%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP140^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:52 AM UTC 2026
% Result : Theorem 2.81s 1.68s
% Output : Refutation 2.81s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP140^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n015.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 08:27:46 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.46 thf(spec,logic,$$dhol).
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % File : ITP140^2 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.46 % Domain : Interactive Theorem Proving
% 0.20/0.46 % Problem : Sledgehammer Paraconsistency problem prob_947__3275214_1
% 0.20/0.46 % Version : Especial.
% 0.20/0.46 % English :
% 0.20/0.46
% 0.20/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.46 % Source : [Des21]
% 0.20/0.46 % Names : Paraconsistency/prob_947__3275214_1 [Des21]
% 0.20/0.46
% 0.20/0.46 % Status : Theorem
% 0.20/0.46 % Rating : 0.00 v7.5.0
% 0.20/0.46 % Syntax : Number of formulae : 343 ( 149 unt; 66 typ; 0 def)
% 0.20/0.46 % Number of atoms : 685 ( 354 equ; 0 cnn)
% 0.20/0.46 % Maximal formula atoms : 16 ( 2 avg)
% 0.20/0.46 % Number of connectives : 4244 ( 114 ~; 6 |; 38 &;3759 @)
% 0.20/0.46 % ( 0 <=>; 327 =>; 0 <=; 0 <~>)
% 0.20/0.46 % Maximal formula depth : 28 ( 8 avg)
% 0.20/0.46 % Number of types : 5 ( 4 usr)
% 0.20/0.46 % Number of type conns : 533 ( 533 >; 0 *; 0 +; 0 <<)
% 0.20/0.46 % Number of symbols : 65 ( 62 usr; 5 con; 0-8 aty)
% 0.20/0.46 % Number of variables : 1340 ( 44 ^;1206 !; 16 ?;1340 :)
% 0.20/0.46 % ( 74 !>; 0 ?*; 0 @-; 0 @+)
% 0.20/0.46 % SPC : TH1_THM_EQU_NAR
% 0.20/0.46
% 0.20/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:24:15.916
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % Could-be-implicit typings (7)
% 0.20/0.46 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv,type,
% 0.20/0.46 paraco415392788lle_tv: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm,type,
% 0.20/0.46 paraco414474393lle_fm: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.20/0.46 product_prod: $tType > $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_String_Ochar,type,
% 0.20/0.46 char: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_List_Olist,type,
% 0.20/0.46 list: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Set_Oset,type,
% 0.20/0.46 set: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Nat_Onat,type,
% 0.20/0.46 nat: $tType ).
% 0.20/0.46
% 0.20/0.46 % Explicit typings (59)
% 0.20/0.46 thf(sy_cl_HOL_Otype,type,
% 0.20/0.46 type:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.20/0.46 ord:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Otop,type,
% 0.20/0.46 top:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.20/0.46 order:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.20/0.46 linorder:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.20/0.46 preorder:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_BNF__Def_OfstOp,type,
% 0.20/0.46 bNF_fstOp:
% 0.20/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_BNF__Def_Opick__middlep,type,
% 0.20/0.46 bNF_pick_middlep:
% 0.20/0.46 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_BNF__Def_OsndOp,type,
% 0.20/0.46 bNF_sndOp:
% 0.20/0.46 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 0.20/0.46 bNF_Greatest_image2:
% 0.20/0.46 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on,type,
% 0.20/0.46 inj_on:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Othe__inv__into,type,
% 0.20/0.46 the_inv_into:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_If,type,
% 0.20/0.46 if:
% 0.20/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.20/0.46 ord_less_eq:
% 0.20/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 0.20/0.46 order_strict_mono:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.20/0.46 top_top:
% 0.20/0.46 !>[A: $tType] : A ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__int,type,
% 0.20/0.46 paraco753463838ge_int: ( nat > nat ) > ( ( list @ char ) > paraco415392788lle_tv ) > ( list @ char ) > paraco415392788lle_tv ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv,type,
% 0.20/0.46 paraco1920534163nge_tv: ( nat > nat ) > paraco415392788lle_tv > paraco415392788lle_tv ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv__rel,type,
% 0.20/0.46 paraco2077297190tv_rel: ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) > ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval,type,
% 0.20/0.46 paraco876059933e_eval: ( ( list @ char ) > paraco415392788lle_tv ) > paraco414474393lle_fm > paraco415392788lle_tv ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H,type,
% 0.20/0.46 paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql,type,
% 0.20/0.46 paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H,type,
% 0.20/0.46 paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H,type,
% 0.20/0.46 paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro,type,
% 0.20/0.46 paraco27778325le_Pro: ( list @ char ) > paraco414474393lle_fm ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth,type,
% 0.20/0.46 paraco251304083_Truth: paraco414474393lle_fm ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Ocase__fm,type,
% 0.20/0.46 paraco1246693743ase_fm:
% 0.20/0.46 !>[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 ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Orec__fm,type,
% 0.20/0.46 paraco815940159rec_fm:
% 0.20/0.46 !>[A: $tType] : ( ( ( list @ char ) > A ) > A > ( paraco414474393lle_fm > A > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A > A > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A > A > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A > A > A ) > paraco414474393lle_fm > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet,type,
% 0.20/0.46 paraco2040174112le_Det: $o > paraco415392788lle_tv ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet,type,
% 0.20/0.46 paraco676387099_Indet: nat > paraco415392788lle_tv ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Ocase__tv,type,
% 0.20/0.46 paraco490622181ase_tv:
% 0.20/0.46 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv,type,
% 0.20/0.46 paraco152590079rec_tv:
% 0.20/0.46 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid,type,
% 0.20/0.46 paraco769098683_valid: paraco414474393lle_fm > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid__in,type,
% 0.20/0.46 paraco2086025920lid_in: ( set @ nat ) > paraco414474393lle_fm > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Product__Type_OPair,type,
% 0.20/0.46 product_Pair:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Product__Type_Oapfst,type,
% 0.20/0.46 product_apfst:
% 0.20/0.46 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Product__Type_Oapsnd,type,
% 0.20/0.46 product_apsnd:
% 0.20/0.46 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Product__Type_Obool_Ocase__bool,type,
% 0.20/0.46 product_case_bool:
% 0.20/0.46 !>[A: $tType] : ( A > A > $o > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Product__Type_Ocurry,type,
% 0.20/0.46 product_curry:
% 0.20/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.20/0.46 produc2004651681e_prod:
% 0.20/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Product__Type_Omap__prod,type,
% 0.20/0.46 product_map_prod:
% 0.20/0.46 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.20/0.46 product_rec_prod:
% 0.20/0.46 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.20/0.46 product_fst:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.20/0.46 product_snd:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.20/0.46 product_swap:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Relation_Oinv__image,type,
% 0.20/0.46 inv_image:
% 0.20/0.46 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Relation_Oinv__imagep,type,
% 0.20/0.46 inv_imagep:
% 0.20/0.46 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Relation_Ototal__on,type,
% 0.20/0.46 total_on:
% 0.20/0.46 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_OCollect,type,
% 0.20/0.46 collect:
% 0.20/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oimage,type,
% 0.20/0.46 image:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Wellfounded_Oaccp,type,
% 0.20/0.46 accp:
% 0.20/0.46 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Wellfounded_Olex__prod,type,
% 0.20/0.46 lex_prod:
% 0.20/0.46 !>[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 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Wellfounded_Owf,type,
% 0.20/0.46 wf:
% 0.20/0.46 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Wfrec_Osame__fst,type,
% 0.20/0.46 same_fst:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_member,type,
% 0.20/0.46 member:
% 0.20/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_f,type,
% 0.20/0.46 f: nat > nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_i,type,
% 0.20/0.46 i: ( list @ char ) > paraco415392788lle_tv ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_p1____,type,
% 0.20/0.46 p1: paraco414474393lle_fm ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_p2____,type,
% 0.20/0.46 p2: paraco414474393lle_fm ).
% 0.20/0.46
% 0.20/0.46 % Relevant facts (256)
% 0.20/0.46 thf(fact_0_a_H_H,axiom,
% 0.20/0.46 ( ( paraco876059933e_eval @ i @ p1 )
% 0.20/0.46 != ( paraco2040174112le_Det @ $true ) ) ).
% 0.20/0.46
% 0.20/0.46 % a''
% 0.20/0.46 thf(fact_1_a,axiom,
% 0.20/0.46 ( ( paraco876059933e_eval @ i @ p2 )
% 0.20/0.46 = ( paraco2040174112le_Det @ $true ) ) ).
% 0.20/0.46
% 0.20/0.46 % a
% 0.20/0.46 thf(fact_2_a_H,axiom,
% 0.20/0.46 ( ( paraco876059933e_eval @ i @ p1 )
% 0.20/0.46 != ( paraco876059933e_eval @ i @ p2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % a'
% 0.20/0.46 thf(fact_3_fm_Oinject_I3_J,axiom,
% 0.20/0.46 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,Y41: paraco414474393lle_fm,Y42: paraco414474393lle_fm] :
% 0.20/0.46 ( ( ( paraco2100061555le_Con @ X41 @ X42 )
% 0.20/0.46 = ( paraco2100061555le_Con @ Y41 @ Y42 ) )
% 0.20/0.46 = ( ( X41 = Y41 )
% 0.20/0.46 & ( X42 = Y42 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.inject(3)
% 0.20/0.46 thf(fact_4_b_H,axiom,
% 0.20/0.46 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
% 0.20/0.46 != ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % b'
% 0.20/0.46 thf(fact_5_eval_Osimps_I4_J,axiom,
% 0.20/0.46 ! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.20/0.46 ( ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.46 = ( paraco876059933e_eval @ I @ Q ) )
% 0.20/0.46 => ( ( paraco876059933e_eval @ I @ ( paraco2100061555le_Con @ P @ Q ) )
% 0.20/0.46 = ( paraco876059933e_eval @ I @ P ) ) )
% 0.20/0.46 & ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.46 != ( paraco876059933e_eval @ I @ Q ) )
% 0.20/0.46 => ( ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.46 = ( paraco2040174112le_Det @ $true ) )
% 0.20/0.46 => ( ( paraco876059933e_eval @ I @ ( paraco2100061555le_Con @ P @ Q ) )
% 0.20/0.46 = ( paraco876059933e_eval @ I @ Q ) ) )
% 0.20/0.46 & ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.46 != ( paraco2040174112le_Det @ $true ) )
% 0.20/0.46 => ( ( ( ( paraco876059933e_eval @ I @ Q )
% 0.20/0.46 = ( paraco2040174112le_Det @ $true ) )
% 0.20/0.46 => ( ( paraco876059933e_eval @ I @ ( paraco2100061555le_Con @ P @ Q ) )
% 0.20/0.46 = ( paraco876059933e_eval @ I @ P ) ) )
% 0.20/0.46 & ( ( ( paraco876059933e_eval @ I @ Q )
% 0.20/0.46 != ( paraco2040174112le_Det @ $true ) )
% 0.20/0.46 => ( ( paraco876059933e_eval @ I @ ( paraco2100061555le_Con @ P @ Q ) )
% 0.20/0.46 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % eval.simps(4)
% 0.20/0.46 thf(fact_6_ih2,axiom,
% 0.20/0.46 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 )
% 0.20/0.46 = ( paraco1920534163nge_tv @ f @ ( paraco876059933e_eval @ i @ p2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ih2
% 0.20/0.46 thf(fact_7_ih1,axiom,
% 0.20/0.46 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
% 0.20/0.46 = ( paraco1920534163nge_tv @ f @ ( paraco876059933e_eval @ i @ p1 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ih1
% 0.20/0.46 thf(fact_8_b_H_H,axiom,
% 0.20/0.46 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
% 0.20/0.46 != ( paraco2040174112le_Det @ $true ) ) ).
% 0.20/0.46
% 0.20/0.46 % b''
% 0.20/0.46 thf(fact_9_tv_Oinject_I1_J,axiom,
% 0.20/0.46 ! [X1: $o,Y1: $o] :
% 0.20/0.46 ( ( ( paraco2040174112le_Det @ X1 )
% 0.20/0.46 = ( paraco2040174112le_Det @ Y1 ) )
% 0.20/0.46 = ( X1 = Y1 ) ) ).
% 0.20/0.46
% 0.20/0.46 % tv.inject(1)
% 0.20/0.46 thf(fact_10_conjunction,axiom,
% 0.20/0.46 ! [P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.20/0.46 ( ( paraco769098683_valid @ ( paraco2100061555le_Con @ P @ Q ) )
% 0.20/0.46 = ( ( paraco769098683_valid @ P )
% 0.20/0.46 & ( paraco769098683_valid @ Q ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % conjunction
% 0.20/0.46 thf(fact_11_conjunction__in,axiom,
% 0.20/0.46 ! [U: set @ nat,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.20/0.46 ( ( paraco2086025920lid_in @ U @ ( paraco2100061555le_Con @ P @ Q ) )
% 0.20/0.46 = ( ( paraco2086025920lid_in @ U @ P )
% 0.20/0.46 & ( paraco2086025920lid_in @ U @ Q ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % conjunction_in
% 0.20/0.46 thf(fact_12_fm_Osimps_I39_J,axiom,
% 0.20/0.46 ! [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] :
% 0.20/0.46 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2100061555le_Con @ X41 @ X42 ) )
% 0.20/0.46 = ( F4 @ X41 @ X42 ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.simps(39)
% 0.20/0.46 thf(fact_13_fm_Odistinct_I13_J,axiom,
% 0.20/0.46 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.20/0.46 ( paraco251304083_Truth
% 0.20/0.46 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.distinct(13)
% 0.20/0.46 thf(fact_14_fm_Osimps_I45_J,axiom,
% 0.20/0.46 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.20/0.46 ( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2100061555le_Con @ X41 @ X42 ) )
% 0.20/0.46 = ( F4 @ X41 @ X42 @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X41 ) @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X42 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.simps(45)
% 0.20/0.46 thf(fact_15_fm_Odistinct_I25_J,axiom,
% 0.20/0.46 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.20/0.46 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 0.20/0.46 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.distinct(25)
% 0.20/0.46 thf(fact_16_fm_Oinject_I4_J,axiom,
% 0.20/0.46 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,Y51: paraco414474393lle_fm,Y52: paraco414474393lle_fm] :
% 0.20/0.46 ( ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 0.20/0.46 = ( paraco2084319816le_Eql @ Y51 @ Y52 ) )
% 0.20/0.46 = ( ( X51 = Y51 )
% 0.20/0.46 & ( X52 = Y52 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.inject(4)
% 0.20/0.46 thf(fact_17_fm_Osimps_I46_J,axiom,
% 0.20/0.46 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.20/0.46 ( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2084319816le_Eql @ X51 @ X52 ) )
% 0.20/0.46 = ( F5 @ X51 @ X52 @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X51 ) @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X52 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.simps(46)
% 0.20/0.46 thf(fact_18_fm_Osimps_I43_J,axiom,
% 0.20/0.46 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A] :
% 0.20/0.46 ( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ paraco251304083_Truth )
% 0.20/0.46 = F2 ) ).
% 0.20/0.46
% 0.20/0.46 % fm.simps(43)
% 0.20/0.46 thf(fact_19_fm_Osimps_I40_J,axiom,
% 0.20/0.46 ! [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] :
% 0.20/0.46 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2084319816le_Eql @ X51 @ X52 ) )
% 0.20/0.46 = ( F5 @ X51 @ X52 ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.simps(40)
% 0.20/0.46 thf(fact_20_fm_Osimps_I37_J,axiom,
% 0.20/0.46 ! [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] :
% 0.20/0.46 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ paraco251304083_Truth )
% 0.20/0.46 = F2 ) ).
% 0.20/0.46
% 0.20/0.46 % fm.simps(37)
% 0.20/0.46 thf(fact_21_fm_Odistinct_I15_J,axiom,
% 0.20/0.46 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.20/0.46 ( paraco251304083_Truth
% 0.20/0.46 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.distinct(15)
% 0.20/0.46 thf(fact_22_change__tv_Osimps_I1_J,axiom,
% 0.20/0.46 ! [F: nat > nat,B2: $o] :
% 0.20/0.46 ( ( paraco1920534163nge_tv @ F @ ( paraco2040174112le_Det @ B2 ) )
% 0.20/0.46 = ( paraco2040174112le_Det @ B2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % change_tv.simps(1)
% 0.20/0.46 thf(fact_23_transfer,axiom,
% 0.20/0.46 ! [U: set @ nat,P: paraco414474393lle_fm] :
% 0.20/0.46 ( ~ ( paraco2086025920lid_in @ U @ P )
% 0.20/0.46 => ~ ( paraco769098683_valid @ P ) ) ).
% 0.20/0.46
% 0.20/0.46 % transfer
% 0.20/0.46 thf(fact_24_change__int__def,axiom,
% 0.20/0.46 ( paraco753463838ge_int
% 0.20/0.46 = ( ^ [F7: nat > nat,I2: ( list @ char ) > paraco415392788lle_tv,S: list @ char] : ( paraco1920534163nge_tv @ F7 @ ( I2 @ S ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % change_int_def
% 0.20/0.46 thf(fact_25_valid__valid__in,axiom,
% 0.20/0.46 ! [P: paraco414474393lle_fm,U: set @ nat] :
% 0.20/0.46 ( ( paraco769098683_valid @ P )
% 0.20/0.46 => ( paraco2086025920lid_in @ U @ P ) ) ).
% 0.20/0.46
% 0.20/0.46 % valid_valid_in
% 0.20/0.46 thf(fact_26_eval__equality__simplify,axiom,
% 0.20/0.46 ! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.20/0.46 ( ( paraco876059933e_eval @ I @ ( paraco2084319816le_Eql @ P @ Q ) )
% 0.20/0.46 = ( paraco2040174112le_Det
% 0.20/0.46 @ ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.46 = ( paraco876059933e_eval @ I @ Q ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % eval_equality_simplify
% 0.20/0.46 thf(fact_27_eval_Osimps_I5_J,axiom,
% 0.20/0.46 ! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.20/0.46 ( ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.46 = ( paraco876059933e_eval @ I @ Q ) )
% 0.20/0.46 => ( ( paraco876059933e_eval @ I @ ( paraco2084319816le_Eql @ P @ Q ) )
% 0.20/0.46 = ( paraco2040174112le_Det @ $true ) ) )
% 0.20/0.46 & ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.46 != ( paraco876059933e_eval @ I @ Q ) )
% 0.20/0.46 => ( ( paraco876059933e_eval @ I @ ( paraco2084319816le_Eql @ P @ Q ) )
% 0.20/0.46 = ( paraco2040174112le_Det @ $false ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % eval.simps(5)
% 0.20/0.46 thf(fact_28_eval_Osimps_I2_J,axiom,
% 0.20/0.46 ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 0.20/0.46 ( ( paraco876059933e_eval @ I @ paraco251304083_Truth )
% 0.20/0.46 = ( paraco2040174112le_Det @ $true ) ) ).
% 0.20/0.46
% 0.20/0.46 % eval.simps(2)
% 0.20/0.46 thf(fact_29_valid__def,axiom,
% 0.20/0.46 ( paraco769098683_valid
% 0.20/0.46 = ( ^ [P2: paraco414474393lle_fm] :
% 0.20/0.46 ! [I2: ( list @ char ) > paraco415392788lle_tv] :
% 0.20/0.46 ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.20/0.46 = ( paraco2040174112le_Det @ $true ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % valid_def
% 0.20/0.46 thf(fact_30_tv_Osimps_I7_J,axiom,
% 0.20/0.46 ! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
% 0.20/0.46 ( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
% 0.20/0.46 = ( F1 @ X1 ) ) ).
% 0.20/0.46
% 0.20/0.46 % tv.simps(7)
% 0.20/0.46 thf(fact_31_change__tv_Oelims,axiom,
% 0.20/0.46 ! [X: nat > nat,Xa: paraco415392788lle_tv,Y: paraco415392788lle_tv] :
% 0.20/0.46 ( ( ( paraco1920534163nge_tv @ X @ Xa )
% 0.20/0.46 = Y )
% 0.20/0.46 => ( ! [B3: $o] :
% 0.20/0.46 ( ( Xa
% 0.20/0.46 = ( paraco2040174112le_Det @ B3 ) )
% 0.20/0.46 => ( Y
% 0.20/0.46 != ( paraco2040174112le_Det @ B3 ) ) )
% 0.20/0.46 => ~ ! [N: nat] :
% 0.20/0.46 ( ( Xa
% 0.20/0.46 = ( paraco676387099_Indet @ N ) )
% 0.20/0.46 => ( Y
% 0.20/0.46 != ( paraco676387099_Indet @ ( X @ N ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % change_tv.elims
% 0.20/0.46 thf(fact_32_tv_Osimps_I5_J,axiom,
% 0.20/0.46 ! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
% 0.20/0.46 ( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
% 0.20/0.46 = ( F1 @ X1 ) ) ).
% 0.20/0.46
% 0.20/0.46 % tv.simps(5)
% 0.20/0.46 thf(fact_33_eval__negation,axiom,
% 0.20/0.46 ! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] :
% 0.20/0.46 ( ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.46 = ( paraco2040174112le_Det @ $false ) )
% 0.20/0.46 => ( ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) )
% 0.20/0.46 = ( paraco2040174112le_Det @ $true ) ) )
% 0.20/0.46 & ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.46 != ( paraco2040174112le_Det @ $false ) )
% 0.20/0.46 => ( ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.46 = ( paraco2040174112le_Det @ $true ) )
% 0.20/0.46 => ( ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) )
% 0.20/0.46 = ( paraco2040174112le_Det @ $false ) ) )
% 0.20/0.46 & ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.46 != ( paraco2040174112le_Det @ $true ) )
% 0.20/0.46 => ( ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) )
% 0.20/0.46 = ( paraco876059933e_eval @ I @ P ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % eval_negation
% 0.20/0.46 thf(fact_34_assms,axiom,
% 0.20/0.46 inj_on @ nat @ nat @ f @ ( top_top @ ( set @ nat ) ) ).
% 0.20/0.46
% 0.20/0.46 % assms
% 0.20/0.46 thf(fact_35_change__tv_Osimps_I2_J,axiom,
% 0.20/0.46 ! [F: nat > nat,N2: nat] :
% 0.20/0.46 ( ( paraco1920534163nge_tv @ F @ ( paraco676387099_Indet @ N2 ) )
% 0.20/0.46 = ( paraco676387099_Indet @ ( F @ N2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % change_tv.simps(2)
% 0.20/0.46 thf(fact_36_fm_Osimps_I41_J,axiom,
% 0.20/0.46 ! [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,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.20/0.46 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco1628874225le_Eql @ X61 @ X62 ) )
% 0.20/0.46 = ( F6 @ X61 @ X62 ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.simps(41)
% 0.20/0.46 thf(fact_37_fm_Osimps_I47_J,axiom,
% 0.20/0.46 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.20/0.46 ( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco1628874225le_Eql @ X61 @ X62 ) )
% 0.20/0.46 = ( F6 @ X61 @ X62 @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X61 ) @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X62 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.simps(47)
% 0.20/0.46 thf(fact_38_fm_Osimps_I38_J,axiom,
% 0.20/0.46 ! [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] :
% 0.20/0.46 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco329115265le_Neg @ X3 ) )
% 0.20/0.46 = ( F3 @ X3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.simps(38)
% 0.20/0.46 thf(fact_39_fm_Odistinct_I17_J,axiom,
% 0.20/0.46 ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.20/0.46 ( paraco251304083_Truth
% 0.20/0.46 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.distinct(17)
% 0.20/0.46 thf(fact_40_fm_Oinject_I2_J,axiom,
% 0.20/0.46 ! [X3: paraco414474393lle_fm,Y3: paraco414474393lle_fm] :
% 0.20/0.46 ( ( ( paraco329115265le_Neg @ X3 )
% 0.20/0.46 = ( paraco329115265le_Neg @ Y3 ) )
% 0.20/0.46 = ( X3 = Y3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.inject(2)
% 0.20/0.46 thf(fact_41_fm_Oinject_I5_J,axiom,
% 0.20/0.46 ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm,Y61: paraco414474393lle_fm,Y62: paraco414474393lle_fm] :
% 0.20/0.46 ( ( ( paraco1628874225le_Eql @ X61 @ X62 )
% 0.20/0.46 = ( paraco1628874225le_Eql @ Y61 @ Y62 ) )
% 0.20/0.46 = ( ( X61 = Y61 )
% 0.20/0.46 & ( X62 = Y62 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.inject(5)
% 0.20/0.46 thf(fact_42_tv_Oinject_I2_J,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat] :
% 0.20/0.46 ( ( ( paraco676387099_Indet @ X2 )
% 0.20/0.46 = ( paraco676387099_Indet @ Y2 ) )
% 0.20/0.46 = ( X2 = Y2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % tv.inject(2)
% 0.20/0.46 thf(fact_43_fm_Odistinct_I23_J,axiom,
% 0.20/0.46 ! [X3: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.20/0.46 ( ( paraco329115265le_Neg @ X3 )
% 0.20/0.46 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 0.20/0.46
% 0.20/0.46 % fm.distinct(23)
% 0.20/0.46 thf(fact_44_tv_Osimps_I6_J,axiom,
% 0.20/0.46 ! [A: $tType,F1: $o > A,F2: nat > A,X2: nat] :
% 0.20/0.46 ( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X2 ) )
% 0.20/0.46 = ( F2 @ X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % tv.simps(6)
% 0.20/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,P3: A > $o] :
% 0.20/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 0.20/0.47 = ( P3 @ A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A: $tType,A3: set @ A] :
% 0.20/0.47 ( ( collect @ A
% 0.20/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 0.20/0.47 = A3 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_47_Collect__cong,axiom,
% 0.20/0.47 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 0.20/0.47 ( ! [X5: A] :
% 0.20/0.47 ( ( P3 @ X5 )
% 0.20/0.47 = ( Q2 @ X5 ) )
% 0.20/0.47 => ( ( collect @ A @ P3 )
% 0.20/0.47 = ( collect @ A @ Q2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_cong
% 0.20/0.47 thf(fact_48_ext,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.20/0.47 ( ! [X5: A] :
% 0.20/0.47 ( ( F @ X5 )
% 0.20/0.47 = ( G @ X5 ) )
% 0.20/0.47 => ( F = G ) ) ).
% 0.20/0.47
% 0.20/0.47 % ext
% 0.20/0.47 thf(fact_49_tv_Osimps_I8_J,axiom,
% 0.20/0.47 ! [A: $tType,F1: $o > A,F2: nat > A,X2: nat] :
% 0.20/0.47 ( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X2 ) )
% 0.20/0.47 = ( F2 @ X2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % tv.simps(8)
% 0.20/0.47 thf(fact_50_eval__equality,axiom,
% 0.20/0.47 ! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.20/0.47 ( ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.47 = ( paraco876059933e_eval @ I @ Q ) )
% 0.20/0.47 => ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
% 0.20/0.47 = ( paraco2040174112le_Det @ $true ) ) )
% 0.20/0.47 & ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.47 != ( paraco876059933e_eval @ I @ Q ) )
% 0.20/0.47 => ( ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.47 = ( paraco2040174112le_Det @ $true ) )
% 0.20/0.47 => ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
% 0.20/0.47 = ( paraco876059933e_eval @ I @ Q ) ) )
% 0.20/0.47 & ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.20/0.47 => ( ( ( ( paraco876059933e_eval @ I @ Q )
% 0.20/0.47 = ( paraco2040174112le_Det @ $true ) )
% 0.20/0.47 => ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
% 0.20/0.47 = ( paraco876059933e_eval @ I @ P ) ) )
% 0.20/0.47 & ( ( ( paraco876059933e_eval @ I @ Q )
% 0.20/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.20/0.47 => ( ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.47 = ( paraco2040174112le_Det @ $false ) )
% 0.20/0.47 => ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
% 0.20/0.47 = ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ Q ) ) ) )
% 0.20/0.47 & ( ( ( paraco876059933e_eval @ I @ P )
% 0.20/0.47 != ( paraco2040174112le_Det @ $false ) )
% 0.20/0.47 => ( ( ( ( paraco876059933e_eval @ I @ Q )
% 0.20/0.47 = ( paraco2040174112le_Det @ $false ) )
% 0.20/0.47 => ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
% 0.20/0.47 = ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) ) ) )
% 0.20/0.47 & ( ( ( paraco876059933e_eval @ I @ Q )
% 0.20/0.47 != ( paraco2040174112le_Det @ $false ) )
% 0.20/0.47 => ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
% 0.20/0.47 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eval_equality
% 0.20/0.47 thf(fact_51_double__negation,axiom,
% 0.20/0.47 ( paraco876059933e_eval
% 0.20/0.47 = ( ^ [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm] : ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ ( paraco329115265le_Neg @ P2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % double_negation
% 0.20/0.47 thf(fact_52_fm_Odistinct_I19_J,axiom,
% 0.20/0.47 ! [X3: paraco414474393lle_fm,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.20/0.47 ( ( paraco329115265le_Neg @ X3 )
% 0.20/0.47 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.distinct(19)
% 0.20/0.47 thf(fact_53_string__tv_Oinduct,axiom,
% 0.20/0.47 ! [P3: paraco415392788lle_tv > $o,A0: paraco415392788lle_tv] :
% 0.20/0.47 ( ( P3 @ ( paraco2040174112le_Det @ $true ) )
% 0.20/0.47 => ( ( P3 @ ( paraco2040174112le_Det @ $false ) )
% 0.20/0.47 => ( ! [N: nat] : ( P3 @ ( paraco676387099_Indet @ N ) )
% 0.20/0.47 => ( P3 @ A0 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % string_tv.induct
% 0.20/0.47 thf(fact_54_change__tv_Oinduct,axiom,
% 0.20/0.47 ! [P3: ( nat > nat ) > paraco415392788lle_tv > $o,A0: nat > nat,A1: paraco415392788lle_tv] :
% 0.20/0.47 ( ! [F8: nat > nat,B3: $o] : ( P3 @ F8 @ ( paraco2040174112le_Det @ B3 ) )
% 0.20/0.47 => ( ! [F8: nat > nat,N: nat] : ( P3 @ F8 @ ( paraco676387099_Indet @ N ) )
% 0.20/0.47 => ( P3 @ A0 @ A1 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % change_tv.induct
% 0.20/0.47 thf(fact_55_string__tv_Ocases,axiom,
% 0.20/0.47 ! [X: paraco415392788lle_tv] :
% 0.20/0.47 ( ( X
% 0.20/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.20/0.47 => ( ( X
% 0.20/0.47 != ( paraco2040174112le_Det @ $false ) )
% 0.20/0.47 => ~ ! [N: nat] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( paraco676387099_Indet @ N ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % string_tv.cases
% 0.20/0.47 thf(fact_56_tv_Oexhaust,axiom,
% 0.20/0.47 ! [Y: paraco415392788lle_tv] :
% 0.20/0.47 ( ! [X12: $o] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( paraco2040174112le_Det @ X12 ) )
% 0.20/0.47 => ~ ! [X22: nat] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( paraco676387099_Indet @ X22 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % tv.exhaust
% 0.20/0.47 thf(fact_57_tv_Oinduct,axiom,
% 0.20/0.47 ! [P3: paraco415392788lle_tv > $o,Tv: paraco415392788lle_tv] :
% 0.20/0.47 ( ! [X5: $o] : ( P3 @ ( paraco2040174112le_Det @ X5 ) )
% 0.20/0.47 => ( ! [X5: nat] : ( P3 @ ( paraco676387099_Indet @ X5 ) )
% 0.20/0.47 => ( P3 @ Tv ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % tv.induct
% 0.20/0.47 thf(fact_58_tv_Odistinct_I1_J,axiom,
% 0.20/0.47 ! [X1: $o,X2: nat] :
% 0.20/0.47 ( ( paraco2040174112le_Det @ X1 )
% 0.20/0.47 != ( paraco676387099_Indet @ X2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % tv.distinct(1)
% 0.20/0.47 thf(fact_59_fm_Odistinct_I21_J,axiom,
% 0.20/0.47 ! [X3: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.20/0.47 ( ( paraco329115265le_Neg @ X3 )
% 0.20/0.47 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.distinct(21)
% 0.20/0.47 thf(fact_60_fm_Odistinct_I11_J,axiom,
% 0.20/0.47 ! [X3: paraco414474393lle_fm] :
% 0.20/0.47 ( paraco251304083_Truth
% 0.20/0.47 != ( paraco329115265le_Neg @ X3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.distinct(11)
% 0.20/0.47 thf(fact_61_fm_Odistinct_I27_J,axiom,
% 0.20/0.47 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.20/0.47 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 0.20/0.47 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.distinct(27)
% 0.20/0.47 thf(fact_62_fm_Osimps_I44_J,axiom,
% 0.20/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X3: paraco414474393lle_fm] :
% 0.20/0.47 ( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco329115265le_Neg @ X3 ) )
% 0.20/0.47 = ( F3 @ X3 @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.simps(44)
% 0.20/0.47 thf(fact_63_fm_Odistinct_I29_J,axiom,
% 0.20/0.47 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.20/0.47 ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 0.20/0.47 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.distinct(29)
% 0.20/0.47 thf(fact_64_iso__tuple__UNIV__I,axiom,
% 0.20/0.47 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % iso_tuple_UNIV_I
% 0.20/0.47 thf(fact_65_UNIV__I,axiom,
% 0.20/0.47 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % UNIV_I
% 0.20/0.47 thf(fact_66_top__apply,axiom,
% 0.20/0.47 ! [C: $tType,D: $tType] :
% 0.20/0.47 ( ( top @ C )
% 0.20/0.47 => ( ( top_top @ ( D > C ) )
% 0.20/0.47 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % top_apply
% 0.20/0.47 thf(fact_67_eval_Osimps_I3_J,axiom,
% 0.20/0.47 ! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] :
% 0.20/0.47 ( ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) )
% 0.20/0.47 = ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I @ P ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eval.simps(3)
% 0.20/0.47 thf(fact_68_inj__def,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.20/0.47 = ( ! [X4: A,Y4: A] :
% 0.20/0.47 ( ( ( F @ X4 )
% 0.20/0.47 = ( F @ Y4 ) )
% 0.20/0.47 => ( X4 = Y4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_def
% 0.20/0.47 thf(fact_69_inj__eq,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.20/0.47 => ( ( ( F @ X )
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 = ( X = Y ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_eq
% 0.20/0.47 thf(fact_70_injI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B] :
% 0.20/0.47 ( ! [X5: A,Y5: A] :
% 0.20/0.47 ( ( ( F @ X5 )
% 0.20/0.47 = ( F @ Y5 ) )
% 0.20/0.47 => ( X5 = Y5 ) )
% 0.20/0.47 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % injI
% 0.20/0.47 thf(fact_71_injD,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.20/0.47 => ( ( ( F @ X )
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 => ( X = Y ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % injD
% 0.20/0.47 thf(fact_72_top__set__def,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( top_top @ ( set @ A ) )
% 0.20/0.47 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % top_set_def
% 0.20/0.47 thf(fact_73_change__tv__injection,axiom,
% 0.20/0.47 ! [F: nat > nat] :
% 0.20/0.47 ( ( inj_on @ nat @ nat @ F @ ( top_top @ ( set @ nat ) ) )
% 0.20/0.47 => ( inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ ( paraco1920534163nge_tv @ F ) @ ( top_top @ ( set @ paraco415392788lle_tv ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % change_tv_injection
% 0.20/0.47 thf(fact_74_UNIV__eq__I,axiom,
% 0.20/0.47 ! [A: $tType,A3: set @ A] :
% 0.20/0.47 ( ! [X5: A] : ( member @ A @ X5 @ A3 )
% 0.20/0.47 => ( ( top_top @ ( set @ A ) )
% 0.20/0.47 = A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % UNIV_eq_I
% 0.20/0.47 thf(fact_75_UNIV__witness,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ? [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % UNIV_witness
% 0.20/0.47 thf(fact_76_inj__onD,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A,Y: A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( ( F @ X )
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 => ( ( member @ A @ X @ A3 )
% 0.20/0.47 => ( ( member @ A @ Y @ A3 )
% 0.20/0.47 => ( X = Y ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_onD
% 0.20/0.47 thf(fact_77_inj__onI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A3: set @ A,F: A > B] :
% 0.20/0.47 ( ! [X5: A,Y5: A] :
% 0.20/0.47 ( ( member @ A @ X5 @ A3 )
% 0.20/0.47 => ( ( member @ A @ Y5 @ A3 )
% 0.20/0.47 => ( ( ( F @ X5 )
% 0.20/0.47 = ( F @ Y5 ) )
% 0.20/0.47 => ( X5 = Y5 ) ) ) )
% 0.20/0.47 => ( inj_on @ A @ B @ F @ A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_onI
% 0.20/0.47 thf(fact_78_inj__on__def,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( inj_on @ A @ B )
% 0.20/0.47 = ( ^ [F7: A > B,A4: set @ A] :
% 0.20/0.47 ! [X4: A] :
% 0.20/0.47 ( ( member @ A @ X4 @ A4 )
% 0.20/0.47 => ! [Y4: A] :
% 0.20/0.47 ( ( member @ A @ Y4 @ A4 )
% 0.20/0.47 => ( ( ( F7 @ X4 )
% 0.20/0.47 = ( F7 @ Y4 ) )
% 0.20/0.47 => ( X4 = Y4 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_def
% 0.20/0.47 thf(fact_79_inj__on__cong,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A3: set @ A,F: A > B,G: A > B] :
% 0.20/0.47 ( ! [A5: A] :
% 0.20/0.47 ( ( member @ A @ A5 @ A3 )
% 0.20/0.47 => ( ( F @ A5 )
% 0.20/0.47 = ( G @ A5 ) ) )
% 0.20/0.47 => ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 = ( inj_on @ A @ B @ G @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_cong
% 0.20/0.47 thf(fact_80_inj__on__eq__iff,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A,Y: A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( member @ A @ X @ A3 )
% 0.20/0.47 => ( ( member @ A @ Y @ A3 )
% 0.20/0.47 => ( ( ( F @ X )
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 = ( X = Y ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_eq_iff
% 0.20/0.47 thf(fact_81_inj__on__contraD,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A,Y: A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( X != Y )
% 0.20/0.47 => ( ( member @ A @ X @ A3 )
% 0.20/0.47 => ( ( member @ A @ Y @ A3 )
% 0.20/0.47 => ( ( F @ X )
% 0.20/0.47 != ( F @ Y ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_contraD
% 0.20/0.47 thf(fact_82_inj__on__inverseI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A3: set @ A,G: B > A,F: A > B] :
% 0.20/0.47 ( ! [X5: A] :
% 0.20/0.47 ( ( member @ A @ X5 @ A3 )
% 0.20/0.47 => ( ( G @ ( F @ X5 ) )
% 0.20/0.47 = X5 ) )
% 0.20/0.47 => ( inj_on @ A @ B @ F @ A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_inverseI
% 0.20/0.47 thf(fact_83_fm_Oinduct,axiom,
% 0.20/0.47 ! [P3: paraco414474393lle_fm > $o,Fm: paraco414474393lle_fm] :
% 0.20/0.47 ( ! [X5: list @ char] : ( P3 @ ( paraco27778325le_Pro @ X5 ) )
% 0.20/0.47 => ( ( P3 @ paraco251304083_Truth )
% 0.20/0.47 => ( ! [X5: paraco414474393lle_fm] :
% 0.20/0.47 ( ( P3 @ X5 )
% 0.20/0.47 => ( P3 @ ( paraco329115265le_Neg @ X5 ) ) )
% 0.20/0.47 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 0.20/0.47 ( ( P3 @ X1a )
% 0.20/0.47 => ( ( P3 @ X22 )
% 0.20/0.47 => ( P3 @ ( paraco2100061555le_Con @ X1a @ X22 ) ) ) )
% 0.20/0.47 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 0.20/0.47 ( ( P3 @ X1a )
% 0.20/0.47 => ( ( P3 @ X22 )
% 0.20/0.47 => ( P3 @ ( paraco2084319816le_Eql @ X1a @ X22 ) ) ) )
% 0.20/0.47 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 0.20/0.47 ( ( P3 @ X1a )
% 0.20/0.47 => ( ( P3 @ X22 )
% 0.20/0.47 => ( P3 @ ( paraco1628874225le_Eql @ X1a @ X22 ) ) ) )
% 0.20/0.47 => ( P3 @ Fm ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.induct
% 0.20/0.47 thf(fact_84_fm_Oexhaust,axiom,
% 0.20/0.47 ! [Y: paraco414474393lle_fm] :
% 0.20/0.47 ( ! [X12: list @ char] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( paraco27778325le_Pro @ X12 ) )
% 0.20/0.47 => ( ( Y != paraco251304083_Truth )
% 0.20/0.47 => ( ! [X32: paraco414474393lle_fm] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( paraco329115265le_Neg @ X32 ) )
% 0.20/0.47 => ( ! [X412: paraco414474393lle_fm,X422: paraco414474393lle_fm] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( paraco2100061555le_Con @ X412 @ X422 ) )
% 0.20/0.47 => ( ! [X512: paraco414474393lle_fm,X522: paraco414474393lle_fm] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( paraco2084319816le_Eql @ X512 @ X522 ) )
% 0.20/0.47 => ~ ! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.exhaust
% 0.20/0.47 thf(fact_85_props_Ocases,axiom,
% 0.20/0.47 ! [X: paraco414474393lle_fm] :
% 0.20/0.47 ( ( X != paraco251304083_Truth )
% 0.20/0.47 => ( ! [S2: list @ char] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( paraco27778325le_Pro @ S2 ) )
% 0.20/0.47 => ( ! [P4: paraco414474393lle_fm] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( paraco329115265le_Neg @ P4 ) )
% 0.20/0.47 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( paraco2100061555le_Con @ P4 @ Q3 ) )
% 0.20/0.47 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( paraco2084319816le_Eql @ P4 @ Q3 ) )
% 0.20/0.47 => ~ ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % props.cases
% 0.20/0.47 thf(fact_86_props_Oinduct,axiom,
% 0.20/0.47 ! [P3: paraco414474393lle_fm > $o,A0: paraco414474393lle_fm] :
% 0.20/0.47 ( ( P3 @ paraco251304083_Truth )
% 0.20/0.47 => ( ! [S2: list @ char] : ( P3 @ ( paraco27778325le_Pro @ S2 ) )
% 0.20/0.47 => ( ! [P4: paraco414474393lle_fm] :
% 0.20/0.47 ( ( P3 @ P4 )
% 0.20/0.47 => ( P3 @ ( paraco329115265le_Neg @ P4 ) ) )
% 0.20/0.47 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.20/0.47 ( ( P3 @ P4 )
% 0.20/0.47 => ( ( P3 @ Q3 )
% 0.20/0.47 => ( P3 @ ( paraco2100061555le_Con @ P4 @ Q3 ) ) ) )
% 0.20/0.47 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.20/0.47 ( ( P3 @ P4 )
% 0.20/0.47 => ( ( P3 @ Q3 )
% 0.20/0.47 => ( P3 @ ( paraco2084319816le_Eql @ P4 @ Q3 ) ) ) )
% 0.20/0.47 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.20/0.47 ( ( P3 @ P4 )
% 0.20/0.47 => ( ( P3 @ Q3 )
% 0.20/0.47 => ( P3 @ ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) )
% 0.20/0.47 => ( P3 @ A0 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % props.induct
% 0.20/0.47 thf(fact_87_old_Obool_Osimps_I4_J,axiom,
% 0.20/0.47 ! [A: $tType,F1: A,F2: A] :
% 0.20/0.47 ( ( product_case_bool @ A @ F1 @ F2 @ $false )
% 0.20/0.47 = F2 ) ).
