ B @ ( G @ X5 ) @ H ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % scomp_scomp
% 0.30/0.47 thf(fact_116_mem__case__prodE,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 0.30/0.47 ( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
% 0.30/0.47 => ~ ! [X: B,Y4: C] :
% 0.30/0.47 ( ( P2
% 0.30/0.47 = ( product_Pair @ B @ C @ X @ Y4 ) )
% 0.30/0.47 => ~ ( member @ A @ Z @ ( C3 @ X @ Y4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_case_prodE
% 0.30/0.47 thf(fact_117_case__prodD,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
% 0.30/0.47 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.30/0.47 => ( F @ A2 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % case_prodD
% 0.30/0.47 thf(fact_118_case__prodE,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C3: A > B > $o,P2: product_prod @ A @ B] :
% 0.30/0.47 ( ( product_case_prod @ A @ B @ $o @ C3 @ P2 )
% 0.30/0.47 => ~ ! [X: A,Y4: B] :
% 0.30/0.47 ( ( P2
% 0.30/0.47 = ( product_Pair @ A @ B @ X @ Y4 ) )
% 0.30/0.47 => ~ ( C3 @ X @ Y4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % case_prodE
% 0.30/0.47 thf(fact_119_case__prodD_H,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,C: $tType,R2: A > B > C > $o,A2: A,B2: B,C3: C] :
% 0.30/0.47 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R2 @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C3 )
% 0.30/0.47 => ( R2 @ A2 @ B2 @ C3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % case_prodD'
% 0.30/0.47 thf(fact_120_case__prodE_H,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P2: product_prod @ A @ B,Z: C] :
% 0.30/0.47 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ Z )
% 0.30/0.47 => ~ ! [X: A,Y4: B] :
% 0.30/0.47 ( ( P2
% 0.30/0.47 = ( product_Pair @ A @ B @ X @ Y4 ) )
% 0.30/0.47 => ~ ( C3 @ X @ Y4 @ Z ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % case_prodE'
% 0.30/0.47 thf(fact_121_scomp__Pair,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,X4: A > ( product_prod @ B @ C )] :
% 0.30/0.47 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C ) )
% 0.30/0.47 = X4 ) ).
% 0.30/0.47
% 0.30/0.47 % scomp_Pair
% 0.30/0.47 thf(fact_122_Pair__scomp,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,X4: C,F: C > A > B] :
% 0.30/0.47 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X4 ) @ F )
% 0.30/0.47 = ( F @ X4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pair_scomp
% 0.30/0.47 thf(fact_123_scomp__def,axiom,
% 0.30/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.47 ( ( product_scomp @ A @ B @ C @ D )
% 0.30/0.47 = ( ^ [F9: A > ( product_prod @ B @ C ),G4: B > C > D,X5: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F9 @ X5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % scomp_def
% 0.30/0.47 thf(fact_124_image__ident,axiom,
% 0.30/0.47 ! [A: $tType,Y6: set @ A] :
% 0.30/0.47 ( ( image @ A @ A
% 0.30/0.47 @ ^ [X5: A] : X5
% 0.30/0.47 @ Y6 )
% 0.30/0.47 = Y6 ) ).
% 0.30/0.47
% 0.30/0.47 % image_ident
% 0.30/0.47 thf(fact_125_image__eqI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B2: A,F: B > A,X4: B,A3: set @ B] :
% 0.30/0.47 ( ( B2
% 0.30/0.47 = ( F @ X4 ) )
% 0.30/0.47 => ( ( member @ B @ X4 @ A3 )
% 0.30/0.47 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_eqI
% 0.30/0.47 thf(fact_126_Sup_OSUP__identity__eq,axiom,
% 0.30/0.47 ! [A: $tType,Sup: ( set @ A ) > A,A3: set @ A] :
% 0.30/0.47 ( ( Sup
% 0.30/0.47 @ ( image @ A @ A
% 0.30/0.47 @ ^ [X5: A] : X5
% 0.30/0.47 @ A3 ) )
% 0.30/0.47 = ( Sup @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sup.SUP_identity_eq
% 0.30/0.47 thf(fact_127_Inf_OINF__identity__eq,axiom,
% 0.30/0.47 ! [A: $tType,Inf: ( set @ A ) > A,A3: set @ A] :
% 0.30/0.47 ( ( Inf
% 0.30/0.47 @ ( image @ A @ A
% 0.30/0.47 @ ^ [X5: A] : X5
% 0.30/0.47 @ A3 ) )
% 0.30/0.47 = ( Inf @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Inf.INF_identity_eq
% 0.30/0.47 thf(fact_128_Compr__image__eq,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
% 0.30/0.47 ( ( collect @ A
% 0.30/0.47 @ ^ [X5: A] :
% 0.30/0.47 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A3 ) )
% 0.30/0.47 & ( P3 @ X5 ) ) )
% 0.30/0.47 = ( image @ B @ A @ F
% 0.30/0.47 @ ( collect @ B
% 0.30/0.47 @ ^ [X5: B] :
% 0.30/0.47 ( ( member @ B @ X5 @ A3 )
% 0.30/0.47 & ( P3 @ ( F @ X5 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Compr_image_eq
% 0.30/0.47 thf(fact_129_split__part,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P3: $o,Q3: A > B > $o] :
% 0.30/0.47 ( ( product_case_prod @ A @ B @ $o
% 0.30/0.47 @ ^ [A6: A,B5: B] :
% 0.30/0.47 ( P3
% 0.30/0.47 & ( Q3 @ A6 @ B5 ) ) )
% 0.30/0.47 = ( ^ [Ab: product_prod @ A @ B] :
% 0.30/0.47 ( P3
% 0.30/0.47 & ( product_case_prod @ A @ B @ $o @ Q3 @ Ab ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % split_part
% 0.30/0.47 thf(fact_130_prod_Odisc__eq__case,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.30/0.47 ( product_case_prod @ A @ B @ $o
% 0.30/0.47 @ ^ [Uu: A,Uv: B] : $true
% 0.30/0.47 @ Prod ) ).
% 0.30/0.47
% 0.30/0.47 % prod.disc_eq_case
% 0.30/0.47 thf(fact_131_Sup_OSUP__cong,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A3: set @ B,B6: set @ B,C5: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 0.30/0.47 ( ( A3 = B6 )
% 0.30/0.47 => ( ! [X: B] :
% 0.30/0.47 ( ( member @ B @ X @ B6 )
% 0.30/0.47 => ( ( C5 @ X )
% 0.30/0.47 = ( D3 @ X ) ) )
% 0.30/0.47 => ( ( Sup @ ( image @ B @ A @ C5 @ A3 ) )
% 0.30/0.47 = ( Sup @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sup.SUP_cong
% 0.30/0.47 thf(fact_132_Inf_OINF__cong,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A3: set @ B,B6: set @ B,C5: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 0.30/0.47 ( ( A3 = B6 )
% 0.30/0.47 => ( ! [X: B] :
% 0.30/0.47 ( ( member @ B @ X @ B6 )
% 0.30/0.47 => ( ( C5 @ X )
% 0.30/0.47 = ( D3 @ X ) ) )
% 0.30/0.47 => ( ( Inf @ ( image @ B @ A @ C5 @ A3 ) )
% 0.30/0.47 = ( Inf @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Inf.INF_cong
% 0.30/0.47 thf(fact_133_rev__image__eqI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X4: A,A3: set @ A,B2: B,F: A > B] :
% 0.30/0.47 ( ( member @ A @ X4 @ A3 )
% 0.30/0.47 => ( ( B2
% 0.30/0.47 = ( F @ X4 ) )
% 0.30/0.47 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rev_image_eqI
% 0.30/0.47 thf(fact_134_ball__imageD,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
% 0.30/0.47 ( ! [X: A] :
% 0.30/0.47 ( ( member @ A @ X @ ( image @ B @ A @ F @ A3 ) )
% 0.30/0.47 => ( P3 @ X ) )
% 0.30/0.47 => ! [X6: B] :
% 0.30/0.47 ( ( member @ B @ X6 @ A3 )
% 0.30/0.47 => ( P3 @ ( F @ X6 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ball_imageD
% 0.30/0.47 thf(fact_135_image__cong,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F: A > B,G: A > B] :
% 0.30/0.47 ( ( M = N2 )
% 0.30/0.47 => ( ! [X: A] :
% 0.30/0.47 ( ( member @ A @ X @ N2 )
% 0.30/0.47 => ( ( F @ X )
% 0.30/0.47 = ( G @ X ) ) )
% 0.30/0.47 => ( ( image @ A @ B @ F @ M )
% 0.30/0.47 = ( image @ A @ B @ G @ N2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_cong
% 0.30/0.47 thf(fact_136_bex__imageD,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
% 0.30/0.47 ( ? [X6: A] :
% 0.30/0.47 ( ( member @ A @ X6 @ ( image @ B @ A @ F @ A3 ) )
% 0.30/0.47 & ( P3 @ X6 ) )
% 0.30/0.47 => ? [X: B] :
% 0.30/0.47 ( ( member @ B @ X @ A3 )
% 0.30/0.47 & ( P3 @ ( F @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bex_imageD
% 0.30/0.47 thf(fact_137_image__iff,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,Z: A,F: B > A,A3: set @ B] :
% 0.30/0.47 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A3 ) )
% 0.30/0.47 = ( ? [X5: B] :
% 0.30/0.47 ( ( member @ B @ X5 @ A3 )
% 0.30/0.47 & ( Z
% 0.30/0.47 = ( F @ X5 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_iff
% 0.30/0.47 thf(fact_138_imageI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X4: A,A3: set @ A,F: A > B] :
% 0.30/0.47 ( ( member @ A @ X4 @ A3 )
% 0.30/0.47 => ( member @ B @ ( F @ X4 ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_139_imageE,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B2: A,F: B > A,A3: set @ B] :
% 0.30/0.47 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) )
% 0.30/0.47 => ~ ! [X: B] :
% 0.30/0.47 ( ( B2
% 0.30/0.47 = ( F @ X ) )
% 0.30/0.47 => ~ ( member @ B @ X @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageE
% 0.30/0.47 thf(fact_140_image__image,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A3: set @ C] :
% 0.30/0.47 ( ( image @ B @ A @ F @ ( image @ C @ B @ G @ A3 ) )
% 0.30/0.47 = ( image @ C @ A
% 0.30/0.47 @ ^ [X5: C] : ( F @ ( G @ X5 ) )
% 0.30/0.47 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_image
% 0.30/0.47 thf(fact_141_image__paired__Times,axiom,
% 0.30/0.47 ! [C: $tType,D: $tType,B: $tType,A: $tType,F: C > A,G: D > B,A3: set @ C,B6: set @ D] :
% 0.30/0.47 ( ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
% 0.30/0.47 @ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
% 0.30/0.47 @ ^ [X5: C,Y5: D] : ( product_Pair @ A @ B @ ( F @ X5 ) @ ( G @ Y5 ) ) )
% 0.30/0.47 @ ( product_Sigma @ C @ D @ A3
% 0.30/0.47 @ ^ [Uu: C] : B6 ) )
% 0.30/0.47 = ( product_Sigma @ A @ B @ ( image @ C @ A @ F @ A3 )
% 0.30/0.47 @ ^ [Uu: A] : ( image @ D @ B @ G @ B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_paired_Times
% 0.30/0.47 thf(fact_142_inv__image__def,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( inv_image @ B @ A )
% 0.30/0.47 = ( ^ [R3: set @ ( product_prod @ B @ B ),F9: A > B] :
% 0.30/0.47 ( collect @ ( product_prod @ A @ A )
% 0.30/0.47 @ ( product_case_prod @ A @ A @ $o
% 0.30/0.47 @ ^ [X5: A,Y5: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F9 @ X5 ) @ ( F9 @ Y5 ) ) @ R3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inv_image_def
% 0.30/0.47 thf(fact_143_same__fst__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( same_fst @ A @ B )
% 0.30/0.47 = ( ^ [P5: A > $o,R4: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.30/0.47 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 0.30/0.47 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 0.30/0.47 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 0.30/0.47 @ ^ [X7: A,Y7: B] :
% 0.30/0.47 ( product_case_prod @ A @ B @ $o
% 0.30/0.47 @ ^ [X5: A,Y5: B] :
% 0.30/0.47 ( ( X7 = X5 )
% 0.30/0.47 & ( P5 @ X5 )
% 0.30/0.47 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y5 ) @ ( R4 @ X5 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % same_fst_def
% 0.30/0.47 thf(fact_144_swap__product,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A3: set @ B,B6: set @ A] :
% 0.30/0.47 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
% 0.30/0.47 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 0.30/0.47 @ ^ [I: B,J: A] : ( product_Pair @ A @ B @ J @ I ) )
% 0.30/0.47 @ ( product_Sigma @ B @ A @ A3
% 0.30/0.47 @ ^ [Uu: B] : B6 ) )
% 0.30/0.47 = ( product_Sigma @ A @ B @ B6
% 0.30/0.47 @ ^ [Uu: A] : A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % swap_product
% 0.30/0.47 thf(fact_145_SigmaI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,B6: A > ( set @ B )] :
% 0.30/0.47 ( ( member @ A @ A2 @ A3 )
% 0.30/0.47 => ( ( member @ B @ B2 @ ( B6 @ A2 ) )
% 0.30/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SigmaI
% 0.30/0.47 thf(fact_146_mem__Sigma__iff,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B6: A > ( set @ B )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
% 0.30/0.47 = ( ( member @ A @ A2 @ A3 )
% 0.30/0.47 & ( member @ B @ B2 @ ( B6 @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_Sigma_iff
% 0.30/0.47 thf(fact_147_in__inv__image,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X4: A,Y: A,R: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y ) @ ( inv_image @ B @ A @ R @ F ) )
% 0.30/0.47 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X4 ) @ ( F @ Y ) ) @ R ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_inv_image
% 0.30/0.47 thf(fact_148_Collect__case__prod,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P3: A > $o,Q3: B > $o] :
% 0.30/0.47 ( ( collect @ ( product_prod @ A @ B )
% 0.30/0.47 @ ( product_case_prod @ A @ B @ $o
% 0.30/0.47 @ ^ [A6: A,B5: B] :
% 0.30/0.47 ( ( P3 @ A6 )
% 0.30/0.47 & ( Q3 @ B5 ) ) ) )
% 0.30/0.47 = ( product_Sigma @ A @ B @ ( collect @ A @ P3 )
% 0.30/0.47 @ ^ [Uu: A] : ( collect @ B @ Q3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_case_prod
% 0.30/0.47 thf(fact_149_same__fstI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P3: A > $o,X4: A,Y8: B,Y: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.30/0.47 ( ( P3 @ X4 )
% 0.30/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y ) @ ( R2 @ X4 ) )
% 0.30/0.47 => ( 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 @ X4 @ Y8 ) @ ( product_Pair @ A @ B @ X4 @ Y ) ) @ ( same_fst @ A @ B @ P3 @ R2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % same_fstI
% 0.30/0.47 thf(fact_150_Collect__case__prod__Sigma,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P3: A > $o,Q3: A > B > $o] :
% 0.30/0.47 ( ( collect @ ( product_prod @ A @ B )
% 0.30/0.47 @ ( product_case_prod @ A @ B @ $o
% 0.30/0.47 @ ^ [X5: A,Y5: B] :
% 0.30/0.47 ( ( P3 @ X5 )
% 0.30/0.47 & ( Q3 @ X5 @ Y5 ) ) ) )
% 0.30/0.47 = ( product_Sigma @ A @ B @ ( collect @ A @ P3 )
% 0.30/0.47 @ ^ [X5: A] : ( collect @ B @ ( Q3 @ X5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_case_prod_Sigma
% 0.30/0.47 thf(fact_151_Sigma__cong,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A3: set @ A,B6: set @ A,C5: A > ( set @ B ),D3: A > ( set @ B )] :
% 0.30/0.47 ( ( A3 = B6 )
% 0.30/0.47 => ( ! [X: A] :
% 0.30/0.47 ( ( member @ A @ X @ B6 )
% 0.30/0.47 => ( ( C5 @ X )
% 0.30/0.47 = ( D3 @ X ) ) )
% 0.30/0.47 => ( ( product_Sigma @ A @ B @ A3 @ C5 )
% 0.30/0.47 = ( product_Sigma @ A @ B @ B6 @ D3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sigma_cong
% 0.30/0.47 thf(fact_152_Times__eq__cancel2,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X4: A,C5: set @ A,A3: set @ B,B6: set @ B] :
% 0.30/0.47 ( ( member @ A @ X4 @ C5 )
% 0.30/0.47 => ( ( ( product_Sigma @ B @ A @ A3
% 0.30/0.47 @ ^ [Uu: B] : C5 )
% 0.30/0.47 = ( product_Sigma @ B @ A @ B6
% 0.30/0.47 @ ^ [Uu: B] : C5 ) )
% 0.30/0.47 = ( A3 = B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Times_eq_cancel2
% 0.30/0.47 thf(fact_153_SigmaE,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C3: product_prod @ A @ B,A3: set @ A,B6: A > ( set @ B )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
% 0.30/0.47 => ~ ! [X: A] :
% 0.30/0.47 ( ( member @ A @ X @ A3 )
% 0.30/0.47 => ! [Y4: B] :
% 0.30/0.47 ( ( member @ B @ Y4 @ ( B6 @ X ) )
% 0.30/0.47 => ( C3
% 0.30/0.47 != ( product_Pair @ A @ B @ X @ Y4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SigmaE
% 0.30/0.47 thf(fact_154_SigmaD1,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B6: A > ( set @ B )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
% 0.30/0.47 => ( member @ A @ A2 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % SigmaD1
% 0.30/0.47 thf(fact_155_SigmaD2,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B6: A > ( set @ B )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
% 0.30/0.47 => ( member @ B @ B2 @ ( B6 @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SigmaD2
% 0.30/0.47 thf(fact_156_SigmaE2,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B6: A > ( set @ B )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
% 0.30/0.47 => ~ ( ( member @ A @ A2 @ A3 )
% 0.30/0.47 => ~ ( member @ B @ B2 @ ( B6 @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SigmaE2
% 0.30/0.47 thf(fact_157_product__swap,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A3: set @ B,B6: set @ A] :
% 0.30/0.47 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
% 0.30/0.47 @ ( product_Sigma @ B @ A @ A3
% 0.30/0.47 @ ^ [Uu: B] : B6 ) )
% 0.30/0.47 = ( product_Sigma @ A @ B @ B6
% 0.30/0.47 @ ^ [Uu: A] : A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % product_swap
% 0.30/0.47 thf(fact_158_rp__inv__image__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( fun_rp_inv_image @ A @ B )
% 0.30/0.47 = ( 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 ) ) ) )
% 0.30/0.47 @ ^ [R4: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A ),F9: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R4 @ F9 ) @ ( inv_image @ A @ B @ S5 @ F9 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rp_inv_image_def
% 0.30/0.47 thf(fact_159_lex__prod__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( lex_prod @ A @ B )
% 0.30/0.47 = ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
% 0.30/0.47 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 0.30/0.47 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 0.30/0.47 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 0.30/0.47 @ ^ [A6: A,B5: B] :
% 0.30/0.47 ( product_case_prod @ A @ B @ $o
% 0.30/0.47 @ ^ [A7: A,B7: B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A7 ) @ Ra )
% 0.30/0.47 | ( ( A6 = A7 )
% 0.30/0.47 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B5 @ B7 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lex_prod_def
% 0.30/0.47 thf(fact_160_Product__Type_Oproduct__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( product_product @ A @ B )
% 0.30/0.47 = ( ^ [A8: set @ A,B8: set @ B] :
% 0.30/0.47 ( product_Sigma @ A @ B @ A8
% 0.30/0.47 @ ^ [Uu: A] : B8 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Product_Type.product_def
% 0.30/0.47 thf(fact_161_in__lex__prod,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 0.30/0.47 ( ( 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 @ A2 @ B2 ) @ ( product_Pair @ A @ B @ A4 @ B3 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
% 0.30/0.47 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A4 ) @ R )
% 0.30/0.47 | ( ( A2 = A4 )
% 0.30/0.47 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_lex_prod
% 0.30/0.47 thf(fact_162_member__product,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X4: product_prod @ A @ B,A3: set @ A,B6: set @ B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ X4 @ ( product_product @ A @ B @ A3 @ B6 ) )
% 0.30/0.47 = ( member @ ( product_prod @ A @ B ) @ X4
% 0.30/0.47 @ ( product_Sigma @ A @ B @ A3
% 0.30/0.47 @ ^ [Uu: A] : B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % member_product
% 0.30/0.47 thf(fact_163_rp__inv__image__rp,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,P3: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
% 0.30/0.47 ( ( fun_reduction_pair @ A @ P3 )
% 0.30/0.47 => ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P3 @ F ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rp_inv_image_rp
% 0.30/0.47 thf(fact_164_map__prod__surj__on,axiom,
% 0.30/0.47 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A3: set @ B,A9: set @ A,G: D > C,B6: set @ D,B9: set @ C] :
% 0.30/0.47 ( ( ( image @ B @ A @ F @ A3 )
% 0.30/0.47 = A9 )
% 0.30/0.47 => ( ( ( image @ D @ C @ G @ B6 )
% 0.30/0.47 = B9 )
% 0.30/0.47 => ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G )
% 0.30/0.47 @ ( product_Sigma @ B @ D @ A3
% 0.30/0.47 @ ^ [Uu: B] : B6 ) )
% 0.30/0.47 = ( product_Sigma @ A @ C @ A9
% 0.30/0.47 @ ^ [Uu: A] : B9 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_prod_surj_on
% 0.30/0.47 thf(fact_165_relation__of__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order_relation_of @ A )
% 0.30/0.47 = ( ^ [P5: A > A > $o,A8: set @ A] :
% 0.30/0.47 ( collect @ ( product_prod @ A @ A )
% 0.30/0.47 @ ( product_case_prod @ A @ A @ $o
% 0.30/0.47 @ ^ [A6: A,B5: A] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 )
% 0.30/0.47 @ ( product_Sigma @ A @ A @ A8
% 0.30/0.47 @ ^ [Uu: A] : A8 ) )
% 0.30/0.47 & ( P5 @ A6 @ B5 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % relation_of_def
% 0.30/0.47 thf(fact_166_map__prod__ident,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( product_map_prod @ A @ A @ B @ B
% 0.30/0.47 @ ^ [X5: A] : X5
% 0.30/0.47 @ ^ [Y5: B] : Y5 )
% 0.30/0.47 = ( ^ [Z2: product_prod @ A @ B] : Z2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_prod_ident
% 0.30/0.47 thf(fact_167_map__prod__simp,axiom,
% 0.30/0.47 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A2: C,B2: D] :
% 0.30/0.47 ( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 0.30/0.47 = ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_prod_simp
% 0.30/0.47 thf(fact_168_map__prod__imageI,axiom,
% 0.30/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),F: A > C,G: B > D] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 0.30/0.47 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A2 ) @ ( G @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) @ R2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_prod_imageI
% 0.30/0.47 thf(fact_169_case__prod__map__prod,axiom,
% 0.30/0.47 ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G: E > C,X4: product_prod @ D @ E] :
% 0.30/0.47 ( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G @ X4 ) )
% 0.30/0.47 = ( product_case_prod @ D @ E @ A
% 0.30/0.47 @ ^ [L: D,R3: E] : ( H @ ( F @ L ) @ ( G @ R3 ) )
% 0.30/0.47 @ X4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % case_prod_map_prod
% 0.30/0.47 thf(fact_170_prod__fun__imageE,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F: C > A,G: D > B,R2: set @ ( product_prod @ C @ D )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G ) @ R2 ) )
% 0.30/0.47 => ~ ! [X: C,Y4: D] :
% 0.30/0.47 ( ( C3
% 0.30/0.47 = ( product_Pair @ A @ B @ ( F @ X ) @ ( G @ Y4 ) ) )
% 0.30/0.47 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X @ Y4 ) @ R2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_fun_imageE
% 0.30/0.47 thf(fact_171_map__prod__def,axiom,
% 0.30/0.47 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 0.30/0.47 ( ( product_map_prod @ A @ C @ B @ D )
% 0.30/0.47 = ( ^ [F9: A > C,G4: B > D] :
% 0.30/0.47 ( product_case_prod @ A @ B @ ( product_prod @ C @ D )
% 0.30/0.47 @ ^ [X5: A,Y5: B] : ( product_Pair @ C @ D @ ( F9 @ X5 ) @ ( G4 @ Y5 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_prod_def
% 0.30/0.47 thf(fact_172_prod_Omap__ident,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.30/0.47 ( ( product_map_prod @ A @ A @ B @ B
% 0.30/0.47 @ ^ [X5: A] : X5
% 0.30/0.47 @ ^ [X5: B] : X5
% 0.30/0.47 @ T2 )
% 0.30/0.47 = T2 ) ).
% 0.30/0.47
% 0.30/0.47 % prod.map_ident
% 0.30/0.47 thf(fact_173_aboveS__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order_aboveS @ A )
% 0.30/0.47 = ( ^ [R3: set @ ( product_prod @ A @ A ),A6: A] :
% 0.30/0.47 ( collect @ A
% 0.30/0.47 @ ^ [B5: A] :
% 0.30/0.47 ( ( B5 != A6 )
% 0.30/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % aboveS_def
% 0.30/0.47 thf(fact_174_above__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order_above @ A )
% 0.30/0.47 = ( ^ [R3: set @ ( product_prod @ A @ A ),A6: A] :
% 0.30/0.47 ( collect @ A
% 0.30/0.47 @ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % above_def
% 0.30/0.47 thf(fact_175_surj__swap,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.30/0.47 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_swap
% 0.30/0.47 thf(fact_176_The__split__eq,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X4: A,Y: B] :
% 0.30/0.47 ( ( the @ ( product_prod @ A @ B )
% 0.30/0.47 @ ( product_case_prod @ A @ B @ $o
% 0.30/0.47 @ ^ [X7: A,Y7: B] :
% 0.30/0.47 ( ( X4 = X7 )
% 0.30/0.47 & ( Y = Y7 ) ) ) )
% 0.30/0.47 = ( product_Pair @ A @ B @ X4 @ Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % The_split_eq
% 0.30/0.47 thf(fact_177_in__inv__imagep,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( inv_imagep @ A @ B )
% 0.30/0.47 = ( ^ [R3: A > A > $o,F9: B > A,X5: B,Y5: B] : ( R3 @ ( F9 @ X5 ) @ ( F9 @ Y5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_inv_imagep
% 0.30/0.47 thf(fact_178_UNIV__Times__UNIV,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 0.30/0.47 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
% 0.30/0.47 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_Times_UNIV
% 0.30/0.47 thf(fact_179_rangeE,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B2: A,F: B > A] :
% 0.30/0.47 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
% 0.30/0.47 => ~ ! [X: B] :
% 0.30/0.47 ( B2
% 0.30/0.47 != ( F @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rangeE
% 0.30/0.47 thf(fact_180_range__composition,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType,F: C > A,G: B > C] :
% 0.30/0.47 ( ( image @ B @ A
% 0.30/0.47 @ ^ [X5: B] : ( F @ ( G @ X5 ) )
% 0.30/0.47 @ ( top_top @ ( set @ B ) ) )
% 0.30/0.47 = ( image @ C @ A @ F @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_composition
% 0.30/0.47 thf(fact_181_UNIV__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( top_top @ ( set @ A ) )
% 0.30/0.47 = ( collect @ A
% 0.30/0.47 @ ^ [X5: A] : $true ) ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_def
% 0.30/0.47 thf(fact_182_rangeI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: B > A,X4: B] : ( member @ A @ ( F @ X4 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rangeI
% 0.30/0.47 thf(fact_183_range__eqI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B2: A,F: B > A,X4: B] :
% 0.30/0.47 ( ( B2
% 0.30/0.47 = ( F @ X4 ) )
% 0.30/0.47 => ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_eqI
% 0.30/0.47 thf(fact_184_map__prod__surj,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
% 0.30/0.47 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.47 = ( top_top @ ( set @ B ) ) )
% 0.30/0.47 => ( ( ( image @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
% 0.30/0.47 = ( top_top @ ( set @ D ) ) )
% 0.30/0.47 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.30/0.47 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_prod_surj
% 0.30/0.47 thf(fact_185_inv__imagep__def,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( inv_imagep @ B @ A )
% 0.30/0.47 = ( ^ [R3: B > B > $o,F9: A > B,X5: A,Y5: A] : ( R3 @ ( F9 @ X5 ) @ ( F9 @ Y5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inv_imagep_def
% 0.30/0.47 thf(fact_186_old_Orec__prod__def,axiom,
% 0.30/0.47 ! [T: $tType,B: $tType,A: $tType] :
% 0.30/0.47 ( ( product_rec_prod @ A @ B @ T )
% 0.30/0.47 = ( ^ [F12: A > B > T,X5: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.rec_prod_def
% 0.30/0.47 thf(fact_187_the__equality,axiom,
% 0.30/0.47 ! [A: $tType,P3: A > $o,A2: A] :
% 0.30/0.47 ( ( P3 @ A2 )
% 0.30/0.47 => ( ! [X: A] :
% 0.30/0.47 ( ( P3 @ X )
% 0.30/0.47 => ( X = A2 ) )
% 0.30/0.47 => ( ( the @ A @ P3 )
% 0.30/0.47 = A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % the_equality
% 0.30/0.47 thf(fact_188_the__eq__trivial,axiom,
% 0.30/0.47 ! [A: $tType,A2: A] :
% 0.30/0.47 ( ( the @ A
% 0.30/0.47 @ ^ [X5: A] : ( X5 = A2 ) )
% 0.30/0.47 = A2 ) ).
% 0.30/0.47
% 0.30/0.47 % the_eq_trivial
% 0.30/0.47 thf(fact_189_the__sym__eq__trivial,axiom,
% 0.30/0.47 ! [A: $tType,X4: A] :
% 0.30/0.47 ( ( the @ A
% 0.30/0.47 @ ( ^ [Y9: A,Z3: A] : ( Y9 = Z3 )
% 0.30/0.47 @ X4 ) )
% 0.30/0.47 = X4 ) ).
% 0.30/0.47
% 0.30/0.47 % the_sym_eq_trivial
% 0.30/0.47 thf(fact_190_the1__equality,axiom,
% 0.30/0.47 ! [A: $tType,P3: A > $o,A2: A] :
% 0.30/0.47 ( ? [X6: A] :
% 0.30/0.47 ( ( P3 @ X6 )
% 0.30/0.47 & ! [Y4: A] :
% 0.30/0.47 ( ( P3 @ Y4 )
% 0.30/0.47 => ( Y4 = X6 ) ) )
% 0.30/0.47 => ( ( P3 @ A2 )
% 0.30/0.47 => ( ( the @ A @ P3 )
% 0.30/0.47 = A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % the1_equality
% 0.30/0.47 thf(fact_191_the1I2,axiom,
% 0.30/0.47 ! [A: $tType,P3: A > $o,Q3: A > $o] :
% 0.30/0.47 ( ? [X6: A] :
% 0.30/0.47 ( ( P3 @ X6 )
% 0.30/0.47 & ! [Y4: A] :
% 0.30/0.47 ( ( P3 @ Y4 )
% 0.30/0.47 => ( Y4 = X6 ) ) )
% 0.30/0.47 => ( ! [X: A] :
% 0.30/0.47 ( ( P3 @ X )
% 0.30/0.47 => ( Q3 @ X ) )
% 0.30/0.47 => ( Q3 @ ( the @ A @ P3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % the1I2
% 0.30/0.47 thf(fact_192_If__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( if @ A )
% 0.30/0.47 = ( ^ [P5: $o,X5: A,Y5: A] :
% 0.30/0.47 ( the @ A
% 0.30/0.47 @ ^ [Z2: A] :
% 0.30/0.47 ( ( P5
% 0.30/0.47 => ( Z2 = X5 ) )
% 0.30/0.47 & ( ~ P5
% 0.30/0.47 => ( Z2 = Y5 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % If_def
% 0.30/0.47 thf(fact_193_theI2,axiom,
% 0.30/0.47 ! [A: $tType,P3: A > $o,A2: A,Q3: A > $o] :
% 0.30/0.47 ( ( P3 @ A2 )
% 0.30/0.47 => ( ! [X: A] :
% 0.30/0.47 ( ( P3 @ X )
% 0.30/0.47 => ( X = A2 ) )
% 0.30/0.47 => ( ! [X: A] :
% 0.30/0.47 ( ( P3 @ X )
% 0.30/0.47 => ( Q3 @ X ) )
% 0.30/0.47 => ( Q3 @ ( the @ A @ P3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % theI2
% 0.30/0.47 thf(fact_194_theI_H,axiom,
% 0.30/0.47 ! [A: $tType,P3: A > $o] :
% 0.30/0.47 ( ? [X6: A] :
% 0.30/0.47 ( ( P3 @ X6 )
% 0.30/0.47 & ! [Y4: A] :
% 0.30/0.47 ( ( P3 @ Y4 )
% 0.30/0.47 => ( Y4 = X6 ) ) )
% 0.30/0.47 => ( P3 @ ( the @ A @ P3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % theI'
% 0.30/0.48 thf(fact_195_theI,axiom,
% 0.30/0.48 ! [A: $tType,P3: A > $o,A2: A] :
% 0.30/0.48 ( ( P3 @ A2 )
% 0.30/0.48 => ( ! [X: A] :
% 0.30/0.48 ( ( P3 @ X )
% 0.30/0.48 => ( X = A2 ) )
% 0.30/0.48 => ( P3 @ ( the @ A @ P3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % theI
% 0.30/0.48 thf(fact_196_surj__def,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,F: B > A] :
% 0.30/0.48 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.48 = ( top_top @ ( set @ A ) ) )
% 0.30/0.48 = ( ! [Y5: A] :
% 0.30/0.48 ? [X5: B] :
% 0.30/0.48 ( Y5
% 0.30/0.48 = ( F @ X5 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % surj_def
% 0.30/0.48 thf(fact_197_surjI,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 0.30/0.48 ( ! [X: A] :
% 0.30/0.48 ( ( G @ ( F @ X ) )
% 0.30/0.48 = X )
% 0.30/0.48 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 0.30/0.48 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % surjI
% 0.30/0.48 thf(fact_198_surjD,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.30/0.48 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.48 = ( top_top @ ( set @ A ) ) )
% 0.30/0.48 => ? [X: B] :
% 0.30/0.48 ( Y
% 0.30/0.48 = ( F @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % surjD
% 0.30/0.48 thf(fact_199_surjE,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.30/0.48 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.48 = ( top_top @ ( set @ A ) ) )
% 0.30/0.48 => ~ ! [X: B] :
% 0.30/0.48 ( Y
% 0.30/0.48 != ( F @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % surjE
% 0.30/0.48 thf(fact_200_old_Orec__unit__def,axiom,
% 0.30/0.48 ! [T: $tType] :
% 0.30/0.48 ( ( product_rec_unit @ T )
% 0.30/0.48 = ( ^ [F12: T,X5: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X5 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % old.rec_unit_def
% 0.30/0.48 thf(fact_201_range__fst,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.30/0.48 = ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % range_fst
% 0.30/0.48 thf(fact_202_fst__map__prod,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X4: product_prod @ C @ D] :
% 0.30/0.48 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X4 ) )
% 0.30/0.48 = ( F @ ( product_fst @ C @ D @ X4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fst_map_prod
% 0.30/0.48 thf(fact_203_fst__eqD,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,X4: A,Y: B,A2: A] :
% 0.30/0.48 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X4 @ Y ) )
% 0.30/0.48 = A2 )
% 0.30/0.48 => ( X4 = A2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % fst_eqD
% 0.30/0.48 thf(fact_204_fst__conv,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,X1: A,X2: B] :
% 0.30/0.48 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 0.30/0.48 = X1 ) ).
% 0.30/0.48
% 0.30/0.48 % fst_conv
% 0.30/0.48 thf(fact_205_fst__def,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( product_fst @ A @ B )
% 0.30/0.48 = ( product_case_prod @ A @ B @ A
% 0.30/0.48 @ ^ [X13: A,X23: B] : X13 ) ) ).
% 0.30/0.48
% 0.30/0.48 % fst_def
% 0.30/0.48 thf(fact_206_eq__fst__iff,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 0.30/0.48 ( ( A2
% 0.30/0.48 = ( product_fst @ A @ B @ P2 ) )
% 0.30/0.48 = ( ? [B5: B] :
% 0.30/0.48 ( P2
% 0.30/0.48 = ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % eq_fst_iff
% 0.30/0.48 thf(fact_207_fstI,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,X4: product_prod @ A @ B,Y: A,Z: B] :
% 0.30/0.48 ( ( X4
% 0.30/0.48 = ( product_Pair @ A @ B @ Y @ Z ) )
% 0.30/0.48 => ( ( product_fst @ A @ B @ X4 )
% 0.30/0.48 = Y ) ) ).
