%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP136^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n021.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 08:55:50 AM UTC 2026
% Result : Theorem 3.04s 1.64s
% Output : Refutation 3.04s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP136^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n021.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:27:23 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.47 ---- Original DTF file ---
% 0.29/0.47 thf(spec,logic,$$dhol).
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % File : ITP136^2 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.47 % Domain : Interactive Theorem Proving
% 0.29/0.47 % Problem : Sledgehammer Paraconsistency problem prob_1104__3276536_1
% 0.29/0.47 % Version : Especial.
% 0.29/0.47 % English :
% 0.29/0.47
% 0.29/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.47 % Source : [Des21]
% 0.29/0.47 % Names : Paraconsistency/prob_1104__3276536_1 [Des21]
% 0.29/0.47
% 0.29/0.47 % Status : Theorem
% 0.29/0.47 % Rating : 0.00 v7.5.0
% 0.29/0.47 % Syntax : Number of formulae : 342 ( 155 unt; 64 typ; 0 def)
% 0.29/0.47 % Number of atoms : 666 ( 347 equ; 0 cnn)
% 0.29/0.47 % Maximal formula atoms : 16 ( 2 avg)
% 0.29/0.47 % Number of connectives : 4097 ( 111 ~; 5 |; 36 &;3642 @)
% 0.29/0.47 % ( 0 <=>; 303 =>; 0 <=; 0 <~>)
% 0.29/0.47 % Maximal formula depth : 28 ( 8 avg)
% 0.29/0.47 % Number of types : 5 ( 4 usr)
% 0.29/0.47 % Number of type conns : 432 ( 432 >; 0 *; 0 +; 0 <<)
% 0.29/0.47 % Number of symbols : 63 ( 60 usr; 5 con; 0-8 aty)
% 0.29/0.47 % Number of variables : 1292 ( 46 ^;1159 !; 16 ?;1292 :)
% 0.29/0.47 % ( 71 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.47 % SPC : TH1_THM_EQU_NAR
% 0.29/0.47
% 0.29/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:24:49.797
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % Could-be-implicit typings (7)
% 0.29/0.47 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv,type,
% 0.29/0.47 paraco415392788lle_tv: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm,type,
% 0.29/0.47 paraco414474393lle_fm: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.29/0.47 product_prod: $tType > $tType > $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_t_String_Ochar,type,
% 0.29/0.47 char: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_t_List_Olist,type,
% 0.29/0.47 list: $tType > $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_t_Set_Oset,type,
% 0.29/0.47 set: $tType > $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_t_Nat_Onat,type,
% 0.29/0.47 nat: $tType ).
% 0.29/0.47
% 0.29/0.47 % Explicit typings (57)
% 0.29/0.47 thf(sy_cl_HOL_Otype,type,
% 0.29/0.47 type:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.29/0.47 ord:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Otop,type,
% 0.29/0.47 top:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.29/0.47 order:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.29/0.47 preorder:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Oorder__top,type,
% 0.29/0.47 order_top:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_BNF__Def_OfstOp,type,
% 0.29/0.47 bNF_fstOp:
% 0.29/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_BNF__Def_Opick__middlep,type,
% 0.29/0.47 bNF_pick_middlep:
% 0.29/0.47 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_BNF__Def_OsndOp,type,
% 0.29/0.47 bNF_sndOp:
% 0.29/0.47 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 0.29/0.47 bNF_Greatest_image2:
% 0.29/0.47 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Fun_Oinj__on,type,
% 0.29/0.47 inj_on:
% 0.29/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Fun_Othe__inv__into,type,
% 0.29/0.47 the_inv_into:
% 0.29/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_If,type,
% 0.29/0.47 if:
% 0.29/0.47 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.47 ord_less_eq:
% 0.29/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.29/0.47 top_top:
% 0.29/0.47 !>[A: $tType] : A ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__int,type,
% 0.29/0.47 paraco753463838ge_int: ( nat > nat ) > ( ( list @ char ) > paraco415392788lle_tv ) > ( list @ char ) > paraco415392788lle_tv ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv,type,
% 0.29/0.47 paraco1920534163nge_tv: ( nat > nat ) > paraco415392788lle_tv > paraco415392788lle_tv ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv__rel,type,
% 0.29/0.47 paraco2077297190tv_rel: ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) > ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval,type,
% 0.29/0.47 paraco876059933e_eval: ( ( list @ char ) > paraco415392788lle_tv ) > paraco414474393lle_fm > paraco415392788lle_tv ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H,type,
% 0.29/0.47 paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql,type,
% 0.29/0.47 paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H,type,
% 0.29/0.47 paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H,type,
% 0.29/0.47 paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro,type,
% 0.29/0.47 paraco27778325le_Pro: ( list @ char ) > paraco414474393lle_fm ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth,type,
% 0.29/0.47 paraco251304083_Truth: paraco414474393lle_fm ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Ocase__fm,type,
% 0.29/0.47 paraco1246693743ase_fm:
% 0.29/0.47 !>[A: $tType] : ( ( ( list @ char ) > A ) > A > ( paraco414474393lle_fm > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A ) > ( paraco414474393lle_fm > paraco414474393lle_fm > A ) > paraco414474393lle_fm > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet,type,
% 0.29/0.47 paraco2040174112le_Det: $o > paraco415392788lle_tv ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet,type,
% 0.29/0.47 paraco676387099_Indet: nat > paraco415392788lle_tv ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Ocase__tv,type,
% 0.29/0.47 paraco490622181ase_tv:
% 0.29/0.47 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv,type,
% 0.29/0.47 paraco152590079rec_tv:
% 0.29/0.47 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid,type,
% 0.29/0.47 paraco769098683_valid: paraco414474393lle_fm > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid__in,type,
% 0.29/0.47 paraco2086025920lid_in: ( set @ nat ) > paraco414474393lle_fm > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.29/0.47 product_Pair:
% 0.29/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_Oapfst,type,
% 0.29/0.47 product_apfst:
% 0.29/0.47 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_Oapsnd,type,
% 0.29/0.47 product_apsnd:
% 0.29/0.47 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_Obool_Ocase__bool,type,
% 0.29/0.47 product_case_bool:
% 0.29/0.47 !>[A: $tType] : ( A > A > $o > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_Ocurry,type,
% 0.29/0.47 product_curry:
% 0.29/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.29/0.47 produc2004651681e_prod:
% 0.29/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_Omap__prod,type,
% 0.29/0.47 product_map_prod:
% 0.29/0.47 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.29/0.47 product_rec_prod:
% 0.29/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.29/0.47 product_fst:
% 0.29/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.29/0.47 product_snd:
% 0.29/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.29/0.47 product_swap:
% 0.29/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Relation_Oinv__image,type,
% 0.29/0.47 inv_image:
% 0.29/0.47 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Relation_Oinv__imagep,type,
% 0.29/0.47 inv_imagep:
% 0.29/0.47 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Relation_Ototal__on,type,
% 0.29/0.47 total_on:
% 0.29/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Set_OCollect,type,
% 0.29/0.47 collect:
% 0.29/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Set_Oimage,type,
% 0.29/0.47 image:
% 0.29/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Wellfounded_Oaccp,type,
% 0.29/0.47 accp:
% 0.29/0.47 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Wellfounded_Olex__prod,type,
% 0.29/0.47 lex_prod:
% 0.29/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Wellfounded_Owf,type,
% 0.29/0.47 wf:
% 0.29/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Wfrec_Osame__fst,type,
% 0.29/0.47 same_fst:
% 0.29/0.47 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_member,type,
% 0.29/0.47 member:
% 0.29/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_f,type,
% 0.29/0.47 f: nat > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_i,type,
% 0.29/0.47 i: ( list @ char ) > paraco415392788lle_tv ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_p1____,type,
% 0.29/0.47 p1: paraco414474393lle_fm ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_p2____,type,
% 0.29/0.47 p2: paraco414474393lle_fm ).
% 0.29/0.47
% 0.29/0.47 % Relevant facts (256)
% 0.29/0.47 thf(fact_0_a_H_H_H,axiom,
% 0.29/0.47 ( ( paraco876059933e_eval @ i @ p2 )
% 0.29/0.47 != ( paraco2040174112le_Det @ $true ) ) ).
% 0.29/0.47
% 0.29/0.47 % a'''
% 0.29/0.47 thf(fact_1_a_H_H,axiom,
% 0.29/0.47 ( ( paraco876059933e_eval @ i @ p1 )
% 0.29/0.47 != ( paraco2040174112le_Det @ $true ) ) ).
% 0.29/0.47
% 0.29/0.47 % a''
% 0.29/0.47 thf(fact_2_a_H_H_H_H,axiom,
% 0.29/0.47 ( ( paraco876059933e_eval @ i @ p1 )
% 0.29/0.47 != ( paraco2040174112le_Det @ $false ) ) ).
% 0.29/0.47
% 0.29/0.47 % a''''
% 0.29/0.47 thf(fact_3_b_H_H,axiom,
% 0.29/0.47 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
% 0.29/0.47 != ( paraco2040174112le_Det @ $true ) ) ).
% 0.29/0.47
% 0.29/0.47 % b''
% 0.29/0.47 thf(fact_4_a,axiom,
% 0.29/0.47 ( ( paraco876059933e_eval @ i @ p2 )
% 0.29/0.47 = ( paraco2040174112le_Det @ $false ) ) ).
% 0.29/0.47
% 0.29/0.47 % a
% 0.29/0.47 thf(fact_5_a_H,axiom,
% 0.29/0.47 ( ( paraco876059933e_eval @ i @ p1 )
% 0.29/0.47 != ( paraco876059933e_eval @ i @ p2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % a'
% 0.29/0.47 thf(fact_6_b,axiom,
% 0.29/0.47 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 )
% 0.29/0.47 = ( paraco2040174112le_Det @ $false ) ) ).
% 0.29/0.47
% 0.29/0.47 % b
% 0.29/0.47 thf(fact_7_b_H,axiom,
% 0.29/0.47 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
% 0.29/0.47 != ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % b'
% 0.29/0.47 thf(fact_8_tv_Oinject_I1_J,axiom,
% 0.29/0.47 ! [X1: $o,Y1: $o] :
% 0.29/0.47 ( ( ( paraco2040174112le_Det @ X1 )
% 0.29/0.47 = ( paraco2040174112le_Det @ Y1 ) )
% 0.29/0.47 = ( X1 = Y1 ) ) ).
% 0.29/0.47
% 0.29/0.47 % tv.inject(1)
% 0.29/0.47 thf(fact_9_ih2,axiom,
% 0.29/0.47 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 )
% 0.29/0.47 = ( paraco1920534163nge_tv @ f @ ( paraco876059933e_eval @ i @ p2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ih2
% 0.29/0.47 thf(fact_10_ih1,axiom,
% 0.29/0.47 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
% 0.29/0.47 = ( paraco1920534163nge_tv @ f @ ( paraco876059933e_eval @ i @ p1 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ih1
% 0.29/0.47 thf(fact_11__092_060open_062change__tv_Af_A_Ieval_Ai_Ap2_J_A_061_ADet_AFalse_092_060close_062,axiom,
% 0.29/0.47 ( ( paraco1920534163nge_tv @ f @ ( paraco876059933e_eval @ i @ p2 ) )
% 0.29/0.47 = ( paraco2040174112le_Det @ $false ) ) ).
% 0.29/0.47
% 0.29/0.47 % \<open>change_tv f (eval i p2) = Det False\<close>
% 0.29/0.47 thf(fact_12_valid__def,axiom,
% 0.29/0.47 ( paraco769098683_valid
% 0.29/0.47 = ( ^ [P: paraco414474393lle_fm] :
% 0.29/0.47 ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 0.29/0.47 ( ( paraco876059933e_eval @ I @ P )
% 0.29/0.47 = ( paraco2040174112le_Det @ $true ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % valid_def
% 0.29/0.47 thf(fact_13_yes,axiom,
% 0.29/0.47 ( ( paraco876059933e_eval @ i @ ( paraco1628874225le_Eql @ p1 @ p2 ) )
% 0.29/0.47 = ( paraco876059933e_eval @ i @ ( paraco329115265le_Neg @ p1 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % yes
% 0.29/0.47 thf(fact_14_eval_Osimps_I2_J,axiom,
% 0.29/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv] :
% 0.29/0.47 ( ( paraco876059933e_eval @ I2 @ paraco251304083_Truth )
% 0.29/0.47 = ( paraco2040174112le_Det @ $true ) ) ).
% 0.29/0.47
% 0.29/0.47 % eval.simps(2)
% 0.29/0.47 thf(fact_15_tv_Osimps_I7_J,axiom,
% 0.29/0.47 ! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
% 0.29/0.47 ( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
% 0.29/0.47 = ( F1 @ X1 ) ) ).
% 0.29/0.47
% 0.29/0.47 % tv.simps(7)
% 0.29/0.47 thf(fact_16_eval_Osimps_I5_J,axiom,
% 0.29/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.29/0.47 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 = ( paraco876059933e_eval @ I2 @ Q ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
% 0.29/0.47 = ( paraco2040174112le_Det @ $true ) ) )
% 0.29/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 != ( paraco876059933e_eval @ I2 @ Q ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
% 0.29/0.47 = ( paraco2040174112le_Det @ $false ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eval.simps(5)
% 0.29/0.47 thf(fact_17_eval__equality__simplify,axiom,
% 0.29/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
% 0.29/0.47 = ( paraco2040174112le_Det
% 0.29/0.47 @ ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 = ( paraco876059933e_eval @ I2 @ Q ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eval_equality_simplify
% 0.29/0.47 thf(fact_18_change__int__def,axiom,
% 0.29/0.47 ( paraco753463838ge_int
% 0.29/0.47 = ( ^ [F: nat > nat,I: ( list @ char ) > paraco415392788lle_tv,S: list @ char] : ( paraco1920534163nge_tv @ F @ ( I @ S ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % change_int_def
% 0.29/0.47 thf(fact_19_eval_Osimps_I4_J,axiom,
% 0.29/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.29/0.47 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 = ( paraco876059933e_eval @ I2 @ Q ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 0.29/0.47 = ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 0.29/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 != ( paraco876059933e_eval @ I2 @ Q ) )
% 0.29/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 = ( paraco2040174112le_Det @ $true ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 0.29/0.47 = ( paraco876059933e_eval @ I2 @ Q ) ) )
% 0.29/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.29/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ Q )
% 0.29/0.47 = ( paraco2040174112le_Det @ $true ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 0.29/0.47 = ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 0.29/0.47 & ( ( ( paraco876059933e_eval @ I2 @ Q )
% 0.29/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 0.29/0.47 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eval.simps(4)
% 0.29/0.47 thf(fact_20_tv_Osimps_I5_J,axiom,
% 0.29/0.47 ! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
% 0.29/0.47 ( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
% 0.29/0.47 = ( F1 @ X1 ) ) ).