% 0.20/0.47
% 0.20/0.47 % old.bool.simps(4)
% 0.20/0.47 thf(fact_88_old_Obool_Osimps_I3_J,axiom,
% 0.20/0.47 ! [A: $tType,F1: A,F2: A] :
% 0.20/0.47 ( ( product_case_bool @ A @ F1 @ F2 @ $true )
% 0.20/0.47 = F1 ) ).
% 0.20/0.47
% 0.20/0.47 % old.bool.simps(3)
% 0.20/0.47 thf(fact_89_bool_Osplit__sel,axiom,
% 0.20/0.47 ! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
% 0.20/0.47 ( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 0.20/0.47 = ( ( Bool
% 0.20/0.47 => ( P3 @ F1 ) )
% 0.20/0.47 & ( ~ Bool
% 0.20/0.47 => ( P3 @ F2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bool.split_sel
% 0.20/0.47 thf(fact_90_fm_Oinject_I1_J,axiom,
% 0.20/0.47 ! [X1: list @ char,Y1: list @ char] :
% 0.20/0.47 ( ( ( paraco27778325le_Pro @ X1 )
% 0.20/0.47 = ( paraco27778325le_Pro @ Y1 ) )
% 0.20/0.47 = ( X1 = Y1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.inject(1)
% 0.20/0.47 thf(fact_91_top1I,axiom,
% 0.20/0.47 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 0.20/0.47
% 0.20/0.47 % top1I
% 0.20/0.47 thf(fact_92_eval_Osimps_I1_J,axiom,
% 0.20/0.47 ! [I: ( list @ char ) > paraco415392788lle_tv,S3: list @ char] :
% 0.20/0.47 ( ( paraco876059933e_eval @ I @ ( paraco27778325le_Pro @ S3 ) )
% 0.20/0.47 = ( I @ S3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % eval.simps(1)
% 0.20/0.47 thf(fact_93_fm_Odistinct_I3_J,axiom,
% 0.20/0.47 ! [X1: list @ char,X3: paraco414474393lle_fm] :
% 0.20/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.20/0.47 != ( paraco329115265le_Neg @ X3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.distinct(3)
% 0.20/0.47 thf(fact_94_fm_Odistinct_I5_J,axiom,
% 0.20/0.47 ! [X1: list @ char,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.20/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.20/0.47 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.distinct(5)
% 0.20/0.47 thf(fact_95_fm_Odistinct_I9_J,axiom,
% 0.20/0.47 ! [X1: list @ char,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.20/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.20/0.47 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.distinct(9)
% 0.20/0.47 thf(fact_96_fm_Odistinct_I7_J,axiom,
% 0.20/0.47 ! [X1: list @ char,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.20/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.20/0.47 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.distinct(7)
% 0.20/0.47 thf(fact_97_fm_Odistinct_I1_J,axiom,
% 0.20/0.47 ! [X1: list @ char] :
% 0.20/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.20/0.47 != paraco251304083_Truth ) ).
% 0.20/0.47
% 0.20/0.47 % fm.distinct(1)
% 0.20/0.47 thf(fact_98_fm_Osimps_I42_J,axiom,
% 0.20/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X1: list @ char] :
% 0.20/0.47 ( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco27778325le_Pro @ X1 ) )
% 0.20/0.47 = ( F1 @ X1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.simps(42)
% 0.20/0.47 thf(fact_99_fm_Osimps_I36_J,axiom,
% 0.20/0.47 ! [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] :
% 0.20/0.47 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco27778325le_Pro @ X1 ) )
% 0.20/0.47 = ( F1 @ X1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % fm.simps(36)
% 0.20/0.47 thf(fact_100_bool_Osplit__sel__asm,axiom,
% 0.20/0.47 ! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
% 0.20/0.47 ( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 0.20/0.47 = ( ~ ( ( Bool
% 0.20/0.47 & ~ ( P3 @ F1 ) )
% 0.20/0.47 | ( ~ Bool
% 0.20/0.47 & ~ ( P3 @ F2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bool.split_sel_asm
% 0.20/0.47 thf(fact_101_bool_Ocase__distrib,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,H: A > B,F1: A,F2: A,Bool: $o] :
% 0.20/0.47 ( ( H @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 0.20/0.47 = ( product_case_bool @ B @ ( H @ F1 ) @ ( H @ F2 ) @ Bool ) ) ).
% 0.20/0.47
% 0.20/0.47 % bool.case_distrib
% 0.20/0.47 thf(fact_102_bool_Ocase__eq__if,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( product_case_bool @ A )
% 0.20/0.47 = ( ^ [F12: A,F22: A,Bool2: $o] : ( if @ A @ Bool2 @ F12 @ F22 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bool.case_eq_if
% 0.20/0.47 thf(fact_103_top__empty__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( top_top @ ( A > $o ) )
% 0.20/0.47 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % top_empty_eq
% 0.20/0.47 thf(fact_104_eval_Ocases,axiom,
% 0.20/0.47 ! [X: product_prod @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm] :
% 0.20/0.47 ( ! [I3: ( list @ char ) > paraco415392788lle_tv,S2: list @ char] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco27778325le_Pro @ S2 ) ) )
% 0.20/0.47 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ paraco251304083_Truth ) )
% 0.20/0.47 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco329115265le_Neg @ P4 ) ) )
% 0.20/0.47 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2100061555le_Con @ P4 @ Q3 ) ) )
% 0.20/0.47 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2084319816le_Eql @ P4 @ Q3 ) ) )
% 0.20/0.47 => ~ ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eval.cases
% 0.20/0.47 thf(fact_105_top__conj_I1_J,axiom,
% 0.20/0.47 ! [A: $tType,X: A,P3: $o] :
% 0.20/0.47 ( ( ( top_top @ ( A > $o ) @ X )
% 0.20/0.47 & P3 )
% 0.20/0.47 = P3 ) ).
% 0.20/0.47
% 0.20/0.47 % top_conj(1)
% 0.20/0.47 thf(fact_106_top__conj_I2_J,axiom,
% 0.20/0.47 ! [A: $tType,P3: $o,X: A] :
% 0.20/0.47 ( ( P3
% 0.20/0.47 & ( top_top @ ( A > $o ) @ X ) )
% 0.20/0.47 = P3 ) ).
% 0.20/0.47
% 0.20/0.47 % top_conj(2)
% 0.20/0.47 thf(fact_107_inj__apsnd,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,F: B > C] :
% 0.20/0.47 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.20/0.47 = ( inj_on @ B @ C @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_apsnd
% 0.20/0.47 thf(fact_108_inj__apfst,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,F: A > C] :
% 0.20/0.47 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.20/0.47 = ( inj_on @ A @ C @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_apfst
% 0.20/0.47 thf(fact_109_old_Oprod_Oinject,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
% 0.20/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.20/0.47 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 0.20/0.47 = ( ( A2 = A6 )
% 0.20/0.47 & ( B2 = B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inject
% 0.20/0.47 thf(fact_110_prod_Oinject,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 0.20/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 0.20/0.47 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.20/0.47 = ( ( X1 = Y1 )
% 0.20/0.47 & ( X2 = Y2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod.inject
% 0.20/0.47 thf(fact_111_apfst__conv,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y: B] :
% 0.20/0.47 ( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y ) )
% 0.20/0.47 = ( product_Pair @ A @ B @ ( F @ X ) @ Y ) ) ).
% 0.20/0.47
% 0.20/0.47 % apfst_conv
% 0.20/0.47 thf(fact_112_apsnd__conv,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y: C] :
% 0.20/0.47 ( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y ) )
% 0.20/0.47 = ( product_Pair @ A @ B @ X @ ( F @ Y ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % apsnd_conv
% 0.20/0.47 thf(fact_113_apsnd__apfst__commute,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P: product_prod @ D @ C] :
% 0.20/0.47 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P ) )
% 0.20/0.47 = ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % apsnd_apfst_commute
% 0.20/0.47 thf(fact_114_old_Oprod_Oinducts,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.20/0.47 ( ! [A5: A,B3: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.20/0.47 => ( P3 @ Prod ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inducts
% 0.20/0.47 thf(fact_115_old_Oprod_Oexhaust,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.20/0.47 ~ ! [A5: A,B3: B] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( product_Pair @ A @ B @ A5 @ B3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.exhaust
% 0.20/0.47 thf(fact_116_prod__induct7,axiom,
% 0.20/0.47 ! [G2: $tType,F9: $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 @ F9 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) )] :
% 0.20/0.47 ( ! [A5: A,B3: B,C2: C,D2: D,E2: E,F8: F9,G3: G2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F9 @ G2 ) @ E2 @ ( product_Pair @ F9 @ G2 @ F8 @ G3 ) ) ) ) ) ) )
% 0.20/0.47 => ( P3 @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_induct7
% 0.20/0.47 thf(fact_117_prod__induct6,axiom,
% 0.20/0.47 ! [F9: $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 @ F9 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) )] :
% 0.20/0.47 ( ! [A5: A,B3: B,C2: C,D2: D,E2: E,F8: F9] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F9 ) @ D2 @ ( product_Pair @ E @ F9 @ E2 @ F8 ) ) ) ) ) )
% 0.20/0.47 => ( P3 @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_induct6
% 0.20/0.47 thf(fact_118_prod__induct5,axiom,
% 0.20/0.47 ! [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,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.20/0.47 ( ! [A5: A,B3: 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 ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.20/0.47 => ( P3 @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_induct5
% 0.20/0.47 thf(fact_119_prod__induct4,axiom,
% 0.20/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.20/0.47 ( ! [A5: A,B3: B,C2: C,D2: D] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.20/0.47 => ( P3 @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_induct4
% 0.20/0.47 thf(fact_120_prod__induct3,axiom,
% 0.20/0.47 ! [C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.20/0.47 ( ! [A5: A,B3: B,C2: C] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C2 ) ) )
% 0.20/0.47 => ( P3 @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_induct3
% 0.20/0.47 thf(fact_121_prod__cases7,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F9: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) )] :
% 0.20/0.47 ~ ! [A5: A,B3: B,C2: C,D2: D,E2: E,F8: F9,G3: G2] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F9 @ G2 ) @ E2 @ ( product_Pair @ F9 @ G2 @ F8 @ G3 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases7
% 0.20/0.47 thf(fact_122_prod__cases6,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F9: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) )] :
% 0.20/0.47 ~ ! [A5: A,B3: B,C2: C,D2: D,E2: E,F8: F9] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F9 ) @ D2 @ ( product_Pair @ E @ F9 @ E2 @ F8 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases6
% 0.20/0.47 thf(fact_123_prod__cases5,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.20/0.47 ~ ! [A5: A,B3: B,C2: C,D2: D,E2: E] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases5
% 0.20/0.47 thf(fact_124_prod__cases4,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.20/0.47 ~ ! [A5: A,B3: B,C2: C,D2: D] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases4
% 0.20/0.47 thf(fact_125_prod__cases3,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.20/0.47 ~ ! [A5: A,B3: B,C2: C] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases3
% 0.20/0.47 thf(fact_126_Pair__inject,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
% 0.20/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.20/0.47 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 0.20/0.47 => ~ ( ( A2 = A6 )
% 0.20/0.47 => ( B2 != B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Pair_inject
% 0.20/0.47 thf(fact_127_prod__cases,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 0.20/0.47 ( ! [A5: A,B3: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.20/0.47 => ( P3 @ P ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases
% 0.20/0.47 thf(fact_128_surj__pair,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 0.20/0.47 ? [X5: A,Y5: B] :
% 0.20/0.47 ( P
% 0.20/0.47 = ( product_Pair @ A @ B @ X5 @ Y5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % surj_pair
% 0.20/0.47 thf(fact_129_old_Oprod_Orec,axiom,
% 0.20/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.20/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.20/0.47 = ( F1 @ A2 @ B2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.rec
% 0.20/0.47 thf(fact_130_internal__case__prod__conv,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 0.20/0.47 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.20/0.47 = ( C3 @ A2 @ B2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % internal_case_prod_conv
% 0.20/0.47 thf(fact_131_prod_Oinj__map,axiom,
% 0.20/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F2: B > D] :
% 0.20/0.47 ( ( inj_on @ A @ C @ F1 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.47 => ( ( inj_on @ B @ D @ F2 @ ( top_top @ ( set @ B ) ) )
% 0.20/0.47 => ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod.inj_map
% 0.20/0.47 thf(fact_132_ssubst__Pair__rhs,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,R: A,S3: B,R2: set @ ( product_prod @ A @ B ),S4: B] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R2 )
% 0.20/0.47 => ( ( S4 = S3 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S4 ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ssubst_Pair_rhs
% 0.20/0.47 thf(fact_133_curry__conv,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.20/0.47 ( ( product_curry @ B @ C @ A )
% 0.20/0.47 = ( ^ [F7: ( product_prod @ B @ C ) > A,A7: B,B5: C] : ( F7 @ ( product_Pair @ B @ C @ A7 @ B5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % curry_conv
% 0.20/0.47 thf(fact_134_curryI,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.20/0.47 ( ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.20/0.47 => ( product_curry @ A @ B @ $o @ F @ A2 @ B2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % curryI
% 0.20/0.47 thf(fact_135_map__prod__simp,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A2: C,B2: D] :
% 0.20/0.47 ( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 0.20/0.47 = ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % map_prod_simp
% 0.20/0.47 thf(fact_136_curryD,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.20/0.47 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 0.20/0.47 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % curryD
% 0.20/0.47 thf(fact_137_curryE,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.20/0.47 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 0.20/0.47 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % curryE
% 0.20/0.47 thf(fact_138_change__tv_Ocases,axiom,
% 0.20/0.47 ! [X: product_prod @ ( nat > nat ) @ paraco415392788lle_tv] :
% 0.20/0.47 ( ! [F8: nat > nat,B3: $o] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ F8 @ ( paraco2040174112le_Det @ B3 ) ) )
% 0.20/0.47 => ~ ! [F8: nat > nat,N: nat] :
% 0.20/0.47 ( X
% 0.20/0.47 != ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ F8 @ ( paraco676387099_Indet @ N ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % change_tv.cases
% 0.20/0.47 thf(fact_139_in__inv__imagep,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( inv_imagep @ A @ B )
% 0.20/0.47 = ( ^ [R3: A > A > $o,F7: B > A,X4: B,Y4: B] : ( R3 @ ( F7 @ X4 ) @ ( F7 @ Y4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % in_inv_imagep
% 0.20/0.47 thf(fact_140_the__inv__f__f,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.20/0.47 => ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X ) )
% 0.20/0.47 = X ) ) ).
% 0.20/0.47
% 0.20/0.47 % the_inv_f_f
% 0.20/0.47 thf(fact_141_change__tv_Opelims,axiom,
% 0.20/0.47 ! [X: nat > nat,Xa: paraco415392788lle_tv,Y: paraco415392788lle_tv] :
% 0.20/0.47 ( ( ( paraco1920534163nge_tv @ X @ Xa )
% 0.20/0.47 = Y )
% 0.20/0.47 => ( ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ Xa ) )
% 0.20/0.47 => ( ! [B3: $o] :
% 0.20/0.47 ( ( Xa
% 0.20/0.47 = ( paraco2040174112le_Det @ B3 ) )
% 0.20/0.47 => ( ( Y
% 0.20/0.47 = ( paraco2040174112le_Det @ B3 ) )
% 0.20/0.47 => ~ ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ ( paraco2040174112le_Det @ B3 ) ) ) ) )
% 0.20/0.47 => ~ ! [N: nat] :
% 0.20/0.47 ( ( Xa
% 0.20/0.47 = ( paraco676387099_Indet @ N ) )
% 0.20/0.47 => ( ( Y
% 0.20/0.47 = ( paraco676387099_Indet @ ( X @ N ) ) )
% 0.20/0.47 => ~ ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ ( paraco676387099_Indet @ N ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % change_tv.pelims
% 0.20/0.47 thf(fact_142_in__inv__image,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R @ F ) )
% 0.20/0.47 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % in_inv_image
% 0.20/0.47 thf(fact_143_the__inv__into__f__f,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( member @ A @ X @ A3 )
% 0.20/0.47 => ( ( the_inv_into @ A @ B @ A3 @ F @ ( F @ X ) )
% 0.20/0.47 = X ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % the_inv_into_f_f
% 0.20/0.47 thf(fact_144_the__inv__into__f__eq,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A,Y: B] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( ( F @ X )
% 0.20/0.47 = Y )
% 0.20/0.47 => ( ( member @ A @ X @ A3 )
% 0.20/0.47 => ( ( the_inv_into @ A @ B @ A3 @ F @ Y )
% 0.20/0.47 = X ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % the_inv_into_f_eq
% 0.20/0.47 thf(fact_145_total__inv__image,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,R: set @ ( product_prod @ B @ B )] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.20/0.47 => ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ R )
% 0.20/0.47 => ( total_on @ A @ ( top_top @ ( set @ A ) ) @ ( inv_image @ B @ A @ R @ F ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % total_inv_image
% 0.20/0.47 thf(fact_146_the__inv__into__onto,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A3 @ F ) @ ( image @ A @ B @ F @ A3 ) )
% 0.20/0.47 = A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % the_inv_into_onto
% 0.20/0.47 thf(fact_147_map__prod__imageI,axiom,
% 0.20/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),F: A > C,G: B > D] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 0.20/0.47 => ( 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 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % map_prod_imageI
% 0.20/0.47 thf(fact_148_inj__swap,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A3 ) ).
% 0.20/0.47
% 0.20/0.47 % inj_swap
% 0.20/0.47 thf(fact_149_image__eqI,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,B2: A,F: B > A,X: B,A3: set @ B] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ X ) )
% 0.20/0.47 => ( ( member @ B @ X @ A3 )
% 0.20/0.47 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_eqI
% 0.20/0.47 thf(fact_150_swap__swap,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 0.20/0.47 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 0.20/0.47 = P ) ).
% 0.20/0.47
% 0.20/0.47 % swap_swap
% 0.20/0.47 thf(fact_151_swap__simp,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,X: B,Y: A] :
% 0.20/0.47 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.20/0.47 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % swap_simp
% 0.20/0.47 thf(fact_152_pair__in__swap__image,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,Y: A,X: B,A3: set @ ( product_prod @ B @ A )] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A3 ) )
% 0.20/0.47 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % pair_in_swap_image
% 0.20/0.47 thf(fact_153_surj__swap,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.20/0.47 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % surj_swap
% 0.20/0.47 thf(fact_154_total__on__def,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( total_on @ A )
% 0.20/0.47 = ( ^ [A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ! [X4: A] :
% 0.20/0.47 ( ( member @ A @ X4 @ A4 )
% 0.20/0.47 => ! [Y4: A] :
% 0.20/0.47 ( ( member @ A @ Y4 @ A4 )
% 0.20/0.47 => ( ( X4 != Y4 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 )
% 0.20/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R3 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % total_on_def
% 0.20/0.47 thf(fact_155_total__onI,axiom,
% 0.20/0.47 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ! [X5: A,Y5: A] :
% 0.20/0.47 ( ( member @ A @ X5 @ A3 )
% 0.20/0.47 => ( ( member @ A @ Y5 @ A3 )
% 0.20/0.47 => ( ( X5 != Y5 )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y5 ) @ R )
% 0.20/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R ) ) ) ) )
% 0.20/0.47 => ( total_on @ A @ A3 @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % total_onI
% 0.20/0.47 thf(fact_156_map__prod__surj,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
% 0.20/0.47 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.20/0.47 = ( top_top @ ( set @ B ) ) )
% 0.20/0.47 => ( ( ( image @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
% 0.20/0.47 = ( top_top @ ( set @ D ) ) )
% 0.20/0.47 => ( ( 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 ) ) ) )
% 0.20/0.47 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % map_prod_surj
% 0.20/0.47 thf(fact_157_range__eqI,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,B2: A,F: B > A,X: B] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ X ) )
% 0.20/0.47 => ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % range_eqI
% 0.20/0.47 thf(fact_158_surj__def,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: B > A] :
% 0.20/0.47 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.20/0.47 = ( top_top @ ( set @ A ) ) )
% 0.20/0.47 = ( ! [Y4: A] :
% 0.20/0.47 ? [X4: B] :
% 0.20/0.47 ( Y4
% 0.20/0.47 = ( F @ X4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % surj_def
% 0.20/0.47 thf(fact_159_rangeI,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % rangeI
% 0.20/0.47 thf(fact_160_surjI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 0.20/0.47 ( ! [X5: A] :
% 0.20/0.47 ( ( G @ ( F @ X5 ) )
% 0.20/0.47 = X5 )
% 0.20/0.47 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 0.20/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % surjI
% 0.20/0.47 thf(fact_161_surjE,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.20/0.47 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.20/0.47 = ( top_top @ ( set @ A ) ) )
% 0.20/0.47 => ~ ! [X5: B] :
% 0.20/0.47 ( Y
% 0.20/0.47 != ( F @ X5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % surjE
% 0.20/0.47 thf(fact_162_surjD,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.20/0.47 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.20/0.47 = ( top_top @ ( set @ A ) ) )
% 0.20/0.47 => ? [X5: B] :
% 0.20/0.47 ( Y
% 0.20/0.47 = ( F @ X5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % surjD
% 0.20/0.47 thf(fact_163_inj__on__image__iff,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A3: set @ A,G: A > B,F: A > A] :
% 0.20/0.47 ( ! [X5: A] :
% 0.20/0.47 ( ( member @ A @ X5 @ A3 )
% 0.20/0.47 => ! [Xa2: A] :
% 0.20/0.47 ( ( member @ A @ Xa2 @ A3 )
% 0.20/0.47 => ( ( ( G @ ( F @ X5 ) )
% 0.20/0.47 = ( G @ ( F @ Xa2 ) ) )
% 0.20/0.47 = ( ( G @ X5 )
% 0.20/0.47 = ( G @ Xa2 ) ) ) ) )
% 0.20/0.47 => ( ( inj_on @ A @ A @ F @ A3 )
% 0.20/0.47 => ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A3 ) )
% 0.20/0.47 = ( inj_on @ A @ B @ G @ A3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_image_iff
% 0.20/0.47 thf(fact_164_imageI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,X: A,A3: set @ A,F: A > B] :
% 0.20/0.47 ( ( member @ A @ X @ A3 )
% 0.20/0.47 => ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % imageI
% 0.20/0.47 thf(fact_165_image__iff,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,Z: A,F: B > A,A3: set @ B] :
% 0.20/0.47 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A3 ) )
% 0.20/0.47 = ( ? [X4: B] :
% 0.20/0.47 ( ( member @ B @ X4 @ A3 )
% 0.20/0.47 & ( Z
% 0.20/0.47 = ( F @ X4 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_iff
% 0.20/0.47 thf(fact_166_bex__imageD,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
% 0.20/0.47 ( ? [X6: A] :
% 0.20/0.47 ( ( member @ A @ X6 @ ( image @ B @ A @ F @ A3 ) )
% 0.20/0.47 & ( P3 @ X6 ) )
% 0.20/0.47 => ? [X5: B] :
% 0.20/0.47 ( ( member @ B @ X5 @ A3 )
% 0.20/0.47 & ( P3 @ ( F @ X5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bex_imageD
% 0.20/0.47 thf(fact_167_image__cong,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,M: set @ A,N3: set @ A,F: A > B,G: A > B] :
% 0.20/0.47 ( ( M = N3 )
% 0.20/0.47 => ( ! [X5: A] :
% 0.20/0.47 ( ( member @ A @ X5 @ N3 )
% 0.20/0.47 => ( ( F @ X5 )
% 0.20/0.47 = ( G @ X5 ) ) )
% 0.20/0.47 => ( ( image @ A @ B @ F @ M )
% 0.20/0.47 = ( image @ A @ B @ G @ N3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_cong
% 0.20/0.47 thf(fact_168_ball__imageD,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
% 0.20/0.47 ( ! [X5: A] :
% 0.20/0.47 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A3 ) )
% 0.20/0.47 => ( P3 @ X5 ) )
% 0.20/0.47 => ! [X6: B] :
% 0.20/0.47 ( ( member @ B @ X6 @ A3 )
% 0.20/0.47 => ( P3 @ ( F @ X6 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ball_imageD
% 0.20/0.47 thf(fact_169_rev__image__eqI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,X: A,A3: set @ A,B2: B,F: A > B] :
% 0.20/0.47 ( ( member @ A @ X @ A3 )
% 0.20/0.47 => ( ( B2
% 0.20/0.47 = ( F @ X ) )
% 0.20/0.47 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % rev_image_eqI
% 0.20/0.47 thf(fact_170_inj__image__mem__iff,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A2: A,A3: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.20/0.47 => ( ( member @ B @ ( F @ A2 ) @ ( image @ A @ B @ F @ A3 ) )
% 0.20/0.47 = ( member @ A @ A2 @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_mem_iff
% 0.20/0.47 thf(fact_171_inj__image__eq__iff,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,B6: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.20/0.47 => ( ( ( image @ A @ B @ F @ A3 )
% 0.20/0.47 = ( image @ A @ B @ F @ B6 ) )
% 0.20/0.47 = ( A3 = B6 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_eq_iff
% 0.20/0.47 thf(fact_172_range__ex1__eq,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,B2: B] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.20/0.47 => ( ( member @ B @ B2 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
% 0.20/0.47 = ( ? [X4: A] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ X4 ) )
% 0.20/0.47 & ! [Y4: A] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ Y4 ) )
% 0.20/0.47 => ( Y4 = X4 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % range_ex1_eq
% 0.20/0.47 thf(fact_173_prod__fun__imageE,axiom,
% 0.20/0.47 ! [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 )] :
% 0.20/0.47 ( ( 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 ) )
% 0.20/0.47 => ~ ! [X5: C,Y5: D] :
% 0.20/0.47 ( ( C3
% 0.20/0.47 = ( product_Pair @ A @ B @ ( F @ X5 ) @ ( G @ Y5 ) ) )
% 0.20/0.47 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X5 @ Y5 ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_fun_imageE
% 0.20/0.47 thf(fact_174_inj__on__the__inv__into,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A3 @ F ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_the_inv_into
% 0.20/0.47 thf(fact_175_f__the__inv__into__f,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: A > B,A3: set @ A,Y: B] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( member @ B @ Y @ ( image @ A @ B @ F @ A3 ) )
% 0.20/0.47 => ( ( F @ ( the_inv_into @ A @ B @ A3 @ F @ Y ) )
% 0.20/0.47 = Y ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % f_the_inv_into_f
% 0.20/0.47 thf(fact_176_total__lex__prod,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
% 0.20/0.47 ( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R )
% 0.20/0.47 => ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ S3 )
% 0.20/0.47 => ( total_on @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( lex_prod @ A @ B @ R @ S3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % total_lex_prod
% 0.20/0.47 thf(fact_177_range__fst,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.20/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % range_fst
% 0.20/0.47 thf(fact_178_range__snd,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.20/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % range_snd
% 0.20/0.47 thf(fact_179_fst__map__prod,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 0.20/0.47 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X ) )
% 0.20/0.47 = ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % fst_map_prod
% 0.20/0.47 thf(fact_180_snd__map__prod,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X: product_prod @ C @ D] :
% 0.20/0.47 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X ) )
% 0.20/0.47 = ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % snd_map_prod
% 0.20/0.47 thf(fact_181_fst__apfst,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
% 0.20/0.47 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
% 0.20/0.47 = ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % fst_apfst
% 0.20/0.47 thf(fact_182_apfst__eq__conv,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G: C > A] :
% 0.20/0.47 ( ( ( product_apfst @ C @ A @ B @ F @ X )
% 0.20/0.47 = ( product_apfst @ C @ A @ B @ G @ X ) )
% 0.20/0.47 = ( ( F @ ( product_fst @ C @ B @ X ) )
% 0.20/0.47 = ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % apfst_eq_conv
% 0.20/0.47 thf(fact_183_snd__apfst,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
% 0.20/0.47 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
% 0.20/0.47 = ( product_snd @ C @ A @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % snd_apfst
% 0.20/0.47 thf(fact_184_fst__apsnd,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
% 0.20/0.47 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
% 0.20/0.47 = ( product_fst @ A @ C @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % fst_apsnd
% 0.20/0.47 thf(fact_185_snd__apsnd,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
% 0.20/0.47 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
% 0.20/0.47 = ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % snd_apsnd
% 0.20/0.47 thf(fact_186_apsnd__eq__conv,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G: C > B] :
% 0.20/0.47 ( ( ( product_apsnd @ C @ B @ A @ F @ X )
% 0.20/0.47 = ( product_apsnd @ C @ B @ A @ G @ X ) )
% 0.20/0.47 = ( ( F @ ( product_snd @ A @ C @ X ) )
% 0.20/0.47 = ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % apsnd_eq_conv
% 0.20/0.47 thf(fact_187_in__lex__prod,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
% 0.20/0.47 ( ( 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 @ A6 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S3 ) )
% 0.20/0.47 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A6 ) @ R )
% 0.20/0.47 | ( ( A2 = A6 )
% 0.20/0.47 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B4 ) @ S3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % in_lex_prod
% 0.20/0.47 thf(fact_188_prod_Ocollapse,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.20/0.47 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.20/0.47 = Prod ) ).
% 0.20/0.47
% 0.20/0.47 % prod.collapse
% 0.20/0.47 thf(fact_189_fst__swap,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 0.20/0.47 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 0.20/0.47 = ( product_snd @ B @ A @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % fst_swap
% 0.20/0.47 thf(fact_190_snd__swap,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 0.20/0.47 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 0.20/0.47 = ( product_fst @ A @ B @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % snd_swap
% 0.20/0.47 thf(fact_191_apsnd__apfst,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,X: product_prod @ D @ C] :
% 0.20/0.47 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ X ) )
% 0.20/0.47 = ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % apsnd_apfst
% 0.20/0.47 thf(fact_192_apfst__apsnd,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 0.20/0.47 ( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G @ X ) )
% 0.20/0.47 = ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % apfst_apsnd
% 0.20/0.47 thf(fact_193_prod_Oswap__def,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( product_swap @ A @ B )
% 0.20/0.47 = ( ^ [P2: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P2 ) @ ( product_fst @ A @ B @ P2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod.swap_def
% 0.20/0.47 thf(fact_194_prod__eq__iff,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( ^ [Y6: product_prod @ A @ B,Z2: product_prod @ A @ B] : ( Y6 = Z2 ) )
% 0.20/0.47 = ( ^ [S: product_prod @ A @ B,T2: product_prod @ A @ B] :
% 0.20/0.47 ( ( ( product_fst @ A @ B @ S )
% 0.20/0.47 = ( product_fst @ A @ B @ T2 ) )
% 0.20/0.47 & ( ( product_snd @ A @ B @ S )
% 0.20/0.47 = ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_eq_iff
% 0.20/0.47 thf(fact_195_prod_Oexpand,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.20/0.47 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.20/0.47 = ( product_fst @ A @ B @ Prod2 ) )
% 0.20/0.47 & ( ( product_snd @ A @ B @ Prod )
% 0.20/0.47 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.20/0.47 => ( Prod = Prod2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod.expand
% 0.20/0.47 thf(fact_196_prod__eqI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,P: product_prod @ A @ B,Q: product_prod @ A @ B] :
% 0.20/0.47 ( ( ( product_fst @ A @ B @ P )
% 0.20/0.47 = ( product_fst @ A @ B @ Q ) )
% 0.20/0.47 => ( ( ( product_snd @ A @ B @ P )
% 0.20/0.47 = ( product_snd @ A @ B @ Q ) )
% 0.20/0.47 => ( P = Q ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_eqI
% 0.20/0.47 thf(fact_197_fst__eqD,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
% 0.20/0.47 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 0.20/0.47 = A2 )
% 0.20/0.47 => ( X = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % fst_eqD
% 0.20/0.47 thf(fact_198_snd__eqD,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
% 0.20/0.47 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.20/0.47 = A2 )
% 0.20/0.47 => ( Y = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % snd_eqD
% 0.20/0.47 thf(fact_199_fst__conv,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,X1: A,X2: B] :
% 0.20/0.47 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 0.20/0.47 = X1 ) ).
% 0.20/0.47
% 0.20/0.47 % fst_conv
% 0.20/0.47 thf(fact_200_snd__conv,axiom,
% 0.20/0.47 ! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
% 0.20/0.47 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
% 0.20/0.47 = X2 ) ).
% 0.20/0.47
% 0.20/0.47 % snd_conv
% 0.20/0.47 thf(fact_201_prod_Oexhaust__sel,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.20/0.47 ( Prod
% 0.20/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod.exhaust_sel
% 0.20/0.47 thf(fact_202_surjective__pairing,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
% 0.20/0.47 ( T3
% 0.20/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % surjective_pairing
% 0.20/0.47 thf(fact_203_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,P3: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
% 0.20/0.47 ( ( P3 @ X @ Y )
% 0.20/0.47 => ( ( A2
% 0.20/0.47 = ( product_Pair @ A @ B @ X @ Y ) )
% 0.20/0.47 => ( P3 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % BNF_Greatest_Fixpoint.subst_Pair
% 0.20/0.47 thf(fact_204_conjI__realizer,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,P3: A > $o,P: A,Q2: B > $o,Q: B] :
% 0.20/0.47 ( ( P3 @ P )
% 0.20/0.47 => ( ( Q2 @ Q )
% 0.20/0.47 => ( ( P3 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q ) ) )
% 0.20/0.47 & ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % conjI_realizer
% 0.20/0.47 thf(fact_205_exI__realizer,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,P3: A > B > $o,Y: A,X: B] :
% 0.20/0.47 ( ( P3 @ Y @ X )
% 0.20/0.47 => ( P3 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % exI_realizer
% 0.20/0.47 thf(fact_206_sndI,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
% 0.20/0.47 ( ( X
% 0.20/0.47 = ( product_Pair @ A @ B @ Y @ Z ) )
% 0.20/0.47 => ( ( product_snd @ A @ B @ X )
% 0.20/0.47 = Z ) ) ).