% 0.30/0.48
% 0.30/0.48 % fstI
% 0.30/0.48 thf(fact_208_fst__image__times,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,B6: set @ B,A3: set @ A] :
% 0.30/0.48 ( ( ( B6
% 0.30/0.48 = ( bot_bot @ ( set @ B ) ) )
% 0.30/0.48 => ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 0.30/0.48 @ ( product_Sigma @ A @ B @ A3
% 0.30/0.48 @ ^ [Uu: A] : B6 ) )
% 0.30/0.48 = ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.48 & ( ( B6
% 0.30/0.48 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.48 => ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 0.30/0.48 @ ( product_Sigma @ A @ B @ A3
% 0.30/0.48 @ ^ [Uu: A] : B6 ) )
% 0.30/0.48 = A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fst_image_times
% 0.30/0.48 thf(fact_209_The__case__prod,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,P3: A > B > $o] :
% 0.30/0.48 ( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P3 ) )
% 0.30/0.48 = ( the @ ( product_prod @ A @ B )
% 0.30/0.48 @ ^ [Xy: product_prod @ A @ B] : ( P3 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % The_case_prod
% 0.30/0.48 thf(fact_210_image__is__empty,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType,F: B > A,A3: set @ B] :
% 0.30/0.48 ( ( ( image @ B @ A @ F @ A3 )
% 0.30/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.48 = ( A3
% 0.30/0.48 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % image_is_empty
% 0.30/0.48 thf(fact_211_empty__is__image,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType,F: B > A,A3: set @ B] :
% 0.30/0.48 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.48 = ( image @ B @ A @ F @ A3 ) )
% 0.30/0.48 = ( A3
% 0.30/0.48 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % empty_is_image
% 0.30/0.48 thf(fact_212_image__empty,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,F: B > A] :
% 0.30/0.48 ( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
% 0.30/0.48 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % image_empty
% 0.30/0.48 thf(fact_213_Sigma__empty1,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,B6: A > ( set @ B )] :
% 0.30/0.48 ( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B6 )
% 0.30/0.48 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Sigma_empty1
% 0.30/0.48 thf(fact_214_snd__map__prod,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X4: product_prod @ C @ D] :
% 0.30/0.48 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X4 ) )
% 0.30/0.48 = ( G @ ( product_snd @ C @ D @ X4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % snd_map_prod
% 0.30/0.48 thf(fact_215_Collect__const,axiom,
% 0.30/0.48 ! [A: $tType,P3: $o] :
% 0.30/0.48 ( ( P3
% 0.30/0.48 => ( ( collect @ A
% 0.30/0.48 @ ^ [S6: A] : P3 )
% 0.30/0.48 = ( top_top @ ( set @ A ) ) ) )
% 0.30/0.48 & ( ~ P3
% 0.30/0.48 => ( ( collect @ A
% 0.30/0.48 @ ^ [S6: A] : P3 )
% 0.30/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_const
% 0.30/0.48 thf(fact_216_Sigma__empty2,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,A3: set @ A] :
% 0.30/0.48 ( ( product_Sigma @ A @ B @ A3
% 0.30/0.48 @ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
% 0.30/0.48 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Sigma_empty2
% 0.30/0.48 thf(fact_217_Times__empty,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType,A3: set @ A,B6: set @ B] :
% 0.30/0.48 ( ( ( product_Sigma @ A @ B @ A3
% 0.30/0.48 @ ^ [Uu: A] : B6 )
% 0.30/0.48 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.30/0.48 = ( ( A3
% 0.30/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.48 | ( B6
% 0.30/0.48 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Times_empty
% 0.30/0.48 thf(fact_218_Collect__const__case__prod,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,P3: $o] :
% 0.30/0.48 ( ( P3
% 0.30/0.48 => ( ( collect @ ( product_prod @ A @ B )
% 0.30/0.48 @ ( product_case_prod @ A @ B @ $o
% 0.30/0.48 @ ^ [A6: A,B5: B] : P3 ) )
% 0.30/0.48 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
% 0.30/0.48 & ( ~ P3
% 0.30/0.48 => ( ( collect @ ( product_prod @ A @ B )
% 0.30/0.48 @ ( product_case_prod @ A @ B @ $o
% 0.30/0.48 @ ^ [A6: A,B5: B] : P3 ) )
% 0.30/0.48 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_const_case_prod
% 0.30/0.48 thf(fact_219_prod_Ocollapse,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.30/0.48 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.30/0.48 = Prod ) ).
% 0.30/0.48
% 0.30/0.48 % prod.collapse
% 0.30/0.48 thf(fact_220_snd__swap,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,X4: product_prod @ A @ B] :
% 0.30/0.48 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X4 ) )
% 0.30/0.48 = ( product_fst @ A @ B @ X4 ) ) ).
% 0.30/0.48
% 0.30/0.48 % snd_swap
% 0.30/0.48 thf(fact_221_fst__swap,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType,X4: product_prod @ B @ A] :
% 0.30/0.48 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X4 ) )
% 0.30/0.48 = ( product_snd @ B @ A @ X4 ) ) ).
% 0.30/0.48
% 0.30/0.48 % fst_swap
% 0.30/0.48 thf(fact_222_range__snd,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.30/0.48 = ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % range_snd
% 0.30/0.48 thf(fact_223_snd__image__times,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,A3: set @ B,B6: set @ A] :
% 0.30/0.48 ( ( ( A3
% 0.30/0.48 = ( bot_bot @ ( set @ B ) ) )
% 0.30/0.48 => ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 0.30/0.48 @ ( product_Sigma @ B @ A @ A3
% 0.30/0.48 @ ^ [Uu: B] : B6 ) )
% 0.30/0.48 = ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.48 & ( ( A3
% 0.30/0.48 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.48 => ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 0.30/0.48 @ ( product_Sigma @ B @ A @ A3
% 0.30/0.48 @ ^ [Uu: B] : B6 ) )
% 0.30/0.48 = B6 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % snd_image_times
% 0.30/0.48 thf(fact_224_sndI,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType,X4: product_prod @ A @ B,Y: A,Z: B] :
% 0.30/0.48 ( ( X4
% 0.30/0.48 = ( product_Pair @ A @ B @ Y @ Z ) )
% 0.30/0.48 => ( ( product_snd @ A @ B @ X4 )
% 0.30/0.48 = Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % sndI
% 0.30/0.48 thf(fact_225_eq__snd__iff,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 0.30/0.48 ( ( B2
% 0.30/0.48 = ( product_snd @ B @ A @ P2 ) )
% 0.30/0.48 = ( ? [A6: B] :
% 0.30/0.48 ( P2
% 0.30/0.48 = ( product_Pair @ B @ A @ A6 @ B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % eq_snd_iff
% 0.30/0.48 thf(fact_226_times__eq__iff,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType,A3: set @ A,B6: set @ B,C5: set @ A,D3: set @ B] :
% 0.30/0.48 ( ( ( product_Sigma @ A @ B @ A3
% 0.30/0.48 @ ^ [Uu: A] : B6 )
% 0.30/0.48 = ( product_Sigma @ A @ B @ C5
% 0.30/0.48 @ ^ [Uu: A] : D3 ) )
% 0.30/0.48 = ( ( ( A3 = C5 )
% 0.30/0.48 & ( B6 = D3 ) )
% 0.30/0.48 | ( ( ( A3
% 0.30/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.48 | ( B6
% 0.30/0.48 = ( bot_bot @ ( set @ B ) ) ) )
% 0.30/0.48 & ( ( C5
% 0.30/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.48 | ( D3
% 0.30/0.48 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % times_eq_iff
% 0.30/0.48 thf(fact_227_empty__def,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( bot_bot @ ( set @ A ) )
% 0.30/0.48 = ( collect @ A
% 0.30/0.48 @ ^ [X5: A] : $false ) ) ).
% 0.30/0.48
% 0.30/0.48 % empty_def
% 0.30/0.48 thf(fact_228_snd__eqD,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,X4: B,Y: A,A2: A] :
% 0.30/0.48 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y ) )
% 0.30/0.48 = A2 )
% 0.30/0.48 => ( Y = A2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % snd_eqD
% 0.30/0.48 thf(fact_229_snd__conv,axiom,
% 0.30/0.48 ! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
% 0.30/0.48 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
% 0.30/0.48 = X2 ) ).
% 0.30/0.48
% 0.30/0.48 % snd_conv
% 0.30/0.48 thf(fact_230_Sigma__empty__iff,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,I4: set @ A,X8: A > ( set @ B )] :
% 0.30/0.48 ( ( ( product_Sigma @ A @ B @ I4 @ X8 )
% 0.30/0.48 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.30/0.48 = ( ! [X5: A] :
% 0.30/0.48 ( ( member @ A @ X5 @ I4 )
% 0.30/0.48 => ( ( X8 @ X5 )
% 0.30/0.48 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Sigma_empty_iff
% 0.30/0.48 thf(fact_231_snd__def,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( product_snd @ A @ B )
% 0.30/0.48 = ( product_case_prod @ A @ B @ B
% 0.30/0.48 @ ^ [X13: A,X23: B] : X23 ) ) ).
% 0.30/0.48
% 0.30/0.48 % snd_def
% 0.30/0.48 thf(fact_232_split__comp__eq,axiom,
% 0.30/0.48 ! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
% 0.30/0.48 ( ( ^ [U: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
% 0.30/0.48 = ( product_case_prod @ D @ B @ C
% 0.30/0.48 @ ^ [X5: D] : ( F @ ( G @ X5 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % split_comp_eq
% 0.30/0.48 thf(fact_233_case__prod__beta,axiom,
% 0.30/0.48 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.48 ( ( product_case_prod @ B @ C @ A )
% 0.30/0.48 = ( ^ [F9: B > C > A,P: product_prod @ B @ C] : ( F9 @ ( product_fst @ B @ C @ P ) @ ( product_snd @ B @ C @ P ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % case_prod_beta
% 0.30/0.48 thf(fact_234_case__prod__beta_H,axiom,
% 0.30/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.48 ( ( product_case_prod @ A @ B @ C )
% 0.30/0.48 = ( ^ [F9: A > B > C,X5: product_prod @ A @ B] : ( F9 @ ( product_fst @ A @ B @ X5 ) @ ( product_snd @ A @ B @ X5 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % case_prod_beta'
% 0.30/0.48 thf(fact_235_prod_Ocase__eq__if,axiom,
% 0.30/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.48 ( ( product_case_prod @ A @ B @ C )
% 0.30/0.48 = ( ^ [F9: A > B > C,Prod2: product_prod @ A @ B] : ( F9 @ ( product_fst @ A @ B @ Prod2 ) @ ( product_snd @ A @ B @ Prod2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % prod.case_eq_if
% 0.30/0.48 thf(fact_236_case__prod__unfold,axiom,
% 0.30/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.48 ( ( product_case_prod @ A @ B @ C )
% 0.30/0.48 = ( ^ [C4: A > B > C,P: product_prod @ A @ B] : ( C4 @ ( product_fst @ A @ B @ P ) @ ( product_snd @ A @ B @ P ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % case_prod_unfold
% 0.30/0.48 thf(fact_237_Product__Type_OCollect__case__prodD,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,X4: product_prod @ A @ B,A3: A > B > $o] :
% 0.30/0.48 ( ( member @ ( product_prod @ A @ B ) @ X4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) )
% 0.30/0.48 => ( A3 @ ( product_fst @ A @ B @ X4 ) @ ( product_snd @ A @ B @ X4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Product_Type.Collect_case_prodD
% 0.30/0.48 thf(fact_238_mem__Times__iff,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType,X4: product_prod @ A @ B,A3: set @ A,B6: set @ B] :
% 0.30/0.48 ( ( member @ ( product_prod @ A @ B ) @ X4
% 0.30/0.48 @ ( product_Sigma @ A @ B @ A3
% 0.30/0.48 @ ^ [Uu: A] : B6 ) )
% 0.30/0.48 = ( ( member @ A @ ( product_fst @ A @ B @ X4 ) @ A3 )
% 0.30/0.48 & ( member @ B @ ( product_snd @ A @ B @ X4 ) @ B6 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_Times_iff
% 0.30/0.48 thf(fact_239_prod_Oexhaust__sel,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.30/0.48 ( Prod
% 0.30/0.48 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % prod.exhaust_sel
% 0.30/0.48 thf(fact_240_surjective__pairing,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.30/0.48 ( T2
% 0.30/0.48 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % surjective_pairing
% 0.30/0.48 thf(fact_241_prod__eq__iff,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ^ [Y9: product_prod @ A @ B,Z3: product_prod @ A @ B] : ( Y9 = Z3 ) )
% 0.30/0.48 = ( ^ [S6: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 0.30/0.48 ( ( ( product_fst @ A @ B @ S6 )
% 0.30/0.48 = ( product_fst @ A @ B @ T3 ) )
% 0.30/0.48 & ( ( product_snd @ A @ B @ S6 )
% 0.30/0.48 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % prod_eq_iff
% 0.30/0.48 thf(fact_242_prod_Oexpand,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod3: product_prod @ A @ B] :
% 0.30/0.48 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.30/0.48 = ( product_fst @ A @ B @ Prod3 ) )
% 0.30/0.48 & ( ( product_snd @ A @ B @ Prod )
% 0.30/0.48 = ( product_snd @ A @ B @ Prod3 ) ) )
% 0.30/0.48 => ( Prod = Prod3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % prod.expand
% 0.30/0.48 thf(fact_243_prod__eqI,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q: product_prod @ A @ B] :
% 0.30/0.48 ( ( ( product_fst @ A @ B @ P2 )
% 0.30/0.48 = ( product_fst @ A @ B @ Q ) )
% 0.30/0.48 => ( ( ( product_snd @ A @ B @ P2 )
% 0.30/0.48 = ( product_snd @ A @ B @ Q ) )
% 0.30/0.48 => ( P2 = Q ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % prod_eqI
% 0.30/0.48 thf(fact_244_prod_Osplit__sel,axiom,
% 0.30/0.48 ! [C: $tType,B: $tType,A: $tType,P3: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.30/0.48 ( ( P3 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.30/0.48 = ( ( Prod
% 0.30/0.48 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.30/0.48 => ( P3 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % prod.split_sel
% 0.30/0.48 thf(fact_245_prod_Osplit__sel__asm,axiom,
% 0.30/0.48 ! [C: $tType,B: $tType,A: $tType,P3: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.30/0.48 ( ( P3 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.30/0.48 = ( ~ ( ( Prod
% 0.30/0.48 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.30/0.48 & ~ ( P3 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % prod.split_sel_asm
% 0.30/0.48 thf(fact_246_scomp__unfold,axiom,
% 0.30/0.48 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.48 ( ( product_scomp @ A @ B @ C @ D )
% 0.30/0.48 = ( ^ [F9: A > ( product_prod @ B @ C ),G4: B > C > D,X5: A] : ( G4 @ ( product_fst @ B @ C @ ( F9 @ X5 ) ) @ ( product_snd @ B @ C @ ( F9 @ X5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % scomp_unfold
% 0.30/0.48 thf(fact_247_prod_Oswap__def,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( product_swap @ A @ B )
% 0.30/0.48 = ( ^ [P: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P ) @ ( product_fst @ A @ B @ P ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % prod.swap_def
% 0.30/0.48 thf(fact_248_fst__image__Sigma,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,A3: set @ A,B6: A > ( set @ B )] :
% 0.30/0.48 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
% 0.30/0.48 = ( collect @ A
% 0.30/0.48 @ ^ [X5: A] :
% 0.30/0.48 ( ( member @ A @ X5 @ A3 )
% 0.30/0.48 & ( ( B6 @ X5 )
% 0.30/0.48 != ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fst_image_Sigma
% 0.30/0.48 thf(fact_249_exE__realizer,axiom,
% 0.30/0.48 ! [C: $tType,A: $tType,B: $tType,P3: A > B > $o,P2: product_prod @ B @ A,Q3: C > $o,F: B > A > C] :
% 0.30/0.48 ( ( P3 @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 0.30/0.48 => ( ! [X: B,Y4: A] :
% 0.30/0.48 ( ( P3 @ Y4 @ X )
% 0.30/0.48 => ( Q3 @ ( F @ X @ Y4 ) ) )
% 0.30/0.48 => ( Q3 @ ( product_case_prod @ B @ A @ C @ F @ P2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % exE_realizer
% 0.30/0.48 thf(fact_250_exI__realizer,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,P3: A > B > $o,Y: A,X4: B] :
% 0.30/0.48 ( ( P3 @ Y @ X4 )
% 0.30/0.48 => ( P3 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % exI_realizer
% 0.30/0.48 thf(fact_251_conjI__realizer,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType,P3: A > $o,P2: A,Q3: B > $o,Q: B] :
% 0.30/0.48 ( ( P3 @ P2 )
% 0.30/0.48 => ( ( Q3 @ Q )
% 0.30/0.48 => ( ( P3 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) )
% 0.30/0.48 & ( Q3 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % conjI_realizer
% 0.30/0.48 thf(fact_252_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,P3: A > B > $o,X4: A,Y: B,A2: product_prod @ A @ B] :
% 0.30/0.48 ( ( P3 @ X4 @ Y )
% 0.30/0.48 => ( ( A2
% 0.30/0.48 = ( product_Pair @ A @ B @ X4 @ Y ) )
% 0.30/0.48 => ( P3 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % BNF_Greatest_Fixpoint.subst_Pair
% 0.30/0.48 thf(fact_253_the__elem__image__unique,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,A3: set @ A,F: A > B,X4: A] :
% 0.30/0.48 ( ( A3
% 0.30/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.48 => ( ! [Y4: A] :
% 0.30/0.48 ( ( member @ A @ Y4 @ A3 )
% 0.30/0.48 => ( ( F @ Y4 )
% 0.30/0.48 = ( F @ X4 ) ) )
% 0.30/0.48 => ( ( the_elem @ B @ ( image @ A @ B @ F @ A3 ) )
% 0.30/0.48 = ( F @ X4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % the_elem_image_unique
% 0.30/0.48 thf(fact_254_subset__fst__snd,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] :
% 0.30/0.48 ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3
% 0.30/0.48 @ ( product_Sigma @ A @ B @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 )
% 0.30/0.48 @ ^ [Uu: A] : ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % subset_fst_snd
% 0.30/0.48 thf(fact_255_Collect__split__mono__strong,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,X8: set @ A,A3: set @ ( product_prod @ A @ B ),Y6: set @ B,P3: A > B > $o,Q3: A > B > $o] :
% 0.30/0.48 ( ( X8
% 0.30/0.48 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) )
% 0.30/0.48 => ( ( Y6
% 0.30/0.48 = ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) )
% 0.30/0.48 => ( ! [X: A] :
% 0.30/0.48 ( ( member @ A @ X @ X8 )
% 0.30/0.48 => ! [Xa: B] :
% 0.30/0.48 ( ( member @ B @ Xa @ Y6 )
% 0.30/0.48 => ( ( P3 @ X @ Xa )
% 0.30/0.48 => ( Q3 @ X @ Xa ) ) ) )
% 0.30/0.48 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P3 ) ) )
% 0.30/0.48 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q3 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_split_mono_strong
% 0.30/0.48
% 0.30/0.48 % Helper facts (3)
% 0.30/0.48 thf(help_If_3_1_T,axiom,
% 0.30/0.48 ! [P3: $o] :
% 0.30/0.48 ( ( P3 = $true )
% 0.30/0.48 | ( P3 = $false ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(help_If_2_1_T,axiom,
% 0.30/0.48 ! [A: $tType,X4: A,Y: A] :
% 0.30/0.48 ( ( if @ A @ $false @ X4 @ Y )
% 0.30/0.48 = Y ) ).
% 0.30/0.48
% 0.30/0.48 thf(help_If_1_1_T,axiom,
% 0.30/0.48 ! [A: $tType,X4: A,Y: A] :
% 0.30/0.48 ( ( if @ A @ $true @ X4 @ Y )
% 0.30/0.48 = X4 ) ).
% 0.30/0.48
% 0.30/0.48 % Conjectures (1)
% 0.30/0.48 thf(conj_0,conjecture,
% 0.30/0.48 ? [I5: ( list @ char ) > paraco415392788lle_tv] :
% 0.30/0.48 ( ( paraco876059933e_eval @ I5 @ p )
% 0.30/0.48 != ( paraco2040174112le_Det @ $true ) ) ).
% 0.30/0.48
% 0.30/0.48 %------------------------------------------------------------------------------
% 0.30/0.48 ------------------------
% 2.58/1.56 ---- Embedded in THF ---
% 2.58/1.56 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.58/1.56 %%% Generated on Tue Feb 24 08:26:39 EST 2026
% 2.58/1.56 %%% using '$$dhol' embedding, version 1.3.0.
% 2.58/1.56 %%% Logic specification used:
% 2.58/1.56 %%% thf(spec, logic, $$dhol).
% 2.58/1.56
% 2.58/1.56 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.g4D8jloWIn/DTF2DTF16534.p
% 2.58/1.56 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv, type, paraco415392788lle_tv: $tType).
% 2.58/1.56 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm, type, paraco414474393lle_fm: $tType).
% 2.58/1.56 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 2.58/1.56 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.58/1.56 thf(ty_t_String_Ochar, type, char: $tType).
% 2.58/1.56 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.58/1.56 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.58/1.56 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.58/1.56 thf(sy_c_Fun__Def_Oreduction__pair, type, fun_reduction_pair: (!> [A:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > $o)))).
% 2.58/1.56 thf(sy_c_Fun__Def_Orp__inv__image, type, fun_rp_inv_image: (!> [A:$tType,B:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))))))).
% 2.58/1.56 thf(sy_c_HOL_OThe, type, the: (!> [A:$tType]: (((A > $o) > A)))).
% 2.58/1.56 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.58/1.56 thf(sy_c_Order__Relation_Oabove, type, order_above: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 2.58/1.56 thf(sy_c_Order__Relation_OaboveS, type, order_aboveS: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 2.58/1.56 thf(sy_c_Order__Relation_Orelation__of, type, order_relation_of: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > (set @ ((product_prod @ A) @ A))))))).
% 2.58/1.56 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.58/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.58/1.56 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.58/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval, type, paraco876059933e_eval: (((list @ char) > paraco415392788lle_tv) > (paraco414474393lle_fm > paraco415392788lle_tv))).
% 2.58/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H, type, paraco2100061555le_Con: (paraco414474393lle_fm > (paraco414474393lle_fm > paraco414474393lle_fm))).
% 2.58/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql, type, paraco2084319816le_Eql: (paraco414474393lle_fm > (paraco414474393lle_fm > paraco414474393lle_fm))).
% 2.58/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H, type, paraco1628874225le_Eql: (paraco414474393lle_fm > (paraco414474393lle_fm > paraco414474393lle_fm))).
% 2.58/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H, type, paraco329115265le_Neg: (paraco414474393lle_fm > paraco414474393lle_fm)).
% 2.58/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro, type, paraco27778325le_Pro: ((list @ char) > paraco414474393lle_fm)).
% 2.58/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth, type, paraco251304083_Truth: paraco414474393lle_fm).
% 2.58/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Ocase__fm, type, paraco1246693743ase_fm: (!> [A:$tType]: ((((list @ char) > A) > (A > ((paraco414474393lle_fm > A) > ((paraco414474393lle_fm > (paraco414474393lle_fm > A)) > ((paraco414474393lle_fm > (paraco414474393lle_fm > A)) > ((paraco414474393lle_fm > (paraco414474393lle_fm > A)) > (paraco414474393lle_fm > A)))))))))).
% 2.58/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet, type, paraco2040174112le_Det: ($o > paraco415392788lle_tv)).
% 2.58/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet, type, paraco676387099_Indet: (nat > paraco415392788lle_tv)).
% 2.58/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Ocase__tv, type, paraco490622181ase_tv: (!> [A:$tType]: ((($o > A) > ((nat > A) > (paraco415392788lle_tv > A)))))).
% 2.58/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv, type, paraco152590079rec_tv: (!> [A:$tType]: ((($o > A) > ((nat > A) > (paraco415392788lle_tv > A)))))).
% 2.58/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid, type, paraco769098683_valid: (paraco414474393lle_fm > $o)).
% 2.58/1.56 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.58/1.56 thf(sy_c_Product__Type_OSigma, type, product_Sigma: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ ((product_prod @ A) @ B))))))).
% 2.58/1.56 thf(sy_c_Product__Type_Obool_Ocase__bool, type, product_case_bool: (!> [A:$tType]: ((A > (A > ($o > A)))))).
% 2.58/1.56 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 2.58/1.56 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.58/1.56 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 2.58/1.56 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.58/1.56 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 > $o)))))).
% 2.58/1.56 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit, type, product_rec_set_unit: (!> [T:$tType]: ((T > (product_unit > (T > $o)))))).
% 2.58/1.56 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit, type, product_rec_unit: (!> [T:$tType]: ((T > (product_unit > T))))).
% 2.58/1.56 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.58/1.56 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 2.58/1.56 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 2.58/1.56 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 2.58/1.56 thf(sy_c_Product__Type_Oproduct, type, product_product: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((product_prod @ A) @ B))))))).
% 2.58/1.56 thf(sy_c_Product__Type_Oscomp, type, product_scomp: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: (((A > ((product_prod @ B) @ C)) > ((B > (C > D)) > (A > D)))))).
% 2.58/1.56 thf(sy_c_Relation_Oinv__image, type, inv_image: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((A > B) > (set @ ((product_prod @ A) @ A))))))).
% 2.58/1.56 thf(sy_c_Relation_Oinv__imagep, type, inv_imagep: (!> [B:$tType,A:$tType]: (((B > (B > $o)) > ((A > B) > (A > (A > $o))))))).
% 2.58/1.56 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.58/1.56 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.58/1.56 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.58/1.56 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)))))))).
% 2.58/1.56 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 2.58/1.56 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.58/1.56 thf(sy_v_p, type, p: paraco414474393lle_fm).
% 2.58/1.56 thf(fact_0_tv_Oinject_I1_J, axiom, (! [X1:$o,Y1:$o]: ((((paraco2040174112le_Det @ X1) = (paraco2040174112le_Det @ Y1)) = (X1 = Y1))))).
% 2.58/1.56 thf(fact_1_assms, axiom, (paraco769098683_valid @ (paraco329115265le_Neg @ p))).
% 2.58/1.56 thf(fact_2_valid__def, axiom, (paraco769098683_valid = (^ [P:paraco414474393lle_fm]: ((! [I:((list @ char) > paraco415392788lle_tv)]: ((((paraco876059933e_eval @ I) @ P) = (paraco2040174112le_Det @ $true)))))))).
% 2.58/1.56 thf(fact_3_eval_Osimps_I2_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv)]: ((((paraco876059933e_eval @ I2) @ paraco251304083_Truth) = (paraco2040174112le_Det @ $true))))).
% 2.58/1.56 thf(fact_4_eval_Osimps_I4_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I2) @ P2) = ((paraco876059933e_eval @ I2) @ Q)) => (((paraco876059933e_eval @ I2) @ ((paraco2100061555le_Con @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ P2))) & ((((paraco876059933e_eval @ I2) @ P2) != ((paraco876059933e_eval @ I2) @ Q)) => (((((paraco876059933e_eval @ I2) @ P2) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco2100061555le_Con @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ Q))) & ((((paraco876059933e_eval @ I2) @ P2) != (paraco2040174112le_Det @ $true)) => (((((paraco876059933e_eval @ I2) @ Q) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco2100061555le_Con @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ P2))) & ((((paraco876059933e_eval @ I2) @ Q) != (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco2100061555le_Con @ P2) @ Q)) = (paraco2040174112le_Det @ $false))))))))))).
% 2.58/1.56 thf(fact_5_eval_Osimps_I5_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I2) @ P2) = ((paraco876059933e_eval @ I2) @ Q)) => (((paraco876059933e_eval @ I2) @ ((paraco2084319816le_Eql @ P2) @ Q)) = (paraco2040174112le_Det @ $true))) & ((((paraco876059933e_eval @ I2) @ P2) != ((paraco876059933e_eval @ I2) @ Q)) => (((paraco876059933e_eval @ I2) @ ((paraco2084319816le_Eql @ P2) @ Q)) = (paraco2040174112le_Det @ $false))))))).
% 2.58/1.56 thf(fact_6_eval__equality__simplify, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((paraco876059933e_eval @ I2) @ ((paraco2084319816le_Eql @ P2) @ Q)) = (paraco2040174112le_Det @ (((paraco876059933e_eval @ I2) @ P2) = ((paraco876059933e_eval @ I2) @ Q))))))).
% 2.58/1.56 thf(fact_7_tv_Osimps_I7_J, axiom, (! [A:$tType,F1:($o > A),F2:(nat > A),X1:$o]: ((((((paraco152590079rec_tv @ A) @ F1) @ F2) @ (paraco2040174112le_Det @ X1)) = (F1 @ X1))))).
% 2.58/1.56 thf(fact_8_eval__negation, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I2) @ P2) = (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P2)) = (paraco2040174112le_Det @ $true))) & ((((paraco876059933e_eval @ I2) @ P2) != (paraco2040174112le_Det @ $false)) => (((((paraco876059933e_eval @ I2) @ P2) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P2)) = (paraco2040174112le_Det @ $false))) & ((((paraco876059933e_eval @ I2) @ P2) != (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P2)) = ((paraco876059933e_eval @ I2) @ P2))))))))).
% 2.58/1.56 thf(fact_9_tv_Osimps_I5_J, axiom, (! [A:$tType,F1:($o > A),F2:(nat > A),X1:$o]: ((((((paraco490622181ase_tv @ A) @ F1) @ F2) @ (paraco2040174112le_Det @ X1)) = (F1 @ X1))))).
% 2.58/1.56 thf(fact_10_eval_Osimps_I1_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),S:(list @ char)]: ((((paraco876059933e_eval @ I2) @ (paraco27778325le_Pro @ S)) = (I2 @ S))))).
% 2.58/1.56 thf(fact_11_tv_Odistinct_I1_J, axiom, (! [X1:$o,X2:nat]: (((paraco2040174112le_Det @ X1) != (paraco676387099_Indet @ X2))))).
% 2.58/1.56 thf(fact_12_tv_Oinduct, axiom, (! [P3:(paraco415392788lle_tv > $o),Tv:paraco415392788lle_tv]: (((! [X:$o]: ((P3 @ (paraco2040174112le_Det @ X)))) => ((! [X:nat]: ((P3 @ (paraco676387099_Indet @ X)))) => (P3 @ Tv)))))).
% 2.58/1.56 thf(fact_13_tv_Oexhaust, axiom, (! [Y:paraco415392788lle_tv]: (((! [X12:$o]: ((Y != (paraco2040174112le_Det @ X12)))) => (~ ((! [X22:nat]: ((Y != (paraco676387099_Indet @ X22)))))))))).
% 2.58/1.56 thf(fact_14_fm_Oinject_I2_J, axiom, (! [X3:paraco414474393lle_fm,Y3:paraco414474393lle_fm]: ((((paraco329115265le_Neg @ X3) = (paraco329115265le_Neg @ Y3)) = (X3 = Y3))))).
% 2.58/1.56 thf(fact_15_tv_Oinject_I2_J, axiom, (! [X2:nat,Y2:nat]: ((((paraco676387099_Indet @ X2) = (paraco676387099_Indet @ Y2)) = (X2 = Y2))))).
% 2.58/1.56 thf(fact_16_fm_Oinject_I4_J, axiom, (! [X51:paraco414474393lle_fm,X52:paraco414474393lle_fm,Y51:paraco414474393lle_fm,Y52:paraco414474393lle_fm]: (((((paraco2084319816le_Eql @ X51) @ X52) = ((paraco2084319816le_Eql @ Y51) @ Y52)) = ((X51 = Y51) & (X52 = Y52)))))).
% 2.58/1.56 thf(fact_17_fm_Oinject_I1_J, axiom, (! [X1:(list @ char),Y1:(list @ char)]: ((((paraco27778325le_Pro @ X1) = (paraco27778325le_Pro @ Y1)) = (X1 = Y1))))).
% 2.58/1.56 thf(fact_18_fm_Oinject_I3_J, axiom, (! [X41:paraco414474393lle_fm,X42:paraco414474393lle_fm,Y41:paraco414474393lle_fm,Y42:paraco414474393lle_fm]: (((((paraco2100061555le_Con @ X41) @ X42) = ((paraco2100061555le_Con @ Y41) @ Y42)) = ((X41 = Y41) & (X42 = Y42)))))).
% 2.58/1.56 thf(fact_19_tv_Osimps_I8_J, axiom, (! [A:$tType,F1:($o > A),F2:(nat > A),X2:nat]: ((((((paraco152590079rec_tv @ A) @ F1) @ F2) @ (paraco676387099_Indet @ X2)) = (F2 @ X2))))).
% 2.58/1.56 thf(fact_20_tv_Osimps_I6_J, axiom, (! [A:$tType,F1:($o > A),F2:(nat > A),X2:nat]: ((((((paraco490622181ase_tv @ A) @ F1) @ F2) @ (paraco676387099_Indet @ X2)) = (F2 @ X2))))).
% 2.58/1.56 thf(fact_21_fm_Odistinct_I25_J, axiom, (! [X41:paraco414474393lle_fm,X42:paraco414474393lle_fm,X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: ((((paraco2100061555le_Con @ X41) @ X42) != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.58/1.56 thf(fact_22_fm_Odistinct_I21_J, axiom, (! [X3:paraco414474393lle_fm,X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: (((paraco329115265le_Neg @ X3) != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.58/1.56 thf(fact_23_fm_Odistinct_I19_J, axiom, (! [X3:paraco414474393lle_fm,X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: (((paraco329115265le_Neg @ X3) != ((paraco2100061555le_Con @ X41) @ X42))))).
% 2.58/1.56 thf(fact_24_fm_Odistinct_I15_J, axiom, (! [X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: ((paraco251304083_Truth != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.58/1.56 thf(fact_25_fm_Odistinct_I13_J, axiom, (! [X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: ((paraco251304083_Truth != ((paraco2100061555le_Con @ X41) @ X42))))).
% 2.58/1.56 thf(fact_26_fm_Odistinct_I11_J, axiom, (! [X3:paraco414474393lle_fm]: ((paraco251304083_Truth != (paraco329115265le_Neg @ X3))))).
% 2.58/1.56 thf(fact_27_fm_Odistinct_I7_J, axiom, (! [X1:(list @ char),X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.58/1.56 thf(fact_28_fm_Odistinct_I5_J, axiom, (! [X1:(list @ char),X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != ((paraco2100061555le_Con @ X41) @ X42))))).
% 2.58/1.56 thf(fact_29_fm_Odistinct_I3_J, axiom, (! [X1:(list @ char),X3:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != (paraco329115265le_Neg @ X3))))).
% 2.58/1.56 thf(fact_30_fm_Odistinct_I1_J, axiom, (! [X1:(list @ char)]: (((paraco27778325le_Pro @ X1) != paraco251304083_Truth)))).
% 2.58/1.56 thf(fact_31_double__negation, axiom, (paraco876059933e_eval = (^ [I:((list @ char) > paraco415392788lle_tv),P:paraco414474393lle_fm]: (((paraco876059933e_eval @ I) @ (paraco329115265le_Neg @ (paraco329115265le_Neg @ P))))))).
% 2.58/1.56 thf(fact_32_string__tv_Oinduct, axiom, (! [P3:(paraco415392788lle_tv > $o),A0:paraco415392788lle_tv]: (((P3 @ (paraco2040174112le_Det @ $true)) => ((P3 @ (paraco2040174112le_Det @ $false)) => ((! [N:nat]: ((P3 @ (paraco676387099_Indet @ N)))) => (P3 @ A0))))))).
% 2.58/1.56 thf(fact_33_string__tv_Ocases, axiom, (! [X4:paraco415392788lle_tv]: (((X4 != (paraco2040174112le_Det @ $true)) => ((X4 != (paraco2040174112le_Det @ $false)) => (~ ((! [N:nat]: ((X4 != (paraco676387099_Indet @ N))))))))))).
% 2.58/1.56 thf(fact_34_fm_Oinduct, axiom, (! [P3:(paraco414474393lle_fm > $o),Fm:paraco414474393lle_fm]: (((! [X:(list @ char)]: ((P3 @ (paraco27778325le_Pro @ X)))) => ((P3 @ paraco251304083_Truth) => ((! [X:paraco414474393lle_fm]: (((P3 @ X) => (P3 @ (paraco329115265le_Neg @ X))))) => ((! [X1a:paraco414474393lle_fm,X22:paraco414474393lle_fm]: (((P3 @ X1a) => ((P3 @ X22) => (P3 @ ((paraco2100061555le_Con @ X1a) @ X22)))))) => ((! [X1a:paraco414474393lle_fm,X22:paraco414474393lle_fm]: (((P3 @ X1a) => ((P3 @ X22) => (P3 @ ((paraco2084319816le_Eql @ X1a) @ X22)))))) => ((! [X1a:paraco414474393lle_fm,X22:paraco414474393lle_fm]: (((P3 @ X1a) => ((P3 @ X22) => (P3 @ ((paraco1628874225le_Eql @ X1a) @ X22)))))) => (P3 @ Fm)))))))))).