% 0.29/0.47
% 0.29/0.47 % tv.simps(5)
% 0.29/0.47 thf(fact_21_eval__negation,axiom,
% 0.29/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm] :
% 0.29/0.47 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 = ( paraco2040174112le_Det @ $false ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 0.29/0.47 = ( paraco2040174112le_Det @ $true ) ) )
% 0.29/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 != ( paraco2040174112le_Det @ $false ) )
% 0.29/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 = ( paraco2040174112le_Det @ $true ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 0.29/0.47 = ( paraco2040174112le_Det @ $false ) ) )
% 0.29/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 0.29/0.47 = ( paraco876059933e_eval @ I2 @ P2 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eval_negation
% 0.29/0.47 thf(fact_22_assms,axiom,
% 0.29/0.47 inj_on @ nat @ nat @ f @ ( top_top @ ( set @ nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % assms
% 0.29/0.47 thf(fact_23_change__tv_Osimps_I1_J,axiom,
% 0.29/0.47 ! [F3: nat > nat,B2: $o] :
% 0.29/0.47 ( ( paraco1920534163nge_tv @ F3 @ ( paraco2040174112le_Det @ B2 ) )
% 0.29/0.47 = ( paraco2040174112le_Det @ B2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % change_tv.simps(1)
% 0.29/0.47 thf(fact_24_fm_Oinject_I2_J,axiom,
% 0.29/0.47 ! [X3: paraco414474393lle_fm,Y3: paraco414474393lle_fm] :
% 0.29/0.47 ( ( ( paraco329115265le_Neg @ X3 )
% 0.29/0.47 = ( paraco329115265le_Neg @ Y3 ) )
% 0.29/0.47 = ( X3 = Y3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.inject(2)
% 0.29/0.47 thf(fact_25_fm_Oinject_I3_J,axiom,
% 0.29/0.47 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,Y41: paraco414474393lle_fm,Y42: paraco414474393lle_fm] :
% 0.29/0.47 ( ( ( paraco2100061555le_Con @ X41 @ X42 )
% 0.29/0.47 = ( paraco2100061555le_Con @ Y41 @ Y42 ) )
% 0.29/0.47 = ( ( X41 = Y41 )
% 0.29/0.47 & ( X42 = Y42 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.inject(3)
% 0.29/0.47 thf(fact_26_fm_Oinject_I5_J,axiom,
% 0.29/0.47 ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm,Y61: paraco414474393lle_fm,Y62: paraco414474393lle_fm] :
% 0.29/0.47 ( ( ( paraco1628874225le_Eql @ X61 @ X62 )
% 0.29/0.47 = ( paraco1628874225le_Eql @ Y61 @ Y62 ) )
% 0.29/0.47 = ( ( X61 = Y61 )
% 0.29/0.47 & ( X62 = Y62 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.inject(5)
% 0.29/0.47 thf(fact_27_fm_Oinject_I4_J,axiom,
% 0.29/0.47 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,Y51: paraco414474393lle_fm,Y52: paraco414474393lle_fm] :
% 0.29/0.47 ( ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 0.29/0.47 = ( paraco2084319816le_Eql @ Y51 @ Y52 ) )
% 0.29/0.47 = ( ( X51 = Y51 )
% 0.29/0.47 & ( X52 = Y52 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.inject(4)
% 0.29/0.47 thf(fact_28_fm_Odistinct_I29_J,axiom,
% 0.29/0.47 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 0.29/0.47 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(29)
% 0.29/0.47 thf(fact_29_fm_Odistinct_I27_J,axiom,
% 0.29/0.47 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 0.29/0.47 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(27)
% 0.29/0.47 thf(fact_30_fm_Odistinct_I25_J,axiom,
% 0.29/0.47 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 0.29/0.47 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(25)
% 0.29/0.47 thf(fact_31_fm_Odistinct_I23_J,axiom,
% 0.29/0.47 ! [X3: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco329115265le_Neg @ X3 )
% 0.29/0.47 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(23)
% 0.29/0.47 thf(fact_32_fm_Odistinct_I21_J,axiom,
% 0.29/0.47 ! [X3: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco329115265le_Neg @ X3 )
% 0.29/0.47 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(21)
% 0.29/0.47 thf(fact_33_fm_Odistinct_I19_J,axiom,
% 0.29/0.47 ! [X3: paraco414474393lle_fm,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco329115265le_Neg @ X3 )
% 0.29/0.47 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(19)
% 0.29/0.47 thf(fact_34_fm_Odistinct_I17_J,axiom,
% 0.29/0.47 ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.29/0.47 ( paraco251304083_Truth
% 0.29/0.47 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(17)
% 0.29/0.47 thf(fact_35_fm_Odistinct_I15_J,axiom,
% 0.29/0.47 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.29/0.47 ( paraco251304083_Truth
% 0.29/0.47 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(15)
% 0.29/0.47 thf(fact_36_fm_Odistinct_I13_J,axiom,
% 0.29/0.47 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.29/0.47 ( paraco251304083_Truth
% 0.29/0.47 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(13)
% 0.29/0.47 thf(fact_37_fm_Odistinct_I11_J,axiom,
% 0.29/0.47 ! [X3: paraco414474393lle_fm] :
% 0.29/0.47 ( paraco251304083_Truth
% 0.29/0.47 != ( paraco329115265le_Neg @ X3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(11)
% 0.29/0.47 thf(fact_38_conjunction,axiom,
% 0.29/0.47 ! [P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco769098683_valid @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 0.29/0.47 = ( ( paraco769098683_valid @ P2 )
% 0.29/0.47 & ( paraco769098683_valid @ Q ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % conjunction
% 0.29/0.47 thf(fact_39_eval__equality,axiom,
% 0.29/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.29/0.47 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 = ( paraco876059933e_eval @ I2 @ Q ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.29/0.47 = ( paraco2040174112le_Det @ $true ) ) )
% 0.29/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 != ( paraco876059933e_eval @ I2 @ Q ) )
% 0.29/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 = ( paraco2040174112le_Det @ $true ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.29/0.47 = ( paraco876059933e_eval @ I2 @ Q ) ) )
% 0.29/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.29/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ Q )
% 0.29/0.47 = ( paraco2040174112le_Det @ $true ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.29/0.47 = ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 0.29/0.47 & ( ( ( paraco876059933e_eval @ I2 @ Q )
% 0.29/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.29/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 = ( paraco2040174112le_Det @ $false ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.29/0.47 = ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ Q ) ) ) )
% 0.29/0.47 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 0.29/0.47 != ( paraco2040174112le_Det @ $false ) )
% 0.29/0.47 => ( ( ( ( paraco876059933e_eval @ I2 @ Q )
% 0.29/0.47 = ( paraco2040174112le_Det @ $false ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.29/0.47 = ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) ) ) )
% 0.29/0.47 & ( ( ( paraco876059933e_eval @ I2 @ Q )
% 0.29/0.47 != ( paraco2040174112le_Det @ $false ) )
% 0.29/0.47 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 0.29/0.47 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eval_equality
% 0.29/0.47 thf(fact_40_double__negation,axiom,
% 0.29/0.47 ( paraco876059933e_eval
% 0.29/0.47 = ( ^ [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] : ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ ( paraco329115265le_Neg @ P ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % double_negation
% 0.29/0.47 thf(fact_41_UNIV__I,axiom,
% 0.29/0.47 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % UNIV_I
% 0.29/0.47 thf(fact_42_iso__tuple__UNIV__I,axiom,
% 0.29/0.47 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % iso_tuple_UNIV_I
% 0.29/0.47 thf(fact_43_top__apply,axiom,
% 0.29/0.47 ! [C: $tType,D: $tType] :
% 0.29/0.47 ( ( top @ C )
% 0.29/0.47 => ( ( top_top @ ( D > C ) )
% 0.29/0.47 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % top_apply
% 0.29/0.47 thf(fact_44_injD,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,X: A,Y: A] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.47 => ( ( ( F3 @ X )
% 0.29/0.47 = ( F3 @ Y ) )
% 0.29/0.47 => ( X = Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % injD
% 0.29/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.29/0.47 ! [A: $tType,A2: A,P3: A > $o] :
% 0.29/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 0.29/0.47 = ( P3 @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % mem_Collect_eq
% 0.29/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.29/0.47 ! [A: $tType,A3: set @ A] :
% 0.29/0.47 ( ( collect @ A
% 0.29/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.29/0.47 = A3 ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_mem_eq
% 0.29/0.47 thf(fact_47_Collect__cong,axiom,
% 0.29/0.47 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 0.29/0.47 ( ! [X4: A] :
% 0.29/0.47 ( ( P3 @ X4 )
% 0.29/0.47 = ( Q2 @ X4 ) )
% 0.29/0.47 => ( ( collect @ A @ P3 )
% 0.29/0.47 = ( collect @ A @ Q2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_cong
% 0.29/0.47 thf(fact_48_ext,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 0.29/0.47 ( ! [X4: A] :
% 0.29/0.47 ( ( F3 @ X4 )
% 0.29/0.47 = ( G @ X4 ) )
% 0.29/0.47 => ( F3 = G ) ) ).
% 0.29/0.47
% 0.29/0.47 % ext
% 0.29/0.47 thf(fact_49_injI,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B] :
% 0.29/0.47 ( ! [X4: A,Y2: A] :
% 0.29/0.47 ( ( ( F3 @ X4 )
% 0.29/0.47 = ( F3 @ Y2 ) )
% 0.29/0.47 => ( X4 = Y2 ) )
% 0.29/0.47 => ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % injI
% 0.29/0.47 thf(fact_50_inj__eq,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,X: A,Y: A] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.47 => ( ( ( F3 @ X )
% 0.29/0.47 = ( F3 @ Y ) )
% 0.29/0.47 = ( X = Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_eq
% 0.29/0.47 thf(fact_51_inj__def,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.47 = ( ! [X2: A,Y4: A] :
% 0.29/0.47 ( ( ( F3 @ X2 )
% 0.29/0.47 = ( F3 @ Y4 ) )
% 0.29/0.47 => ( X2 = Y4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_def
% 0.29/0.47 thf(fact_52_fm_Oinduct,axiom,
% 0.29/0.47 ! [P3: paraco414474393lle_fm > $o,Fm: paraco414474393lle_fm] :
% 0.29/0.47 ( ! [X4: list @ char] : ( P3 @ ( paraco27778325le_Pro @ X4 ) )
% 0.29/0.47 => ( ( P3 @ paraco251304083_Truth )
% 0.29/0.47 => ( ! [X4: paraco414474393lle_fm] :
% 0.29/0.47 ( ( P3 @ X4 )
% 0.29/0.47 => ( P3 @ ( paraco329115265le_Neg @ X4 ) ) )
% 0.29/0.47 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 0.29/0.47 ( ( P3 @ X1a )
% 0.29/0.47 => ( ( P3 @ X22 )
% 0.29/0.47 => ( P3 @ ( paraco2100061555le_Con @ X1a @ X22 ) ) ) )
% 0.29/0.47 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 0.29/0.47 ( ( P3 @ X1a )
% 0.29/0.47 => ( ( P3 @ X22 )
% 0.29/0.47 => ( P3 @ ( paraco2084319816le_Eql @ X1a @ X22 ) ) ) )
% 0.29/0.47 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 0.29/0.47 ( ( P3 @ X1a )
% 0.29/0.47 => ( ( P3 @ X22 )
% 0.29/0.47 => ( P3 @ ( paraco1628874225le_Eql @ X1a @ X22 ) ) ) )
% 0.29/0.47 => ( P3 @ Fm ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.induct
% 0.29/0.47 thf(fact_53_fm_Oexhaust,axiom,
% 0.29/0.47 ! [Y: paraco414474393lle_fm] :
% 0.29/0.47 ( ! [X12: list @ char] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( paraco27778325le_Pro @ X12 ) )
% 0.29/0.47 => ( ( Y != paraco251304083_Truth )
% 0.29/0.47 => ( ! [X32: paraco414474393lle_fm] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( paraco329115265le_Neg @ X32 ) )
% 0.29/0.47 => ( ! [X412: paraco414474393lle_fm,X422: paraco414474393lle_fm] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( paraco2100061555le_Con @ X412 @ X422 ) )
% 0.29/0.47 => ( ! [X512: paraco414474393lle_fm,X522: paraco414474393lle_fm] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( paraco2084319816le_Eql @ X512 @ X522 ) )
% 0.29/0.47 => ~ ! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.exhaust
% 0.29/0.47 thf(fact_54_props_Ocases,axiom,
% 0.29/0.47 ! [X: paraco414474393lle_fm] :
% 0.29/0.47 ( ( X != paraco251304083_Truth )
% 0.29/0.47 => ( ! [S2: list @ char] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( paraco27778325le_Pro @ S2 ) )
% 0.29/0.47 => ( ! [P4: paraco414474393lle_fm] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( paraco329115265le_Neg @ P4 ) )
% 0.29/0.47 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( paraco2100061555le_Con @ P4 @ Q3 ) )
% 0.29/0.47 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( paraco2084319816le_Eql @ P4 @ Q3 ) )
% 0.29/0.47 => ~ ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % props.cases
% 0.29/0.47 thf(fact_55_fm_Oinject_I1_J,axiom,
% 0.29/0.47 ! [X1: list @ char,Y1: list @ char] :
% 0.29/0.47 ( ( ( paraco27778325le_Pro @ X1 )
% 0.29/0.47 = ( paraco27778325le_Pro @ Y1 ) )
% 0.29/0.47 = ( X1 = Y1 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.inject(1)
% 0.29/0.47 thf(fact_56_top__set__def,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( top_top @ ( set @ A ) )
% 0.29/0.47 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % top_set_def
% 0.29/0.47 thf(fact_57_eval_Osimps_I1_J,axiom,
% 0.29/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,S3: list @ char] :
% 0.29/0.47 ( ( paraco876059933e_eval @ I2 @ ( paraco27778325le_Pro @ S3 ) )
% 0.29/0.47 = ( I2 @ S3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % eval.simps(1)
% 0.29/0.47 thf(fact_58_fm_Odistinct_I3_J,axiom,
% 0.29/0.47 ! [X1: list @ char,X3: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.29/0.47 != ( paraco329115265le_Neg @ X3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(3)
% 0.29/0.47 thf(fact_59_fm_Odistinct_I5_J,axiom,
% 0.29/0.47 ! [X1: list @ char,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.29/0.47 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(5)
% 0.29/0.47 thf(fact_60_fm_Odistinct_I9_J,axiom,
% 0.29/0.47 ! [X1: list @ char,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.29/0.47 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(9)
% 0.29/0.47 thf(fact_61_fm_Odistinct_I7_J,axiom,
% 0.29/0.47 ! [X1: list @ char,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.29/0.47 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(7)
% 0.29/0.47 thf(fact_62_fm_Odistinct_I1_J,axiom,
% 0.29/0.47 ! [X1: list @ char] :
% 0.29/0.47 ( ( paraco27778325le_Pro @ X1 )
% 0.29/0.47 != paraco251304083_Truth ) ).
% 0.29/0.47
% 0.29/0.47 % fm.distinct(1)
% 0.29/0.47 thf(fact_63_change__tv__injection,axiom,
% 0.29/0.47 ! [F3: nat > nat] :
% 0.29/0.47 ( ( inj_on @ nat @ nat @ F3 @ ( top_top @ ( set @ nat ) ) )
% 0.29/0.47 => ( inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ ( paraco1920534163nge_tv @ F3 ) @ ( top_top @ ( set @ paraco415392788lle_tv ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % change_tv_injection
% 0.29/0.47 thf(fact_64_UNIV__witness,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % UNIV_witness
% 0.29/0.47 thf(fact_65_UNIV__eq__I,axiom,
% 0.29/0.47 ! [A: $tType,A3: set @ A] :
% 0.29/0.47 ( ! [X4: A] : ( member @ A @ X4 @ A3 )
% 0.29/0.47 => ( ( top_top @ ( set @ A ) )
% 0.29/0.47 = A3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % UNIV_eq_I
% 0.29/0.47 thf(fact_66_inj__on__inverseI,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,A3: set @ A,G: B > A,F3: A > B] :
% 0.29/0.47 ( ! [X4: A] :
% 0.29/0.47 ( ( member @ A @ X4 @ A3 )
% 0.29/0.47 => ( ( G @ ( F3 @ X4 ) )
% 0.29/0.47 = X4 ) )
% 0.29/0.47 => ( inj_on @ A @ B @ F3 @ A3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_inverseI
% 0.29/0.47 thf(fact_67_inj__on__contraD,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X: A,Y: A] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.29/0.47 => ( ( X != Y )
% 0.29/0.47 => ( ( member @ A @ X @ A3 )
% 0.29/0.47 => ( ( member @ A @ Y @ A3 )
% 0.29/0.47 => ( ( F3 @ X )
% 0.29/0.47 != ( F3 @ Y ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_contraD
% 0.29/0.47 thf(fact_68_inj__on__eq__iff,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X: A,Y: A] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.29/0.47 => ( ( member @ A @ X @ A3 )
% 0.29/0.47 => ( ( member @ A @ Y @ A3 )
% 0.29/0.47 => ( ( ( F3 @ X )
% 0.29/0.47 = ( F3 @ Y ) )
% 0.29/0.47 = ( X = Y ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_eq_iff
% 0.29/0.47 thf(fact_69_inj__on__cong,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,A3: set @ A,F3: A > B,G: A > B] :
% 0.29/0.47 ( ! [A4: A] :
% 0.29/0.47 ( ( member @ A @ A4 @ A3 )
% 0.29/0.47 => ( ( F3 @ A4 )
% 0.29/0.47 = ( G @ A4 ) ) )
% 0.29/0.47 => ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.29/0.47 = ( inj_on @ A @ B @ G @ A3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_cong
% 0.29/0.47 thf(fact_70_inj__on__def,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType] :
% 0.29/0.47 ( ( inj_on @ A @ B )
% 0.29/0.47 = ( ^ [F: A > B,A5: set @ A] :
% 0.29/0.47 ! [X2: A] :
% 0.29/0.47 ( ( member @ A @ X2 @ A5 )
% 0.29/0.47 => ! [Y4: A] :
% 0.29/0.47 ( ( member @ A @ Y4 @ A5 )
% 0.29/0.47 => ( ( ( F @ X2 )
% 0.29/0.47 = ( F @ Y4 ) )
% 0.29/0.47 => ( X2 = Y4 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_def
% 0.29/0.47 thf(fact_71_inj__onI,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,A3: set @ A,F3: A > B] :
% 0.29/0.47 ( ! [X4: A,Y2: A] :
% 0.29/0.47 ( ( member @ A @ X4 @ A3 )
% 0.29/0.47 => ( ( member @ A @ Y2 @ A3 )
% 0.29/0.47 => ( ( ( F3 @ X4 )
% 0.29/0.47 = ( F3 @ Y2 ) )
% 0.29/0.47 => ( X4 = Y2 ) ) ) )
% 0.29/0.47 => ( inj_on @ A @ B @ F3 @ A3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_onI
% 0.29/0.47 thf(fact_72_inj__onD,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X: A,Y: A] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.29/0.47 => ( ( ( F3 @ X )
% 0.29/0.47 = ( F3 @ Y ) )
% 0.29/0.47 => ( ( member @ A @ X @ A3 )
% 0.29/0.47 => ( ( member @ A @ Y @ A3 )
% 0.29/0.47 => ( X = Y ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_onD
% 0.29/0.47 thf(fact_73_props_Oinduct,axiom,
% 0.29/0.47 ! [P3: paraco414474393lle_fm > $o,A0: paraco414474393lle_fm] :
% 0.29/0.47 ( ( P3 @ paraco251304083_Truth )
% 0.29/0.47 => ( ! [S2: list @ char] : ( P3 @ ( paraco27778325le_Pro @ S2 ) )
% 0.29/0.47 => ( ! [P4: paraco414474393lle_fm] :
% 0.29/0.47 ( ( P3 @ P4 )
% 0.29/0.47 => ( P3 @ ( paraco329115265le_Neg @ P4 ) ) )
% 0.29/0.47 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.29/0.47 ( ( P3 @ P4 )
% 0.29/0.47 => ( ( P3 @ Q3 )
% 0.29/0.47 => ( P3 @ ( paraco2100061555le_Con @ P4 @ Q3 ) ) ) )
% 0.29/0.47 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.29/0.47 ( ( P3 @ P4 )
% 0.29/0.47 => ( ( P3 @ Q3 )
% 0.29/0.47 => ( P3 @ ( paraco2084319816le_Eql @ P4 @ Q3 ) ) ) )
% 0.29/0.47 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.29/0.47 ( ( P3 @ P4 )
% 0.29/0.47 => ( ( P3 @ Q3 )
% 0.29/0.47 => ( P3 @ ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) )
% 0.29/0.47 => ( P3 @ A0 ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % props.induct
% 0.29/0.47 thf(fact_74_eval_Ocases,axiom,
% 0.29/0.47 ! [X: product_prod @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm] :
% 0.29/0.47 ( ! [I3: ( list @ char ) > paraco415392788lle_tv,S2: list @ char] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco27778325le_Pro @ S2 ) ) )
% 0.29/0.47 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ paraco251304083_Truth ) )
% 0.29/0.47 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco329115265le_Neg @ P4 ) ) )
% 0.29/0.47 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2100061555le_Con @ P4 @ Q3 ) ) )
% 0.29/0.47 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2084319816le_Eql @ P4 @ Q3 ) ) )
% 0.29/0.47 => ~ ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eval.cases
% 0.29/0.47 thf(fact_75_top__empty__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( top_top @ ( A > $o ) )
% 0.29/0.47 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % top_empty_eq
% 0.29/0.47 thf(fact_76_change__tv_Oelims,axiom,
% 0.29/0.47 ! [X: nat > nat,Xa: paraco415392788lle_tv,Y: paraco415392788lle_tv] :
% 0.29/0.47 ( ( ( paraco1920534163nge_tv @ X @ Xa )
% 0.29/0.47 = Y )
% 0.29/0.47 => ( ! [B3: $o] :
% 0.29/0.47 ( ( Xa
% 0.29/0.47 = ( paraco2040174112le_Det @ B3 ) )
% 0.29/0.47 => ( Y
% 0.29/0.47 != ( paraco2040174112le_Det @ B3 ) ) )
% 0.29/0.47 => ~ ! [N: nat] :
% 0.29/0.47 ( ( Xa
% 0.29/0.47 = ( paraco676387099_Indet @ N ) )
% 0.29/0.47 => ( Y
% 0.29/0.47 != ( paraco676387099_Indet @ ( X @ N ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % change_tv.elims
% 0.29/0.47 thf(fact_77_valid__valid__in,axiom,
% 0.29/0.47 ! [P2: paraco414474393lle_fm,U: set @ nat] :
% 0.29/0.47 ( ( paraco769098683_valid @ P2 )
% 0.29/0.47 => ( paraco2086025920lid_in @ U @ P2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % valid_valid_in
% 0.29/0.47 thf(fact_78_transfer,axiom,
% 0.29/0.47 ! [U: set @ nat,P2: paraco414474393lle_fm] :
% 0.29/0.47 ( ~ ( paraco2086025920lid_in @ U @ P2 )
% 0.29/0.47 => ~ ( paraco769098683_valid @ P2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % transfer
% 0.29/0.47 thf(fact_79_the__inv__f__f,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,X: A] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.47 => ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ ( F3 @ X ) )
% 0.29/0.47 = X ) ) ).