% 0.20/0.47
% 0.20/0.47 % sndI
% 0.20/0.47 thf(fact_207_eq__snd__iff,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,B2: A,P: product_prod @ B @ A] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( product_snd @ B @ A @ P ) )
% 0.20/0.47 = ( ? [A7: B] :
% 0.20/0.47 ( P
% 0.20/0.47 = ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eq_snd_iff
% 0.20/0.47 thf(fact_208_eq__fst__iff,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,A2: A,P: product_prod @ A @ B] :
% 0.20/0.47 ( ( A2
% 0.20/0.47 = ( product_fst @ A @ B @ P ) )
% 0.20/0.47 = ( ? [B5: B] :
% 0.20/0.47 ( P
% 0.20/0.47 = ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eq_fst_iff
% 0.20/0.47 thf(fact_209_fstI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
% 0.20/0.47 ( ( X
% 0.20/0.47 = ( product_Pair @ A @ B @ Y @ Z ) )
% 0.20/0.47 => ( ( product_fst @ A @ B @ X )
% 0.20/0.47 = Y ) ) ).
% 0.20/0.47
% 0.20/0.47 % fstI
% 0.20/0.47 thf(fact_210_same__fstI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,P3: A > $o,X: A,Y7: B,Y: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.20/0.47 ( ( P3 @ X )
% 0.20/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R2 @ X ) )
% 0.20/0.47 => ( 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 @ X @ Y7 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P3 @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % same_fstI
% 0.20/0.47 thf(fact_211_wf__map__prod__image,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
% 0.20/0.47 ( ( wf @ A @ R )
% 0.20/0.47 => ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.20/0.47 => ( wf @ B @ ( image @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( product_map_prod @ A @ B @ A @ B @ F @ F ) @ R ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % wf_map_prod_image
% 0.20/0.47 thf(fact_212_wf__def,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( wf @ A )
% 0.20/0.47 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ! [P5: A > $o] :
% 0.20/0.47 ( ! [X4: A] :
% 0.20/0.47 ( ! [Y4: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R3 )
% 0.20/0.47 => ( P5 @ Y4 ) )
% 0.20/0.47 => ( P5 @ X4 ) )
% 0.20/0.47 => ! [X7: A] : ( P5 @ X7 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % wf_def
% 0.20/0.47 thf(fact_213_wfE__min,axiom,
% 0.20/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Q2: set @ A] :
% 0.20/0.47 ( ( wf @ A @ R2 )
% 0.20/0.47 => ( ( member @ A @ X @ Q2 )
% 0.20/0.47 => ~ ! [Z3: A] :
% 0.20/0.47 ( ( member @ A @ Z3 @ Q2 )
% 0.20/0.47 => ~ ! [Y8: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y8 @ Z3 ) @ R2 )
% 0.20/0.47 => ~ ( member @ A @ Y8 @ Q2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % wfE_min
% 0.20/0.47 thf(fact_214_wfI__min,axiom,
% 0.20/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ! [X5: A,Q4: set @ A] :
% 0.20/0.47 ( ( member @ A @ X5 @ Q4 )
% 0.20/0.47 => ? [Xa3: A] :
% 0.20/0.47 ( ( member @ A @ Xa3 @ Q4 )
% 0.20/0.47 & ! [Y5: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Xa3 ) @ R2 )
% 0.20/0.47 => ~ ( member @ A @ Y5 @ Q4 ) ) ) )
% 0.20/0.47 => ( wf @ A @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % wfI_min
% 0.20/0.47 thf(fact_215_wfUNIVI,axiom,
% 0.20/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ( ! [P6: A > $o,X5: A] :
% 0.20/0.47 ( ! [Xa3: A] :
% 0.20/0.47 ( ! [Y5: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Xa3 ) @ R )
% 0.20/0.47 => ( P6 @ Y5 ) )
% 0.20/0.47 => ( P6 @ Xa3 ) )
% 0.20/0.47 => ( P6 @ X5 ) )
% 0.20/0.47 => ( wf @ A @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % wfUNIVI
% 0.20/0.47 thf(fact_216_wf__asym,axiom,
% 0.20/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X: A] :
% 0.20/0.47 ( ( wf @ A @ R )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X ) @ R )
% 0.20/0.47 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ R ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % wf_asym
% 0.20/0.47 thf(fact_217_wf__induct,axiom,
% 0.20/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P3: A > $o,A2: A] :
% 0.20/0.47 ( ( wf @ A @ R )
% 0.20/0.47 => ( ! [X5: A] :
% 0.20/0.47 ( ! [Y8: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y8 @ X5 ) @ R )
% 0.20/0.47 => ( P3 @ Y8 ) )
% 0.20/0.47 => ( P3 @ X5 ) )
% 0.20/0.47 => ( P3 @ A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % wf_induct
% 0.20/0.47 thf(fact_218_wf__irrefl,axiom,
% 0.20/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
% 0.20/0.47 ( ( wf @ A @ R )
% 0.20/0.47 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % wf_irrefl
% 0.20/0.47 thf(fact_219_wf__not__sym,axiom,
% 0.20/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X: A] :
% 0.20/0.47 ( ( wf @ A @ R )
% 0.20/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X ) @ R )
% 0.20/0.47 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ R ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % wf_not_sym
% 0.20/0.47 thf(fact_220_wf__not__refl,axiom,
% 0.20/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
% 0.20/0.47 ( ( wf @ A @ R )
% 0.20/0.47 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % wf_not_refl
% 0.20/0.47 thf(fact_221_wf__eq__minimal,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( wf @ A )
% 0.20/0.47 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.20/0.47 ! [Q5: set @ A] :
% 0.20/0.47 ( ? [X4: A] : ( member @ A @ X4 @ Q5 )
% 0.20/0.47 => ? [X4: A] :
% 0.20/0.47 ( ( member @ A @ X4 @ Q5 )
% 0.20/0.47 & ! [Y4: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R3 )
% 0.20/0.47 => ~ ( member @ A @ Y4 @ Q5 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % wf_eq_minimal
% 0.20/0.47 thf(fact_222_wf__induct__rule,axiom,
% 0.20/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P3: A > $o,A2: A] :
% 0.20/0.47 ( ( wf @ A @ R )
% 0.20/0.47 => ( ! [X5: A] :
% 0.20/0.47 ( ! [Y8: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y8 @ X5 ) @ R )
% 0.20/0.47 => ( P3 @ Y8 ) )
% 0.20/0.47 => ( P3 @ X5 ) )
% 0.20/0.47 => ( P3 @ A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % wf_induct_rule
% 0.20/0.47 thf(fact_223_dependent__wf__choice,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P3: ( A > B ) > A > B > $o] :
% 0.20/0.47 ( ( wf @ A @ R2 )
% 0.20/0.47 => ( ! [F8: A > B,G3: A > B,X5: A,R4: B] :
% 0.20/0.47 ( ! [Z4: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X5 ) @ R2 )
% 0.20/0.47 => ( ( F8 @ Z4 )
% 0.20/0.47 = ( G3 @ Z4 ) ) )
% 0.20/0.47 => ( ( P3 @ F8 @ X5 @ R4 )
% 0.20/0.47 = ( P3 @ G3 @ X5 @ R4 ) ) )
% 0.20/0.47 => ( ! [X5: A,F8: A > B] :
% 0.20/0.47 ( ! [Y8: A] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y8 @ X5 ) @ R2 )
% 0.20/0.47 => ( P3 @ F8 @ Y8 @ ( F8 @ Y8 ) ) )
% 0.20/0.47 => ? [X_1: B] : ( P3 @ F8 @ X5 @ X_1 ) )
% 0.20/0.47 => ? [F8: A > B] :
% 0.20/0.47 ! [X6: A] : ( P3 @ F8 @ X6 @ ( F8 @ X6 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dependent_wf_choice
% 0.20/0.47 thf(fact_224_sndOp__def,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,C: $tType] :
% 0.20/0.47 ( ( bNF_sndOp @ C @ A @ B )
% 0.20/0.47 = ( ^ [P5: C > A > $o,Q5: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P5 @ Q5 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sndOp_def
% 0.20/0.47 thf(fact_225_fstOp__def,axiom,
% 0.20/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.20/0.47 ( ( bNF_fstOp @ A @ B @ C )
% 0.20/0.47 = ( ^ [P5: A > B > $o,Q5: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P5 @ Q5 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % fstOp_def
% 0.20/0.47 thf(fact_226_image2__eqI,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C3: C,G: B > C,A3: set @ B] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ X ) )
% 0.20/0.47 => ( ( C3
% 0.20/0.47 = ( G @ X ) )
% 0.20/0.47 => ( ( member @ B @ X @ A3 )
% 0.20/0.47 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F @ G ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image2_eqI
% 0.20/0.47 thf(fact_227_the__inv__into__into,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: B,B6: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( member @ B @ X @ ( image @ A @ B @ F @ A3 ) )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.20/0.47 => ( member @ A @ ( the_inv_into @ A @ B @ A3 @ F @ X ) @ B6 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % the_inv_into_into
% 0.20/0.47 thf(fact_228_strict__mono__inv,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType] :
% 0.20/0.47 ( ( ( linorder @ B )
% 0.20/0.47 & ( linorder @ A ) )
% 0.20/0.47 => ! [F: A > B,G: B > A] :
% 0.20/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.20/0.47 => ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.20/0.47 = ( top_top @ ( set @ B ) ) )
% 0.20/0.47 => ( ! [X5: A] :
% 0.20/0.47 ( ( G @ ( F @ X5 ) )
% 0.20/0.47 = X5 )
% 0.20/0.47 => ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % strict_mono_inv
% 0.20/0.47 thf(fact_229_order__refl,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( preorder @ A )
% 0.20/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_refl
% 0.20/0.47 thf(fact_230_subset__antisym,axiom,
% 0.20/0.47 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
% 0.20/0.47 => ( A3 = B6 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_antisym
% 0.20/0.47 thf(fact_231_subsetI,axiom,
% 0.20/0.47 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.20/0.47 ( ! [X5: A] :
% 0.20/0.47 ( ( member @ A @ X5 @ A3 )
% 0.20/0.47 => ( member @ A @ X5 @ B6 ) )
% 0.20/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% 0.20/0.47
% 0.20/0.47 % subsetI
% 0.20/0.47 thf(fact_232_subrelI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.20/0.47 ( ! [X5: A,Y5: B] :
% 0.20/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) @ R )
% 0.20/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) @ S3 ) )
% 0.20/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % subrelI
% 0.20/0.47 thf(fact_233_inj__on__image__mem__iff,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,B6: set @ A,A2: A,A3: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ B6 )
% 0.20/0.47 => ( ( member @ A @ A2 @ B6 )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.20/0.47 => ( ( member @ B @ ( F @ A2 ) @ ( image @ A @ B @ F @ A3 ) )
% 0.20/0.47 = ( member @ A @ A2 @ A3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_image_mem_iff
% 0.20/0.47 thf(fact_234_inj__on__image__eq__iff,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,C4: set @ A,A3: set @ A,B6: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ C4 )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ C4 )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.20/0.47 => ( ( ( image @ A @ B @ F @ A3 )
% 0.20/0.47 = ( image @ A @ B @ F @ B6 ) )
% 0.20/0.47 = ( A3 = B6 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_image_eq_iff
% 0.20/0.47 thf(fact_235_inj__on__subset,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,B6: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
% 0.20/0.47 => ( inj_on @ A @ B @ F @ B6 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_subset
% 0.20/0.47 thf(fact_236_subset__inj__on,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,B6: set @ A,A3: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ B6 )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.20/0.47 => ( inj_on @ A @ B @ F @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_inj_on
% 0.20/0.47 thf(fact_237_strict__mono__less__eq,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( ( linorder @ A )
% 0.20/0.47 & ( order @ B ) )
% 0.20/0.47 => ! [F: A > B,X: A,Y: A] :
% 0.20/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.20/0.47 => ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) )
% 0.20/0.47 = ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % strict_mono_less_eq
% 0.20/0.47 thf(fact_238_dual__order_Oantisym,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [B2: A,A2: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.20/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.antisym
% 0.20/0.47 thf(fact_239_dual__order_Oeq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ( ( ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 0.20/0.47 = ( ^ [A7: A,B5: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ B5 @ A7 )
% 0.20/0.47 & ( ord_less_eq @ A @ A7 @ B5 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.eq_iff
% 0.20/0.47 thf(fact_240_strict__mono__eq,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( ( linorder @ A )
% 0.20/0.47 & ( order @ B ) )
% 0.20/0.47 => ! [F: A > B,X: A,Y: A] :
% 0.20/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.20/0.47 => ( ( ( F @ X )
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 = ( X = Y ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % strict_mono_eq
% 0.20/0.47 thf(fact_241_dual__order_Otrans,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [B2: A,A2: A,C3: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 0.20/0.47 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.trans
% 0.20/0.47 thf(fact_242_linorder__wlog,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [P3: A > A > $o,A2: A,B2: A] :
% 0.20/0.47 ( ! [A5: A,B3: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.20/0.47 => ( P3 @ A5 @ B3 ) )
% 0.20/0.47 => ( ! [A5: A,B3: A] :
% 0.20/0.47 ( ( P3 @ B3 @ A5 )
% 0.20/0.47 => ( P3 @ A5 @ B3 ) )
% 0.20/0.47 => ( P3 @ A2 @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % linorder_wlog
% 0.20/0.47 thf(fact_243_dual__order_Orefl,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.refl
% 0.20/0.47 thf(fact_244_order__trans,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( preorder @ A )
% 0.20/0.47 => ! [X: A,Y: A,Z: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.20/0.47 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.20/0.47 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_trans
% 0.20/0.47 thf(fact_245_order__class_Oorder_Oantisym,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [A2: A,B2: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.20/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_class.order.antisym
% 0.20/0.47 thf(fact_246_ord__le__eq__trans,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ord @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C3: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.20/0.47 => ( ( B2 = C3 )
% 0.20/0.47 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ord_le_eq_trans
% 0.20/0.47 thf(fact_247_ord__eq__le__trans,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ord @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C3: A] :
% 0.20/0.47 ( ( A2 = B2 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.20/0.47 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ord_eq_le_trans
% 0.20/0.47 thf(fact_248_order__class_Oorder_Oeq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ( ( ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 0.20/0.47 = ( ^ [A7: A,B5: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A7 @ B5 )
% 0.20/0.47 & ( ord_less_eq @ A @ B5 @ A7 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_class.order.eq_iff
% 0.20/0.47 thf(fact_249_antisym__conv,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [Y: A,X: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.20/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.20/0.47 = ( X = Y ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % antisym_conv
% 0.20/0.47 thf(fact_250_le__cases3,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [X: A,Y: A,Z: A] :
% 0.20/0.47 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.20/0.47 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.20/0.47 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.20/0.47 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.20/0.47 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.20/0.47 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.20/0.47 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.20/0.47 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.20/0.47 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.20/0.47 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.20/0.47 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.20/0.47 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_cases3
% 0.20/0.47 thf(fact_251_order_Otrans,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C3: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.20/0.47 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order.trans
% 0.20/0.47 thf(fact_252_le__cases,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [X: A,Y: A] :
% 0.20/0.47 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.20/0.47 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_cases
% 0.20/0.47 thf(fact_253_eq__refl,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( preorder @ A )
% 0.20/0.47 => ! [X: A,Y: A] :
% 0.20/0.47 ( ( X = Y )
% 0.20/0.47 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eq_refl
% 0.20/0.47 thf(fact_254_linear,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [X: A,Y: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.20/0.47 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % linear
% 0.20/0.47 thf(fact_255_antisym,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [X: A,Y: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.20/0.47 => ( ( ord_less_eq @ A @ Y @ X )
% 0.20/0.47 => ( X = Y ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % antisym
% 0.20/0.47
% 0.20/0.47 % Type constructors (17)
% 0.20/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.20/0.47 ! [A8: $tType,A9: $tType] :
% 0.20/0.47 ( ( preorder @ A9 )
% 0.20/0.47 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.20/0.47 ! [A8: $tType,A9: $tType] :
% 0.20/0.47 ( ( order @ A9 )
% 0.20/0.47 => ( order @ ( A8 > A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Orderings_Otop,axiom,
% 0.20/0.47 ! [A8: $tType,A9: $tType] :
% 0.20/0.47 ( ( top @ A9 )
% 0.20/0.47 => ( top @ ( A8 > A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.20/0.47 ! [A8: $tType,A9: $tType] :
% 0.20/0.47 ( ( ord @ A9 )
% 0.20/0.47 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.20/0.47 preorder @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.20/0.47 linorder @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.20/0.47 order @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.20/0.47 ord @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.20/0.47 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.20/0.47 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Orderings_Otop_6,axiom,
% 0.20/0.47 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 0.20/0.47 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
% 0.20/0.47 preorder @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_9,axiom,
% 0.20/0.47 linorder @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_10,axiom,
% 0.20/0.47 order @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Otop_11,axiom,
% 0.20/0.47 top @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 0.20/0.47 ord @ $o ).
% 0.20/0.47
% 0.20/0.47 % Helper facts (3)
% 0.20/0.47 thf(help_If_3_1_T,axiom,
% 0.20/0.47 ! [P3: $o] :
% 0.20/0.47 ( ( P3 = $true )
% 0.20/0.47 | ( P3 = $false ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(help_If_2_1_T,axiom,
% 0.20/0.47 ! [A: $tType,X: A,Y: A] :
% 0.20/0.47 ( ( if @ A @ $false @ X @ Y )
% 0.20/0.47 = Y ) ).
% 0.20/0.47
% 0.20/0.47 thf(help_If_1_1_T,axiom,
% 0.20/0.47 ! [A: $tType,X: A,Y: A] :
% 0.20/0.47 ( ( if @ A @ $true @ X @ Y )
% 0.20/0.47 = X ) ).
% 0.20/0.47
% 0.20/0.47 % Conjectures (1)
% 0.20/0.47 thf(conj_0,conjecture,
% 0.20/0.47 ( ( paraco876059933e_eval @ i @ ( paraco2100061555le_Con @ p1 @ p2 ) )
% 0.20/0.47 = ( paraco876059933e_eval @ i @ p1 ) ) ).
% 0.20/0.47
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 ------------------------
% 2.75/1.51 ---- Embedded in THF ---
% 2.75/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.75/1.51 %%% Generated on Tue Feb 24 08:27:47 EST 2026
% 2.75/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 2.75/1.51 %%% Logic specification used:
% 2.75/1.51 %%% thf(spec, logic, $$dhol).
% 2.75/1.51
% 2.75/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.cVWy2Pj8hg/DTF2DTF31276.p
% 2.75/1.51 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv, type, paraco415392788lle_tv: $tType).
% 2.75/1.51 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm, type, paraco414474393lle_fm: $tType).
% 2.75/1.51 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.75/1.51 thf(ty_t_String_Ochar, type, char: $tType).
% 2.75/1.51 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.75/1.51 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.75/1.51 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.75/1.51 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.75/1.51 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.75/1.51 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.75/1.51 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.75/1.51 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.75/1.51 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.75/1.51 thf(sy_c_BNF__Def_OfstOp, type, bNF_fstOp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (((product_prod @ A) @ C) > ((product_prod @ A) @ B))))))).
% 2.75/1.51 thf(sy_c_BNF__Def_Opick__middlep, type, bNF_pick_middlep: (!> [B:$tType,A:$tType,C:$tType]: (((B > (A > $o)) > ((A > (C > $o)) > (B > (C > A))))))).
% 2.75/1.51 thf(sy_c_BNF__Def_OsndOp, type, bNF_sndOp: (!> [C:$tType,A:$tType,B:$tType]: (((C > (A > $o)) > ((A > (B > $o)) > (((product_prod @ C) @ B) > ((product_prod @ A) @ B))))))).
% 2.75/1.51 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2, type, bNF_Greatest_image2: (!> [C:$tType,A:$tType,B:$tType]: (((set @ C) > ((C > A) > ((C > B) > (set @ ((product_prod @ A) @ B)))))))).
% 2.75/1.51 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 2.75/1.51 thf(sy_c_Fun_Othe__inv__into, type, the_inv_into: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (B > A)))))).
% 2.75/1.51 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.75/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.75/1.51 thf(sy_c_Orderings_Oorder__class_Ostrict__mono, type, order_strict_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.75/1.51 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__int, type, paraco753463838ge_int: ((nat > nat) > (((list @ char) > paraco415392788lle_tv) > ((list @ char) > paraco415392788lle_tv)))).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv, type, paraco1920534163nge_tv: ((nat > nat) > (paraco415392788lle_tv > paraco415392788lle_tv))).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv__rel, type, paraco2077297190tv_rel: (((product_prod @ (nat > nat)) @ paraco415392788lle_tv) > (((product_prod @ (nat > nat)) @ paraco415392788lle_tv) > $o))).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval, type, paraco876059933e_eval: (((list @ char) > paraco415392788lle_tv) > (paraco414474393lle_fm > paraco415392788lle_tv))).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H, type, paraco2100061555le_Con: (paraco414474393lle_fm > (paraco414474393lle_fm > paraco414474393lle_fm))).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql, type, paraco2084319816le_Eql: (paraco414474393lle_fm > (paraco414474393lle_fm > paraco414474393lle_fm))).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H, type, paraco1628874225le_Eql: (paraco414474393lle_fm > (paraco414474393lle_fm > paraco414474393lle_fm))).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H, type, paraco329115265le_Neg: (paraco414474393lle_fm > paraco414474393lle_fm)).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro, type, paraco27778325le_Pro: ((list @ char) > paraco414474393lle_fm)).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth, type, paraco251304083_Truth: paraco414474393lle_fm).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Ocase__fm, type, paraco1246693743ase_fm: (!> [A:$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)))))))))).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Orec__fm, type, paraco815940159rec_fm: (!> [A:$tType]: ((((list @ char) > A) > (A > ((paraco414474393lle_fm > (A > A)) > ((paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))) > ((paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))) > ((paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))) > (paraco414474393lle_fm > A)))))))))).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet, type, paraco2040174112le_Det: ($o > paraco415392788lle_tv)).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet, type, paraco676387099_Indet: (nat > paraco415392788lle_tv)).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Ocase__tv, type, paraco490622181ase_tv: (!> [A:$tType]: ((($o > A) > ((nat > A) > (paraco415392788lle_tv > A)))))).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv, type, paraco152590079rec_tv: (!> [A:$tType]: ((($o > A) > ((nat > A) > (paraco415392788lle_tv > A)))))).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid, type, paraco769098683_valid: (paraco414474393lle_fm > $o)).
% 2.75/1.51 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid__in, type, paraco2086025920lid_in: ((set @ nat) > (paraco414474393lle_fm > $o))).
% 2.75/1.51 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.75/1.51 thf(sy_c_Product__Type_Oapfst, type, product_apfst: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > (((product_prod @ A) @ B) > ((product_prod @ C) @ B)))))).
% 2.75/1.51 thf(sy_c_Product__Type_Oapsnd, type, product_apsnd: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((product_prod @ A) @ B) > ((product_prod @ A) @ C)))))).
% 2.75/1.51 thf(sy_c_Product__Type_Obool_Ocase__bool, type, product_case_bool: (!> [A:$tType]: ((A > (A > ($o > A)))))).
% 2.75/1.51 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 2.75/1.51 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.75/1.51 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 2.75/1.51 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.75/1.51 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 2.75/1.51 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 2.75/1.51 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 2.75/1.51 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))))))).
% 2.75/1.51 thf(sy_c_Relation_Oinv__imagep, type, inv_imagep: (!> [B:$tType,A:$tType]: (((B > (B > $o)) > ((A > B) > (A > (A > $o))))))).
% 2.75/1.51 thf(sy_c_Relation_Ototal__on, type, total_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.75/1.51 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.75/1.51 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.75/1.51 thf(sy_c_Wellfounded_Oaccp, type, accp: (!> [A:$tType]: (((A > (A > $o)) > (A > $o))))).
% 2.75/1.51 thf(sy_c_Wellfounded_Olex__prod, type, lex_prod: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 2.75/1.51 thf(sy_c_Wellfounded_Owf, type, wf: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 2.75/1.51 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 2.75/1.51 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.75/1.51 thf(sy_v_f, type, f: (nat > nat)).
% 2.75/1.51 thf(sy_v_i, type, i: ((list @ char) > paraco415392788lle_tv)).
% 2.75/1.51 thf(sy_v_p1____, type, p1: paraco414474393lle_fm).
% 2.75/1.51 thf(sy_v_p2____, type, p2: paraco414474393lle_fm).
% 2.75/1.51 thf(fact_0_a_H_H, axiom, (((paraco876059933e_eval @ i) @ p1) != (paraco2040174112le_Det @ $true))).
% 2.75/1.51 thf(fact_1_a, axiom, (((paraco876059933e_eval @ i) @ p2) = (paraco2040174112le_Det @ $true))).
% 2.75/1.51 thf(fact_2_a_H, axiom, (((paraco876059933e_eval @ i) @ p1) != ((paraco876059933e_eval @ i) @ p2))).
% 2.75/1.51 thf(fact_3_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)))))).
% 2.75/1.51 thf(fact_4_b_H, axiom, (((paraco876059933e_eval @ ((paraco753463838ge_int @ f) @ i)) @ p1) != ((paraco876059933e_eval @ ((paraco753463838ge_int @ f) @ i)) @ p2))).
% 2.75/1.51 thf(fact_5_eval_Osimps_I4_J, axiom, (! [I:((list @ char) > paraco415392788lle_tv),P:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I) @ P) = ((paraco876059933e_eval @ I) @ Q)) => (((paraco876059933e_eval @ I) @ ((paraco2100061555le_Con @ P) @ Q)) = ((paraco876059933e_eval @ I) @ P))) & ((((paraco876059933e_eval @ I) @ P) != ((paraco876059933e_eval @ I) @ Q)) => (((((paraco876059933e_eval @ I) @ P) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I) @ ((paraco2100061555le_Con @ P) @ Q)) = ((paraco876059933e_eval @ I) @ Q))) & ((((paraco876059933e_eval @ I) @ P) != (paraco2040174112le_Det @ $true)) => (((((paraco876059933e_eval @ I) @ Q) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I) @ ((paraco2100061555le_Con @ P) @ Q)) = ((paraco876059933e_eval @ I) @ P))) & ((((paraco876059933e_eval @ I) @ Q) != (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I) @ ((paraco2100061555le_Con @ P) @ Q)) = (paraco2040174112le_Det @ $false))))))))))).
% 2.75/1.51 thf(fact_6_ih2, axiom, (((paraco876059933e_eval @ ((paraco753463838ge_int @ f) @ i)) @ p2) = ((paraco1920534163nge_tv @ f) @ ((paraco876059933e_eval @ i) @ p2)))).
% 2.75/1.51 thf(fact_7_ih1, axiom, (((paraco876059933e_eval @ ((paraco753463838ge_int @ f) @ i)) @ p1) = ((paraco1920534163nge_tv @ f) @ ((paraco876059933e_eval @ i) @ p1)))).
% 2.75/1.51 thf(fact_8_b_H_H, axiom, (((paraco876059933e_eval @ ((paraco753463838ge_int @ f) @ i)) @ p1) != (paraco2040174112le_Det @ $true))).
% 2.75/1.51 thf(fact_9_tv_Oinject_I1_J, axiom, (! [X1:$o,Y1:$o]: ((((paraco2040174112le_Det @ X1) = (paraco2040174112le_Det @ Y1)) = (X1 = Y1))))).
% 2.75/1.51 thf(fact_10_conjunction, axiom, (! [P:paraco414474393lle_fm,Q:paraco414474393lle_fm]: (((paraco769098683_valid @ ((paraco2100061555le_Con @ P) @ Q)) = ((paraco769098683_valid @ P) & (paraco769098683_valid @ Q)))))).
% 2.75/1.51 thf(fact_11_conjunction__in, axiom, (! [U:(set @ nat),P:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((paraco2086025920lid_in @ U) @ ((paraco2100061555le_Con @ P) @ Q)) = (((paraco2086025920lid_in @ U) @ P) & ((paraco2086025920lid_in @ U) @ Q)))))).
% 2.75/1.51 thf(fact_12_fm_Osimps_I39_J, axiom, (! [A:$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))))).
% 2.75/1.51 thf(fact_13_fm_Odistinct_I13_J, axiom, (! [X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: ((paraco251304083_Truth != ((paraco2100061555le_Con @ X41) @ X42))))).
% 2.75/1.51 thf(fact_14_fm_Osimps_I45_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > (A > A)),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: ((((((((((paraco815940159rec_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ ((paraco2100061555le_Con @ X41) @ X42)) = ((((F4 @ X41) @ X42) @ ((((((((paraco815940159rec_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ X41)) @ ((((((((paraco815940159rec_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ X42)))))).
% 2.75/1.51 thf(fact_15_fm_Odistinct_I25_J, axiom, (! [X41:paraco414474393lle_fm,X42:paraco414474393lle_fm,X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: ((((paraco2100061555le_Con @ X41) @ X42) != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.75/1.51 thf(fact_16_fm_Oinject_I4_J, axiom, (! [X51:paraco414474393lle_fm,X52:paraco414474393lle_fm,Y51:paraco414474393lle_fm,Y52:paraco414474393lle_fm]: (((((paraco2084319816le_Eql @ X51) @ X52) = ((paraco2084319816le_Eql @ Y51) @ Y52)) = ((X51 = Y51) & (X52 = Y52)))))).
% 2.75/1.51 thf(fact_17_fm_Osimps_I46_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > (A > A)),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: ((((((((((paraco815940159rec_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ ((paraco2084319816le_Eql @ X51) @ X52)) = ((((F5 @ X51) @ X52) @ ((((((((paraco815940159rec_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ X51)) @ ((((((((paraco815940159rec_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ X52)))))).
% 2.75/1.51 thf(fact_18_fm_Osimps_I43_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > (A > A)),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A))))]: ((((((((((paraco815940159rec_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ paraco251304083_Truth) = F2)))).
% 2.75/1.51 thf(fact_19_fm_Osimps_I40_J, axiom, (! [A:$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))))).
% 2.75/1.51 thf(fact_20_fm_Osimps_I37_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > A),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > A))]: ((((((((((paraco1246693743ase_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ paraco251304083_Truth) = F2)))).
% 2.75/1.51 thf(fact_21_fm_Odistinct_I15_J, axiom, (! [X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: ((paraco251304083_Truth != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.75/1.51 thf(fact_22_change__tv_Osimps_I1_J, axiom, (! [F:(nat > nat),B2:$o]: ((((paraco1920534163nge_tv @ F) @ (paraco2040174112le_Det @ B2)) = (paraco2040174112le_Det @ B2))))).
% 2.75/1.51 thf(fact_23_transfer, axiom, (! [U:(set @ nat),P:paraco414474393lle_fm]: (((~ (((paraco2086025920lid_in @ U) @ P))) => (~ ((paraco769098683_valid @ P))))))).
% 2.75/1.51 thf(fact_24_change__int__def, axiom, (paraco753463838ge_int = (^ [F7:(nat > nat),I2:((list @ char) > paraco415392788lle_tv),S:(list @ char)]: (((paraco1920534163nge_tv @ F7) @ (I2 @ S)))))).
% 2.75/1.51 thf(fact_25_valid__valid__in, axiom, (! [P:paraco414474393lle_fm,U:(set @ nat)]: (((paraco769098683_valid @ P) => ((paraco2086025920lid_in @ U) @ P))))).
% 2.75/1.51 thf(fact_26_eval__equality__simplify, axiom, (! [I:((list @ char) > paraco415392788lle_tv),P:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((paraco876059933e_eval @ I) @ ((paraco2084319816le_Eql @ P) @ Q)) = (paraco2040174112le_Det @ (((paraco876059933e_eval @ I) @ P) = ((paraco876059933e_eval @ I) @ Q))))))).
% 2.75/1.51 thf(fact_27_eval_Osimps_I5_J, axiom, (! [I:((list @ char) > paraco415392788lle_tv),P:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I) @ P) = ((paraco876059933e_eval @ I) @ Q)) => (((paraco876059933e_eval @ I) @ ((paraco2084319816le_Eql @ P) @ Q)) = (paraco2040174112le_Det @ $true))) & ((((paraco876059933e_eval @ I) @ P) != ((paraco876059933e_eval @ I) @ Q)) => (((paraco876059933e_eval @ I) @ ((paraco2084319816le_Eql @ P) @ Q)) = (paraco2040174112le_Det @ $false))))))).
% 2.75/1.51 thf(fact_28_eval_Osimps_I2_J, axiom, (! [I:((list @ char) > paraco415392788lle_tv)]: ((((paraco876059933e_eval @ I) @ paraco251304083_Truth) = (paraco2040174112le_Det @ $true))))).
% 2.75/1.51 thf(fact_29_valid__def, axiom, (paraco769098683_valid = (^ [P2:paraco414474393lle_fm]: ((! [I2:((list @ char) > paraco415392788lle_tv)]: ((((paraco876059933e_eval @ I2) @ P2) = (paraco2040174112le_Det @ $true)))))))).
% 2.75/1.51 thf(fact_30_tv_Osimps_I7_J, axiom, (! [A:$tType,F1:($o > A),F2:(nat > A),X1:$o]: ((((((paraco152590079rec_tv @ A) @ F1) @ F2) @ (paraco2040174112le_Det @ X1)) = (F1 @ X1))))).
% 2.75/1.51 thf(fact_31_change__tv_Oelims, axiom, (! [X:(nat > nat),Xa:paraco415392788lle_tv,Y:paraco415392788lle_tv]: (((((paraco1920534163nge_tv @ X) @ Xa) = Y) => ((! [B3:$o]: (((Xa = (paraco2040174112le_Det @ B3)) => (Y != (paraco2040174112le_Det @ B3))))) => (~ ((! [N:nat]: (((Xa = (paraco676387099_Indet @ N)) => (Y != (paraco676387099_Indet @ (X @ N))))))))))))).
% 2.75/1.51 thf(fact_32_tv_Osimps_I5_J, axiom, (! [A:$tType,F1:($o > A),F2:(nat > A),X1:$o]: ((((((paraco490622181ase_tv @ A) @ F1) @ F2) @ (paraco2040174112le_Det @ X1)) = (F1 @ X1))))).
% 2.75/1.51 thf(fact_33_eval__negation, axiom, (! [I:((list @ char) > paraco415392788lle_tv),P:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I) @ P) = (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I) @ (paraco329115265le_Neg @ P)) = (paraco2040174112le_Det @ $true))) & ((((paraco876059933e_eval @ I) @ P) != (paraco2040174112le_Det @ $false)) => (((((paraco876059933e_eval @ I) @ P) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I) @ (paraco329115265le_Neg @ P)) = (paraco2040174112le_Det @ $false))) & ((((paraco876059933e_eval @ I) @ P) != (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I) @ (paraco329115265le_Neg @ P)) = ((paraco876059933e_eval @ I) @ P))))))))).
% 2.75/1.51 thf(fact_34_assms, axiom, ((((inj_on @ nat) @ nat) @ f) @ (top_top @ (set @ nat)))).
% 2.75/1.51 thf(fact_35_change__tv_Osimps_I2_J, axiom, (! [F:(nat > nat),N2:nat]: ((((paraco1920534163nge_tv @ F) @ (paraco676387099_Indet @ N2)) = (paraco676387099_Indet @ (F @ N2)))))).
% 2.75/1.51 thf(fact_36_fm_Osimps_I41_J, axiom, (! [A:$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)),X61:paraco414474393lle_fm,X62:paraco414474393lle_fm]: ((((((((((paraco1246693743ase_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ ((paraco1628874225le_Eql @ X61) @ X62)) = ((F6 @ X61) @ X62))))).
% 2.75/1.51 thf(fact_37_fm_Osimps_I47_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > (A > A)),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),X61:paraco414474393lle_fm,X62:paraco414474393lle_fm]: ((((((((((paraco815940159rec_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ ((paraco1628874225le_Eql @ X61) @ X62)) = ((((F6 @ X61) @ X62) @ ((((((((paraco815940159rec_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ X61)) @ ((((((((paraco815940159rec_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ X62)))))).
% 2.75/1.51 thf(fact_38_fm_Osimps_I38_J, axiom, (! [A:$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))))).
% 2.75/1.51 thf(fact_39_fm_Odistinct_I17_J, axiom, (! [X61:paraco414474393lle_fm,X62:paraco414474393lle_fm]: ((paraco251304083_Truth != ((paraco1628874225le_Eql @ X61) @ X62))))).
% 2.75/1.51 thf(fact_40_fm_Oinject_I2_J, axiom, (! [X3:paraco414474393lle_fm,Y3:paraco414474393lle_fm]: ((((paraco329115265le_Neg @ X3) = (paraco329115265le_Neg @ Y3)) = (X3 = Y3))))).
% 2.75/1.51 thf(fact_41_fm_Oinject_I5_J, axiom, (! [X61:paraco414474393lle_fm,X62:paraco414474393lle_fm,Y61:paraco414474393lle_fm,Y62:paraco414474393lle_fm]: (((((paraco1628874225le_Eql @ X61) @ X62) = ((paraco1628874225le_Eql @ Y61) @ Y62)) = ((X61 = Y61) & (X62 = Y62)))))).
% 2.75/1.51 thf(fact_42_tv_Oinject_I2_J, axiom, (! [X2:nat,Y2:nat]: ((((paraco676387099_Indet @ X2) = (paraco676387099_Indet @ Y2)) = (X2 = Y2))))).
% 2.75/1.51 thf(fact_43_fm_Odistinct_I23_J, axiom, (! [X3:paraco414474393lle_fm,X61:paraco414474393lle_fm,X62:paraco414474393lle_fm]: (((paraco329115265le_Neg @ X3) != ((paraco1628874225le_Eql @ X61) @ X62))))).
% 2.75/1.51 thf(fact_44_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))))).
% 2.75/1.51 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P3:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P3)) = (P3 @ A2))))).
% 2.75/1.51 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A3)))) = A3)))).
% 2.75/1.51 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P3:(A > $o),Q2:(A > $o)]: (((! [X5:A]: (((P3 @ X5) = (Q2 @ X5)))) => (((collect @ A) @ P3) = ((collect @ A) @ Q2)))))).