% 2.58/1.56 thf(fact_35_fm_Oexhaust, axiom, (! [Y:paraco414474393lle_fm]: (((! [X12:(list @ char)]: ((Y != (paraco27778325le_Pro @ X12)))) => ((Y != paraco251304083_Truth) => ((! [X32:paraco414474393lle_fm]: ((Y != (paraco329115265le_Neg @ X32)))) => ((! [X412:paraco414474393lle_fm,X422:paraco414474393lle_fm]: ((Y != ((paraco2100061555le_Con @ X412) @ X422)))) => ((! [X512:paraco414474393lle_fm,X522:paraco414474393lle_fm]: ((Y != ((paraco2084319816le_Eql @ X512) @ X522)))) => (~ ((! [X61:paraco414474393lle_fm,X62:paraco414474393lle_fm]: ((Y != ((paraco1628874225le_Eql @ X61) @ X62)))))))))))))).
% 2.58/1.56 thf(fact_36_eval_Osimps_I3_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm]: ((((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P2)) = ((((paraco490622181ase_tv @ paraco415392788lle_tv) @ (((product_case_bool @ paraco415392788lle_tv) @ (paraco2040174112le_Det @ $false)) @ (paraco2040174112le_Det @ $true))) @ paraco676387099_Indet) @ ((paraco876059933e_eval @ I2) @ P2)))))).
% 2.58/1.56 thf(fact_37_eval__equality, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I2) @ P2) = ((paraco876059933e_eval @ I2) @ Q)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = (paraco2040174112le_Det @ $true))) & ((((paraco876059933e_eval @ I2) @ P2) != ((paraco876059933e_eval @ I2) @ Q)) => (((((paraco876059933e_eval @ I2) @ P2) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ Q))) & ((((paraco876059933e_eval @ I2) @ P2) != (paraco2040174112le_Det @ $true)) => (((((paraco876059933e_eval @ I2) @ Q) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ P2))) & ((((paraco876059933e_eval @ I2) @ Q) != (paraco2040174112le_Det @ $true)) => (((((paraco876059933e_eval @ I2) @ P2) = (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ Q)))) & ((((paraco876059933e_eval @ I2) @ P2) != (paraco2040174112le_Det @ $false)) => (((((paraco876059933e_eval @ I2) @ Q) = (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P2)))) & ((((paraco876059933e_eval @ I2) @ Q) != (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = (paraco2040174112le_Det @ $false))))))))))))))).
% 2.58/1.56 thf(fact_38_eval_Ocases, axiom, (! [X4:((product_prod @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm)]: (((! [I3:((list @ char) > paraco415392788lle_tv),S2:(list @ char)]: ((X4 != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ (paraco27778325le_Pro @ S2))))) => ((! [I3:((list @ char) > paraco415392788lle_tv)]: ((X4 != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ paraco251304083_Truth)))) => ((! [I3:((list @ char) > paraco415392788lle_tv),P4:paraco414474393lle_fm]: ((X4 != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ (paraco329115265le_Neg @ P4))))) => ((! [I3:((list @ char) > paraco415392788lle_tv),P4:paraco414474393lle_fm,Q2:paraco414474393lle_fm]: ((X4 != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ ((paraco2100061555le_Con @ P4) @ Q2))))) => ((! [I3:((list @ char) > paraco415392788lle_tv),P4:paraco414474393lle_fm,Q2:paraco414474393lle_fm]: ((X4 != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ ((paraco2084319816le_Eql @ P4) @ Q2))))) => (~ ((! [I3:((list @ char) > paraco415392788lle_tv),P4:paraco414474393lle_fm,Q2:paraco414474393lle_fm]: ((X4 != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ ((paraco1628874225le_Eql @ P4) @ Q2))))))))))))))).
% 2.58/1.56 thf(fact_39_fm_Osimps_I37_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > A),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > A))]: ((((((((((paraco1246693743ase_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ paraco251304083_Truth) = F2)))).
% 2.58/1.56 thf(fact_40_fm_Osimps_I39_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > A),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: ((((((((((paraco1246693743ase_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ ((paraco2100061555le_Con @ X41) @ X42)) = ((F4 @ X41) @ X42))))).
% 2.58/1.56 thf(fact_41_fm_Osimps_I36_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > A),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),X1:(list @ char)]: ((((((((((paraco1246693743ase_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ (paraco27778325le_Pro @ X1)) = (F1 @ X1))))).
% 2.58/1.56 thf(fact_42_fm_Osimps_I40_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > A),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: ((((((((((paraco1246693743ase_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ ((paraco2084319816le_Eql @ X51) @ X52)) = ((F5 @ X51) @ X52))))).
% 2.58/1.56 thf(fact_43_fm_Osimps_I38_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > A),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),X3:paraco414474393lle_fm]: ((((((((((paraco1246693743ase_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ (paraco329115265le_Neg @ X3)) = (F3 @ X3))))).
% 2.58/1.56 thf(fact_44_fm_Odistinct_I17_J, axiom, (! [X612:paraco414474393lle_fm,X622:paraco414474393lle_fm]: ((paraco251304083_Truth != ((paraco1628874225le_Eql @ X612) @ X622))))).
% 2.58/1.56 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P3:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P3)) = (P3 @ A2))))).
% 2.58/1.56 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X5:A]: ((((member @ A) @ X5) @ A3)))) = A3)))).
% 2.58/1.56 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P3:(A > $o),Q3:(A > $o)]: (((! [X:A]: (((P3 @ X) = (Q3 @ X)))) => (((collect @ A) @ P3) = ((collect @ A) @ Q3)))))).
% 2.58/1.56 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X:A]: (((F @ X) = (G @ X)))) => (F = G))))).
% 2.58/1.56 thf(fact_49_fm_Oinject_I5_J, axiom, (! [X612:paraco414474393lle_fm,X622:paraco414474393lle_fm,Y61:paraco414474393lle_fm,Y62:paraco414474393lle_fm]: (((((paraco1628874225le_Eql @ X612) @ X622) = ((paraco1628874225le_Eql @ Y61) @ Y62)) = ((X612 = Y61) & (X622 = Y62)))))).
% 2.58/1.56 thf(fact_50_fm_Osimps_I41_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > A),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),X612:paraco414474393lle_fm,X622:paraco414474393lle_fm]: ((((((((((paraco1246693743ase_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ ((paraco1628874225le_Eql @ X612) @ X622)) = ((F6 @ X612) @ X622))))).
% 2.58/1.56 thf(fact_51_fm_Odistinct_I23_J, axiom, (! [X3:paraco414474393lle_fm,X612:paraco414474393lle_fm,X622:paraco414474393lle_fm]: (((paraco329115265le_Neg @ X3) != ((paraco1628874225le_Eql @ X612) @ X622))))).
% 2.58/1.56 thf(fact_52_fm_Odistinct_I29_J, axiom, (! [X51:paraco414474393lle_fm,X52:paraco414474393lle_fm,X612:paraco414474393lle_fm,X622:paraco414474393lle_fm]: ((((paraco2084319816le_Eql @ X51) @ X52) != ((paraco1628874225le_Eql @ X612) @ X622))))).
% 2.58/1.56 thf(fact_53_fm_Odistinct_I9_J, axiom, (! [X1:(list @ char),X612:paraco414474393lle_fm,X622:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != ((paraco1628874225le_Eql @ X612) @ X622))))).
% 2.58/1.56 thf(fact_54_fm_Odistinct_I27_J, axiom, (! [X41:paraco414474393lle_fm,X42:paraco414474393lle_fm,X612:paraco414474393lle_fm,X622:paraco414474393lle_fm]: ((((paraco2100061555le_Con @ X41) @ X42) != ((paraco1628874225le_Eql @ X612) @ X622))))).
% 2.58/1.56 thf(fact_55_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A4:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A4) @ B3)) = ((A2 = A4) & (B2 = B3)))))).
% 2.58/1.56 thf(fact_56_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X2:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X2) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 2.58/1.56 thf(fact_57_bool_Osplit__sel__asm, axiom, (! [A:$tType,P3:(A > $o),F1:A,F2:A,Bool:$o]: (((P3 @ ((((product_case_bool @ A) @ F1) @ F2) @ Bool)) = (~ (((Bool & (~ ((P3 @ F1)))) | ((~ (Bool)) & (~ ((P3 @ F2))))))))))).
% 2.58/1.56 thf(fact_58_bool_Ocase__distrib, axiom, (! [A:$tType,B:$tType,H:(A > B),F1:A,F2:A,Bool:$o]: (((H @ ((((product_case_bool @ A) @ F1) @ F2) @ Bool)) = ((((product_case_bool @ B) @ (H @ F1)) @ (H @ F2)) @ Bool))))).
% 2.58/1.56 thf(fact_59_bool_Ocase__eq__if, axiom, (! [A:$tType]: (((product_case_bool @ A) = (^ [F12:A,F22:A,Bool2:$o]: (((((if @ A) @ Bool2) @ F12) @ F22))))))).
% 2.58/1.56 thf(fact_60_bool_Osplit__sel, axiom, (! [A:$tType,P3:(A > $o),F1:A,F2:A,Bool:$o]: (((P3 @ ((((product_case_bool @ A) @ F1) @ F2) @ Bool)) = ((Bool => (P3 @ F1)) & ((~ (Bool)) => (P3 @ F2))))))).
% 2.58/1.56 thf(fact_61_old_Obool_Osimps_I3_J, axiom, (! [A:$tType,F1:A,F2:A]: ((((((product_case_bool @ A) @ F1) @ F2) @ $true) = F1)))).
% 2.58/1.56 thf(fact_62_old_Obool_Osimps_I4_J, axiom, (! [A:$tType,F1:A,F2:A]: ((((((product_case_bool @ A) @ F1) @ F2) @ $false) = F2)))).
% 2.58/1.56 thf(fact_63_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X:A,Y4:B]: ((P2 = ((((product_Pair @ A) @ B) @ X) @ Y4))))))).
% 2.58/1.56 thf(fact_64_prod__cases, axiom, (! [B:$tType,A:$tType,P3:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B4:B]: ((P3 @ ((((product_Pair @ A) @ B) @ A5) @ B4)))) => (P3 @ P2))))).
% 2.58/1.56 thf(fact_65_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A4:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A4) @ B3)) => (~ (((A2 = A4) => (B2 != B3)))))))).
% 2.58/1.56 thf(fact_66_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B4:B,C2:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B4) @ C2)))))))))).
% 2.58/1.56 thf(fact_67_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B4:B,C2:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 2.58/1.56 thf(fact_68_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A5:A,B4:B,C2:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.58/1.56 thf(fact_69_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F7:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F7)))))]: ((~ ((! [A5:A,B4:B,C2:C,D2:D,E2:E,F8:F7]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F7))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F7)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F7))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F7)) @ D2) @ ((((product_Pair @ E) @ F7) @ E2) @ F8))))))))))))).
% 2.58/1.56 thf(fact_70_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F7:$tType,G2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F7) @ G2))))))]: ((~ ((! [A5:A,B4:B,C2:C,D2:D,E2:E,F8:F7,G3:G2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F7) @ G2)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F7) @ G2))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F7) @ G2)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F7) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F7) @ G2)) @ E2) @ ((((product_Pair @ F7) @ G2) @ F8) @ G3)))))))))))))).
% 2.58/1.56 thf(fact_71_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B4:B,C2:C]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B4) @ C2))))) => (P3 @ X4))))).
% 2.58/1.56 thf(fact_72_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B4:B,C2:C,D2:D]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P3 @ X4))))).
% 2.58/1.56 thf(fact_73_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A5:A,B4:B,C2:C,D2:D,E2:E]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P3 @ X4))))).
% 2.58/1.56 thf(fact_74_prod__induct6, axiom, (! [F7:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F7))))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F7)))))]: (((! [A5:A,B4:B,C2:C,D2:D,E2:E,F8:F7]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F7))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F7)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F7))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F7)) @ D2) @ ((((product_Pair @ E) @ F7) @ E2) @ F8)))))))) => (P3 @ X4))))).
% 2.58/1.56 thf(fact_75_prod__induct7, axiom, (! [G2:$tType,F7:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F7) @ G2)))))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F7) @ G2))))))]: (((! [A5:A,B4:B,C2:C,D2:D,E2:E,F8:F7,G3:G2]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F7) @ G2)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F7) @ G2))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F7) @ G2)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F7) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F7) @ G2)) @ E2) @ ((((product_Pair @ F7) @ G2) @ F8) @ G3))))))))) => (P3 @ X4))))).
% 2.58/1.56 thf(fact_76_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B4:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B4))))))))).
% 2.58/1.56 thf(fact_77_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P3:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B4:B]: ((P3 @ ((((product_Pair @ A) @ B) @ A5) @ B4)))) => (P3 @ Prod))))).
% 2.58/1.56 thf(fact_78_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 2.58/1.56 thf(fact_79_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C3 @ A2) @ B2))))).
% 2.58/1.56 thf(fact_80_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R:A,S:B,R2:(set @ ((product_prod @ A) @ B)),S3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S)) @ R2) => ((S3 = S) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S3)) @ R2)))))).
% 2.58/1.56 thf(fact_81_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F9:(((product_prod @ B) @ C) > A),A6:B,B5:C]: ((F9 @ ((((product_Pair @ B) @ C) @ A6) @ B5)))))))).
% 2.58/1.56 thf(fact_82_curryI, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: (((F @ ((((product_Pair @ A) @ B) @ A2) @ B2)) => ((((((product_curry @ A) @ B) @ $o) @ F) @ A2) @ B2))))).
% 2.58/1.56 thf(fact_83_swap__simp, axiom, (! [A:$tType,B:$tType,X4:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X4) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X4))))).
% 2.58/1.56 thf(fact_84_eval_Osimps_I6_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I2) @ P2) = ((paraco876059933e_eval @ I2) @ Q)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = (paraco2040174112le_Det @ $true))) & ((((paraco876059933e_eval @ I2) @ P2) != ((paraco876059933e_eval @ I2) @ Q)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = (((((product_case_prod @ paraco415392788lle_tv) @ paraco415392788lle_tv) @ paraco415392788lle_tv) @ (^ [X5:paraco415392788lle_tv,Y5:paraco415392788lle_tv]: (((((paraco490622181ase_tv @ paraco415392788lle_tv) @ (((product_case_bool @ paraco415392788lle_tv) @ ((paraco876059933e_eval @ I2) @ Q)) @ ((((paraco490622181ase_tv @ paraco415392788lle_tv) @ (((product_case_bool @ paraco415392788lle_tv) @ ((paraco876059933e_eval @ I2) @ P2)) @ ((((paraco490622181ase_tv @ paraco415392788lle_tv) @ (((product_case_bool @ paraco415392788lle_tv) @ (paraco2040174112le_Det @ $false)) @ (paraco2040174112le_Det @ $true))) @ paraco676387099_Indet) @ ((paraco876059933e_eval @ I2) @ Q)))) @ (^ [Nat:nat]: (((((paraco490622181ase_tv @ paraco415392788lle_tv) @ (((product_case_bool @ paraco415392788lle_tv) @ (paraco2040174112le_Det @ $false)) @ (paraco2040174112le_Det @ $true))) @ paraco676387099_Indet) @ ((paraco876059933e_eval @ I2) @ Q))))) @ Y5))) @ (^ [Nat:nat]: (((((paraco490622181ase_tv @ paraco415392788lle_tv) @ (((product_case_bool @ paraco415392788lle_tv) @ ((paraco876059933e_eval @ I2) @ P2)) @ ((((paraco490622181ase_tv @ paraco415392788lle_tv) @ (((product_case_bool @ paraco415392788lle_tv) @ (paraco2040174112le_Det @ $false)) @ (paraco2040174112le_Det @ $true))) @ paraco676387099_Indet) @ ((paraco876059933e_eval @ I2) @ P2)))) @ (^ [Nata:nat]: ((paraco2040174112le_Det @ $false)))) @ Y5)))) @ X5)))) @ ((((product_Pair @ paraco415392788lle_tv) @ paraco415392788lle_tv) @ ((paraco876059933e_eval @ I2) @ P2)) @ ((paraco876059933e_eval @ I2) @ Q))))))))).
% 2.58/1.56 thf(fact_85_swap__swap, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P2)) = P2)))).
% 2.58/1.56 thf(fact_86_case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),A2:B,B2:C]: (((((((product_case_prod @ B) @ C) @ A) @ F) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((F @ A2) @ B2))))).
% 2.58/1.56 thf(fact_87_case__prod__curry, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ ((((product_curry @ A) @ B) @ C) @ F)) = F)))).
% 2.58/1.56 thf(fact_88_curry__case__prod, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C))]: ((((((product_curry @ A) @ B) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F)) = F)))).
% 2.58/1.56 thf(fact_89_case__swap, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > (B > A)),P2:((product_prod @ C) @ B)]: (((((((product_case_prod @ B) @ C) @ A) @ (^ [Y5:B,X5:C]: (((F @ X5) @ Y5)))) @ (((product_swap @ C) @ B) @ P2)) = (((((product_case_prod @ C) @ B) @ A) @ F) @ P2))))).
% 2.58/1.56 thf(fact_90_cond__case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C)),G:(((product_prod @ A) @ B) > C)]: (((! [X:A,Y4:B]: ((((F @ X) @ Y4) = (G @ ((((product_Pair @ A) @ B) @ X) @ Y4))))) => (((((product_case_prod @ A) @ B) @ C) @ F) = G))))).
% 2.58/1.56 thf(fact_91_case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ (^ [X5:A,Y5:B]: ((F @ ((((product_Pair @ A) @ B) @ X5) @ Y5))))) = F)))).
% 2.58/1.56 thf(fact_92_case__prodE2, axiom, (! [B:$tType,A:$tType,C:$tType,Q3:(A > $o),P3:(B > (C > A)),Z:((product_prod @ B) @ C)]: (((Q3 @ (((((product_case_prod @ B) @ C) @ A) @ P3) @ Z)) => (~ ((! [X:B,Y4:C]: (((Z = ((((product_Pair @ B) @ C) @ X) @ Y4)) => (~ ((Q3 @ ((P3 @ X) @ Y4))))))))))))).
% 2.58/1.56 thf(fact_93_curry__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_curry @ A) @ B) @ C) = (^ [C4:(((product_prod @ A) @ B) > C),X5:A,Y5:B]: ((C4 @ ((((product_Pair @ A) @ B) @ X5) @ Y5)))))))).
% 2.58/1.56 thf(fact_94_curry__K, axiom, (! [B:$tType,C:$tType,A:$tType,C3:C]: ((((((product_curry @ A) @ B) @ C) @ (^ [X5:((product_prod @ A) @ B)]: (C3))) = (^ [X5:A,Y5:B]: (C3)))))).
% 2.58/1.56 thf(fact_95_prod_Ocase__distrib, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,H:(C > D),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((H @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = (((((product_case_prod @ A) @ B) @ D) @ (^ [X13:A,X23:B]: ((H @ ((F @ X13) @ X23))))) @ Prod))))).
% 2.58/1.56 thf(fact_96_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc2004651681e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 2.58/1.56 thf(fact_97_tv_Ocase__distrib, axiom, (! [B:$tType,A:$tType,H:(A > B),F1:($o > A),F2:(nat > A),Tv:paraco415392788lle_tv]: (((H @ ((((paraco490622181ase_tv @ A) @ F1) @ F2) @ Tv)) = ((((paraco490622181ase_tv @ B) @ (^ [X5:$o]: ((H @ (F1 @ X5))))) @ (^ [X5:nat]: ((H @ (F2 @ X5))))) @ Tv))))).
% 2.58/1.56 thf(fact_98_fm_Ocase__distrib, axiom, (! [A:$tType,B:$tType,H:(A > B),F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > A),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),Fm:paraco414474393lle_fm]: (((H @ ((((((((paraco1246693743ase_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ Fm)) = ((((((((paraco1246693743ase_fm @ B) @ (^ [X5:(list @ char)]: ((H @ (F1 @ X5))))) @ (H @ F2)) @ (^ [X5:paraco414474393lle_fm]: ((H @ (F3 @ X5))))) @ (^ [X13:paraco414474393lle_fm,X23:paraco414474393lle_fm]: ((H @ ((F4 @ X13) @ X23))))) @ (^ [X13:paraco414474393lle_fm,X23:paraco414474393lle_fm]: ((H @ ((F5 @ X13) @ X23))))) @ (^ [X13:paraco414474393lle_fm,X23:paraco414474393lle_fm]: ((H @ ((F6 @ X13) @ X23))))) @ Fm))))).
% 2.58/1.56 thf(fact_99_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > (B > C)),X1:A,X2:B]: (((((((product_case_prod @ A) @ B) @ C) @ F) @ ((((product_Pair @ A) @ B) @ X1) @ X2)) = ((F @ X1) @ X2))))).
% 2.58/1.56 thf(fact_100_curryE, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A2) @ B2) => (F @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 2.58/1.56 thf(fact_101_curryD, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A2) @ B2) => (F @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 2.58/1.56 thf(fact_102_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q:((product_prod @ A) @ B),F:(A > (B > C)),G:(A > (B > C)),P2:((product_prod @ A) @ B)]: (((! [X:A,Y4:B]: (((((((product_Pair @ A) @ B) @ X) @ Y4) = Q) => (((F @ X) @ Y4) = ((G @ X) @ Y4))))) => ((P2 = Q) => ((((((product_case_prod @ A) @ B) @ C) @ F) @ P2) = (((((product_case_prod @ A) @ B) @ C) @ G) @ Q))))))).
% 2.58/1.56 thf(fact_103_case__bool__if, axiom, (! [A:$tType]: (((product_case_bool @ A) = (^ [X5:A,Y5:A,P5:$o]: (((((if @ A) @ P5) @ X5) @ Y5))))))).
% 2.58/1.56 thf(fact_104_case__prod__app, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ (D > A)) = (^ [F9:(B > (C > (D > A))),X5:((product_prod @ B) @ C),Y5:D]: ((((((product_case_prod @ B) @ C) @ A) @ (^ [L:B,R3:C]: ((((F9 @ L) @ R3) @ Y5)))) @ X5))))))).
% 2.58/1.56 thf(fact_105_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S4:(set @ ((product_prod @ A) @ B))]: ((((^ [X5:A,Y5:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y5)) @ R2))) = (^ [X5:A,Y5:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y5)) @ S4)))) = (R2 = S4))))).
% 2.58/1.56 thf(fact_106_pair__imageI, axiom, (! [C:$tType,B:$tType,A:$tType,A2:A,B2:B,A3:(set @ ((product_prod @ A) @ B)),F:(A > (B > C))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ A3) => (((member @ C) @ ((F @ A2) @ B2)) @ ((((image @ ((product_prod @ A) @ B)) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F)) @ A3)))))).
% 2.58/1.56 thf(fact_107_scomp__apply, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: ((((((product_scomp @ B) @ C) @ D) @ A) = (^ [F9:(B > ((product_prod @ C) @ D)),G4:(C > (D > A)),X5:B]: ((((((product_case_prod @ C) @ D) @ A) @ G4) @ (F9 @ X5)))))))).
% 2.58/1.56 thf(fact_108_case__prodI, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A2:A,B2:B]: ((((F @ A2) @ B2) => (((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 2.58/1.56 thf(fact_109_case__prodI2, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),C3:(A > (B > $o))]: (((! [A5:A,B4:B]: (((P2 = ((((product_Pair @ A) @ B) @ A5) @ B4)) => ((C3 @ A5) @ B4)))) => (((((product_case_prod @ A) @ B) @ $o) @ C3) @ P2))))).
% 2.58/1.56 thf(fact_110_mem__case__prodI, axiom, (! [A:$tType,B:$tType,C:$tType,Z:A,C3:(B > (C > (set @ A))),A2:B,B2:C]: (((((member @ A) @ Z) @ ((C3 @ A2) @ B2)) => (((member @ A) @ Z) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B2))))))).
% 2.58/1.56 thf(fact_111_mem__case__prodI2, axiom, (! [C:$tType,B:$tType,A:$tType,P2:((product_prod @ A) @ B),Z:C,C3:(A > (B > (set @ C)))]: (((! [A5:A,B4:B]: (((P2 = ((((product_Pair @ A) @ B) @ A5) @ B4)) => (((member @ C) @ Z) @ ((C3 @ A5) @ B4))))) => (((member @ C) @ Z) @ (((((product_case_prod @ A) @ B) @ (set @ C)) @ C3) @ P2)))))).
% 2.58/1.56 thf(fact_112_case__prodI2_H, axiom, (! [A:$tType,B:$tType,C:$tType,P2:((product_prod @ A) @ B),C3:(A > (B > (C > $o))),X4:C]: (((! [A5:A,B4:B]: (((((((product_Pair @ A) @ B) @ A5) @ B4) = P2) => (((C3 @ A5) @ B4) @ X4)))) => ((((((product_case_prod @ A) @ B) @ (C > $o)) @ C3) @ P2) @ X4))))).
% 2.58/1.56 thf(fact_113_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y:A,X4:B,A3:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y) @ X4)) @ ((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A3)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X4) @ Y)) @ A3))))).
% 2.58/1.56 thf(fact_114_case__prod__Pair__iden, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) @ P2) = P2)))).
% 2.58/1.56 thf(fact_115_scomp__scomp, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F7:$tType,E:$tType,F:(A > ((product_prod @ E) @ F7)),G:(E > (F7 > ((product_prod @ C) @ D))),H:(C > (D > B))]: ((((((((product_scomp @ A) @ C) @ D) @ B) @ ((((((product_scomp @ A) @ E) @ F7) @ ((product_prod @ C) @ D)) @ F) @ G)) @ H) = ((((((product_scomp @ A) @ E) @ F7) @ B) @ F) @ (^ [X5:E]: (((((((product_scomp @ F7) @ C) @ D) @ B) @ (G @ X5)) @ H)))))))).
% 2.58/1.56 thf(fact_116_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z:A,C3:(B > (C > (set @ A))),P2:((product_prod @ B) @ C)]: (((((member @ A) @ Z) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C3) @ P2)) => (~ ((! [X:B,Y4:C]: (((P2 = ((((product_Pair @ B) @ C) @ X) @ Y4)) => (~ ((((member @ A) @ Z) @ ((C3 @ X) @ Y4))))))))))))).
% 2.58/1.56 thf(fact_117_case__prodD, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A2:A,B2:B]: (((((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) => ((F @ A2) @ B2))))).
% 2.58/1.56 thf(fact_118_case__prodE, axiom, (! [A:$tType,B:$tType,C3:(A > (B > $o)),P2:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ $o) @ C3) @ P2) => (~ ((! [X:A,Y4:B]: (((P2 = ((((product_Pair @ A) @ B) @ X) @ Y4)) => (~ (((C3 @ X) @ Y4)))))))))))).
% 2.58/1.56 thf(fact_119_case__prodD_H, axiom, (! [B:$tType,A:$tType,C:$tType,R2:(A > (B > (C > $o))),A2:A,B2:B,C3:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ R2) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ C3) => (((R2 @ A2) @ B2) @ C3))))).
% 2.58/1.56 thf(fact_120_case__prodE_H, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(A > (B > (C > $o))),P2:((product_prod @ A) @ B),Z:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ C3) @ P2) @ Z) => (~ ((! [X:A,Y4:B]: (((P2 = ((((product_Pair @ A) @ B) @ X) @ Y4)) => (~ ((((C3 @ X) @ Y4) @ Z)))))))))))).
% 2.58/1.56 thf(fact_121_scomp__Pair, axiom, (! [C:$tType,B:$tType,A:$tType,X4:(A > ((product_prod @ B) @ C))]: ((((((((product_scomp @ A) @ B) @ C) @ ((product_prod @ B) @ C)) @ X4) @ ((product_Pair @ B) @ C)) = X4)))).
% 2.58/1.56 thf(fact_122_Pair__scomp, axiom, (! [A:$tType,B:$tType,C:$tType,X4:C,F:(C > (A > B))]: ((((((((product_scomp @ A) @ C) @ A) @ B) @ (((product_Pair @ C) @ A) @ X4)) @ F) = (F @ X4))))).
% 2.58/1.56 thf(fact_123_scomp__def, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((((product_scomp @ A) @ B) @ C) @ D) = (^ [F9:(A > ((product_prod @ B) @ C)),G4:(B > (C > D)),X5:A]: ((((((product_case_prod @ B) @ C) @ D) @ G4) @ (F9 @ X5)))))))).
% 2.58/1.56 thf(fact_124_image__ident, axiom, (! [A:$tType,Y6:(set @ A)]: ((((((image @ A) @ A) @ (^ [X5:A]: (X5))) @ Y6) = Y6)))).
% 2.58/1.56 thf(fact_125_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),X4:B,A3:(set @ B)]: (((B2 = (F @ X4)) => ((((member @ B) @ X4) @ A3) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ A3))))))).
% 2.58/1.56 thf(fact_126_Sup_OSUP__identity__eq, axiom, (! [A:$tType,Sup:((set @ A) > A),A3:(set @ A)]: (((Sup @ ((((image @ A) @ A) @ (^ [X5:A]: (X5))) @ A3)) = (Sup @ A3))))).
% 2.58/1.56 thf(fact_127_Inf_OINF__identity__eq, axiom, (! [A:$tType,Inf:((set @ A) > A),A3:(set @ A)]: (((Inf @ ((((image @ A) @ A) @ (^ [X5:A]: (X5))) @ A3)) = (Inf @ A3))))).
% 2.58/1.56 thf(fact_128_Compr__image__eq, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B),P3:(A > $o)]: ((((collect @ A) @ (^ [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F) @ A3)) & (P3 @ X5))))) = ((((image @ B) @ A) @ F) @ ((collect @ B) @ (^ [X5:B]: (((((member @ B) @ X5) @ A3) & (P3 @ (F @ X5))))))))))).
% 2.58/1.56 thf(fact_129_split__part, axiom, (! [B:$tType,A:$tType,P3:$o,Q3:(A > (B > $o))]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [A6:A,B5:B]: ((P3 & ((Q3 @ A6) @ B5))))) = (^ [Ab:((product_prod @ A) @ B)]: ((P3 & (((((product_case_prod @ A) @ B) @ $o) @ Q3) @ Ab)))))))).
% 2.58/1.56 thf(fact_130_prod_Odisc__eq__case, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [Uu:A,Uv:B]: ($true))) @ Prod)))).
% 2.58/1.56 thf(fact_131_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A3:(set @ B),B6:(set @ B),C5:(B > A),D3:(B > A),Sup:((set @ A) > A)]: (((A3 = B6) => ((! [X:B]: (((((member @ B) @ X) @ B6) => ((C5 @ X) = (D3 @ X))))) => ((Sup @ ((((image @ B) @ A) @ C5) @ A3)) = (Sup @ ((((image @ B) @ A) @ D3) @ B6)))))))).
% 2.58/1.56 thf(fact_132_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A3:(set @ B),B6:(set @ B),C5:(B > A),D3:(B > A),Inf:((set @ A) > A)]: (((A3 = B6) => ((! [X:B]: (((((member @ B) @ X) @ B6) => ((C5 @ X) = (D3 @ X))))) => ((Inf @ ((((image @ B) @ A) @ C5) @ A3)) = (Inf @ ((((image @ B) @ A) @ D3) @ B6)))))))).
% 2.58/1.56 thf(fact_133_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X4:A,A3:(set @ A),B2:B,F:(A > B)]: (((((member @ A) @ X4) @ A3) => ((B2 = (F @ X4)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F) @ A3))))))).
% 2.58/1.56 thf(fact_134_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B),P3:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((((image @ B) @ A) @ F) @ A3)) => (P3 @ X)))) => (! [X6:B]: (((((member @ B) @ X6) @ A3) => (P3 @ (F @ X6))))))))).
% 2.58/1.56 thf(fact_135_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N2:(set @ A),F:(A > B),G:(A > B)]: (((M = N2) => ((! [X:A]: (((((member @ A) @ X) @ N2) => ((F @ X) = (G @ X))))) => (((((image @ A) @ B) @ F) @ M) = ((((image @ A) @ B) @ G) @ N2))))))).
% 2.58/1.56 thf(fact_136_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B),P3:(A > $o)]: (((? [X6:A]: (((((member @ A) @ X6) @ ((((image @ B) @ A) @ F) @ A3)) & (P3 @ X6)))) => (? [X:B]: (((((member @ B) @ X) @ A3) & (P3 @ (F @ X))))))))).
% 2.58/1.56 thf(fact_137_image__iff, axiom, (! [A:$tType,B:$tType,Z:A,F:(B > A),A3:(set @ B)]: (((((member @ A) @ Z) @ ((((image @ B) @ A) @ F) @ A3)) = (? [X5:B]: (((((member @ B) @ X5) @ A3) & (Z = (F @ X5))))))))).
% 2.58/1.56 thf(fact_138_imageI, axiom, (! [B:$tType,A:$tType,X4:A,A3:(set @ A),F:(A > B)]: (((((member @ A) @ X4) @ A3) => (((member @ B) @ (F @ X4)) @ ((((image @ A) @ B) @ F) @ A3)))))).
% 2.58/1.56 thf(fact_139_imageE, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),A3:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ A3)) => (~ ((! [X:B]: (((B2 = (F @ X)) => (~ ((((member @ B) @ X) @ A3)))))))))))).
% 2.58/1.56 thf(fact_140_image__image, axiom, (! [A:$tType,B:$tType,C:$tType,F:(B > A),G:(C > B),A3:(set @ C)]: ((((((image @ B) @ A) @ F) @ ((((image @ C) @ B) @ G) @ A3)) = ((((image @ C) @ A) @ (^ [X5:C]: ((F @ (G @ X5))))) @ A3))))).
% 2.58/1.56 thf(fact_141_image__paired__Times, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,F:(C > A),G:(D > B),A3:(set @ C),B6:(set @ D)]: ((((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ C) @ D) @ ((product_prod @ A) @ B)) @ (^ [X5:C,Y5:D]: (((((product_Pair @ A) @ B) @ (F @ X5)) @ (G @ Y5)))))) @ ((((product_Sigma @ C) @ D) @ A3) @ (^ [Uu:C]: (B6)))) = ((((product_Sigma @ A) @ B) @ ((((image @ C) @ A) @ F) @ A3)) @ (^ [Uu:A]: (((((image @ D) @ B) @ G) @ B6)))))))).
% 2.58/1.56 thf(fact_142_inv__image__def, axiom, (! [A:$tType,B:$tType]: ((((inv_image @ B) @ A) = (^ [R3:(set @ ((product_prod @ B) @ B)),F9:(A > B)]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X5:A,Y5:A]: ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F9 @ X5)) @ (F9 @ Y5))) @ R3))))))))))).
% 2.58/1.56 thf(fact_143_same__fst__def, axiom, (! [B:$tType,A:$tType]: ((((same_fst @ A) @ B) = (^ [P5:(A > $o),R4:(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)) @ (^ [X7:A,Y7:B]: (((((product_case_prod @ A) @ B) @ $o) @ (^ [X5:A,Y5:B]: (((X7 = X5) & ((P5 @ X5) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y7) @ Y5)) @ (R4 @ X5)))))))))))))))))).
% 2.58/1.56 thf(fact_144_swap__product, axiom, (! [B:$tType,A:$tType,A3:(set @ B),B6:(set @ A)]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ B) @ A) @ ((product_prod @ A) @ B)) @ (^ [I:B,J:A]: (((((product_Pair @ A) @ B) @ J) @ I))))) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu:B]: (B6)))) = ((((product_Sigma @ A) @ B) @ B6) @ (^ [Uu:A]: (A3))))))).
% 2.58/1.56 thf(fact_145_SigmaI, axiom, (! [B:$tType,A:$tType,A2:A,A3:(set @ A),B2:B,B6:(A > (set @ B))]: (((((member @ A) @ A2) @ A3) => ((((member @ B) @ B2) @ (B6 @ A2)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B6))))))).
% 2.58/1.56 thf(fact_146_mem__Sigma__iff, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,A3:(set @ A),B6:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B6)) = ((((member @ A) @ A2) @ A3) & (((member @ B) @ B2) @ (B6 @ A2))))))).
% 2.58/1.56 thf(fact_147_in__inv__image, axiom, (! [A:$tType,B:$tType,X4:A,Y:A,R:(set @ ((product_prod @ B) @ B)),F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y)) @ ((((inv_image @ B) @ A) @ R) @ F)) = (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X4)) @ (F @ Y))) @ R))))).