% 0.29/0.47
% 0.29/0.47 % the_inv_f_f
% 0.29/0.47 thf(fact_80_conjunction__in,axiom,
% 0.29/0.47 ! [U: set @ nat,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco2086025920lid_in @ U @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 0.29/0.47 = ( ( paraco2086025920lid_in @ U @ P2 )
% 0.29/0.47 & ( paraco2086025920lid_in @ U @ Q ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % conjunction_in
% 0.29/0.47 thf(fact_81_fm_Osimps_I37_J,axiom,
% 0.29/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F32: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A] :
% 0.29/0.47 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F32 @ F4 @ F5 @ F6 @ paraco251304083_Truth )
% 0.29/0.47 = F2 ) ).
% 0.29/0.47
% 0.29/0.47 % fm.simps(37)
% 0.29/0.47 thf(fact_82_tv_Oinject_I2_J,axiom,
% 0.29/0.47 ! [X23: nat,Y22: nat] :
% 0.29/0.47 ( ( ( paraco676387099_Indet @ X23 )
% 0.29/0.47 = ( paraco676387099_Indet @ Y22 ) )
% 0.29/0.47 = ( X23 = Y22 ) ) ).
% 0.29/0.47
% 0.29/0.47 % tv.inject(2)
% 0.29/0.47 thf(fact_83_top1I,axiom,
% 0.29/0.47 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 0.29/0.47
% 0.29/0.47 % top1I
% 0.29/0.47 thf(fact_84_string__tv_Oinduct,axiom,
% 0.29/0.47 ! [P3: paraco415392788lle_tv > $o,A0: paraco415392788lle_tv] :
% 0.29/0.47 ( ( P3 @ ( paraco2040174112le_Det @ $true ) )
% 0.29/0.47 => ( ( P3 @ ( paraco2040174112le_Det @ $false ) )
% 0.29/0.47 => ( ! [N: nat] : ( P3 @ ( paraco676387099_Indet @ N ) )
% 0.29/0.47 => ( P3 @ A0 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % string_tv.induct
% 0.29/0.47 thf(fact_85_change__tv_Oinduct,axiom,
% 0.29/0.47 ! [P3: ( nat > nat ) > paraco415392788lle_tv > $o,A0: nat > nat,A1: paraco415392788lle_tv] :
% 0.29/0.47 ( ! [F7: nat > nat,B3: $o] : ( P3 @ F7 @ ( paraco2040174112le_Det @ B3 ) )
% 0.29/0.47 => ( ! [F7: nat > nat,N: nat] : ( P3 @ F7 @ ( paraco676387099_Indet @ N ) )
% 0.29/0.47 => ( P3 @ A0 @ A1 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % change_tv.induct
% 0.29/0.47 thf(fact_86_string__tv_Ocases,axiom,
% 0.29/0.47 ! [X: paraco415392788lle_tv] :
% 0.29/0.47 ( ( X
% 0.29/0.47 != ( paraco2040174112le_Det @ $true ) )
% 0.29/0.47 => ( ( X
% 0.29/0.47 != ( paraco2040174112le_Det @ $false ) )
% 0.29/0.47 => ~ ! [N: nat] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( paraco676387099_Indet @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % string_tv.cases
% 0.29/0.47 thf(fact_87_tv_Oexhaust,axiom,
% 0.29/0.47 ! [Y: paraco415392788lle_tv] :
% 0.29/0.47 ( ! [X12: $o] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( paraco2040174112le_Det @ X12 ) )
% 0.29/0.47 => ~ ! [X22: nat] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( paraco676387099_Indet @ X22 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % tv.exhaust
% 0.29/0.47 thf(fact_88_tv_Oinduct,axiom,
% 0.29/0.47 ! [P3: paraco415392788lle_tv > $o,Tv: paraco415392788lle_tv] :
% 0.29/0.47 ( ! [X4: $o] : ( P3 @ ( paraco2040174112le_Det @ X4 ) )
% 0.29/0.47 => ( ! [X4: nat] : ( P3 @ ( paraco676387099_Indet @ X4 ) )
% 0.29/0.47 => ( P3 @ Tv ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % tv.induct
% 0.29/0.47 thf(fact_89_tv_Odistinct_I1_J,axiom,
% 0.29/0.47 ! [X1: $o,X23: nat] :
% 0.29/0.47 ( ( paraco2040174112le_Det @ X1 )
% 0.29/0.47 != ( paraco676387099_Indet @ X23 ) ) ).
% 0.29/0.47
% 0.29/0.47 % tv.distinct(1)
% 0.29/0.47 thf(fact_90_the__inv__into__f__f,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X: A] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.29/0.47 => ( ( member @ A @ X @ A3 )
% 0.29/0.47 => ( ( the_inv_into @ A @ B @ A3 @ F3 @ ( F3 @ X ) )
% 0.29/0.47 = X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % the_inv_into_f_f
% 0.29/0.47 thf(fact_91_the__inv__into__f__eq,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X: A,Y: B] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.29/0.47 => ( ( ( F3 @ X )
% 0.29/0.47 = Y )
% 0.29/0.47 => ( ( member @ A @ X @ A3 )
% 0.29/0.47 => ( ( the_inv_into @ A @ B @ A3 @ F3 @ Y )
% 0.29/0.47 = X ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % the_inv_into_f_eq
% 0.29/0.47 thf(fact_92_tv_Osimps_I6_J,axiom,
% 0.29/0.47 ! [A: $tType,F1: $o > A,F2: nat > A,X23: nat] :
% 0.29/0.47 ( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X23 ) )
% 0.29/0.47 = ( F2 @ X23 ) ) ).
% 0.29/0.47
% 0.29/0.47 % tv.simps(6)
% 0.29/0.47 thf(fact_93_change__tv_Osimps_I2_J,axiom,
% 0.29/0.47 ! [F3: nat > nat,N2: nat] :
% 0.29/0.47 ( ( paraco1920534163nge_tv @ F3 @ ( paraco676387099_Indet @ N2 ) )
% 0.29/0.47 = ( paraco676387099_Indet @ ( F3 @ N2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % change_tv.simps(2)
% 0.29/0.47 thf(fact_94_fm_Osimps_I36_J,axiom,
% 0.29/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F32: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X1: list @ char] :
% 0.29/0.47 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F32 @ F4 @ F5 @ F6 @ ( paraco27778325le_Pro @ X1 ) )
% 0.29/0.47 = ( F1 @ X1 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.simps(36)
% 0.29/0.47 thf(fact_95_tv_Osimps_I8_J,axiom,
% 0.29/0.47 ! [A: $tType,F1: $o > A,F2: nat > A,X23: nat] :
% 0.29/0.47 ( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X23 ) )
% 0.29/0.47 = ( F2 @ X23 ) ) ).
% 0.29/0.47
% 0.29/0.47 % tv.simps(8)
% 0.29/0.47 thf(fact_96_fm_Osimps_I38_J,axiom,
% 0.29/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F32: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X3: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F32 @ F4 @ F5 @ F6 @ ( paraco329115265le_Neg @ X3 ) )
% 0.29/0.47 = ( F32 @ X3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.simps(38)
% 0.29/0.47 thf(fact_97_fm_Osimps_I39_J,axiom,
% 0.29/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F32: 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] :
% 0.29/0.47 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F32 @ F4 @ F5 @ F6 @ ( paraco2100061555le_Con @ X41 @ X42 ) )
% 0.29/0.47 = ( F4 @ X41 @ X42 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.simps(39)
% 0.29/0.47 thf(fact_98_fm_Osimps_I41_J,axiom,
% 0.29/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F32: 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] :
% 0.29/0.47 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F32 @ F4 @ F5 @ F6 @ ( paraco1628874225le_Eql @ X61 @ X62 ) )
% 0.29/0.47 = ( F6 @ X61 @ X62 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.simps(41)
% 0.29/0.47 thf(fact_99_fm_Osimps_I40_J,axiom,
% 0.29/0.47 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F32: 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] :
% 0.29/0.47 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F32 @ F4 @ F5 @ F6 @ ( paraco2084319816le_Eql @ X51 @ X52 ) )
% 0.29/0.47 = ( F5 @ X51 @ X52 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fm.simps(40)
% 0.29/0.47 thf(fact_100_prod_Oinject,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,X1: A,X23: B,Y1: A,Y22: B] :
% 0.29/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X23 )
% 0.29/0.47 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.29/0.47 = ( ( X1 = Y1 )
% 0.29/0.47 & ( X23 = Y22 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod.inject
% 0.29/0.47 thf(fact_101_old_Oprod_Oinject,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
% 0.29/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.29/0.47 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 0.29/0.47 = ( ( A2 = A6 )
% 0.29/0.47 & ( B2 = B4 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % old.prod.inject
% 0.29/0.47 thf(fact_102_eval_Osimps_I3_J,axiom,
% 0.29/0.47 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm] :
% 0.29/0.47 ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 0.29/0.47 = ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ P2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eval.simps(3)
% 0.29/0.47 thf(fact_103_top__conj_I2_J,axiom,
% 0.29/0.47 ! [A: $tType,P3: $o,X: A] :
% 0.29/0.47 ( ( P3
% 0.29/0.47 & ( top_top @ ( A > $o ) @ X ) )
% 0.29/0.47 = P3 ) ).
% 0.29/0.47
% 0.29/0.47 % top_conj(2)
% 0.29/0.47 thf(fact_104_top__conj_I1_J,axiom,
% 0.29/0.47 ! [A: $tType,X: A,P3: $o] :
% 0.29/0.47 ( ( ( top_top @ ( A > $o ) @ X )
% 0.29/0.47 & P3 )
% 0.29/0.47 = P3 ) ).
% 0.29/0.47
% 0.29/0.47 % top_conj(1)
% 0.29/0.47 thf(fact_105_surj__pair,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.29/0.47 ? [X4: A,Y2: B] :
% 0.29/0.47 ( P2
% 0.29/0.47 = ( product_Pair @ A @ B @ X4 @ Y2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % surj_pair
% 0.29/0.47 thf(fact_106_bool_Osplit__sel__asm,axiom,
% 0.29/0.47 ! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
% 0.29/0.47 ( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 0.29/0.47 = ( ~ ( ( Bool
% 0.29/0.47 & ~ ( P3 @ F1 ) )
% 0.29/0.47 | ( ~ Bool
% 0.29/0.47 & ~ ( P3 @ F2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bool.split_sel_asm
% 0.29/0.47 thf(fact_107_bool_Ocase__distrib,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,H: A > B,F1: A,F2: A,Bool: $o] :
% 0.29/0.47 ( ( H @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 0.29/0.47 = ( product_case_bool @ B @ ( H @ F1 ) @ ( H @ F2 ) @ Bool ) ) ).
% 0.29/0.47
% 0.29/0.47 % bool.case_distrib
% 0.29/0.47 thf(fact_108_bool_Ocase__eq__if,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( product_case_bool @ A )
% 0.29/0.47 = ( ^ [F12: A,F22: A,Bool2: $o] : ( if @ A @ Bool2 @ F12 @ F22 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bool.case_eq_if
% 0.29/0.47 thf(fact_109_bool_Osplit__sel,axiom,
% 0.29/0.47 ! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
% 0.29/0.47 ( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 0.29/0.47 = ( ( Bool
% 0.29/0.47 => ( P3 @ F1 ) )
% 0.29/0.47 & ( ~ Bool
% 0.29/0.47 => ( P3 @ F2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bool.split_sel
% 0.29/0.47 thf(fact_110_old_Obool_Osimps_I3_J,axiom,
% 0.29/0.47 ! [A: $tType,F1: A,F2: A] :
% 0.29/0.47 ( ( product_case_bool @ A @ F1 @ F2 @ $true )
% 0.29/0.47 = F1 ) ).
% 0.29/0.47
% 0.29/0.47 % old.bool.simps(3)
% 0.29/0.47 thf(fact_111_old_Obool_Osimps_I4_J,axiom,
% 0.29/0.47 ! [A: $tType,F1: A,F2: A] :
% 0.29/0.47 ( ( product_case_bool @ A @ F1 @ F2 @ $false )
% 0.29/0.47 = F2 ) ).
% 0.29/0.47
% 0.29/0.47 % old.bool.simps(4)
% 0.29/0.47 thf(fact_112_change__tv_Ocases,axiom,
% 0.29/0.47 ! [X: product_prod @ ( nat > nat ) @ paraco415392788lle_tv] :
% 0.29/0.47 ( ! [F7: nat > nat,B3: $o] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ F7 @ ( paraco2040174112le_Det @ B3 ) ) )
% 0.29/0.47 => ~ ! [F7: nat > nat,N: nat] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ F7 @ ( paraco676387099_Indet @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % change_tv.cases
% 0.29/0.47 thf(fact_113_old_Oprod_Oinducts,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.29/0.47 ( ! [A4: A,B3: B] : ( P3 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
% 0.29/0.47 => ( P3 @ Prod ) ) ).
% 0.29/0.47
% 0.29/0.47 % old.prod.inducts
% 0.29/0.47 thf(fact_114_old_Oprod_Oexhaust,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.29/0.47 ~ ! [A4: A,B3: B] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( product_Pair @ A @ B @ A4 @ B3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % old.prod.exhaust
% 0.29/0.47 thf(fact_115_prod__induct7,axiom,
% 0.29/0.47 ! [G2: $tType,F8: $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 @ F8 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) ) )] :
% 0.29/0.47 ( ! [A4: A,B3: B,C2: C,D2: D,E2: E,F7: F8,G3: G2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F8 @ G2 ) @ E2 @ ( product_Pair @ F8 @ G2 @ F7 @ G3 ) ) ) ) ) ) )
% 0.29/0.47 => ( P3 @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_induct7
% 0.29/0.47 thf(fact_116_prod__induct6,axiom,
% 0.29/0.47 ! [F8: $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 @ F8 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) ) )] :
% 0.29/0.47 ( ! [A4: A,B3: B,C2: C,D2: D,E2: E,F7: F8] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F8 ) @ D2 @ ( product_Pair @ E @ F8 @ E2 @ F7 ) ) ) ) ) )
% 0.29/0.47 => ( P3 @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_induct6
% 0.29/0.47 thf(fact_117_prod__induct5,axiom,
% 0.29/0.47 ! [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 ) ) )] :
% 0.29/0.47 ( ! [A4: A,B3: B,C2: C,D2: D,E2: E] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( 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 ) ) ) ) )
% 0.29/0.47 => ( P3 @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_induct5
% 0.29/0.47 thf(fact_118_prod__induct4,axiom,
% 0.29/0.47 ! [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 ) )] :
% 0.29/0.47 ( ! [A4: A,B3: B,C2: C,D2: D] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.29/0.47 => ( P3 @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_induct4
% 0.29/0.47 thf(fact_119_prod__induct3,axiom,
% 0.29/0.47 ! [C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.29/0.47 ( ! [A4: A,B3: B,C2: C] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B3 @ C2 ) ) )
% 0.29/0.47 => ( P3 @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_induct3
% 0.29/0.47 thf(fact_120_prod__cases7,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F8: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) ) )] :
% 0.29/0.47 ~ ! [A4: A,B3: B,C2: C,D2: D,E2: E,F7: F8,G3: G2] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F8 @ G2 ) @ E2 @ ( product_Pair @ F8 @ G2 @ F7 @ G3 ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_cases7
% 0.29/0.47 thf(fact_121_prod__cases6,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F8: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) ) )] :
% 0.29/0.47 ~ ! [A4: A,B3: B,C2: C,D2: D,E2: E,F7: F8] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F8 ) @ D2 @ ( product_Pair @ E @ F8 @ E2 @ F7 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_cases6
% 0.29/0.47 thf(fact_122_prod__cases5,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.29/0.47 ~ ! [A4: A,B3: B,C2: C,D2: D,E2: E] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( 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 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_cases5
% 0.29/0.47 thf(fact_123_prod__cases4,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.29/0.47 ~ ! [A4: A,B3: B,C2: C,D2: D] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_cases4
% 0.29/0.47 thf(fact_124_prod__cases3,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.29/0.47 ~ ! [A4: A,B3: B,C2: C] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B3 @ C2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_cases3
% 0.29/0.47 thf(fact_125_Pair__inject,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
% 0.29/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.29/0.47 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 0.29/0.47 => ~ ( ( A2 = A6 )
% 0.29/0.47 => ( B2 != B4 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Pair_inject
% 0.29/0.47 thf(fact_126_prod__cases,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.29/0.47 ( ! [A4: A,B3: B] : ( P3 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
% 0.29/0.47 => ( P3 @ P2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_cases
% 0.29/0.47 thf(fact_127_old_Oprod_Orec,axiom,
% 0.29/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.29/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.29/0.47 = ( F1 @ A2 @ B2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % old.prod.rec
% 0.29/0.47 thf(fact_128_internal__case__prod__conv,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 0.29/0.47 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.29/0.47 = ( C3 @ A2 @ B2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % internal_case_prod_conv
% 0.29/0.47 thf(fact_129_inj__apsnd,axiom,
% 0.29/0.47 ! [A: $tType,C: $tType,B: $tType,F3: B > C] :
% 0.29/0.47 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.29/0.47 = ( inj_on @ B @ C @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_apsnd
% 0.29/0.47 thf(fact_130_inj__apfst,axiom,
% 0.29/0.47 ! [B: $tType,C: $tType,A: $tType,F3: A > C] :
% 0.29/0.47 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.29/0.47 = ( inj_on @ A @ C @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_apfst
% 0.29/0.47 thf(fact_131_ssubst__Pair__rhs,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,R: A,S3: B,R2: set @ ( product_prod @ A @ B ),S4: B] :
% 0.29/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R2 )
% 0.29/0.47 => ( ( S4 = S3 )
% 0.29/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S4 ) @ R2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ssubst_Pair_rhs
% 0.29/0.47 thf(fact_132_apfst__conv,axiom,
% 0.29/0.47 ! [C: $tType,A: $tType,B: $tType,F3: C > A,X: C,Y: B] :
% 0.29/0.47 ( ( product_apfst @ C @ A @ B @ F3 @ ( product_Pair @ C @ B @ X @ Y ) )
% 0.29/0.47 = ( product_Pair @ A @ B @ ( F3 @ X ) @ Y ) ) ).