% 2.75/1.51 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X5:A]: (((F @ X5) = (G @ X5)))) => (F = G))))).
% 2.75/1.51 thf(fact_49_tv_Osimps_I8_J, axiom, (! [A:$tType,F1:($o > A),F2:(nat > A),X2:nat]: ((((((paraco152590079rec_tv @ A) @ F1) @ F2) @ (paraco676387099_Indet @ X2)) = (F2 @ X2))))).
% 2.75/1.51 thf(fact_50_eval__equality, axiom, (! [I:((list @ char) > paraco415392788lle_tv),P:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I) @ P) = ((paraco876059933e_eval @ I) @ Q)) => (((paraco876059933e_eval @ I) @ ((paraco1628874225le_Eql @ P) @ Q)) = (paraco2040174112le_Det @ $true))) & ((((paraco876059933e_eval @ I) @ P) != ((paraco876059933e_eval @ I) @ Q)) => (((((paraco876059933e_eval @ I) @ P) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I) @ ((paraco1628874225le_Eql @ P) @ Q)) = ((paraco876059933e_eval @ I) @ Q))) & ((((paraco876059933e_eval @ I) @ P) != (paraco2040174112le_Det @ $true)) => (((((paraco876059933e_eval @ I) @ Q) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I) @ ((paraco1628874225le_Eql @ P) @ Q)) = ((paraco876059933e_eval @ I) @ P))) & ((((paraco876059933e_eval @ I) @ Q) != (paraco2040174112le_Det @ $true)) => (((((paraco876059933e_eval @ I) @ P) = (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I) @ ((paraco1628874225le_Eql @ P) @ Q)) = ((paraco876059933e_eval @ I) @ (paraco329115265le_Neg @ Q)))) & ((((paraco876059933e_eval @ I) @ P) != (paraco2040174112le_Det @ $false)) => (((((paraco876059933e_eval @ I) @ Q) = (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I) @ ((paraco1628874225le_Eql @ P) @ Q)) = ((paraco876059933e_eval @ I) @ (paraco329115265le_Neg @ P)))) & ((((paraco876059933e_eval @ I) @ Q) != (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I) @ ((paraco1628874225le_Eql @ P) @ Q)) = (paraco2040174112le_Det @ $false))))))))))))))).
% 2.75/1.51 thf(fact_51_double__negation, axiom, (paraco876059933e_eval = (^ [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm]: (((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ (paraco329115265le_Neg @ P2))))))).
% 2.75/1.51 thf(fact_52_fm_Odistinct_I19_J, axiom, (! [X3:paraco414474393lle_fm,X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: (((paraco329115265le_Neg @ X3) != ((paraco2100061555le_Con @ X41) @ X42))))).
% 2.75/1.51 thf(fact_53_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))))))).
% 2.75/1.51 thf(fact_54_change__tv_Oinduct, axiom, (! [P3:((nat > nat) > (paraco415392788lle_tv > $o)),A0:(nat > nat),A1:paraco415392788lle_tv]: (((! [F8:(nat > nat),B3:$o]: (((P3 @ F8) @ (paraco2040174112le_Det @ B3)))) => ((! [F8:(nat > nat),N:nat]: (((P3 @ F8) @ (paraco676387099_Indet @ N)))) => ((P3 @ A0) @ A1)))))).
% 2.75/1.51 thf(fact_55_string__tv_Ocases, axiom, (! [X:paraco415392788lle_tv]: (((X != (paraco2040174112le_Det @ $true)) => ((X != (paraco2040174112le_Det @ $false)) => (~ ((! [N:nat]: ((X != (paraco676387099_Indet @ N))))))))))).
% 2.75/1.51 thf(fact_56_tv_Oexhaust, axiom, (! [Y:paraco415392788lle_tv]: (((! [X12:$o]: ((Y != (paraco2040174112le_Det @ X12)))) => (~ ((! [X22:nat]: ((Y != (paraco676387099_Indet @ X22)))))))))).
% 2.75/1.51 thf(fact_57_tv_Oinduct, axiom, (! [P3:(paraco415392788lle_tv > $o),Tv:paraco415392788lle_tv]: (((! [X5:$o]: ((P3 @ (paraco2040174112le_Det @ X5)))) => ((! [X5:nat]: ((P3 @ (paraco676387099_Indet @ X5)))) => (P3 @ Tv)))))).
% 2.75/1.51 thf(fact_58_tv_Odistinct_I1_J, axiom, (! [X1:$o,X2:nat]: (((paraco2040174112le_Det @ X1) != (paraco676387099_Indet @ X2))))).
% 2.75/1.51 thf(fact_59_fm_Odistinct_I21_J, axiom, (! [X3:paraco414474393lle_fm,X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: (((paraco329115265le_Neg @ X3) != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.75/1.51 thf(fact_60_fm_Odistinct_I11_J, axiom, (! [X3:paraco414474393lle_fm]: ((paraco251304083_Truth != (paraco329115265le_Neg @ X3))))).
% 2.75/1.51 thf(fact_61_fm_Odistinct_I27_J, axiom, (! [X41:paraco414474393lle_fm,X42:paraco414474393lle_fm,X61:paraco414474393lle_fm,X62:paraco414474393lle_fm]: ((((paraco2100061555le_Con @ X41) @ X42) != ((paraco1628874225le_Eql @ X61) @ X62))))).
% 2.75/1.51 thf(fact_62_fm_Osimps_I44_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > (A > A)),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),X3:paraco414474393lle_fm]: ((((((((((paraco815940159rec_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ (paraco329115265le_Neg @ X3)) = ((F3 @ X3) @ ((((((((paraco815940159rec_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ X3)))))).
% 2.75/1.51 thf(fact_63_fm_Odistinct_I29_J, axiom, (! [X51:paraco414474393lle_fm,X52:paraco414474393lle_fm,X61:paraco414474393lle_fm,X62:paraco414474393lle_fm]: ((((paraco2084319816le_Eql @ X51) @ X52) != ((paraco1628874225le_Eql @ X61) @ X62))))).
% 2.75/1.51 thf(fact_64_iso__tuple__UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.75/1.51 thf(fact_65_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.75/1.51 thf(fact_66_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X4:D]: ((top_top @ C)))))))).
% 2.75/1.51 thf(fact_67_eval_Osimps_I3_J, axiom, (! [I:((list @ char) > paraco415392788lle_tv),P:paraco414474393lle_fm]: ((((paraco876059933e_eval @ I) @ (paraco329115265le_Neg @ P)) = ((((paraco490622181ase_tv @ paraco415392788lle_tv) @ (((product_case_bool @ paraco415392788lle_tv) @ (paraco2040174112le_Det @ $false)) @ (paraco2040174112le_Det @ $true))) @ paraco676387099_Indet) @ ((paraco876059933e_eval @ I) @ P)))))).
% 2.75/1.51 thf(fact_68_inj__def, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) = (! [X4:A,Y4:A]: ((((F @ X4) = (F @ Y4)) => (X4 = Y4)))))))).
% 2.75/1.51 thf(fact_69_inj__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((F @ X) = (F @ Y)) = (X = Y)))))).
% 2.75/1.51 thf(fact_70_injI, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((! [X5:A,Y5:A]: ((((F @ X5) = (F @ Y5)) => (X5 = Y5)))) => ((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))))))).
% 2.75/1.51 thf(fact_71_injD, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((F @ X) = (F @ Y)) => (X = Y)))))).
% 2.75/1.51 thf(fact_72_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.75/1.51 thf(fact_73_change__tv__injection, axiom, (! [F:(nat > nat)]: ((((((inj_on @ nat) @ nat) @ F) @ (top_top @ (set @ nat))) => ((((inj_on @ paraco415392788lle_tv) @ paraco415392788lle_tv) @ (paraco1920534163nge_tv @ F)) @ (top_top @ (set @ paraco415392788lle_tv))))))).
% 2.75/1.51 thf(fact_74_UNIV__eq__I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X5:A]: ((((member @ A) @ X5) @ A3))) => ((top_top @ (set @ A)) = A3))))).
% 2.75/1.51 thf(fact_75_UNIV__witness, axiom, (! [A:$tType]: ((? [X5:A]: ((((member @ A) @ X5) @ (top_top @ (set @ A)))))))).
% 2.75/1.51 thf(fact_76_inj__onD, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ A3) => (((F @ X) = (F @ Y)) => ((((member @ A) @ X) @ A3) => ((((member @ A) @ Y) @ A3) => (X = Y)))))))).
% 2.75/1.51 thf(fact_77_inj__onI, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F:(A > B)]: (((! [X5:A,Y5:A]: (((((member @ A) @ X5) @ A3) => ((((member @ A) @ Y5) @ A3) => (((F @ X5) = (F @ Y5)) => (X5 = Y5)))))) => ((((inj_on @ A) @ B) @ F) @ A3))))).
% 2.75/1.51 thf(fact_78_inj__on__def, axiom, (! [B:$tType,A:$tType]: ((((inj_on @ A) @ B) = (^ [F7:(A > B),A4:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A4) => (! [Y4:A]: (((((member @ A) @ Y4) @ A4) => (((F7 @ X4) = (F7 @ Y4)) => (X4 = Y4)))))))))))))).
% 2.75/1.51 thf(fact_79_inj__on__cong, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F:(A > B),G:(A > B)]: (((! [A5:A]: (((((member @ A) @ A5) @ A3) => ((F @ A5) = (G @ A5))))) => (((((inj_on @ A) @ B) @ F) @ A3) = ((((inj_on @ A) @ B) @ G) @ A3)))))).
% 2.75/1.51 thf(fact_80_inj__on__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ A3) => ((((member @ A) @ X) @ A3) => ((((member @ A) @ Y) @ A3) => (((F @ X) = (F @ Y)) = (X = Y)))))))).
% 2.75/1.51 thf(fact_81_inj__on__contraD, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ A3) => ((X != Y) => ((((member @ A) @ X) @ A3) => ((((member @ A) @ Y) @ A3) => ((F @ X) != (F @ Y))))))))).
% 2.75/1.51 thf(fact_82_inj__on__inverseI, axiom, (! [B:$tType,A:$tType,A3:(set @ A),G:(B > A),F:(A > B)]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => ((G @ (F @ X5)) = X5)))) => ((((inj_on @ A) @ B) @ F) @ A3))))).
% 2.75/1.51 thf(fact_83_fm_Oinduct, axiom, (! [P3:(paraco414474393lle_fm > $o),Fm:paraco414474393lle_fm]: (((! [X5:(list @ char)]: ((P3 @ (paraco27778325le_Pro @ X5)))) => ((P3 @ paraco251304083_Truth) => ((! [X5:paraco414474393lle_fm]: (((P3 @ X5) => (P3 @ (paraco329115265le_Neg @ X5))))) => ((! [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)))))))))).
% 2.75/1.51 thf(fact_84_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)))) => (~ ((! [X612:paraco414474393lle_fm,X622:paraco414474393lle_fm]: ((Y != ((paraco1628874225le_Eql @ X612) @ X622)))))))))))))).
% 2.75/1.51 thf(fact_85_props_Ocases, axiom, (! [X:paraco414474393lle_fm]: (((X != paraco251304083_Truth) => ((! [S2:(list @ char)]: ((X != (paraco27778325le_Pro @ S2)))) => ((! [P4:paraco414474393lle_fm]: ((X != (paraco329115265le_Neg @ P4)))) => ((! [P4:paraco414474393lle_fm,Q3:paraco414474393lle_fm]: ((X != ((paraco2100061555le_Con @ P4) @ Q3)))) => ((! [P4:paraco414474393lle_fm,Q3:paraco414474393lle_fm]: ((X != ((paraco2084319816le_Eql @ P4) @ Q3)))) => (~ ((! [P4:paraco414474393lle_fm,Q3:paraco414474393lle_fm]: ((X != ((paraco1628874225le_Eql @ P4) @ Q3)))))))))))))).
% 2.75/1.51 thf(fact_86_props_Oinduct, axiom, (! [P3:(paraco414474393lle_fm > $o),A0:paraco414474393lle_fm]: (((P3 @ paraco251304083_Truth) => ((! [S2:(list @ char)]: ((P3 @ (paraco27778325le_Pro @ S2)))) => ((! [P4:paraco414474393lle_fm]: (((P3 @ P4) => (P3 @ (paraco329115265le_Neg @ P4))))) => ((! [P4:paraco414474393lle_fm,Q3:paraco414474393lle_fm]: (((P3 @ P4) => ((P3 @ Q3) => (P3 @ ((paraco2100061555le_Con @ P4) @ Q3)))))) => ((! [P4:paraco414474393lle_fm,Q3:paraco414474393lle_fm]: (((P3 @ P4) => ((P3 @ Q3) => (P3 @ ((paraco2084319816le_Eql @ P4) @ Q3)))))) => ((! [P4:paraco414474393lle_fm,Q3:paraco414474393lle_fm]: (((P3 @ P4) => ((P3 @ Q3) => (P3 @ ((paraco1628874225le_Eql @ P4) @ Q3)))))) => (P3 @ A0)))))))))).
% 2.75/1.51 thf(fact_87_old_Obool_Osimps_I4_J, axiom, (! [A:$tType,F1:A,F2:A]: ((((((product_case_bool @ A) @ F1) @ F2) @ $false) = F2)))).
% 2.75/1.51 thf(fact_88_old_Obool_Osimps_I3_J, axiom, (! [A:$tType,F1:A,F2:A]: ((((((product_case_bool @ A) @ F1) @ F2) @ $true) = F1)))).
% 2.75/1.51 thf(fact_89_bool_Osplit__sel, axiom, (! [A:$tType,P3:(A > $o),F1:A,F2:A,Bool:$o]: (((P3 @ ((((product_case_bool @ A) @ F1) @ F2) @ Bool)) = ((Bool => (P3 @ F1)) & ((~ (Bool)) => (P3 @ F2))))))).
% 2.75/1.51 thf(fact_90_fm_Oinject_I1_J, axiom, (! [X1:(list @ char),Y1:(list @ char)]: ((((paraco27778325le_Pro @ X1) = (paraco27778325le_Pro @ Y1)) = (X1 = Y1))))).
% 2.75/1.51 thf(fact_91_top1I, axiom, (! [A:$tType,X:A]: (((top_top @ (A > $o)) @ X)))).
% 2.75/1.51 thf(fact_92_eval_Osimps_I1_J, axiom, (! [I:((list @ char) > paraco415392788lle_tv),S3:(list @ char)]: ((((paraco876059933e_eval @ I) @ (paraco27778325le_Pro @ S3)) = (I @ S3))))).
% 2.75/1.51 thf(fact_93_fm_Odistinct_I3_J, axiom, (! [X1:(list @ char),X3:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != (paraco329115265le_Neg @ X3))))).
% 2.75/1.51 thf(fact_94_fm_Odistinct_I5_J, axiom, (! [X1:(list @ char),X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != ((paraco2100061555le_Con @ X41) @ X42))))).
% 2.75/1.51 thf(fact_95_fm_Odistinct_I9_J, axiom, (! [X1:(list @ char),X61:paraco414474393lle_fm,X62:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != ((paraco1628874225le_Eql @ X61) @ X62))))).
% 2.75/1.51 thf(fact_96_fm_Odistinct_I7_J, axiom, (! [X1:(list @ char),X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.75/1.51 thf(fact_97_fm_Odistinct_I1_J, axiom, (! [X1:(list @ char)]: (((paraco27778325le_Pro @ X1) != paraco251304083_Truth)))).
% 2.75/1.51 thf(fact_98_fm_Osimps_I42_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > (A > A)),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > (A > (A > A)))),X1:(list @ char)]: ((((((((((paraco815940159rec_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ (paraco27778325le_Pro @ X1)) = (F1 @ X1))))).
% 2.75/1.51 thf(fact_99_fm_Osimps_I36_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F3:(paraco414474393lle_fm > A),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),X1:(list @ char)]: ((((((((((paraco1246693743ase_fm @ A) @ F1) @ F2) @ F3) @ F4) @ F5) @ F6) @ (paraco27778325le_Pro @ X1)) = (F1 @ X1))))).
% 2.75/1.51 thf(fact_100_bool_Osplit__sel__asm, axiom, (! [A:$tType,P3:(A > $o),F1:A,F2:A,Bool:$o]: (((P3 @ ((((product_case_bool @ A) @ F1) @ F2) @ Bool)) = (~ (((Bool & (~ ((P3 @ F1)))) | ((~ (Bool)) & (~ ((P3 @ F2))))))))))).
% 2.75/1.51 thf(fact_101_bool_Ocase__distrib, axiom, (! [A:$tType,B:$tType,H:(A > B),F1:A,F2:A,Bool:$o]: (((H @ ((((product_case_bool @ A) @ F1) @ F2) @ Bool)) = ((((product_case_bool @ B) @ (H @ F1)) @ (H @ F2)) @ Bool))))).
% 2.75/1.51 thf(fact_102_bool_Ocase__eq__if, axiom, (! [A:$tType]: (((product_case_bool @ A) = (^ [F12:A,F22:A,Bool2:$o]: (((((if @ A) @ Bool2) @ F12) @ F22))))))).
% 2.75/1.51 thf(fact_103_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X4:A]: ((((member @ A) @ X4) @ (top_top @ (set @ A))))))))).
% 2.75/1.51 thf(fact_104_eval_Ocases, axiom, (! [X:((product_prod @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm)]: (((! [I3:((list @ char) > paraco415392788lle_tv),S2:(list @ char)]: ((X != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ (paraco27778325le_Pro @ S2))))) => ((! [I3:((list @ char) > paraco415392788lle_tv)]: ((X != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ paraco251304083_Truth)))) => ((! [I3:((list @ char) > paraco415392788lle_tv),P4:paraco414474393lle_fm]: ((X != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ (paraco329115265le_Neg @ P4))))) => ((! [I3:((list @ char) > paraco415392788lle_tv),P4:paraco414474393lle_fm,Q3:paraco414474393lle_fm]: ((X != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ ((paraco2100061555le_Con @ P4) @ Q3))))) => ((! [I3:((list @ char) > paraco415392788lle_tv),P4:paraco414474393lle_fm,Q3:paraco414474393lle_fm]: ((X != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ ((paraco2084319816le_Eql @ P4) @ Q3))))) => (~ ((! [I3:((list @ char) > paraco415392788lle_tv),P4:paraco414474393lle_fm,Q3:paraco414474393lle_fm]: ((X != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ ((paraco1628874225le_Eql @ P4) @ Q3))))))))))))))).
% 2.75/1.51 thf(fact_105_top__conj_I1_J, axiom, (! [A:$tType,X:A,P3:$o]: (((((top_top @ (A > $o)) @ X) & P3) = P3)))).
% 2.75/1.51 thf(fact_106_top__conj_I2_J, axiom, (! [A:$tType,P3:$o,X:A]: (((P3 & ((top_top @ (A > $o)) @ X)) = P3)))).
% 2.75/1.51 thf(fact_107_inj__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F:(B > C)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = ((((inj_on @ B) @ C) @ F) @ (top_top @ (set @ B))))))).
% 2.75/1.51 thf(fact_108_inj__apfst, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > C)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = ((((inj_on @ A) @ C) @ F) @ (top_top @ (set @ A))))))).
% 2.75/1.51 thf(fact_109_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A6:A,B4:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A6) @ B4)) = ((A2 = A6) & (B2 = B4)))))).
% 2.75/1.51 thf(fact_110_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X2:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X2) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 2.75/1.51 thf(fact_111_apfst__conv, axiom, (! [C:$tType,A:$tType,B:$tType,F:(C > A),X:C,Y:B]: (((((((product_apfst @ C) @ A) @ B) @ F) @ ((((product_Pair @ C) @ B) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ (F @ X)) @ Y))))).
% 2.75/1.51 thf(fact_112_apsnd__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > B),X:A,Y:C]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ ((((product_Pair @ A) @ C) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ X) @ (F @ Y)))))).
% 2.75/1.51 thf(fact_113_apsnd__apfst__commute, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F:(C > B),G:(D > A),P:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ (((((product_apfst @ D) @ A) @ C) @ G) @ P)) = (((((product_apfst @ D) @ A) @ B) @ G) @ (((((product_apsnd @ C) @ B) @ D) @ F) @ P)))))).
% 2.75/1.51 thf(fact_114_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P3:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B3:B]: ((P3 @ ((((product_Pair @ A) @ B) @ A5) @ B3)))) => (P3 @ Prod))))).
% 2.75/1.51 thf(fact_115_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B3:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B3))))))))).
% 2.75/1.51 thf(fact_116_prod__induct7, axiom, (! [G2:$tType,F9:$tType,E:$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 @ F9) @ G2)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F9) @ G2))))))]: (((! [A5:A,B3:B,C2:C,D2:D,E2:E,F8:F9,G3:G2]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F9) @ G2)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F9) @ G2))))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F9) @ G2)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F9) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F9) @ G2)) @ E2) @ ((((product_Pair @ F9) @ G2) @ F8) @ G3))))))))) => (P3 @ X))))).
% 2.75/1.51 thf(fact_117_prod__induct6, axiom, (! [F9:$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) @ F9))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F9)))))]: (((! [A5:A,B3:B,C2:C,D2:D,E2:E,F8:F9]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F9))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F9)))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F9))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F9)) @ D2) @ ((((product_Pair @ E) @ F9) @ E2) @ F8)))))))) => (P3 @ X))))).
% 2.75/1.51 thf(fact_118_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A5:A,B3:B,C2:C,D2:D,E2:E]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P3 @ X))))).
% 2.75/1.51 thf(fact_119_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B3:B,C2:C,D2:D]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B3) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P3 @ X))))).
% 2.75/1.51 thf(fact_120_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B3:B,C2:C]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B3) @ C2))))) => (P3 @ X))))).
% 2.75/1.51 thf(fact_121_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F9:$tType,G2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F9) @ G2))))))]: ((~ ((! [A5:A,B3:B,C2:C,D2:D,E2:E,F8:F9,G3:G2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F9) @ G2)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F9) @ G2))))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F9) @ G2)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F9) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F9) @ G2)) @ E2) @ ((((product_Pair @ F9) @ G2) @ F8) @ G3)))))))))))))).
% 2.75/1.51 thf(fact_122_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F9:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F9)))))]: ((~ ((! [A5:A,B3:B,C2:C,D2:D,E2:E,F8:F9]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F9))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F9)))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F9))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F9)) @ D2) @ ((((product_Pair @ E) @ F9) @ E2) @ F8))))))))))))).
% 2.75/1.51 thf(fact_123_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A5:A,B3:B,C2:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.75/1.51 thf(fact_124_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B3:B,C2:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B3) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 2.75/1.51 thf(fact_125_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B3:B,C2:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B3) @ C2)))))))))).
% 2.75/1.51 thf(fact_126_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A6:A,B4:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A6) @ B4)) => (~ (((A2 = A6) => (B2 != B4)))))))).
% 2.75/1.51 thf(fact_127_prod__cases, axiom, (! [B:$tType,A:$tType,P3:(((product_prod @ A) @ B) > $o),P:((product_prod @ A) @ B)]: (((! [A5:A,B3:B]: ((P3 @ ((((product_Pair @ A) @ B) @ A5) @ B3)))) => (P3 @ P))))).
% 2.75/1.51 thf(fact_128_surj__pair, axiom, (! [A:$tType,B:$tType,P:((product_prod @ A) @ B)]: ((? [X5:A,Y5:B]: ((P = ((((product_Pair @ A) @ B) @ X5) @ Y5))))))).
% 2.75/1.51 thf(fact_129_old_Oprod_Orec, axiom, (! [A:$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))))).
% 2.75/1.51 thf(fact_130_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C3 @ A2) @ B2))))).
% 2.75/1.51 thf(fact_131_prod_Oinj__map, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,F1:(A > C),F2:(B > D)]: ((((((inj_on @ A) @ C) @ F1) @ (top_top @ (set @ A))) => (((((inj_on @ B) @ D) @ F2) @ (top_top @ (set @ B))) => ((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F1) @ F2)) @ (top_top @ (set @ ((product_prod @ A) @ B))))))))).
% 2.75/1.51 thf(fact_132_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R:A,S3:B,R2:(set @ ((product_prod @ A) @ B)),S4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S3)) @ R2) => ((S4 = S3) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S4)) @ R2)))))).
% 2.75/1.51 thf(fact_133_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F7:(((product_prod @ B) @ C) > A),A7:B,B5:C]: ((F7 @ ((((product_Pair @ B) @ C) @ A7) @ B5)))))))).
% 2.75/1.51 thf(fact_134_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) @ $o) @ F) @ A2) @ B2))))).
% 2.75/1.51 thf(fact_135_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G:(D > B),A2:C,B2:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G) @ ((((product_Pair @ C) @ D) @ A2) @ B2)) = ((((product_Pair @ A) @ B) @ (F @ A2)) @ (G @ B2)))))).
% 2.75/1.51 thf(fact_136_curryD, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A2) @ B2) => (F @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 2.75/1.51 thf(fact_137_curryE, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A2) @ B2) => (F @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 2.75/1.51 thf(fact_138_change__tv_Ocases, axiom, (! [X:((product_prod @ (nat > nat)) @ paraco415392788lle_tv)]: (((! [F8:(nat > nat),B3:$o]: ((X != ((((product_Pair @ (nat > nat)) @ paraco415392788lle_tv) @ F8) @ (paraco2040174112le_Det @ B3))))) => (~ ((! [F8:(nat > nat),N:nat]: ((X != ((((product_Pair @ (nat > nat)) @ paraco415392788lle_tv) @ F8) @ (paraco676387099_Indet @ N))))))))))).
% 2.75/1.51 thf(fact_139_in__inv__imagep, axiom, (! [B:$tType,A:$tType]: ((((inv_imagep @ A) @ B) = (^ [R3:(A > (A > $o)),F7:(B > A),X4:B,Y4:B]: (((R3 @ (F7 @ X4)) @ (F7 @ Y4)))))))).
% 2.75/1.51 thf(fact_140_the__inv__f__f, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((the_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F) @ (F @ X)) = X))))).
% 2.75/1.51 thf(fact_141_change__tv_Opelims, axiom, (! [X:(nat > nat),Xa:paraco415392788lle_tv,Y:paraco415392788lle_tv]: (((((paraco1920534163nge_tv @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ (nat > nat)) @ paraco415392788lle_tv)) @ paraco2077297190tv_rel) @ ((((product_Pair @ (nat > nat)) @ paraco415392788lle_tv) @ X) @ Xa)) => ((! [B3:$o]: (((Xa = (paraco2040174112le_Det @ B3)) => ((Y = (paraco2040174112le_Det @ B3)) => (~ ((((accp @ ((product_prod @ (nat > nat)) @ paraco415392788lle_tv)) @ paraco2077297190tv_rel) @ ((((product_Pair @ (nat > nat)) @ paraco415392788lle_tv) @ X) @ (paraco2040174112le_Det @ B3))))))))) => (~ ((! [N:nat]: (((Xa = (paraco676387099_Indet @ N)) => ((Y = (paraco676387099_Indet @ (X @ N))) => (~ ((((accp @ ((product_prod @ (nat > nat)) @ paraco415392788lle_tv)) @ paraco2077297190tv_rel) @ ((((product_Pair @ (nat > nat)) @ paraco415392788lle_tv) @ X) @ (paraco676387099_Indet @ N))))))))))))))))).
% 2.75/1.51 thf(fact_142_in__inv__image, axiom, (! [A:$tType,B:$tType,X:A,Y:A,R:(set @ ((product_prod @ B) @ B)),F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((((inv_image @ B) @ A) @ R) @ F)) = (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X)) @ (F @ Y))) @ R))))).
% 2.75/1.51 thf(fact_143_the__inv__into__f__f, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A),X:A]: ((((((inj_on @ A) @ B) @ F) @ A3) => ((((member @ A) @ X) @ A3) => ((((((the_inv_into @ A) @ B) @ A3) @ F) @ (F @ X)) = X)))))).
% 2.75/1.51 thf(fact_144_the__inv__into__f__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A),X:A,Y:B]: ((((((inj_on @ A) @ B) @ F) @ A3) => (((F @ X) = Y) => ((((member @ A) @ X) @ A3) => ((((((the_inv_into @ A) @ B) @ A3) @ F) @ Y) = X))))))).
% 2.75/1.51 thf(fact_145_total__inv__image, axiom, (! [B:$tType,A:$tType,F:(A > B),R:(set @ ((product_prod @ B) @ B))]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((total_on @ B) @ (top_top @ (set @ B))) @ R) => (((total_on @ A) @ (top_top @ (set @ A))) @ ((((inv_image @ B) @ A) @ R) @ F))))))).
% 2.75/1.51 thf(fact_146_the__inv__into__onto, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A3) => (((((image @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A3) @ F)) @ ((((image @ A) @ B) @ F) @ A3)) = A3))))).
% 2.75/1.51 thf(fact_147_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)))))).
% 2.75/1.51 thf(fact_148_inj__swap, axiom, (! [B:$tType,A:$tType,A3:(set @ ((product_prod @ A) @ B))]: (((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ B) @ A)) @ ((product_swap @ A) @ B)) @ A3)))).
% 2.75/1.51 thf(fact_149_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),X:B,A3:(set @ B)]: (((B2 = (F @ X)) => ((((member @ B) @ X) @ A3) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ A3))))))).
% 2.75/1.51 thf(fact_150_swap__swap, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P)) = P)))).
% 2.75/1.51 thf(fact_151_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X))))).
% 2.75/1.51 thf(fact_152_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y:A,X:B,A3:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y) @ X)) @ ((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A3)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y)) @ A3))))).
% 2.75/1.51 thf(fact_153_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))))))).
% 2.75/1.51 thf(fact_154_total__on__def, axiom, (! [A:$tType]: (((total_on @ A) = (^ [A4:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: ((! [X4:A]: (((((member @ A) @ X4) @ A4) => (! [Y4:A]: (((((member @ A) @ Y4) @ A4) => ((X4 != Y4) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ R3) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X4)) @ R3))))))))))))))).
% 2.75/1.51 thf(fact_155_total__onI, axiom, (! [A:$tType,A3:(set @ A),R:(set @ ((product_prod @ A) @ A))]: (((! [X5:A,Y5:A]: (((((member @ A) @ X5) @ A3) => ((((member @ A) @ Y5) @ A3) => ((X5 != Y5) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Y5)) @ R) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ X5)) @ R))))))) => (((total_on @ A) @ A3) @ R))))).
% 2.75/1.51 thf(fact_156_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))))))))).
% 2.75/1.51 thf(fact_157_range__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),X:B]: (((B2 = (F @ X)) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 2.75/1.51 thf(fact_158_surj__def, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y4:A]: ((? [X4:B]: ((Y4 = (F @ X4)))))))))).
% 2.75/1.51 thf(fact_159_rangeI, axiom, (! [A:$tType,B:$tType,F:(B > A),X:B]: ((((member @ A) @ (F @ X)) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B))))))).
% 2.75/1.51 thf(fact_160_surjI, axiom, (! [B:$tType,A:$tType,G:(B > A),F:(A > B)]: (((! [X5:A]: (((G @ (F @ X5)) = X5))) => (((((image @ B) @ A) @ G) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.75/1.51 thf(fact_161_surjE, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X5:B]: ((Y != (F @ X5)))))))))).
% 2.75/1.51 thf(fact_162_surjD, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X5:B]: ((Y = (F @ X5)))))))).
% 2.75/1.51 thf(fact_163_inj__on__image__iff, axiom, (! [B:$tType,A:$tType,A3:(set @ A),G:(A > B),F:(A > A)]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (! [Xa2:A]: (((((member @ A) @ Xa2) @ A3) => (((G @ (F @ X5)) = (G @ (F @ Xa2))) = ((G @ X5) = (G @ Xa2))))))))) => (((((inj_on @ A) @ A) @ F) @ A3) => (((((inj_on @ A) @ B) @ G) @ ((((image @ A) @ A) @ F) @ A3)) = ((((inj_on @ A) @ B) @ G) @ A3))))))).
% 2.75/1.51 thf(fact_164_imageI, axiom, (! [B:$tType,A:$tType,X:A,A3:(set @ A),F:(A > B)]: (((((member @ A) @ X) @ A3) => (((member @ B) @ (F @ X)) @ ((((image @ A) @ B) @ F) @ A3)))))).
% 2.75/1.51 thf(fact_165_image__iff, axiom, (! [A:$tType,B:$tType,Z:A,F:(B > A),A3:(set @ B)]: (((((member @ A) @ Z) @ ((((image @ B) @ A) @ F) @ A3)) = (? [X4:B]: (((((member @ B) @ X4) @ A3) & (Z = (F @ X4))))))))).
% 2.75/1.51 thf(fact_166_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B),P3:(A > $o)]: (((? [X6:A]: (((((member @ A) @ X6) @ ((((image @ B) @ A) @ F) @ A3)) & (P3 @ X6)))) => (? [X5:B]: (((((member @ B) @ X5) @ A3) & (P3 @ (F @ X5))))))))).
% 2.75/1.51 thf(fact_167_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N3:(set @ A),F:(A > B),G:(A > B)]: (((M = N3) => ((! [X5:A]: (((((member @ A) @ X5) @ N3) => ((F @ X5) = (G @ X5))))) => (((((image @ A) @ B) @ F) @ M) = ((((image @ A) @ B) @ G) @ N3))))))).
% 2.75/1.51 thf(fact_168_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B),P3:(A > $o)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F) @ A3)) => (P3 @ X5)))) => (! [X6:B]: (((((member @ B) @ X6) @ A3) => (P3 @ (F @ X6))))))))).
% 2.75/1.51 thf(fact_169_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X:A,A3:(set @ A),B2:B,F:(A > B)]: (((((member @ A) @ X) @ A3) => ((B2 = (F @ X)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F) @ A3))))))).
% 2.75/1.51 thf(fact_170_inj__image__mem__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:A,A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((member @ B) @ (F @ A2)) @ ((((image @ A) @ B) @ F) @ A3)) = (((member @ A) @ A2) @ A3)))))).
% 2.75/1.51 thf(fact_171_inj__image__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((image @ A) @ B) @ F) @ A3) = ((((image @ A) @ B) @ F) @ B6)) = (A3 = B6)))))).
% 2.75/1.51 thf(fact_172_range__ex1__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),B2:B]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((member @ B) @ B2) @ ((((image @ A) @ B) @ F) @ (top_top @ (set @ A)))) = (? [X4:A]: (((B2 = (F @ X4)) & (! [Y4:A]: (((B2 = (F @ Y4)) => (Y4 = X4)))))))))))).
% 2.75/1.51 thf(fact_173_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C3:((product_prod @ A) @ B),F:(C > A),G:(D > B),R2:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C3) @ ((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G)) @ R2)) => (~ ((! [X5:C,Y5:D]: (((C3 = ((((product_Pair @ A) @ B) @ (F @ X5)) @ (G @ Y5))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X5) @ Y5)) @ R2)))))))))))).
% 2.75/1.51 thf(fact_174_inj__on__the__inv__into, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A3) => ((((inj_on @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A3) @ F)) @ ((((image @ A) @ B) @ F) @ A3)))))).
% 2.75/1.51 thf(fact_175_f__the__inv__into__f, axiom, (! [A:$tType,B:$tType,F:(A > B),A3:(set @ A),Y:B]: ((((((inj_on @ A) @ B) @ F) @ A3) => ((((member @ B) @ Y) @ ((((image @ A) @ B) @ F) @ A3)) => ((F @ (((((the_inv_into @ A) @ B) @ A3) @ F) @ Y)) = Y)))))).
% 2.75/1.51 thf(fact_176_total__lex__prod, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ B) @ B))]: (((((total_on @ A) @ (top_top @ (set @ A))) @ R) => ((((total_on @ B) @ (top_top @ (set @ B))) @ S3) => (((total_on @ ((product_prod @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) @ ((((lex_prod @ A) @ B) @ R) @ S3))))))).
% 2.75/1.51 thf(fact_177_range__fst, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = (top_top @ (set @ A)))))).
% 2.75/1.51 thf(fact_178_range__snd, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ A)))))).
% 2.75/1.51 thf(fact_179_fst__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > A),G:(D > B),X:((product_prod @ C) @ D)]: (((((product_fst @ A) @ B) @ (((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G) @ X)) = (F @ (((product_fst @ C) @ D) @ X)))))).
% 2.75/1.51 thf(fact_180_snd__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > B),G:(D > A),X:((product_prod @ C) @ D)]: (((((product_snd @ B) @ A) @ (((((((product_map_prod @ C) @ B) @ D) @ A) @ F) @ G) @ X)) = (G @ (((product_snd @ C) @ D) @ X)))))).
% 2.75/1.51 thf(fact_181_fst__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > A),X:((product_prod @ C) @ B)]: (((((product_fst @ A) @ B) @ (((((product_apfst @ C) @ A) @ B) @ F) @ X)) = (F @ (((product_fst @ C) @ B) @ X)))))).
% 2.75/1.51 thf(fact_182_apfst__eq__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > A),X:((product_prod @ C) @ B),G:(C > A)]: ((((((((product_apfst @ C) @ A) @ B) @ F) @ X) = (((((product_apfst @ C) @ A) @ B) @ G) @ X)) = ((F @ (((product_fst @ C) @ B) @ X)) = (G @ (((product_fst @ C) @ B) @ X))))))).
% 2.75/1.51 thf(fact_183_snd__apfst, axiom, (! [B:$tType,A:$tType,C:$tType,F:(C > B),X:((product_prod @ C) @ A)]: (((((product_snd @ B) @ A) @ (((((product_apfst @ C) @ B) @ A) @ F) @ X)) = (((product_snd @ C) @ A) @ X))))).
% 2.75/1.51 thf(fact_184_fst__apsnd, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),X:((product_prod @ A) @ C)]: (((((product_fst @ A) @ B) @ (((((product_apsnd @ C) @ B) @ A) @ F) @ X)) = (((product_fst @ A) @ C) @ X))))).
% 2.75/1.51 thf(fact_185_snd__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F:(C > A),X:((product_prod @ B) @ C)]: (((((product_snd @ B) @ A) @ (((((product_apsnd @ C) @ A) @ B) @ F) @ X)) = (F @ (((product_snd @ B) @ C) @ X)))))).
% 2.75/1.51 thf(fact_186_apsnd__eq__conv, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),X:((product_prod @ A) @ C),G:(C > B)]: ((((((((product_apsnd @ C) @ B) @ A) @ F) @ X) = (((((product_apsnd @ C) @ B) @ A) @ G) @ X)) = ((F @ (((product_snd @ A) @ C) @ X)) = (G @ (((product_snd @ A) @ C) @ X))))))).