% 2.58/1.56 thf(fact_148_Collect__case__prod, axiom, (! [B:$tType,A:$tType,P3:(A > $o),Q3:(B > $o)]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A6:A,B5:B]: (((P3 @ A6) & (Q3 @ B5)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P3)) @ (^ [Uu:A]: (((collect @ B) @ Q3)))))))).
% 2.58/1.56 thf(fact_149_same__fstI, axiom, (! [B:$tType,A:$tType,P3:(A > $o),X4:A,Y8:B,Y:B,R2:(A > (set @ ((product_prod @ B) @ B)))]: (((P3 @ X4) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y8) @ Y)) @ (R2 @ X4)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y8)) @ ((((product_Pair @ A) @ B) @ X4) @ Y))) @ ((((same_fst @ A) @ B) @ P3) @ R2))))))).
% 2.58/1.56 thf(fact_150_Collect__case__prod__Sigma, axiom, (! [B:$tType,A:$tType,P3:(A > $o),Q3:(A > (B > $o))]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X5:A,Y5:B]: (((P3 @ X5) & ((Q3 @ X5) @ Y5)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P3)) @ (^ [X5:A]: (((collect @ B) @ (Q3 @ X5))))))))).
% 2.58/1.56 thf(fact_151_Sigma__cong, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B6:(set @ A),C5:(A > (set @ B)),D3:(A > (set @ B))]: (((A3 = B6) => ((! [X:A]: (((((member @ A) @ X) @ B6) => ((C5 @ X) = (D3 @ X))))) => (((((product_Sigma @ A) @ B) @ A3) @ C5) = ((((product_Sigma @ A) @ B) @ B6) @ D3))))))).
% 2.58/1.56 thf(fact_152_Times__eq__cancel2, axiom, (! [A:$tType,B:$tType,X4:A,C5:(set @ A),A3:(set @ B),B6:(set @ B)]: (((((member @ A) @ X4) @ C5) => ((((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu:B]: (C5))) = ((((product_Sigma @ B) @ A) @ B6) @ (^ [Uu:B]: (C5)))) = (A3 = B6)))))).
% 2.58/1.56 thf(fact_153_SigmaE, axiom, (! [A:$tType,B:$tType,C3:((product_prod @ A) @ B),A3:(set @ A),B6:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ C3) @ ((((product_Sigma @ A) @ B) @ A3) @ B6)) => (~ ((! [X:A]: (((((member @ A) @ X) @ A3) => (! [Y4:B]: (((((member @ B) @ Y4) @ (B6 @ X)) => (C3 != ((((product_Pair @ A) @ B) @ X) @ Y4)))))))))))))).
% 2.58/1.56 thf(fact_154_SigmaD1, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,A3:(set @ A),B6:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B6)) => (((member @ A) @ A2) @ A3))))).
% 2.58/1.56 thf(fact_155_SigmaD2, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,A3:(set @ A),B6:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B6)) => (((member @ B) @ B2) @ (B6 @ A2)))))).
% 2.58/1.56 thf(fact_156_SigmaE2, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,A3:(set @ A),B6:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B6)) => (~ (((((member @ A) @ A2) @ A3) => (~ ((((member @ B) @ B2) @ (B6 @ A2))))))))))).
% 2.58/1.56 thf(fact_157_product__swap, axiom, (! [B:$tType,A:$tType,A3:(set @ B),B6:(set @ A)]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu:B]: (B6)))) = ((((product_Sigma @ A) @ B) @ B6) @ (^ [Uu:A]: (A3))))))).
% 2.58/1.56 thf(fact_158_rp__inv__image__def, axiom, (! [B:$tType,A:$tType]: ((((fun_rp_inv_image @ A) @ B) = ((((product_case_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))))) @ (^ [R4:(set @ ((product_prod @ A) @ A)),S5:(set @ ((product_prod @ A) @ A)),F9:(B > A)]: (((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ ((((inv_image @ A) @ B) @ R4) @ F9)) @ ((((inv_image @ A) @ B) @ S5) @ F9))))))))).
% 2.58/1.56 thf(fact_159_lex__prod__def, axiom, (! [B:$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)) @ (^ [A6:A,B5:B]: (((((product_case_prod @ A) @ B) @ $o) @ (^ [A7:A,B7:B]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A6) @ A7)) @ Ra) | ((A6 = A7) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B5) @ B7)) @ Rb))))))))))))))))).
% 2.58/1.56 thf(fact_160_Product__Type_Oproduct__def, axiom, (! [B:$tType,A:$tType]: ((((product_product @ A) @ B) = (^ [A8:(set @ A),B8:(set @ B)]: (((((product_Sigma @ A) @ B) @ A8) @ (^ [Uu:A]: (B8))))))))).
% 2.58/1.56 thf(fact_161_in__lex__prod, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A4:A,B3:B,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Pair @ A) @ B) @ A4) @ B3))) @ ((((lex_prod @ A) @ B) @ R) @ S)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A4)) @ R) | ((A2 = A4) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B2) @ B3)) @ S))))))).
% 2.58/1.56 thf(fact_162_member__product, axiom, (! [B:$tType,A:$tType,X4:((product_prod @ A) @ B),A3:(set @ A),B6:(set @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X4) @ ((((product_product @ A) @ B) @ A3) @ B6)) = (((member @ ((product_prod @ A) @ B)) @ X4) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B6)))))))).
% 2.58/1.56 thf(fact_163_rp__inv__image__rp, axiom, (! [A:$tType,B:$tType,P3:((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))),F:(B > A)]: ((((fun_reduction_pair @ A) @ P3) => ((fun_reduction_pair @ B) @ ((((fun_rp_inv_image @ A) @ B) @ P3) @ F)))))).
% 2.58/1.56 thf(fact_164_map__prod__surj__on, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F:(B > A),A3:(set @ B),A9:(set @ A),G:(D > C),B6:(set @ D),B9:(set @ C)]: (((((((image @ B) @ A) @ F) @ A3) = A9) => ((((((image @ D) @ C) @ G) @ B6) = B9) => (((((image @ ((product_prod @ B) @ D)) @ ((product_prod @ A) @ C)) @ ((((((product_map_prod @ B) @ A) @ D) @ C) @ F) @ G)) @ ((((product_Sigma @ B) @ D) @ A3) @ (^ [Uu:B]: (B6)))) = ((((product_Sigma @ A) @ C) @ A9) @ (^ [Uu:A]: (B9))))))))).
% 2.58/1.56 thf(fact_165_relation__of__def, axiom, (! [A:$tType]: (((order_relation_of @ A) = (^ [P5:(A > (A > $o)),A8:(set @ A)]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [A6:A,B5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A6) @ B5)) @ ((((product_Sigma @ A) @ A) @ A8) @ (^ [Uu:A]: (A8)))) & ((P5 @ A6) @ B5)))))))))))).
% 2.58/1.56 thf(fact_166_map__prod__ident, axiom, (! [B:$tType,A:$tType]: ((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X5:A]: (X5))) @ (^ [Y5:B]: (Y5))) = (^ [Z2:((product_prod @ A) @ B)]: (Z2)))))).
% 2.58/1.56 thf(fact_167_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G:(D > B),A2:C,B2:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G) @ ((((product_Pair @ C) @ D) @ A2) @ B2)) = ((((product_Pair @ A) @ B) @ (F @ A2)) @ (G @ B2)))))).
% 2.58/1.56 thf(fact_168_map__prod__imageI, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,A2:A,B2:B,R2:(set @ ((product_prod @ A) @ B)),F:(A > C),G:(B > D)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R2) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F @ A2)) @ (G @ B2))) @ ((((image @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F) @ G)) @ R2)))))).
% 2.58/1.56 thf(fact_169_case__prod__map__prod, axiom, (! [C:$tType,A:$tType,B:$tType,E:$tType,D:$tType,H:(B > (C > A)),F:(D > B),G:(E > C),X4:((product_prod @ D) @ E)]: (((((((product_case_prod @ B) @ C) @ A) @ H) @ (((((((product_map_prod @ D) @ B) @ E) @ C) @ F) @ G) @ X4)) = (((((product_case_prod @ D) @ E) @ A) @ (^ [L:D,R3:E]: (((H @ (F @ L)) @ (G @ R3))))) @ X4))))).
% 2.58/1.56 thf(fact_170_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C3:((product_prod @ A) @ B),F:(C > A),G:(D > B),R2:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C3) @ ((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G)) @ R2)) => (~ ((! [X:C,Y4:D]: (((C3 = ((((product_Pair @ A) @ B) @ (F @ X)) @ (G @ Y4))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X) @ Y4)) @ R2)))))))))))).
% 2.58/1.56 thf(fact_171_map__prod__def, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType]: ((((((product_map_prod @ A) @ C) @ B) @ D) = (^ [F9:(A > C),G4:(B > D)]: (((((product_case_prod @ A) @ B) @ ((product_prod @ C) @ D)) @ (^ [X5:A,Y5:B]: (((((product_Pair @ C) @ D) @ (F9 @ X5)) @ (G4 @ Y5))))))))))).
% 2.58/1.56 thf(fact_172_prod_Omap__ident, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: (((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X5:A]: (X5))) @ (^ [X5:B]: (X5))) @ T2) = T2)))).
% 2.58/1.56 thf(fact_173_aboveS__def, axiom, (! [A:$tType]: (((order_aboveS @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),A6:A]: (((collect @ A) @ (^ [B5:A]: (((B5 != A6) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A6) @ B5)) @ R3))))))))))).
% 2.58/1.56 thf(fact_174_above__def, axiom, (! [A:$tType]: (((order_above @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),A6:A]: (((collect @ A) @ (^ [B5:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A6) @ B5)) @ R3)))))))))).
% 2.58/1.56 thf(fact_175_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))))))).
% 2.58/1.56 thf(fact_176_The__split__eq, axiom, (! [A:$tType,B:$tType,X4:A,Y:B]: ((((the @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X7:A,Y7:B]: (((X4 = X7) & (Y = Y7)))))) = ((((product_Pair @ A) @ B) @ X4) @ Y))))).
% 2.58/1.56 thf(fact_177_in__inv__imagep, axiom, (! [B:$tType,A:$tType]: ((((inv_imagep @ A) @ B) = (^ [R3:(A > (A > $o)),F9:(B > A),X5:B,Y5:B]: (((R3 @ (F9 @ X5)) @ (F9 @ Y5)))))))).
% 2.58/1.56 thf(fact_178_UNIV__Times__UNIV, axiom, (! [B:$tType,A:$tType]: ((((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu:A]: ((top_top @ (set @ B))))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 2.58/1.56 thf(fact_179_rangeE, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))) => (~ ((! [X:B]: ((B2 != (F @ X)))))))))).
% 2.58/1.56 thf(fact_180_range__composition, axiom, (! [A:$tType,C:$tType,B:$tType,F:(C > A),G:(B > C)]: ((((((image @ B) @ A) @ (^ [X5:B]: ((F @ (G @ X5))))) @ (top_top @ (set @ B))) = ((((image @ C) @ A) @ F) @ ((((image @ B) @ C) @ G) @ (top_top @ (set @ B)))))))).
% 2.58/1.56 thf(fact_181_UNIV__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (^ [X5:A]: ($true))))))).
% 2.58/1.56 thf(fact_182_rangeI, axiom, (! [A:$tType,B:$tType,F:(B > A),X4:B]: ((((member @ A) @ (F @ X4)) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B))))))).
% 2.58/1.56 thf(fact_183_range__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),X4:B]: (((B2 = (F @ X4)) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 2.58/1.56 thf(fact_184_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F:(A > B),G:(C > D)]: (((((((image @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image @ C) @ D) @ G) @ (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) @ G)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 2.58/1.56 thf(fact_185_inv__imagep__def, axiom, (! [A:$tType,B:$tType]: ((((inv_imagep @ B) @ A) = (^ [R3:(B > (B > $o)),F9:(A > B),X5:A,Y5:A]: (((R3 @ (F9 @ X5)) @ (F9 @ Y5)))))))).
% 2.58/1.56 thf(fact_186_old_Orec__prod__def, axiom, (! [T:$tType,B:$tType,A:$tType]: (((((product_rec_prod @ A) @ B) @ T) = (^ [F12:(A > (B > T)),X5:((product_prod @ A) @ B)]: (((the @ T) @ (((((product_rec_set_prod @ A) @ B) @ T) @ F12) @ X5)))))))).
% 2.58/1.56 thf(fact_187_the__equality, axiom, (! [A:$tType,P3:(A > $o),A2:A]: (((P3 @ A2) => ((! [X:A]: (((P3 @ X) => (X = A2)))) => (((the @ A) @ P3) = A2)))))).
% 2.58/1.56 thf(fact_188_the__eq__trivial, axiom, (! [A:$tType,A2:A]: ((((the @ A) @ (^ [X5:A]: ((X5 = A2)))) = A2)))).
% 2.58/1.56 thf(fact_189_the__sym__eq__trivial, axiom, (! [A:$tType,X4:A]: ((((the @ A) @ ((^ [Y9:A,Z3:A]: ((Y9 = Z3))) @ X4)) = X4)))).
% 2.58/1.56 thf(fact_190_the1__equality, axiom, (! [A:$tType,P3:(A > $o),A2:A]: (((? [X6:A]: (((P3 @ X6) & (! [Y4:A]: (((P3 @ Y4) => (Y4 = X6))))))) => ((P3 @ A2) => (((the @ A) @ P3) = A2)))))).
% 2.58/1.56 thf(fact_191_the1I2, axiom, (! [A:$tType,P3:(A > $o),Q3:(A > $o)]: (((? [X6:A]: (((P3 @ X6) & (! [Y4:A]: (((P3 @ Y4) => (Y4 = X6))))))) => ((! [X:A]: (((P3 @ X) => (Q3 @ X)))) => (Q3 @ ((the @ A) @ P3))))))).
% 2.58/1.56 thf(fact_192_If__def, axiom, (! [A:$tType]: (((if @ A) = (^ [P5:$o,X5:A,Y5:A]: (((the @ A) @ (^ [Z2:A]: (((P5 => (Z2 = X5)) & ((~ (P5)) => (Z2 = Y5)))))))))))).
% 2.58/1.56 thf(fact_193_theI2, axiom, (! [A:$tType,P3:(A > $o),A2:A,Q3:(A > $o)]: (((P3 @ A2) => ((! [X:A]: (((P3 @ X) => (X = A2)))) => ((! [X:A]: (((P3 @ X) => (Q3 @ X)))) => (Q3 @ ((the @ A) @ P3)))))))).
% 2.58/1.56 thf(fact_194_theI_H, axiom, (! [A:$tType,P3:(A > $o)]: (((? [X6:A]: (((P3 @ X6) & (! [Y4:A]: (((P3 @ Y4) => (Y4 = X6))))))) => (P3 @ ((the @ A) @ P3)))))).
% 2.58/1.56 thf(fact_195_theI, axiom, (! [A:$tType,P3:(A > $o),A2:A]: (((P3 @ A2) => ((! [X:A]: (((P3 @ X) => (X = A2)))) => (P3 @ ((the @ A) @ P3))))))).
% 2.58/1.56 thf(fact_196_surj__def, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y5:A]: ((? [X5:B]: ((Y5 = (F @ X5)))))))))).
% 2.58/1.56 thf(fact_197_surjI, axiom, (! [B:$tType,A:$tType,G:(B > A),F:(A > B)]: (((! [X:A]: (((G @ (F @ X)) = X))) => (((((image @ B) @ A) @ G) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.58/1.56 thf(fact_198_surjD, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X:B]: ((Y = (F @ X)))))))).
% 2.58/1.56 thf(fact_199_surjE, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X:B]: ((Y != (F @ X)))))))))).
% 2.58/1.56 thf(fact_200_old_Orec__unit__def, axiom, (! [T:$tType]: (((product_rec_unit @ T) = (^ [F12:T,X5:product_unit]: (((the @ T) @ (((product_rec_set_unit @ T) @ F12) @ X5)))))))).
% 2.58/1.56 thf(fact_201_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)))))).
% 2.58/1.56 thf(fact_202_fst__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > A),G:(D > B),X4:((product_prod @ C) @ D)]: (((((product_fst @ A) @ B) @ (((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G) @ X4)) = (F @ (((product_fst @ C) @ D) @ X4)))))).
% 2.58/1.56 thf(fact_203_fst__eqD, axiom, (! [B:$tType,A:$tType,X4:A,Y:B,A2:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X4) @ Y)) = A2) => (X4 = A2))))).
% 2.58/1.56 thf(fact_204_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X2:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X2)) = X1)))).
% 2.58/1.56 thf(fact_205_fst__def, axiom, (! [B:$tType,A:$tType]: ((((product_fst @ A) @ B) = ((((product_case_prod @ A) @ B) @ A) @ (^ [X13:A,X23:B]: (X13))))))).
% 2.58/1.56 thf(fact_206_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A2:A,P2:((product_prod @ A) @ B)]: (((A2 = (((product_fst @ A) @ B) @ P2)) = (? [B5:B]: ((P2 = ((((product_Pair @ A) @ B) @ A2) @ B5)))))))).
% 2.58/1.56 thf(fact_207_fstI, axiom, (! [B:$tType,A:$tType,X4:((product_prod @ A) @ B),Y:A,Z:B]: (((X4 = ((((product_Pair @ A) @ B) @ Y) @ Z)) => ((((product_fst @ A) @ B) @ X4) = Y))))).
% 2.58/1.56 thf(fact_208_fst__image__times, axiom, (! [B:$tType,A:$tType,B6:(set @ B),A3:(set @ A)]: ((((B6 = (bot_bot @ (set @ B))) => (((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B6)))) = (bot_bot @ (set @ A)))) & ((B6 != (bot_bot @ (set @ B))) => (((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B6)))) = A3)))))).
% 2.58/1.56 thf(fact_209_The__case__prod, axiom, (! [B:$tType,A:$tType,P3:(A > (B > $o))]: ((((the @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P3)) = ((the @ ((product_prod @ A) @ B)) @ (^ [Xy:((product_prod @ A) @ B)]: (((P3 @ (((product_fst @ A) @ B) @ Xy)) @ (((product_snd @ A) @ B) @ Xy))))))))).
% 2.58/1.56 thf(fact_210_image__is__empty, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B)]: (((((((image @ B) @ A) @ F) @ A3) = (bot_bot @ (set @ A))) = (A3 = (bot_bot @ (set @ B))))))).
% 2.58/1.56 thf(fact_211_empty__is__image, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B)]: ((((bot_bot @ (set @ A)) = ((((image @ B) @ A) @ F) @ A3)) = (A3 = (bot_bot @ (set @ B))))))).
% 2.58/1.56 thf(fact_212_image__empty, axiom, (! [B:$tType,A:$tType,F:(B > A)]: ((((((image @ B) @ A) @ F) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))).
% 2.58/1.56 thf(fact_213_Sigma__empty1, axiom, (! [B:$tType,A:$tType,B6:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ (bot_bot @ (set @ A))) @ B6) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 2.58/1.56 thf(fact_214_snd__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > B),G:(D > A),X4:((product_prod @ C) @ D)]: (((((product_snd @ B) @ A) @ (((((((product_map_prod @ C) @ B) @ D) @ A) @ F) @ G) @ X4)) = (G @ (((product_snd @ C) @ D) @ X4)))))).
% 2.58/1.56 thf(fact_215_Collect__const, axiom, (! [A:$tType,P3:$o]: (((P3 => (((collect @ A) @ (^ [S6:A]: (P3))) = (top_top @ (set @ A)))) & ((~ (P3)) => (((collect @ A) @ (^ [S6:A]: (P3))) = (bot_bot @ (set @ A)))))))).
% 2.58/1.56 thf(fact_216_Sigma__empty2, axiom, (! [B:$tType,A:$tType,A3:(set @ A)]: ((((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: ((bot_bot @ (set @ B))))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 2.58/1.56 thf(fact_217_Times__empty, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B6:(set @ B)]: (((((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B6))) = (bot_bot @ (set @ ((product_prod @ A) @ B)))) = ((A3 = (bot_bot @ (set @ A))) | (B6 = (bot_bot @ (set @ B)))))))).
% 2.58/1.56 thf(fact_218_Collect__const__case__prod, axiom, (! [B:$tType,A:$tType,P3:$o]: (((P3 => (((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A6:A,B5:B]: (P3)))) = (top_top @ (set @ ((product_prod @ A) @ B))))) & ((~ (P3)) => (((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A6:A,B5:B]: (P3)))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))))).
% 2.58/1.56 thf(fact_219_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 2.58/1.56 thf(fact_220_snd__swap, axiom, (! [B:$tType,A:$tType,X4:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X4)) = (((product_fst @ A) @ B) @ X4))))).
% 2.58/1.56 thf(fact_221_fst__swap, axiom, (! [A:$tType,B:$tType,X4:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X4)) = (((product_snd @ B) @ A) @ X4))))).
% 2.58/1.56 thf(fact_222_range__snd, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ A)))))).
% 2.58/1.56 thf(fact_223_snd__image__times, axiom, (! [B:$tType,A:$tType,A3:(set @ B),B6:(set @ A)]: ((((A3 = (bot_bot @ (set @ B))) => (((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu:B]: (B6)))) = (bot_bot @ (set @ A)))) & ((A3 != (bot_bot @ (set @ B))) => (((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu:B]: (B6)))) = B6)))))).
% 2.58/1.56 thf(fact_224_sndI, axiom, (! [A:$tType,B:$tType,X4:((product_prod @ A) @ B),Y:A,Z:B]: (((X4 = ((((product_Pair @ A) @ B) @ Y) @ Z)) => ((((product_snd @ A) @ B) @ X4) = Z))))).
% 2.58/1.56 thf(fact_225_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B2:A,P2:((product_prod @ B) @ A)]: (((B2 = (((product_snd @ B) @ A) @ P2)) = (? [A6:B]: ((P2 = ((((product_Pair @ B) @ A) @ A6) @ B2)))))))).
% 2.58/1.56 thf(fact_226_times__eq__iff, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B6:(set @ B),C5:(set @ A),D3:(set @ B)]: (((((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B6))) = ((((product_Sigma @ A) @ B) @ C5) @ (^ [Uu:A]: (D3)))) = (((A3 = C5) & (B6 = D3)) | (((A3 = (bot_bot @ (set @ A))) | (B6 = (bot_bot @ (set @ B)))) & ((C5 = (bot_bot @ (set @ A))) | (D3 = (bot_bot @ (set @ B)))))))))).
% 2.58/1.56 thf(fact_227_empty__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (^ [X5:A]: ($false))))))).
% 2.58/1.56 thf(fact_228_snd__eqD, axiom, (! [B:$tType,A:$tType,X4:B,Y:A,A2:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X4) @ Y)) = A2) => (Y = A2))))).
% 2.58/1.56 thf(fact_229_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X2:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X2)) = X2)))).
% 2.58/1.56 thf(fact_230_Sigma__empty__iff, axiom, (! [B:$tType,A:$tType,I4:(set @ A),X8:(A > (set @ B))]: (((((((product_Sigma @ A) @ B) @ I4) @ X8) = (bot_bot @ (set @ ((product_prod @ A) @ B)))) = (! [X5:A]: (((((member @ A) @ X5) @ I4) => ((X8 @ X5) = (bot_bot @ (set @ B)))))))))).
% 2.58/1.56 thf(fact_231_snd__def, axiom, (! [B:$tType,A:$tType]: ((((product_snd @ A) @ B) = ((((product_case_prod @ A) @ B) @ B) @ (^ [X13:A,X23:B]: (X23))))))).
% 2.58/1.56 thf(fact_232_split__comp__eq, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,F:(A > (B > C)),G:(D > A)]: (((^ [U:((product_prod @ D) @ B)]: (((F @ (G @ (((product_fst @ D) @ B) @ U))) @ (((product_snd @ D) @ B) @ U)))) = ((((product_case_prod @ D) @ B) @ C) @ (^ [X5:D]: ((F @ (G @ X5))))))))).
% 2.58/1.56 thf(fact_233_case__prod__beta, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ A) = (^ [F9:(B > (C > A)),P:((product_prod @ B) @ C)]: (((F9 @ (((product_fst @ B) @ C) @ P)) @ (((product_snd @ B) @ C) @ P)))))))).
% 2.58/1.56 thf(fact_234_case__prod__beta_H, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F9:(A > (B > C)),X5:((product_prod @ A) @ B)]: (((F9 @ (((product_fst @ A) @ B) @ X5)) @ (((product_snd @ A) @ B) @ X5)))))))).
% 2.58/1.56 thf(fact_235_prod_Ocase__eq__if, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F9:(A > (B > C)),Prod2:((product_prod @ A) @ B)]: (((F9 @ (((product_fst @ A) @ B) @ Prod2)) @ (((product_snd @ A) @ B) @ Prod2)))))))).
% 2.58/1.56 thf(fact_236_case__prod__unfold, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [C4:(A > (B > C)),P:((product_prod @ A) @ B)]: (((C4 @ (((product_fst @ A) @ B) @ P)) @ (((product_snd @ A) @ B) @ P)))))))).
% 2.58/1.56 thf(fact_237_Product__Type_OCollect__case__prodD, axiom, (! [B:$tType,A:$tType,X4:((product_prod @ A) @ B),A3:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ X4) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A3))) => ((A3 @ (((product_fst @ A) @ B) @ X4)) @ (((product_snd @ A) @ B) @ X4)))))).
% 2.58/1.56 thf(fact_238_mem__Times__iff, axiom, (! [A:$tType,B:$tType,X4:((product_prod @ A) @ B),A3:(set @ A),B6:(set @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X4) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B6)))) = ((((member @ A) @ (((product_fst @ A) @ B) @ X4)) @ A3) & (((member @ B) @ (((product_snd @ A) @ B) @ X4)) @ B6)))))).
% 2.58/1.56 thf(fact_239_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 2.58/1.56 thf(fact_240_surjective__pairing, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: ((T2 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T2)) @ (((product_snd @ A) @ B) @ T2)))))).
% 2.58/1.56 thf(fact_241_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y9:((product_prod @ A) @ B),Z3:((product_prod @ A) @ B)]: ((Y9 = Z3))) = (^ [S6:((product_prod @ A) @ B),T3:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S6) = (((product_fst @ A) @ B) @ T3)) & ((((product_snd @ A) @ B) @ S6) = (((product_snd @ A) @ B) @ T3))))))))).
% 2.58/1.56 thf(fact_242_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod3:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod3)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod3))) => (Prod = Prod3))))).
% 2.58/1.56 thf(fact_243_prod__eqI, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),Q:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P2) = (((product_fst @ A) @ B) @ Q)) => (((((product_snd @ A) @ B) @ P2) = (((product_snd @ A) @ B) @ Q)) => (P2 = Q)))))).
% 2.58/1.56 thf(fact_244_prod_Osplit__sel, axiom, (! [C:$tType,B:$tType,A:$tType,P3:(C > $o),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P3 @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) => (P3 @ ((F @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))).
% 2.58/1.56 thf(fact_245_prod_Osplit__sel__asm, axiom, (! [C:$tType,B:$tType,A:$tType,P3:(C > $o),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P3 @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = (~ (((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) & (~ ((P3 @ ((F @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))))))).
% 2.58/1.56 thf(fact_246_scomp__unfold, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((((product_scomp @ A) @ B) @ C) @ D) = (^ [F9:(A > ((product_prod @ B) @ C)),G4:(B > (C > D)),X5:A]: (((G4 @ (((product_fst @ B) @ C) @ (F9 @ X5))) @ (((product_snd @ B) @ C) @ (F9 @ X5))))))))).
% 2.58/1.56 thf(fact_247_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P)) @ (((product_fst @ A) @ B) @ P)))))))).
% 2.58/1.56 thf(fact_248_fst__image__Sigma, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B6:(A > (set @ B))]: ((((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ B6)) = ((collect @ A) @ (^ [X5:A]: (((((member @ A) @ X5) @ A3) & ((B6 @ X5) != (bot_bot @ (set @ B))))))))))).
% 2.58/1.56 thf(fact_249_exE__realizer, axiom, (! [C:$tType,A:$tType,B:$tType,P3:(A > (B > $o)),P2:((product_prod @ B) @ A),Q3:(C > $o),F:(B > (A > C))]: ((((P3 @ (((product_snd @ B) @ A) @ P2)) @ (((product_fst @ B) @ A) @ P2)) => ((! [X:B,Y4:A]: ((((P3 @ Y4) @ X) => (Q3 @ ((F @ X) @ Y4))))) => (Q3 @ (((((product_case_prod @ B) @ A) @ C) @ F) @ P2))))))).
% 2.58/1.56 thf(fact_250_exI__realizer, axiom, (! [B:$tType,A:$tType,P3:(A > (B > $o)),Y:A,X4:B]: ((((P3 @ Y) @ X4) => ((P3 @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X4) @ Y))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X4) @ Y))))))).
% 2.58/1.56 thf(fact_251_conjI__realizer, axiom, (! [A:$tType,B:$tType,P3:(A > $o),P2:A,Q3:(B > $o),Q:B]: (((P3 @ P2) => ((Q3 @ Q) => ((P3 @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q))) & (Q3 @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q))))))))).
% 2.58/1.56 thf(fact_252_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P3:(A > (B > $o)),X4:A,Y:B,A2:((product_prod @ A) @ B)]: ((((P3 @ X4) @ Y) => ((A2 = ((((product_Pair @ A) @ B) @ X4) @ Y)) => ((P3 @ (((product_fst @ A) @ B) @ A2)) @ (((product_snd @ A) @ B) @ A2))))))).
% 2.58/1.56 thf(fact_253_the__elem__image__unique, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F:(A > B),X4:A]: (((A3 != (bot_bot @ (set @ A))) => ((! [Y4:A]: (((((member @ A) @ Y4) @ A3) => ((F @ Y4) = (F @ X4))))) => (((the_elem @ B) @ ((((image @ A) @ B) @ F) @ A3)) = (F @ X4))))))).
% 2.58/1.56 thf(fact_254_subset__fst__snd, axiom, (! [B:$tType,A:$tType,A3:(set @ ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A3) @ ((((product_Sigma @ A) @ B) @ ((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A3)) @ (^ [Uu:A]: (((((image @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A3)))))))).
% 2.58/1.56 thf(fact_255_Collect__split__mono__strong, axiom, (! [B:$tType,A:$tType,X8:(set @ A),A3:(set @ ((product_prod @ A) @ B)),Y6:(set @ B),P3:(A > (B > $o)),Q3:(A > (B > $o))]: (((X8 = ((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A3)) => ((Y6 = ((((image @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A3)) => ((! [X:A]: (((((member @ A) @ X) @ X8) => (! [Xa:B]: (((((member @ B) @ Xa) @ Y6) => (((P3 @ X) @ Xa) => ((Q3 @ X) @ Xa)))))))) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A3) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P3))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A3) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ Q3)))))))))).
% 2.58/1.56 thf(help_If_3_1_T, axiom, (! [P3:$o]: (((P3 = $true) | (P3 = $false))))).
% 2.58/1.56 thf(help_If_2_1_T, axiom, (! [A:$tType,X4:A,Y:A]: ((((((if @ A) @ $false) @ X4) @ Y) = Y)))).
% 2.58/1.56 thf(help_If_1_1_T, axiom, (! [A:$tType,X4:A,Y:A]: ((((((if @ A) @ $true) @ X4) @ Y) = X4)))).
% 2.58/1.56 thf(conj_0, conjecture, (? [I5:((list @ char) > paraco415392788lle_tv)]: ((((paraco876059933e_eval @ I5) @ p) != (paraco2040174112le_Det @ $true))))).
% 2.58/1.56 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.g4D8jloWIn/DTF2DTF16534.p
% 2.58/1.56 ------------------------
% 2.58/1.59 ---- Cleaned THF ---
% 2.58/1.59 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv,type,
% 2.58/1.59 paraco415392788lle_tv: $tType ).
% 2.58/1.59
% 2.58/1.59 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm,type,
% 2.58/1.59 paraco414474393lle_fm: $tType ).
% 2.58/1.59
% 2.58/1.59 thf(ty_t_Product__Type_Ounit,type,
% 2.58/1.59 product_unit: $tType ).
% 2.58/1.59
% 2.58/1.59 thf(ty_t_Product__Type_Oprod,type,
% 2.58/1.59 product_prod: $tType > $tType > $tType ).
% 2.58/1.59
% 2.58/1.59 thf(ty_t_String_Ochar,type,
% 2.58/1.59 char: $tType ).
% 2.58/1.59
% 2.58/1.59 thf(ty_t_List_Olist,type,
% 2.58/1.59 list: $tType > $tType ).
% 2.58/1.59
% 2.58/1.59 thf(ty_t_Set_Oset,type,
% 2.58/1.59 set: $tType > $tType ).
% 2.58/1.59
% 2.58/1.59 thf(ty_t_Nat_Onat,type,
% 2.58/1.59 nat: $tType ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Fun__Def_Oreduction__pair,type,
% 2.58/1.59 fun_reduction_pair:
% 2.58/1.59 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 2.58/1.59 fun_rp_inv_image:
% 2.58/1.59 !>[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 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_HOL_OThe,type,
% 2.58/1.59 the:
% 2.58/1.59 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_If,type,
% 2.58/1.59 if:
% 2.58/1.59 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Order__Relation_Oabove,type,
% 2.58/1.59 order_above:
% 2.58/1.59 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Order__Relation_OaboveS,type,
% 2.58/1.59 order_aboveS:
% 2.58/1.59 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Order__Relation_Orelation__of,type,
% 2.58/1.59 order_relation_of:
% 2.58/1.59 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.58/1.59 bot_bot:
% 2.58/1.59 !>[A: $tType] : A ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.58/1.59 ord_less_eq:
% 2.58/1.59 !>[A: $tType] : ( A > A > $o ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.58/1.59 top_top:
% 2.58/1.59 !>[A: $tType] : A ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval,type,
% 2.58/1.59 paraco876059933e_eval: ( ( list @ char ) > paraco415392788lle_tv ) > paraco414474393lle_fm > paraco415392788lle_tv ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H,type,
% 2.58/1.59 paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql,type,
% 2.58/1.59 paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H,type,
% 2.58/1.59 paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H,type,
% 2.58/1.59 paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro,type,
% 2.58/1.59 paraco27778325le_Pro: ( list @ char ) > paraco414474393lle_fm ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth,type,
% 2.58/1.59 paraco251304083_Truth: paraco414474393lle_fm ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Ocase__fm,type,
% 2.58/1.59 paraco1246693743ase_fm:
% 2.58/1.59 !>[A: $tType] : ( ( ( list @ char ) > A ) > A > ( paraco414474393lle_fm > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A ) > paraco414474393lle_fm > A ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet,type,
% 2.58/1.59 paraco2040174112le_Det: $o > paraco415392788lle_tv ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet,type,
% 2.58/1.59 paraco676387099_Indet: nat > paraco415392788lle_tv ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Ocase__tv,type,
% 2.58/1.59 paraco490622181ase_tv:
% 2.58/1.59 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv,type,
% 2.58/1.59 paraco152590079rec_tv:
% 2.58/1.59 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid,type,
% 2.58/1.59 paraco769098683_valid: paraco414474393lle_fm > $o ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_OPair,type,
% 2.58/1.59 product_Pair:
% 2.58/1.59 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_OSigma,type,
% 2.58/1.59 product_Sigma:
% 2.58/1.59 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Obool_Ocase__bool,type,
% 2.58/1.59 product_case_bool:
% 2.58/1.59 !>[A: $tType] : ( A > A > $o > A ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Ocurry,type,
% 2.58/1.59 product_curry:
% 2.58/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.58/1.59 produc2004651681e_prod:
% 2.58/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Omap__prod,type,
% 2.58/1.59 product_map_prod:
% 2.58/1.59 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.58/1.59 product_rec_prod:
% 2.58/1.59 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 2.58/1.59 product_rec_set_prod:
% 2.58/1.59 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 2.58/1.59 product_rec_set_unit:
% 2.58/1.59 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 2.58/1.59 product_rec_unit:
% 2.58/1.59 !>[T: $tType] : ( T > product_unit > T ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 2.58/1.59 product_case_prod:
% 2.58/1.59 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 2.58/1.59 product_fst:
% 2.58/1.59 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 2.58/1.59 product_snd:
% 2.58/1.59 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.58/1.59 product_swap:
% 2.58/1.59 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Oproduct,type,
% 2.58/1.59 product_product:
% 2.58/1.59 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Product__Type_Oscomp,type,
% 2.58/1.59 product_scomp:
% 2.58/1.59 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Relation_Oinv__image,type,
% 2.58/1.59 inv_image:
% 2.58/1.59 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Relation_Oinv__imagep,type,
% 2.58/1.59 inv_imagep:
% 2.58/1.59 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Set_OCollect,type,
% 2.58/1.59 collect:
% 2.58/1.59 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Set_Oimage,type,
% 2.58/1.59 image:
% 2.58/1.59 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Set_Othe__elem,type,
% 2.58/1.59 the_elem:
% 2.58/1.59 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Wellfounded_Olex__prod,type,
% 2.58/1.59 lex_prod:
% 2.58/1.59 !>[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 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Wfrec_Osame__fst,type,
% 2.58/1.59 same_fst:
% 2.58/1.59 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_member,type,
% 2.58/1.59 member:
% 2.58/1.59 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_v_p,type,
% 2.58/1.59 p: paraco414474393lle_fm ).