% 0.29/0.47
% 0.29/0.47 % apfst_conv
% 0.29/0.47 thf(fact_133_apsnd__conv,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,C: $tType,F3: C > B,X: A,Y: C] :
% 0.29/0.47 ( ( product_apsnd @ C @ B @ A @ F3 @ ( product_Pair @ A @ C @ X @ Y ) )
% 0.29/0.47 = ( product_Pair @ A @ B @ X @ ( F3 @ Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % apsnd_conv
% 0.29/0.47 thf(fact_134_apsnd__apfst__commute,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: C > B,G: D > A,P2: product_prod @ D @ C] :
% 0.29/0.47 ( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apfst @ D @ A @ C @ G @ P2 ) )
% 0.29/0.47 = ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F3 @ P2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % apsnd_apfst_commute
% 0.29/0.47 thf(fact_135_change__tv_Opelims,axiom,
% 0.29/0.47 ! [X: nat > nat,Xa: paraco415392788lle_tv,Y: paraco415392788lle_tv] :
% 0.29/0.47 ( ( ( paraco1920534163nge_tv @ X @ Xa )
% 0.29/0.47 = Y )
% 0.29/0.47 => ( ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ Xa ) )
% 0.29/0.47 => ( ! [B3: $o] :
% 0.29/0.47 ( ( Xa
% 0.29/0.47 = ( paraco2040174112le_Det @ B3 ) )
% 0.29/0.47 => ( ( Y
% 0.29/0.47 = ( paraco2040174112le_Det @ B3 ) )
% 0.29/0.47 => ~ ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ ( paraco2040174112le_Det @ B3 ) ) ) ) )
% 0.29/0.47 => ~ ! [N: nat] :
% 0.29/0.47 ( ( Xa
% 0.29/0.47 = ( paraco676387099_Indet @ N ) )
% 0.29/0.47 => ( ( Y
% 0.29/0.47 = ( paraco676387099_Indet @ ( X @ N ) ) )
% 0.29/0.47 => ~ ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ ( paraco676387099_Indet @ N ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % change_tv.pelims
% 0.29/0.47 thf(fact_136_prod_Oinj__map,axiom,
% 0.29/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F2: B > D] :
% 0.29/0.47 ( ( inj_on @ A @ C @ F1 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.47 => ( ( inj_on @ B @ D @ F2 @ ( top_top @ ( set @ B ) ) )
% 0.29/0.47 => ( 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 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod.inj_map
% 0.29/0.47 thf(fact_137_curry__conv,axiom,
% 0.29/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.29/0.47 ( ( product_curry @ B @ C @ A )
% 0.29/0.47 = ( ^ [F: ( product_prod @ B @ C ) > A,A7: B,B5: C] : ( F @ ( product_Pair @ B @ C @ A7 @ B5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % curry_conv
% 0.29/0.47 thf(fact_138_in__inv__imagep,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType] :
% 0.29/0.47 ( ( inv_imagep @ A @ B )
% 0.29/0.47 = ( ^ [R3: A > A > $o,F: B > A,X2: B,Y4: B] : ( R3 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % in_inv_imagep
% 0.29/0.47 thf(fact_139_curryI,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.29/0.47 ( ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.29/0.47 => ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % curryI
% 0.29/0.47 thf(fact_140_map__prod__simp,axiom,
% 0.29/0.47 ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G: D > B,A2: C,B2: D] :
% 0.29/0.47 ( ( product_map_prod @ C @ A @ D @ B @ F3 @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 0.29/0.47 = ( product_Pair @ A @ B @ ( F3 @ A2 ) @ ( G @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % map_prod_simp
% 0.29/0.47 thf(fact_141_curryD,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.29/0.47 ( ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 )
% 0.29/0.47 => ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % curryD
% 0.29/0.47 thf(fact_142_curryE,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.29/0.47 ( ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 )
% 0.29/0.47 => ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % curryE
% 0.29/0.47 thf(fact_143_the__inv__into__onto,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.29/0.47 => ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A3 @ F3 ) @ ( image @ A @ B @ F3 @ A3 ) )
% 0.29/0.47 = A3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % the_inv_into_onto
% 0.29/0.47 thf(fact_144_in__inv__image,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ B @ B ),F3: A > B] :
% 0.29/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R @ F3 ) )
% 0.29/0.47 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) @ R ) ) ).
% 0.29/0.47
% 0.29/0.47 % in_inv_image
% 0.29/0.47 thf(fact_145_map__prod__imageI,axiom,
% 0.29/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),F3: A > C,G: B > D] :
% 0.29/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 0.29/0.47 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F3 @ A2 ) @ ( G @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G ) @ R2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % map_prod_imageI
% 0.29/0.47 thf(fact_146_inj__swap,axiom,
% 0.29/0.47 ! [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 ) ).
% 0.29/0.47
% 0.29/0.47 % inj_swap
% 0.29/0.47 thf(fact_147_image__eqI,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,B2: A,F3: B > A,X: B,A3: set @ B] :
% 0.29/0.47 ( ( B2
% 0.29/0.47 = ( F3 @ X ) )
% 0.29/0.47 => ( ( member @ B @ X @ A3 )
% 0.29/0.47 => ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_eqI
% 0.29/0.47 thf(fact_148_swap__swap,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.29/0.47 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 0.29/0.47 = P2 ) ).
% 0.29/0.47
% 0.29/0.47 % swap_swap
% 0.29/0.47 thf(fact_149_swap__simp,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,X: B,Y: A] :
% 0.29/0.47 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.29/0.47 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % swap_simp
% 0.29/0.47 thf(fact_150_pair__in__swap__image,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,Y: A,X: B,A3: set @ ( product_prod @ B @ A )] :
% 0.29/0.47 ( ( 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 ) )
% 0.29/0.47 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % pair_in_swap_image
% 0.29/0.47 thf(fact_151_surj__swap,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType] :
% 0.29/0.47 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.29/0.47 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % surj_swap
% 0.29/0.47 thf(fact_152_map__prod__surj,axiom,
% 0.29/0.47 ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: A > B,G: C > D] :
% 0.29/0.47 ( ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.47 = ( top_top @ ( set @ B ) ) )
% 0.29/0.47 => ( ( ( image @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
% 0.29/0.47 = ( top_top @ ( set @ D ) ) )
% 0.29/0.47 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.29/0.47 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % map_prod_surj
% 0.29/0.47 thf(fact_153_rev__image__eqI,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,X: A,A3: set @ A,B2: B,F3: A > B] :
% 0.29/0.47 ( ( member @ A @ X @ A3 )
% 0.29/0.47 => ( ( B2
% 0.29/0.47 = ( F3 @ X ) )
% 0.29/0.47 => ( member @ B @ B2 @ ( image @ A @ B @ F3 @ A3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rev_image_eqI
% 0.29/0.47 thf(fact_154_ball__imageD,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P3: A > $o] :
% 0.29/0.47 ( ! [X4: A] :
% 0.29/0.47 ( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A3 ) )
% 0.29/0.47 => ( P3 @ X4 ) )
% 0.29/0.47 => ! [X5: B] :
% 0.29/0.47 ( ( member @ B @ X5 @ A3 )
% 0.29/0.47 => ( P3 @ ( F3 @ X5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ball_imageD
% 0.29/0.47 thf(fact_155_image__cong,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,M: set @ A,N3: set @ A,F3: A > B,G: A > B] :
% 0.29/0.47 ( ( M = N3 )
% 0.29/0.47 => ( ! [X4: A] :
% 0.29/0.47 ( ( member @ A @ X4 @ N3 )
% 0.29/0.47 => ( ( F3 @ X4 )
% 0.29/0.47 = ( G @ X4 ) ) )
% 0.29/0.47 => ( ( image @ A @ B @ F3 @ M )
% 0.29/0.47 = ( image @ A @ B @ G @ N3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_cong
% 0.29/0.47 thf(fact_156_bex__imageD,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P3: A > $o] :
% 0.29/0.47 ( ? [X5: A] :
% 0.29/0.47 ( ( member @ A @ X5 @ ( image @ B @ A @ F3 @ A3 ) )
% 0.29/0.47 & ( P3 @ X5 ) )
% 0.29/0.47 => ? [X4: B] :
% 0.29/0.47 ( ( member @ B @ X4 @ A3 )
% 0.29/0.47 & ( P3 @ ( F3 @ X4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bex_imageD
% 0.29/0.47 thf(fact_157_image__iff,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,Z: A,F3: B > A,A3: set @ B] :
% 0.29/0.47 ( ( member @ A @ Z @ ( image @ B @ A @ F3 @ A3 ) )
% 0.29/0.47 = ( ? [X2: B] :
% 0.29/0.47 ( ( member @ B @ X2 @ A3 )
% 0.29/0.47 & ( Z
% 0.29/0.47 = ( F3 @ X2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_iff
% 0.29/0.47 thf(fact_158_imageI,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,X: A,A3: set @ A,F3: A > B] :
% 0.29/0.47 ( ( member @ A @ X @ A3 )
% 0.29/0.47 => ( member @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % imageI
% 0.29/0.47 thf(fact_159_inj__on__image__iff,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,A3: set @ A,G: A > B,F3: A > A] :
% 0.29/0.47 ( ! [X4: A] :
% 0.29/0.47 ( ( member @ A @ X4 @ A3 )
% 0.29/0.47 => ! [Xa2: A] :
% 0.29/0.47 ( ( member @ A @ Xa2 @ A3 )
% 0.29/0.47 => ( ( ( G @ ( F3 @ X4 ) )
% 0.29/0.47 = ( G @ ( F3 @ Xa2 ) ) )
% 0.29/0.47 = ( ( G @ X4 )
% 0.29/0.47 = ( G @ Xa2 ) ) ) ) )
% 0.29/0.47 => ( ( inj_on @ A @ A @ F3 @ A3 )
% 0.29/0.47 => ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F3 @ A3 ) )
% 0.29/0.47 = ( inj_on @ A @ B @ G @ A3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_image_iff
% 0.29/0.47 thf(fact_160_surjD,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,F3: B > A,Y: A] :
% 0.29/0.47 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.29/0.47 = ( top_top @ ( set @ A ) ) )
% 0.29/0.47 => ? [X4: B] :
% 0.29/0.47 ( Y
% 0.29/0.47 = ( F3 @ X4 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % surjD
% 0.29/0.47 thf(fact_161_surjE,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,F3: B > A,Y: A] :
% 0.29/0.47 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.29/0.47 = ( top_top @ ( set @ A ) ) )
% 0.29/0.47 => ~ ! [X4: B] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( F3 @ X4 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % surjE
% 0.29/0.47 thf(fact_162_surjI,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,G: B > A,F3: A > B] :
% 0.29/0.47 ( ! [X4: A] :
% 0.29/0.47 ( ( G @ ( F3 @ X4 ) )
% 0.29/0.47 = X4 )
% 0.29/0.47 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 0.29/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % surjI
% 0.29/0.47 thf(fact_163_rangeI,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,F3: B > A,X: B] : ( member @ A @ ( F3 @ X ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rangeI
% 0.29/0.47 thf(fact_164_surj__def,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: B > A] :
% 0.29/0.47 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.29/0.47 = ( top_top @ ( set @ A ) ) )
% 0.29/0.47 = ( ! [Y4: A] :
% 0.29/0.47 ? [X2: B] :
% 0.29/0.47 ( Y4
% 0.29/0.47 = ( F3 @ X2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % surj_def
% 0.29/0.47 thf(fact_165_range__eqI,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,B2: A,F3: B > A,X: B] :
% 0.29/0.47 ( ( B2
% 0.29/0.47 = ( F3 @ X ) )
% 0.29/0.47 => ( member @ A @ B2 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % range_eqI
% 0.29/0.47 thf(fact_166_range__ex1__eq,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,B2: B] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.47 => ( ( member @ B @ B2 @ ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
% 0.29/0.47 = ( ? [X2: A] :
% 0.29/0.47 ( ( B2
% 0.29/0.47 = ( F3 @ X2 ) )
% 0.29/0.47 & ! [Y4: A] :
% 0.29/0.47 ( ( B2
% 0.29/0.47 = ( F3 @ Y4 ) )
% 0.29/0.47 => ( Y4 = X2 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % range_ex1_eq
% 0.29/0.47 thf(fact_167_inj__image__eq__iff,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B6: set @ A] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.47 => ( ( ( image @ A @ B @ F3 @ A3 )
% 0.29/0.47 = ( image @ A @ B @ F3 @ B6 ) )
% 0.29/0.47 = ( A3 = B6 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_image_eq_iff
% 0.29/0.47 thf(fact_168_inj__image__mem__iff,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,A2: A,A3: set @ A] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.47 => ( ( member @ B @ ( F3 @ A2 ) @ ( image @ A @ B @ F3 @ A3 ) )
% 0.29/0.47 = ( member @ A @ A2 @ A3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_image_mem_iff
% 0.29/0.47 thf(fact_169_prod__fun__imageE,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F3: C > A,G: D > B,R2: set @ ( product_prod @ C @ D )] :
% 0.29/0.47 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G ) @ R2 ) )
% 0.29/0.47 => ~ ! [X4: C,Y2: D] :
% 0.29/0.47 ( ( C3
% 0.29/0.47 = ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G @ Y2 ) ) )
% 0.29/0.47 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X4 @ Y2 ) @ R2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_fun_imageE
% 0.29/0.47 thf(fact_170_f__the__inv__into__f,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,F3: A > B,A3: set @ A,Y: B] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.29/0.47 => ( ( member @ B @ Y @ ( image @ A @ B @ F3 @ A3 ) )
% 0.29/0.47 => ( ( F3 @ ( the_inv_into @ A @ B @ A3 @ F3 @ Y ) )
% 0.29/0.47 = Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % f_the_inv_into_f
% 0.29/0.47 thf(fact_171_inj__on__the__inv__into,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.29/0.47 => ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A3 @ F3 ) @ ( image @ A @ B @ F3 @ A3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_the_inv_into
% 0.29/0.47 thf(fact_172_total__inv__image,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,F3: A > B,R: set @ ( product_prod @ B @ B )] :
% 0.29/0.47 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.47 => ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ R )
% 0.29/0.47 => ( total_on @ A @ ( top_top @ ( set @ A ) ) @ ( inv_image @ B @ A @ R @ F3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % total_inv_image
% 0.29/0.47 thf(fact_173_range__fst,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType] :
% 0.29/0.47 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.29/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % range_fst
% 0.29/0.47 thf(fact_174_range__snd,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType] :
% 0.29/0.47 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.29/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % range_snd
% 0.29/0.47 thf(fact_175_fst__map__prod,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > A,G: D > B,X: product_prod @ C @ D] :
% 0.29/0.47 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G @ X ) )
% 0.29/0.47 = ( F3 @ ( product_fst @ C @ D @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % fst_map_prod
% 0.29/0.47 thf(fact_176_snd__map__prod,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > B,G: D > A,X: product_prod @ C @ D] :
% 0.29/0.47 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F3 @ G @ X ) )
% 0.29/0.47 = ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % snd_map_prod
% 0.29/0.47 thf(fact_177_apfst__eq__conv,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,C: $tType,F3: C > A,X: product_prod @ C @ B,G: C > A] :
% 0.29/0.47 ( ( ( product_apfst @ C @ A @ B @ F3 @ X )
% 0.29/0.47 = ( product_apfst @ C @ A @ B @ G @ X ) )
% 0.29/0.47 = ( ( F3 @ ( product_fst @ C @ B @ X ) )
% 0.29/0.47 = ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % apfst_eq_conv
% 0.29/0.47 thf(fact_178_fst__apfst,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,C: $tType,F3: C > A,X: product_prod @ C @ B] :
% 0.29/0.47 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F3 @ X ) )
% 0.29/0.47 = ( F3 @ ( product_fst @ C @ B @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % fst_apfst
% 0.29/0.47 thf(fact_179_snd__apfst,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,C: $tType,F3: C > B,X: product_prod @ C @ A] :
% 0.29/0.47 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F3 @ X ) )
% 0.29/0.47 = ( product_snd @ C @ A @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % snd_apfst
% 0.29/0.47 thf(fact_180_fst__apsnd,axiom,
% 0.29/0.47 ! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C] :
% 0.29/0.47 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F3 @ X ) )
% 0.29/0.47 = ( product_fst @ A @ C @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % fst_apsnd
% 0.29/0.47 thf(fact_181_apsnd__eq__conv,axiom,
% 0.29/0.47 ! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C,G: C > B] :
% 0.29/0.47 ( ( ( product_apsnd @ C @ B @ A @ F3 @ X )
% 0.29/0.47 = ( product_apsnd @ C @ B @ A @ G @ X ) )
% 0.29/0.47 = ( ( F3 @ ( product_snd @ A @ C @ X ) )
% 0.29/0.47 = ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % apsnd_eq_conv
% 0.29/0.47 thf(fact_182_snd__apsnd,axiom,
% 0.29/0.47 ! [A: $tType,C: $tType,B: $tType,F3: C > A,X: product_prod @ B @ C] :
% 0.29/0.47 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F3 @ X ) )
% 0.29/0.47 = ( F3 @ ( product_snd @ B @ C @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % snd_apsnd
% 0.29/0.47 thf(fact_183_prod_Ocollapse,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.29/0.47 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.29/0.47 = Prod ) ).
% 0.29/0.47
% 0.29/0.47 % prod.collapse
% 0.29/0.47 thf(fact_184_snd__swap,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 0.29/0.47 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 0.29/0.47 = ( product_fst @ A @ B @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % snd_swap
% 0.29/0.47 thf(fact_185_fst__swap,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 0.29/0.47 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 0.29/0.47 = ( product_snd @ B @ A @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % fst_swap
% 0.29/0.47 thf(fact_186_apsnd__apfst,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: C > B,G: D > A,X: product_prod @ D @ C] :
% 0.29/0.47 ( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apfst @ D @ A @ C @ G @ X ) )
% 0.29/0.47 = ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F3 @ ( product_snd @ D @ C @ X ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % apsnd_apfst
% 0.29/0.47 thf(fact_187_apfst__apsnd,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,D: $tType,C: $tType,F3: C > A,G: D > B,X: product_prod @ C @ D] :
% 0.29/0.47 ( ( product_apfst @ C @ A @ B @ F3 @ ( product_apsnd @ D @ B @ C @ G @ X ) )
% 0.29/0.47 = ( product_Pair @ A @ B @ ( F3 @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % apfst_apsnd
% 0.29/0.47 thf(fact_188_prod_Oswap__def,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType] :
% 0.29/0.47 ( ( product_swap @ A @ B )
% 0.29/0.47 = ( ^ [P: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P ) @ ( product_fst @ A @ B @ P ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod.swap_def
% 0.29/0.47 thf(fact_189_fst__eqD,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
% 0.29/0.47 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 0.29/0.47 = A2 )
% 0.29/0.47 => ( X = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % fst_eqD
% 0.29/0.47 thf(fact_190_snd__eqD,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
% 0.29/0.47 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.29/0.47 = A2 )
% 0.29/0.47 => ( Y = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % snd_eqD
% 0.29/0.47 thf(fact_191_fst__conv,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,X1: A,X23: B] :
% 0.29/0.47 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X23 ) )
% 0.29/0.47 = X1 ) ).
% 0.29/0.47
% 0.29/0.47 % fst_conv
% 0.29/0.47 thf(fact_192_snd__conv,axiom,
% 0.29/0.47 ! [Aa: $tType,A: $tType,X1: Aa,X23: A] :
% 0.29/0.47 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X23 ) )
% 0.29/0.47 = X23 ) ).