% 2.75/1.51 thf(fact_187_in__lex__prod, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A6:A,B4:B,R:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Pair @ A) @ B) @ A6) @ B4))) @ ((((lex_prod @ A) @ B) @ R) @ S3)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A6)) @ R) | ((A2 = A6) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B2) @ B4)) @ S3))))))).
% 2.75/1.51 thf(fact_188_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 2.75/1.51 thf(fact_189_fst__swap, axiom, (! [A:$tType,B:$tType,X:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X)) = (((product_snd @ B) @ A) @ X))))).
% 2.75/1.51 thf(fact_190_snd__swap, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X)) = (((product_fst @ A) @ B) @ X))))).
% 2.75/1.51 thf(fact_191_apsnd__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F:(C > B),G:(D > A),X:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ (((((product_apfst @ D) @ A) @ C) @ G) @ X)) = ((((product_Pair @ A) @ B) @ (G @ (((product_fst @ D) @ C) @ X))) @ (F @ (((product_snd @ D) @ C) @ X))))))).
% 2.75/1.51 thf(fact_192_apfst__apsnd, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,F:(C > A),G:(D > B),X:((product_prod @ C) @ D)]: (((((((product_apfst @ C) @ A) @ B) @ F) @ (((((product_apsnd @ D) @ B) @ C) @ G) @ X)) = ((((product_Pair @ A) @ B) @ (F @ (((product_fst @ C) @ D) @ X))) @ (G @ (((product_snd @ C) @ D) @ X))))))).
% 2.75/1.51 thf(fact_193_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P2:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P2)) @ (((product_fst @ A) @ B) @ P2)))))))).
% 2.75/1.51 thf(fact_194_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y6:((product_prod @ A) @ B),Z2:((product_prod @ A) @ B)]: ((Y6 = Z2))) = (^ [S:((product_prod @ A) @ B),T2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S) = (((product_fst @ A) @ B) @ T2)) & ((((product_snd @ A) @ B) @ S) = (((product_snd @ A) @ B) @ T2))))))))).
% 2.75/1.51 thf(fact_195_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))).
% 2.75/1.51 thf(fact_196_prod__eqI, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B),Q:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P) = (((product_fst @ A) @ B) @ Q)) => (((((product_snd @ A) @ B) @ P) = (((product_snd @ A) @ B) @ Q)) => (P = Q)))))).
% 2.75/1.51 thf(fact_197_fst__eqD, axiom, (! [B:$tType,A:$tType,X:A,Y:B,A2:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X) @ Y)) = A2) => (X = A2))))).
% 2.75/1.51 thf(fact_198_snd__eqD, axiom, (! [B:$tType,A:$tType,X:B,Y:A,A2:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = A2) => (Y = A2))))).
% 2.75/1.51 thf(fact_199_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X2:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X2)) = X1)))).
% 2.75/1.51 thf(fact_200_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X2:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X2)) = X2)))).
% 2.75/1.51 thf(fact_201_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 2.75/1.51 thf(fact_202_surjective__pairing, axiom, (! [B:$tType,A:$tType,T3:((product_prod @ A) @ B)]: ((T3 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T3)) @ (((product_snd @ A) @ B) @ T3)))))).
% 2.75/1.51 thf(fact_203_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P3:(A > (B > $o)),X:A,Y:B,A2:((product_prod @ A) @ B)]: ((((P3 @ X) @ Y) => ((A2 = ((((product_Pair @ A) @ B) @ X) @ Y)) => ((P3 @ (((product_fst @ A) @ B) @ A2)) @ (((product_snd @ A) @ B) @ A2))))))).
% 2.75/1.51 thf(fact_204_conjI__realizer, axiom, (! [A:$tType,B:$tType,P3:(A > $o),P:A,Q2:(B > $o),Q:B]: (((P3 @ P) => ((Q2 @ Q) => ((P3 @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P) @ Q))) & (Q2 @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P) @ Q))))))))).
% 2.75/1.51 thf(fact_205_exI__realizer, axiom, (! [B:$tType,A:$tType,P3:(A > (B > $o)),Y:A,X:B]: ((((P3 @ Y) @ X) => ((P3 @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))))))).
% 2.75/1.51 thf(fact_206_sndI, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),Y:A,Z:B]: (((X = ((((product_Pair @ A) @ B) @ Y) @ Z)) => ((((product_snd @ A) @ B) @ X) = Z))))).
% 2.75/1.51 thf(fact_207_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B2:A,P:((product_prod @ B) @ A)]: (((B2 = (((product_snd @ B) @ A) @ P)) = (? [A7:B]: ((P = ((((product_Pair @ B) @ A) @ A7) @ B2)))))))).
% 2.75/1.51 thf(fact_208_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A2:A,P:((product_prod @ A) @ B)]: (((A2 = (((product_fst @ A) @ B) @ P)) = (? [B5:B]: ((P = ((((product_Pair @ A) @ B) @ A2) @ B5)))))))).
% 2.75/1.51 thf(fact_209_fstI, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),Y:A,Z:B]: (((X = ((((product_Pair @ A) @ B) @ Y) @ Z)) => ((((product_fst @ A) @ B) @ X) = Y))))).
% 2.75/1.51 thf(fact_210_same__fstI, axiom, (! [B:$tType,A:$tType,P3:(A > $o),X:A,Y7:B,Y:B,R2:(A > (set @ ((product_prod @ B) @ B)))]: (((P3 @ X) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y7) @ Y)) @ (R2 @ X)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y7)) @ ((((product_Pair @ A) @ B) @ X) @ Y))) @ ((((same_fst @ A) @ B) @ P3) @ R2))))))).
% 2.75/1.51 thf(fact_211_wf__map__prod__image, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),F:(A > B)]: ((((wf @ A) @ R) => (((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((wf @ B) @ ((((image @ ((product_prod @ A) @ A)) @ ((product_prod @ B) @ B)) @ ((((((product_map_prod @ A) @ B) @ A) @ B) @ F) @ F)) @ R))))))).
% 2.75/1.51 thf(fact_212_wf__def, axiom, (! [A:$tType]: (((wf @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [P5:(A > $o)]: (((! [X4:A]: (((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X4)) @ R3) => (P5 @ Y4)))) => (P5 @ X4)))) => (! [X7:A]: ((P5 @ X7)))))))))))).
% 2.75/1.51 thf(fact_213_wfE__min, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),X:A,Q2:(set @ A)]: ((((wf @ A) @ R2) => ((((member @ A) @ X) @ Q2) => (~ ((! [Z3:A]: (((((member @ A) @ Z3) @ Q2) => (~ ((! [Y8:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y8) @ Z3)) @ R2) => (~ ((((member @ A) @ Y8) @ Q2)))))))))))))))))).
% 2.75/1.51 thf(fact_214_wfI__min, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((! [X5:A,Q4:(set @ A)]: (((((member @ A) @ X5) @ Q4) => (? [Xa3:A]: (((((member @ A) @ Xa3) @ Q4) & (! [Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ Xa3)) @ R2) => (~ ((((member @ A) @ Y5) @ Q4)))))))))))) => ((wf @ A) @ R2))))).
% 2.75/1.51 thf(fact_215_wfUNIVI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((! [P6:(A > $o),X5:A]: (((! [Xa3:A]: (((! [Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ Xa3)) @ R) => (P6 @ Y5)))) => (P6 @ Xa3)))) => (P6 @ X5)))) => ((wf @ A) @ R))))).
% 2.75/1.51 thf(fact_216_wf__asym, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A2:A,X:A]: ((((wf @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ X)) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ A2)) @ R)))))))).
% 2.75/1.51 thf(fact_217_wf__induct, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),P3:(A > $o),A2:A]: ((((wf @ A) @ R) => ((! [X5:A]: (((! [Y8:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y8) @ X5)) @ R) => (P3 @ Y8)))) => (P3 @ X5)))) => (P3 @ A2)))))).
% 2.75/1.51 thf(fact_218_wf__irrefl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A2:A]: ((((wf @ A) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R))))))).
% 2.75/1.51 thf(fact_219_wf__not__sym, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A2:A,X:A]: ((((wf @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ X)) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ A2)) @ R)))))))).
% 2.75/1.51 thf(fact_220_wf__not__refl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A2:A]: ((((wf @ A) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R))))))).
% 2.75/1.51 thf(fact_221_wf__eq__minimal, axiom, (! [A:$tType]: (((wf @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [Q5:(set @ A)]: (((? [X4:A]: ((((member @ A) @ X4) @ Q5))) => (? [X4:A]: (((((member @ A) @ X4) @ Q5) & (! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X4)) @ R3) => (~ ((((member @ A) @ Y4) @ Q5)))))))))))))))))).
% 2.75/1.51 thf(fact_222_wf__induct__rule, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),P3:(A > $o),A2:A]: ((((wf @ A) @ R) => ((! [X5:A]: (((! [Y8:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y8) @ X5)) @ R) => (P3 @ Y8)))) => (P3 @ X5)))) => (P3 @ A2)))))).
% 2.75/1.51 thf(fact_223_dependent__wf__choice, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ A)),P3:((A > B) > (A > (B > $o)))]: ((((wf @ A) @ R2) => ((! [F8:(A > B),G3:(A > B),X5:A,R4:B]: (((! [Z4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z4) @ X5)) @ R2) => ((F8 @ Z4) = (G3 @ Z4))))) => ((((P3 @ F8) @ X5) @ R4) = (((P3 @ G3) @ X5) @ R4))))) => ((! [X5:A,F8:(A > B)]: (((! [Y8:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y8) @ X5)) @ R2) => (((P3 @ F8) @ Y8) @ (F8 @ Y8))))) => (? [X_1:B]: ((((P3 @ F8) @ X5) @ X_1)))))) => (? [F8:(A > B)]: ((! [X6:A]: ((((P3 @ F8) @ X6) @ (F8 @ X6)))))))))))).
% 2.75/1.51 thf(fact_224_sndOp__def, axiom, (! [B:$tType,A:$tType,C:$tType]: (((((bNF_sndOp @ C) @ A) @ B) = (^ [P5:(C > (A > $o)),Q5:(A > (B > $o)),Ac:((product_prod @ C) @ B)]: (((((product_Pair @ A) @ B) @ (((((((bNF_pick_middlep @ C) @ A) @ B) @ P5) @ Q5) @ (((product_fst @ C) @ B) @ Ac)) @ (((product_snd @ C) @ B) @ Ac))) @ (((product_snd @ C) @ B) @ Ac)))))))).
% 2.75/1.51 thf(fact_225_fstOp__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((bNF_fstOp @ A) @ B) @ C) = (^ [P5:(A > (B > $o)),Q5:(B > (C > $o)),Ac:((product_prod @ A) @ C)]: (((((product_Pair @ A) @ B) @ (((product_fst @ A) @ C) @ Ac)) @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P5) @ Q5) @ (((product_fst @ A) @ C) @ Ac)) @ (((product_snd @ A) @ C) @ Ac))))))))).
% 2.75/1.51 thf(fact_226_image2__eqI, axiom, (! [A:$tType,C:$tType,B:$tType,B2:A,F:(B > A),X:B,C3:C,G:(B > C),A3:(set @ B)]: (((B2 = (F @ X)) => ((C3 = (G @ X)) => ((((member @ B) @ X) @ A3) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ B2) @ C3)) @ ((((((bNF_Greatest_image2 @ B) @ A) @ C) @ A3) @ F) @ G)))))))).
% 2.75/1.51 thf(fact_227_the__inv__into__into, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A),X:B,B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A3) => ((((member @ B) @ X) @ ((((image @ A) @ B) @ F) @ A3)) => ((((ord_less_eq @ (set @ A)) @ A3) @ B6) => (((member @ A) @ (((((the_inv_into @ A) @ B) @ A3) @ F) @ X)) @ B6))))))).
% 2.75/1.51 thf(fact_228_strict__mono__inv, axiom, (! [A:$tType,B:$tType]: ((((linorder @ B) & (linorder @ A)) => (! [F:(A > B),G:(B > A)]: (((((order_strict_mono @ A) @ B) @ F) => ((((((image @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((! [X5:A]: (((G @ (F @ X5)) = X5))) => (((order_strict_mono @ B) @ A) @ G)))))))))).
% 2.75/1.51 thf(fact_229_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.75/1.51 thf(fact_230_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ A3) => (A3 = B6)))))).
% 2.75/1.51 thf(fact_231_subsetI, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((member @ A) @ X5) @ B6)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B6))))).
% 2.75/1.51 thf(fact_232_subrelI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: (((! [X5:A,Y5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y5)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y5)) @ S3)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S3))))).
% 2.75/1.51 thf(fact_233_inj__on__image__mem__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),B6:(set @ A),A2:A,A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B6) => ((((member @ A) @ A2) @ B6) => ((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((member @ B) @ (F @ A2)) @ ((((image @ A) @ B) @ F) @ A3)) = (((member @ A) @ A2) @ A3)))))))).
% 2.75/1.51 thf(fact_234_inj__on__image__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),C4:(set @ A),A3:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ C4) => ((((ord_less_eq @ (set @ A)) @ A3) @ C4) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => ((((((image @ A) @ B) @ F) @ A3) = ((((image @ A) @ B) @ F) @ B6)) = (A3 = B6)))))))).
% 2.75/1.51 thf(fact_235_inj__on__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A3) => ((((ord_less_eq @ (set @ A)) @ B6) @ A3) => ((((inj_on @ A) @ B) @ F) @ B6)))))).
% 2.75/1.51 thf(fact_236_subset__inj__on, axiom, (! [B:$tType,A:$tType,F:(A > B),B6:(set @ A),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B6) => ((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((inj_on @ A) @ B) @ F) @ A3)))))).
% 2.75/1.51 thf(fact_237_strict__mono__less__eq, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),X:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less_eq @ B) @ (F @ X)) @ (F @ Y)) = (((ord_less_eq @ A) @ X) @ Y))))))))).
% 2.75/1.51 thf(fact_238_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.75/1.51 thf(fact_239_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y6:A,Z2:A]: ((Y6 = Z2))) = (^ [A7:A,B5:A]: (((((ord_less_eq @ A) @ B5) @ A7) & (((ord_less_eq @ A) @ A7) @ B5))))))))).
% 2.75/1.51 thf(fact_240_strict__mono__eq, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),X:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F) => (((F @ X) = (F @ Y)) = (X = Y))))))))).
% 2.75/1.51 thf(fact_241_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C3:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C3) @ B2) => (((ord_less_eq @ A) @ C3) @ A2))))))))).
% 2.75/1.51 thf(fact_242_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > (A > $o)),A2:A,B2:A]: (((! [A5:A,B3:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((P3 @ A5) @ B3)))) => ((! [A5:A,B3:A]: ((((P3 @ B3) @ A5) => ((P3 @ A5) @ B3)))) => ((P3 @ A2) @ B2))))))))).
% 2.75/1.51 thf(fact_243_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.75/1.51 thf(fact_244_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.75/1.51 thf(fact_245_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.75/1.51 thf(fact_246_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.75/1.51 thf(fact_247_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C3:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.75/1.51 thf(fact_248_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y6:A,Z2:A]: ((Y6 = Z2))) = (^ [A7:A,B5:A]: (((((ord_less_eq @ A) @ A7) @ B5) & (((ord_less_eq @ A) @ B5) @ A7))))))))).
% 2.75/1.51 thf(fact_249_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.75/1.51 thf(fact_250_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z)))) => (((((ord_less_eq @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X)))) => (~ (((((ord_less_eq @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.75/1.51 thf(fact_251_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.75/1.51 thf(fact_252_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.75/1.51 thf(fact_253_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.75/1.51 thf(fact_254_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.75/1.51 thf(fact_255_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.75/1.51 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 2.75/1.51 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 2.75/1.51 thf(tcon_fun___Orderings_Otop, axiom, (! [A8:$tType,A9:$tType]: (((top @ A9) => (top @ (A8 > A9)))))).
% 2.75/1.51 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 2.75/1.51 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, (preorder @ nat)).
% 2.75/1.51 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.75/1.51 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, (order @ nat)).
% 2.75/1.51 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, (ord @ nat)).
% 2.75/1.51 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 2.75/1.51 thf(tcon_Set_Oset___Orderings_Oorder_5, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 2.75/1.51 thf(tcon_Set_Oset___Orderings_Otop_6, axiom, (! [A8:$tType]: ((top @ (set @ A8))))).
% 2.75/1.51 thf(tcon_Set_Oset___Orderings_Oord_7, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 2.75/1.51 thf(tcon_HOL_Obool___Orderings_Opreorder_8, axiom, (preorder @ $o)).
% 2.75/1.51 thf(tcon_HOL_Obool___Orderings_Olinorder_9, axiom, (linorder @ $o)).
% 2.75/1.51 thf(tcon_HOL_Obool___Orderings_Oorder_10, axiom, (order @ $o)).
% 2.75/1.51 thf(tcon_HOL_Obool___Orderings_Otop_11, axiom, (top @ $o)).
% 2.75/1.51 thf(tcon_HOL_Obool___Orderings_Oord_12, axiom, (ord @ $o)).
% 2.75/1.51 thf(help_If_3_1_T, axiom, (! [P3:$o]: (((P3 = $true) | (P3 = $false))))).
% 2.75/1.51 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.75/1.51 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.75/1.51 thf(conj_0, conjecture, (((paraco876059933e_eval @ i) @ ((paraco2100061555le_Con @ p1) @ p2)) = ((paraco876059933e_eval @ i) @ p1))).
% 2.75/1.51 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.cVWy2Pj8hg/DTF2DTF31276.p
% 2.75/1.51 ------------------------
% 2.75/1.54 ---- Cleaned THF ---
% 2.75/1.54 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv,type,
% 2.75/1.54 paraco415392788lle_tv: $tType ).
% 2.75/1.54
% 2.75/1.54 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm,type,
% 2.75/1.54 paraco414474393lle_fm: $tType ).
% 2.75/1.54
% 2.75/1.54 thf(ty_t_Product__Type_Oprod,type,
% 2.75/1.54 product_prod: $tType > $tType > $tType ).
% 2.75/1.54
% 2.75/1.54 thf(ty_t_String_Ochar,type,
% 2.75/1.54 char: $tType ).
% 2.75/1.54
% 2.75/1.54 thf(ty_t_List_Olist,type,
% 2.75/1.54 list: $tType > $tType ).
% 2.75/1.54
% 2.75/1.54 thf(ty_t_Set_Oset,type,
% 2.75/1.54 set: $tType > $tType ).
% 2.75/1.54
% 2.75/1.54 thf(ty_t_Nat_Onat,type,
% 2.75/1.54 nat: $tType ).
% 2.75/1.54
% 2.75/1.54 thf(sy_cl_HOL_Otype,type,
% 2.75/1.54 type:
% 2.75/1.54 !>[A: $tType] : $o ).
% 2.75/1.54
% 2.75/1.54 thf(sy_cl_Orderings_Oord,type,
% 2.75/1.54 ord:
% 2.75/1.54 !>[A: $tType] : $o ).
% 2.75/1.54
% 2.75/1.54 thf(sy_cl_Orderings_Otop,type,
% 2.75/1.54 top:
% 2.75/1.54 !>[A: $tType] : $o ).
% 2.75/1.54
% 2.75/1.54 thf(sy_cl_Orderings_Oorder,type,
% 2.75/1.54 order:
% 2.75/1.54 !>[A: $tType] : $o ).
% 2.75/1.54
% 2.75/1.54 thf(sy_cl_Orderings_Olinorder,type,
% 2.75/1.54 linorder:
% 2.75/1.54 !>[A: $tType] : $o ).
% 2.75/1.54
% 2.75/1.54 thf(sy_cl_Orderings_Opreorder,type,
% 2.75/1.54 preorder:
% 2.75/1.54 !>[A: $tType] : $o ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_BNF__Def_OfstOp,type,
% 2.75/1.54 bNF_fstOp:
% 2.75/1.54 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_BNF__Def_Opick__middlep,type,
% 2.75/1.54 bNF_pick_middlep:
% 2.75/1.54 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_BNF__Def_OsndOp,type,
% 2.75/1.54 bNF_sndOp:
% 2.75/1.54 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 2.75/1.54 bNF_Greatest_image2:
% 2.75/1.54 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Fun_Oinj__on,type,
% 2.75/1.54 inj_on:
% 2.75/1.54 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Fun_Othe__inv__into,type,
% 2.75/1.54 the_inv_into:
% 2.75/1.54 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_If,type,
% 2.75/1.54 if:
% 2.75/1.54 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.75/1.54 ord_less_eq:
% 2.75/1.54 !>[A: $tType] : ( A > A > $o ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 2.75/1.54 order_strict_mono:
% 2.75/1.54 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.75/1.54 top_top:
% 2.75/1.54 !>[A: $tType] : A ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__int,type,
% 2.75/1.54 paraco753463838ge_int: ( nat > nat ) > ( ( list @ char ) > paraco415392788lle_tv ) > ( list @ char ) > paraco415392788lle_tv ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv,type,
% 2.75/1.54 paraco1920534163nge_tv: ( nat > nat ) > paraco415392788lle_tv > paraco415392788lle_tv ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv__rel,type,
% 2.75/1.54 paraco2077297190tv_rel: ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) > ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) > $o ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval,type,
% 2.75/1.54 paraco876059933e_eval: ( ( list @ char ) > paraco415392788lle_tv ) > paraco414474393lle_fm > paraco415392788lle_tv ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H,type,
% 2.75/1.54 paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql,type,
% 2.75/1.54 paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H,type,
% 2.75/1.54 paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H,type,
% 2.75/1.54 paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro,type,
% 2.75/1.54 paraco27778325le_Pro: ( list @ char ) > paraco414474393lle_fm ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth,type,
% 2.75/1.54 paraco251304083_Truth: paraco414474393lle_fm ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Ocase__fm,type,
% 2.75/1.54 paraco1246693743ase_fm:
% 2.75/1.54 !>[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 ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Orec__fm,type,
% 2.75/1.54 paraco815940159rec_fm:
% 2.75/1.54 !>[A: $tType] : ( ( ( list @ char ) > A ) > A > ( paraco414474393lle_fm > A > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A > A > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A > A > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A > A > A ) > paraco414474393lle_fm > A ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet,type,
% 2.75/1.54 paraco2040174112le_Det: $o > paraco415392788lle_tv ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet,type,
% 2.75/1.54 paraco676387099_Indet: nat > paraco415392788lle_tv ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Ocase__tv,type,
% 2.75/1.54 paraco490622181ase_tv:
% 2.75/1.54 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv,type,
% 2.75/1.54 paraco152590079rec_tv:
% 2.75/1.54 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid,type,
% 2.75/1.54 paraco769098683_valid: paraco414474393lle_fm > $o ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid__in,type,
% 2.75/1.54 paraco2086025920lid_in: ( set @ nat ) > paraco414474393lle_fm > $o ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Product__Type_OPair,type,
% 2.75/1.54 product_Pair:
% 2.75/1.54 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Product__Type_Oapfst,type,
% 2.75/1.54 product_apfst:
% 2.75/1.54 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Product__Type_Oapsnd,type,
% 2.75/1.54 product_apsnd:
% 2.75/1.54 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Product__Type_Obool_Ocase__bool,type,
% 2.75/1.54 product_case_bool:
% 2.75/1.54 !>[A: $tType] : ( A > A > $o > A ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Product__Type_Ocurry,type,
% 2.75/1.54 product_curry:
% 2.75/1.54 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.75/1.54 produc2004651681e_prod:
% 2.75/1.54 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Product__Type_Omap__prod,type,
% 2.75/1.54 product_map_prod:
% 2.75/1.54 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.75/1.54 product_rec_prod:
% 2.75/1.54 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 2.75/1.54 product_fst:
% 2.75/1.54 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 2.75/1.54 product_snd:
% 2.75/1.54 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.75/1.54 product_swap:
% 2.75/1.54 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Relation_Oinv__image,type,
% 2.75/1.54 inv_image:
% 2.75/1.54 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Relation_Oinv__imagep,type,
% 2.75/1.54 inv_imagep:
% 2.75/1.54 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Relation_Ototal__on,type,
% 2.75/1.54 total_on:
% 2.75/1.54 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Set_OCollect,type,
% 2.75/1.54 collect:
% 2.75/1.54 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Set_Oimage,type,
% 2.75/1.54 image:
% 2.75/1.54 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Wellfounded_Oaccp,type,
% 2.75/1.54 accp:
% 2.75/1.54 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Wellfounded_Olex__prod,type,
% 2.75/1.54 lex_prod:
% 2.75/1.54 !>[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 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Wellfounded_Owf,type,
% 2.75/1.54 wf:
% 2.75/1.54 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_Wfrec_Osame__fst,type,
% 2.75/1.54 same_fst:
% 2.75/1.54 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_c_member,type,
% 2.75/1.54 member:
% 2.75/1.54 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.75/1.54
% 2.75/1.54 thf(sy_v_f,type,
% 2.75/1.54 f: nat > nat ).
% 2.75/1.54
% 2.75/1.54 thf(sy_v_i,type,
% 2.75/1.54 i: ( list @ char ) > paraco415392788lle_tv ).
% 2.75/1.54
% 2.75/1.54 thf(sy_v_p1____,type,
% 2.75/1.54 p1: paraco414474393lle_fm ).
% 2.75/1.54
% 2.75/1.54 thf(sy_v_p2____,type,
% 2.75/1.54 p2: paraco414474393lle_fm ).
% 2.75/1.54
% 2.75/1.54 thf(fact_0_a_H_H,axiom,
% 2.75/1.54 ( ( paraco876059933e_eval @ i @ p1 )
% 2.75/1.54 != ( paraco2040174112le_Det @ $true ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_1_a,axiom,
% 2.75/1.54 ( ( paraco876059933e_eval @ i @ p2 )
% 2.75/1.54 = ( paraco2040174112le_Det @ $true ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_2_a_H,axiom,
% 2.75/1.54 ( ( paraco876059933e_eval @ i @ p1 )
% 2.75/1.54 != ( paraco876059933e_eval @ i @ p2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_3_fm_Oinject_I3_J,axiom,
% 2.75/1.54 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,Y41: paraco414474393lle_fm,Y42: paraco414474393lle_fm] :
% 2.75/1.54 ( ( ( paraco2100061555le_Con @ X41 @ X42 )
% 2.75/1.54 = ( paraco2100061555le_Con @ Y41 @ Y42 ) )
% 2.75/1.54 = ( ( X41 = Y41 )
% 2.75/1.54 & ( X42 = Y42 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_4_b_H,axiom,
% 2.75/1.54 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
% 2.75/1.54 != ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_5_eval_Osimps_I4_J,axiom,
% 2.75/1.54 ! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.75/1.54 ( ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 = ( paraco876059933e_eval @ I @ Q ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco2100061555le_Con @ P @ Q ) )
% 2.75/1.54 = ( paraco876059933e_eval @ I @ P ) ) )
% 2.75/1.54 & ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 != ( paraco876059933e_eval @ I @ Q ) )
% 2.75/1.54 => ( ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 = ( paraco2040174112le_Det @ $true ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco2100061555le_Con @ P @ Q ) )
% 2.75/1.54 = ( paraco876059933e_eval @ I @ Q ) ) )
% 2.75/1.54 & ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 != ( paraco2040174112le_Det @ $true ) )
% 2.75/1.54 => ( ( ( ( paraco876059933e_eval @ I @ Q )
% 2.75/1.54 = ( paraco2040174112le_Det @ $true ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco2100061555le_Con @ P @ Q ) )
% 2.75/1.54 = ( paraco876059933e_eval @ I @ P ) ) )
% 2.75/1.54 & ( ( ( paraco876059933e_eval @ I @ Q )
% 2.75/1.54 != ( paraco2040174112le_Det @ $true ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco2100061555le_Con @ P @ Q ) )
% 2.75/1.54 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_6_ih2,axiom,
% 2.75/1.54 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 )
% 2.75/1.54 = ( paraco1920534163nge_tv @ f @ ( paraco876059933e_eval @ i @ p2 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_7_ih1,axiom,
% 2.75/1.54 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
% 2.75/1.54 = ( paraco1920534163nge_tv @ f @ ( paraco876059933e_eval @ i @ p1 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_8_b_H_H,axiom,
% 2.75/1.54 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
% 2.75/1.54 != ( paraco2040174112le_Det @ $true ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_9_tv_Oinject_I1_J,axiom,
% 2.75/1.54 ! [X1: $o,Y1: $o] :
% 2.75/1.54 ( ( ( paraco2040174112le_Det @ X1 )
% 2.75/1.54 = ( paraco2040174112le_Det @ Y1 ) )
% 2.75/1.54 = ( X1 = Y1 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_10_conjunction,axiom,
% 2.75/1.54 ! [P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco769098683_valid @ ( paraco2100061555le_Con @ P @ Q ) )
% 2.75/1.54 = ( ( paraco769098683_valid @ P )
% 2.75/1.54 & ( paraco769098683_valid @ Q ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_11_conjunction__in,axiom,
% 2.75/1.54 ! [U: set @ nat,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco2086025920lid_in @ U @ ( paraco2100061555le_Con @ P @ Q ) )
% 2.75/1.54 = ( ( paraco2086025920lid_in @ U @ P )
% 2.75/1.54 & ( paraco2086025920lid_in @ U @ Q ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_12_fm_Osimps_I39_J,axiom,
% 2.75/1.54 ! [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] :
% 2.75/1.54 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2100061555le_Con @ X41 @ X42 ) )
% 2.75/1.54 = ( F4 @ X41 @ X42 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_13_fm_Odistinct_I13_J,axiom,
% 2.75/1.54 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.75/1.54 ( paraco251304083_Truth
% 2.75/1.54 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_14_fm_Osimps_I45_J,axiom,
% 2.75/1.54 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2100061555le_Con @ X41 @ X42 ) )
% 2.75/1.54 = ( F4 @ X41 @ X42 @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X41 ) @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X42 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_15_fm_Odistinct_I25_J,axiom,
% 2.75/1.54 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 2.75/1.54 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_16_fm_Oinject_I4_J,axiom,
% 2.75/1.54 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,Y51: paraco414474393lle_fm,Y52: paraco414474393lle_fm] :
% 2.75/1.54 ( ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 2.75/1.54 = ( paraco2084319816le_Eql @ Y51 @ Y52 ) )
% 2.75/1.54 = ( ( X51 = Y51 )
% 2.75/1.54 & ( X52 = Y52 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_17_fm_Osimps_I46_J,axiom,
% 2.75/1.54 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2084319816le_Eql @ X51 @ X52 ) )
% 2.75/1.54 = ( F5 @ X51 @ X52 @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X51 ) @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X52 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_18_fm_Osimps_I43_J,axiom,
% 2.75/1.54 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A] :
% 2.75/1.54 ( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ paraco251304083_Truth )
% 2.75/1.54 = F2 ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_19_fm_Osimps_I40_J,axiom,
% 2.75/1.54 ! [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] :
% 2.75/1.54 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco2084319816le_Eql @ X51 @ X52 ) )
% 2.75/1.54 = ( F5 @ X51 @ X52 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_20_fm_Osimps_I37_J,axiom,
% 2.75/1.54 ! [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] :
% 2.75/1.54 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ paraco251304083_Truth )
% 2.75/1.54 = F2 ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_21_fm_Odistinct_I15_J,axiom,
% 2.75/1.54 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.75/1.54 ( paraco251304083_Truth
% 2.75/1.54 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_22_change__tv_Osimps_I1_J,axiom,
% 2.75/1.54 ! [F: nat > nat,B2: $o] :
% 2.75/1.54 ( ( paraco1920534163nge_tv @ F @ ( paraco2040174112le_Det @ B2 ) )
% 2.75/1.54 = ( paraco2040174112le_Det @ B2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_23_transfer,axiom,
% 2.75/1.54 ! [U: set @ nat,P: paraco414474393lle_fm] :
% 2.75/1.54 ( ~ ( paraco2086025920lid_in @ U @ P )
% 2.75/1.54 => ~ ( paraco769098683_valid @ P ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_24_change__int__def,axiom,
% 2.75/1.54 ( paraco753463838ge_int
% 2.75/1.54 = ( ^ [F7: nat > nat,I2: ( list @ char ) > paraco415392788lle_tv,S: list @ char] : ( paraco1920534163nge_tv @ F7 @ ( I2 @ S ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_25_valid__valid__in,axiom,
% 2.75/1.54 ! [P: paraco414474393lle_fm,U: set @ nat] :
% 2.75/1.54 ( ( paraco769098683_valid @ P )
% 2.75/1.54 => ( paraco2086025920lid_in @ U @ P ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_26_eval__equality__simplify,axiom,
% 2.75/1.54 ! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco876059933e_eval @ I @ ( paraco2084319816le_Eql @ P @ Q ) )
% 2.75/1.54 = ( paraco2040174112le_Det
% 2.75/1.54 @ ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 = ( paraco876059933e_eval @ I @ Q ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_27_eval_Osimps_I5_J,axiom,
% 2.75/1.54 ! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.75/1.54 ( ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 = ( paraco876059933e_eval @ I @ Q ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco2084319816le_Eql @ P @ Q ) )
% 2.75/1.54 = ( paraco2040174112le_Det @ $true ) ) )
% 2.75/1.54 & ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 != ( paraco876059933e_eval @ I @ Q ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco2084319816le_Eql @ P @ Q ) )
% 2.75/1.54 = ( paraco2040174112le_Det @ $false ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_28_eval_Osimps_I2_J,axiom,
% 2.75/1.54 ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 2.75/1.54 ( ( paraco876059933e_eval @ I @ paraco251304083_Truth )
% 2.75/1.54 = ( paraco2040174112le_Det @ $true ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_29_valid__def,axiom,
% 2.75/1.54 ( paraco769098683_valid
% 2.75/1.54 = ( ^ [P2: paraco414474393lle_fm] :
% 2.75/1.54 ! [I2: ( list @ char ) > paraco415392788lle_tv] :
% 2.75/1.54 ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.75/1.54 = ( paraco2040174112le_Det @ $true ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_30_tv_Osimps_I7_J,axiom,
% 2.75/1.54 ! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
% 2.75/1.54 ( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
% 2.75/1.54 = ( F1 @ X1 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_31_change__tv_Oelims,axiom,
% 2.75/1.54 ! [X: nat > nat,Xa: paraco415392788lle_tv,Y: paraco415392788lle_tv] :
% 2.75/1.54 ( ( ( paraco1920534163nge_tv @ X @ Xa )
% 2.75/1.54 = Y )
% 2.75/1.54 => ( ! [B3: $o] :
% 2.75/1.54 ( ( Xa
% 2.75/1.54 = ( paraco2040174112le_Det @ B3 ) )
% 2.75/1.54 => ( Y
% 2.75/1.54 != ( paraco2040174112le_Det @ B3 ) ) )
% 2.75/1.54 => ~ ! [N: nat] :
% 2.75/1.54 ( ( Xa
% 2.75/1.54 = ( paraco676387099_Indet @ N ) )
% 2.75/1.54 => ( Y
% 2.75/1.54 != ( paraco676387099_Indet @ ( X @ N ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_32_tv_Osimps_I5_J,axiom,
% 2.75/1.54 ! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
% 2.75/1.54 ( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
% 2.75/1.54 = ( F1 @ X1 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_33_eval__negation,axiom,
% 2.75/1.54 ! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] :
% 2.75/1.54 ( ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 = ( paraco2040174112le_Det @ $false ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) )
% 2.75/1.54 = ( paraco2040174112le_Det @ $true ) ) )
% 2.75/1.54 & ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 != ( paraco2040174112le_Det @ $false ) )
% 2.75/1.54 => ( ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 = ( paraco2040174112le_Det @ $true ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) )
% 2.75/1.54 = ( paraco2040174112le_Det @ $false ) ) )
% 2.75/1.54 & ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 != ( paraco2040174112le_Det @ $true ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) )
% 2.75/1.54 = ( paraco876059933e_eval @ I @ P ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_34_assms,axiom,
% 2.75/1.54 inj_on @ nat @ nat @ f @ ( top_top @ ( set @ nat ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_35_change__tv_Osimps_I2_J,axiom,
% 2.75/1.54 ! [F: nat > nat,N2: nat] :
% 2.75/1.54 ( ( paraco1920534163nge_tv @ F @ ( paraco676387099_Indet @ N2 ) )
% 2.75/1.54 = ( paraco676387099_Indet @ ( F @ N2 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_36_fm_Osimps_I41_J,axiom,
% 2.75/1.54 ! [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,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco1628874225le_Eql @ X61 @ X62 ) )
% 2.75/1.54 = ( F6 @ X61 @ X62 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_37_fm_Osimps_I47_J,axiom,
% 2.75/1.54 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco1628874225le_Eql @ X61 @ X62 ) )
% 2.75/1.54 = ( F6 @ X61 @ X62 @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X61 ) @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X62 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_38_fm_Osimps_I38_J,axiom,
% 2.75/1.54 ! [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] :
% 2.75/1.54 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco329115265le_Neg @ X3 ) )
% 2.75/1.54 = ( F3 @ X3 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_39_fm_Odistinct_I17_J,axiom,
% 2.75/1.54 ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.75/1.54 ( paraco251304083_Truth
% 2.75/1.54 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_40_fm_Oinject_I2_J,axiom,
% 2.75/1.54 ! [X3: paraco414474393lle_fm,Y3: paraco414474393lle_fm] :
% 2.75/1.54 ( ( ( paraco329115265le_Neg @ X3 )
% 2.75/1.54 = ( paraco329115265le_Neg @ Y3 ) )
% 2.75/1.54 = ( X3 = Y3 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_41_fm_Oinject_I5_J,axiom,
% 2.75/1.54 ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm,Y61: paraco414474393lle_fm,Y62: paraco414474393lle_fm] :
% 2.75/1.54 ( ( ( paraco1628874225le_Eql @ X61 @ X62 )
% 2.75/1.54 = ( paraco1628874225le_Eql @ Y61 @ Y62 ) )
% 2.75/1.54 = ( ( X61 = Y61 )
% 2.75/1.54 & ( X62 = Y62 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_42_tv_Oinject_I2_J,axiom,
% 2.75/1.54 ! [X2: nat,Y2: nat] :
% 2.75/1.54 ( ( ( paraco676387099_Indet @ X2 )
% 2.75/1.54 = ( paraco676387099_Indet @ Y2 ) )
% 2.75/1.54 = ( X2 = Y2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_43_fm_Odistinct_I23_J,axiom,
% 2.75/1.54 ! [X3: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco329115265le_Neg @ X3 )
% 2.75/1.54 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_44_tv_Osimps_I6_J,axiom,
% 2.75/1.54 ! [A: $tType,F1: $o > A,F2: nat > A,X2: nat] :
% 2.75/1.54 ( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X2 ) )
% 2.75/1.54 = ( F2 @ X2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_45_mem__Collect__eq,axiom,
% 2.75/1.54 ! [A: $tType,A2: A,P3: A > $o] :
% 2.75/1.54 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 2.75/1.54 = ( P3 @ A2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_46_Collect__mem__eq,axiom,
% 2.75/1.54 ! [A: $tType,A3: set @ A] :
% 2.75/1.54 ( ( collect @ A
% 2.75/1.54 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 2.75/1.54 = A3 ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_47_Collect__cong,axiom,
% 2.75/1.54 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 2.75/1.54 ( ! [X5: A] :
% 2.75/1.54 ( ( P3 @ X5 )
% 2.75/1.54 = ( Q2 @ X5 ) )
% 2.75/1.54 => ( ( collect @ A @ P3 )
% 2.75/1.54 = ( collect @ A @ Q2 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_48_ext,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.75/1.54 ( ! [X5: A] :