% 2.58/1.59
% 2.58/1.59 thf(fact_0_tv_Oinject_I1_J,axiom,
% 2.58/1.59 ! [X1: $o,Y1: $o] :
% 2.58/1.59 ( ( ( paraco2040174112le_Det @ X1 )
% 2.58/1.59 = ( paraco2040174112le_Det @ Y1 ) )
% 2.58/1.59 = ( X1 = Y1 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_1_assms,axiom,
% 2.58/1.59 paraco769098683_valid @ ( paraco329115265le_Neg @ p ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_2_valid__def,axiom,
% 2.58/1.59 ( paraco769098683_valid
% 2.58/1.59 = ( ^ [P: paraco414474393lle_fm] :
% 2.58/1.59 ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 2.58/1.59 ( ( paraco876059933e_eval @ I @ P )
% 2.58/1.59 = ( paraco2040174112le_Det @ $true ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_3_eval_Osimps_I2_J,axiom,
% 2.58/1.59 ! [I2: ( list @ char ) > paraco415392788lle_tv] :
% 2.58/1.59 ( ( paraco876059933e_eval @ I2 @ paraco251304083_Truth )
% 2.58/1.59 = ( paraco2040174112le_Det @ $true ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_4_eval_Osimps_I4_J,axiom,
% 2.58/1.59 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.58/1.59 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 = ( paraco876059933e_eval @ I2 @ Q ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 2.58/1.59 = ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 2.58/1.59 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 != ( paraco876059933e_eval @ I2 @ Q ) )
% 2.58/1.59 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 = ( paraco2040174112le_Det @ $true ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 2.58/1.59 = ( paraco876059933e_eval @ I2 @ Q ) ) )
% 2.58/1.59 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 != ( paraco2040174112le_Det @ $true ) )
% 2.58/1.59 => ( ( ( ( paraco876059933e_eval @ I2 @ Q )
% 2.58/1.59 = ( paraco2040174112le_Det @ $true ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 2.58/1.59 = ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 2.58/1.59 & ( ( ( paraco876059933e_eval @ I2 @ Q )
% 2.58/1.59 != ( paraco2040174112le_Det @ $true ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 2.58/1.59 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_5_eval_Osimps_I5_J,axiom,
% 2.58/1.59 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.58/1.59 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 = ( paraco876059933e_eval @ I2 @ Q ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
% 2.58/1.59 = ( paraco2040174112le_Det @ $true ) ) )
% 2.58/1.59 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 != ( paraco876059933e_eval @ I2 @ Q ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
% 2.58/1.59 = ( paraco2040174112le_Det @ $false ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_6_eval__equality__simplify,axiom,
% 2.58/1.59 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
% 2.58/1.59 = ( paraco2040174112le_Det
% 2.58/1.59 @ ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 = ( paraco876059933e_eval @ I2 @ Q ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_7_tv_Osimps_I7_J,axiom,
% 2.58/1.59 ! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
% 2.58/1.59 ( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
% 2.58/1.59 = ( F1 @ X1 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_8_eval__negation,axiom,
% 2.58/1.59 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm] :
% 2.58/1.59 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 = ( paraco2040174112le_Det @ $false ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 2.58/1.59 = ( paraco2040174112le_Det @ $true ) ) )
% 2.58/1.59 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 != ( paraco2040174112le_Det @ $false ) )
% 2.58/1.59 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 = ( paraco2040174112le_Det @ $true ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 2.58/1.59 = ( paraco2040174112le_Det @ $false ) ) )
% 2.58/1.59 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 != ( paraco2040174112le_Det @ $true ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 2.58/1.59 = ( paraco876059933e_eval @ I2 @ P2 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_9_tv_Osimps_I5_J,axiom,
% 2.58/1.59 ! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
% 2.58/1.59 ( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
% 2.58/1.59 = ( F1 @ X1 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_10_eval_Osimps_I1_J,axiom,
% 2.58/1.59 ! [I2: ( list @ char ) > paraco415392788lle_tv,S: list @ char] :
% 2.58/1.59 ( ( paraco876059933e_eval @ I2 @ ( paraco27778325le_Pro @ S ) )
% 2.58/1.59 = ( I2 @ S ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_11_tv_Odistinct_I1_J,axiom,
% 2.58/1.59 ! [X1: $o,X2: nat] :
% 2.58/1.59 ( ( paraco2040174112le_Det @ X1 )
% 2.58/1.59 != ( paraco676387099_Indet @ X2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_12_tv_Oinduct,axiom,
% 2.58/1.59 ! [P3: paraco415392788lle_tv > $o,Tv: paraco415392788lle_tv] :
% 2.58/1.59 ( ! [X: $o] : ( P3 @ ( paraco2040174112le_Det @ X ) )
% 2.58/1.59 => ( ! [X: nat] : ( P3 @ ( paraco676387099_Indet @ X ) )
% 2.58/1.59 => ( P3 @ Tv ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_13_tv_Oexhaust,axiom,
% 2.58/1.59 ! [Y: paraco415392788lle_tv] :
% 2.58/1.59 ( ! [X12: $o] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( paraco2040174112le_Det @ X12 ) )
% 2.58/1.59 => ~ ! [X22: nat] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( paraco676387099_Indet @ X22 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_14_fm_Oinject_I2_J,axiom,
% 2.58/1.59 ! [X3: paraco414474393lle_fm,Y3: paraco414474393lle_fm] :
% 2.58/1.59 ( ( ( paraco329115265le_Neg @ X3 )
% 2.58/1.59 = ( paraco329115265le_Neg @ Y3 ) )
% 2.58/1.59 = ( X3 = Y3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_15_tv_Oinject_I2_J,axiom,
% 2.58/1.59 ! [X2: nat,Y2: nat] :
% 2.58/1.59 ( ( ( paraco676387099_Indet @ X2 )
% 2.58/1.59 = ( paraco676387099_Indet @ Y2 ) )
% 2.58/1.59 = ( X2 = Y2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_16_fm_Oinject_I4_J,axiom,
% 2.58/1.59 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,Y51: paraco414474393lle_fm,Y52: paraco414474393lle_fm] :
% 2.58/1.59 ( ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 2.58/1.59 = ( paraco2084319816le_Eql @ Y51 @ Y52 ) )
% 2.58/1.59 = ( ( X51 = Y51 )
% 2.58/1.59 & ( X52 = Y52 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_17_fm_Oinject_I1_J,axiom,
% 2.58/1.59 ! [X1: list @ char,Y1: list @ char] :
% 2.58/1.59 ( ( ( paraco27778325le_Pro @ X1 )
% 2.58/1.59 = ( paraco27778325le_Pro @ Y1 ) )
% 2.58/1.59 = ( X1 = Y1 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_18_fm_Oinject_I3_J,axiom,
% 2.58/1.59 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,Y41: paraco414474393lle_fm,Y42: paraco414474393lle_fm] :
% 2.58/1.59 ( ( ( paraco2100061555le_Con @ X41 @ X42 )
% 2.58/1.59 = ( paraco2100061555le_Con @ Y41 @ Y42 ) )
% 2.58/1.59 = ( ( X41 = Y41 )
% 2.58/1.59 & ( X42 = Y42 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_19_tv_Osimps_I8_J,axiom,
% 2.58/1.59 ! [A: $tType,F1: $o > A,F2: nat > A,X2: nat] :
% 2.58/1.59 ( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X2 ) )
% 2.58/1.59 = ( F2 @ X2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_20_tv_Osimps_I6_J,axiom,
% 2.58/1.59 ! [A: $tType,F1: $o > A,F2: nat > A,X2: nat] :
% 2.58/1.59 ( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X2 ) )
% 2.58/1.59 = ( F2 @ X2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_21_fm_Odistinct_I25_J,axiom,
% 2.58/1.59 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 2.58/1.59 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_22_fm_Odistinct_I21_J,axiom,
% 2.58/1.59 ! [X3: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco329115265le_Neg @ X3 )
% 2.58/1.59 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_23_fm_Odistinct_I19_J,axiom,
% 2.58/1.59 ! [X3: paraco414474393lle_fm,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco329115265le_Neg @ X3 )
% 2.58/1.59 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_24_fm_Odistinct_I15_J,axiom,
% 2.58/1.59 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.58/1.59 ( paraco251304083_Truth
% 2.58/1.59 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_25_fm_Odistinct_I13_J,axiom,
% 2.58/1.59 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.58/1.59 ( paraco251304083_Truth
% 2.58/1.59 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_26_fm_Odistinct_I11_J,axiom,
% 2.58/1.59 ! [X3: paraco414474393lle_fm] :
% 2.58/1.59 ( paraco251304083_Truth
% 2.58/1.59 != ( paraco329115265le_Neg @ X3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_27_fm_Odistinct_I7_J,axiom,
% 2.58/1.59 ! [X1: list @ char,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco27778325le_Pro @ X1 )
% 2.58/1.59 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_28_fm_Odistinct_I5_J,axiom,
% 2.58/1.59 ! [X1: list @ char,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco27778325le_Pro @ X1 )
% 2.58/1.59 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_29_fm_Odistinct_I3_J,axiom,
% 2.58/1.59 ! [X1: list @ char,X3: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco27778325le_Pro @ X1 )
% 2.58/1.59 != ( paraco329115265le_Neg @ X3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_30_fm_Odistinct_I1_J,axiom,
% 2.58/1.59 ! [X1: list @ char] :
% 2.58/1.59 ( ( paraco27778325le_Pro @ X1 )
% 2.58/1.59 != paraco251304083_Truth ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_31_double__negation,axiom,
% 2.58/1.59 ( paraco876059933e_eval
% 2.58/1.59 = ( ^ [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] : ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ ( paraco329115265le_Neg @ P ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_32_string__tv_Oinduct,axiom,
% 2.58/1.59 ! [P3: paraco415392788lle_tv > $o,A0: paraco415392788lle_tv] :
% 2.58/1.59 ( ( P3 @ ( paraco2040174112le_Det @ $true ) )
% 2.58/1.59 => ( ( P3 @ ( paraco2040174112le_Det @ $false ) )
% 2.58/1.59 => ( ! [N: nat] : ( P3 @ ( paraco676387099_Indet @ N ) )
% 2.58/1.59 => ( P3 @ A0 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_33_string__tv_Ocases,axiom,
% 2.58/1.59 ! [X4: paraco415392788lle_tv] :
% 2.58/1.59 ( ( X4
% 2.58/1.59 != ( paraco2040174112le_Det @ $true ) )
% 2.58/1.59 => ( ( X4
% 2.58/1.59 != ( paraco2040174112le_Det @ $false ) )
% 2.58/1.59 => ~ ! [N: nat] :
% 2.58/1.59 ( X4
% 2.58/1.59 != ( paraco676387099_Indet @ N ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_34_fm_Oinduct,axiom,
% 2.58/1.59 ! [P3: paraco414474393lle_fm > $o,Fm: paraco414474393lle_fm] :
% 2.58/1.59 ( ! [X: list @ char] : ( P3 @ ( paraco27778325le_Pro @ X ) )
% 2.58/1.59 => ( ( P3 @ paraco251304083_Truth )
% 2.58/1.59 => ( ! [X: paraco414474393lle_fm] :
% 2.58/1.59 ( ( P3 @ X )
% 2.58/1.59 => ( P3 @ ( paraco329115265le_Neg @ X ) ) )
% 2.58/1.59 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 2.58/1.59 ( ( P3 @ X1a )
% 2.58/1.59 => ( ( P3 @ X22 )
% 2.58/1.59 => ( P3 @ ( paraco2100061555le_Con @ X1a @ X22 ) ) ) )
% 2.58/1.59 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 2.58/1.59 ( ( P3 @ X1a )
% 2.58/1.59 => ( ( P3 @ X22 )
% 2.58/1.59 => ( P3 @ ( paraco2084319816le_Eql @ X1a @ X22 ) ) ) )
% 2.58/1.59 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 2.58/1.59 ( ( P3 @ X1a )
% 2.58/1.59 => ( ( P3 @ X22 )
% 2.58/1.59 => ( P3 @ ( paraco1628874225le_Eql @ X1a @ X22 ) ) ) )
% 2.58/1.59 => ( P3 @ Fm ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_35_fm_Oexhaust,axiom,
% 2.58/1.59 ! [Y: paraco414474393lle_fm] :
% 2.58/1.59 ( ! [X12: list @ char] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( paraco27778325le_Pro @ X12 ) )
% 2.58/1.59 => ( ( Y != paraco251304083_Truth )
% 2.58/1.59 => ( ! [X32: paraco414474393lle_fm] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( paraco329115265le_Neg @ X32 ) )
% 2.58/1.59 => ( ! [X412: paraco414474393lle_fm,X422: paraco414474393lle_fm] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( paraco2100061555le_Con @ X412 @ X422 ) )
% 2.58/1.59 => ( ! [X512: paraco414474393lle_fm,X522: paraco414474393lle_fm] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( paraco2084319816le_Eql @ X512 @ X522 ) )
% 2.58/1.59 => ~ ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_36_eval_Osimps_I3_J,axiom,
% 2.58/1.59 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 2.58/1.59 = ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ P2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_37_eval__equality,axiom,
% 2.58/1.59 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.58/1.59 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 = ( paraco876059933e_eval @ I2 @ Q ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.58/1.59 = ( paraco2040174112le_Det @ $true ) ) )
% 2.58/1.59 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 != ( paraco876059933e_eval @ I2 @ Q ) )
% 2.58/1.59 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 = ( paraco2040174112le_Det @ $true ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.58/1.59 = ( paraco876059933e_eval @ I2 @ Q ) ) )
% 2.58/1.59 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 != ( paraco2040174112le_Det @ $true ) )
% 2.58/1.59 => ( ( ( ( paraco876059933e_eval @ I2 @ Q )
% 2.58/1.59 = ( paraco2040174112le_Det @ $true ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.58/1.59 = ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 2.58/1.59 & ( ( ( paraco876059933e_eval @ I2 @ Q )
% 2.58/1.59 != ( paraco2040174112le_Det @ $true ) )
% 2.58/1.59 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 = ( paraco2040174112le_Det @ $false ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.58/1.59 = ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ Q ) ) ) )
% 2.58/1.59 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 != ( paraco2040174112le_Det @ $false ) )
% 2.58/1.59 => ( ( ( ( paraco876059933e_eval @ I2 @ Q )
% 2.58/1.59 = ( paraco2040174112le_Det @ $false ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.58/1.59 = ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) ) ) )
% 2.58/1.59 & ( ( ( paraco876059933e_eval @ I2 @ Q )
% 2.58/1.59 != ( paraco2040174112le_Det @ $false ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.58/1.59 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_38_eval_Ocases,axiom,
% 2.58/1.59 ! [X4: product_prod @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm] :
% 2.58/1.59 ( ! [I3: ( list @ char ) > paraco415392788lle_tv,S2: list @ char] :
% 2.58/1.59 ( X4
% 2.58/1.59 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco27778325le_Pro @ S2 ) ) )
% 2.58/1.59 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv] :
% 2.58/1.59 ( X4
% 2.58/1.59 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ paraco251304083_Truth ) )
% 2.58/1.59 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm] :
% 2.58/1.59 ( X4
% 2.58/1.59 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco329115265le_Neg @ P4 ) ) )
% 2.58/1.59 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 2.58/1.59 ( X4
% 2.58/1.59 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2100061555le_Con @ P4 @ Q2 ) ) )
% 2.58/1.59 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 2.58/1.59 ( X4
% 2.58/1.59 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2084319816le_Eql @ P4 @ Q2 ) ) )
% 2.58/1.59 => ~ ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 2.58/1.59 ( X4
% 2.58/1.59 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco1628874225le_Eql @ P4 @ Q2 ) ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_39_fm_Osimps_I37_J,axiom,
% 2.58/1.59 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A] :
% 2.58/1.59 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ paraco251304083_Truth )
% 2.58/1.59 = F2 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_40_fm_Osimps_I39_J,axiom,
% 2.58/1.59 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2100061555le_Con @ X41 @ X42 ) )
% 2.58/1.59 = ( F4 @ X41 @ X42 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_41_fm_Osimps_I36_J,axiom,
% 2.58/1.59 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X1: list @ char] :
% 2.58/1.59 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco27778325le_Pro @ X1 ) )
% 2.58/1.59 = ( F1 @ X1 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_42_fm_Osimps_I40_J,axiom,
% 2.58/1.59 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2084319816le_Eql @ X51 @ X52 ) )
% 2.58/1.59 = ( F5 @ X51 @ X52 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_43_fm_Osimps_I38_J,axiom,
% 2.58/1.59 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X3: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco329115265le_Neg @ X3 ) )
% 2.58/1.59 = ( F3 @ X3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_44_fm_Odistinct_I17_J,axiom,
% 2.58/1.59 ! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 2.58/1.59 ( paraco251304083_Truth
% 2.58/1.59 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_45_mem__Collect__eq,axiom,
% 2.58/1.59 ! [A: $tType,A2: A,P3: A > $o] :
% 2.58/1.59 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 2.58/1.59 = ( P3 @ A2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_46_Collect__mem__eq,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A] :
% 2.58/1.59 ( ( collect @ A
% 2.58/1.59 @ ^ [X5: A] : ( member @ A @ X5 @ A3 ) )
% 2.58/1.59 = A3 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_47_Collect__cong,axiom,
% 2.58/1.59 ! [A: $tType,P3: A > $o,Q3: A > $o] :
% 2.58/1.59 ( ! [X: A] :
% 2.58/1.59 ( ( P3 @ X )
% 2.58/1.59 = ( Q3 @ X ) )
% 2.58/1.59 => ( ( collect @ A @ P3 )
% 2.58/1.59 = ( collect @ A @ Q3 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_48_ext,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.58/1.59 ( ! [X: A] :
% 2.58/1.59 ( ( F @ X )
% 2.58/1.59 = ( G @ X ) )
% 2.58/1.59 => ( F = G ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_49_fm_Oinject_I5_J,axiom,
% 2.58/1.59 ! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm,Y61: paraco414474393lle_fm,Y62: paraco414474393lle_fm] :
% 2.58/1.59 ( ( ( paraco1628874225le_Eql @ X612 @ X622 )
% 2.58/1.59 = ( paraco1628874225le_Eql @ Y61 @ Y62 ) )
% 2.58/1.59 = ( ( X612 = Y61 )
% 2.58/1.59 & ( X622 = Y62 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_50_fm_Osimps_I41_J,axiom,
% 2.58/1.59 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco1628874225le_Eql @ X612 @ X622 ) )
% 2.58/1.59 = ( F6 @ X612 @ X622 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_51_fm_Odistinct_I23_J,axiom,
% 2.58/1.59 ! [X3: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco329115265le_Neg @ X3 )
% 2.58/1.59 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_52_fm_Odistinct_I29_J,axiom,
% 2.58/1.59 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 2.58/1.59 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_53_fm_Odistinct_I9_J,axiom,
% 2.58/1.59 ! [X1: list @ char,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco27778325le_Pro @ X1 )
% 2.58/1.59 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_54_fm_Odistinct_I27_J,axiom,
% 2.58/1.59 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 2.58/1.59 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 2.58/1.59 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_55_old_Oprod_Oinject,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
% 2.58/1.59 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.58/1.59 = ( product_Pair @ A @ B @ A4 @ B3 ) )
% 2.58/1.59 = ( ( A2 = A4 )
% 2.58/1.59 & ( B2 = B3 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_56_prod_Oinject,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 2.58/1.59 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 2.58/1.59 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.58/1.59 = ( ( X1 = Y1 )
% 2.58/1.59 & ( X2 = Y2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_57_bool_Osplit__sel__asm,axiom,
% 2.58/1.59 ! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
% 2.58/1.59 ( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 2.58/1.59 = ( ~ ( ( Bool
% 2.58/1.59 & ~ ( P3 @ F1 ) )
% 2.58/1.59 | ( ~ Bool
% 2.58/1.59 & ~ ( P3 @ F2 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_58_bool_Ocase__distrib,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,H: A > B,F1: A,F2: A,Bool: $o] :
% 2.58/1.59 ( ( H @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 2.58/1.59 = ( product_case_bool @ B @ ( H @ F1 ) @ ( H @ F2 ) @ Bool ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_59_bool_Ocase__eq__if,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( product_case_bool @ A )
% 2.58/1.59 = ( ^ [F12: A,F22: A,Bool2: $o] : ( if @ A @ Bool2 @ F12 @ F22 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_60_bool_Osplit__sel,axiom,
% 2.58/1.59 ! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
% 2.58/1.59 ( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 2.58/1.59 = ( ( Bool
% 2.58/1.59 => ( P3 @ F1 ) )
% 2.58/1.59 & ( ~ Bool
% 2.58/1.59 => ( P3 @ F2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_61_old_Obool_Osimps_I3_J,axiom,
% 2.58/1.59 ! [A: $tType,F1: A,F2: A] :
% 2.58/1.59 ( ( product_case_bool @ A @ F1 @ F2 @ $true )
% 2.58/1.59 = F1 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_62_old_Obool_Osimps_I4_J,axiom,
% 2.58/1.59 ! [A: $tType,F1: A,F2: A] :
% 2.58/1.59 ( ( product_case_bool @ A @ F1 @ F2 @ $false )
% 2.58/1.59 = F2 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_63_surj__pair,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.58/1.59 ? [X: A,Y4: B] :
% 2.58/1.59 ( P2
% 2.58/1.59 = ( product_Pair @ A @ B @ X @ Y4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_64_prod__cases,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.58/1.59 ( ! [A5: A,B4: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 2.58/1.59 => ( P3 @ P2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_65_Pair__inject,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
% 2.58/1.59 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.58/1.59 = ( product_Pair @ A @ B @ A4 @ B3 ) )
% 2.58/1.59 => ~ ( ( A2 = A4 )
% 2.58/1.59 => ( B2 != B3 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_66_prod__cases3,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.58/1.59 ~ ! [A5: A,B4: B,C2: C] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_67_prod__cases4,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.58/1.59 ~ ! [A5: A,B4: B,C2: C,D2: D] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_68_prod__cases5,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.58/1.59 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_69_prod__cases6,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F7: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) )] :
% 2.58/1.59 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F8: F7] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F7 ) @ D2 @ ( product_Pair @ E @ F7 @ E2 @ F8 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_70_prod__cases7,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F7: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) )] :
% 2.58/1.59 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F8: F7,G3: G2] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F7 @ G2 ) @ E2 @ ( product_Pair @ F7 @ G2 @ F8 @ G3 ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_71_prod__induct3,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.58/1.59 ( ! [A5: A,B4: B,C2: C] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 2.58/1.59 => ( P3 @ X4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_72_prod__induct4,axiom,
% 2.58/1.59 ! [D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.58/1.59 ( ! [A5: A,B4: B,C2: C,D2: D] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 2.58/1.59 => ( P3 @ X4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_73_prod__induct5,axiom,
% 2.58/1.59 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.58/1.59 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 2.58/1.59 => ( P3 @ X4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_74_prod__induct6,axiom,
% 2.58/1.59 ! [F7: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) )] :
% 2.58/1.59 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F8: F7] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F7 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F7 ) @ D2 @ ( product_Pair @ E @ F7 @ E2 @ F8 ) ) ) ) ) )
% 2.58/1.59 => ( P3 @ X4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_75_prod__induct7,axiom,
% 2.58/1.59 ! [G2: $tType,F7: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) )] :
% 2.58/1.59 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F8: F7,G3: G2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F7 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F7 @ G2 ) @ E2 @ ( product_Pair @ F7 @ G2 @ F8 @ G3 ) ) ) ) ) ) )
% 2.58/1.59 => ( P3 @ X4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_76_old_Oprod_Oexhaust,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.58/1.59 ~ ! [A5: A,B4: B] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_77_old_Oprod_Oinducts,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.58/1.59 ( ! [A5: A,B4: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 2.58/1.59 => ( P3 @ Prod ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_78_old_Oprod_Orec,axiom,
% 2.58/1.59 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.58/1.59 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.58/1.59 = ( F1 @ A2 @ B2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_79_internal__case__prod__conv,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 2.58/1.59 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 2.58/1.59 = ( C3 @ A2 @ B2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_80_ssubst__Pair__rhs,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S3: B] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
% 2.58/1.59 => ( ( S3 = S )
% 2.58/1.59 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_81_curry__conv,axiom,
% 2.58/1.59 ! [A: $tType,C: $tType,B: $tType] :
% 2.58/1.59 ( ( product_curry @ B @ C @ A )
% 2.58/1.59 = ( ^ [F9: ( product_prod @ B @ C ) > A,A6: B,B5: C] : ( F9 @ ( product_Pair @ B @ C @ A6 @ B5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_82_curryI,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.58/1.59 ( ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.58/1.59 => ( product_curry @ A @ B @ $o @ F @ A2 @ B2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_83_swap__simp,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,X4: B,Y: A] :
% 2.58/1.59 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y ) )
% 2.58/1.59 = ( product_Pair @ A @ B @ Y @ X4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_84_eval_Osimps_I6_J,axiom,
% 2.58/1.59 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.58/1.59 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 = ( paraco876059933e_eval @ I2 @ Q ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.58/1.59 = ( paraco2040174112le_Det @ $true ) ) )
% 2.58/1.59 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.58/1.59 != ( paraco876059933e_eval @ I2 @ Q ) )
% 2.58/1.59 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.58/1.59 = ( product_case_prod @ paraco415392788lle_tv @ paraco415392788lle_tv @ paraco415392788lle_tv
% 2.58/1.59 @ ^ [X5: paraco415392788lle_tv,Y5: paraco415392788lle_tv] :
% 2.58/1.59 ( paraco490622181ase_tv @ paraco415392788lle_tv
% 2.58/1.59 @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco876059933e_eval @ I2 @ Q )
% 2.58/1.59 @ ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco876059933e_eval @ I2 @ P2 ) @ ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ Q ) ) )
% 2.58/1.59 @ ^ [Nat: nat] : ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ Q ) )
% 2.58/1.59 @ Y5 ) )
% 2.58/1.59 @ ^ [Nat: nat] :
% 2.58/1.59 ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco876059933e_eval @ I2 @ P2 ) @ ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 2.58/1.59 @ ^ [Nata: nat] : ( paraco2040174112le_Det @ $false )
% 2.58/1.59 @ Y5 )
% 2.58/1.59 @ X5 )
% 2.58/1.59 @ ( product_Pair @ paraco415392788lle_tv @ paraco415392788lle_tv @ ( paraco876059933e_eval @ I2 @ P2 ) @ ( paraco876059933e_eval @ I2 @ Q ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_85_swap__swap,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 2.58/1.59 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 2.58/1.59 = P2 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_86_case__prod__conv,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
% 2.58/1.59 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 2.58/1.59 = ( F @ A2 @ B2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_87_case__prod__curry,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 2.58/1.59 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 2.58/1.59 = F ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_88_curry__case__prod,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 2.58/1.59 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
% 2.58/1.59 = F ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_89_case__swap,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,C: $tType,F: C > B > A,P2: product_prod @ C @ B] :
% 2.58/1.59 ( ( product_case_prod @ B @ C @ A
% 2.58/1.59 @ ^ [Y5: B,X5: C] : ( F @ X5 @ Y5 )
% 2.58/1.59 @ ( product_swap @ C @ B @ P2 ) )
% 2.58/1.59 = ( product_case_prod @ C @ B @ A @ F @ P2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_90_cond__case__prod__eta,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
% 2.58/1.59 ( ! [X: A,Y4: B] :
% 2.58/1.59 ( ( F @ X @ Y4 )
% 2.58/1.59 = ( G @ ( product_Pair @ A @ B @ X @ Y4 ) ) )
% 2.58/1.59 => ( ( product_case_prod @ A @ B @ C @ F )
% 2.58/1.59 = G ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_91_case__prod__eta,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 2.58/1.59 ( ( product_case_prod @ A @ B @ C
% 2.58/1.59 @ ^ [X5: A,Y5: B] : ( F @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) )
% 2.58/1.59 = F ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_92_case__prodE2,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,C: $tType,Q3: A > $o,P3: B > C > A,Z: product_prod @ B @ C] :
% 2.58/1.59 ( ( Q3 @ ( product_case_prod @ B @ C @ A @ P3 @ Z ) )
% 2.58/1.59 => ~ ! [X: B,Y4: C] :
% 2.58/1.59 ( ( Z
% 2.58/1.59 = ( product_Pair @ B @ C @ X @ Y4 ) )
% 2.58/1.59 => ~ ( Q3 @ ( P3 @ X @ Y4 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_93_curry__def,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.58/1.59 ( ( product_curry @ A @ B @ C )
% 2.58/1.59 = ( ^ [C4: ( product_prod @ A @ B ) > C,X5: A,Y5: B] : ( C4 @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_94_curry__K,axiom,
% 2.58/1.59 ! [B: $tType,C: $tType,A: $tType,C3: C] :
% 2.58/1.59 ( ( product_curry @ A @ B @ C
% 2.58/1.59 @ ^ [X5: product_prod @ A @ B] : C3 )
% 2.58/1.59 = ( ^ [X5: A,Y5: B] : C3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_95_prod_Ocase__distrib,axiom,
% 2.58/1.59 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
% 2.58/1.59 ( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 2.58/1.59 = ( product_case_prod @ A @ B @ D
% 2.58/1.59 @ ^ [X13: A,X23: B] : ( H @ ( F @ X13 @ X23 ) )
% 2.58/1.59 @ Prod ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_96_internal__case__prod__def,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.58/1.59 ( ( produc2004651681e_prod @ A @ B @ C )
% 2.58/1.59 = ( product_case_prod @ A @ B @ C ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_97_tv_Ocase__distrib,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,H: A > B,F1: $o > A,F2: nat > A,Tv: paraco415392788lle_tv] :
% 2.58/1.59 ( ( H @ ( paraco490622181ase_tv @ A @ F1 @ F2 @ Tv ) )
% 2.58/1.59 = ( paraco490622181ase_tv @ B
% 2.58/1.59 @ ^ [X5: $o] : ( H @ ( F1 @ X5 ) )
% 2.58/1.59 @ ^ [X5: nat] : ( H @ ( F2 @ X5 ) )
% 2.58/1.59 @ Tv ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_98_fm_Ocase__distrib,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,H: A > B,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,Fm: paraco414474393lle_fm] :
% 2.58/1.59 ( ( H @ ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ Fm ) )
% 2.58/1.59 = ( paraco1246693743ase_fm @ B
% 2.58/1.59 @ ^ [X5: list @ char] : ( H @ ( F1 @ X5 ) )
% 2.58/1.59 @ ( H @ F2 )
% 2.58/1.59 @ ^ [X5: paraco414474393lle_fm] : ( H @ ( F3 @ X5 ) )
% 2.58/1.59 @ ^ [X13: paraco414474393lle_fm,X23: paraco414474393lle_fm] : ( H @ ( F4 @ X13 @ X23 ) )
% 2.58/1.59 @ ^ [X13: paraco414474393lle_fm,X23: paraco414474393lle_fm] : ( H @ ( F5 @ X13 @ X23 ) )
% 2.58/1.59 @ ^ [X13: paraco414474393lle_fm,X23: paraco414474393lle_fm] : ( H @ ( F6 @ X13 @ X23 ) )
% 2.58/1.59 @ Fm ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_99_old_Oprod_Ocase,axiom,
% 2.58/1.59 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X2: B] :
% 2.58/1.59 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 2.58/1.59 = ( F @ X1 @ X2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_100_curryE,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.58/1.59 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 2.58/1.59 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_101_curryD,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.58/1.59 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 2.58/1.59 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_102_split__cong,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType,Q: product_prod @ A @ B,F: A > B > C,G: A > B > C,P2: product_prod @ A @ B] :
% 2.58/1.59 ( ! [X: A,Y4: B] :
% 2.58/1.59 ( ( ( product_Pair @ A @ B @ X @ Y4 )
% 2.58/1.59 = Q )
% 2.58/1.59 => ( ( F @ X @ Y4 )
% 2.58/1.59 = ( G @ X @ Y4 ) ) )
% 2.58/1.59 => ( ( P2 = Q )
% 2.58/1.59 => ( ( product_case_prod @ A @ B @ C @ F @ P2 )
% 2.58/1.59 = ( product_case_prod @ A @ B @ C @ G @ Q ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_103_case__bool__if,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( product_case_bool @ A )
% 2.58/1.59 = ( ^ [X5: A,Y5: A,P5: $o] : ( if @ A @ P5 @ X5 @ Y5 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_104_case__prod__app,axiom,
% 2.58/1.59 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 2.58/1.59 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 2.58/1.59 = ( ^ [F9: B > C > D > A,X5: product_prod @ B @ C,Y5: D] :
% 2.58/1.59 ( product_case_prod @ B @ C @ A
% 2.58/1.59 @ ^ [L: B,R3: C] : ( F9 @ L @ R3 @ Y5 )
% 2.58/1.59 @ X5 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_105_pred__equals__eq2,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
% 2.58/1.59 ( ( ( ^ [X5: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) @ R2 ) )
% 2.58/1.59 = ( ^ [X5: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) @ S4 ) ) )
% 2.58/1.59 = ( R2 = S4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_106_pair__imageI,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set @ ( product_prod @ A @ B ),F: A > B > C] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A3 )
% 2.58/1.59 => ( member @ C @ ( F @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A3 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_107_scomp__apply,axiom,
% 2.58/1.59 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 2.58/1.59 ( ( product_scomp @ B @ C @ D @ A )
% 2.58/1.59 = ( ^ [F9: B > ( product_prod @ C @ D ),G4: C > D > A,X5: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F9 @ X5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_108_case__prodI,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
% 2.58/1.59 ( ( F @ A2 @ B2 )
% 2.58/1.59 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_109_case__prodI2,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,C3: A > B > $o] :
% 2.58/1.59 ( ! [A5: A,B4: B] :
% 2.58/1.59 ( ( P2
% 2.58/1.59 = ( product_Pair @ A @ B @ A5 @ B4 ) )
% 2.58/1.59 => ( C3 @ A5 @ B4 ) )
% 2.58/1.59 => ( product_case_prod @ A @ B @ $o @ C3 @ P2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_110_mem__case__prodI,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),A2: B,B2: C] :
% 2.58/1.59 ( ( member @ A @ Z @ ( C3 @ A2 @ B2 ) )