% 0.29/0.47
% 0.29/0.47 % snd_conv
% 0.29/0.47 thf(fact_193_prod_Oexhaust__sel,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.29/0.47 ( Prod
% 0.29/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod.exhaust_sel
% 0.29/0.47 thf(fact_194_surjective__pairing,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.29/0.47 ( T2
% 0.29/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % surjective_pairing
% 0.29/0.47 thf(fact_195_total__onI,axiom,
% 0.29/0.47 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.29/0.47 ( ! [X4: A,Y2: A] :
% 0.29/0.47 ( ( member @ A @ X4 @ A3 )
% 0.29/0.47 => ( ( member @ A @ Y2 @ A3 )
% 0.29/0.47 => ( ( X4 != Y2 )
% 0.29/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y2 ) @ R )
% 0.29/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X4 ) @ R ) ) ) ) )
% 0.29/0.47 => ( total_on @ A @ A3 @ R ) ) ).
% 0.29/0.47
% 0.29/0.47 % total_onI
% 0.29/0.47 thf(fact_196_total__on__def,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( total_on @ A )
% 0.29/0.47 = ( ^ [A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.29/0.47 ! [X2: A] :
% 0.29/0.47 ( ( member @ A @ X2 @ A5 )
% 0.29/0.47 => ! [Y4: A] :
% 0.29/0.47 ( ( member @ A @ Y4 @ A5 )
% 0.29/0.47 => ( ( X2 != Y4 )
% 0.29/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R3 )
% 0.29/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R3 ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % total_on_def
% 0.29/0.47 thf(fact_197_prod__eqI,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q: product_prod @ A @ B] :
% 0.29/0.47 ( ( ( product_fst @ A @ B @ P2 )
% 0.29/0.47 = ( product_fst @ A @ B @ Q ) )
% 0.29/0.47 => ( ( ( product_snd @ A @ B @ P2 )
% 0.29/0.47 = ( product_snd @ A @ B @ Q ) )
% 0.29/0.47 => ( P2 = Q ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_eqI
% 0.29/0.47 thf(fact_198_prod_Oexpand,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.29/0.47 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.29/0.47 = ( product_fst @ A @ B @ Prod2 ) )
% 0.29/0.47 & ( ( product_snd @ A @ B @ Prod )
% 0.29/0.47 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.29/0.47 => ( Prod = Prod2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod.expand
% 0.29/0.47 thf(fact_199_prod__eq__iff,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType] :
% 0.29/0.47 ( ( ^ [Y5: product_prod @ A @ B,Z2: product_prod @ A @ B] : ( Y5 = Z2 ) )
% 0.29/0.47 = ( ^ [S: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 0.29/0.47 ( ( ( product_fst @ A @ B @ S )
% 0.29/0.47 = ( product_fst @ A @ B @ T3 ) )
% 0.29/0.47 & ( ( product_snd @ A @ B @ S )
% 0.29/0.47 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % prod_eq_iff
% 0.29/0.47 thf(fact_200_exI__realizer,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,P3: A > B > $o,Y: A,X: B] :
% 0.29/0.47 ( ( P3 @ Y @ X )
% 0.29/0.47 => ( P3 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % exI_realizer
% 0.29/0.47 thf(fact_201_conjI__realizer,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,P3: A > $o,P2: A,Q2: B > $o,Q: B] :
% 0.29/0.47 ( ( P3 @ P2 )
% 0.29/0.47 => ( ( Q2 @ Q )
% 0.29/0.47 => ( ( P3 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) )
% 0.29/0.47 & ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % conjI_realizer
% 0.29/0.47 thf(fact_202_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,P3: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
% 0.29/0.47 ( ( P3 @ X @ Y )
% 0.29/0.47 => ( ( A2
% 0.29/0.47 = ( product_Pair @ A @ B @ X @ Y ) )
% 0.29/0.47 => ( P3 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % BNF_Greatest_Fixpoint.subst_Pair
% 0.29/0.47 thf(fact_203_eq__snd__iff,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 0.29/0.47 ( ( B2
% 0.29/0.47 = ( product_snd @ B @ A @ P2 ) )
% 0.29/0.47 = ( ? [A7: B] :
% 0.29/0.47 ( P2
% 0.29/0.47 = ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eq_snd_iff
% 0.29/0.47 thf(fact_204_sndI,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
% 0.29/0.47 ( ( X
% 0.29/0.47 = ( product_Pair @ A @ B @ Y @ Z ) )
% 0.29/0.47 => ( ( product_snd @ A @ B @ X )
% 0.29/0.47 = Z ) ) ).
% 0.29/0.47
% 0.29/0.47 % sndI
% 0.29/0.47 thf(fact_205_fstI,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
% 0.29/0.47 ( ( X
% 0.29/0.47 = ( product_Pair @ A @ B @ Y @ Z ) )
% 0.29/0.47 => ( ( product_fst @ A @ B @ X )
% 0.29/0.47 = Y ) ) ).
% 0.29/0.47
% 0.29/0.47 % fstI
% 0.29/0.47 thf(fact_206_eq__fst__iff,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 0.29/0.47 ( ( A2
% 0.29/0.47 = ( product_fst @ A @ B @ P2 ) )
% 0.29/0.47 = ( ? [B5: B] :
% 0.29/0.47 ( P2
% 0.29/0.47 = ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eq_fst_iff
% 0.29/0.47 thf(fact_207_total__lex__prod,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
% 0.29/0.47 ( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R )
% 0.29/0.47 => ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ S3 )
% 0.29/0.47 => ( total_on @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( lex_prod @ A @ B @ R @ S3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % total_lex_prod
% 0.29/0.47 thf(fact_208_sndOp__def,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,C: $tType] :
% 0.29/0.47 ( ( bNF_sndOp @ C @ A @ B )
% 0.29/0.47 = ( ^ [P5: C > A > $o,Q4: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P5 @ Q4 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sndOp_def
% 0.29/0.47 thf(fact_209_in__lex__prod,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
% 0.29/0.47 ( ( 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 ) )
% 0.29/0.47 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A6 ) @ R )
% 0.29/0.47 | ( ( A2 = A6 )
% 0.29/0.47 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B4 ) @ S3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % in_lex_prod
% 0.29/0.47 thf(fact_210_fstOp__def,axiom,
% 0.29/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.29/0.47 ( ( bNF_fstOp @ A @ B @ C )
% 0.29/0.47 = ( ^ [P5: A > B > $o,Q4: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P5 @ Q4 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % fstOp_def
% 0.29/0.47 thf(fact_211_wf__map__prod__image,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F3: A > B] :
% 0.29/0.47 ( ( wf @ A @ R )
% 0.29/0.47 => ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.47 => ( wf @ B @ ( image @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( product_map_prod @ A @ B @ A @ B @ F3 @ F3 ) @ R ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % wf_map_prod_image
% 0.29/0.47 thf(fact_212_wf__induct__rule,axiom,
% 0.29/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P3: A > $o,A2: A] :
% 0.29/0.47 ( ( wf @ A @ R )
% 0.29/0.47 => ( ! [X4: A] :
% 0.29/0.47 ( ! [Y6: A] :
% 0.29/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X4 ) @ R )
% 0.29/0.47 => ( P3 @ Y6 ) )
% 0.29/0.47 => ( P3 @ X4 ) )
% 0.29/0.47 => ( P3 @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % wf_induct_rule
% 0.29/0.47 thf(fact_213_wf__eq__minimal,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( wf @ A )
% 0.29/0.47 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.29/0.47 ! [Q4: set @ A] :
% 0.29/0.47 ( ? [X2: A] : ( member @ A @ X2 @ Q4 )
% 0.29/0.47 => ? [X2: A] :
% 0.29/0.47 ( ( member @ A @ X2 @ Q4 )
% 0.29/0.47 & ! [Y4: A] :
% 0.29/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R3 )
% 0.29/0.47 => ~ ( member @ A @ Y4 @ Q4 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % wf_eq_minimal
% 0.29/0.47 thf(fact_214_wf__not__refl,axiom,
% 0.29/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
% 0.29/0.47 ( ( wf @ A @ R )
% 0.29/0.47 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% 0.29/0.47
% 0.29/0.47 % wf_not_refl
% 0.29/0.47 thf(fact_215_wf__not__sym,axiom,
% 0.29/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X: A] :
% 0.29/0.47 ( ( wf @ A @ R )
% 0.29/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X ) @ R )
% 0.29/0.47 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ R ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % wf_not_sym
% 0.29/0.47 thf(fact_216_wf__irrefl,axiom,
% 0.29/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
% 0.29/0.47 ( ( wf @ A @ R )
% 0.29/0.47 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% 0.29/0.47
% 0.29/0.47 % wf_irrefl
% 0.29/0.47 thf(fact_217_wf__induct,axiom,
% 0.29/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P3: A > $o,A2: A] :
% 0.29/0.47 ( ( wf @ A @ R )
% 0.29/0.47 => ( ! [X4: A] :
% 0.29/0.47 ( ! [Y6: A] :
% 0.29/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X4 ) @ R )
% 0.29/0.47 => ( P3 @ Y6 ) )
% 0.29/0.47 => ( P3 @ X4 ) )
% 0.29/0.47 => ( P3 @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % wf_induct
% 0.29/0.47 thf(fact_218_wf__asym,axiom,
% 0.29/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X: A] :
% 0.29/0.47 ( ( wf @ A @ R )
% 0.29/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X ) @ R )
% 0.29/0.47 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ R ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % wf_asym
% 0.29/0.47 thf(fact_219_wfUNIVI,axiom,
% 0.29/0.47 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.29/0.47 ( ! [P6: A > $o,X4: A] :
% 0.29/0.47 ( ! [Xa3: A] :
% 0.29/0.47 ( ! [Y2: A] :
% 0.29/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa3 ) @ R )
% 0.29/0.47 => ( P6 @ Y2 ) )
% 0.29/0.47 => ( P6 @ Xa3 ) )
% 0.29/0.47 => ( P6 @ X4 ) )
% 0.29/0.47 => ( wf @ A @ R ) ) ).
% 0.29/0.47
% 0.29/0.47 % wfUNIVI
% 0.29/0.47 thf(fact_220_wfI__min,axiom,
% 0.29/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.29/0.47 ( ! [X4: A,Q5: set @ A] :
% 0.29/0.47 ( ( member @ A @ X4 @ Q5 )
% 0.29/0.47 => ? [Xa3: A] :
% 0.29/0.47 ( ( member @ A @ Xa3 @ Q5 )
% 0.29/0.47 & ! [Y2: A] :
% 0.29/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa3 ) @ R2 )
% 0.29/0.47 => ~ ( member @ A @ Y2 @ Q5 ) ) ) )
% 0.29/0.47 => ( wf @ A @ R2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % wfI_min
% 0.29/0.47 thf(fact_221_wfE__min,axiom,
% 0.29/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Q2: set @ A] :
% 0.29/0.47 ( ( wf @ A @ R2 )
% 0.29/0.47 => ( ( member @ A @ X @ Q2 )
% 0.29/0.47 => ~ ! [Z3: A] :
% 0.29/0.47 ( ( member @ A @ Z3 @ Q2 )
% 0.29/0.47 => ~ ! [Y6: A] :
% 0.29/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R2 )
% 0.29/0.47 => ~ ( member @ A @ Y6 @ Q2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % wfE_min
% 0.29/0.47 thf(fact_222_wf__def,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( wf @ A )
% 0.29/0.47 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.29/0.47 ! [P5: A > $o] :
% 0.29/0.47 ( ! [X2: A] :
% 0.29/0.47 ( ! [Y4: A] :
% 0.29/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R3 )
% 0.29/0.47 => ( P5 @ Y4 ) )
% 0.29/0.47 => ( P5 @ X2 ) )
% 0.29/0.47 => ! [X6: A] : ( P5 @ X6 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % wf_def
% 0.29/0.47 thf(fact_223_dependent__wf__choice,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P3: ( A > B ) > A > B > $o] :
% 0.29/0.47 ( ( wf @ A @ R2 )
% 0.29/0.47 => ( ! [F7: A > B,G3: A > B,X4: A,R4: B] :
% 0.29/0.47 ( ! [Z4: A] :
% 0.29/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X4 ) @ R2 )
% 0.29/0.47 => ( ( F7 @ Z4 )
% 0.29/0.47 = ( G3 @ Z4 ) ) )
% 0.29/0.47 => ( ( P3 @ F7 @ X4 @ R4 )
% 0.29/0.47 = ( P3 @ G3 @ X4 @ R4 ) ) )
% 0.29/0.47 => ( ! [X4: A,F7: A > B] :
% 0.29/0.47 ( ! [Y6: A] :
% 0.29/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X4 ) @ R2 )
% 0.29/0.47 => ( P3 @ F7 @ Y6 @ ( F7 @ Y6 ) ) )
% 0.29/0.47 => ? [X_1: B] : ( P3 @ F7 @ X4 @ X_1 ) )
% 0.29/0.47 => ? [F7: A > B] :
% 0.29/0.47 ! [X5: A] : ( P3 @ F7 @ X5 @ ( F7 @ X5 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dependent_wf_choice
% 0.29/0.47 thf(fact_224_same__fstI,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType,P3: A > $o,X: A,Y7: B,Y: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.29/0.47 ( ( P3 @ X )
% 0.29/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R2 @ X ) )
% 0.29/0.47 => ( 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 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % same_fstI
% 0.29/0.47 thf(fact_225_image2__eqI,axiom,
% 0.29/0.47 ! [A: $tType,C: $tType,B: $tType,B2: A,F3: B > A,X: B,C3: C,G: B > C,A3: set @ B] :
% 0.29/0.47 ( ( B2
% 0.29/0.47 = ( F3 @ X ) )
% 0.29/0.47 => ( ( C3
% 0.29/0.47 = ( G @ X ) )
% 0.29/0.47 => ( ( member @ B @ X @ A3 )
% 0.29/0.47 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F3 @ G ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % image2_eqI
% 0.29/0.48 thf(fact_226_the__inv__into__into,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X: B,B6: set @ A] :
% 0.29/0.48 ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.29/0.48 => ( ( member @ B @ X @ ( image @ A @ B @ F3 @ A3 ) )
% 0.29/0.48 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.29/0.48 => ( member @ A @ ( the_inv_into @ A @ B @ A3 @ F3 @ X ) @ B6 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % the_inv_into_into
% 0.29/0.48 thf(fact_227_order__refl,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( preorder @ A )
% 0.29/0.48 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_refl
% 0.29/0.48 thf(fact_228_subsetI,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.29/0.48 ( ! [X4: A] :
% 0.29/0.48 ( ( member @ A @ X4 @ A3 )
% 0.29/0.48 => ( member @ A @ X4 @ B6 ) )
% 0.29/0.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% 0.29/0.48
% 0.29/0.48 % subsetI
% 0.29/0.48 thf(fact_229_subset__antisym,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.29/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
% 0.29/0.48 => ( A3 = B6 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_antisym
% 0.29/0.48 thf(fact_230_inj__on__image__eq__iff,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType,F3: A > B,C4: set @ A,A3: set @ A,B6: set @ A] :
% 0.29/0.48 ( ( inj_on @ A @ B @ F3 @ C4 )
% 0.29/0.48 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ C4 )
% 0.29/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.29/0.48 => ( ( ( image @ A @ B @ F3 @ A3 )
% 0.29/0.48 = ( image @ A @ B @ F3 @ B6 ) )
% 0.29/0.48 = ( A3 = B6 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % inj_on_image_eq_iff
% 0.29/0.48 thf(fact_231_inj__on__image__mem__iff,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType,F3: A > B,B6: set @ A,A2: A,A3: set @ A] :
% 0.29/0.48 ( ( inj_on @ A @ B @ F3 @ B6 )
% 0.29/0.48 => ( ( member @ A @ A2 @ B6 )
% 0.29/0.48 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.29/0.48 => ( ( member @ B @ ( F3 @ A2 ) @ ( image @ A @ B @ F3 @ A3 ) )
% 0.29/0.48 = ( member @ A @ A2 @ A3 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % inj_on_image_mem_iff
% 0.29/0.48 thf(fact_232_top_Oextremum__uniqueI,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( order_top @ A )
% 0.29/0.48 => ! [A2: A] :
% 0.29/0.48 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.29/0.48 => ( A2
% 0.29/0.48 = ( top_top @ A ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % top.extremum_uniqueI
% 0.29/0.48 thf(fact_233_top_Oextremum__unique,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( order_top @ A )
% 0.29/0.48 => ! [A2: A] :
% 0.29/0.48 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.29/0.48 = ( A2
% 0.29/0.48 = ( top_top @ A ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % top.extremum_unique
% 0.29/0.48 thf(fact_234_top__greatest,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( order_top @ A )
% 0.29/0.48 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % top_greatest
% 0.29/0.48 thf(fact_235_subset__UNIV,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_UNIV
% 0.29/0.48 thf(fact_236_inj__on__subset,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B6: set @ A] :
% 0.29/0.48 ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.29/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
% 0.29/0.48 => ( inj_on @ A @ B @ F3 @ B6 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % inj_on_subset
% 0.29/0.48 thf(fact_237_subset__inj__on,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType,F3: A > B,B6: set @ A,A3: set @ A] :
% 0.29/0.48 ( ( inj_on @ A @ B @ F3 @ B6 )
% 0.29/0.48 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.29/0.48 => ( inj_on @ A @ B @ F3 @ A3 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_inj_on
% 0.29/0.48 thf(fact_238_in__mono,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B6: set @ A,X: A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.29/0.48 => ( ( member @ A @ X @ A3 )
% 0.29/0.48 => ( member @ A @ X @ B6 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % in_mono
% 0.29/0.48 thf(fact_239_subsetD,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B6: set @ A,C3: A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.29/0.48 => ( ( member @ A @ C3 @ A3 )
% 0.29/0.48 => ( member @ A @ C3 @ B6 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subsetD
% 0.29/0.48 thf(fact_240_equalityE,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.29/0.48 ( ( A3 = B6 )
% 0.29/0.48 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.29/0.48 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % equalityE
% 0.29/0.48 thf(fact_241_subset__eq,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.48 = ( ^ [A5: set @ A,B7: set @ A] :
% 0.29/0.48 ! [X2: A] :
% 0.29/0.48 ( ( member @ A @ X2 @ A5 )
% 0.29/0.48 => ( member @ A @ X2 @ B7 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_eq
% 0.29/0.48 thf(fact_242_equalityD1,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.29/0.48 ( ( A3 = B6 )
% 0.29/0.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% 0.29/0.48
% 0.29/0.48 % equalityD1
% 0.29/0.48 thf(fact_243_equalityD2,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 0.29/0.48 ( ( A3 = B6 )
% 0.29/0.48 => ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ).