% 2.75/1.54 ( ( F @ X5 )
% 2.75/1.54 = ( G @ X5 ) )
% 2.75/1.54 => ( F = G ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_49_tv_Osimps_I8_J,axiom,
% 2.75/1.54 ! [A: $tType,F1: $o > A,F2: nat > A,X2: nat] :
% 2.75/1.54 ( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X2 ) )
% 2.75/1.54 = ( F2 @ X2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_50_eval__equality,axiom,
% 2.75/1.54 ! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.75/1.54 ( ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 = ( paraco876059933e_eval @ I @ Q ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
% 2.75/1.54 = ( paraco2040174112le_Det @ $true ) ) )
% 2.75/1.54 & ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 != ( paraco876059933e_eval @ I @ Q ) )
% 2.75/1.54 => ( ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 = ( paraco2040174112le_Det @ $true ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
% 2.75/1.54 = ( paraco876059933e_eval @ I @ Q ) ) )
% 2.75/1.54 & ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 != ( paraco2040174112le_Det @ $true ) )
% 2.75/1.54 => ( ( ( ( paraco876059933e_eval @ I @ Q )
% 2.75/1.54 = ( paraco2040174112le_Det @ $true ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
% 2.75/1.54 = ( paraco876059933e_eval @ I @ P ) ) )
% 2.75/1.54 & ( ( ( paraco876059933e_eval @ I @ Q )
% 2.75/1.54 != ( paraco2040174112le_Det @ $true ) )
% 2.75/1.54 => ( ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 = ( paraco2040174112le_Det @ $false ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
% 2.75/1.54 = ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ Q ) ) ) )
% 2.75/1.54 & ( ( ( paraco876059933e_eval @ I @ P )
% 2.75/1.54 != ( paraco2040174112le_Det @ $false ) )
% 2.75/1.54 => ( ( ( ( paraco876059933e_eval @ I @ Q )
% 2.75/1.54 = ( paraco2040174112le_Det @ $false ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
% 2.75/1.54 = ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) ) ) )
% 2.75/1.54 & ( ( ( paraco876059933e_eval @ I @ Q )
% 2.75/1.54 != ( paraco2040174112le_Det @ $false ) )
% 2.75/1.54 => ( ( paraco876059933e_eval @ I @ ( paraco1628874225le_Eql @ P @ Q ) )
% 2.75/1.54 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_51_double__negation,axiom,
% 2.75/1.54 ( paraco876059933e_eval
% 2.75/1.54 = ( ^ [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm] : ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ ( paraco329115265le_Neg @ P2 ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_52_fm_Odistinct_I19_J,axiom,
% 2.75/1.54 ! [X3: paraco414474393lle_fm,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco329115265le_Neg @ X3 )
% 2.75/1.54 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_53_string__tv_Oinduct,axiom,
% 2.75/1.54 ! [P3: paraco415392788lle_tv > $o,A0: paraco415392788lle_tv] :
% 2.75/1.54 ( ( P3 @ ( paraco2040174112le_Det @ $true ) )
% 2.75/1.54 => ( ( P3 @ ( paraco2040174112le_Det @ $false ) )
% 2.75/1.54 => ( ! [N: nat] : ( P3 @ ( paraco676387099_Indet @ N ) )
% 2.75/1.54 => ( P3 @ A0 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_54_change__tv_Oinduct,axiom,
% 2.75/1.54 ! [P3: ( nat > nat ) > paraco415392788lle_tv > $o,A0: nat > nat,A1: paraco415392788lle_tv] :
% 2.75/1.54 ( ! [F8: nat > nat,B3: $o] : ( P3 @ F8 @ ( paraco2040174112le_Det @ B3 ) )
% 2.75/1.54 => ( ! [F8: nat > nat,N: nat] : ( P3 @ F8 @ ( paraco676387099_Indet @ N ) )
% 2.75/1.54 => ( P3 @ A0 @ A1 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_55_string__tv_Ocases,axiom,
% 2.75/1.54 ! [X: paraco415392788lle_tv] :
% 2.75/1.54 ( ( X
% 2.75/1.54 != ( paraco2040174112le_Det @ $true ) )
% 2.75/1.54 => ( ( X
% 2.75/1.54 != ( paraco2040174112le_Det @ $false ) )
% 2.75/1.54 => ~ ! [N: nat] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( paraco676387099_Indet @ N ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_56_tv_Oexhaust,axiom,
% 2.75/1.54 ! [Y: paraco415392788lle_tv] :
% 2.75/1.54 ( ! [X12: $o] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( paraco2040174112le_Det @ X12 ) )
% 2.75/1.54 => ~ ! [X22: nat] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( paraco676387099_Indet @ X22 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_57_tv_Oinduct,axiom,
% 2.75/1.54 ! [P3: paraco415392788lle_tv > $o,Tv: paraco415392788lle_tv] :
% 2.75/1.54 ( ! [X5: $o] : ( P3 @ ( paraco2040174112le_Det @ X5 ) )
% 2.75/1.54 => ( ! [X5: nat] : ( P3 @ ( paraco676387099_Indet @ X5 ) )
% 2.75/1.54 => ( P3 @ Tv ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_58_tv_Odistinct_I1_J,axiom,
% 2.75/1.54 ! [X1: $o,X2: nat] :
% 2.75/1.54 ( ( paraco2040174112le_Det @ X1 )
% 2.75/1.54 != ( paraco676387099_Indet @ X2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_59_fm_Odistinct_I21_J,axiom,
% 2.75/1.54 ! [X3: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco329115265le_Neg @ X3 )
% 2.75/1.54 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_60_fm_Odistinct_I11_J,axiom,
% 2.75/1.54 ! [X3: paraco414474393lle_fm] :
% 2.75/1.54 ( paraco251304083_Truth
% 2.75/1.54 != ( paraco329115265le_Neg @ X3 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_61_fm_Odistinct_I27_J,axiom,
% 2.75/1.54 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 2.75/1.54 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_62_fm_Osimps_I44_J,axiom,
% 2.75/1.54 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X3: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco329115265le_Neg @ X3 ) )
% 2.75/1.54 = ( F3 @ X3 @ ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ X3 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_63_fm_Odistinct_I29_J,axiom,
% 2.75/1.54 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 2.75/1.54 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_64_iso__tuple__UNIV__I,axiom,
% 2.75/1.54 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_65_UNIV__I,axiom,
% 2.75/1.54 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_66_top__apply,axiom,
% 2.75/1.54 ! [C: $tType,D: $tType] :
% 2.75/1.54 ( ( top @ C )
% 2.75/1.54 => ( ( top_top @ ( D > C ) )
% 2.75/1.54 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_67_eval_Osimps_I3_J,axiom,
% 2.75/1.54 ! [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ P ) )
% 2.75/1.54 = ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I @ P ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_68_inj__def,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.75/1.54 = ( ! [X4: A,Y4: A] :
% 2.75/1.54 ( ( ( F @ X4 )
% 2.75/1.54 = ( F @ Y4 ) )
% 2.75/1.54 => ( X4 = Y4 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_69_inj__eq,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.75/1.54 => ( ( ( F @ X )
% 2.75/1.54 = ( F @ Y ) )
% 2.75/1.54 = ( X = Y ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_70_injI,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B] :
% 2.75/1.54 ( ! [X5: A,Y5: A] :
% 2.75/1.54 ( ( ( F @ X5 )
% 2.75/1.54 = ( F @ Y5 ) )
% 2.75/1.54 => ( X5 = Y5 ) )
% 2.75/1.54 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_71_injD,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.75/1.54 => ( ( ( F @ X )
% 2.75/1.54 = ( F @ Y ) )
% 2.75/1.54 => ( X = Y ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_72_top__set__def,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( top_top @ ( set @ A ) )
% 2.75/1.54 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_73_change__tv__injection,axiom,
% 2.75/1.54 ! [F: nat > nat] :
% 2.75/1.54 ( ( inj_on @ nat @ nat @ F @ ( top_top @ ( set @ nat ) ) )
% 2.75/1.54 => ( inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ ( paraco1920534163nge_tv @ F ) @ ( top_top @ ( set @ paraco415392788lle_tv ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_74_UNIV__eq__I,axiom,
% 2.75/1.54 ! [A: $tType,A3: set @ A] :
% 2.75/1.54 ( ! [X5: A] : ( member @ A @ X5 @ A3 )
% 2.75/1.54 => ( ( top_top @ ( set @ A ) )
% 2.75/1.54 = A3 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_75_UNIV__witness,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ? [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_76_inj__onD,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A,Y: A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ A3 )
% 2.75/1.54 => ( ( ( F @ X )
% 2.75/1.54 = ( F @ Y ) )
% 2.75/1.54 => ( ( member @ A @ X @ A3 )
% 2.75/1.54 => ( ( member @ A @ Y @ A3 )
% 2.75/1.54 => ( X = Y ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_77_inj__onI,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,A3: set @ A,F: A > B] :
% 2.75/1.54 ( ! [X5: A,Y5: A] :
% 2.75/1.54 ( ( member @ A @ X5 @ A3 )
% 2.75/1.54 => ( ( member @ A @ Y5 @ A3 )
% 2.75/1.54 => ( ( ( F @ X5 )
% 2.75/1.54 = ( F @ Y5 ) )
% 2.75/1.54 => ( X5 = Y5 ) ) ) )
% 2.75/1.54 => ( inj_on @ A @ B @ F @ A3 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_78_inj__on__def,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType] :
% 2.75/1.54 ( ( inj_on @ A @ B )
% 2.75/1.54 = ( ^ [F7: A > B,A4: set @ A] :
% 2.75/1.54 ! [X4: A] :
% 2.75/1.54 ( ( member @ A @ X4 @ A4 )
% 2.75/1.54 => ! [Y4: A] :
% 2.75/1.54 ( ( member @ A @ Y4 @ A4 )
% 2.75/1.54 => ( ( ( F7 @ X4 )
% 2.75/1.54 = ( F7 @ Y4 ) )
% 2.75/1.54 => ( X4 = Y4 ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_79_inj__on__cong,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,A3: set @ A,F: A > B,G: A > B] :
% 2.75/1.54 ( ! [A5: A] :
% 2.75/1.54 ( ( member @ A @ A5 @ A3 )
% 2.75/1.54 => ( ( F @ A5 )
% 2.75/1.54 = ( G @ A5 ) ) )
% 2.75/1.54 => ( ( inj_on @ A @ B @ F @ A3 )
% 2.75/1.54 = ( inj_on @ A @ B @ G @ A3 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_80_inj__on__eq__iff,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A,Y: A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ A3 )
% 2.75/1.54 => ( ( member @ A @ X @ A3 )
% 2.75/1.54 => ( ( member @ A @ Y @ A3 )
% 2.75/1.54 => ( ( ( F @ X )
% 2.75/1.54 = ( F @ Y ) )
% 2.75/1.54 = ( X = Y ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_81_inj__on__contraD,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A,Y: A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ A3 )
% 2.75/1.54 => ( ( X != Y )
% 2.75/1.54 => ( ( member @ A @ X @ A3 )
% 2.75/1.54 => ( ( member @ A @ Y @ A3 )
% 2.75/1.54 => ( ( F @ X )
% 2.75/1.54 != ( F @ Y ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_82_inj__on__inverseI,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,A3: set @ A,G: B > A,F: A > B] :
% 2.75/1.54 ( ! [X5: A] :
% 2.75/1.54 ( ( member @ A @ X5 @ A3 )
% 2.75/1.54 => ( ( G @ ( F @ X5 ) )
% 2.75/1.54 = X5 ) )
% 2.75/1.54 => ( inj_on @ A @ B @ F @ A3 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_83_fm_Oinduct,axiom,
% 2.75/1.54 ! [P3: paraco414474393lle_fm > $o,Fm: paraco414474393lle_fm] :
% 2.75/1.54 ( ! [X5: list @ char] : ( P3 @ ( paraco27778325le_Pro @ X5 ) )
% 2.75/1.54 => ( ( P3 @ paraco251304083_Truth )
% 2.75/1.54 => ( ! [X5: paraco414474393lle_fm] :
% 2.75/1.54 ( ( P3 @ X5 )
% 2.75/1.54 => ( P3 @ ( paraco329115265le_Neg @ X5 ) ) )
% 2.75/1.54 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 2.75/1.54 ( ( P3 @ X1a )
% 2.75/1.54 => ( ( P3 @ X22 )
% 2.75/1.54 => ( P3 @ ( paraco2100061555le_Con @ X1a @ X22 ) ) ) )
% 2.75/1.54 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 2.75/1.54 ( ( P3 @ X1a )
% 2.75/1.54 => ( ( P3 @ X22 )
% 2.75/1.54 => ( P3 @ ( paraco2084319816le_Eql @ X1a @ X22 ) ) ) )
% 2.75/1.54 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 2.75/1.54 ( ( P3 @ X1a )
% 2.75/1.54 => ( ( P3 @ X22 )
% 2.75/1.54 => ( P3 @ ( paraco1628874225le_Eql @ X1a @ X22 ) ) ) )
% 2.75/1.54 => ( P3 @ Fm ) ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_84_fm_Oexhaust,axiom,
% 2.75/1.54 ! [Y: paraco414474393lle_fm] :
% 2.75/1.54 ( ! [X12: list @ char] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( paraco27778325le_Pro @ X12 ) )
% 2.75/1.54 => ( ( Y != paraco251304083_Truth )
% 2.75/1.54 => ( ! [X32: paraco414474393lle_fm] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( paraco329115265le_Neg @ X32 ) )
% 2.75/1.54 => ( ! [X412: paraco414474393lle_fm,X422: paraco414474393lle_fm] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( paraco2100061555le_Con @ X412 @ X422 ) )
% 2.75/1.54 => ( ! [X512: paraco414474393lle_fm,X522: paraco414474393lle_fm] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( paraco2084319816le_Eql @ X512 @ X522 ) )
% 2.75/1.54 => ~ ! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_85_props_Ocases,axiom,
% 2.75/1.54 ! [X: paraco414474393lle_fm] :
% 2.75/1.54 ( ( X != paraco251304083_Truth )
% 2.75/1.54 => ( ! [S2: list @ char] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( paraco27778325le_Pro @ S2 ) )
% 2.75/1.54 => ( ! [P4: paraco414474393lle_fm] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( paraco329115265le_Neg @ P4 ) )
% 2.75/1.54 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( paraco2100061555le_Con @ P4 @ Q3 ) )
% 2.75/1.54 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( paraco2084319816le_Eql @ P4 @ Q3 ) )
% 2.75/1.54 => ~ ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_86_props_Oinduct,axiom,
% 2.75/1.54 ! [P3: paraco414474393lle_fm > $o,A0: paraco414474393lle_fm] :
% 2.75/1.54 ( ( P3 @ paraco251304083_Truth )
% 2.75/1.54 => ( ! [S2: list @ char] : ( P3 @ ( paraco27778325le_Pro @ S2 ) )
% 2.75/1.54 => ( ! [P4: paraco414474393lle_fm] :
% 2.75/1.54 ( ( P3 @ P4 )
% 2.75/1.54 => ( P3 @ ( paraco329115265le_Neg @ P4 ) ) )
% 2.75/1.54 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.75/1.54 ( ( P3 @ P4 )
% 2.75/1.54 => ( ( P3 @ Q3 )
% 2.75/1.54 => ( P3 @ ( paraco2100061555le_Con @ P4 @ Q3 ) ) ) )
% 2.75/1.54 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.75/1.54 ( ( P3 @ P4 )
% 2.75/1.54 => ( ( P3 @ Q3 )
% 2.75/1.54 => ( P3 @ ( paraco2084319816le_Eql @ P4 @ Q3 ) ) ) )
% 2.75/1.54 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.75/1.54 ( ( P3 @ P4 )
% 2.75/1.54 => ( ( P3 @ Q3 )
% 2.75/1.54 => ( P3 @ ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) )
% 2.75/1.54 => ( P3 @ A0 ) ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_87_old_Obool_Osimps_I4_J,axiom,
% 2.75/1.54 ! [A: $tType,F1: A,F2: A] :
% 2.75/1.54 ( ( product_case_bool @ A @ F1 @ F2 @ $false )
% 2.75/1.54 = F2 ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_88_old_Obool_Osimps_I3_J,axiom,
% 2.75/1.54 ! [A: $tType,F1: A,F2: A] :
% 2.75/1.54 ( ( product_case_bool @ A @ F1 @ F2 @ $true )
% 2.75/1.54 = F1 ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_89_bool_Osplit__sel,axiom,
% 2.75/1.54 ! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
% 2.75/1.54 ( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 2.75/1.54 = ( ( Bool
% 2.75/1.54 => ( P3 @ F1 ) )
% 2.75/1.54 & ( ~ Bool
% 2.75/1.54 => ( P3 @ F2 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_90_fm_Oinject_I1_J,axiom,
% 2.75/1.54 ! [X1: list @ char,Y1: list @ char] :
% 2.75/1.54 ( ( ( paraco27778325le_Pro @ X1 )
% 2.75/1.54 = ( paraco27778325le_Pro @ Y1 ) )
% 2.75/1.54 = ( X1 = Y1 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_91_top1I,axiom,
% 2.75/1.54 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_92_eval_Osimps_I1_J,axiom,
% 2.75/1.54 ! [I: ( list @ char ) > paraco415392788lle_tv,S3: list @ char] :
% 2.75/1.54 ( ( paraco876059933e_eval @ I @ ( paraco27778325le_Pro @ S3 ) )
% 2.75/1.54 = ( I @ S3 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_93_fm_Odistinct_I3_J,axiom,
% 2.75/1.54 ! [X1: list @ char,X3: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco27778325le_Pro @ X1 )
% 2.75/1.54 != ( paraco329115265le_Neg @ X3 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_94_fm_Odistinct_I5_J,axiom,
% 2.75/1.54 ! [X1: list @ char,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco27778325le_Pro @ X1 )
% 2.75/1.54 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_95_fm_Odistinct_I9_J,axiom,
% 2.75/1.54 ! [X1: list @ char,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco27778325le_Pro @ X1 )
% 2.75/1.54 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_96_fm_Odistinct_I7_J,axiom,
% 2.75/1.54 ! [X1: list @ char,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.75/1.54 ( ( paraco27778325le_Pro @ X1 )
% 2.75/1.54 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_97_fm_Odistinct_I1_J,axiom,
% 2.75/1.54 ! [X1: list @ char] :
% 2.75/1.54 ( ( paraco27778325le_Pro @ X1 )
% 2.75/1.54 != paraco251304083_Truth ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_98_fm_Osimps_I42_J,axiom,
% 2.75/1.54 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F3: paraco414474393lle_fm > A > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A > A > A,X1: list @ char] :
% 2.75/1.54 ( ( paraco815940159rec_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco27778325le_Pro @ X1 ) )
% 2.75/1.54 = ( F1 @ X1 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_99_fm_Osimps_I36_J,axiom,
% 2.75/1.54 ! [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] :
% 2.75/1.54 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( paraco27778325le_Pro @ X1 ) )
% 2.75/1.54 = ( F1 @ X1 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_100_bool_Osplit__sel__asm,axiom,
% 2.75/1.54 ! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
% 2.75/1.54 ( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 2.75/1.54 = ( ~ ( ( Bool
% 2.75/1.54 & ~ ( P3 @ F1 ) )
% 2.75/1.54 | ( ~ Bool
% 2.75/1.54 & ~ ( P3 @ F2 ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_101_bool_Ocase__distrib,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,H: A > B,F1: A,F2: A,Bool: $o] :
% 2.75/1.54 ( ( H @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 2.75/1.54 = ( product_case_bool @ B @ ( H @ F1 ) @ ( H @ F2 ) @ Bool ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_102_bool_Ocase__eq__if,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( product_case_bool @ A )
% 2.75/1.54 = ( ^ [F12: A,F22: A,Bool2: $o] : ( if @ A @ Bool2 @ F12 @ F22 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_103_top__empty__eq,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( top_top @ ( A > $o ) )
% 2.75/1.54 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_104_eval_Ocases,axiom,
% 2.75/1.54 ! [X: product_prod @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm] :
% 2.75/1.54 ( ! [I3: ( list @ char ) > paraco415392788lle_tv,S2: list @ char] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco27778325le_Pro @ S2 ) ) )
% 2.75/1.54 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ paraco251304083_Truth ) )
% 2.75/1.54 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco329115265le_Neg @ P4 ) ) )
% 2.75/1.54 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2100061555le_Con @ P4 @ Q3 ) ) )
% 2.75/1.54 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2084319816le_Eql @ P4 @ Q3 ) ) )
% 2.75/1.54 => ~ ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_105_top__conj_I1_J,axiom,
% 2.75/1.54 ! [A: $tType,X: A,P3: $o] :
% 2.75/1.54 ( ( ( top_top @ ( A > $o ) @ X )
% 2.75/1.54 & P3 )
% 2.75/1.54 = P3 ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_106_top__conj_I2_J,axiom,
% 2.75/1.54 ! [A: $tType,P3: $o,X: A] :
% 2.75/1.54 ( ( P3
% 2.75/1.54 & ( top_top @ ( A > $o ) @ X ) )
% 2.75/1.54 = P3 ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_107_inj__apsnd,axiom,
% 2.75/1.54 ! [A: $tType,C: $tType,B: $tType,F: B > C] :
% 2.75/1.54 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.75/1.54 = ( inj_on @ B @ C @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_108_inj__apfst,axiom,
% 2.75/1.54 ! [B: $tType,C: $tType,A: $tType,F: A > C] :
% 2.75/1.54 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.75/1.54 = ( inj_on @ A @ C @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_109_old_Oprod_Oinject,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
% 2.75/1.54 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.75/1.54 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 2.75/1.54 = ( ( A2 = A6 )
% 2.75/1.54 & ( B2 = B4 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_110_prod_Oinject,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 2.75/1.54 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 2.75/1.54 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.75/1.54 = ( ( X1 = Y1 )
% 2.75/1.54 & ( X2 = Y2 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_111_apfst__conv,axiom,
% 2.75/1.54 ! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y: B] :
% 2.75/1.54 ( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y ) )
% 2.75/1.54 = ( product_Pair @ A @ B @ ( F @ X ) @ Y ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_112_apsnd__conv,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y: C] :
% 2.75/1.54 ( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y ) )
% 2.75/1.54 = ( product_Pair @ A @ B @ X @ ( F @ Y ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_113_apsnd__apfst__commute,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P: product_prod @ D @ C] :
% 2.75/1.54 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P ) )
% 2.75/1.54 = ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_114_old_Oprod_Oinducts,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.75/1.54 ( ! [A5: A,B3: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 2.75/1.54 => ( P3 @ Prod ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_115_old_Oprod_Oexhaust,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.75/1.54 ~ ! [A5: A,B3: B] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( product_Pair @ A @ B @ A5 @ B3 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_116_prod__induct7,axiom,
% 2.75/1.54 ! [G2: $tType,F9: $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 @ F9 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) )] :
% 2.75/1.54 ( ! [A5: A,B3: B,C2: C,D2: D,E2: E,F8: F9,G3: G2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F9 @ G2 ) @ E2 @ ( product_Pair @ F9 @ G2 @ F8 @ G3 ) ) ) ) ) ) )
% 2.75/1.54 => ( P3 @ X ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_117_prod__induct6,axiom,
% 2.75/1.54 ! [F9: $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 @ F9 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) )] :
% 2.75/1.54 ( ! [A5: A,B3: B,C2: C,D2: D,E2: E,F8: F9] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F9 ) @ D2 @ ( product_Pair @ E @ F9 @ E2 @ F8 ) ) ) ) ) )
% 2.75/1.54 => ( P3 @ X ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_118_prod__induct5,axiom,
% 2.75/1.54 ! [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,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.75/1.54 ( ! [A5: A,B3: 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 ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 2.75/1.54 => ( P3 @ X ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_119_prod__induct4,axiom,
% 2.75/1.54 ! [D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.75/1.54 ( ! [A5: A,B3: B,C2: C,D2: D] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 2.75/1.54 => ( P3 @ X ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_120_prod__induct3,axiom,
% 2.75/1.54 ! [C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.75/1.54 ( ! [A5: A,B3: B,C2: C] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C2 ) ) )
% 2.75/1.54 => ( P3 @ X ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_121_prod__cases7,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F9: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) )] :
% 2.75/1.54 ~ ! [A5: A,B3: B,C2: C,D2: D,E2: E,F8: F9,G3: G2] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F9 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F9 @ G2 ) @ E2 @ ( product_Pair @ F9 @ G2 @ F8 @ G3 ) ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_122_prod__cases6,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F9: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) )] :
% 2.75/1.54 ~ ! [A5: A,B3: B,C2: C,D2: D,E2: E,F8: F9] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F9 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F9 ) @ D2 @ ( product_Pair @ E @ F9 @ E2 @ F8 ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_123_prod__cases5,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.75/1.54 ~ ! [A5: A,B3: B,C2: C,D2: D,E2: E] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_124_prod__cases4,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.75/1.54 ~ ! [A5: A,B3: B,C2: C,D2: D] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_125_prod__cases3,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.75/1.54 ~ ! [A5: A,B3: B,C2: C] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C2 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_126_Pair__inject,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
% 2.75/1.54 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.75/1.54 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 2.75/1.54 => ~ ( ( A2 = A6 )
% 2.75/1.54 => ( B2 != B4 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_127_prod__cases,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 2.75/1.54 ( ! [A5: A,B3: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 2.75/1.54 => ( P3 @ P ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_128_surj__pair,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 2.75/1.54 ? [X5: A,Y5: B] :
% 2.75/1.54 ( P
% 2.75/1.54 = ( product_Pair @ A @ B @ X5 @ Y5 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_129_old_Oprod_Orec,axiom,
% 2.75/1.54 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.75/1.54 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.75/1.54 = ( F1 @ A2 @ B2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_130_internal__case__prod__conv,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 2.75/1.54 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 2.75/1.54 = ( C3 @ A2 @ B2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_131_prod_Oinj__map,axiom,
% 2.75/1.54 ! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F2: B > D] :
% 2.75/1.54 ( ( inj_on @ A @ C @ F1 @ ( top_top @ ( set @ A ) ) )
% 2.75/1.54 => ( ( inj_on @ B @ D @ F2 @ ( top_top @ ( set @ B ) ) )
% 2.75/1.54 => ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_132_ssubst__Pair__rhs,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,R: A,S3: B,R2: set @ ( product_prod @ A @ B ),S4: B] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R2 )
% 2.75/1.54 => ( ( S4 = S3 )
% 2.75/1.54 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S4 ) @ R2 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_133_curry__conv,axiom,
% 2.75/1.54 ! [A: $tType,C: $tType,B: $tType] :
% 2.75/1.54 ( ( product_curry @ B @ C @ A )
% 2.75/1.54 = ( ^ [F7: ( product_prod @ B @ C ) > A,A7: B,B5: C] : ( F7 @ ( product_Pair @ B @ C @ A7 @ B5 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_134_curryI,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.75/1.54 ( ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.75/1.54 => ( product_curry @ A @ B @ $o @ F @ A2 @ B2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_135_map__prod__simp,axiom,
% 2.75/1.54 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A2: C,B2: D] :
% 2.75/1.54 ( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 2.75/1.54 = ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_136_curryD,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.75/1.54 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 2.75/1.54 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_137_curryE,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.75/1.54 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 2.75/1.54 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_138_change__tv_Ocases,axiom,
% 2.75/1.54 ! [X: product_prod @ ( nat > nat ) @ paraco415392788lle_tv] :
% 2.75/1.54 ( ! [F8: nat > nat,B3: $o] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ F8 @ ( paraco2040174112le_Det @ B3 ) ) )
% 2.75/1.54 => ~ ! [F8: nat > nat,N: nat] :
% 2.75/1.54 ( X
% 2.75/1.54 != ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ F8 @ ( paraco676387099_Indet @ N ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_139_in__inv__imagep,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType] :
% 2.75/1.54 ( ( inv_imagep @ A @ B )
% 2.75/1.54 = ( ^ [R3: A > A > $o,F7: B > A,X4: B,Y4: B] : ( R3 @ ( F7 @ X4 ) @ ( F7 @ Y4 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_140_the__inv__f__f,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.75/1.54 => ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X ) )
% 2.75/1.54 = X ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_141_change__tv_Opelims,axiom,
% 2.75/1.54 ! [X: nat > nat,Xa: paraco415392788lle_tv,Y: paraco415392788lle_tv] :
% 2.75/1.54 ( ( ( paraco1920534163nge_tv @ X @ Xa )
% 2.75/1.54 = Y )
% 2.75/1.54 => ( ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ Xa ) )
% 2.75/1.54 => ( ! [B3: $o] :
% 2.75/1.54 ( ( Xa
% 2.75/1.54 = ( paraco2040174112le_Det @ B3 ) )
% 2.75/1.54 => ( ( Y
% 2.75/1.54 = ( paraco2040174112le_Det @ B3 ) )
% 2.75/1.54 => ~ ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ ( paraco2040174112le_Det @ B3 ) ) ) ) )
% 2.75/1.54 => ~ ! [N: nat] :
% 2.75/1.54 ( ( Xa
% 2.75/1.54 = ( paraco676387099_Indet @ N ) )
% 2.75/1.54 => ( ( Y
% 2.75/1.54 = ( paraco676387099_Indet @ ( X @ N ) ) )
% 2.75/1.54 => ~ ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ ( paraco676387099_Indet @ N ) ) ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_142_in__inv__image,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ B @ B ),F: A > B] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R @ F ) )
% 2.75/1.54 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_143_the__inv__into__f__f,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ A3 )
% 2.75/1.54 => ( ( member @ A @ X @ A3 )
% 2.75/1.54 => ( ( the_inv_into @ A @ B @ A3 @ F @ ( F @ X ) )
% 2.75/1.54 = X ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_144_the__inv__into__f__eq,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: A,Y: B] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ A3 )
% 2.75/1.54 => ( ( ( F @ X )
% 2.75/1.54 = Y )
% 2.75/1.54 => ( ( member @ A @ X @ A3 )
% 2.75/1.54 => ( ( the_inv_into @ A @ B @ A3 @ F @ Y )
% 2.75/1.54 = X ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_145_total__inv__image,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,R: set @ ( product_prod @ B @ B )] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.75/1.54 => ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ R )
% 2.75/1.54 => ( total_on @ A @ ( top_top @ ( set @ A ) ) @ ( inv_image @ B @ A @ R @ F ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_146_the__inv__into__onto,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,A3: set @ A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ A3 )
% 2.75/1.54 => ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A3 @ F ) @ ( image @ A @ B @ F @ A3 ) )
% 2.75/1.54 = A3 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_147_map__prod__imageI,axiom,
% 2.75/1.54 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),F: A > C,G: B > D] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 2.75/1.54 => ( 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 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_148_inj__swap,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A3 ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_149_image__eqI,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,B2: A,F: B > A,X: B,A3: set @ B] :
% 2.75/1.54 ( ( B2
% 2.75/1.54 = ( F @ X ) )
% 2.75/1.54 => ( ( member @ B @ X @ A3 )
% 2.75/1.54 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_150_swap__swap,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 2.75/1.54 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 2.75/1.54 = P ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_151_swap__simp,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,X: B,Y: A] :
% 2.75/1.54 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 2.75/1.54 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_152_pair__in__swap__image,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,Y: A,X: B,A3: set @ ( product_prod @ B @ A )] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A3 ) )
% 2.75/1.54 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A3 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_153_surj__swap,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType] :
% 2.75/1.54 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.75/1.54 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_154_total__on__def,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( total_on @ A )
% 2.75/1.54 = ( ^ [A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 2.75/1.54 ! [X4: A] :
% 2.75/1.54 ( ( member @ A @ X4 @ A4 )
% 2.75/1.54 => ! [Y4: A] :
% 2.75/1.54 ( ( member @ A @ Y4 @ A4 )
% 2.75/1.54 => ( ( X4 != Y4 )
% 2.75/1.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 )
% 2.75/1.54 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R3 ) ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_155_total__onI,axiom,
% 2.75/1.54 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A )] :
% 2.75/1.54 ( ! [X5: A,Y5: A] :
% 2.75/1.54 ( ( member @ A @ X5 @ A3 )
% 2.75/1.54 => ( ( member @ A @ Y5 @ A3 )
% 2.75/1.54 => ( ( X5 != Y5 )
% 2.75/1.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y5 ) @ R )
% 2.75/1.54 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R ) ) ) ) )
% 2.75/1.54 => ( total_on @ A @ A3 @ R ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_156_map__prod__surj,axiom,
% 2.75/1.54 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
% 2.75/1.54 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.75/1.54 = ( top_top @ ( set @ B ) ) )
% 2.75/1.54 => ( ( ( image @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
% 2.75/1.54 = ( top_top @ ( set @ D ) ) )
% 2.75/1.54 => ( ( 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 ) ) ) )
% 2.75/1.54 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_157_range__eqI,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,B2: A,F: B > A,X: B] :
% 2.75/1.54 ( ( B2
% 2.75/1.54 = ( F @ X ) )
% 2.75/1.54 => ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_158_surj__def,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: B > A] :
% 2.75/1.54 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.75/1.54 = ( top_top @ ( set @ A ) ) )
% 2.75/1.54 = ( ! [Y4: A] :
% 2.75/1.54 ? [X4: B] :
% 2.75/1.54 ( Y4
% 2.75/1.54 = ( F @ X4 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_159_rangeI,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_160_surjI,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 2.75/1.54 ( ! [X5: A] :
% 2.75/1.54 ( ( G @ ( F @ X5 ) )
% 2.75/1.54 = X5 )
% 2.75/1.54 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 2.75/1.54 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_161_surjE,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 2.75/1.54 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.75/1.54 = ( top_top @ ( set @ A ) ) )
% 2.75/1.54 => ~ ! [X5: B] :
% 2.75/1.54 ( Y
% 2.75/1.54 != ( F @ X5 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_162_surjD,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 2.75/1.54 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.75/1.54 = ( top_top @ ( set @ A ) ) )
% 2.75/1.54 => ? [X5: B] :
% 2.75/1.54 ( Y
% 2.75/1.54 = ( F @ X5 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_163_inj__on__image__iff,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,A3: set @ A,G: A > B,F: A > A] :
% 2.75/1.54 ( ! [X5: A] :
% 2.75/1.54 ( ( member @ A @ X5 @ A3 )
% 2.75/1.54 => ! [Xa2: A] :
% 2.75/1.54 ( ( member @ A @ Xa2 @ A3 )
% 2.75/1.54 => ( ( ( G @ ( F @ X5 ) )
% 2.75/1.54 = ( G @ ( F @ Xa2 ) ) )
% 2.75/1.54 = ( ( G @ X5 )
% 2.75/1.54 = ( G @ Xa2 ) ) ) ) )
% 2.75/1.54 => ( ( inj_on @ A @ A @ F @ A3 )
% 2.75/1.54 => ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A3 ) )
% 2.75/1.54 = ( inj_on @ A @ B @ G @ A3 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_164_imageI,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,X: A,A3: set @ A,F: A > B] :
% 2.75/1.54 ( ( member @ A @ X @ A3 )
% 2.75/1.54 => ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_165_image__iff,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,Z: A,F: B > A,A3: set @ B] :
% 2.75/1.54 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A3 ) )
% 2.75/1.54 = ( ? [X4: B] :
% 2.75/1.54 ( ( member @ B @ X4 @ A3 )
% 2.75/1.54 & ( Z
% 2.75/1.54 = ( F @ X4 ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_166_bex__imageD,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
% 2.75/1.54 ( ? [X6: A] :
% 2.75/1.54 ( ( member @ A @ X6 @ ( image @ B @ A @ F @ A3 ) )
% 2.75/1.54 & ( P3 @ X6 ) )
% 2.75/1.54 => ? [X5: B] :
% 2.75/1.54 ( ( member @ B @ X5 @ A3 )
% 2.75/1.54 & ( P3 @ ( F @ X5 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_167_image__cong,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,M: set @ A,N3: set @ A,F: A > B,G: A > B] :
% 2.75/1.54 ( ( M = N3 )
% 2.75/1.54 => ( ! [X5: A] :
% 2.75/1.54 ( ( member @ A @ X5 @ N3 )
% 2.75/1.54 => ( ( F @ X5 )
% 2.75/1.54 = ( G @ X5 ) ) )
% 2.75/1.54 => ( ( image @ A @ B @ F @ M )
% 2.75/1.54 = ( image @ A @ B @ G @ N3 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_168_ball__imageD,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P3: A > $o] :
% 2.75/1.54 ( ! [X5: A] :
% 2.75/1.54 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A3 ) )
% 2.75/1.54 => ( P3 @ X5 ) )
% 2.75/1.54 => ! [X6: B] :
% 2.75/1.54 ( ( member @ B @ X6 @ A3 )
% 2.75/1.54 => ( P3 @ ( F @ X6 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_169_rev__image__eqI,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,X: A,A3: set @ A,B2: B,F: A > B] :
% 2.75/1.54 ( ( member @ A @ X @ A3 )
% 2.75/1.54 => ( ( B2
% 2.75/1.54 = ( F @ X ) )
% 2.75/1.54 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_170_inj__image__mem__iff,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,A2: A,A3: set @ A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.75/1.54 => ( ( member @ B @ ( F @ A2 ) @ ( image @ A @ B @ F @ A3 ) )
% 2.75/1.54 = ( member @ A @ A2 @ A3 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_171_inj__image__eq__iff,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,B6: set @ A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.75/1.54 => ( ( ( image @ A @ B @ F @ A3 )
% 2.75/1.54 = ( image @ A @ B @ F @ B6 ) )
% 2.75/1.54 = ( A3 = B6 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_172_range__ex1__eq,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,B2: B] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.75/1.54 => ( ( member @ B @ B2 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
% 2.75/1.54 = ( ? [X4: A] :
% 2.75/1.54 ( ( B2
% 2.75/1.54 = ( F @ X4 ) )
% 2.75/1.54 & ! [Y4: A] :
% 2.75/1.54 ( ( B2
% 2.75/1.54 = ( F @ Y4 ) )
% 2.75/1.54 => ( Y4 = X4 ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_173_prod__fun__imageE,axiom,
% 2.75/1.54 ! [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 )] :
% 2.75/1.54 ( ( 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 ) )