% 2.58/1.59 => ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_111_mem__case__prodI2,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z: C,C3: A > B > ( set @ C )] :
% 2.58/1.59 ( ! [A5: A,B4: B] :
% 2.58/1.59 ( ( P2
% 2.58/1.59 = ( product_Pair @ A @ B @ A5 @ B4 ) )
% 2.58/1.59 => ( member @ C @ Z @ ( C3 @ A5 @ B4 ) ) )
% 2.58/1.59 => ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_112_case__prodI2_H,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C3: A > B > C > $o,X4: C] :
% 2.58/1.59 ( ! [A5: A,B4: B] :
% 2.58/1.59 ( ( ( product_Pair @ A @ B @ A5 @ B4 )
% 2.58/1.59 = P2 )
% 2.58/1.59 => ( C3 @ A5 @ B4 @ X4 ) )
% 2.58/1.59 => ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ X4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_113_pair__in__swap__image,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,Y: A,X4: B,A3: set @ ( product_prod @ B @ A )] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X4 ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A3 ) )
% 2.58/1.59 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y ) @ A3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_114_case__prod__Pair__iden,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 2.58/1.59 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
% 2.58/1.59 = P2 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_115_scomp__scomp,axiom,
% 2.58/1.59 ! [A: $tType,C: $tType,D: $tType,B: $tType,F7: $tType,E: $tType,F: A > ( product_prod @ E @ F7 ),G: E > F7 > ( product_prod @ C @ D ),H: C > D > B] :
% 2.58/1.59 ( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F7 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
% 2.58/1.59 = ( product_scomp @ A @ E @ F7 @ B @ F
% 2.58/1.59 @ ^ [X5: E] : ( product_scomp @ F7 @ C @ D @ B @ ( G @ X5 ) @ H ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_116_mem__case__prodE,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 2.58/1.59 ( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
% 2.58/1.59 => ~ ! [X: B,Y4: C] :
% 2.58/1.59 ( ( P2
% 2.58/1.59 = ( product_Pair @ B @ C @ X @ Y4 ) )
% 2.58/1.59 => ~ ( member @ A @ Z @ ( C3 @ X @ Y4 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_117_case__prodD,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
% 2.58/1.59 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.58/1.59 => ( F @ A2 @ B2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_118_case__prodE,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,C3: A > B > $o,P2: product_prod @ A @ B] :
% 2.58/1.59 ( ( product_case_prod @ A @ B @ $o @ C3 @ P2 )
% 2.58/1.59 => ~ ! [X: A,Y4: B] :
% 2.58/1.59 ( ( P2
% 2.58/1.59 = ( product_Pair @ A @ B @ X @ Y4 ) )
% 2.58/1.59 => ~ ( C3 @ X @ Y4 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_119_case__prodD_H,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,C: $tType,R2: A > B > C > $o,A2: A,B2: B,C3: C] :
% 2.58/1.59 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R2 @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C3 )
% 2.58/1.59 => ( R2 @ A2 @ B2 @ C3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_120_case__prodE_H,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P2: product_prod @ A @ B,Z: C] :
% 2.58/1.59 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ Z )
% 2.58/1.59 => ~ ! [X: A,Y4: B] :
% 2.58/1.59 ( ( P2
% 2.58/1.59 = ( product_Pair @ A @ B @ X @ Y4 ) )
% 2.58/1.59 => ~ ( C3 @ X @ Y4 @ Z ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_121_scomp__Pair,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType,X4: A > ( product_prod @ B @ C )] :
% 2.58/1.59 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C ) )
% 2.58/1.59 = X4 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_122_Pair__scomp,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,C: $tType,X4: C,F: C > A > B] :
% 2.58/1.59 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X4 ) @ F )
% 2.58/1.59 = ( F @ X4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_123_scomp__def,axiom,
% 2.58/1.59 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 2.58/1.59 ( ( product_scomp @ A @ B @ C @ D )
% 2.58/1.59 = ( ^ [F9: A > ( product_prod @ B @ C ),G4: B > C > D,X5: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F9 @ X5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_124_image__ident,axiom,
% 2.58/1.59 ! [A: $tType,Y6: set @ A] :
% 2.58/1.59 ( ( image @ A @ A
% 2.58/1.59 @ ^ [X5: A] : X5
% 2.58/1.59 @ Y6 )
% 2.58/1.59 = Y6 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_125_image__eqI,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,B2: A,F: B > A,X4: B,A3: set @ B] :
% 2.58/1.59 ( ( B2
% 2.58/1.59 = ( F @ X4 ) )
% 2.58/1.59 => ( ( member @ B @ X4 @ A3 )
% 2.58/1.59 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_126_Sup_OSUP__identity__eq,axiom,
% 2.58/1.59 ! [A: $tType,Sup: ( set @ A ) > A,A3: set @ A] :
% 2.58/1.59 ( ( Sup
% 2.58/1.59 @ ( image @ A @ A
% 2.58/1.59 @ ^ [X5: A] : X5
% 2.58/1.59 @ A3 ) )
% 2.58/1.59 = ( Sup @ A3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_127_Inf_OINF__identity__eq,axiom,
% 2.58/1.59 ! [A: $tType,Inf: ( set @ A ) > A,A3: set @ A] :
% 2.58/1.59 ( ( Inf
% 2.58/1.59 @ ( image @ A @ A
% 2.58/1.59 @ ^ [X5: A] : X5
% 2.58/1.59 @ A3 ) )
% 2.58/1.59 = ( Inf @ A3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_128_Compr__image__eq,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
% 2.58/1.59 ( ( collect @ A
% 2.58/1.59 @ ^ [X5: A] :
% 2.58/1.59 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A3 ) )
% 2.58/1.59 & ( P3 @ X5 ) ) )
% 2.58/1.59 = ( image @ B @ A @ F
% 2.58/1.59 @ ( collect @ B
% 2.58/1.59 @ ^ [X5: B] :
% 2.58/1.59 ( ( member @ B @ X5 @ A3 )
% 2.58/1.59 & ( P3 @ ( F @ X5 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_129_split__part,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P3: $o,Q3: A > B > $o] :
% 2.58/1.59 ( ( product_case_prod @ A @ B @ $o
% 2.58/1.59 @ ^ [A6: A,B5: B] :
% 2.58/1.59 ( P3
% 2.58/1.59 & ( Q3 @ A6 @ B5 ) ) )
% 2.58/1.59 = ( ^ [Ab: product_prod @ A @ B] :
% 2.58/1.59 ( P3
% 2.58/1.59 & ( product_case_prod @ A @ B @ $o @ Q3 @ Ab ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_130_prod_Odisc__eq__case,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.58/1.59 ( product_case_prod @ A @ B @ $o
% 2.58/1.59 @ ^ [Uu: A,Uv: B] : $true
% 2.58/1.59 @ Prod ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_131_Sup_OSUP__cong,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,A3: set @ B,B6: set @ B,C5: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 2.58/1.59 ( ( A3 = B6 )
% 2.58/1.59 => ( ! [X: B] :
% 2.58/1.59 ( ( member @ B @ X @ B6 )
% 2.58/1.59 => ( ( C5 @ X )
% 2.58/1.59 = ( D3 @ X ) ) )
% 2.58/1.59 => ( ( Sup @ ( image @ B @ A @ C5 @ A3 ) )
% 2.58/1.59 = ( Sup @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_132_Inf_OINF__cong,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,A3: set @ B,B6: set @ B,C5: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 2.58/1.59 ( ( A3 = B6 )
% 2.58/1.59 => ( ! [X: B] :
% 2.58/1.59 ( ( member @ B @ X @ B6 )
% 2.58/1.59 => ( ( C5 @ X )
% 2.58/1.59 = ( D3 @ X ) ) )
% 2.58/1.59 => ( ( Inf @ ( image @ B @ A @ C5 @ A3 ) )
% 2.58/1.59 = ( Inf @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_133_rev__image__eqI,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,X4: A,A3: set @ A,B2: B,F: A > B] :
% 2.58/1.59 ( ( member @ A @ X4 @ A3 )
% 2.58/1.59 => ( ( B2
% 2.58/1.59 = ( F @ X4 ) )
% 2.58/1.59 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_134_ball__imageD,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
% 2.58/1.59 ( ! [X: A] :
% 2.58/1.59 ( ( member @ A @ X @ ( image @ B @ A @ F @ A3 ) )
% 2.58/1.59 => ( P3 @ X ) )
% 2.58/1.59 => ! [X6: B] :
% 2.58/1.59 ( ( member @ B @ X6 @ A3 )
% 2.58/1.59 => ( P3 @ ( F @ X6 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_135_image__cong,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F: A > B,G: A > B] :
% 2.58/1.59 ( ( M = N2 )
% 2.58/1.59 => ( ! [X: A] :
% 2.58/1.59 ( ( member @ A @ X @ N2 )
% 2.58/1.59 => ( ( F @ X )
% 2.58/1.59 = ( G @ X ) ) )
% 2.58/1.59 => ( ( image @ A @ B @ F @ M )
% 2.58/1.59 = ( image @ A @ B @ G @ N2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_136_bex__imageD,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
% 2.58/1.59 ( ? [X6: A] :
% 2.58/1.59 ( ( member @ A @ X6 @ ( image @ B @ A @ F @ A3 ) )
% 2.58/1.59 & ( P3 @ X6 ) )
% 2.58/1.59 => ? [X: B] :
% 2.58/1.59 ( ( member @ B @ X @ A3 )
% 2.58/1.59 & ( P3 @ ( F @ X ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_137_image__iff,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,Z: A,F: B > A,A3: set @ B] :
% 2.58/1.59 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A3 ) )
% 2.58/1.59 = ( ? [X5: B] :
% 2.58/1.59 ( ( member @ B @ X5 @ A3 )
% 2.58/1.59 & ( Z
% 2.58/1.59 = ( F @ X5 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_138_imageI,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,X4: A,A3: set @ A,F: A > B] :
% 2.58/1.59 ( ( member @ A @ X4 @ A3 )
% 2.58/1.59 => ( member @ B @ ( F @ X4 ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_139_imageE,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,B2: A,F: B > A,A3: set @ B] :
% 2.58/1.59 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) )
% 2.58/1.59 => ~ ! [X: B] :
% 2.58/1.59 ( ( B2
% 2.58/1.59 = ( F @ X ) )
% 2.58/1.59 => ~ ( member @ B @ X @ A3 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_140_image__image,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A3: set @ C] :
% 2.58/1.59 ( ( image @ B @ A @ F @ ( image @ C @ B @ G @ A3 ) )
% 2.58/1.59 = ( image @ C @ A
% 2.58/1.59 @ ^ [X5: C] : ( F @ ( G @ X5 ) )
% 2.58/1.59 @ A3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_141_image__paired__Times,axiom,
% 2.58/1.59 ! [C: $tType,D: $tType,B: $tType,A: $tType,F: C > A,G: D > B,A3: set @ C,B6: set @ D] :
% 2.58/1.59 ( ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
% 2.58/1.59 @ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
% 2.58/1.59 @ ^ [X5: C,Y5: D] : ( product_Pair @ A @ B @ ( F @ X5 ) @ ( G @ Y5 ) ) )
% 2.58/1.59 @ ( product_Sigma @ C @ D @ A3
% 2.58/1.59 @ ^ [Uu: C] : B6 ) )
% 2.58/1.59 = ( product_Sigma @ A @ B @ ( image @ C @ A @ F @ A3 )
% 2.58/1.59 @ ^ [Uu: A] : ( image @ D @ B @ G @ B6 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_142_inv__image__def,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType] :
% 2.58/1.59 ( ( inv_image @ B @ A )
% 2.58/1.59 = ( ^ [R3: set @ ( product_prod @ B @ B ),F9: A > B] :
% 2.58/1.59 ( collect @ ( product_prod @ A @ A )
% 2.58/1.59 @ ( product_case_prod @ A @ A @ $o
% 2.58/1.59 @ ^ [X5: A,Y5: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F9 @ X5 ) @ ( F9 @ Y5 ) ) @ R3 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_143_same__fst__def,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( same_fst @ A @ B )
% 2.58/1.59 = ( ^ [P5: A > $o,R4: A > ( set @ ( product_prod @ B @ B ) )] :
% 2.58/1.59 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 2.58/1.59 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 2.58/1.59 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 2.58/1.59 @ ^ [X7: A,Y7: B] :
% 2.58/1.59 ( product_case_prod @ A @ B @ $o
% 2.58/1.59 @ ^ [X5: A,Y5: B] :
% 2.58/1.59 ( ( X7 = X5 )
% 2.58/1.59 & ( P5 @ X5 )
% 2.58/1.59 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y5 ) @ ( R4 @ X5 ) ) ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_144_swap__product,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,A3: set @ B,B6: set @ A] :
% 2.58/1.59 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
% 2.58/1.59 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 2.58/1.59 @ ^ [I: B,J: A] : ( product_Pair @ A @ B @ J @ I ) )
% 2.58/1.59 @ ( product_Sigma @ B @ A @ A3
% 2.58/1.59 @ ^ [Uu: B] : B6 ) )
% 2.58/1.59 = ( product_Sigma @ A @ B @ B6
% 2.58/1.59 @ ^ [Uu: A] : A3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_145_SigmaI,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,B6: A > ( set @ B )] :
% 2.58/1.59 ( ( member @ A @ A2 @ A3 )
% 2.58/1.59 => ( ( member @ B @ B2 @ ( B6 @ A2 ) )
% 2.58/1.59 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_146_mem__Sigma__iff,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B6: A > ( set @ B )] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
% 2.58/1.59 = ( ( member @ A @ A2 @ A3 )
% 2.58/1.59 & ( member @ B @ B2 @ ( B6 @ A2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_147_in__inv__image,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,X4: A,Y: A,R: set @ ( product_prod @ B @ B ),F: A > B] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y ) @ ( inv_image @ B @ A @ R @ F ) )
% 2.58/1.59 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X4 ) @ ( F @ Y ) ) @ R ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_148_Collect__case__prod,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P3: A > $o,Q3: B > $o] :
% 2.58/1.59 ( ( collect @ ( product_prod @ A @ B )
% 2.58/1.59 @ ( product_case_prod @ A @ B @ $o
% 2.58/1.59 @ ^ [A6: A,B5: B] :
% 2.58/1.59 ( ( P3 @ A6 )
% 2.58/1.59 & ( Q3 @ B5 ) ) ) )
% 2.58/1.59 = ( product_Sigma @ A @ B @ ( collect @ A @ P3 )
% 2.58/1.59 @ ^ [Uu: A] : ( collect @ B @ Q3 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_149_same__fstI,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P3: A > $o,X4: A,Y8: B,Y: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 2.58/1.59 ( ( P3 @ X4 )
% 2.58/1.59 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y ) @ ( R2 @ X4 ) )
% 2.58/1.59 => ( 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 @ X4 @ Y8 ) @ ( product_Pair @ A @ B @ X4 @ Y ) ) @ ( same_fst @ A @ B @ P3 @ R2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_150_Collect__case__prod__Sigma,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P3: A > $o,Q3: A > B > $o] :
% 2.58/1.59 ( ( collect @ ( product_prod @ A @ B )
% 2.58/1.59 @ ( product_case_prod @ A @ B @ $o
% 2.58/1.59 @ ^ [X5: A,Y5: B] :
% 2.58/1.59 ( ( P3 @ X5 )
% 2.58/1.59 & ( Q3 @ X5 @ Y5 ) ) ) )
% 2.58/1.59 = ( product_Sigma @ A @ B @ ( collect @ A @ P3 )
% 2.58/1.59 @ ^ [X5: A] : ( collect @ B @ ( Q3 @ X5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_151_Sigma__cong,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,A3: set @ A,B6: set @ A,C5: A > ( set @ B ),D3: A > ( set @ B )] :
% 2.58/1.59 ( ( A3 = B6 )
% 2.58/1.59 => ( ! [X: A] :
% 2.58/1.59 ( ( member @ A @ X @ B6 )
% 2.58/1.59 => ( ( C5 @ X )
% 2.58/1.59 = ( D3 @ X ) ) )
% 2.58/1.59 => ( ( product_Sigma @ A @ B @ A3 @ C5 )
% 2.58/1.59 = ( product_Sigma @ A @ B @ B6 @ D3 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_152_Times__eq__cancel2,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,X4: A,C5: set @ A,A3: set @ B,B6: set @ B] :
% 2.58/1.59 ( ( member @ A @ X4 @ C5 )
% 2.58/1.59 => ( ( ( product_Sigma @ B @ A @ A3
% 2.58/1.59 @ ^ [Uu: B] : C5 )
% 2.58/1.59 = ( product_Sigma @ B @ A @ B6
% 2.58/1.59 @ ^ [Uu: B] : C5 ) )
% 2.58/1.59 = ( A3 = B6 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_153_SigmaE,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,C3: product_prod @ A @ B,A3: set @ A,B6: A > ( set @ B )] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
% 2.58/1.59 => ~ ! [X: A] :
% 2.58/1.59 ( ( member @ A @ X @ A3 )
% 2.58/1.59 => ! [Y4: B] :
% 2.58/1.59 ( ( member @ B @ Y4 @ ( B6 @ X ) )
% 2.58/1.59 => ( C3
% 2.58/1.59 != ( product_Pair @ A @ B @ X @ Y4 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_154_SigmaD1,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B6: A > ( set @ B )] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
% 2.58/1.59 => ( member @ A @ A2 @ A3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_155_SigmaD2,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B6: A > ( set @ B )] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
% 2.58/1.59 => ( member @ B @ B2 @ ( B6 @ A2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_156_SigmaE2,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B6: A > ( set @ B )] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
% 2.58/1.59 => ~ ( ( member @ A @ A2 @ A3 )
% 2.58/1.59 => ~ ( member @ B @ B2 @ ( B6 @ A2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_157_product__swap,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,A3: set @ B,B6: set @ A] :
% 2.58/1.59 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
% 2.58/1.59 @ ( product_Sigma @ B @ A @ A3
% 2.58/1.59 @ ^ [Uu: B] : B6 ) )
% 2.58/1.59 = ( product_Sigma @ A @ B @ B6
% 2.58/1.59 @ ^ [Uu: A] : A3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_158_rp__inv__image__def,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( fun_rp_inv_image @ A @ B )
% 2.58/1.59 = ( 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 ) ) ) )
% 2.58/1.59 @ ^ [R4: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A ),F9: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R4 @ F9 ) @ ( inv_image @ A @ B @ S5 @ F9 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_159_lex__prod__def,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( lex_prod @ A @ B )
% 2.58/1.59 = ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
% 2.58/1.59 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 2.58/1.59 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 2.58/1.59 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 2.58/1.59 @ ^ [A6: A,B5: B] :
% 2.58/1.59 ( product_case_prod @ A @ B @ $o
% 2.58/1.59 @ ^ [A7: A,B7: B] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A7 ) @ Ra )
% 2.58/1.59 | ( ( A6 = A7 )
% 2.58/1.59 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B5 @ B7 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_160_Product__Type_Oproduct__def,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( product_product @ A @ B )
% 2.58/1.59 = ( ^ [A8: set @ A,B8: set @ B] :
% 2.58/1.59 ( product_Sigma @ A @ B @ A8
% 2.58/1.59 @ ^ [Uu: A] : B8 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_161_in__lex__prod,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 2.58/1.59 ( ( 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 @ A2 @ B2 ) @ ( product_Pair @ A @ B @ A4 @ B3 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
% 2.58/1.59 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A4 ) @ R )
% 2.58/1.59 | ( ( A2 = A4 )
% 2.58/1.59 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_162_member__product,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,X4: product_prod @ A @ B,A3: set @ A,B6: set @ B] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ B ) @ X4 @ ( product_product @ A @ B @ A3 @ B6 ) )
% 2.58/1.59 = ( member @ ( product_prod @ A @ B ) @ X4
% 2.58/1.59 @ ( product_Sigma @ A @ B @ A3
% 2.58/1.59 @ ^ [Uu: A] : B6 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_163_rp__inv__image__rp,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,P3: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
% 2.58/1.59 ( ( fun_reduction_pair @ A @ P3 )
% 2.58/1.59 => ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P3 @ F ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_164_map__prod__surj__on,axiom,
% 2.58/1.59 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A3: set @ B,A9: set @ A,G: D > C,B6: set @ D,B9: set @ C] :
% 2.58/1.59 ( ( ( image @ B @ A @ F @ A3 )
% 2.58/1.59 = A9 )
% 2.58/1.59 => ( ( ( image @ D @ C @ G @ B6 )
% 2.58/1.59 = B9 )
% 2.58/1.59 => ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G )
% 2.58/1.59 @ ( product_Sigma @ B @ D @ A3
% 2.58/1.59 @ ^ [Uu: B] : B6 ) )
% 2.58/1.59 = ( product_Sigma @ A @ C @ A9
% 2.58/1.59 @ ^ [Uu: A] : B9 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_165_relation__of__def,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( order_relation_of @ A )
% 2.58/1.59 = ( ^ [P5: A > A > $o,A8: set @ A] :
% 2.58/1.59 ( collect @ ( product_prod @ A @ A )
% 2.58/1.59 @ ( product_case_prod @ A @ A @ $o
% 2.58/1.59 @ ^ [A6: A,B5: A] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 )
% 2.58/1.59 @ ( product_Sigma @ A @ A @ A8
% 2.58/1.59 @ ^ [Uu: A] : A8 ) )
% 2.58/1.59 & ( P5 @ A6 @ B5 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_166_map__prod__ident,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( product_map_prod @ A @ A @ B @ B
% 2.58/1.59 @ ^ [X5: A] : X5
% 2.58/1.59 @ ^ [Y5: B] : Y5 )
% 2.58/1.59 = ( ^ [Z2: product_prod @ A @ B] : Z2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_167_map__prod__simp,axiom,
% 2.58/1.59 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A2: C,B2: D] :
% 2.58/1.59 ( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 2.58/1.59 = ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_168_map__prod__imageI,axiom,
% 2.58/1.59 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),F: A > C,G: B > D] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 2.58/1.59 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A2 ) @ ( G @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) @ R2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_169_case__prod__map__prod,axiom,
% 2.58/1.59 ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G: E > C,X4: product_prod @ D @ E] :
% 2.58/1.59 ( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G @ X4 ) )
% 2.58/1.59 = ( product_case_prod @ D @ E @ A
% 2.58/1.59 @ ^ [L: D,R3: E] : ( H @ ( F @ L ) @ ( G @ R3 ) )
% 2.58/1.59 @ X4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_170_prod__fun__imageE,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F: C > A,G: D > B,R2: set @ ( product_prod @ C @ D )] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G ) @ R2 ) )
% 2.58/1.59 => ~ ! [X: C,Y4: D] :
% 2.58/1.59 ( ( C3
% 2.58/1.59 = ( product_Pair @ A @ B @ ( F @ X ) @ ( G @ Y4 ) ) )
% 2.58/1.59 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X @ Y4 ) @ R2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_171_map__prod__def,axiom,
% 2.58/1.59 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 2.58/1.59 ( ( product_map_prod @ A @ C @ B @ D )
% 2.58/1.59 = ( ^ [F9: A > C,G4: B > D] :
% 2.58/1.59 ( product_case_prod @ A @ B @ ( product_prod @ C @ D )
% 2.58/1.59 @ ^ [X5: A,Y5: B] : ( product_Pair @ C @ D @ ( F9 @ X5 ) @ ( G4 @ Y5 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_172_prod_Omap__ident,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 2.58/1.59 ( ( product_map_prod @ A @ A @ B @ B
% 2.58/1.59 @ ^ [X5: A] : X5
% 2.58/1.59 @ ^ [X5: B] : X5
% 2.58/1.59 @ T2 )
% 2.58/1.59 = T2 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_173_aboveS__def,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( order_aboveS @ A )
% 2.58/1.59 = ( ^ [R3: set @ ( product_prod @ A @ A ),A6: A] :
% 2.58/1.59 ( collect @ A
% 2.58/1.59 @ ^ [B5: A] :
% 2.58/1.59 ( ( B5 != A6 )
% 2.58/1.59 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R3 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_174_above__def,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( order_above @ A )
% 2.58/1.59 = ( ^ [R3: set @ ( product_prod @ A @ A ),A6: A] :
% 2.58/1.59 ( collect @ A
% 2.58/1.59 @ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R3 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_175_surj__swap,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.58/1.59 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_176_The__split__eq,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,X4: A,Y: B] :
% 2.58/1.59 ( ( the @ ( product_prod @ A @ B )
% 2.58/1.59 @ ( product_case_prod @ A @ B @ $o
% 2.58/1.59 @ ^ [X7: A,Y7: B] :
% 2.58/1.59 ( ( X4 = X7 )
% 2.58/1.59 & ( Y = Y7 ) ) ) )
% 2.58/1.59 = ( product_Pair @ A @ B @ X4 @ Y ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_177_in__inv__imagep,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( inv_imagep @ A @ B )
% 2.58/1.59 = ( ^ [R3: A > A > $o,F9: B > A,X5: B,Y5: B] : ( R3 @ ( F9 @ X5 ) @ ( F9 @ Y5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_178_UNIV__Times__UNIV,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 2.58/1.59 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
% 2.58/1.59 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_179_rangeE,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,B2: A,F: B > A] :
% 2.58/1.59 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
% 2.58/1.59 => ~ ! [X: B] :
% 2.58/1.59 ( B2
% 2.58/1.59 != ( F @ X ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_180_range__composition,axiom,
% 2.58/1.59 ! [A: $tType,C: $tType,B: $tType,F: C > A,G: B > C] :
% 2.58/1.59 ( ( image @ B @ A
% 2.58/1.59 @ ^ [X5: B] : ( F @ ( G @ X5 ) )
% 2.58/1.59 @ ( top_top @ ( set @ B ) ) )
% 2.58/1.59 = ( image @ C @ A @ F @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_181_UNIV__def,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( top_top @ ( set @ A ) )
% 2.58/1.59 = ( collect @ A
% 2.58/1.59 @ ^ [X5: A] : $true ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_182_rangeI,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,F: B > A,X4: B] : ( member @ A @ ( F @ X4 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_183_range__eqI,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,B2: A,F: B > A,X4: B] :
% 2.58/1.59 ( ( B2
% 2.58/1.59 = ( F @ X4 ) )
% 2.58/1.59 => ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_184_map__prod__surj,axiom,
% 2.58/1.59 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
% 2.58/1.59 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.58/1.59 = ( top_top @ ( set @ B ) ) )
% 2.58/1.59 => ( ( ( image @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
% 2.58/1.59 = ( top_top @ ( set @ D ) ) )
% 2.58/1.59 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 2.58/1.59 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_185_inv__imagep__def,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType] :
% 2.58/1.59 ( ( inv_imagep @ B @ A )
% 2.58/1.59 = ( ^ [R3: B > B > $o,F9: A > B,X5: A,Y5: A] : ( R3 @ ( F9 @ X5 ) @ ( F9 @ Y5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_186_old_Orec__prod__def,axiom,
% 2.58/1.59 ! [T: $tType,B: $tType,A: $tType] :
% 2.58/1.59 ( ( product_rec_prod @ A @ B @ T )
% 2.58/1.59 = ( ^ [F12: A > B > T,X5: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_187_the__equality,axiom,
% 2.58/1.59 ! [A: $tType,P3: A > $o,A2: A] :
% 2.58/1.59 ( ( P3 @ A2 )
% 2.58/1.59 => ( ! [X: A] :
% 2.58/1.59 ( ( P3 @ X )
% 2.58/1.59 => ( X = A2 ) )
% 2.58/1.59 => ( ( the @ A @ P3 )
% 2.58/1.59 = A2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_188_the__eq__trivial,axiom,
% 2.58/1.59 ! [A: $tType,A2: A] :
% 2.58/1.59 ( ( the @ A
% 2.58/1.59 @ ^ [X5: A] : ( X5 = A2 ) )
% 2.58/1.59 = A2 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_189_the__sym__eq__trivial,axiom,
% 2.58/1.59 ! [A: $tType,X4: A] :
% 2.58/1.59 ( ( the @ A
% 2.58/1.59 @ ( ^ [Y9: A,Z3: A] : ( Y9 = Z3 )
% 2.58/1.59 @ X4 ) )
% 2.58/1.59 = X4 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_190_the1__equality,axiom,
% 2.58/1.59 ! [A: $tType,P3: A > $o,A2: A] :
% 2.58/1.59 ( ? [X6: A] :
% 2.58/1.59 ( ( P3 @ X6 )
% 2.58/1.59 & ! [Y4: A] :
% 2.58/1.59 ( ( P3 @ Y4 )
% 2.58/1.59 => ( Y4 = X6 ) ) )
% 2.58/1.59 => ( ( P3 @ A2 )
% 2.58/1.59 => ( ( the @ A @ P3 )
% 2.58/1.59 = A2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_191_the1I2,axiom,
% 2.58/1.59 ! [A: $tType,P3: A > $o,Q3: A > $o] :
% 2.58/1.59 ( ? [X6: A] :
% 2.58/1.59 ( ( P3 @ X6 )
% 2.58/1.59 & ! [Y4: A] :
% 2.58/1.59 ( ( P3 @ Y4 )
% 2.58/1.59 => ( Y4 = X6 ) ) )
% 2.58/1.59 => ( ! [X: A] :
% 2.58/1.59 ( ( P3 @ X )
% 2.58/1.59 => ( Q3 @ X ) )
% 2.58/1.59 => ( Q3 @ ( the @ A @ P3 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_192_If__def,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( if @ A )
% 2.58/1.59 = ( ^ [P5: $o,X5: A,Y5: A] :
% 2.58/1.59 ( the @ A
% 2.58/1.59 @ ^ [Z2: A] :
% 2.58/1.59 ( ( P5
% 2.58/1.59 => ( Z2 = X5 ) )
% 2.58/1.59 & ( ~ P5
% 2.58/1.59 => ( Z2 = Y5 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_193_theI2,axiom,
% 2.58/1.59 ! [A: $tType,P3: A > $o,A2: A,Q3: A > $o] :
% 2.58/1.59 ( ( P3 @ A2 )
% 2.58/1.59 => ( ! [X: A] :
% 2.58/1.59 ( ( P3 @ X )
% 2.58/1.59 => ( X = A2 ) )
% 2.58/1.59 => ( ! [X: A] :
% 2.58/1.59 ( ( P3 @ X )
% 2.58/1.59 => ( Q3 @ X ) )
% 2.58/1.59 => ( Q3 @ ( the @ A @ P3 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_194_theI_H,axiom,
% 2.58/1.59 ! [A: $tType,P3: A > $o] :
% 2.58/1.59 ( ? [X6: A] :
% 2.58/1.59 ( ( P3 @ X6 )
% 2.58/1.59 & ! [Y4: A] :
% 2.58/1.59 ( ( P3 @ Y4 )
% 2.58/1.59 => ( Y4 = X6 ) ) )
% 2.58/1.59 => ( P3 @ ( the @ A @ P3 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_195_theI,axiom,
% 2.58/1.59 ! [A: $tType,P3: A > $o,A2: A] :
% 2.58/1.59 ( ( P3 @ A2 )
% 2.58/1.59 => ( ! [X: A] :
% 2.58/1.59 ( ( P3 @ X )
% 2.58/1.59 => ( X = A2 ) )
% 2.58/1.59 => ( P3 @ ( the @ A @ P3 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_196_surj__def,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,F: B > A] :
% 2.58/1.59 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.58/1.59 = ( top_top @ ( set @ A ) ) )
% 2.58/1.59 = ( ! [Y5: A] :
% 2.58/1.59 ? [X5: B] :
% 2.58/1.59 ( Y5
% 2.58/1.59 = ( F @ X5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_197_surjI,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 2.58/1.59 ( ! [X: A] :
% 2.58/1.59 ( ( G @ ( F @ X ) )
% 2.58/1.59 = X )
% 2.58/1.59 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 2.58/1.59 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_198_surjD,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 2.58/1.59 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.58/1.59 = ( top_top @ ( set @ A ) ) )
% 2.58/1.59 => ? [X: B] :
% 2.58/1.59 ( Y
% 2.58/1.59 = ( F @ X ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_199_surjE,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 2.58/1.59 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.58/1.59 = ( top_top @ ( set @ A ) ) )
% 2.58/1.59 => ~ ! [X: B] :
% 2.58/1.59 ( Y
% 2.58/1.59 != ( F @ X ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_200_old_Orec__unit__def,axiom,
% 2.58/1.59 ! [T: $tType] :
% 2.58/1.59 ( ( product_rec_unit @ T )
% 2.58/1.59 = ( ^ [F12: T,X5: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_201_range__fst,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.58/1.59 = ( top_top @ ( set @ A ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_202_fst__map__prod,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X4: product_prod @ C @ D] :
% 2.58/1.59 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X4 ) )
% 2.58/1.59 = ( F @ ( product_fst @ C @ D @ X4 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_203_fst__eqD,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,X4: A,Y: B,A2: A] :
% 2.58/1.59 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X4 @ Y ) )
% 2.58/1.59 = A2 )
% 2.58/1.59 => ( X4 = A2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_204_fst__conv,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,X1: A,X2: B] :
% 2.58/1.59 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 2.58/1.59 = X1 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_205_fst__def,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( product_fst @ A @ B )
% 2.58/1.59 = ( product_case_prod @ A @ B @ A
% 2.58/1.59 @ ^ [X13: A,X23: B] : X13 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_206_eq__fst__iff,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 2.58/1.59 ( ( A2
% 2.58/1.59 = ( product_fst @ A @ B @ P2 ) )
% 2.58/1.59 = ( ? [B5: B] :
% 2.58/1.59 ( P2
% 2.58/1.59 = ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_207_fstI,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,X4: product_prod @ A @ B,Y: A,Z: B] :
% 2.58/1.59 ( ( X4
% 2.58/1.59 = ( product_Pair @ A @ B @ Y @ Z ) )
% 2.58/1.59 => ( ( product_fst @ A @ B @ X4 )
% 2.58/1.59 = Y ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_208_fst__image__times,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,B6: set @ B,A3: set @ A] :
% 2.58/1.59 ( ( ( B6
% 2.58/1.59 = ( bot_bot @ ( set @ B ) ) )
% 2.58/1.59 => ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 2.58/1.59 @ ( product_Sigma @ A @ B @ A3
% 2.58/1.59 @ ^ [Uu: A] : B6 ) )
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) ) )
% 2.58/1.59 & ( ( B6
% 2.58/1.59 != ( bot_bot @ ( set @ B ) ) )
% 2.58/1.59 => ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 2.58/1.59 @ ( product_Sigma @ A @ B @ A3
% 2.58/1.59 @ ^ [Uu: A] : B6 ) )
% 2.58/1.59 = A3 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_209_The__case__prod,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P3: A > B > $o] :
% 2.58/1.59 ( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P3 ) )
% 2.58/1.59 = ( the @ ( product_prod @ A @ B )
% 2.58/1.59 @ ^ [Xy: product_prod @ A @ B] : ( P3 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_210_image__is__empty,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,F: B > A,A3: set @ B] :
% 2.58/1.59 ( ( ( image @ B @ A @ F @ A3 )
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) )
% 2.58/1.59 = ( A3
% 2.58/1.59 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_211_empty__is__image,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,F: B > A,A3: set @ B] :
% 2.58/1.59 ( ( ( bot_bot @ ( set @ A ) )
% 2.58/1.59 = ( image @ B @ A @ F @ A3 ) )
% 2.58/1.59 = ( A3
% 2.58/1.59 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_212_image__empty,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,F: B > A] :
% 2.58/1.59 ( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_213_Sigma__empty1,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,B6: A > ( set @ B )] :
% 2.58/1.59 ( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B6 )
% 2.58/1.59 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_214_snd__map__prod,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X4: product_prod @ C @ D] :
% 2.58/1.59 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X4 ) )
% 2.58/1.59 = ( G @ ( product_snd @ C @ D @ X4 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_215_Collect__const,axiom,
% 2.58/1.59 ! [A: $tType,P3: $o] :
% 2.58/1.59 ( ( P3
% 2.58/1.59 => ( ( collect @ A
% 2.58/1.59 @ ^ [S6: A] : P3 )
% 2.58/1.59 = ( top_top @ ( set @ A ) ) ) )
% 2.58/1.59 & ( ~ P3
% 2.58/1.59 => ( ( collect @ A
% 2.58/1.59 @ ^ [S6: A] : P3 )
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_216_Sigma__empty2,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,A3: set @ A] :
% 2.58/1.59 ( ( product_Sigma @ A @ B @ A3
% 2.58/1.59 @ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
% 2.58/1.59 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_217_Times__empty,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,A3: set @ A,B6: set @ B] :
% 2.58/1.59 ( ( ( product_Sigma @ A @ B @ A3
% 2.58/1.59 @ ^ [Uu: A] : B6 )
% 2.58/1.59 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.58/1.59 = ( ( A3
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) )
% 2.58/1.59 | ( B6
% 2.58/1.59 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_218_Collect__const__case__prod,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P3: $o] :
% 2.58/1.59 ( ( P3
% 2.58/1.59 => ( ( collect @ ( product_prod @ A @ B )
% 2.58/1.59 @ ( product_case_prod @ A @ B @ $o
% 2.58/1.59 @ ^ [A6: A,B5: B] : P3 ) )
% 2.58/1.59 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
% 2.58/1.59 & ( ~ P3
% 2.58/1.59 => ( ( collect @ ( product_prod @ A @ B )
% 2.58/1.59 @ ( product_case_prod @ A @ B @ $o
% 2.58/1.59 @ ^ [A6: A,B5: B] : P3 ) )
% 2.58/1.59 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_219_prod_Ocollapse,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.58/1.59 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 2.58/1.59 = Prod ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_220_snd__swap,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,X4: product_prod @ A @ B] :
% 2.58/1.59 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X4 ) )
% 2.58/1.59 = ( product_fst @ A @ B @ X4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_221_fst__swap,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,X4: product_prod @ B @ A] :