% 0.29/0.48
% 0.29/0.48 % equalityD2
% 0.29/0.48 thf(fact_244_subset__iff,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.48 = ( ^ [A5: set @ A,B7: set @ A] :
% 0.29/0.48 ! [T3: A] :
% 0.29/0.48 ( ( member @ A @ T3 @ A5 )
% 0.29/0.48 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_iff
% 0.29/0.48 thf(fact_245_subset__refl,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.29/0.48
% 0.29/0.48 % subset_refl
% 0.29/0.48 thf(fact_246_Collect__mono,axiom,
% 0.29/0.48 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 0.29/0.48 ( ! [X4: A] :
% 0.29/0.48 ( ( P3 @ X4 )
% 0.29/0.48 => ( Q2 @ X4 ) )
% 0.29/0.48 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_mono
% 0.29/0.48 thf(fact_247_subset__trans,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B6: set @ A,C4: set @ A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.29/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.29/0.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_trans
% 0.29/0.48 thf(fact_248_le__funD,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType] :
% 0.29/0.48 ( ( ord @ B )
% 0.29/0.48 => ! [F3: A > B,G: A > B,X: A] :
% 0.29/0.48 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 0.29/0.48 => ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % le_funD
% 0.29/0.48 thf(fact_249_le__funE,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType] :
% 0.29/0.48 ( ( ord @ B )
% 0.29/0.48 => ! [F3: A > B,G: A > B,X: A] :
% 0.29/0.48 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 0.29/0.48 => ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % le_funE
% 0.29/0.48 thf(fact_250_le__funI,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType] :
% 0.29/0.48 ( ( ord @ B )
% 0.29/0.48 => ! [F3: A > B,G: A > B] :
% 0.29/0.48 ( ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
% 0.29/0.48 => ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % le_funI
% 0.29/0.48 thf(fact_251_set__eq__subset,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( ^ [Y5: set @ A,Z2: set @ A] : ( Y5 = Z2 ) )
% 0.29/0.48 = ( ^ [A5: set @ A,B7: set @ A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
% 0.29/0.48 & ( ord_less_eq @ ( set @ A ) @ B7 @ A5 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % set_eq_subset
% 0.29/0.48 thf(fact_252_le__fun__def,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType] :
% 0.29/0.48 ( ( ord @ B )
% 0.29/0.48 => ( ( ord_less_eq @ ( A > B ) )
% 0.29/0.48 = ( ^ [F: A > B,G4: A > B] :
% 0.29/0.48 ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G4 @ X2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % le_fun_def
% 0.29/0.48 thf(fact_253_Collect__mono__iff,axiom,
% 0.29/0.48 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q2 ) )
% 0.29/0.48 = ( ! [X2: A] :
% 0.29/0.48 ( ( P3 @ X2 )
% 0.29/0.48 => ( Q2 @ X2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_mono_iff
% 0.29/0.48 thf(fact_254_order__subst1,axiom,
% 0.29/0.48 ! [A: $tType,B: $tType] :
% 0.29/0.48 ( ( ( order @ B )
% 0.29/0.48 & ( order @ A ) )
% 0.29/0.48 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 0.29/0.48 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 0.29/0.48 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 0.29/0.48 => ( ! [X4: B,Y2: B] :
% 0.29/0.48 ( ( ord_less_eq @ B @ X4 @ Y2 )
% 0.29/0.48 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y2 ) ) )
% 0.29/0.48 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_subst1
% 0.29/0.48 thf(fact_255_order__subst2,axiom,
% 0.29/0.48 ! [A: $tType,C: $tType] :
% 0.29/0.48 ( ( ( order @ C )
% 0.29/0.48 & ( order @ A ) )
% 0.29/0.48 => ! [A2: A,B2: A,F3: A > C,C3: C] :
% 0.29/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.48 => ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
% 0.29/0.48 => ( ! [X4: A,Y2: A] :
% 0.29/0.48 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.29/0.48 => ( ord_less_eq @ C @ ( F3 @ X4 ) @ ( F3 @ Y2 ) ) )
% 0.29/0.48 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_subst2
% 0.29/0.48
% 0.29/0.48 % Type constructors (18)
% 0.29/0.48 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.29/0.48 ! [A8: $tType,A9: $tType] :
% 0.29/0.48 ( ( order_top @ A9 )
% 0.29/0.48 => ( order_top @ ( A8 > A9 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.29/0.48 ! [A8: $tType,A9: $tType] :
% 0.29/0.48 ( ( preorder @ A9 )
% 0.29/0.48 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.29/0.48 ! [A8: $tType,A9: $tType] :
% 0.29/0.48 ( ( order @ A9 )
% 0.29/0.48 => ( order @ ( A8 > A9 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_fun___Orderings_Otop,axiom,
% 0.29/0.48 ! [A8: $tType,A9: $tType] :
% 0.29/0.48 ( ( top @ A9 )
% 0.29/0.48 => ( top @ ( A8 > A9 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_fun___Orderings_Oord,axiom,
% 0.29/0.48 ! [A8: $tType,A9: $tType] :
% 0.29/0.48 ( ( ord @ A9 )
% 0.29/0.48 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.29/0.48 preorder @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.29/0.48 order @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.29/0.48 ord @ nat ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Set_Oset___Orderings_Oorder__top_4,axiom,
% 0.29/0.48 ! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 0.29/0.48 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.29/0.48 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
% 0.29/0.48 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 0.29/0.48 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_HOL_Obool___Orderings_Oorder__top_9,axiom,
% 0.29/0.48 order_top @ $o ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 0.29/0.48 preorder @ $o ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 0.29/0.48 order @ $o ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_HOL_Obool___Orderings_Otop_12,axiom,
% 0.29/0.48 top @ $o ).
% 0.29/0.48
% 0.29/0.48 thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
% 0.29/0.48 ord @ $o ).
% 0.29/0.48
% 0.29/0.48 % Helper facts (3)
% 0.29/0.48 thf(help_If_3_1_T,axiom,
% 0.29/0.48 ! [P3: $o] :
% 0.29/0.48 ( ( P3 = $true )
% 0.29/0.48 | ( P3 = $false ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(help_If_2_1_T,axiom,
% 0.29/0.48 ! [A: $tType,X: A,Y: A] :
% 0.29/0.48 ( ( if @ A @ $false @ X @ Y )
% 0.29/0.48 = Y ) ).
% 0.29/0.48
% 0.29/0.48 thf(help_If_1_1_T,axiom,
% 0.29/0.48 ! [A: $tType,X: A,Y: A] :
% 0.29/0.48 ( ( if @ A @ $true @ X @ Y )
% 0.29/0.48 = X ) ).
% 0.29/0.48
% 0.29/0.48 % Conjectures (1)
% 0.29/0.48 thf(conj_0,conjecture,
% 0.29/0.48 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 )
% 0.29/0.48 != ( paraco2040174112le_Det @ $true ) ) ).
% 0.29/0.48
% 0.29/0.48 %------------------------------------------------------------------------------
% 0.29/0.48 ------------------------
% 2.96/1.53 ---- Embedded in THF ---
% 2.96/1.53 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.96/1.53 %%% Generated on Tue Feb 24 08:27:24 EST 2026
% 2.96/1.53 %%% using '$$dhol' embedding, version 1.3.0.
% 2.96/1.53 %%% Logic specification used:
% 2.96/1.53 %%% thf(spec, logic, $$dhol).
% 2.96/1.53
% 2.96/1.53 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.xGjE5LYYhO/DTF2DTF24296.p
% 2.96/1.53 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv, type, paraco415392788lle_tv: $tType).
% 2.96/1.53 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm, type, paraco414474393lle_fm: $tType).
% 2.96/1.53 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.96/1.53 thf(ty_t_String_Ochar, type, char: $tType).
% 2.96/1.53 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.96/1.53 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.96/1.53 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.96/1.53 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.96/1.53 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.96/1.53 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.96/1.53 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.96/1.53 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.96/1.53 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 2.96/1.53 thf(sy_c_BNF__Def_OfstOp, type, bNF_fstOp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (((product_prod @ A) @ C) > ((product_prod @ A) @ B))))))).
% 2.96/1.53 thf(sy_c_BNF__Def_Opick__middlep, type, bNF_pick_middlep: (!> [B:$tType,A:$tType,C:$tType]: (((B > (A > $o)) > ((A > (C > $o)) > (B > (C > A))))))).
% 2.96/1.53 thf(sy_c_BNF__Def_OsndOp, type, bNF_sndOp: (!> [C:$tType,A:$tType,B:$tType]: (((C > (A > $o)) > ((A > (B > $o)) > (((product_prod @ C) @ B) > ((product_prod @ A) @ B))))))).
% 2.96/1.53 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2, type, bNF_Greatest_image2: (!> [C:$tType,A:$tType,B:$tType]: (((set @ C) > ((C > A) > ((C > B) > (set @ ((product_prod @ A) @ B)))))))).
% 2.96/1.53 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 2.96/1.53 thf(sy_c_Fun_Othe__inv__into, type, the_inv_into: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (B > A)))))).
% 2.96/1.53 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.96/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.96/1.53 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__int, type, paraco753463838ge_int: ((nat > nat) > (((list @ char) > paraco415392788lle_tv) > ((list @ char) > paraco415392788lle_tv)))).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv, type, paraco1920534163nge_tv: ((nat > nat) > (paraco415392788lle_tv > paraco415392788lle_tv))).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv__rel, type, paraco2077297190tv_rel: (((product_prod @ (nat > nat)) @ paraco415392788lle_tv) > (((product_prod @ (nat > nat)) @ paraco415392788lle_tv) > $o))).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval, type, paraco876059933e_eval: (((list @ char) > paraco415392788lle_tv) > (paraco414474393lle_fm > paraco415392788lle_tv))).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H, type, paraco2100061555le_Con: (paraco414474393lle_fm > (paraco414474393lle_fm > paraco414474393lle_fm))).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql, type, paraco2084319816le_Eql: (paraco414474393lle_fm > (paraco414474393lle_fm > paraco414474393lle_fm))).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H, type, paraco1628874225le_Eql: (paraco414474393lle_fm > (paraco414474393lle_fm > paraco414474393lle_fm))).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H, type, paraco329115265le_Neg: (paraco414474393lle_fm > paraco414474393lle_fm)).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro, type, paraco27778325le_Pro: ((list @ char) > paraco414474393lle_fm)).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth, type, paraco251304083_Truth: paraco414474393lle_fm).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Ocase__fm, type, paraco1246693743ase_fm: (!> [A:$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)))))))))).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet, type, paraco2040174112le_Det: ($o > paraco415392788lle_tv)).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet, type, paraco676387099_Indet: (nat > paraco415392788lle_tv)).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Ocase__tv, type, paraco490622181ase_tv: (!> [A:$tType]: ((($o > A) > ((nat > A) > (paraco415392788lle_tv > A)))))).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv, type, paraco152590079rec_tv: (!> [A:$tType]: ((($o > A) > ((nat > A) > (paraco415392788lle_tv > A)))))).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid, type, paraco769098683_valid: (paraco414474393lle_fm > $o)).
% 2.96/1.53 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid__in, type, paraco2086025920lid_in: ((set @ nat) > (paraco414474393lle_fm > $o))).
% 2.96/1.53 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.96/1.53 thf(sy_c_Product__Type_Oapfst, type, product_apfst: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > (((product_prod @ A) @ B) > ((product_prod @ C) @ B)))))).
% 2.96/1.53 thf(sy_c_Product__Type_Oapsnd, type, product_apsnd: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((product_prod @ A) @ B) > ((product_prod @ A) @ C)))))).
% 2.96/1.53 thf(sy_c_Product__Type_Obool_Ocase__bool, type, product_case_bool: (!> [A:$tType]: ((A > (A > ($o > A)))))).
% 2.96/1.53 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 2.96/1.53 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.96/1.53 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 2.96/1.53 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.96/1.53 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 2.96/1.53 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 2.96/1.53 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 2.96/1.53 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))))))).
% 2.96/1.53 thf(sy_c_Relation_Oinv__imagep, type, inv_imagep: (!> [B:$tType,A:$tType]: (((B > (B > $o)) > ((A > B) > (A > (A > $o))))))).
% 2.96/1.53 thf(sy_c_Relation_Ototal__on, type, total_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.96/1.53 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.96/1.53 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.96/1.53 thf(sy_c_Wellfounded_Oaccp, type, accp: (!> [A:$tType]: (((A > (A > $o)) > (A > $o))))).
% 2.96/1.53 thf(sy_c_Wellfounded_Olex__prod, type, lex_prod: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 2.96/1.53 thf(sy_c_Wellfounded_Owf, type, wf: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 2.96/1.53 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 2.96/1.53 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.96/1.53 thf(sy_v_f, type, f: (nat > nat)).
% 2.96/1.53 thf(sy_v_i, type, i: ((list @ char) > paraco415392788lle_tv)).
% 2.96/1.53 thf(sy_v_p1____, type, p1: paraco414474393lle_fm).
% 2.96/1.53 thf(sy_v_p2____, type, p2: paraco414474393lle_fm).
% 2.96/1.53 thf(fact_0_a_H_H_H, axiom, (((paraco876059933e_eval @ i) @ p2) != (paraco2040174112le_Det @ $true))).
% 2.96/1.53 thf(fact_1_a_H_H, axiom, (((paraco876059933e_eval @ i) @ p1) != (paraco2040174112le_Det @ $true))).
% 2.96/1.53 thf(fact_2_a_H_H_H_H, axiom, (((paraco876059933e_eval @ i) @ p1) != (paraco2040174112le_Det @ $false))).
% 2.96/1.53 thf(fact_3_b_H_H, axiom, (((paraco876059933e_eval @ ((paraco753463838ge_int @ f) @ i)) @ p1) != (paraco2040174112le_Det @ $true))).
% 2.96/1.53 thf(fact_4_a, axiom, (((paraco876059933e_eval @ i) @ p2) = (paraco2040174112le_Det @ $false))).
% 2.96/1.53 thf(fact_5_a_H, axiom, (((paraco876059933e_eval @ i) @ p1) != ((paraco876059933e_eval @ i) @ p2))).
% 2.96/1.53 thf(fact_6_b, axiom, (((paraco876059933e_eval @ ((paraco753463838ge_int @ f) @ i)) @ p2) = (paraco2040174112le_Det @ $false))).
% 2.96/1.53 thf(fact_7_b_H, axiom, (((paraco876059933e_eval @ ((paraco753463838ge_int @ f) @ i)) @ p1) != ((paraco876059933e_eval @ ((paraco753463838ge_int @ f) @ i)) @ p2))).
% 2.96/1.53 thf(fact_8_tv_Oinject_I1_J, axiom, (! [X1:$o,Y1:$o]: ((((paraco2040174112le_Det @ X1) = (paraco2040174112le_Det @ Y1)) = (X1 = Y1))))).
% 2.96/1.53 thf(fact_9_ih2, axiom, (((paraco876059933e_eval @ ((paraco753463838ge_int @ f) @ i)) @ p2) = ((paraco1920534163nge_tv @ f) @ ((paraco876059933e_eval @ i) @ p2)))).
% 2.96/1.53 thf(fact_10_ih1, axiom, (((paraco876059933e_eval @ ((paraco753463838ge_int @ f) @ i)) @ p1) = ((paraco1920534163nge_tv @ f) @ ((paraco876059933e_eval @ i) @ p1)))).
% 2.96/1.53 thf(fact_11__092_060open_062change__tv_Af_A_Ieval_Ai_Ap2_J_A_061_ADet_AFalse_092_060close_062, axiom, (((paraco1920534163nge_tv @ f) @ ((paraco876059933e_eval @ i) @ p2)) = (paraco2040174112le_Det @ $false))).
% 2.96/1.53 thf(fact_12_valid__def, axiom, (paraco769098683_valid = (^ [P:paraco414474393lle_fm]: ((! [I:((list @ char) > paraco415392788lle_tv)]: ((((paraco876059933e_eval @ I) @ P) = (paraco2040174112le_Det @ $true)))))))).
% 2.96/1.53 thf(fact_13_yes, axiom, (((paraco876059933e_eval @ i) @ ((paraco1628874225le_Eql @ p1) @ p2)) = ((paraco876059933e_eval @ i) @ (paraco329115265le_Neg @ p1)))).
% 2.96/1.53 thf(fact_14_eval_Osimps_I2_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv)]: ((((paraco876059933e_eval @ I2) @ paraco251304083_Truth) = (paraco2040174112le_Det @ $true))))).
% 2.96/1.53 thf(fact_15_tv_Osimps_I7_J, axiom, (! [A:$tType,F1:($o > A),F2:(nat > A),X1:$o]: ((((((paraco152590079rec_tv @ A) @ F1) @ F2) @ (paraco2040174112le_Det @ X1)) = (F1 @ X1))))).
% 2.96/1.53 thf(fact_16_eval_Osimps_I5_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I2) @ P2) = ((paraco876059933e_eval @ I2) @ Q)) => (((paraco876059933e_eval @ I2) @ ((paraco2084319816le_Eql @ P2) @ Q)) = (paraco2040174112le_Det @ $true))) & ((((paraco876059933e_eval @ I2) @ P2) != ((paraco876059933e_eval @ I2) @ Q)) => (((paraco876059933e_eval @ I2) @ ((paraco2084319816le_Eql @ P2) @ Q)) = (paraco2040174112le_Det @ $false))))))).
% 2.96/1.53 thf(fact_17_eval__equality__simplify, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((paraco876059933e_eval @ I2) @ ((paraco2084319816le_Eql @ P2) @ Q)) = (paraco2040174112le_Det @ (((paraco876059933e_eval @ I2) @ P2) = ((paraco876059933e_eval @ I2) @ Q))))))).
% 2.96/1.53 thf(fact_18_change__int__def, axiom, (paraco753463838ge_int = (^ [F:(nat > nat),I:((list @ char) > paraco415392788lle_tv),S:(list @ char)]: (((paraco1920534163nge_tv @ F) @ (I @ S)))))).
% 2.96/1.53 thf(fact_19_eval_Osimps_I4_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I2) @ P2) = ((paraco876059933e_eval @ I2) @ Q)) => (((paraco876059933e_eval @ I2) @ ((paraco2100061555le_Con @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ P2))) & ((((paraco876059933e_eval @ I2) @ P2) != ((paraco876059933e_eval @ I2) @ Q)) => (((((paraco876059933e_eval @ I2) @ P2) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco2100061555le_Con @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ Q))) & ((((paraco876059933e_eval @ I2) @ P2) != (paraco2040174112le_Det @ $true)) => (((((paraco876059933e_eval @ I2) @ Q) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco2100061555le_Con @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ P2))) & ((((paraco876059933e_eval @ I2) @ Q) != (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco2100061555le_Con @ P2) @ Q)) = (paraco2040174112le_Det @ $false))))))))))).
% 2.96/1.53 thf(fact_20_tv_Osimps_I5_J, axiom, (! [A:$tType,F1:($o > A),F2:(nat > A),X1:$o]: ((((((paraco490622181ase_tv @ A) @ F1) @ F2) @ (paraco2040174112le_Det @ X1)) = (F1 @ X1))))).