% 2.75/1.54 => ~ ! [X5: C,Y5: D] :
% 2.75/1.54 ( ( C3
% 2.75/1.54 = ( product_Pair @ A @ B @ ( F @ X5 ) @ ( G @ Y5 ) ) )
% 2.75/1.54 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X5 @ Y5 ) @ R2 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_174_inj__on__the__inv__into,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,A3: set @ A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ A3 )
% 2.75/1.54 => ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A3 @ F ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_175_f__the__inv__into__f,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,F: A > B,A3: set @ A,Y: B] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ A3 )
% 2.75/1.54 => ( ( member @ B @ Y @ ( image @ A @ B @ F @ A3 ) )
% 2.75/1.54 => ( ( F @ ( the_inv_into @ A @ B @ A3 @ F @ Y ) )
% 2.75/1.54 = Y ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_176_total__lex__prod,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
% 2.75/1.54 ( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R )
% 2.75/1.54 => ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ S3 )
% 2.75/1.54 => ( total_on @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( lex_prod @ A @ B @ R @ S3 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_177_range__fst,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType] :
% 2.75/1.54 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.75/1.54 = ( top_top @ ( set @ A ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_178_range__snd,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType] :
% 2.75/1.54 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.75/1.54 = ( top_top @ ( set @ A ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_179_fst__map__prod,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 2.75/1.54 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X ) )
% 2.75/1.54 = ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_180_snd__map__prod,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X: product_prod @ C @ D] :
% 2.75/1.54 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X ) )
% 2.75/1.54 = ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_181_fst__apfst,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
% 2.75/1.54 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
% 2.75/1.54 = ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_182_apfst__eq__conv,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G: C > A] :
% 2.75/1.54 ( ( ( product_apfst @ C @ A @ B @ F @ X )
% 2.75/1.54 = ( product_apfst @ C @ A @ B @ G @ X ) )
% 2.75/1.54 = ( ( F @ ( product_fst @ C @ B @ X ) )
% 2.75/1.54 = ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_183_snd__apfst,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
% 2.75/1.54 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
% 2.75/1.54 = ( product_snd @ C @ A @ X ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_184_fst__apsnd,axiom,
% 2.75/1.54 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
% 2.75/1.54 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
% 2.75/1.54 = ( product_fst @ A @ C @ X ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_185_snd__apsnd,axiom,
% 2.75/1.54 ! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
% 2.75/1.54 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
% 2.75/1.54 = ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_186_apsnd__eq__conv,axiom,
% 2.75/1.54 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G: C > B] :
% 2.75/1.54 ( ( ( product_apsnd @ C @ B @ A @ F @ X )
% 2.75/1.54 = ( product_apsnd @ C @ B @ A @ G @ X ) )
% 2.75/1.54 = ( ( F @ ( product_snd @ A @ C @ X ) )
% 2.75/1.54 = ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_187_in__lex__prod,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
% 2.75/1.54 ( ( 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 @ A6 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S3 ) )
% 2.75/1.54 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A6 ) @ R )
% 2.75/1.54 | ( ( A2 = A6 )
% 2.75/1.54 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B4 ) @ S3 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_188_prod_Ocollapse,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.75/1.54 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 2.75/1.54 = Prod ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_189_fst__swap,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 2.75/1.54 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 2.75/1.54 = ( product_snd @ B @ A @ X ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_190_snd__swap,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 2.75/1.54 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 2.75/1.54 = ( product_fst @ A @ B @ X ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_191_apsnd__apfst,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,X: product_prod @ D @ C] :
% 2.75/1.54 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ X ) )
% 2.75/1.54 = ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_192_apfst__apsnd,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 2.75/1.54 ( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G @ X ) )
% 2.75/1.54 = ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_193_prod_Oswap__def,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType] :
% 2.75/1.54 ( ( product_swap @ A @ B )
% 2.75/1.54 = ( ^ [P2: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P2 ) @ ( product_fst @ A @ B @ P2 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_194_prod__eq__iff,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType] :
% 2.75/1.54 ( ( ^ [Y6: product_prod @ A @ B,Z2: product_prod @ A @ B] : ( Y6 = Z2 ) )
% 2.75/1.54 = ( ^ [S: product_prod @ A @ B,T2: product_prod @ A @ B] :
% 2.75/1.54 ( ( ( product_fst @ A @ B @ S )
% 2.75/1.54 = ( product_fst @ A @ B @ T2 ) )
% 2.75/1.54 & ( ( product_snd @ A @ B @ S )
% 2.75/1.54 = ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_195_prod_Oexpand,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 2.75/1.54 ( ( ( ( product_fst @ A @ B @ Prod )
% 2.75/1.54 = ( product_fst @ A @ B @ Prod2 ) )
% 2.75/1.54 & ( ( product_snd @ A @ B @ Prod )
% 2.75/1.54 = ( product_snd @ A @ B @ Prod2 ) ) )
% 2.75/1.54 => ( Prod = Prod2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_196_prod__eqI,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,P: product_prod @ A @ B,Q: product_prod @ A @ B] :
% 2.75/1.54 ( ( ( product_fst @ A @ B @ P )
% 2.75/1.54 = ( product_fst @ A @ B @ Q ) )
% 2.75/1.54 => ( ( ( product_snd @ A @ B @ P )
% 2.75/1.54 = ( product_snd @ A @ B @ Q ) )
% 2.75/1.54 => ( P = Q ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_197_fst__eqD,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
% 2.75/1.54 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 2.75/1.54 = A2 )
% 2.75/1.54 => ( X = A2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_198_snd__eqD,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
% 2.75/1.54 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 2.75/1.54 = A2 )
% 2.75/1.54 => ( Y = A2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_199_fst__conv,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,X1: A,X2: B] :
% 2.75/1.54 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 2.75/1.54 = X1 ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_200_snd__conv,axiom,
% 2.75/1.54 ! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
% 2.75/1.54 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
% 2.75/1.54 = X2 ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_201_prod_Oexhaust__sel,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.75/1.54 ( Prod
% 2.75/1.54 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_202_surjective__pairing,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
% 2.75/1.54 ( T3
% 2.75/1.54 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_203_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,P3: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
% 2.75/1.54 ( ( P3 @ X @ Y )
% 2.75/1.54 => ( ( A2
% 2.75/1.54 = ( product_Pair @ A @ B @ X @ Y ) )
% 2.75/1.54 => ( P3 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_204_conjI__realizer,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,P3: A > $o,P: A,Q2: B > $o,Q: B] :
% 2.75/1.54 ( ( P3 @ P )
% 2.75/1.54 => ( ( Q2 @ Q )
% 2.75/1.54 => ( ( P3 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q ) ) )
% 2.75/1.54 & ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_205_exI__realizer,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,P3: A > B > $o,Y: A,X: B] :
% 2.75/1.54 ( ( P3 @ Y @ X )
% 2.75/1.54 => ( P3 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_206_sndI,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
% 2.75/1.54 ( ( X
% 2.75/1.54 = ( product_Pair @ A @ B @ Y @ Z ) )
% 2.75/1.54 => ( ( product_snd @ A @ B @ X )
% 2.75/1.54 = Z ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_207_eq__snd__iff,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,B2: A,P: product_prod @ B @ A] :
% 2.75/1.54 ( ( B2
% 2.75/1.54 = ( product_snd @ B @ A @ P ) )
% 2.75/1.54 = ( ? [A7: B] :
% 2.75/1.54 ( P
% 2.75/1.54 = ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_208_eq__fst__iff,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType,A2: A,P: product_prod @ A @ B] :
% 2.75/1.54 ( ( A2
% 2.75/1.54 = ( product_fst @ A @ B @ P ) )
% 2.75/1.54 = ( ? [B5: B] :
% 2.75/1.54 ( P
% 2.75/1.54 = ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_209_fstI,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
% 2.75/1.54 ( ( X
% 2.75/1.54 = ( product_Pair @ A @ B @ Y @ Z ) )
% 2.75/1.54 => ( ( product_fst @ A @ B @ X )
% 2.75/1.54 = Y ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_210_same__fstI,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,P3: A > $o,X: A,Y7: B,Y: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 2.75/1.54 ( ( P3 @ X )
% 2.75/1.54 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R2 @ X ) )
% 2.75/1.54 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y7 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P3 @ R2 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_211_wf__map__prod__image,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
% 2.75/1.54 ( ( wf @ A @ R )
% 2.75/1.54 => ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.75/1.54 => ( wf @ B @ ( image @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( product_map_prod @ A @ B @ A @ B @ F @ F ) @ R ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_212_wf__def,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( wf @ A )
% 2.75/1.54 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 2.75/1.54 ! [P5: A > $o] :
% 2.75/1.54 ( ! [X4: A] :
% 2.75/1.54 ( ! [Y4: A] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R3 )
% 2.75/1.54 => ( P5 @ Y4 ) )
% 2.75/1.54 => ( P5 @ X4 ) )
% 2.75/1.54 => ! [X7: A] : ( P5 @ X7 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_213_wfE__min,axiom,
% 2.75/1.54 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Q2: set @ A] :
% 2.75/1.54 ( ( wf @ A @ R2 )
% 2.75/1.54 => ( ( member @ A @ X @ Q2 )
% 2.75/1.54 => ~ ! [Z3: A] :
% 2.75/1.54 ( ( member @ A @ Z3 @ Q2 )
% 2.75/1.54 => ~ ! [Y8: A] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y8 @ Z3 ) @ R2 )
% 2.75/1.54 => ~ ( member @ A @ Y8 @ Q2 ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_214_wfI__min,axiom,
% 2.75/1.54 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 2.75/1.54 ( ! [X5: A,Q4: set @ A] :
% 2.75/1.54 ( ( member @ A @ X5 @ Q4 )
% 2.75/1.54 => ? [Xa3: A] :
% 2.75/1.54 ( ( member @ A @ Xa3 @ Q4 )
% 2.75/1.54 & ! [Y5: A] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Xa3 ) @ R2 )
% 2.75/1.54 => ~ ( member @ A @ Y5 @ Q4 ) ) ) )
% 2.75/1.54 => ( wf @ A @ R2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_215_wfUNIVI,axiom,
% 2.75/1.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 2.75/1.54 ( ! [P6: A > $o,X5: A] :
% 2.75/1.54 ( ! [Xa3: A] :
% 2.75/1.54 ( ! [Y5: A] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Xa3 ) @ R )
% 2.75/1.54 => ( P6 @ Y5 ) )
% 2.75/1.54 => ( P6 @ Xa3 ) )
% 2.75/1.54 => ( P6 @ X5 ) )
% 2.75/1.54 => ( wf @ A @ R ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_216_wf__asym,axiom,
% 2.75/1.54 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X: A] :
% 2.75/1.54 ( ( wf @ A @ R )
% 2.75/1.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X ) @ R )
% 2.75/1.54 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ R ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_217_wf__induct,axiom,
% 2.75/1.54 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P3: A > $o,A2: A] :
% 2.75/1.54 ( ( wf @ A @ R )
% 2.75/1.54 => ( ! [X5: A] :
% 2.75/1.54 ( ! [Y8: A] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y8 @ X5 ) @ R )
% 2.75/1.54 => ( P3 @ Y8 ) )
% 2.75/1.54 => ( P3 @ X5 ) )
% 2.75/1.54 => ( P3 @ A2 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_218_wf__irrefl,axiom,
% 2.75/1.54 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
% 2.75/1.54 ( ( wf @ A @ R )
% 2.75/1.54 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_219_wf__not__sym,axiom,
% 2.75/1.54 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X: A] :
% 2.75/1.54 ( ( wf @ A @ R )
% 2.75/1.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X ) @ R )
% 2.75/1.54 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ R ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_220_wf__not__refl,axiom,
% 2.75/1.54 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
% 2.75/1.54 ( ( wf @ A @ R )
% 2.75/1.54 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_221_wf__eq__minimal,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( wf @ A )
% 2.75/1.54 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 2.75/1.54 ! [Q5: set @ A] :
% 2.75/1.54 ( ? [X4: A] : ( member @ A @ X4 @ Q5 )
% 2.75/1.54 => ? [X4: A] :
% 2.75/1.54 ( ( member @ A @ X4 @ Q5 )
% 2.75/1.54 & ! [Y4: A] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R3 )
% 2.75/1.54 => ~ ( member @ A @ Y4 @ Q5 ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_222_wf__induct__rule,axiom,
% 2.75/1.54 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P3: A > $o,A2: A] :
% 2.75/1.54 ( ( wf @ A @ R )
% 2.75/1.54 => ( ! [X5: A] :
% 2.75/1.54 ( ! [Y8: A] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y8 @ X5 ) @ R )
% 2.75/1.54 => ( P3 @ Y8 ) )
% 2.75/1.54 => ( P3 @ X5 ) )
% 2.75/1.54 => ( P3 @ A2 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_223_dependent__wf__choice,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P3: ( A > B ) > A > B > $o] :
% 2.75/1.54 ( ( wf @ A @ R2 )
% 2.75/1.54 => ( ! [F8: A > B,G3: A > B,X5: A,R4: B] :
% 2.75/1.54 ( ! [Z4: A] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X5 ) @ R2 )
% 2.75/1.54 => ( ( F8 @ Z4 )
% 2.75/1.54 = ( G3 @ Z4 ) ) )
% 2.75/1.54 => ( ( P3 @ F8 @ X5 @ R4 )
% 2.75/1.54 = ( P3 @ G3 @ X5 @ R4 ) ) )
% 2.75/1.54 => ( ! [X5: A,F8: A > B] :
% 2.75/1.54 ( ! [Y8: A] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y8 @ X5 ) @ R2 )
% 2.75/1.54 => ( P3 @ F8 @ Y8 @ ( F8 @ Y8 ) ) )
% 2.75/1.54 => ? [X_1: B] : ( P3 @ F8 @ X5 @ X_1 ) )
% 2.75/1.54 => ? [F8: A > B] :
% 2.75/1.54 ! [X6: A] : ( P3 @ F8 @ X6 @ ( F8 @ X6 ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_224_sndOp__def,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,C: $tType] :
% 2.75/1.54 ( ( bNF_sndOp @ C @ A @ B )
% 2.75/1.54 = ( ^ [P5: C > A > $o,Q5: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P5 @ Q5 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_225_fstOp__def,axiom,
% 2.75/1.54 ! [C: $tType,B: $tType,A: $tType] :
% 2.75/1.54 ( ( bNF_fstOp @ A @ B @ C )
% 2.75/1.54 = ( ^ [P5: A > B > $o,Q5: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P5 @ Q5 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_226_image2__eqI,axiom,
% 2.75/1.54 ! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C3: C,G: B > C,A3: set @ B] :
% 2.75/1.54 ( ( B2
% 2.75/1.54 = ( F @ X ) )
% 2.75/1.54 => ( ( C3
% 2.75/1.54 = ( G @ X ) )
% 2.75/1.54 => ( ( member @ B @ X @ A3 )
% 2.75/1.54 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F @ G ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_227_the__inv__into__into,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X: B,B6: set @ A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ A3 )
% 2.75/1.54 => ( ( member @ B @ X @ ( image @ A @ B @ F @ A3 ) )
% 2.75/1.54 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 2.75/1.54 => ( member @ A @ ( the_inv_into @ A @ B @ A3 @ F @ X ) @ B6 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_228_strict__mono__inv,axiom,
% 2.75/1.54 ! [A: $tType,B: $tType] :
% 2.75/1.54 ( ( ( linorder @ B )
% 2.75/1.54 & ( linorder @ A ) )
% 2.75/1.54 => ! [F: A > B,G: B > A] :
% 2.75/1.54 ( ( order_strict_mono @ A @ B @ F )
% 2.75/1.54 => ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.75/1.54 = ( top_top @ ( set @ B ) ) )
% 2.75/1.54 => ( ! [X5: A] :
% 2.75/1.54 ( ( G @ ( F @ X5 ) )
% 2.75/1.54 = X5 )
% 2.75/1.54 => ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_229_order__refl,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( preorder @ A )
% 2.75/1.54 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_230_subset__antisym,axiom,
% 2.75/1.54 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 2.75/1.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 2.75/1.54 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
% 2.75/1.54 => ( A3 = B6 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_231_subsetI,axiom,
% 2.75/1.54 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 2.75/1.54 ( ! [X5: A] :
% 2.75/1.54 ( ( member @ A @ X5 @ A3 )
% 2.75/1.54 => ( member @ A @ X5 @ B6 ) )
% 2.75/1.54 => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_232_subrelI,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 2.75/1.54 ( ! [X5: A,Y5: B] :
% 2.75/1.54 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) @ R )
% 2.75/1.54 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) @ S3 ) )
% 2.75/1.54 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_233_inj__on__image__mem__iff,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,B6: set @ A,A2: A,A3: set @ A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ B6 )
% 2.75/1.54 => ( ( member @ A @ A2 @ B6 )
% 2.75/1.54 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 2.75/1.54 => ( ( member @ B @ ( F @ A2 ) @ ( image @ A @ B @ F @ A3 ) )
% 2.75/1.54 = ( member @ A @ A2 @ A3 ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_234_inj__on__image__eq__iff,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,C4: set @ A,A3: set @ A,B6: set @ A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ C4 )
% 2.75/1.54 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ C4 )
% 2.75/1.54 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 2.75/1.54 => ( ( ( image @ A @ B @ F @ A3 )
% 2.75/1.54 = ( image @ A @ B @ F @ B6 ) )
% 2.75/1.54 = ( A3 = B6 ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_235_inj__on__subset,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,B6: set @ A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ A3 )
% 2.75/1.54 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
% 2.75/1.54 => ( inj_on @ A @ B @ F @ B6 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_236_subset__inj__on,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType,F: A > B,B6: set @ A,A3: set @ A] :
% 2.75/1.54 ( ( inj_on @ A @ B @ F @ B6 )
% 2.75/1.54 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 2.75/1.54 => ( inj_on @ A @ B @ F @ A3 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_237_strict__mono__less__eq,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType] :
% 2.75/1.54 ( ( ( linorder @ A )
% 2.75/1.54 & ( order @ B ) )
% 2.75/1.54 => ! [F: A > B,X: A,Y: A] :
% 2.75/1.54 ( ( order_strict_mono @ A @ B @ F )
% 2.75/1.54 => ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) )
% 2.75/1.54 = ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_238_dual__order_Oantisym,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( order @ A )
% 2.75/1.54 => ! [B2: A,A2: A] :
% 2.75/1.54 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.75/1.54 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.75/1.54 => ( A2 = B2 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_239_dual__order_Oeq__iff,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( order @ A )
% 2.75/1.54 => ( ( ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 2.75/1.54 = ( ^ [A7: A,B5: A] :
% 2.75/1.54 ( ( ord_less_eq @ A @ B5 @ A7 )
% 2.75/1.54 & ( ord_less_eq @ A @ A7 @ B5 ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_240_strict__mono__eq,axiom,
% 2.75/1.54 ! [B: $tType,A: $tType] :
% 2.75/1.54 ( ( ( linorder @ A )
% 2.75/1.54 & ( order @ B ) )
% 2.75/1.54 => ! [F: A > B,X: A,Y: A] :
% 2.75/1.54 ( ( order_strict_mono @ A @ B @ F )
% 2.75/1.54 => ( ( ( F @ X )
% 2.75/1.54 = ( F @ Y ) )
% 2.75/1.54 = ( X = Y ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_241_dual__order_Otrans,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( order @ A )
% 2.75/1.54 => ! [B2: A,A2: A,C3: A] :
% 2.75/1.54 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.75/1.54 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 2.75/1.54 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_242_linorder__wlog,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( linorder @ A )
% 2.75/1.54 => ! [P3: A > A > $o,A2: A,B2: A] :
% 2.75/1.54 ( ! [A5: A,B3: A] :
% 2.75/1.54 ( ( ord_less_eq @ A @ A5 @ B3 )
% 2.75/1.54 => ( P3 @ A5 @ B3 ) )
% 2.75/1.54 => ( ! [A5: A,B3: A] :
% 2.75/1.54 ( ( P3 @ B3 @ A5 )
% 2.75/1.54 => ( P3 @ A5 @ B3 ) )
% 2.75/1.54 => ( P3 @ A2 @ B2 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_243_dual__order_Orefl,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( order @ A )
% 2.75/1.54 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_244_order__trans,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( preorder @ A )
% 2.75/1.54 => ! [X: A,Y: A,Z: A] :
% 2.75/1.54 ( ( ord_less_eq @ A @ X @ Y )
% 2.75/1.54 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.75/1.54 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_245_order__class_Oorder_Oantisym,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( order @ A )
% 2.75/1.54 => ! [A2: A,B2: A] :
% 2.75/1.54 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.75/1.54 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.75/1.54 => ( A2 = B2 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_246_ord__le__eq__trans,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( ord @ A )
% 2.75/1.54 => ! [A2: A,B2: A,C3: A] :
% 2.75/1.54 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.75/1.54 => ( ( B2 = C3 )
% 2.75/1.54 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_247_ord__eq__le__trans,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( ord @ A )
% 2.75/1.54 => ! [A2: A,B2: A,C3: A] :
% 2.75/1.54 ( ( A2 = B2 )
% 2.75/1.54 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 2.75/1.54 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_248_order__class_Oorder_Oeq__iff,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( order @ A )
% 2.75/1.54 => ( ( ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 2.75/1.54 = ( ^ [A7: A,B5: A] :
% 2.75/1.54 ( ( ord_less_eq @ A @ A7 @ B5 )
% 2.75/1.54 & ( ord_less_eq @ A @ B5 @ A7 ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_249_antisym__conv,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( order @ A )
% 2.75/1.54 => ! [Y: A,X: A] :
% 2.75/1.54 ( ( ord_less_eq @ A @ Y @ X )
% 2.75/1.54 => ( ( ord_less_eq @ A @ X @ Y )
% 2.75/1.54 = ( X = Y ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_250_le__cases3,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( linorder @ A )
% 2.75/1.54 => ! [X: A,Y: A,Z: A] :
% 2.75/1.54 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.75/1.54 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 2.75/1.54 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.75/1.54 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 2.75/1.54 => ( ( ( ord_less_eq @ A @ X @ Z )
% 2.75/1.54 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 2.75/1.54 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 2.75/1.54 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.75/1.54 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 2.75/1.54 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 2.75/1.54 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 2.75/1.54 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_251_order_Otrans,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( order @ A )
% 2.75/1.54 => ! [A2: A,B2: A,C3: A] :
% 2.75/1.54 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.75/1.54 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 2.75/1.54 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_252_le__cases,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( linorder @ A )
% 2.75/1.54 => ! [X: A,Y: A] :
% 2.75/1.54 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.75/1.54 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_253_eq__refl,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( preorder @ A )
% 2.75/1.54 => ! [X: A,Y: A] :
% 2.75/1.54 ( ( X = Y )
% 2.75/1.54 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_254_linear,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( linorder @ A )
% 2.75/1.54 => ! [X: A,Y: A] :
% 2.75/1.54 ( ( ord_less_eq @ A @ X @ Y )
% 2.75/1.54 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(fact_255_antisym,axiom,
% 2.75/1.54 ! [A: $tType] :
% 2.75/1.54 ( ( order @ A )
% 2.75/1.54 => ! [X: A,Y: A] :
% 2.75/1.54 ( ( ord_less_eq @ A @ X @ Y )
% 2.75/1.54 => ( ( ord_less_eq @ A @ Y @ X )
% 2.75/1.54 => ( X = Y ) ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.75/1.54 ! [A8: $tType,A9: $tType] :
% 2.75/1.54 ( ( preorder @ A9 )
% 2.75/1.54 => ( preorder @ ( A8 > A9 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.75/1.54 ! [A8: $tType,A9: $tType] :
% 2.75/1.54 ( ( order @ A9 )
% 2.75/1.54 => ( order @ ( A8 > A9 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_fun___Orderings_Otop,axiom,
% 2.75/1.54 ! [A8: $tType,A9: $tType] :
% 2.75/1.54 ( ( top @ A9 )
% 2.75/1.54 => ( top @ ( A8 > A9 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_fun___Orderings_Oord,axiom,
% 2.75/1.54 ! [A8: $tType,A9: $tType] :
% 2.75/1.54 ( ( ord @ A9 )
% 2.75/1.54 => ( ord @ ( A8 > A9 ) ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.75/1.54 preorder @ nat ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.75/1.54 linorder @ nat ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.75/1.54 order @ nat ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.75/1.54 ord @ nat ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 2.75/1.54 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 2.75/1.54 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_Set_Oset___Orderings_Otop_6,axiom,
% 2.75/1.54 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 2.75/1.54 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
% 2.75/1.54 preorder @ $o ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_HOL_Obool___Orderings_Olinorder_9,axiom,
% 2.75/1.54 linorder @ $o ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_HOL_Obool___Orderings_Oorder_10,axiom,
% 2.75/1.54 order @ $o ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_HOL_Obool___Orderings_Otop_11,axiom,
% 2.75/1.54 top @ $o ).
% 2.75/1.54
% 2.75/1.54 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 2.75/1.54 ord @ $o ).
% 2.75/1.54
% 2.75/1.54 thf(help_If_3_1_T,axiom,
% 2.75/1.54 ! [P3: $o] :
% 2.75/1.54 ( ( P3 = $true )
% 2.75/1.54 | ( P3 = $false ) ) ).
% 2.75/1.54
% 2.75/1.54 thf(help_If_2_1_T,axiom,
% 2.75/1.54 ! [A: $tType,X: A,Y: A] :
% 2.75/1.54 ( ( if @ A @ $false @ X @ Y )
% 2.75/1.54 = Y ) ).
% 2.75/1.54
% 2.75/1.54 thf(help_If_1_1_T,axiom,
% 2.75/1.54 ! [A: $tType,X: A,Y: A] :
% 2.75/1.54 ( ( if @ A @ $true @ X @ Y )
% 2.75/1.54 = X ) ).
% 2.75/1.54
% 2.75/1.54 thf(conj_0,conjecture,
% 2.75/1.54 ( ( paraco876059933e_eval @ i @ ( paraco2100061555le_Con @ p1 @ p2 ) )
% 2.75/1.54 = ( paraco876059933e_eval @ i @ p1 ) ) ).
% 2.75/1.54 ------------------------
% 2.75/1.56 % (31425)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_31276 for (2999ds/27Mi)
% 2.75/1.56 % (31427)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_31276 for (2999ds/2Mi)
% 2.75/1.56 % (31428)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_31276 for (2999ds/275Mi)
% 2.75/1.56 % (31423)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/183Mi)
% 2.75/1.56 % (31426)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/2Mi)
% 2.75/1.56 % (31427)Instruction limit reached!
% 2.75/1.56 % (31427)------------------------------
% 2.75/1.56 % (31427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.56 % (31427)Termination reason: Unknown
% 2.75/1.56 % (31427)Termination phase: shuffling
% 2.75/1.56
% 2.75/1.56 % (31427)Memory used [KB]: 1407
% 2.75/1.56 % (31427)Time elapsed: 0.002 s
% 2.75/1.56 % (31427)Instructions burned: 3 (million)
% 2.75/1.56 % (31427)------------------------------
% 2.75/1.56 % (31427)------------------------------
% 2.75/1.56 % (31426)Instruction limit reached!
% 2.75/1.56 % (31426)------------------------------
% 2.75/1.56 % (31426)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.56 % (31426)Termination reason: Unknown
% 2.75/1.56 % (31426)Termination phase: shuffling
% 2.75/1.56
% 2.75/1.56 % (31426)Memory used [KB]: 1407
% 2.75/1.56 % (31426)Time elapsed: 0.002 s
% 2.75/1.56 % (31426)Instructions burned: 3 (million)
% 2.75/1.56 % (31426)------------------------------
% 2.75/1.56 % (31426)------------------------------
% 2.75/1.56 % (31429)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/18Mi)
% 2.75/1.56 % (31424)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_31276 for (2999ds/4Mi)
% 2.75/1.57 % (31424)Instruction limit reached!
% 2.75/1.57 % (31424)------------------------------
% 2.75/1.57 % (31424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.57 % (31424)Termination reason: Unknown
% 2.75/1.57 % (31424)Termination phase: shuffling
% 2.75/1.57
% 2.75/1.57 % (31424)Memory used [KB]: 1407
% 2.75/1.57 % (31424)Time elapsed: 0.003 s
% 2.75/1.57 % (31424)Instructions burned: 5 (million)
% 2.75/1.57 % (31424)------------------------------
% 2.75/1.57 % (31424)------------------------------
% 2.75/1.57 % (31425)Instruction limit reached!
% 2.75/1.57 % (31425)------------------------------
% 2.75/1.57 % (31425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.57 % (31425)Termination reason: Unknown
% 2.75/1.57 % (31425)Termination phase: Property scanning
% 2.75/1.57
% 2.75/1.57 % (31425)Memory used [KB]: 1918
% 2.75/1.57 % (31425)Time elapsed: 0.010 s
% 2.75/1.57 % (31425)Instructions burned: 27 (million)
% 2.75/1.57 % (31425)------------------------------
% 2.75/1.57 % (31425)------------------------------
% 2.75/1.57 % (31429)Instruction limit reached!
% 2.75/1.57 % (31429)------------------------------
% 2.75/1.57 % (31429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.57 % (31429)Termination reason: Unknown
% 2.75/1.57 % (31429)Termination phase: shuffling
% 2.75/1.57
% 2.75/1.57 % (31429)Memory used [KB]: 1791
% 2.75/1.57 % (31429)Time elapsed: 0.009 s
% 2.75/1.57 % (31429)Instructions burned: 20 (million)
% 2.75/1.57 % (31429)------------------------------
% 2.75/1.57 % (31429)------------------------------
% 2.75/1.57 % (31432)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/3Mi)
% 2.75/1.57 % (31432)Instruction limit reached!
% 2.75/1.57 % (31432)------------------------------
% 2.75/1.57 % (31432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.57 % (31432)Termination reason: Unknown
% 2.75/1.57 % (31432)Termination phase: shuffling
% 2.75/1.57
% 2.75/1.57 % (31432)Memory used [KB]: 1407
% 2.75/1.57 % (31432)Time elapsed: 0.002 s
% 2.75/1.57 % (31432)Instructions burned: 3 (million)
% 2.75/1.57 % (31432)------------------------------
% 2.75/1.57 % (31432)------------------------------
% 2.75/1.57 % (31433)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_31276 for (2999ds/37Mi)
% 2.75/1.58 % (31428)Refutation not found, incomplete strategy
% 2.75/1.58 % (31428)------------------------------
% 2.75/1.58 % (31428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.58 % (31428)Termination reason: Refutation not found, incomplete strategy
% 2.75/1.58
% 2.75/1.58
% 2.75/1.58 % (31428)Memory used [KB]: 6396
% 2.75/1.58 % (31428)Time elapsed: 0.014 s
% 2.75/1.58 % (31428)Instructions burned: 39 (million)
% 2.75/1.58 % (31428)------------------------------
% 2.75/1.58 % (31428)------------------------------
% 2.75/1.58 % (31434)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_31276 for (2999ds/15Mi)
% 2.75/1.58 % (31435)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_31276 for (2999ds/3Mi)
% 2.75/1.58 % (31435)Instruction limit reached!
% 2.75/1.58 % (31435)------------------------------
% 2.75/1.58 % (31435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.58 % (31435)Termination reason: Unknown
% 2.75/1.58 % (31435)Termination phase: shuffling
% 2.75/1.58
% 2.75/1.58 % (31435)Memory used [KB]: 1407
% 2.75/1.58 % (31435)Time elapsed: 0.002 s
% 2.75/1.58 % (31435)Instructions burned: 3 (million)
% 2.75/1.58 % (31435)------------------------------
% 2.75/1.58 % (31435)------------------------------
% 2.75/1.58 % (31434)Instruction limit reached!
% 2.75/1.58 % (31434)------------------------------
% 2.75/1.58 % (31434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.58 % (31434)Termination reason: Unknown
% 2.75/1.58 % (31434)Termination phase: shuffling
% 2.75/1.58
% 2.75/1.58 % (31434)Memory used [KB]: 1791
% 2.75/1.58 % (31434)Time elapsed: 0.010 s
% 2.75/1.58 % (31434)Instructions burned: 17 (million)
% 2.75/1.58 % (31434)------------------------------
% 2.75/1.58 % (31434)------------------------------
% 2.75/1.59 % (31436)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_31276 for (2999ds/1041Mi)
% 2.75/1.59 % (31437)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/7Mi)
% 2.75/1.59 % (31446)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_31276 for (2999ds/3Mi)
% 2.75/1.59 % (31446)Instruction limit reached!
% 2.75/1.59 % (31446)------------------------------
% 2.75/1.59 % (31446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.59 % (31446)Termination reason: Unknown
% 2.75/1.59 % (31446)Termination phase: shuffling
% 2.75/1.59
% 2.75/1.59 % (31446)Memory used [KB]: 1407
% 2.75/1.59 % (31446)Time elapsed: 0.002 s
% 2.75/1.59 % (31446)Instructions burned: 3 (million)
% 2.75/1.59 % (31446)------------------------------
% 2.75/1.59 % (31446)------------------------------
% 2.75/1.59 % (31433)Instruction limit reached!
% 2.75/1.59 % (31433)------------------------------
% 2.75/1.59 % (31433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.59 % (31433)Termination reason: Unknown
% 2.75/1.59 % (31433)Termination phase: shuffling
% 2.75/1.59
% 2.75/1.59 % (31433)Memory used [KB]: 2174
% 2.75/1.59 % (31433)Time elapsed: 0.020 s
% 2.75/1.59 % (31433)Instructions burned: 37 (million)
% 2.75/1.59 % (31433)------------------------------
% 2.75/1.59 % (31433)------------------------------
% 2.75/1.59 % (31445)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/16Mi)
% 2.75/1.59 % (31437)Instruction limit reached!
% 2.75/1.59 % (31437)------------------------------
% 2.75/1.59 % (31437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.59 % (31437)Termination reason: Unknown
% 2.75/1.59 % (31437)Termination phase: shuffling
% 2.75/1.59
% 2.75/1.59 % (31437)Memory used [KB]: 1535
% 2.75/1.59 % (31437)Time elapsed: 0.005 s
% 2.75/1.59 % (31437)Instructions burned: 7 (million)
% 2.75/1.59 % (31437)------------------------------
% 2.75/1.59 % (31437)------------------------------
% 2.81/1.60 % (31445)Instruction limit reached!
% 2.81/1.60 % (31445)------------------------------
% 2.81/1.60 % (31445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.60 % (31445)Termination reason: Unknown
% 2.81/1.60 % (31445)Termination phase: shuffling
% 2.81/1.60
% 2.81/1.60 % (31445)Memory used [KB]: 1791
% 2.81/1.60 % (31445)Time elapsed: 0.007 s
% 2.81/1.60 % (31445)Instructions burned: 16 (million)
% 2.81/1.60 % (31445)------------------------------
% 2.81/1.60 % (31445)------------------------------
% 2.81/1.60 % (31447)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_31276 for (2999ds/3Mi)
% 2.81/1.60 % (31447)Instruction limit reached!
% 2.81/1.60 % (31447)------------------------------
% 2.81/1.60 % (31447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.60 % (31447)Termination reason: Unknown
% 2.81/1.60 % (31447)Termination phase: shuffling
% 2.81/1.60
% 2.81/1.60 % (31447)Memory used [KB]: 1407
% 2.81/1.60 % (31447)Time elapsed: 0.003 s
% 2.81/1.60 % (31447)Instructions burned: 4 (million)
% 2.81/1.60 % (31447)------------------------------
% 2.81/1.60 % (31447)------------------------------
% 2.81/1.61 % (31452)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/4Mi)
% 2.81/1.61 % (31448)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/7Mi)
% 2.81/1.61 % (31452)Instruction limit reached!
% 2.81/1.61 % (31452)------------------------------
% 2.81/1.61 % (31452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.61 % (31452)Termination reason: Unknown
% 2.81/1.61 % (31452)Termination phase: shuffling
% 2.81/1.61
% 2.81/1.61 % (31452)Memory used [KB]: 1407
% 2.81/1.61 % (31452)Time elapsed: 0.003 s
% 2.81/1.61 % (31452)Instructions burned: 5 (million)
% 2.81/1.61 % (31452)------------------------------
% 2.81/1.61 % (31452)------------------------------
% 2.81/1.61 % (31450)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/3Mi)
% 2.81/1.61 % (31454)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/18Mi)
% 2.81/1.61 % (31450)Instruction limit reached!
% 2.81/1.61 % (31450)------------------------------
% 2.81/1.61 % (31450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.61 % (31450)Termination reason: Unknown
% 2.81/1.61 % (31450)Termination phase: shuffling
% 2.81/1.61
% 2.81/1.61 % (31450)Memory used [KB]: 1407
% 2.81/1.61 % (31450)Time elapsed: 0.003 s
% 2.81/1.61 % (31450)Instructions burned: 3 (million)
% 2.81/1.61 % (31450)------------------------------
% 2.81/1.61 % (31450)------------------------------
% 2.81/1.61 % (31448)Instruction limit reached!
% 2.81/1.61 % (31448)------------------------------
% 2.81/1.61 % (31448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.61 % (31448)Termination reason: Unknown
% 2.81/1.61 % (31448)Termination phase: shuffling
% 2.81/1.61
% 2.81/1.61 % (31448)Memory used [KB]: 1535
% 2.81/1.61 % (31448)Time elapsed: 0.005 s
% 2.81/1.61 % (31448)Instructions burned: 7 (million)
% 2.81/1.61 % (31448)------------------------------
% 2.81/1.61 % (31448)------------------------------
% 2.81/1.62 % (31454)Instruction limit reached!