% 2.58/1.59 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X4 ) )
% 2.58/1.59 = ( product_snd @ B @ A @ X4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_222_range__snd,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.58/1.59 = ( top_top @ ( set @ A ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_223_snd__image__times,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,A3: set @ B,B6: set @ A] :
% 2.58/1.59 ( ( ( A3
% 2.58/1.59 = ( bot_bot @ ( set @ B ) ) )
% 2.58/1.59 => ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 2.58/1.59 @ ( product_Sigma @ B @ A @ A3
% 2.58/1.59 @ ^ [Uu: B] : B6 ) )
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) ) )
% 2.58/1.59 & ( ( A3
% 2.58/1.59 != ( bot_bot @ ( set @ B ) ) )
% 2.58/1.59 => ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 2.58/1.59 @ ( product_Sigma @ B @ A @ A3
% 2.58/1.59 @ ^ [Uu: B] : B6 ) )
% 2.58/1.59 = B6 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_224_sndI,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,X4: product_prod @ A @ B,Y: A,Z: B] :
% 2.58/1.59 ( ( X4
% 2.58/1.59 = ( product_Pair @ A @ B @ Y @ Z ) )
% 2.58/1.59 => ( ( product_snd @ A @ B @ X4 )
% 2.58/1.59 = Z ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_225_eq__snd__iff,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 2.58/1.59 ( ( B2
% 2.58/1.59 = ( product_snd @ B @ A @ P2 ) )
% 2.58/1.59 = ( ? [A6: B] :
% 2.58/1.59 ( P2
% 2.58/1.59 = ( product_Pair @ B @ A @ A6 @ B2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_226_times__eq__iff,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,A3: set @ A,B6: set @ B,C5: set @ A,D3: set @ B] :
% 2.58/1.59 ( ( ( product_Sigma @ A @ B @ A3
% 2.58/1.59 @ ^ [Uu: A] : B6 )
% 2.58/1.59 = ( product_Sigma @ A @ B @ C5
% 2.58/1.59 @ ^ [Uu: A] : D3 ) )
% 2.58/1.59 = ( ( ( A3 = C5 )
% 2.58/1.59 & ( B6 = D3 ) )
% 2.58/1.59 | ( ( ( A3
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) )
% 2.58/1.59 | ( B6
% 2.58/1.59 = ( bot_bot @ ( set @ B ) ) ) )
% 2.58/1.59 & ( ( C5
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) )
% 2.58/1.59 | ( D3
% 2.58/1.59 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_227_empty__def,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( bot_bot @ ( set @ A ) )
% 2.58/1.59 = ( collect @ A
% 2.58/1.59 @ ^ [X5: A] : $false ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_228_snd__eqD,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,X4: B,Y: A,A2: A] :
% 2.58/1.59 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y ) )
% 2.58/1.59 = A2 )
% 2.58/1.59 => ( Y = A2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_229_snd__conv,axiom,
% 2.58/1.59 ! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
% 2.58/1.59 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
% 2.58/1.59 = X2 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_230_Sigma__empty__iff,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,I4: set @ A,X8: A > ( set @ B )] :
% 2.58/1.59 ( ( ( product_Sigma @ A @ B @ I4 @ X8 )
% 2.58/1.59 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.58/1.59 = ( ! [X5: A] :
% 2.58/1.59 ( ( member @ A @ X5 @ I4 )
% 2.58/1.59 => ( ( X8 @ X5 )
% 2.58/1.59 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_231_snd__def,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( product_snd @ A @ B )
% 2.58/1.59 = ( product_case_prod @ A @ B @ B
% 2.58/1.59 @ ^ [X13: A,X23: B] : X23 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_232_split__comp__eq,axiom,
% 2.58/1.59 ! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
% 2.58/1.59 ( ( ^ [U: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
% 2.58/1.59 = ( product_case_prod @ D @ B @ C
% 2.58/1.59 @ ^ [X5: D] : ( F @ ( G @ X5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_233_case__prod__beta,axiom,
% 2.58/1.59 ! [A: $tType,C: $tType,B: $tType] :
% 2.58/1.59 ( ( product_case_prod @ B @ C @ A )
% 2.58/1.59 = ( ^ [F9: B > C > A,P: product_prod @ B @ C] : ( F9 @ ( product_fst @ B @ C @ P ) @ ( product_snd @ B @ C @ P ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_234_case__prod__beta_H,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.58/1.59 ( ( product_case_prod @ A @ B @ C )
% 2.58/1.59 = ( ^ [F9: A > B > C,X5: product_prod @ A @ B] : ( F9 @ ( product_fst @ A @ B @ X5 ) @ ( product_snd @ A @ B @ X5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_235_prod_Ocase__eq__if,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.58/1.59 ( ( product_case_prod @ A @ B @ C )
% 2.58/1.59 = ( ^ [F9: A > B > C,Prod2: product_prod @ A @ B] : ( F9 @ ( product_fst @ A @ B @ Prod2 ) @ ( product_snd @ A @ B @ Prod2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_236_case__prod__unfold,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType] :
% 2.58/1.59 ( ( product_case_prod @ A @ B @ C )
% 2.58/1.59 = ( ^ [C4: A > B > C,P: product_prod @ A @ B] : ( C4 @ ( product_fst @ A @ B @ P ) @ ( product_snd @ A @ B @ P ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_237_Product__Type_OCollect__case__prodD,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,X4: product_prod @ A @ B,A3: A > B > $o] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ B ) @ X4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) )
% 2.58/1.59 => ( A3 @ ( product_fst @ A @ B @ X4 ) @ ( product_snd @ A @ B @ X4 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_238_mem__Times__iff,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,X4: product_prod @ A @ B,A3: set @ A,B6: set @ B] :
% 2.58/1.59 ( ( member @ ( product_prod @ A @ B ) @ X4
% 2.58/1.59 @ ( product_Sigma @ A @ B @ A3
% 2.58/1.59 @ ^ [Uu: A] : B6 ) )
% 2.58/1.59 = ( ( member @ A @ ( product_fst @ A @ B @ X4 ) @ A3 )
% 2.58/1.59 & ( member @ B @ ( product_snd @ A @ B @ X4 ) @ B6 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_239_prod_Oexhaust__sel,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.58/1.59 ( Prod
% 2.58/1.59 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_240_surjective__pairing,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 2.58/1.59 ( T2
% 2.58/1.59 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_241_prod__eq__iff,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( ^ [Y9: product_prod @ A @ B,Z3: product_prod @ A @ B] : ( Y9 = Z3 ) )
% 2.58/1.59 = ( ^ [S6: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 2.58/1.59 ( ( ( product_fst @ A @ B @ S6 )
% 2.58/1.59 = ( product_fst @ A @ B @ T3 ) )
% 2.58/1.59 & ( ( product_snd @ A @ B @ S6 )
% 2.58/1.59 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_242_prod_Oexpand,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod3: product_prod @ A @ B] :
% 2.58/1.59 ( ( ( ( product_fst @ A @ B @ Prod )
% 2.58/1.59 = ( product_fst @ A @ B @ Prod3 ) )
% 2.58/1.59 & ( ( product_snd @ A @ B @ Prod )
% 2.58/1.59 = ( product_snd @ A @ B @ Prod3 ) ) )
% 2.58/1.59 => ( Prod = Prod3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_243_prod__eqI,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q: product_prod @ A @ B] :
% 2.58/1.59 ( ( ( product_fst @ A @ B @ P2 )
% 2.58/1.59 = ( product_fst @ A @ B @ Q ) )
% 2.58/1.59 => ( ( ( product_snd @ A @ B @ P2 )
% 2.58/1.59 = ( product_snd @ A @ B @ Q ) )
% 2.58/1.59 => ( P2 = Q ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_244_prod_Osplit__sel,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType,P3: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 2.58/1.59 ( ( P3 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 2.58/1.59 = ( ( Prod
% 2.58/1.59 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 2.58/1.59 => ( P3 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_245_prod_Osplit__sel__asm,axiom,
% 2.58/1.59 ! [C: $tType,B: $tType,A: $tType,P3: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 2.58/1.59 ( ( P3 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 2.58/1.59 = ( ~ ( ( Prod
% 2.58/1.59 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 2.58/1.59 & ~ ( P3 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_246_scomp__unfold,axiom,
% 2.58/1.59 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 2.58/1.59 ( ( product_scomp @ A @ B @ C @ D )
% 2.58/1.59 = ( ^ [F9: A > ( product_prod @ B @ C ),G4: B > C > D,X5: A] : ( G4 @ ( product_fst @ B @ C @ ( F9 @ X5 ) ) @ ( product_snd @ B @ C @ ( F9 @ X5 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_247_prod_Oswap__def,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( product_swap @ A @ B )
% 2.58/1.59 = ( ^ [P: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P ) @ ( product_fst @ A @ B @ P ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_248_fst__image__Sigma,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,A3: set @ A,B6: A > ( set @ B )] :
% 2.58/1.59 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B6 ) )
% 2.58/1.59 = ( collect @ A
% 2.58/1.59 @ ^ [X5: A] :
% 2.58/1.59 ( ( member @ A @ X5 @ A3 )
% 2.58/1.59 & ( ( B6 @ X5 )
% 2.58/1.59 != ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_249_exE__realizer,axiom,
% 2.58/1.59 ! [C: $tType,A: $tType,B: $tType,P3: A > B > $o,P2: product_prod @ B @ A,Q3: C > $o,F: B > A > C] :
% 2.58/1.59 ( ( P3 @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 2.58/1.59 => ( ! [X: B,Y4: A] :
% 2.58/1.59 ( ( P3 @ Y4 @ X )
% 2.58/1.59 => ( Q3 @ ( F @ X @ Y4 ) ) )
% 2.58/1.59 => ( Q3 @ ( product_case_prod @ B @ A @ C @ F @ P2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_250_exI__realizer,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P3: A > B > $o,Y: A,X4: B] :
% 2.58/1.59 ( ( P3 @ Y @ X4 )
% 2.58/1.59 => ( P3 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_251_conjI__realizer,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,P3: A > $o,P2: A,Q3: B > $o,Q: B] :
% 2.58/1.59 ( ( P3 @ P2 )
% 2.58/1.59 => ( ( Q3 @ Q )
% 2.58/1.59 => ( ( P3 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) )
% 2.58/1.59 & ( Q3 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_252_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P3: A > B > $o,X4: A,Y: B,A2: product_prod @ A @ B] :
% 2.58/1.59 ( ( P3 @ X4 @ Y )
% 2.58/1.59 => ( ( A2
% 2.58/1.59 = ( product_Pair @ A @ B @ X4 @ Y ) )
% 2.58/1.59 => ( P3 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_253_the__elem__image__unique,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,A3: set @ A,F: A > B,X4: A] :
% 2.58/1.59 ( ( A3
% 2.58/1.59 != ( bot_bot @ ( set @ A ) ) )
% 2.58/1.59 => ( ! [Y4: A] :
% 2.58/1.59 ( ( member @ A @ Y4 @ A3 )
% 2.58/1.59 => ( ( F @ Y4 )
% 2.58/1.59 = ( F @ X4 ) ) )
% 2.58/1.59 => ( ( the_elem @ B @ ( image @ A @ B @ F @ A3 ) )
% 2.58/1.59 = ( F @ X4 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_254_subset__fst__snd,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] :
% 2.58/1.59 ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3
% 2.58/1.59 @ ( product_Sigma @ A @ B @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 )
% 2.58/1.59 @ ^ [Uu: A] : ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_255_Collect__split__mono__strong,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,X8: set @ A,A3: set @ ( product_prod @ A @ B ),Y6: set @ B,P3: A > B > $o,Q3: A > B > $o] :
% 2.58/1.59 ( ( X8
% 2.58/1.59 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) )
% 2.58/1.59 => ( ( Y6
% 2.58/1.59 = ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) )
% 2.58/1.59 => ( ! [X: A] :
% 2.58/1.59 ( ( member @ A @ X @ X8 )
% 2.58/1.59 => ! [Xa: B] :
% 2.58/1.59 ( ( member @ B @ Xa @ Y6 )
% 2.58/1.59 => ( ( P3 @ X @ Xa )
% 2.58/1.59 => ( Q3 @ X @ Xa ) ) ) )
% 2.58/1.59 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P3 ) ) )
% 2.58/1.59 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q3 ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(help_If_3_1_T,axiom,
% 2.58/1.59 ! [P3: $o] :
% 2.58/1.59 ( ( P3 = $true )
% 2.58/1.59 | ( P3 = $false ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(help_If_2_1_T,axiom,
% 2.58/1.59 ! [A: $tType,X4: A,Y: A] :
% 2.58/1.59 ( ( if @ A @ $false @ X4 @ Y )
% 2.58/1.59 = Y ) ).
% 2.58/1.59
% 2.58/1.59 thf(help_If_1_1_T,axiom,
% 2.58/1.59 ! [A: $tType,X4: A,Y: A] :
% 2.58/1.59 ( ( if @ A @ $true @ X4 @ Y )
% 2.58/1.59 = X4 ) ).
% 2.58/1.59
% 2.58/1.59 thf(conj_0,conjecture,
% 2.58/1.59 ? [I5: ( list @ char ) > paraco415392788lle_tv] :
% 2.58/1.59 ( ( paraco876059933e_eval @ I5 @ p )
% 2.58/1.59 != ( paraco2040174112le_Det @ $true ) ) ).
% 2.58/1.59 ------------------------
% 2.77/1.62 % (16863)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_16534 for (2999ds/183Mi)
% 2.77/1.63 % (16869)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_16534 for (2999ds/18Mi)
% 2.77/1.63 % (16864)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_16534 for (2999ds/4Mi)
% 2.77/1.63 % (16869)Instruction limit reached!
% 2.77/1.63 % (16869)------------------------------
% 2.77/1.63 % (16869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.63 % (16869)Termination reason: Unknown
% 2.77/1.63 % (16869)Termination phase: shuffling
% 2.77/1.63
% 2.77/1.63 % (16869)Memory used [KB]: 1918
% 2.77/1.63 % (16869)Time elapsed: 0.008 s
% 2.77/1.63 % (16869)Instructions burned: 19 (million)
% 2.77/1.63 % (16869)------------------------------
% 2.77/1.63 % (16869)------------------------------
% 2.77/1.63 % (16865)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_16534 for (2999ds/27Mi)
% 2.77/1.63 % (16866)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_16534 for (2999ds/2Mi)
% 2.77/1.63 % (16867)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_16534 for (2999ds/2Mi)
% 2.77/1.63 % (16868)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_16534 for (2999ds/275Mi)
% 2.77/1.63 % (16866)Instruction limit reached!
% 2.77/1.63 % (16866)------------------------------
% 2.77/1.63 % (16866)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.63 % (16866)Termination reason: Unknown
% 2.77/1.63 % (16866)Termination phase: shuffling
% 2.77/1.63
% 2.77/1.63 % (16866)Memory used [KB]: 1407
% 2.77/1.63 % (16866)Time elapsed: 0.004 s
% 2.77/1.63 % (16866)Instructions burned: 3 (million)
% 2.77/1.63 % (16866)------------------------------
% 2.77/1.63 % (16866)------------------------------
% 2.77/1.63 % (16867)Instruction limit reached!
% 2.77/1.63 % (16867)------------------------------
% 2.77/1.63 % (16867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.63 % (16867)Termination reason: Unknown
% 2.77/1.63 % (16867)Termination phase: shuffling
% 2.77/1.63
% 2.77/1.63 % (16867)Memory used [KB]: 1407
% 2.77/1.63 % (16867)Time elapsed: 0.004 s
% 2.77/1.63 % (16867)Instructions burned: 3 (million)
% 2.77/1.63 % (16867)------------------------------
% 2.77/1.63 % (16867)------------------------------
% 2.77/1.63 % (16864)Instruction limit reached!
% 2.77/1.63 % (16864)------------------------------
% 2.77/1.63 % (16864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.63 % (16864)Termination reason: Unknown
% 2.77/1.63 % (16864)Termination phase: shuffling
% 2.77/1.63
% 2.77/1.63 % (16864)Memory used [KB]: 1535
% 2.77/1.63 % (16864)Time elapsed: 0.005 s
% 2.77/1.63 % (16864)Instructions burned: 5 (million)
% 2.77/1.63 % (16864)------------------------------
% 2.77/1.63 % (16864)------------------------------
% 2.77/1.64 % (16870)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_16534 for (2999ds/3Mi)
% 2.77/1.64 % (16870)Instruction limit reached!
% 2.77/1.64 % (16870)------------------------------
% 2.77/1.64 % (16870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.64 % (16870)Termination reason: Unknown
% 2.77/1.64 % (16870)Termination phase: shuffling
% 2.77/1.64
% 2.77/1.64 % (16870)Memory used [KB]: 1407
% 2.77/1.64 % (16870)Time elapsed: 0.003 s
% 2.77/1.64 % (16870)Instructions burned: 3 (million)
% 2.77/1.64 % (16870)------------------------------
% 2.77/1.64 % (16870)------------------------------
% 2.77/1.65 % (16865)Instruction limit reached!
% 2.77/1.65 % (16865)------------------------------
% 2.77/1.65 % (16865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.65 % (16865)Termination reason: Unknown
% 2.77/1.65 % (16865)Termination phase: Property scanning
% 2.77/1.65
% 2.77/1.65 % (16865)Memory used [KB]: 2046
% 2.77/1.65 % (16865)Time elapsed: 0.019 s
% 2.77/1.65 % (16865)Instructions burned: 27 (million)
% 2.77/1.65 % (16865)------------------------------
% 2.77/1.65 % (16865)------------------------------
% 2.77/1.65 % (16871)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_16534 for (2999ds/37Mi)
% 2.77/1.65 % (16872)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_16534 for (2999ds/15Mi)
% 2.77/1.65 % (16873)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_16534 for (2999ds/3Mi)
% 2.77/1.65 % (16874)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_16534 for (2999ds/1041Mi)
% 2.77/1.65 % (16873)Instruction limit reached!
% 2.77/1.65 % (16873)------------------------------
% 2.77/1.65 % (16873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.65 % (16873)Termination reason: Unknown
% 2.77/1.65 % (16873)Termination phase: shuffling
% 2.77/1.65
% 2.77/1.65 % (16873)Memory used [KB]: 1407
% 2.77/1.65 % (16873)Time elapsed: 0.004 s
% 2.77/1.65 % (16873)Instructions burned: 3 (million)
% 2.77/1.65 % (16873)------------------------------
% 2.77/1.65 % (16873)------------------------------
% 2.77/1.66 % (16868)Refutation not found, incomplete strategy
% 2.77/1.66 % (16868)------------------------------
% 2.77/1.66 % (16868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.66 % (16868)Termination reason: Refutation not found, incomplete strategy
% 2.77/1.66
% 2.77/1.66
% 2.77/1.66 % (16868)Memory used [KB]: 6652
% 2.77/1.66 % (16868)Time elapsed: 0.031 s
% 2.77/1.66 % (16868)Instructions burned: 46 (million)
% 2.77/1.66 % (16868)------------------------------
% 2.77/1.66 % (16868)------------------------------
% 2.77/1.66 % (16872)Instruction limit reached!
% 2.77/1.66 % (16872)------------------------------
% 2.77/1.66 % (16872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.66 % (16872)Termination reason: Unknown
% 2.77/1.66 % (16872)Termination phase: shuffling
% 2.77/1.66
% 2.77/1.66 % (16872)Memory used [KB]: 1791
% 2.77/1.66 % (16872)Time elapsed: 0.012 s
% 2.77/1.66 % (16872)Instructions burned: 16 (million)
% 2.77/1.66 % (16872)------------------------------
% 2.77/1.66 % (16872)------------------------------
% 2.77/1.67 % (16875)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_16534 for (2999ds/7Mi)
% 2.77/1.67 % (16875)Instruction limit reached!
% 2.77/1.67 % (16875)------------------------------
% 2.77/1.67 % (16875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.67 % (16875)Termination reason: Unknown
% 2.77/1.67 % (16875)Termination phase: shuffling
% 2.77/1.67
% 2.77/1.67 % (16875)Memory used [KB]: 1535
% 2.77/1.67 % (16875)Time elapsed: 0.006 s
% 2.77/1.67 % (16875)Instructions burned: 8 (million)
% 2.77/1.67 % (16875)------------------------------
% 2.77/1.67 % (16875)------------------------------
% 2.77/1.67 % (16876)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_16534 for (2999ds/16Mi)
% 2.77/1.67 % (16871)Instruction limit reached!
% 2.77/1.67 % (16871)------------------------------
% 2.77/1.67 % (16871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.67 % (16871)Termination reason: Unknown
% 2.77/1.67 % (16871)Termination phase: shuffling
% 2.77/1.67
% 2.77/1.67 % (16871)Memory used [KB]: 2174
% 2.77/1.67 % (16871)Time elapsed: 0.022 s
% 2.77/1.67 % (16871)Instructions burned: 37 (million)
% 2.77/1.67 % (16871)------------------------------
% 2.77/1.67 % (16871)------------------------------
% 2.77/1.68 % (16876)Instruction limit reached!
% 2.77/1.68 % (16876)------------------------------
% 2.77/1.68 % (16876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.68 % (16876)Termination reason: Unknown
% 2.77/1.68 % (16876)Termination phase: shuffling
% 2.77/1.68
% 2.77/1.68 % (16876)Memory used [KB]: 1791
% 2.77/1.68 % (16876)Time elapsed: 0.007 s
% 2.77/1.68 % (16876)Instructions burned: 16 (million)
% 2.77/1.68 % (16876)------------------------------
% 2.77/1.68 % (16876)------------------------------
% 2.77/1.68 % (16877)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_16534 for (2999ds/3Mi)
% 2.77/1.68 % (16878)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_16534 for (2999ds/3Mi)
% 2.77/1.68 % (16863)Instruction limit reached!
% 2.77/1.68 % (16863)------------------------------
% 2.77/1.68 % (16863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.68 % (16863)Termination reason: Unknown
% 2.77/1.68 % (16863)Termination phase: Saturation
% 2.77/1.68
% 2.77/1.68 % (16863)Memory used [KB]: 7675
% 2.77/1.68 % (16863)Time elapsed: 0.058 s
% 2.77/1.68 % (16863)Instructions burned: 185 (million)
% 2.77/1.68 % (16863)------------------------------
% 2.77/1.68 % (16863)------------------------------
% 2.77/1.68 % (16879)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_16534 for (2999ds/7Mi)
% 2.77/1.68 % (16877)Instruction limit reached!
% 2.77/1.68 % (16877)------------------------------
% 2.77/1.68 % (16877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.68 % (16877)Termination reason: Unknown
% 2.77/1.68 % (16877)Termination phase: shuffling
% 2.77/1.68
% 2.77/1.68 % (16877)Memory used [KB]: 1407
% 2.77/1.68 % (16877)Time elapsed: 0.004 s
% 2.77/1.68 % (16877)Instructions burned: 3 (million)
% 2.77/1.68 % (16877)------------------------------
% 2.77/1.68 % (16877)------------------------------
% 2.77/1.68 % (16878)Instruction limit reached!
% 2.77/1.68 % (16878)------------------------------
% 2.77/1.68 % (16878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.68 % (16878)Termination reason: Unknown
% 2.77/1.68 % (16878)Termination phase: shuffling
% 2.77/1.68
% 2.77/1.68 % (16878)Memory used [KB]: 1407
% 2.77/1.68 % (16878)Time elapsed: 0.004 s
% 2.77/1.68 % (16878)Instructions burned: 3 (million)
% 2.77/1.68 % (16878)------------------------------
% 2.77/1.68 % (16878)------------------------------
% 2.77/1.68 % (16879)Instruction limit reached!
% 2.77/1.68 % (16879)------------------------------
% 2.77/1.68 % (16879)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.68 % (16879)Termination reason: Unknown
% 2.77/1.68 % (16879)Termination phase: shuffling
% 2.77/1.68
% 2.77/1.68 % (16879)Memory used [KB]: 1535
% 2.77/1.68 % (16879)Time elapsed: 0.004 s
% 2.77/1.68 % (16879)Instructions burned: 8 (million)
% 2.77/1.68 % (16879)------------------------------
% 2.77/1.68 % (16879)------------------------------
% 2.77/1.69 % (16881)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_16534 for (2999ds/4Mi)
% 2.77/1.69 % (16881)Instruction limit reached!
% 2.77/1.69 % (16881)------------------------------
% 2.77/1.69 % (16881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.69 % (16881)Termination reason: Unknown
% 2.77/1.69 % (16881)Termination phase: shuffling
% 2.77/1.69
% 2.77/1.69 % (16881)Memory used [KB]: 1535
% 2.77/1.69 % (16881)Time elapsed: 0.003 s
% 2.77/1.69 % (16881)Instructions burned: 4 (million)
% 2.77/1.69 % (16881)------------------------------
% 2.77/1.69 % (16881)------------------------------
% 2.77/1.69 % (16880)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_16534 for (2999ds/3Mi)
% 2.77/1.69 % (16880)Instruction limit reached!
% 2.77/1.69 % (16880)------------------------------
% 2.77/1.69 % (16880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.69 % (16880)Termination reason: Unknown
% 2.77/1.69 % (16880)Termination phase: shuffling
% 2.77/1.69
% 2.77/1.69 % (16880)Memory used [KB]: 1535
% 2.77/1.69 % (16880)Time elapsed: 0.004 s
% 2.77/1.69 % (16880)Instructions burned: 4 (million)
% 2.77/1.69 % (16880)------------------------------
% 2.77/1.69 % (16880)------------------------------
% 2.77/1.69 % (16885)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_16534 for (2999ds/902Mi)
% 2.77/1.70 % (16883)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_16534 for (2999ds/710Mi)
% 2.77/1.70 % (16886)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_16534 for (2999ds/21Mi)
% 2.77/1.70 % (16882)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_16534 for (2999ds/18Mi)
% 2.77/1.70 % (16884)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_16534 for (2999ds/6Mi)
% 2.77/1.70 % (16884)Instruction limit reached!
% 2.77/1.70 % (16884)------------------------------
% 2.77/1.70 % (16884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.70 % (16884)Termination reason: Unknown
% 2.77/1.70 % (16884)Termination phase: shuffling
% 2.77/1.70
% 2.77/1.70 % (16884)Memory used [KB]: 1535
% 2.77/1.70 % (16884)Time elapsed: 0.006 s
% 2.77/1.70 % (16884)Instructions burned: 7 (million)
% 2.77/1.70 % (16884)------------------------------
% 2.77/1.70 % (16884)------------------------------
% 2.77/1.71 % (16886)Instruction limit reached!
% 2.77/1.71 % (16886)------------------------------
% 2.77/1.71 % (16886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.71 % (16886)Termination reason: Unknown
% 2.77/1.71 % (16886)Termination phase: shuffling
% 2.77/1.71
% 2.77/1.71 % (16886)Memory used [KB]: 1918
% 2.77/1.71 % (16886)Time elapsed: 0.010 s
% 2.77/1.71 % (16886)Instructions burned: 23 (million)
% 2.77/1.71 % (16886)------------------------------
% 2.77/1.71 % (16886)------------------------------
% 2.77/1.71 % (16887)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_16534 for (2999ds/5Mi)
% 2.77/1.71 % (16882)Instruction limit reached!
% 2.77/1.71 % (16882)------------------------------
% 2.77/1.71 % (16882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.71 % (16882)Termination reason: Unknown
% 2.77/1.71 % (16882)Termination phase: shuffling
% 2.77/1.71
% 2.77/1.71 % (16882)Memory used [KB]: 1791
% 2.77/1.71 % (16882)Time elapsed: 0.014 s
% 2.77/1.71 % (16882)Instructions burned: 19 (million)
% 2.77/1.71 % (16882)------------------------------
% 2.77/1.71 % (16882)------------------------------
% 2.77/1.71 % (16887)Instruction limit reached!
% 2.77/1.71 % (16887)------------------------------
% 2.77/1.71 % (16887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.71 % (16887)Termination reason: Unknown
% 2.77/1.71 % (16887)Termination phase: shuffling
% 2.77/1.71
% 2.77/1.71 % (16887)Memory used [KB]: 1535
% 2.77/1.71 % (16887)Time elapsed: 0.005 s
% 2.77/1.71 % (16887)Instructions burned: 6 (million)
% 2.77/1.71 % (16887)------------------------------
% 2.77/1.71 % (16887)------------------------------
% 2.77/1.72 % (16889)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_16534 for (2998ds/377Mi)
% 2.77/1.72 % (16888)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_16534 for (2998ds/6Mi)
% 2.77/1.72 % (16888)Instruction limit reached!
% 2.77/1.72 % (16888)------------------------------
% 2.77/1.72 % (16888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.72 % (16888)Termination reason: Unknown
% 2.77/1.72 % (16888)Termination phase: shuffling
% 2.77/1.72
% 2.77/1.72 % (16888)Memory used [KB]: 1663
% 2.77/1.72 % (16888)Time elapsed: 0.005 s
% 2.77/1.72 % (16888)Instructions burned: 7 (million)
% 2.77/1.72 % (16888)------------------------------
% 2.77/1.72 % (16888)------------------------------
% 2.77/1.73 % (16890)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_16534 for (2998ds/779Mi)
% 2.77/1.73 % (16891)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_16534 for (2998ds/19Mi)
% 2.77/1.73 % (16892)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_16534 for (2998ds/879Mi)
% 2.77/1.74 % (16891)Instruction limit reached!
% 2.77/1.74 % (16891)------------------------------
% 2.77/1.74 % (16891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.74 % (16891)Termination reason: Unknown
% 2.77/1.74 % (16891)Termination phase: shuffling
% 2.77/1.74
% 2.77/1.74 % (16891)Memory used [KB]: 1791
% 2.77/1.74 % (16891)Time elapsed: 0.012 s
% 2.77/1.74 % (16891)Instructions burned: 19 (million)
% 2.77/1.74 % (16891)------------------------------
% 2.77/1.74 % (16891)------------------------------
% 2.77/1.75 % (16893)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_16534 for (2998ds/17Mi)
% 2.77/1.76 % (16893)Instruction limit reached!
% 2.77/1.76 % (16893)------------------------------
% 2.77/1.76 % (16893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.76 % (16893)Termination reason: Unknown
% 2.77/1.76 % (16893)Termination phase: shuffling
% 2.77/1.76
% 2.77/1.76 % (16893)Memory used [KB]: 1791
% 2.77/1.76 % (16893)Time elapsed: 0.009 s
% 2.77/1.76 % (16893)Instructions burned: 17 (million)
% 2.77/1.76 % (16893)------------------------------
% 2.77/1.76 % (16893)------------------------------
% 2.77/1.77 % (16894)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_16534 for (2998ds/3Mi)
% 2.77/1.77 % (16894)Instruction limit reached!
% 2.77/1.77 % (16894)------------------------------
% 2.77/1.77 % (16894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.77 % (16894)Termination reason: Unknown
% 2.77/1.77 % (16894)Termination phase: shuffling
% 2.77/1.77
% 2.77/1.77 % (16894)Memory used [KB]: 1535
% 2.77/1.77 % (16894)Time elapsed: 0.003 s
% 2.77/1.77 % (16894)Instructions burned: 3 (million)
% 2.77/1.77 % (16894)------------------------------
% 2.77/1.77 % (16894)------------------------------
% 2.77/1.78 % (16895)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_16534 for (2998ds/30Mi)
% 2.77/1.79 % (16895)Instruction limit reached!
% 2.77/1.79 % (16895)------------------------------
% 2.77/1.79 % (16895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.79 % (16895)Termination reason: Unknown
% 2.77/1.79 % (16895)Termination phase: shuffling
% 2.77/1.79
% 2.77/1.79 % (16895)Memory used [KB]: 2046
% 2.77/1.79 % (16895)Time elapsed: 0.013 s
% 2.77/1.79 % (16895)Instructions burned: 31 (million)
% 2.77/1.79 % (16895)------------------------------
% 2.77/1.79 % (16895)------------------------------
% 2.77/1.80 % (16896)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_16534 for (2997ds/127Mi)
% 2.77/1.84 % (16892)First to succeed.
% 2.77/1.84 % (16892)Refutation found. Thanks to Tanya!
% 2.77/1.84 % SZS status Theorem for DTF2THF_16534
% 2.77/1.84 % SZS output start Proof for DTF2THF_16534
% 2.77/1.84 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 2.77/1.84 thf(type_def_6, type, set: ($tType) > $tType).
% 2.77/1.84 thf(type_def_8, type, char: $tType).
% 2.77/1.84 thf(type_def_9, type, list: ($tType) > $tType).
% 2.77/1.84 thf(type_def_10, type, paraco415392788lle_tv: $tType).
% 2.77/1.84 thf(type_def_11, type, paraco414474393lle_fm: $tType).
% 2.77/1.84 thf(type_def_12, type, nat: $tType).
% 2.77/1.84 thf(type_def_13, type, product_unit: $tType).
% 2.77/1.84 thf(func_def_0, type, paraco415392788lle_tv: $tType).
% 2.77/1.84 thf(func_def_1, type, paraco414474393lle_fm: $tType).
% 2.77/1.84 thf(func_def_2, type, product_unit: $tType).
% 2.77/1.84 thf(func_def_3, type, product_prod: ($tType * $tType) > $tType).
% 2.77/1.84 thf(func_def_4, type, char: $tType).
% 2.77/1.84 thf(func_def_5, type, list: ($tType) > $tType).
% 2.77/1.84 thf(func_def_6, type, set: ($tType) > $tType).
% 2.77/1.84 thf(func_def_7, type, nat: $tType).
% 2.77/1.84 thf(func_def_8, type, fun_reduction_pair: !>[X0: $tType]:(product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X0 @ X0 > $o)).
% 2.77/1.84 thf(func_def_9, 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)).
% 2.77/1.84 thf(func_def_10, type, the: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.77/1.84 thf(func_def_11, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.77/1.84 thf(func_def_12, type, order_above: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 2.77/1.84 thf(func_def_13, type, order_aboveS: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 2.77/1.84 thf(func_def_14, type, order_relation_of: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > set @ product_prod @ X0 @ X0)).
% 2.77/1.84 thf(func_def_15, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.77/1.84 thf(func_def_16, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.77/1.84 thf(func_def_17, type, top_top: !>[X0: $tType]:(X0)).
% 2.77/1.84 thf(func_def_18, type, paraco876059933e_eval: (list @ char > paraco415392788lle_tv) > paraco414474393lle_fm > paraco415392788lle_tv).
% 2.77/1.84 thf(func_def_19, type, paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_20, type, paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_21, type, paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_22, type, paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_23, type, paraco27778325le_Pro: list @ char > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_24, type, paraco251304083_Truth: paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_25, type, paraco1246693743ase_fm: !>[X0: $tType]:((list @ char > X0) > X0 > (paraco414474393lle_fm > X0) > (paraco414474393lle_fm > paraco414474393lle_fm > X0) > (paraco414474393lle_fm > paraco414474393lle_fm > X0) > (paraco414474393lle_fm > paraco414474393lle_fm > X0) > paraco414474393lle_fm > X0)).
% 2.77/1.84 thf(func_def_26, type, paraco2040174112le_Det: $o > paraco415392788lle_tv).
% 2.77/1.84 thf(func_def_27, type, paraco676387099_Indet: nat > paraco415392788lle_tv).
% 2.77/1.84 thf(func_def_28, type, paraco490622181ase_tv: !>[X0: $tType]:(($o > X0) > (nat > X0) > paraco415392788lle_tv > X0)).
% 2.77/1.84 thf(func_def_29, type, paraco152590079rec_tv: !>[X0: $tType]:(($o > X0) > (nat > X0) > paraco415392788lle_tv > X0)).
% 2.77/1.84 thf(func_def_30, type, paraco769098683_valid: paraco414474393lle_fm > $o).
% 2.77/1.84 thf(func_def_31, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.77/1.84 thf(func_def_32, type, product_Sigma: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ product_prod @ X0 @ X1)).
% 2.77/1.84 thf(func_def_33, type, product_case_bool: !>[X0: $tType]:(X0 > X0 > $o > X0)).
% 2.77/1.84 thf(func_def_34, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 2.77/1.84 thf(func_def_35, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.77/1.84 thf(func_def_36, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 2.77/1.84 thf(func_def_37, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.77/1.84 thf(func_def_38, type, product_rec_set_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2 > $o)).
% 2.77/1.84 thf(func_def_39, type, product_rec_set_unit: !>[X0: $tType]:(X0 > product_unit > X0 > $o)).
% 2.77/1.84 thf(func_def_40, type, product_rec_unit: !>[X0: $tType]:(X0 > product_unit > X0)).
% 2.77/1.84 thf(func_def_41, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.77/1.84 thf(func_def_42, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.77/1.84 thf(func_def_43, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.77/1.84 thf(func_def_44, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 2.77/1.84 thf(func_def_45, type, product_product: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ product_prod @ X0 @ X1)).
% 2.77/1.84 thf(func_def_46, type, product_scomp: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > product_prod @ X1 @ X2) > (X1 > X2 > X3) > X0 > X3)).
% 2.77/1.84 thf(func_def_47, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 2.77/1.84 thf(func_def_48, type, inv_imagep: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > X1 > X1 > $o)).
% 2.77/1.84 thf(func_def_49, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.77/1.84 thf(func_def_50, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 2.77/1.84 thf(func_def_51, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 2.77/1.84 thf(func_def_52, 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)).
% 2.77/1.84 thf(func_def_53, 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)).
% 2.77/1.84 thf(func_def_54, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.77/1.84 thf(func_def_55, type, p: paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_73, type, sP0: (paraco414474393lle_fm > $o) > $o).
% 2.77/1.84 thf(func_def_74, type, sP1: (paraco414474393lle_fm > $o) > $o).
% 2.77/1.84 thf(func_def_75, type, sK2: paraco414474393lle_fm > list @ char).