% 2.96/1.53 thf(fact_21_eval__negation, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I2) @ P2) = (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P2)) = (paraco2040174112le_Det @ $true))) & ((((paraco876059933e_eval @ I2) @ P2) != (paraco2040174112le_Det @ $false)) => (((((paraco876059933e_eval @ I2) @ P2) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P2)) = (paraco2040174112le_Det @ $false))) & ((((paraco876059933e_eval @ I2) @ P2) != (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P2)) = ((paraco876059933e_eval @ I2) @ P2))))))))).
% 2.96/1.53 thf(fact_22_assms, axiom, ((((inj_on @ nat) @ nat) @ f) @ (top_top @ (set @ nat)))).
% 2.96/1.53 thf(fact_23_change__tv_Osimps_I1_J, axiom, (! [F3:(nat > nat),B2:$o]: ((((paraco1920534163nge_tv @ F3) @ (paraco2040174112le_Det @ B2)) = (paraco2040174112le_Det @ B2))))).
% 2.96/1.53 thf(fact_24_fm_Oinject_I2_J, axiom, (! [X3:paraco414474393lle_fm,Y3:paraco414474393lle_fm]: ((((paraco329115265le_Neg @ X3) = (paraco329115265le_Neg @ Y3)) = (X3 = Y3))))).
% 2.96/1.53 thf(fact_25_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)))))).
% 2.96/1.53 thf(fact_26_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)))))).
% 2.96/1.53 thf(fact_27_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)))))).
% 2.96/1.53 thf(fact_28_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))))).
% 2.96/1.53 thf(fact_29_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))))).
% 2.96/1.53 thf(fact_30_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))))).
% 2.96/1.53 thf(fact_31_fm_Odistinct_I23_J, axiom, (! [X3:paraco414474393lle_fm,X61:paraco414474393lle_fm,X62:paraco414474393lle_fm]: (((paraco329115265le_Neg @ X3) != ((paraco1628874225le_Eql @ X61) @ X62))))).
% 2.96/1.53 thf(fact_32_fm_Odistinct_I21_J, axiom, (! [X3:paraco414474393lle_fm,X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: (((paraco329115265le_Neg @ X3) != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.96/1.53 thf(fact_33_fm_Odistinct_I19_J, axiom, (! [X3:paraco414474393lle_fm,X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: (((paraco329115265le_Neg @ X3) != ((paraco2100061555le_Con @ X41) @ X42))))).
% 2.96/1.53 thf(fact_34_fm_Odistinct_I17_J, axiom, (! [X61:paraco414474393lle_fm,X62:paraco414474393lle_fm]: ((paraco251304083_Truth != ((paraco1628874225le_Eql @ X61) @ X62))))).
% 2.96/1.53 thf(fact_35_fm_Odistinct_I15_J, axiom, (! [X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: ((paraco251304083_Truth != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.96/1.53 thf(fact_36_fm_Odistinct_I13_J, axiom, (! [X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: ((paraco251304083_Truth != ((paraco2100061555le_Con @ X41) @ X42))))).
% 2.96/1.53 thf(fact_37_fm_Odistinct_I11_J, axiom, (! [X3:paraco414474393lle_fm]: ((paraco251304083_Truth != (paraco329115265le_Neg @ X3))))).
% 2.96/1.53 thf(fact_38_conjunction, axiom, (! [P2:paraco414474393lle_fm,Q:paraco414474393lle_fm]: (((paraco769098683_valid @ ((paraco2100061555le_Con @ P2) @ Q)) = ((paraco769098683_valid @ P2) & (paraco769098683_valid @ Q)))))).
% 2.96/1.53 thf(fact_39_eval__equality, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I2) @ P2) = ((paraco876059933e_eval @ I2) @ Q)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = (paraco2040174112le_Det @ $true))) & ((((paraco876059933e_eval @ I2) @ P2) != ((paraco876059933e_eval @ I2) @ Q)) => (((((paraco876059933e_eval @ I2) @ P2) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ Q))) & ((((paraco876059933e_eval @ I2) @ P2) != (paraco2040174112le_Det @ $true)) => (((((paraco876059933e_eval @ I2) @ Q) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ P2))) & ((((paraco876059933e_eval @ I2) @ Q) != (paraco2040174112le_Det @ $true)) => (((((paraco876059933e_eval @ I2) @ P2) = (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ Q)))) & ((((paraco876059933e_eval @ I2) @ P2) != (paraco2040174112le_Det @ $false)) => (((((paraco876059933e_eval @ I2) @ Q) = (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = ((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P2)))) & ((((paraco876059933e_eval @ I2) @ Q) != (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P2) @ Q)) = (paraco2040174112le_Det @ $false))))))))))))))).
% 2.96/1.53 thf(fact_40_double__negation, axiom, (paraco876059933e_eval = (^ [I:((list @ char) > paraco415392788lle_tv),P:paraco414474393lle_fm]: (((paraco876059933e_eval @ I) @ (paraco329115265le_Neg @ (paraco329115265le_Neg @ P))))))).
% 2.96/1.53 thf(fact_41_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.96/1.53 thf(fact_42_iso__tuple__UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.96/1.53 thf(fact_43_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X2:D]: ((top_top @ C)))))))).
% 2.96/1.53 thf(fact_44_injD, axiom, (! [B:$tType,A:$tType,F3:(A > B),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => (((F3 @ X) = (F3 @ Y)) => (X = Y)))))).
% 2.96/1.53 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P3:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P3)) = (P3 @ A2))))).
% 2.96/1.53 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A3)))) = A3)))).
% 2.96/1.53 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P3:(A > $o),Q2:(A > $o)]: (((! [X4:A]: (((P3 @ X4) = (Q2 @ X4)))) => (((collect @ A) @ P3) = ((collect @ A) @ Q2)))))).
% 2.96/1.53 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G:(A > B)]: (((! [X4:A]: (((F3 @ X4) = (G @ X4)))) => (F3 = G))))).
% 2.96/1.53 thf(fact_49_injI, axiom, (! [B:$tType,A:$tType,F3:(A > B)]: (((! [X4:A,Y2:A]: ((((F3 @ X4) = (F3 @ Y2)) => (X4 = Y2)))) => ((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))))))).
% 2.96/1.53 thf(fact_50_inj__eq, axiom, (! [B:$tType,A:$tType,F3:(A > B),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => (((F3 @ X) = (F3 @ Y)) = (X = Y)))))).
% 2.96/1.53 thf(fact_51_inj__def, axiom, (! [B:$tType,A:$tType,F3:(A > B)]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) = (! [X2:A,Y4:A]: ((((F3 @ X2) = (F3 @ Y4)) => (X2 = Y4)))))))).
% 2.96/1.53 thf(fact_52_fm_Oinduct, axiom, (! [P3:(paraco414474393lle_fm > $o),Fm:paraco414474393lle_fm]: (((! [X4:(list @ char)]: ((P3 @ (paraco27778325le_Pro @ X4)))) => ((P3 @ paraco251304083_Truth) => ((! [X4:paraco414474393lle_fm]: (((P3 @ X4) => (P3 @ (paraco329115265le_Neg @ X4))))) => ((! [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)))))))))).
% 2.96/1.53 thf(fact_53_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)))))))))))))).
% 2.96/1.53 thf(fact_54_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)))))))))))))).
% 2.96/1.53 thf(fact_55_fm_Oinject_I1_J, axiom, (! [X1:(list @ char),Y1:(list @ char)]: ((((paraco27778325le_Pro @ X1) = (paraco27778325le_Pro @ Y1)) = (X1 = Y1))))).
% 2.96/1.53 thf(fact_56_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.96/1.53 thf(fact_57_eval_Osimps_I1_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),S3:(list @ char)]: ((((paraco876059933e_eval @ I2) @ (paraco27778325le_Pro @ S3)) = (I2 @ S3))))).
% 2.96/1.53 thf(fact_58_fm_Odistinct_I3_J, axiom, (! [X1:(list @ char),X3:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != (paraco329115265le_Neg @ X3))))).
% 2.96/1.53 thf(fact_59_fm_Odistinct_I5_J, axiom, (! [X1:(list @ char),X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != ((paraco2100061555le_Con @ X41) @ X42))))).
% 2.96/1.53 thf(fact_60_fm_Odistinct_I9_J, axiom, (! [X1:(list @ char),X61:paraco414474393lle_fm,X62:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != ((paraco1628874225le_Eql @ X61) @ X62))))).
% 2.96/1.53 thf(fact_61_fm_Odistinct_I7_J, axiom, (! [X1:(list @ char),X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.96/1.53 thf(fact_62_fm_Odistinct_I1_J, axiom, (! [X1:(list @ char)]: (((paraco27778325le_Pro @ X1) != paraco251304083_Truth)))).
% 2.96/1.53 thf(fact_63_change__tv__injection, axiom, (! [F3:(nat > nat)]: ((((((inj_on @ nat) @ nat) @ F3) @ (top_top @ (set @ nat))) => ((((inj_on @ paraco415392788lle_tv) @ paraco415392788lle_tv) @ (paraco1920534163nge_tv @ F3)) @ (top_top @ (set @ paraco415392788lle_tv))))))).
% 2.96/1.53 thf(fact_64_UNIV__witness, axiom, (! [A:$tType]: ((? [X4:A]: ((((member @ A) @ X4) @ (top_top @ (set @ A)))))))).
% 2.96/1.53 thf(fact_65_UNIV__eq__I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X4:A]: ((((member @ A) @ X4) @ A3))) => ((top_top @ (set @ A)) = A3))))).
% 2.96/1.53 thf(fact_66_inj__on__inverseI, axiom, (! [B:$tType,A:$tType,A3:(set @ A),G:(B > A),F3:(A > B)]: (((! [X4:A]: (((((member @ A) @ X4) @ A3) => ((G @ (F3 @ X4)) = X4)))) => ((((inj_on @ A) @ B) @ F3) @ A3))))).
% 2.96/1.53 thf(fact_67_inj__on__contraD, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F3) @ A3) => ((X != Y) => ((((member @ A) @ X) @ A3) => ((((member @ A) @ Y) @ A3) => ((F3 @ X) != (F3 @ Y))))))))).
% 2.96/1.53 thf(fact_68_inj__on__eq__iff, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F3) @ A3) => ((((member @ A) @ X) @ A3) => ((((member @ A) @ Y) @ A3) => (((F3 @ X) = (F3 @ Y)) = (X = Y)))))))).
% 2.96/1.53 thf(fact_69_inj__on__cong, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F3:(A > B),G:(A > B)]: (((! [A4:A]: (((((member @ A) @ A4) @ A3) => ((F3 @ A4) = (G @ A4))))) => (((((inj_on @ A) @ B) @ F3) @ A3) = ((((inj_on @ A) @ B) @ G) @ A3)))))).
% 2.96/1.53 thf(fact_70_inj__on__def, axiom, (! [B:$tType,A:$tType]: ((((inj_on @ A) @ B) = (^ [F:(A > B),A5:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A5) => (! [Y4:A]: (((((member @ A) @ Y4) @ A5) => (((F @ X2) = (F @ Y4)) => (X2 = Y4)))))))))))))).
% 2.96/1.53 thf(fact_71_inj__onI, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F3:(A > B)]: (((! [X4:A,Y2:A]: (((((member @ A) @ X4) @ A3) => ((((member @ A) @ Y2) @ A3) => (((F3 @ X4) = (F3 @ Y2)) => (X4 = Y2)))))) => ((((inj_on @ A) @ B) @ F3) @ A3))))).
% 2.96/1.53 thf(fact_72_inj__onD, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F3) @ A3) => (((F3 @ X) = (F3 @ Y)) => ((((member @ A) @ X) @ A3) => ((((member @ A) @ Y) @ A3) => (X = Y)))))))).
% 2.96/1.53 thf(fact_73_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)))))))))).
% 2.96/1.53 thf(fact_74_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))))))))))))))).
% 2.96/1.53 thf(fact_75_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A))))))))).
% 2.96/1.53 thf(fact_76_change__tv_Oelims, axiom, (! [X:(nat > nat),Xa:paraco415392788lle_tv,Y:paraco415392788lle_tv]: (((((paraco1920534163nge_tv @ X) @ Xa) = Y) => ((! [B3:$o]: (((Xa = (paraco2040174112le_Det @ B3)) => (Y != (paraco2040174112le_Det @ B3))))) => (~ ((! [N:nat]: (((Xa = (paraco676387099_Indet @ N)) => (Y != (paraco676387099_Indet @ (X @ N))))))))))))).
% 2.96/1.53 thf(fact_77_valid__valid__in, axiom, (! [P2:paraco414474393lle_fm,U:(set @ nat)]: (((paraco769098683_valid @ P2) => ((paraco2086025920lid_in @ U) @ P2))))).
% 2.96/1.53 thf(fact_78_transfer, axiom, (! [U:(set @ nat),P2:paraco414474393lle_fm]: (((~ (((paraco2086025920lid_in @ U) @ P2))) => (~ ((paraco769098683_valid @ P2))))))).
% 2.96/1.53 thf(fact_79_the__inv__f__f, axiom, (! [B:$tType,A:$tType,F3:(A > B),X:A]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((((((the_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F3) @ (F3 @ X)) = X))))).
% 2.96/1.53 thf(fact_80_conjunction__in, axiom, (! [U:(set @ nat),P2:paraco414474393lle_fm,Q:paraco414474393lle_fm]: ((((paraco2086025920lid_in @ U) @ ((paraco2100061555le_Con @ P2) @ Q)) = (((paraco2086025920lid_in @ U) @ P2) & ((paraco2086025920lid_in @ U) @ Q)))))).
% 2.96/1.53 thf(fact_81_fm_Osimps_I37_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F32:(paraco414474393lle_fm > A),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > A))]: ((((((((((paraco1246693743ase_fm @ A) @ F1) @ F2) @ F32) @ F4) @ F5) @ F6) @ paraco251304083_Truth) = F2)))).
% 2.96/1.53 thf(fact_82_tv_Oinject_I2_J, axiom, (! [X23:nat,Y22:nat]: ((((paraco676387099_Indet @ X23) = (paraco676387099_Indet @ Y22)) = (X23 = Y22))))).
% 2.96/1.53 thf(fact_83_top1I, axiom, (! [A:$tType,X:A]: (((top_top @ (A > $o)) @ X)))).
% 2.96/1.53 thf(fact_84_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))))))).
% 2.96/1.53 thf(fact_85_change__tv_Oinduct, axiom, (! [P3:((nat > nat) > (paraco415392788lle_tv > $o)),A0:(nat > nat),A1:paraco415392788lle_tv]: (((! [F7:(nat > nat),B3:$o]: (((P3 @ F7) @ (paraco2040174112le_Det @ B3)))) => ((! [F7:(nat > nat),N:nat]: (((P3 @ F7) @ (paraco676387099_Indet @ N)))) => ((P3 @ A0) @ A1)))))).
% 2.96/1.53 thf(fact_86_string__tv_Ocases, axiom, (! [X:paraco415392788lle_tv]: (((X != (paraco2040174112le_Det @ $true)) => ((X != (paraco2040174112le_Det @ $false)) => (~ ((! [N:nat]: ((X != (paraco676387099_Indet @ N))))))))))).
% 2.96/1.53 thf(fact_87_tv_Oexhaust, axiom, (! [Y:paraco415392788lle_tv]: (((! [X12:$o]: ((Y != (paraco2040174112le_Det @ X12)))) => (~ ((! [X22:nat]: ((Y != (paraco676387099_Indet @ X22)))))))))).
% 2.96/1.53 thf(fact_88_tv_Oinduct, axiom, (! [P3:(paraco415392788lle_tv > $o),Tv:paraco415392788lle_tv]: (((! [X4:$o]: ((P3 @ (paraco2040174112le_Det @ X4)))) => ((! [X4:nat]: ((P3 @ (paraco676387099_Indet @ X4)))) => (P3 @ Tv)))))).
% 2.96/1.53 thf(fact_89_tv_Odistinct_I1_J, axiom, (! [X1:$o,X23:nat]: (((paraco2040174112le_Det @ X1) != (paraco676387099_Indet @ X23))))).
% 2.96/1.53 thf(fact_90_the__inv__into__f__f, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),X:A]: ((((((inj_on @ A) @ B) @ F3) @ A3) => ((((member @ A) @ X) @ A3) => ((((((the_inv_into @ A) @ B) @ A3) @ F3) @ (F3 @ X)) = X)))))).
% 2.96/1.53 thf(fact_91_the__inv__into__f__eq, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),X:A,Y:B]: ((((((inj_on @ A) @ B) @ F3) @ A3) => (((F3 @ X) = Y) => ((((member @ A) @ X) @ A3) => ((((((the_inv_into @ A) @ B) @ A3) @ F3) @ Y) = X))))))).
% 2.96/1.53 thf(fact_92_tv_Osimps_I6_J, axiom, (! [A:$tType,F1:($o > A),F2:(nat > A),X23:nat]: ((((((paraco490622181ase_tv @ A) @ F1) @ F2) @ (paraco676387099_Indet @ X23)) = (F2 @ X23))))).
% 2.96/1.53 thf(fact_93_change__tv_Osimps_I2_J, axiom, (! [F3:(nat > nat),N2:nat]: ((((paraco1920534163nge_tv @ F3) @ (paraco676387099_Indet @ N2)) = (paraco676387099_Indet @ (F3 @ N2)))))).
% 2.96/1.53 thf(fact_94_fm_Osimps_I36_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F32:(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) @ F32) @ F4) @ F5) @ F6) @ (paraco27778325le_Pro @ X1)) = (F1 @ X1))))).
% 2.96/1.53 thf(fact_95_tv_Osimps_I8_J, axiom, (! [A:$tType,F1:($o > A),F2:(nat > A),X23:nat]: ((((((paraco152590079rec_tv @ A) @ F1) @ F2) @ (paraco676387099_Indet @ X23)) = (F2 @ X23))))).
% 2.96/1.53 thf(fact_96_fm_Osimps_I38_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F32:(paraco414474393lle_fm > A),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),X3:paraco414474393lle_fm]: ((((((((((paraco1246693743ase_fm @ A) @ F1) @ F2) @ F32) @ F4) @ F5) @ F6) @ (paraco329115265le_Neg @ X3)) = (F32 @ X3))))).
% 2.96/1.53 thf(fact_97_fm_Osimps_I39_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F32:(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) @ F32) @ F4) @ F5) @ F6) @ ((paraco2100061555le_Con @ X41) @ X42)) = ((F4 @ X41) @ X42))))).
% 2.96/1.53 thf(fact_98_fm_Osimps_I41_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F32:(paraco414474393lle_fm > A),F4:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F5:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),F6:(paraco414474393lle_fm > (paraco414474393lle_fm > A)),X61:paraco414474393lle_fm,X62:paraco414474393lle_fm]: ((((((((((paraco1246693743ase_fm @ A) @ F1) @ F2) @ F32) @ F4) @ F5) @ F6) @ ((paraco1628874225le_Eql @ X61) @ X62)) = ((F6 @ X61) @ X62))))).