% 2.81/1.62 % (31454)------------------------------
% 2.81/1.62 % (31454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.62 % (31454)Termination reason: Unknown
% 2.81/1.62 % (31454)Termination phase: shuffling
% 2.81/1.62
% 2.81/1.62 % (31454)Memory used [KB]: 1791
% 2.81/1.62 % (31454)Time elapsed: 0.011 s
% 2.81/1.62 % (31454)Instructions burned: 19 (million)
% 2.81/1.62 % (31454)------------------------------
% 2.81/1.62 % (31454)------------------------------
% 2.81/1.62 % (31460)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_31276 for (2999ds/902Mi)
% 2.81/1.62 % (31458)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/6Mi)
% 2.81/1.63 % (31458)Instruction limit reached!
% 2.81/1.63 % (31458)------------------------------
% 2.81/1.63 % (31458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.63 % (31458)Termination reason: Unknown
% 2.81/1.63 % (31458)Termination phase: shuffling
% 2.81/1.63
% 2.81/1.63 % (31458)Memory used [KB]: 1535
% 2.81/1.63 % (31458)Time elapsed: 0.005 s
% 2.81/1.63 % (31458)Instructions burned: 7 (million)
% 2.81/1.63 % (31458)------------------------------
% 2.81/1.63 % (31458)------------------------------
% 2.81/1.63 % (31462)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_31276 for (2999ds/21Mi)
% 2.81/1.63 % (31456)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/710Mi)
% 2.81/1.64 % (31464)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/5Mi)
% 2.81/1.64 % (31462)Instruction limit reached!
% 2.81/1.64 % (31462)------------------------------
% 2.81/1.64 % (31462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.64 % (31462)Termination reason: Unknown
% 2.81/1.64 % (31462)Termination phase: shuffling
% 2.81/1.64
% 2.81/1.64 % (31462)Memory used [KB]: 1918
% 2.81/1.64 % (31462)Time elapsed: 0.012 s
% 2.81/1.64 % (31462)Instructions burned: 22 (million)
% 2.81/1.64 % (31462)------------------------------
% 2.81/1.64 % (31462)------------------------------
% 2.81/1.64 % (31464)Instruction limit reached!
% 2.81/1.64 % (31464)------------------------------
% 2.81/1.64 % (31464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.64 % (31464)Termination reason: Unknown
% 2.81/1.64 % (31464)Termination phase: shuffling
% 2.81/1.64
% 2.81/1.64 % (31464)Memory used [KB]: 1407
% 2.81/1.64 % (31464)Time elapsed: 0.004 s
% 2.81/1.64 % (31464)Instructions burned: 5 (million)
% 2.81/1.64 % (31464)------------------------------
% 2.81/1.64 % (31464)------------------------------
% 2.81/1.64 % (31465)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_31276 for (2999ds/6Mi)
% 2.81/1.64 % (31465)Instruction limit reached!
% 2.81/1.64 % (31465)------------------------------
% 2.81/1.64 % (31465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.64 % (31465)Termination reason: Unknown
% 2.81/1.64 % (31465)Termination phase: shuffling
% 2.81/1.64
% 2.81/1.64 % (31465)Memory used [KB]: 1535
% 2.81/1.64 % (31465)Time elapsed: 0.003 s
% 2.81/1.64 % (31465)Instructions burned: 6 (million)
% 2.81/1.64 % (31465)------------------------------
% 2.81/1.64 % (31465)------------------------------
% 2.81/1.65 % (31423)Instruction limit reached!
% 2.81/1.65 % (31423)------------------------------
% 2.81/1.65 % (31423)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.65 % (31423)Termination reason: Unknown
% 2.81/1.65 % (31423)Termination phase: Saturation
% 2.81/1.65
% 2.81/1.65 % (31423)Memory used [KB]: 7291
% 2.81/1.65 % (31423)Time elapsed: 0.088 s
% 2.81/1.65 % (31423)Instructions burned: 183 (million)
% 2.81/1.65 % (31423)------------------------------
% 2.81/1.65 % (31423)------------------------------
% 2.81/1.65 % (31470)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/377Mi)
% 2.81/1.65 % (31471)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_31276 for (2999ds/779Mi)
% 2.81/1.66 % (31472)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_31276 for (2999ds/19Mi)
% 2.81/1.66 % (31472)Instruction limit reached!
% 2.81/1.66 % (31472)------------------------------
% 2.81/1.66 % (31472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.66 % (31472)Termination reason: Unknown
% 2.81/1.66 % (31472)Termination phase: shuffling
% 2.81/1.66
% 2.81/1.66 % (31472)Memory used [KB]: 1791
% 2.81/1.66 % (31472)Time elapsed: 0.010 s
% 2.81/1.66 % (31472)Instructions burned: 19 (million)
% 2.81/1.66 % (31472)------------------------------
% 2.81/1.66 % (31472)------------------------------
% 2.81/1.67 % (31460)First to succeed.
% 2.81/1.68 % (31476)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_31276 for (2999ds/879Mi)
% 2.81/1.68 % (31460)Refutation found. Thanks to Tanya!
% 2.81/1.68 % SZS status Theorem for DTF2THF_31276
% 2.81/1.68 % SZS output start Proof for DTF2THF_31276
% 2.81/1.68 thf(type_def_6, type, product_prod: ($tType * $tType) > $tType).
% 2.81/1.68 thf(type_def_7, type, set: ($tType) > $tType).
% 2.81/1.68 thf(type_def_8, type, nat: $tType).
% 2.81/1.68 thf(type_def_9, type, char: $tType).
% 2.81/1.68 thf(type_def_10, type, list: ($tType) > $tType).
% 2.81/1.68 thf(type_def_11, type, paraco415392788lle_tv: $tType).
% 2.81/1.68 thf(type_def_12, type, paraco414474393lle_fm: $tType).
% 2.81/1.68 thf(func_def_0, type, paraco415392788lle_tv: $tType).
% 2.81/1.68 thf(func_def_1, type, paraco414474393lle_fm: $tType).
% 2.81/1.68 thf(func_def_2, type, product_prod: ($tType * $tType) > $tType).
% 2.81/1.68 thf(func_def_3, type, char: $tType).
% 2.81/1.68 thf(func_def_4, type, list: ($tType) > $tType).
% 2.81/1.68 thf(func_def_5, type, set: ($tType) > $tType).
% 2.81/1.68 thf(func_def_6, type, nat: $tType).
% 2.81/1.68 thf(func_def_7, type, type: !>[X0: $tType]:($o)).
% 2.81/1.68 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 2.81/1.68 thf(func_def_9, type, top: !>[X0: $tType]:($o)).
% 2.81/1.68 thf(func_def_10, type, order: !>[X0: $tType]:($o)).
% 2.81/1.68 thf(func_def_11, type, linorder: !>[X0: $tType]:($o)).
% 2.81/1.68 thf(func_def_12, type, preorder: !>[X0: $tType]:($o)).
% 2.81/1.68 thf(func_def_13, type, bNF_fstOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X0 @ X1)).
% 2.81/1.68 thf(func_def_14, type, bNF_pick_middlep: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > X1)).
% 2.81/1.68 thf(func_def_15, type, bNF_sndOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 2.81/1.68 thf(func_def_16, type, bNF_Greatest_image2: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > X1) > (X0 > X2) > set @ product_prod @ X1 @ X2)).
% 2.81/1.68 thf(func_def_17, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 2.81/1.68 thf(func_def_18, type, the_inv_into: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X1 > X0)).
% 2.81/1.68 thf(func_def_19, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.81/1.68 thf(func_def_20, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.81/1.68 thf(func_def_21, type, order_strict_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.81/1.68 thf(func_def_22, type, top_top: !>[X0: $tType]:(X0)).
% 2.81/1.68 thf(func_def_23, type, paraco753463838ge_int: (nat > nat) > (list @ char > paraco415392788lle_tv) > list @ char > paraco415392788lle_tv).
% 2.81/1.68 thf(func_def_24, type, paraco1920534163nge_tv: (nat > nat) > paraco415392788lle_tv > paraco415392788lle_tv).
% 2.81/1.68 thf(func_def_25, type, paraco2077297190tv_rel: product_prod @ (nat > nat) @ paraco415392788lle_tv > product_prod @ (nat > nat) @ paraco415392788lle_tv > $o).
% 2.81/1.68 thf(func_def_26, type, paraco876059933e_eval: (list @ char > paraco415392788lle_tv) > paraco414474393lle_fm > paraco415392788lle_tv).
% 2.81/1.68 thf(func_def_27, type, paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_28, type, paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_29, type, paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_30, type, paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_31, type, paraco27778325le_Pro: list @ char > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_32, type, paraco251304083_Truth: paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_33, 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)).
% 2.81/1.68 thf(func_def_34, type, paraco815940159rec_fm: !>[X0: $tType]:((list @ char > X0) > X0 > (paraco414474393lle_fm > X0 > X0) > (paraco414474393lle_fm > paraco414474393lle_fm > X0 > X0 > X0) > (paraco414474393lle_fm > paraco414474393lle_fm > X0 > X0 > X0) > (paraco414474393lle_fm > paraco414474393lle_fm > X0 > X0 > X0) > paraco414474393lle_fm > X0)).
% 2.81/1.68 thf(func_def_35, type, paraco2040174112le_Det: $o > paraco415392788lle_tv).
% 2.81/1.68 thf(func_def_36, type, paraco676387099_Indet: nat > paraco415392788lle_tv).
% 2.81/1.68 thf(func_def_37, type, paraco490622181ase_tv: !>[X0: $tType]:(($o > X0) > (nat > X0) > paraco415392788lle_tv > X0)).
% 2.81/1.68 thf(func_def_38, type, paraco152590079rec_tv: !>[X0: $tType]:(($o > X0) > (nat > X0) > paraco415392788lle_tv > X0)).
% 2.81/1.68 thf(func_def_39, type, paraco769098683_valid: paraco414474393lle_fm > $o).
% 2.81/1.68 thf(func_def_40, type, paraco2086025920lid_in: set @ nat > paraco414474393lle_fm > $o).
% 2.81/1.68 thf(func_def_41, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.81/1.68 thf(func_def_42, type, product_apfst: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 2.81/1.68 thf(func_def_43, type, product_apsnd: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X2 @ X0 > product_prod @ X2 @ X1)).
% 2.81/1.68 thf(func_def_44, type, product_case_bool: !>[X0: $tType]:(X0 > X0 > $o > X0)).
% 2.81/1.68 thf(func_def_45, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 2.81/1.68 thf(func_def_46, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.81/1.68 thf(func_def_47, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 2.81/1.68 thf(func_def_48, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.81/1.68 thf(func_def_49, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.81/1.68 thf(func_def_50, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.81/1.68 thf(func_def_51, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 2.81/1.68 thf(func_def_52, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 2.81/1.68 thf(func_def_53, type, inv_imagep: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > X1 > X1 > $o)).
% 2.81/1.68 thf(func_def_54, type, total_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.81/1.68 thf(func_def_55, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.81/1.68 thf(func_def_56, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 2.81/1.68 thf(func_def_57, type, accp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > $o)).
% 2.81/1.68 thf(func_def_58, 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)).
% 2.81/1.68 thf(func_def_59, type, wf: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 2.81/1.68 thf(func_def_60, 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)).
% 2.81/1.68 thf(func_def_61, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.81/1.68 thf(func_def_62, type, f: nat > nat).
% 2.81/1.68 thf(func_def_63, type, i: list @ char > paraco415392788lle_tv).
% 2.81/1.68 thf(func_def_64, type, p1: paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_65, type, p2: paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_81, type, sP0: (paraco414474393lle_fm > $o) > $o).
% 2.81/1.68 thf(func_def_82, type, sP1: (paraco414474393lle_fm > $o) > $o).
% 2.81/1.68 thf(func_def_83, type, sP2: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.81/1.68 thf(func_def_84, type, sP3: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.81/1.68 thf(func_def_85, type, sP4: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.81/1.68 thf(func_def_86, type, sP5: (paraco414474393lle_fm > $o) > $o).
% 2.81/1.68 thf(func_def_87, type, sP6: (paraco414474393lle_fm > $o) > $o).
% 2.81/1.68 thf(func_def_88, type, sP7: paraco415392788lle_tv > (nat > nat) > paraco415392788lle_tv > $o).
% 2.81/1.68 thf(func_def_89, type, sK8: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_90, type, sK9: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_91, type, sK10: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_92, type, sK11: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_93, type, sK12: (paraco414474393lle_fm > $o) > list @ char).
% 2.81/1.68 thf(func_def_94, type, sK13: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_95, type, sK14: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_96, type, sK15: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_97, type, sK16: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 2.81/1.68 thf(func_def_98, type, sK17: ((nat > nat) > paraco415392788lle_tv > $o) > nat > nat).
% 2.81/1.68 thf(func_def_99, type, sK18: ((nat > nat) > paraco415392788lle_tv > $o) > nat).
% 2.81/1.68 thf(func_def_100, type, sK19: ((nat > nat) > paraco415392788lle_tv > $o) > nat > nat).
% 2.81/1.68 thf(func_def_101, type, sK20: ((nat > nat) > paraco415392788lle_tv > $o) > $o).
% 2.81/1.68 thf(func_def_102, type, sK21: !>[X0: $tType]:(set @ X0 > X0)).
% 2.81/1.68 thf(func_def_103, type, sK22: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 2.81/1.68 thf(func_def_104, type, sK23: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 2.81/1.68 thf(func_def_105, type, sK24: !>[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)).
% 2.81/1.68 thf(func_def_106, type, sK25: !>[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)).
% 2.81/1.68 thf(func_def_107, type, sK26: !>[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)).
% 2.81/1.68 thf(func_def_108, type, sK27: !>[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)).
% 2.81/1.68 thf(func_def_109, type, sK28: !>[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)).
% 2.81/1.68 thf(func_def_110, type, sK29: !>[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)).
% 2.81/1.68 thf(func_def_111, type, sK30: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.81/1.68 thf(func_def_112, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 2.81/1.68 thf(func_def_113, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 2.81/1.68 thf(func_def_114, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 2.81/1.68 thf(func_def_115, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 2.81/1.68 thf(func_def_116, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 2.81/1.68 thf(func_def_117, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 2.81/1.68 thf(func_def_118, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 2.81/1.68 thf(func_def_119, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 2.81/1.68 thf(func_def_120, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 2.81/1.68 thf(func_def_121, type, sK40: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > X0)).
% 2.81/1.68 thf(func_def_122, type, sK41: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > X0)).
% 2.81/1.68 thf(func_def_123, type, sK42: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.81/1.68 thf(func_def_124, type, sK43: !>[X0: $tType]:((X0 > $o) > set @ product_prod @ X0 @ X0 > X0)).
% 2.81/1.68 thf(func_def_125, type, sK44: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X0)).
% 2.81/1.68 thf(func_def_126, type, sK45: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.81/1.68 thf(func_def_127, type, sK46: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.81/1.68 thf(func_def_128, type, sK47: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > X0)).
% 2.81/1.68 thf(func_def_129, type, sK48: product_prod @ (nat > nat) @ paraco415392788lle_tv > nat > nat).
% 2.81/1.68 thf(func_def_130, type, sK49: product_prod @ (nat > nat) @ paraco415392788lle_tv > nat).
% 2.81/1.68 thf(func_def_131, type, sK50: product_prod @ (nat > nat) @ paraco415392788lle_tv > nat > nat).
% 2.81/1.68 thf(func_def_132, type, sK51: product_prod @ (nat > nat) @ paraco415392788lle_tv > $o).
% 2.81/1.68 thf(func_def_133, type, sK52: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X1)).
% 2.81/1.68 thf(func_def_134, type, sK53: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X1)).
% 2.81/1.68 thf(func_def_135, type, sK54: (paraco415392788lle_tv > $o) > nat).
% 2.81/1.68 thf(func_def_136, type, sK55: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X1)).
% 2.81/1.68 thf(func_def_137, type, sK56: (paraco415392788lle_tv > $o) > nat).
% 2.81/1.68 thf(func_def_138, type, sK57: (paraco415392788lle_tv > $o) > $o).
% 2.81/1.68 thf(func_def_139, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.81/1.68 thf(func_def_140, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.81/1.68 thf(func_def_141, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.81/1.68 thf(func_def_142, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.81/1.68 thf(func_def_143, type, sK62: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 2.81/1.68 thf(func_def_144, type, sK63: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(set @ product_prod @ X3 @ X2 > (X2 > X0) > (X3 > X1) > product_prod @ X1 @ X0 > X3)).
% 2.81/1.68 thf(func_def_145, type, sK64: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(set @ product_prod @ X3 @ X2 > (X2 > X0) > (X3 > X1) > product_prod @ X1 @ X0 > X2)).
% 2.81/1.68 thf(func_def_146, type, sK65: !>[X0: $tType, X1: $tType]:(X1 > (X0 > X1) > X0)).
% 2.81/1.68 thf(func_def_147, type, sK66: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1)).
% 2.81/1.68 thf(func_def_148, type, sK67: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.81/1.68 thf(func_def_149, type, sK68: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.81/1.68 thf(func_def_150, type, sK69: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.81/1.68 thf(func_def_151, type, sK70: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.81/1.68 thf(func_def_152, type, sK71: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.81/1.68 thf(func_def_153, type, sK72: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.81/1.68 thf(func_def_154, type, sK73: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0 > X1)).
% 2.81/1.68 thf(func_def_155, type, sK74: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 2.81/1.68 thf(func_def_156, type, sK75: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 2.81/1.68 thf(func_def_157, type, sK76: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 2.81/1.68 thf(func_def_158, type, sK77: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 2.81/1.68 thf(func_def_159, type, sK78: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 2.81/1.68 thf(func_def_160, type, sK79: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 2.81/1.68 thf(func_def_161, type, sK80: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 2.81/1.68 thf(func_def_162, type, sK81: !>[X0: $tType]:((X0 > $o) > set @ product_prod @ X0 @ X0 > X0)).
% 2.81/1.68 thf(func_def_163, type, sK82: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.81/1.68 thf(func_def_164, type, sK83: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.81/1.68 thf(func_def_165, type, sK84: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 2.81/1.68 thf(func_def_166, type, sK85: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 2.81/1.68 thf(func_def_167, type, sK86: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 2.81/1.68 thf(func_def_168, type, sK87: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 2.81/1.68 thf(func_def_169, type, sK88: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_170, type, sK89: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_171, type, sK90: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_172, type, sK91: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_173, type, sK92: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_174, type, sK93: paraco414474393lle_fm > list @ char).
% 2.81/1.68 thf(func_def_175, type, sK94: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_176, type, sK95: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_177, type, sK96: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.81/1.68 thf(func_def_178, type, sK97: !>[X0: $tType, X1: $tType]:(X1 > X0 > (X1 > X0) > X1)).
% 2.81/1.68 thf(func_def_179, type, sK98: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 2.81/1.68 thf(func_def_180, type, sK99: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_181, type, sK100: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_182, type, sK101: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_183, type, sK102: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_184, type, sK103: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_185, type, sK104: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_186, type, sK105: (paraco414474393lle_fm > $o) > list @ char).
% 2.81/1.68 thf(func_def_187, type, sK106: (paraco414474393lle_fm > $o) > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_188, type, sK107: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char > paraco415392788lle_tv).
% 2.81/1.68 thf(func_def_189, type, sK108: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_190, type, sK109: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char > paraco415392788lle_tv).
% 2.81/1.68 thf(func_def_191, type, sK110: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_192, type, sK111: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_193, type, sK112: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char > paraco415392788lle_tv).
% 2.81/1.68 thf(func_def_194, type, sK113: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char > paraco415392788lle_tv).
% 2.81/1.68 thf(func_def_195, type, sK114: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char).
% 2.81/1.68 thf(func_def_196, type, sK115: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char > paraco415392788lle_tv).
% 2.81/1.68 thf(func_def_197, type, sK116: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_198, type, sK117: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_199, type, sK118: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > list @ char > paraco415392788lle_tv).
% 2.81/1.68 thf(func_def_200, type, sK119: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_201, type, sK120: product_prod @ (list @ char > paraco415392788lle_tv) @ paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_202, type, sK121: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.81/1.68 thf(func_def_203, type, sK122: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.81/1.68 thf(func_def_204, type, sK123: !>[X0: $tType, X1: $tType]:((X1 > X1) > (X1 > X0) > set @ X1 > X1)).
% 2.81/1.68 thf(func_def_205, type, sK124: !>[X0: $tType, X1: $tType]:((X1 > X1) > (X1 > X0) > set @ X1 > X1)).
% 2.81/1.68 thf(func_def_206, type, sK125: paraco415392788lle_tv > (nat > nat) > paraco415392788lle_tv > nat).
% 2.81/1.68 thf(func_def_207, type, sK126: paraco415392788lle_tv > paraco415392788lle_tv > (nat > nat) > $o).
% 2.81/1.68 thf(func_def_208, type, sK127: !>[X0: $tType]:(X0)).
% 2.81/1.68 thf(func_def_209, type, sK128: paraco415392788lle_tv > nat).
% 2.81/1.68 thf(func_def_210, type, sK129: paraco415392788lle_tv > $o).
% 2.81/1.68 thf(func_def_211, type, sK130: !>[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)).
% 2.81/1.68 thf(func_def_212, type, sK131: !>[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)).
% 2.81/1.68 thf(func_def_213, type, sK132: !>[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)).
% 2.81/1.68 thf(func_def_214, type, sK133: !>[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)).
% 2.81/1.68 thf(func_def_215, type, sK134: !>[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)).
% 2.81/1.68 thf(func_def_216, type, sK135: !>[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)).
% 2.81/1.68 thf(func_def_217, type, sK136: !>[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)).
% 2.81/1.68 thf(func_def_218, type, sK137: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.81/1.68 thf(func_def_219, type, sK138: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.81/1.68 thf(func_def_220, type, sK139: paraco415392788lle_tv > nat).
% 2.81/1.68 thf(func_def_221, type, sK140: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 2.81/1.68 thf(func_def_222, type, sK141: !>[X0: $tType, X1: $tType]:(((X1 > X0) > X1 > X0 > $o) > set @ product_prod @ X1 @ X1 > X1 > X0)).
% 2.81/1.68 thf(func_def_223, type, sK142: !>[X0: $tType, X1: $tType]:(((X1 > X0) > X1 > X0 > $o) > set @ product_prod @ X1 @ X1 > X1 > X0)).
% 2.81/1.68 thf(func_def_224, type, sK143: !>[X0: $tType, X1: $tType]:(((X1 > X0) > X1 > X0 > $o) > set @ product_prod @ X1 @ X1 > X1)).
% 2.81/1.68 thf(func_def_225, type, sK144: !>[X0: $tType, X1: $tType]:(((X1 > X0) > X1 > X0 > $o) > set @ product_prod @ X1 @ X1 > X0)).
% 2.81/1.68 thf(func_def_226, type, sK145: !>[X0: $tType, X1: $tType]:(((X1 > X0) > X1 > X0 > $o) > X1 > X0)).
% 2.81/1.68 thf(func_def_227, type, sK146: !>[X0: $tType, X1: $tType]:(((X1 > X0) > X1 > X0 > $o) > set @ product_prod @ X1 @ X1 > X1)).
% 2.81/1.68 thf(func_def_228, type, sK147: !>[X0: $tType, X1: $tType]:(((X1 > X0) > X1 > X0 > $o) > set @ product_prod @ X1 @ X1 > X1 > X0)).
% 2.81/1.68 thf(func_def_229, type, sK148: !>[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)).
% 2.81/1.68 thf(func_def_230, type, sK149: !>[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)).
% 2.81/1.68 thf(func_def_231, type, sK150: !>[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)).
% 2.81/1.68 thf(func_def_232, type, sK151: !>[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)).
% 2.81/1.68 thf(func_def_233, type, sK152: !>[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)).
% 2.81/1.68 thf(func_def_234, type, sK153: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > set @ X1 > X1)).
% 2.81/1.68 thf(func_def_235, type, sK154: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 2.81/1.68 thf(func_def_236, type, sK155: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 2.81/1.68 thf(func_def_237, type, sK156: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 2.81/1.68 thf(func_def_238, type, sK157: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 2.81/1.68 thf(func_def_239, type, sK158: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 2.81/1.68 thf(func_def_240, type, sK159: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_241, type, sK160: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_242, type, sK161: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_243, type, sK162: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_244, type, sK163: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_245, type, sK164: paraco414474393lle_fm > list @ char).
% 2.81/1.68 thf(func_def_246, type, sK165: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_247, type, sK166: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.81/1.68 thf(func_def_248, type, sK167: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.81/1.68 thf(func_def_249, type, sK168: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.81/1.68 thf(func_def_250, type, sK169: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0)).
% 2.81/1.68 thf(func_def_251, type, sK170: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0)).
% 2.81/1.68 thf(func_def_252, type, sK171: !>[X0: $tType]:(X0 > set @ product_prod @ X0 @ X0 > X0)).
% 2.81/1.68 thf(func_def_253, type, sK172: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > X1)).
% 2.81/1.68 thf(func_def_254, type, sK173: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.81/1.68 thf(func_def_255, type, sK174: !>[X0: $tType, X1: $tType]:(product_prod @ X1 @ X0 > X0 > X1)).
% 2.81/1.68 thf(func_def_256, type, sK175: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > $o)).
% 2.81/1.68 thf(func_def_257, type, sK176: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0)).
% 2.81/1.68 thf(func_def_258, type, sK177: !>[X0: $tType]:(X0 > set @ product_prod @ X0 @ X0 > X0)).
% 2.81/1.68 thf(func_def_259, type, sK178: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > X0)).
% 2.81/1.68 thf(func_def_260, type, sK179: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.81/1.68 thf(func_def_261, type, sK180: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.81/1.68 thf(func_def_262, type, sK181: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.81/1.68 thf(func_def_263, type, sK182: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.81/1.68 thf(func_def_264, type, sK183: paraco415392788lle_tv > paraco415392788lle_tv > $o).
% 2.81/1.68 thf(func_def_265, type, sK184: paraco415392788lle_tv > paraco415392788lle_tv > (nat > nat) > nat).
% 2.81/1.68 thf(func_def_267, type, ph186: !>[X0: $tType]:(X0)).
% 2.81/1.68 thf(f1709,plain,(
% 2.81/1.68 $false),
% 2.81/1.68 inference(avatar_sat_refutation,[status(thm)],[f1696,f1698,f1708])).
% 2.81/1.68 thf(f1708,plain,(
% 2.81/1.68 spl185_1),
% 2.81/1.68 inference(avatar_contradiction_clause,[status(thm)],[f1707])).
% 2.81/1.68 thf(f1707,plain,(
% 2.81/1.68 $false | spl185_1),
% 2.81/1.68 inference(trivial_inequality_removal,[status(thm)],[f1706])).
% 2.81/1.68 thf(f1706,plain,(
% 2.81/1.68 ((paraco2040174112le_Det @ $true) != (paraco2040174112le_Det @ $true)) | spl185_1),
% 2.81/1.68 inference(superposition,[status(thm)],[f1691,f1381])).
% 2.81/1.68 thf(f1381,plain,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p2) = (paraco2040174112le_Det @ $true))),
% 2.81/1.68 inference(cnf_transformation,[status(thm)],[f356])).
% 2.81/1.68 thf(f356,plain,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p2) = (paraco2040174112le_Det @ $true))),
% 2.81/1.68 inference(fool_elimination,[status(thm)],[f2])).
% 2.81/1.68 thf(f2,axiom,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p2) = (paraco2040174112le_Det @ $true))),
% 2.81/1.68 file('/export/starexec/sandbox2/tmp/tmp.cVWy2Pj8hg/DTF2THF_31276.p',fact_1_a)).
% 2.81/1.68 thf(f1691,plain,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p2) != (paraco2040174112le_Det @ $true)) | spl185_1),
% 2.81/1.68 inference(avatar_component_clause,[status(thm)],[f1689])).
% 2.81/1.68 thf(f1689,definition,(
% 2.81/1.68 spl185_1 <=> ((paraco876059933e_eval @ i @ p2) = (paraco2040174112le_Det @ $true))),
% 2.81/1.68 introduced(definition,[new_symbols(naming,[spl185_1])],[avatar_definition])).
% 2.81/1.68 thf(f1698,plain,(
% 2.81/1.68 ~spl185_2),
% 2.81/1.68 inference(avatar_split_clause,[status(thm)],[f1697,f1693])).
% 2.81/1.68 thf(f1693,definition,(
% 2.81/1.68 spl185_2 <=> ((paraco876059933e_eval @ i @ p1) = (paraco2040174112le_Det @ $true))),
% 2.81/1.68 introduced(definition,[new_symbols(naming,[spl185_2])],[avatar_definition])).
% 2.81/1.68 thf(f1697,plain,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p1) != (paraco2040174112le_Det @ $true))),
% 2.81/1.68 inference(forward_demodulation,[status(thm)],[f1684,f1381])).
% 2.81/1.68 thf(f1684,plain,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p1) != (paraco876059933e_eval @ i @ p2))),
% 2.81/1.68 inference(trivial_inequality_removal,[status(thm)],[f1683])).
% 2.81/1.68 thf(f1683,plain,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p1) != (paraco876059933e_eval @ i @ p2)) | ((paraco876059933e_eval @ i @ p1) != (paraco876059933e_eval @ i @ p1))),
% 2.81/1.68 inference(superposition,[status(thm)],[f1484,f1552])).
% 2.81/1.68 thf(f1552,plain,(
% 2.81/1.68 ( ! [X2 : paraco414474393lle_fm,X0 : list @ char > paraco415392788lle_tv,X1 : paraco414474393lle_fm] : (((paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)) = (paraco876059933e_eval @ X0 @ X1)) | ((paraco876059933e_eval @ X0 @ X2) != (paraco876059933e_eval @ X0 @ X1))) )),
% 2.81/1.68 inference(cnf_transformation,[status(thm)],[f861])).
% 2.81/1.68 thf(f861,plain,(
% 2.81/1.68 ! [X0 : list @ char > paraco415392788lle_tv,X1 : paraco414474393lle_fm,X2 : paraco414474393lle_fm] : ((((paraco876059933e_eval @ X0 @ X2) = (paraco876059933e_eval @ X0 @ X1)) | ((((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X0 @ X1)) | ((((paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)) = (paraco876059933e_eval @ X0 @ X1)) | ((paraco876059933e_eval @ X0 @ X2) != (paraco2040174112le_Det @ $true))) & (((paraco876059933e_eval @ X0 @ X2) = (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)) = (paraco2040174112le_Det @ $false))))) & (((paraco2040174112le_Det @ $true) != (paraco876059933e_eval @ X0 @ X1)) | ((paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)) = (paraco876059933e_eval @ X0 @ X2))))) & (((paraco876059933e_eval @ X0 @ X2) != (paraco876059933e_eval @ X0 @ X1)) | ((paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)) = (paraco876059933e_eval @ X0 @ X1))))),
% 2.81/1.68 inference(ennf_transformation,[status(thm)],[f556])).
% 2.81/1.68 thf(f556,plain,(
% 2.81/1.68 ! [X0 : list @ char > paraco415392788lle_tv,X1 : paraco414474393lle_fm,X2 : paraco414474393lle_fm] : ((((paraco876059933e_eval @ X0 @ X2) != (paraco876059933e_eval @ X0 @ X1)) => ((((paraco2040174112le_Det @ $true) != (paraco876059933e_eval @ X0 @ X1)) => ((((paraco876059933e_eval @ X0 @ X2) != (paraco2040174112le_Det @ $true)) => ((paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)) = (paraco2040174112le_Det @ $false))) & (((paraco876059933e_eval @ X0 @ X2) = (paraco2040174112le_Det @ $true)) => ((paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)) = (paraco876059933e_eval @ X0 @ X1))))) & (((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X0 @ X1)) => ((paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)) = (paraco876059933e_eval @ X0 @ X2))))) & (((paraco876059933e_eval @ X0 @ X2) = (paraco876059933e_eval @ X0 @ X1)) => ((paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)) = (paraco876059933e_eval @ X0 @ X1))))),
% 2.81/1.68 inference(fool_elimination,[status(thm)],[f555])).
% 2.81/1.68 thf(f555,plain,(
% 2.81/1.68 ! [X0 : list @ char > paraco415392788lle_tv,X1 : paraco414474393lle_fm,X2 : paraco414474393lle_fm] : ((((paraco876059933e_eval @ X0 @ X2) = (paraco876059933e_eval @ X0 @ X1)) => ((paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)) = (paraco876059933e_eval @ X0 @ X1))) & (((paraco876059933e_eval @ X0 @ X2) != (paraco876059933e_eval @ X0 @ X1)) => ((((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X0 @ X1)) => ((paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)) = (paraco876059933e_eval @ X0 @ X2))) & (((paraco2040174112le_Det @ $true) != (paraco876059933e_eval @ X0 @ X1)) => ((((paraco876059933e_eval @ X0 @ X2) = (paraco2040174112le_Det @ $true)) => ((paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)) = (paraco876059933e_eval @ X0 @ X1))) & (((paraco876059933e_eval @ X0 @ X2) != (paraco2040174112le_Det @ $true)) => ((paraco2040174112le_Det @ $false) = (paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)))))))))),
% 2.81/1.68 inference(rectify,[status(thm)],[f6])).
% 2.81/1.68 thf(f6,axiom,(
% 2.81/1.68 ! [X9 : list @ char > paraco415392788lle_tv,X10 : paraco414474393lle_fm,X11 : paraco414474393lle_fm] : ((((paraco876059933e_eval @ X9 @ X10) = (paraco876059933e_eval @ X9 @ X11)) => ((paraco876059933e_eval @ X9 @ X10) = (paraco876059933e_eval @ X9 @ (paraco2100061555le_Con @ X10 @ X11)))) & (((paraco876059933e_eval @ X9 @ X10) != (paraco876059933e_eval @ X9 @ X11)) => ((((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X9 @ X10)) => ((paraco876059933e_eval @ X9 @ X11) = (paraco876059933e_eval @ X9 @ (paraco2100061555le_Con @ X10 @ X11)))) & (((paraco2040174112le_Det @ $true) != (paraco876059933e_eval @ X9 @ X10)) => ((((paraco876059933e_eval @ X9 @ X11) = (paraco2040174112le_Det @ $true)) => ((paraco876059933e_eval @ X9 @ X10) = (paraco876059933e_eval @ X9 @ (paraco2100061555le_Con @ X10 @ X11)))) & (((paraco876059933e_eval @ X9 @ X11) != (paraco2040174112le_Det @ $true)) => ((paraco2040174112le_Det @ $false) = (paraco876059933e_eval @ X9 @ (paraco2100061555le_Con @ X10 @ X11)))))))))),
% 2.81/1.68 file('/export/starexec/sandbox2/tmp/tmp.cVWy2Pj8hg/DTF2THF_31276.p',fact_5_eval_Osimps_I4_J)).
% 2.81/1.68 thf(f1484,plain,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p1) != (paraco876059933e_eval @ i @ (paraco2100061555le_Con @ p1 @ p2)))),
% 2.81/1.68 inference(cnf_transformation,[status(thm)],[f739])).
% 2.81/1.68 thf(f739,plain,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p1) != (paraco876059933e_eval @ i @ (paraco2100061555le_Con @ p1 @ p2)))),
% 2.81/1.68 inference(flattening,[status(thm)],[f278])).
% 2.81/1.68 thf(f278,negated_conjecture,(
% 2.81/1.68 ~((paraco876059933e_eval @ i @ p1) = (paraco876059933e_eval @ i @ (paraco2100061555le_Con @ p1 @ p2)))),
% 2.81/1.68 inference(negated_conjecture,[status(cth)],[f277])).
% 2.81/1.68 thf(f277,conjecture,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p1) = (paraco876059933e_eval @ i @ (paraco2100061555le_Con @ p1 @ p2)))),
% 2.81/1.68 file('/export/starexec/sandbox2/tmp/tmp.cVWy2Pj8hg/DTF2THF_31276.p',conj_0)).
% 2.81/1.68 thf(f1696,plain,(
% 2.81/1.68 ~spl185_1 | spl185_2),
% 2.81/1.68 inference(avatar_split_clause,[status(thm)],[f1687,f1693,f1689])).
% 2.81/1.68 thf(f1687,plain,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p1) = (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ i @ p2) != (paraco2040174112le_Det @ $true))),
% 2.81/1.68 inference(duplicate_literal_removal,[status(thm)],[f1686])).
% 2.81/1.68 thf(f1686,plain,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p1) = (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ i @ p2) != (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ i @ p1) = (paraco2040174112le_Det @ $true))),
% 2.81/1.68 inference(forward_demodulation,[status(thm)],[f1685,f1381])).
% 2.81/1.68 thf(f1685,plain,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p1) = (paraco876059933e_eval @ i @ p2)) | ((paraco876059933e_eval @ i @ p1) = (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ i @ p2) != (paraco2040174112le_Det @ $true))),
% 2.81/1.68 inference(trivial_inequality_removal,[status(thm)],[f1682])).
% 2.81/1.68 thf(f1682,plain,(
% 2.81/1.68 ((paraco876059933e_eval @ i @ p2) != (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ i @ p1) != (paraco876059933e_eval @ i @ p1)) | ((paraco876059933e_eval @ i @ p1) = (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ i @ p1) = (paraco876059933e_eval @ i @ p2))),
% 2.81/1.68 inference(superposition,[status(thm)],[f1484,f1555])).
% 2.81/1.68 thf(f1555,plain,(
% 2.81/1.68 ( ! [X2 : paraco414474393lle_fm,X0 : list @ char > paraco415392788lle_tv,X1 : paraco414474393lle_fm] : (((paraco876059933e_eval @ X0 @ (paraco2100061555le_Con @ X1 @ X2)) = (paraco876059933e_eval @ X0 @ X1)) | ((paraco876059933e_eval @ X0 @ X2) = (paraco876059933e_eval @ X0 @ X1)) | ((paraco876059933e_eval @ X0 @ X2) != (paraco2040174112le_Det @ $true)) | ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X0 @ X1))) )),
% 2.81/1.68 inference(cnf_transformation,[status(thm)],[f861])).
% 2.81/1.68 % SZS output end Proof for DTF2THF_31276
% 2.81/1.68 % (31460)------------------------------
% 2.81/1.68 % (31460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.68 % (31460)Termination reason: Refutation
% 2.81/1.68
% 2.81/1.68 % (31460)Memory used [KB]: 7547
% 2.81/1.68 % (31460)Time elapsed: 0.055 s
% 2.81/1.68 % (31460)Instructions burned: 142 (million)
% 2.81/1.68 % (31460)------------------------------
% 2.81/1.68 % (31460)------------------------------
% 2.81/1.68 % (31422)Success in time 0.121 s
%------------------------------------------------------------------------------