% 2.77/1.84 thf(func_def_76, type, sK3: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_77, type, sK4: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_78, type, sK5: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_79, type, sK6: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_80, type, sK7: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_81, type, sK8: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_82, type, sK9: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_83, type, sK10: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.77/1.84 thf(func_def_84, type, sK11: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.77/1.84 thf(func_def_85, type, sK12: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.77/1.84 thf(func_def_86, type, sK13: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.77/1.84 thf(func_def_87, type, sK14: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.77/1.84 thf(func_def_88, type, sK15: (paraco415392788lle_tv > $o) > nat).
% 2.77/1.84 thf(func_def_89, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.77/1.84 thf(func_def_90, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.77/1.84 thf(func_def_91, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.77/1.84 thf(func_def_92, type, sK19: !>[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)).
% 2.77/1.84 thf(func_def_93, type, sK20: !>[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)).
% 2.77/1.84 thf(func_def_94, type, sK21: !>[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)).
% 2.77/1.84 thf(func_def_95, type, sK22: !>[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)).
% 2.77/1.84 thf(func_def_96, type, sK23: !>[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)).
% 2.77/1.84 thf(func_def_97, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.77/1.84 thf(func_def_98, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.77/1.84 thf(func_def_99, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.77/1.84 thf(func_def_100, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.77/1.84 thf(func_def_101, type, sK28: paraco415392788lle_tv > $o).
% 2.77/1.84 thf(func_def_102, type, sK29: paraco415392788lle_tv > nat).
% 2.77/1.84 thf(func_def_103, type, sK30: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char > paraco415392788lle_tv).
% 2.77/1.84 thf(func_def_104, type, sK31: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char > paraco415392788lle_tv).
% 2.77/1.84 thf(func_def_105, type, sK32: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_106, type, sK33: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_107, type, sK34: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char > paraco415392788lle_tv).
% 2.77/1.84 thf(func_def_108, type, sK35: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_109, type, sK36: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_110, type, sK37: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char > paraco415392788lle_tv).
% 2.77/1.84 thf(func_def_111, type, sK38: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char).
% 2.77/1.84 thf(func_def_112, type, sK39: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char > paraco415392788lle_tv).
% 2.77/1.84 thf(func_def_113, type, sK40: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_114, type, sK41: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char > paraco415392788lle_tv).
% 2.77/1.84 thf(func_def_115, type, sK42: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_116, type, sK43: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_117, type, sK44: (paraco415392788lle_tv > $o) > $o).
% 2.77/1.84 thf(func_def_118, type, sK45: (paraco415392788lle_tv > $o) > nat).
% 2.77/1.84 thf(func_def_119, type, sK46: !>[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)).
% 2.77/1.84 thf(func_def_120, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.77/1.84 thf(func_def_121, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.77/1.84 thf(func_def_122, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.77/1.84 thf(func_def_123, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.77/1.84 thf(func_def_124, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.77/1.84 thf(func_def_125, type, sK52: paraco415392788lle_tv > nat).
% 2.77/1.84 thf(func_def_126, type, sK53: !>[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)).
% 2.77/1.84 thf(func_def_127, type, sK54: !>[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)).
% 2.77/1.84 thf(func_def_128, type, sK55: !>[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)).
% 2.77/1.84 thf(func_def_129, type, sK56: !>[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)).
% 2.77/1.84 thf(func_def_130, type, sK57: !>[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)).
% 2.77/1.84 thf(func_def_131, type, sK58: !>[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)).
% 2.77/1.84 thf(func_def_132, type, sK59: !>[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)).
% 2.77/1.84 thf(func_def_133, type, sK60: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_134, type, sK61: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_135, type, sK62: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_136, type, sK63: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_137, type, sK64: (paraco414474393lle_fm > $o) > list @ char).
% 2.77/1.84 thf(func_def_138, type, sK65: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_139, type, sK66: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_140, type, sK67: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.77/1.84 thf(func_def_142, type, ph69: !>[X0: $tType]:(X0)).
% 2.77/1.84 thf(f1611,plain,(
% 2.77/1.84 $false),
% 2.77/1.84 inference(avatar_sat_refutation,[status(thm)],[f950,f1173,f1290,f1389,f1610])).
% 2.77/1.84 thf(f1610,plain,(
% 2.77/1.84 ~spl68_9),
% 2.77/1.84 inference(avatar_contradiction_clause,[status(thm)],[f1609])).
% 2.77/1.84 thf(f1609,plain,(
% 2.77/1.84 $false | ~spl68_9),
% 2.77/1.84 inference(subsumption_resolution,[status(thm)],[f1460,f1122])).
% 2.77/1.84 thf(f1122,plain,(
% 2.77/1.84 ( ! [X1 : $o] : (($false = X1)) ) | ~spl68_9),
% 2.77/1.84 inference(avatar_component_clause,[status(thm)],[f1121])).
% 2.77/1.84 thf(f1121,definition,(
% 2.77/1.84 spl68_9 <=> ! [X1 : $o] : ($false = X1)),
% 2.77/1.84 introduced(definition,[new_symbols(naming,[spl68_9])],[avatar_definition])).
% 2.77/1.84 thf(f1460,plain,(
% 2.77/1.84 ($true != $false) | ~spl68_9),
% 2.77/1.84 inference(forward_demodulation,[status(thm)],[f1441,f1122])).
% 2.77/1.84 thf(f1441,plain,(
% 2.77/1.84 ( ! [X1 : $tType,X0 : $tType,X3 : $tType,X2 : $tType,X4 : product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o] : (($true != (X4 @ (product_Pair @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 @ (sK24 @ X0 @ X1 @ X2 @ X3 @ X4) @ (product_Pair @ X2 @ product_prod @ X1 @ X0 @ (sK25 @ X0 @ X1 @ X2 @ X3 @ X4) @ (product_Pair @ X1 @ X0 @ (sK26 @ X0 @ X1 @ X2 @ X3 @ X4) @ (sK27 @ X0 @ X1 @ X2 @ X3 @ X4))))))) ) | ~spl68_9),
% 2.77/1.84 inference(trivial_inequality_removal,[status(thm)],[f1425])).
% 2.77/1.84 thf(f1425,plain,(
% 2.77/1.84 ( ! [X1 : $tType,X0 : $tType,X3 : $tType,X2 : $tType,X4 : product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o] : (($true != (X4 @ (product_Pair @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 @ (sK24 @ X0 @ X1 @ X2 @ X3 @ X4) @ (product_Pair @ X2 @ product_prod @ X1 @ X0 @ (sK25 @ X0 @ X1 @ X2 @ X3 @ X4) @ (product_Pair @ X1 @ X0 @ (sK26 @ X0 @ X1 @ X2 @ X3 @ X4) @ (sK27 @ X0 @ X1 @ X2 @ X3 @ X4)))))) | ($true = $false)) ) | ~spl68_9),
% 2.77/1.84 inference(backward_demodulation,[status(thm)],[f771,f1122])).
% 2.77/1.84 thf(f771,plain,(
% 2.77/1.84 ( ! [X1 : $tType,X0 : $tType,X3 : $tType,X2 : $tType,X4 : product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o,X5 : product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0] : (($true != (X4 @ (product_Pair @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 @ (sK24 @ X0 @ X1 @ X2 @ X3 @ X4) @ (product_Pair @ X2 @ product_prod @ X1 @ X0 @ (sK25 @ X0 @ X1 @ X2 @ X3 @ X4) @ (product_Pair @ X1 @ X0 @ (sK26 @ X0 @ X1 @ X2 @ X3 @ X4) @ (sK27 @ X0 @ X1 @ X2 @ X3 @ X4)))))) | ($true = (X4 @ X5))) )),
% 2.77/1.84 inference(cnf_transformation,[status(thm)],[f698])).
% 2.77/1.84 thf(f698,plain,(
% 2.77/1.84 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : $tType,X4 : product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o,X5 : product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0] : (($true != (X4 @ (product_Pair @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 @ (sK24 @ X0 @ X1 @ X2 @ X3 @ X4) @ (product_Pair @ X2 @ product_prod @ X1 @ X0 @ (sK25 @ X0 @ X1 @ X2 @ X3 @ X4) @ (product_Pair @ X1 @ X0 @ (sK26 @ X0 @ X1 @ X2 @ X3 @ X4) @ (sK27 @ X0 @ X1 @ X2 @ X3 @ X4)))))) | ($true = (X4 @ X5)))),
% 2.77/1.84 inference(skolemisation,[status(esa),new_symbols(skolem,[sK24,sK25,sK26,sK27])],[f668,f697])).
% 2.77/1.84 thf(f697,plain,(
% 2.77/1.84 ! [X3 : $tType,X2 : $tType,X1 : $tType,X0 : $tType,X4 : product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o] : (? [X6 : X3,X7 : X2,X8 : X1,X9 : X0] : ($true != (X4 @ (product_Pair @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 @ X6 @ (product_Pair @ X2 @ product_prod @ X1 @ X0 @ X7 @ (product_Pair @ X1 @ X0 @ X8 @ X9))))) => ($true != (X4 @ (product_Pair @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 @ (sK24 @ X0 @ X1 @ X2 @ X3 @ X4) @ (product_Pair @ X2 @ product_prod @ X1 @ X0 @ (sK25 @ X0 @ X1 @ X2 @ X3 @ X4) @ (product_Pair @ X1 @ X0 @ (sK26 @ X0 @ X1 @ X2 @ X3 @ X4) @ (sK27 @ X0 @ X1 @ X2 @ X3 @ X4)))))))),
% 2.77/1.84 introduced(definition,[],[choice_axiom])).
% 2.77/1.84 thf(f668,plain,(
% 2.77/1.84 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : $tType,X4 : product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o,X5 : product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0] : (? [X6 : X3,X7 : X2,X8 : X1,X9 : X0] : ($true != (X4 @ (product_Pair @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 @ X6 @ (product_Pair @ X2 @ product_prod @ X1 @ X0 @ X7 @ (product_Pair @ X1 @ X0 @ X8 @ X9))))) | ($true = (X4 @ X5)))),
% 2.77/1.84 inference(ennf_transformation,[status(thm)],[f406])).
% 2.77/1.84 thf(f406,plain,(
% 2.77/1.84 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : $tType,X4 : product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o,X5 : product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0] : (! [X6 : X3,X7 : X2,X8 : X1,X9 : X0] : ($true = (X4 @ (product_Pair @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 @ X6 @ (product_Pair @ X2 @ product_prod @ X1 @ X0 @ X7 @ (product_Pair @ X1 @ X0 @ X8 @ X9))))) => ($true = (X4 @ X5)))),
% 2.77/1.84 inference(fool_elimination,[status(thm)],[f405])).
% 2.77/1.84 thf(f405,plain,(
% 2.77/1.84 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : $tType,X4 : product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o,X5 : product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0] : (! [X6 : X3,X7 : X2,X8 : X1,X9 : X0] : (X4 @ (product_Pair @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 @ X6 @ (product_Pair @ X2 @ product_prod @ X1 @ X0 @ X7 @ (product_Pair @ X1 @ X0 @ X8 @ X9)))) => (X4 @ X5))),
% 2.77/1.84 inference(rectify,[status(thm)],[f73])).
% 2.77/1.84 thf(f73,axiom,(
% 2.77/1.84 ! [X3 : $tType,X2 : $tType,X1 : $tType,X0 : $tType,X16 : product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > $o,X35 : product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3] : (! [X72 : X0,X73 : X1,X74 : X2,X75 : X3] : (X16 @ (product_Pair @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 @ X72 @ (product_Pair @ X1 @ product_prod @ X2 @ X3 @ X73 @ (product_Pair @ X2 @ X3 @ X74 @ X75)))) => (X16 @ X35))),
% 2.77/1.84 file('/export/starexec/sandbox/tmp/tmp.g4D8jloWIn/DTF2THF_16534.p',fact_72_prod__induct4)).
% 2.77/1.84 thf(f1389,plain,(
% 2.77/1.84 spl68_7 | spl68_9 | ~spl68_12),
% 2.77/1.84 inference(avatar_split_clause,[status(thm)],[f1388,f1281,f1121,f1115])).
% 2.77/1.84 thf(f1115,definition,(
% 2.77/1.84 spl68_7 <=> ! [X2 : $o] : ($true = X2)),
% 2.77/1.84 introduced(definition,[new_symbols(naming,[spl68_7])],[avatar_definition])).
% 2.77/1.84 thf(f1281,definition,(
% 2.77/1.84 spl68_12 <=> ! [X3 : $o] : (((paraco2040174112le_Det @ $false) = (paraco2040174112le_Det @ X3)) | ($false = X3))),
% 2.77/1.84 introduced(definition,[new_symbols(naming,[spl68_12])],[avatar_definition])).
% 2.77/1.84 thf(f1388,plain,(
% 2.77/1.84 ( ! [X0 : $o,X1 : $o] : (($true = X0) | ($false = X1)) ) | ~spl68_12),
% 2.77/1.84 inference(subsumption_resolution,[status(thm)],[f1370,f859])).
% 2.77/1.84 thf(f859,plain,(
% 2.77/1.84 ( ! [X0 : $o,X1 : $o] : (($true = X0) | ((paraco2040174112le_Det @ X0) != (paraco2040174112le_Det @ X1)) | ($false = X1)) )),
% 2.77/1.84 inference(equality_proxy_clausification,[status(thm)],[f856])).
% 2.77/1.84 thf(f856,plain,(
% 2.77/1.84 ( ! [X0 : $o,X1 : $o] : ((((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1)) = $false) | ($true = X0) | ($false = X1)) )),
% 2.77/1.84 inference(binary_proxy_clausification,[status(thm)],[f855])).
% 2.77/1.84 thf(f855,plain,(
% 2.77/1.84 ( ! [X0 : $o,X1 : $o] : ((X0 = X1) | (((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1)) = $false)) )),
% 2.77/1.84 inference(equality_proxy_clausification,[status(thm)],[f849])).
% 2.77/1.84 thf(f849,plain,(
% 2.77/1.84 ( ! [X0 : $o,X1 : $o] : (((X0 = X1) = $true) | (((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1)) = $false)) )),
% 2.77/1.84 inference(binary_proxy_clausification,[status(thm)],[f765])).
% 2.77/1.84 thf(f765,plain,(
% 2.77/1.84 ( ! [X0 : $o,X1 : $o] : (((X0 = X1) = ((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1)))) )),
% 2.77/1.84 inference(cnf_transformation,[status(thm)],[f382])).
% 2.77/1.84 thf(f382,plain,(
% 2.77/1.84 ! [X0 : $o,X1 : $o] : ((X0 = X1) = ((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1)))),
% 2.77/1.84 inference(fool_elimination,[status(thm)],[f381])).
% 2.77/1.84 thf(f381,plain,(
% 2.77/1.84 ! [X0 : $o,X1 : $o] : ((X0 = X1) = ((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1)))),
% 2.77/1.84 inference(rectify,[status(thm)],[f1])).
% 2.77/1.84 thf(f1,axiom,(
% 2.77/1.84 ! [X5 : $o,X6 : $o] : ((X5 = X6) = ((paraco2040174112le_Det @ X5) = (paraco2040174112le_Det @ X6)))),
% 2.77/1.84 file('/export/starexec/sandbox/tmp/tmp.g4D8jloWIn/DTF2THF_16534.p',fact_0_tv_Oinject_I1_J)).
% 2.77/1.84 thf(f1370,plain,(
% 2.77/1.84 ( ! [X0 : $o,X1 : $o] : (($true = X0) | ($false = X1) | ((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1))) ) | ~spl68_12),
% 2.77/1.84 inference(trivial_inequality_removal,[status(thm)],[f1354])).
% 2.77/1.84 thf(f1354,plain,(
% 2.77/1.84 ( ! [X0 : $o,X1 : $o] : (($true = $false) | ($true = X0) | ($false = X1) | ((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1))) ) | ~spl68_12),
% 2.77/1.84 inference(superposition,[status(thm)],[f1282,f853])).
% 2.77/1.84 thf(f853,plain,(
% 2.77/1.84 ( ! [X0 : $o,X1 : $o] : (((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1)) | ($true = X0) | ($true = X1)) )),
% 2.77/1.84 inference(binary_proxy_clausification,[status(thm)],[f852])).
% 2.77/1.84 thf(f852,plain,(
% 2.77/1.84 ( ! [X0 : $o,X1 : $o] : (((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1)) | (X0 != X1)) )),
% 2.77/1.84 inference(equality_proxy_clausification,[status(thm)],[f851])).
% 2.77/1.84 thf(f851,plain,(
% 2.77/1.84 ( ! [X0 : $o,X1 : $o] : (($true = ((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1))) | (X0 != X1)) )),
% 2.77/1.84 inference(equality_proxy_clausification,[status(thm)],[f850])).
% 2.77/1.84 thf(f850,plain,(
% 2.77/1.84 ( ! [X0 : $o,X1 : $o] : (((X0 = X1) = $false) | ($true = ((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1)))) )),
% 2.77/1.84 inference(binary_proxy_clausification,[status(thm)],[f765])).
% 2.77/1.84 thf(f1282,plain,(
% 2.77/1.84 ( ! [X3 : $o] : (((paraco2040174112le_Det @ $false) = (paraco2040174112le_Det @ X3)) | ($false = X3)) ) | ~spl68_12),
% 2.77/1.84 inference(avatar_component_clause,[status(thm)],[f1281])).
% 2.77/1.84 thf(f1290,plain,(
% 2.77/1.84 spl68_12 | ~spl68_2),
% 2.77/1.84 inference(avatar_split_clause,[status(thm)],[f1101,f893,f1281])).
% 2.77/1.84 thf(f893,definition,(
% 2.77/1.84 spl68_2 <=> ((paraco2040174112le_Det @ $false) = (paraco2040174112le_Det @ $true))),
% 2.77/1.84 introduced(definition,[new_symbols(naming,[spl68_2])],[avatar_definition])).
% 2.77/1.84 thf(f1101,plain,(
% 2.77/1.84 ( ! [X0 : $o] : (($false = X0) | ((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ $false))) ) | ~spl68_2),
% 2.77/1.84 inference(trivial_inequality_removal,[status(thm)],[f1094])).
% 2.77/1.84 thf(f1094,plain,(
% 2.77/1.84 ( ! [X0 : $o] : (($true = $false) | ((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ $false)) | ($false = X0)) ) | ~spl68_2),
% 2.77/1.84 inference(superposition,[status(thm)],[f894,f854])).
% 2.77/1.84 thf(f854,plain,(
% 2.77/1.84 ( ! [X0 : $o,X1 : $o] : (((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1)) | ($false = X0) | ($false = X1)) )),
% 2.77/1.84 inference(binary_proxy_clausification,[status(thm)],[f852])).
% 2.77/1.84 thf(f894,plain,(
% 2.77/1.84 ((paraco2040174112le_Det @ $false) = (paraco2040174112le_Det @ $true)) | ~spl68_2),
% 2.77/1.84 inference(avatar_component_clause,[status(thm)],[f893])).
% 2.77/1.84 thf(f1173,plain,(
% 2.77/1.84 ~spl68_7),
% 2.77/1.84 inference(avatar_contradiction_clause,[status(thm)],[f1172])).
% 2.77/1.84 thf(f1172,plain,(
% 2.77/1.84 $false | ~spl68_7),
% 2.77/1.84 inference(subsumption_resolution,[status(thm)],[f1161,f1116])).
% 2.77/1.84 thf(f1116,plain,(
% 2.77/1.84 ( ! [X2 : $o] : (($true = X2)) ) | ~spl68_7),
% 2.77/1.84 inference(avatar_component_clause,[status(thm)],[f1115])).
% 2.77/1.84 thf(f1161,plain,(
% 2.77/1.84 ( ! [X0 : paraco414474393lle_fm > $o] : (($true != (X0 @ (paraco2084319816le_Eql @ (sK62 @ X0) @ (sK63 @ X0))))) ) | ~spl68_7),
% 2.77/1.84 inference(trivial_inequality_removal,[status(thm)],[f1150])).
% 2.77/1.84 thf(f1150,plain,(
% 2.77/1.84 ( ! [X0 : paraco414474393lle_fm > $o] : (($true != $true) | ($true != (X0 @ (paraco2084319816le_Eql @ (sK62 @ X0) @ (sK63 @ X0))))) ) | ~spl68_7),
% 2.77/1.84 inference(backward_demodulation,[status(thm)],[f807,f1116])).
% 2.77/1.84 thf(f807,plain,(
% 2.77/1.84 ( ! [X0 : paraco414474393lle_fm > $o] : (($true != (X0 @ (paraco2084319816le_Eql @ (sK62 @ X0) @ (sK63 @ X0)))) | ($true != (sP0 @ X0))) )),
% 2.77/1.84 inference(cnf_transformation,[status(thm)],[f726])).
% 2.77/1.84 thf(f726,plain,(
% 2.77/1.84 ! [X0 : paraco414474393lle_fm > $o] : ((($true = (X0 @ (sK63 @ X0))) & ($true != (X0 @ (paraco2084319816le_Eql @ (sK62 @ X0) @ (sK63 @ X0)))) & ($true = (X0 @ (sK62 @ X0)))) | ($true != (sP0 @ X0)))),
% 2.77/1.84 inference(skolemisation,[status(esa),new_symbols(skolem,[sK62,sK63])],[f724,f725])).
% 2.77/1.84 thf(f725,plain,(
% 2.77/1.84 ! [X0 : paraco414474393lle_fm > $o] : (? [X1 : paraco414474393lle_fm,X2 : paraco414474393lle_fm] : (($true = (X0 @ X2)) & ($true != (X0 @ (paraco2084319816le_Eql @ X1 @ X2))) & ($true = (X0 @ X1))) => (($true = (X0 @ (sK63 @ X0))) & ($true != (X0 @ (paraco2084319816le_Eql @ (sK62 @ X0) @ (sK63 @ X0)))) & ($true = (X0 @ (sK62 @ X0)))))),
% 2.77/1.84 introduced(definition,[],[choice_axiom])).
% 2.77/1.84 thf(f724,plain,(
% 2.77/1.84 ! [X0 : paraco414474393lle_fm > $o] : (? [X1 : paraco414474393lle_fm,X2 : paraco414474393lle_fm] : (($true = (X0 @ X2)) & ($true != (X0 @ (paraco2084319816le_Eql @ X1 @ X2))) & ($true = (X0 @ X1))) | ($true != (sP0 @ X0)))),
% 2.77/1.84 inference(rectify,[status(thm)],[f723])).
% 2.77/1.84 thf(f723,plain,(
% 2.77/1.84 ! [X0 : paraco414474393lle_fm > $o] : (? [X6 : paraco414474393lle_fm,X7 : paraco414474393lle_fm] : (($true = (X0 @ X7)) & ($true != (X0 @ (paraco2084319816le_Eql @ X6 @ X7))) & ($true = (X0 @ X6))) | ($true != (sP0 @ X0)))),
% 2.77/1.84 inference(nnf_transformation,[status(thm)],[f675])).
% 2.77/1.84 thf(f675,plain,(
% 2.77/1.84 ! [X0 : paraco414474393lle_fm > $o] : (? [X6 : paraco414474393lle_fm,X7 : paraco414474393lle_fm] : (($true = (X0 @ X7)) & ($true != (X0 @ (paraco2084319816le_Eql @ X6 @ X7))) & ($true = (X0 @ X6))) | ~($true = (sP0 @ X0)))),
% 2.77/1.84 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 2.77/1.84 thf(f950,plain,(
% 2.77/1.84 spl68_2),
% 2.77/1.84 inference(avatar_contradiction_clause,[status(thm)],[f949])).
% 2.77/1.84 thf(f949,plain,(
% 2.77/1.84 $false | spl68_2),
% 2.77/1.84 inference(subsumption_resolution,[status(thm)],[f948,f895])).
% 2.77/1.84 thf(f895,plain,(
% 2.77/1.84 ((paraco2040174112le_Det @ $false) != (paraco2040174112le_Det @ $true)) | spl68_2),
% 2.77/1.84 inference(avatar_component_clause,[status(thm)],[f893])).
% 2.77/1.84 thf(f948,plain,(
% 2.77/1.84 ((paraco2040174112le_Det @ $false) = (paraco2040174112le_Det @ $true)) | spl68_2),
% 2.77/1.84 inference(backward_demodulation,[status(thm)],[f863,f947])).
% 2.77/1.84 thf(f947,plain,(
% 2.77/1.84 ( ! [X0 : list @ char > paraco415392788lle_tv] : (((paraco876059933e_eval @ X0 @ (paraco329115265le_Neg @ p)) = (paraco2040174112le_Det @ $false))) ) | spl68_2),
% 2.77/1.84 inference(subsumption_resolution,[status(thm)],[f946,f895])).
% 2.77/1.84 thf(f946,plain,(
% 2.77/1.84 ( ! [X0 : list @ char > paraco415392788lle_tv] : (((paraco876059933e_eval @ X0 @ (paraco329115265le_Neg @ p)) = (paraco2040174112le_Det @ $false)) | ((paraco2040174112le_Det @ $false) = (paraco2040174112le_Det @ $true))) )),
% 2.77/1.84 inference(trivial_inequality_removal,[status(thm)],[f942])).
% 2.77/1.84 thf(f942,plain,(
% 2.77/1.84 ( ! [X0 : list @ char > paraco415392788lle_tv] : (((paraco2040174112le_Det @ $true) != (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ X0 @ (paraco329115265le_Neg @ p)) = (paraco2040174112le_Det @ $false)) | ((paraco2040174112le_Det @ $false) = (paraco2040174112le_Det @ $true))) )),
% 2.77/1.84 inference(superposition,[status(thm)],[f792,f747])).
% 2.77/1.84 thf(f747,plain,(
% 2.77/1.84 ( ! [X0 : list @ char > paraco415392788lle_tv] : (((paraco876059933e_eval @ X0 @ p) = (paraco2040174112le_Det @ $true))) )),
% 2.77/1.84 inference(cnf_transformation,[status(thm)],[f661])).
% 2.77/1.84 thf(f661,plain,(
% 2.77/1.84 ! [X0 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X0 @ p) = (paraco2040174112le_Det @ $true))),
% 2.77/1.84 inference(ennf_transformation,[status(thm)],[f312])).
% 2.77/1.84 thf(f312,plain,(
% 2.77/1.84 ~? [X0 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X0 @ p) != (paraco2040174112le_Det @ $true))),
% 2.77/1.84 inference(fool_elimination,[status(thm)],[f311])).
% 2.77/1.84 thf(f311,plain,(
% 2.77/1.84 ~? [X0 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X0 @ p) != (paraco2040174112le_Det @ $true))),
% 2.77/1.84 inference(rectify,[status(thm)],[f261])).
% 2.77/1.84 thf(f261,negated_conjecture,(
% 2.77/1.84 ~? [X142 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X142 @ p) != (paraco2040174112le_Det @ $true))),
% 2.77/1.84 inference(negated_conjecture,[status(cth)],[f260])).
% 2.77/1.84 thf(f260,conjecture,(
% 2.77/1.84 ? [X142 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X142 @ p) != (paraco2040174112le_Det @ $true))),
% 2.77/1.84 file('/export/starexec/sandbox/tmp/tmp.g4D8jloWIn/DTF2THF_16534.p',conj_0)).
% 2.77/1.84 thf(f792,plain,(
% 2.77/1.84 ( ! [X0 : list @ char > paraco415392788lle_tv,X1 : paraco414474393lle_fm] : (((paraco876059933e_eval @ X0 @ X1) != (paraco2040174112le_Det @ $true)) | ((paraco2040174112le_Det @ $false) = (paraco876059933e_eval @ X0 @ X1)) | ((paraco876059933e_eval @ X0 @ (paraco329115265le_Neg @ X1)) = (paraco2040174112le_Det @ $false))) )),
% 2.77/1.84 inference(cnf_transformation,[status(thm)],[f669])).
% 2.77/1.84 thf(f669,plain,(
% 2.77/1.84 ! [X0 : list @ char > paraco415392788lle_tv,X1 : paraco414474393lle_fm] : ((((paraco876059933e_eval @ X0 @ (paraco329115265le_Neg @ X1)) = (paraco2040174112le_Det @ $true)) | ((paraco2040174112le_Det @ $false) != (paraco876059933e_eval @ X0 @ X1))) & (((paraco2040174112le_Det @ $false) = (paraco876059933e_eval @ X0 @ X1)) | ((((paraco876059933e_eval @ X0 @ X1) = (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ X0 @ (paraco329115265le_Neg @ X1)) = (paraco876059933e_eval @ X0 @ X1))) & (((paraco876059933e_eval @ X0 @ X1) != (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ X0 @ (paraco329115265le_Neg @ X1)) = (paraco2040174112le_Det @ $false))))))),
% 2.77/1.84 inference(ennf_transformation,[status(thm)],[f284])).
% 2.77/1.84 thf(f284,plain,(
% 2.77/1.84 ! [X0 : list @ char > paraco415392788lle_tv,X1 : paraco414474393lle_fm] : ((((paraco2040174112le_Det @ $false) != (paraco876059933e_eval @ X0 @ X1)) => ((((paraco876059933e_eval @ X0 @ X1) != (paraco2040174112le_Det @ $true)) => ((paraco876059933e_eval @ X0 @ (paraco329115265le_Neg @ X1)) = (paraco876059933e_eval @ X0 @ X1))) & (((paraco876059933e_eval @ X0 @ X1) = (paraco2040174112le_Det @ $true)) => ((paraco876059933e_eval @ X0 @ (paraco329115265le_Neg @ X1)) = (paraco2040174112le_Det @ $false))))) & (((paraco2040174112le_Det @ $false) = (paraco876059933e_eval @ X0 @ X1)) => ((paraco876059933e_eval @ X0 @ (paraco329115265le_Neg @ X1)) = (paraco2040174112le_Det @ $true))))),
% 2.77/1.84 inference(fool_elimination,[status(thm)],[f283])).
% 2.77/1.84 thf(f283,plain,(
% 2.77/1.84 ! [X0 : list @ char > paraco415392788lle_tv,X1 : paraco414474393lle_fm] : ((((paraco876059933e_eval @ X0 @ X1) != (paraco2040174112le_Det @ $false)) => ((((paraco876059933e_eval @ X0 @ X1) = (paraco2040174112le_Det @ $true)) => ((paraco2040174112le_Det @ $false) = (paraco876059933e_eval @ X0 @ (paraco329115265le_Neg @ X1)))) & (((paraco876059933e_eval @ X0 @ X1) != (paraco2040174112le_Det @ $true)) => ((paraco876059933e_eval @ X0 @ (paraco329115265le_Neg @ X1)) = (paraco876059933e_eval @ X0 @ X1))))) & (((paraco2040174112le_Det @ $false) = (paraco876059933e_eval @ X0 @ X1)) => ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X0 @ (paraco329115265le_Neg @ X1)))))),
% 2.77/1.84 inference(rectify,[status(thm)],[f9])).
% 2.77/1.84 thf(f9,axiom,(
% 2.77/1.84 ! [X9 : list @ char > paraco415392788lle_tv,X10 : paraco414474393lle_fm] : ((((paraco876059933e_eval @ X9 @ X10) != (paraco2040174112le_Det @ $false)) => ((((paraco876059933e_eval @ X9 @ X10) = (paraco2040174112le_Det @ $true)) => ((paraco2040174112le_Det @ $false) = (paraco876059933e_eval @ X9 @ (paraco329115265le_Neg @ X10)))) & (((paraco876059933e_eval @ X9 @ X10) != (paraco2040174112le_Det @ $true)) => ((paraco876059933e_eval @ X9 @ X10) = (paraco876059933e_eval @ X9 @ (paraco329115265le_Neg @ X10)))))) & (((paraco2040174112le_Det @ $false) = (paraco876059933e_eval @ X9 @ X10)) => ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X9 @ (paraco329115265le_Neg @ X10)))))),
% 2.77/1.84 file('/export/starexec/sandbox/tmp/tmp.g4D8jloWIn/DTF2THF_16534.p',fact_8_eval__negation)).
% 2.77/1.84 thf(f863,plain,(
% 2.77/1.84 ( ! [X1 : list @ char > paraco415392788lle_tv] : (((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X1 @ (paraco329115265le_Neg @ p)))) )),
% 2.77/1.84 inference(equality_proxy_clausification,[status(thm)],[f862])).
% 2.77/1.84 thf(f862,plain,(
% 2.77/1.84 ( ! [X1 : list @ char > paraco415392788lle_tv] : (($true = ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X1 @ (paraco329115265le_Neg @ p))))) )),
% 2.77/1.84 inference(beta_eta_normalization,[status(thm)],[f861])).
% 2.77/1.84 thf(f861,plain,(
% 2.77/1.84 ( ! [X1 : list @ char > paraco415392788lle_tv] : (($true = ((^[Y0 : list @ char > paraco415392788lle_tv]: ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ Y0 @ (paraco329115265le_Neg @ p)))) @ X1))) )),
% 2.77/1.84 inference(pi_clausification,[status(thm)],[f860])).
% 2.77/1.84 thf(f860,plain,(
% 2.77/1.84 ($true = (!! @ (list @ char > paraco415392788lle_tv) @ (^[Y0 : list @ char > paraco415392788lle_tv]: ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ Y0 @ (paraco329115265le_Neg @ p))))))),
% 2.77/1.84 inference(beta_eta_normalization,[status(thm)],[f818])).
% 2.77/1.84 thf(f818,plain,(
% 2.77/1.84 ($true = ((^[Y0 : paraco414474393lle_fm]: (!! @ (list @ char > paraco415392788lle_tv) @ (^[Y1 : list @ char > paraco415392788lle_tv]: ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ Y1 @ Y0))))) @ (paraco329115265le_Neg @ p)))),
% 2.77/1.84 inference(definition_unfolding,[status(thm)],[f756,f773])).
% 2.77/1.84 thf(f773,plain,(
% 2.77/1.84 (paraco769098683_valid = (^[Y0 : paraco414474393lle_fm]: (!! @ (list @ char > paraco415392788lle_tv) @ (^[Y1 : list @ char > paraco415392788lle_tv]: ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ Y1 @ Y0))))))),
% 2.77/1.84 inference(cnf_transformation,[status(thm)],[f430])).
% 2.77/1.84 thf(f430,plain,(
% 2.77/1.84 (paraco769098683_valid = (^[Y0 : paraco414474393lle_fm]: (!! @ (list @ char > paraco415392788lle_tv) @ (^[Y1 : list @ char > paraco415392788lle_tv]: ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ Y1 @ Y0))))))),
% 2.77/1.84 inference(fool_elimination,[status(thm)],[f429])).
% 2.77/1.84 thf(f429,plain,(
% 2.77/1.84 ((^[X0 : paraco414474393lle_fm] : (! [X1 : list @ char > paraco415392788lle_tv] : ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X1 @ X0)))) = paraco769098683_valid)),
% 2.77/1.84 inference(rectify,[status(thm)],[f3])).
% 2.77/1.84 thf(f3,axiom,(
% 2.77/1.84 ((^[X7 : paraco414474393lle_fm] : (! [X8 : list @ char > paraco415392788lle_tv] : ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X8 @ X7)))) = paraco769098683_valid)),
% 2.77/1.84 file('/export/starexec/sandbox/tmp/tmp.g4D8jloWIn/DTF2THF_16534.p',fact_2_valid__def)).
% 2.77/1.84 thf(f756,plain,(
% 2.77/1.84 ((paraco769098683_valid @ (paraco329115265le_Neg @ p)) = $true)),
% 2.77/1.84 inference(cnf_transformation,[status(thm)],[f594])).
% 2.77/1.84 thf(f594,plain,(
% 2.77/1.84 ((paraco769098683_valid @ (paraco329115265le_Neg @ p)) = $true)),
% 2.77/1.84 inference(fool_elimination,[status(thm)],[f593])).
% 2.77/1.84 thf(f593,plain,(
% 2.77/1.84 (paraco769098683_valid @ (paraco329115265le_Neg @ p))),
% 2.77/1.84 inference(rectify,[status(thm)],[f2])).
% 2.77/1.84 thf(f2,axiom,(
% 2.77/1.84 (paraco769098683_valid @ (paraco329115265le_Neg @ p))),
% 2.77/1.84 file('/export/starexec/sandbox/tmp/tmp.g4D8jloWIn/DTF2THF_16534.p',fact_1_assms)).
% 2.77/1.84 % SZS output end Proof for DTF2THF_16534
% 2.77/1.84 % (16892)------------------------------
% 2.77/1.84 % (16892)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.84 % (16892)Termination reason: Refutation
% 2.77/1.84
% 2.77/1.84 % (16892)Memory used [KB]: 7675
% 2.77/1.84 % (16892)Time elapsed: 0.107 s
% 2.77/1.84 % (16892)Instructions burned: 306 (million)
% 2.77/1.84 % (16892)------------------------------
% 2.77/1.84 % (16892)------------------------------
% 2.77/1.84 % (16862)Success in time 0.257 s
%------------------------------------------------------------------------------