% 2.96/1.53 thf(fact_99_fm_Osimps_I40_J, axiom, (! [A:$tType,F1:((list @ char) > A),F2:A,F32:(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) @ F32) @ F4) @ F5) @ F6) @ ((paraco2084319816le_Eql @ X51) @ X52)) = ((F5 @ X51) @ X52))))).
% 2.96/1.53 thf(fact_100_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X23:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X23) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X23 = Y22)))))).
% 2.96/1.53 thf(fact_101_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)))))).
% 2.96/1.53 thf(fact_102_eval_Osimps_I3_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P2:paraco414474393lle_fm]: ((((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P2)) = ((((paraco490622181ase_tv @ paraco415392788lle_tv) @ (((product_case_bool @ paraco415392788lle_tv) @ (paraco2040174112le_Det @ $false)) @ (paraco2040174112le_Det @ $true))) @ paraco676387099_Indet) @ ((paraco876059933e_eval @ I2) @ P2)))))).
% 2.96/1.53 thf(fact_103_top__conj_I2_J, axiom, (! [A:$tType,P3:$o,X:A]: (((P3 & ((top_top @ (A > $o)) @ X)) = P3)))).
% 2.96/1.53 thf(fact_104_top__conj_I1_J, axiom, (! [A:$tType,X:A,P3:$o]: (((((top_top @ (A > $o)) @ X) & P3) = P3)))).
% 2.96/1.53 thf(fact_105_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X4:A,Y2:B]: ((P2 = ((((product_Pair @ A) @ B) @ X4) @ Y2))))))).
% 2.96/1.53 thf(fact_106_bool_Osplit__sel__asm, axiom, (! [A:$tType,P3:(A > $o),F1:A,F2:A,Bool:$o]: (((P3 @ ((((product_case_bool @ A) @ F1) @ F2) @ Bool)) = (~ (((Bool & (~ ((P3 @ F1)))) | ((~ (Bool)) & (~ ((P3 @ F2))))))))))).
% 2.96/1.53 thf(fact_107_bool_Ocase__distrib, axiom, (! [A:$tType,B:$tType,H:(A > B),F1:A,F2:A,Bool:$o]: (((H @ ((((product_case_bool @ A) @ F1) @ F2) @ Bool)) = ((((product_case_bool @ B) @ (H @ F1)) @ (H @ F2)) @ Bool))))).
% 2.96/1.53 thf(fact_108_bool_Ocase__eq__if, axiom, (! [A:$tType]: (((product_case_bool @ A) = (^ [F12:A,F22:A,Bool2:$o]: (((((if @ A) @ Bool2) @ F12) @ F22))))))).
% 2.96/1.53 thf(fact_109_bool_Osplit__sel, axiom, (! [A:$tType,P3:(A > $o),F1:A,F2:A,Bool:$o]: (((P3 @ ((((product_case_bool @ A) @ F1) @ F2) @ Bool)) = ((Bool => (P3 @ F1)) & ((~ (Bool)) => (P3 @ F2))))))).
% 2.96/1.53 thf(fact_110_old_Obool_Osimps_I3_J, axiom, (! [A:$tType,F1:A,F2:A]: ((((((product_case_bool @ A) @ F1) @ F2) @ $true) = F1)))).
% 2.96/1.53 thf(fact_111_old_Obool_Osimps_I4_J, axiom, (! [A:$tType,F1:A,F2:A]: ((((((product_case_bool @ A) @ F1) @ F2) @ $false) = F2)))).
% 2.96/1.53 thf(fact_112_change__tv_Ocases, axiom, (! [X:((product_prod @ (nat > nat)) @ paraco415392788lle_tv)]: (((! [F7:(nat > nat),B3:$o]: ((X != ((((product_Pair @ (nat > nat)) @ paraco415392788lle_tv) @ F7) @ (paraco2040174112le_Det @ B3))))) => (~ ((! [F7:(nat > nat),N:nat]: ((X != ((((product_Pair @ (nat > nat)) @ paraco415392788lle_tv) @ F7) @ (paraco676387099_Indet @ N))))))))))).
% 2.96/1.53 thf(fact_113_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P3:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A4:A,B3:B]: ((P3 @ ((((product_Pair @ A) @ B) @ A4) @ B3)))) => (P3 @ Prod))))).
% 2.96/1.53 thf(fact_114_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A4:A,B3:B]: ((Y != ((((product_Pair @ A) @ B) @ A4) @ B3))))))))).
% 2.96/1.53 thf(fact_115_prod__induct7, axiom, (! [G2:$tType,F8:$tType,E:$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 @ F8) @ G2)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F8) @ G2))))))]: (((! [A4:A,B3:B,C2:C,D2:D,E2:E,F7:F8,G3:G2]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F8) @ G2)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F8) @ G2))))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F8) @ G2)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F8) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F8) @ G2)) @ E2) @ ((((product_Pair @ F8) @ G2) @ F7) @ G3))))))))) => (P3 @ X))))).
% 2.96/1.53 thf(fact_116_prod__induct6, axiom, (! [F8:$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) @ F8))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F8)))))]: (((! [A4:A,B3:B,C2:C,D2:D,E2:E,F7:F8]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F8))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F8)))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F8))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F8)) @ D2) @ ((((product_Pair @ E) @ F8) @ E2) @ F7)))))))) => (P3 @ X))))).
% 2.96/1.53 thf(fact_117_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A4:A,B3:B,C2:C,D2:D,E2:E]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P3 @ X))))).
% 2.96/1.53 thf(fact_118_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A4:A,B3:B,C2:C,D2:D]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B3) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P3 @ X))))).
% 2.96/1.53 thf(fact_119_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A4:A,B3:B,C2:C]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B3) @ C2))))) => (P3 @ X))))).
% 2.96/1.53 thf(fact_120_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F8:$tType,G2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F8) @ G2))))))]: ((~ ((! [A4:A,B3:B,C2:C,D2:D,E2:E,F7:F8,G3:G2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F8) @ G2)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F8) @ G2))))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F8) @ G2)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F8) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F8) @ G2)) @ E2) @ ((((product_Pair @ F8) @ G2) @ F7) @ G3)))))))))))))).
% 2.96/1.53 thf(fact_121_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F8:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F8)))))]: ((~ ((! [A4:A,B3:B,C2:C,D2:D,E2:E,F7:F8]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F8))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F8)))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F8))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F8)) @ D2) @ ((((product_Pair @ E) @ F8) @ E2) @ F7))))))))))))).
% 2.96/1.53 thf(fact_122_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A4:A,B3:B,C2:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.96/1.53 thf(fact_123_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A4:A,B3:B,C2:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B3) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 2.96/1.53 thf(fact_124_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A4:A,B3:B,C2:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B3) @ C2)))))))))).
% 2.96/1.53 thf(fact_125_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)))))))).
% 2.96/1.53 thf(fact_126_prod__cases, axiom, (! [B:$tType,A:$tType,P3:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A4:A,B3:B]: ((P3 @ ((((product_Pair @ A) @ B) @ A4) @ B3)))) => (P3 @ P2))))).
% 2.96/1.53 thf(fact_127_old_Oprod_Orec, axiom, (! [A:$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))))).
% 2.96/1.53 thf(fact_128_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C3 @ A2) @ B2))))).
% 2.96/1.53 thf(fact_129_inj__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F3:(B > C)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F3)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = ((((inj_on @ B) @ C) @ F3) @ (top_top @ (set @ B))))))).
% 2.96/1.53 thf(fact_130_inj__apfst, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(A > C)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F3)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = ((((inj_on @ A) @ C) @ F3) @ (top_top @ (set @ A))))))).
% 2.96/1.53 thf(fact_131_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R:A,S3:B,R2:(set @ ((product_prod @ A) @ B)),S4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S3)) @ R2) => ((S4 = S3) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S4)) @ R2)))))).
% 2.96/1.53 thf(fact_132_apfst__conv, axiom, (! [C:$tType,A:$tType,B:$tType,F3:(C > A),X:C,Y:B]: (((((((product_apfst @ C) @ A) @ B) @ F3) @ ((((product_Pair @ C) @ B) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ (F3 @ X)) @ Y))))).
% 2.96/1.53 thf(fact_133_apsnd__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F3:(C > B),X:A,Y:C]: (((((((product_apsnd @ C) @ B) @ A) @ F3) @ ((((product_Pair @ A) @ C) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ X) @ (F3 @ Y)))))).
% 2.96/1.53 thf(fact_134_apsnd__apfst__commute, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F3:(C > B),G:(D > A),P2:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F3) @ (((((product_apfst @ D) @ A) @ C) @ G) @ P2)) = (((((product_apfst @ D) @ A) @ B) @ G) @ (((((product_apsnd @ C) @ B) @ D) @ F3) @ P2)))))).
% 2.96/1.53 thf(fact_135_change__tv_Opelims, axiom, (! [X:(nat > nat),Xa:paraco415392788lle_tv,Y:paraco415392788lle_tv]: (((((paraco1920534163nge_tv @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ (nat > nat)) @ paraco415392788lle_tv)) @ paraco2077297190tv_rel) @ ((((product_Pair @ (nat > nat)) @ paraco415392788lle_tv) @ X) @ Xa)) => ((! [B3:$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))))))))))))))))).
% 2.96/1.53 thf(fact_136_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))))))))).
% 2.96/1.53 thf(fact_137_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F:(((product_prod @ B) @ C) > A),A7:B,B5:C]: ((F @ ((((product_Pair @ B) @ C) @ A7) @ B5)))))))).
% 2.96/1.53 thf(fact_138_in__inv__imagep, axiom, (! [B:$tType,A:$tType]: ((((inv_imagep @ A) @ B) = (^ [R3:(A > (A > $o)),F:(B > A),X2:B,Y4:B]: (((R3 @ (F @ X2)) @ (F @ Y4)))))))).
% 2.96/1.53 thf(fact_139_curryI, axiom, (! [A:$tType,B:$tType,F3:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: (((F3 @ ((((product_Pair @ A) @ B) @ A2) @ B2)) => ((((((product_curry @ A) @ B) @ $o) @ F3) @ A2) @ B2))))).
% 2.96/1.53 thf(fact_140_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F3:(C > A),G:(D > B),A2:C,B2:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F3) @ G) @ ((((product_Pair @ C) @ D) @ A2) @ B2)) = ((((product_Pair @ A) @ B) @ (F3 @ A2)) @ (G @ B2)))))).
% 2.96/1.53 thf(fact_141_curryD, axiom, (! [A:$tType,B:$tType,F3:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F3) @ A2) @ B2) => (F3 @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 2.96/1.53 thf(fact_142_curryE, axiom, (! [A:$tType,B:$tType,F3:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F3) @ A2) @ B2) => (F3 @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 2.96/1.53 thf(fact_143_the__inv__into__onto, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ A3) => (((((image @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A3) @ F3)) @ ((((image @ A) @ B) @ F3) @ A3)) = A3))))).
% 2.96/1.53 thf(fact_144_in__inv__image, axiom, (! [A:$tType,B:$tType,X:A,Y:A,R:(set @ ((product_prod @ B) @ B)),F3:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((((inv_image @ B) @ A) @ R) @ F3)) = (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F3 @ X)) @ (F3 @ Y))) @ R))))).
% 2.96/1.53 thf(fact_145_map__prod__imageI, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,A2:A,B2:B,R2:(set @ ((product_prod @ A) @ B)),F3:(A > C),G:(B > D)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R2) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F3 @ A2)) @ (G @ B2))) @ ((((image @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F3) @ G)) @ R2)))))).
% 2.96/1.53 thf(fact_146_inj__swap, axiom, (! [B:$tType,A:$tType,A3:(set @ ((product_prod @ A) @ B))]: (((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ B) @ A)) @ ((product_swap @ A) @ B)) @ A3)))).
% 2.96/1.53 thf(fact_147_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F3:(B > A),X:B,A3:(set @ B)]: (((B2 = (F3 @ X)) => ((((member @ B) @ X) @ A3) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F3) @ A3))))))).
% 2.96/1.53 thf(fact_148_swap__swap, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P2)) = P2)))).
% 2.96/1.53 thf(fact_149_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X))))).
% 2.96/1.53 thf(fact_150_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y:A,X:B,A3:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y) @ X)) @ ((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A3)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y)) @ A3))))).
% 2.96/1.53 thf(fact_151_surj__swap, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 2.96/1.53 thf(fact_152_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F3:(A > B),G:(C > D)]: (((((((image @ A) @ B) @ F3) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image @ C) @ D) @ G) @ (top_top @ (set @ C))) = (top_top @ (set @ D))) => (((((image @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F3) @ G)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 2.96/1.53 thf(fact_153_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X:A,A3:(set @ A),B2:B,F3:(A > B)]: (((((member @ A) @ X) @ A3) => ((B2 = (F3 @ X)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F3) @ A3))))))).
% 2.96/1.53 thf(fact_154_ball__imageD, axiom, (! [A:$tType,B:$tType,F3:(B > A),A3:(set @ B),P3:(A > $o)]: (((! [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F3) @ A3)) => (P3 @ X4)))) => (! [X5:B]: (((((member @ B) @ X5) @ A3) => (P3 @ (F3 @ X5))))))))).
% 2.96/1.53 thf(fact_155_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N3:(set @ A),F3:(A > B),G:(A > B)]: (((M = N3) => ((! [X4:A]: (((((member @ A) @ X4) @ N3) => ((F3 @ X4) = (G @ X4))))) => (((((image @ A) @ B) @ F3) @ M) = ((((image @ A) @ B) @ G) @ N3))))))).
% 2.96/1.53 thf(fact_156_bex__imageD, axiom, (! [A:$tType,B:$tType,F3:(B > A),A3:(set @ B),P3:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F3) @ A3)) & (P3 @ X5)))) => (? [X4:B]: (((((member @ B) @ X4) @ A3) & (P3 @ (F3 @ X4))))))))).
% 2.96/1.53 thf(fact_157_image__iff, axiom, (! [A:$tType,B:$tType,Z:A,F3:(B > A),A3:(set @ B)]: (((((member @ A) @ Z) @ ((((image @ B) @ A) @ F3) @ A3)) = (? [X2:B]: (((((member @ B) @ X2) @ A3) & (Z = (F3 @ X2))))))))).
% 2.96/1.53 thf(fact_158_imageI, axiom, (! [B:$tType,A:$tType,X:A,A3:(set @ A),F3:(A > B)]: (((((member @ A) @ X) @ A3) => (((member @ B) @ (F3 @ X)) @ ((((image @ A) @ B) @ F3) @ A3)))))).
% 2.96/1.53 thf(fact_159_inj__on__image__iff, axiom, (! [B:$tType,A:$tType,A3:(set @ A),G:(A > B),F3:(A > A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A3) => (! [Xa2:A]: (((((member @ A) @ Xa2) @ A3) => (((G @ (F3 @ X4)) = (G @ (F3 @ Xa2))) = ((G @ X4) = (G @ Xa2))))))))) => (((((inj_on @ A) @ A) @ F3) @ A3) => (((((inj_on @ A) @ B) @ G) @ ((((image @ A) @ A) @ F3) @ A3)) = ((((inj_on @ A) @ B) @ G) @ A3))))))).
% 2.96/1.53 thf(fact_160_surjD, axiom, (! [A:$tType,B:$tType,F3:(B > A),Y:A]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X4:B]: ((Y = (F3 @ X4)))))))).
% 2.96/1.53 thf(fact_161_surjE, axiom, (! [A:$tType,B:$tType,F3:(B > A),Y:A]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X4:B]: ((Y != (F3 @ X4)))))))))).
% 2.96/1.53 thf(fact_162_surjI, axiom, (! [B:$tType,A:$tType,G:(B > A),F3:(A > B)]: (((! [X4:A]: (((G @ (F3 @ X4)) = X4))) => (((((image @ B) @ A) @ G) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.96/1.53 thf(fact_163_rangeI, axiom, (! [A:$tType,B:$tType,F3:(B > A),X:B]: ((((member @ A) @ (F3 @ X)) @ ((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))))))).
% 2.96/1.53 thf(fact_164_surj__def, axiom, (! [B:$tType,A:$tType,F3:(B > A)]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y4:A]: ((? [X2:B]: ((Y4 = (F3 @ X2)))))))))).
% 2.96/1.53 thf(fact_165_range__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F3:(B > A),X:B]: (((B2 = (F3 @ X)) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F3) @ (top_top @ (set @ B)))))))).
% 2.96/1.53 thf(fact_166_range__ex1__eq, axiom, (! [B:$tType,A:$tType,F3:(A > B),B2:B]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((((member @ B) @ B2) @ ((((image @ A) @ B) @ F3) @ (top_top @ (set @ A)))) = (? [X2:A]: (((B2 = (F3 @ X2)) & (! [Y4:A]: (((B2 = (F3 @ Y4)) => (Y4 = X2)))))))))))).
% 2.96/1.53 thf(fact_167_inj__image__eq__iff, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((((((image @ A) @ B) @ F3) @ A3) = ((((image @ A) @ B) @ F3) @ B6)) = (A3 = B6)))))).
% 2.96/1.53 thf(fact_168_inj__image__mem__iff, axiom, (! [B:$tType,A:$tType,F3:(A > B),A2:A,A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((((member @ B) @ (F3 @ A2)) @ ((((image @ A) @ B) @ F3) @ A3)) = (((member @ A) @ A2) @ A3)))))).
% 2.96/1.53 thf(fact_169_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C3:((product_prod @ A) @ B),F3:(C > A),G:(D > B),R2:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C3) @ ((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F3) @ G)) @ R2)) => (~ ((! [X4:C,Y2:D]: (((C3 = ((((product_Pair @ A) @ B) @ (F3 @ X4)) @ (G @ Y2))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X4) @ Y2)) @ R2)))))))))))).
% 2.96/1.53 thf(fact_170_f__the__inv__into__f, axiom, (! [A:$tType,B:$tType,F3:(A > B),A3:(set @ A),Y:B]: ((((((inj_on @ A) @ B) @ F3) @ A3) => ((((member @ B) @ Y) @ ((((image @ A) @ B) @ F3) @ A3)) => ((F3 @ (((((the_inv_into @ A) @ B) @ A3) @ F3) @ Y)) = Y)))))).
% 2.96/1.53 thf(fact_171_inj__on__the__inv__into, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ A3) => ((((inj_on @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A3) @ F3)) @ ((((image @ A) @ B) @ F3) @ A3)))))).
% 2.96/1.53 thf(fact_172_total__inv__image, axiom, (! [B:$tType,A:$tType,F3:(A > B),R:(set @ ((product_prod @ B) @ B))]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((((total_on @ B) @ (top_top @ (set @ B))) @ R) => (((total_on @ A) @ (top_top @ (set @ A))) @ ((((inv_image @ B) @ A) @ R) @ F3))))))).
% 2.96/1.53 thf(fact_173_range__fst, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = (top_top @ (set @ A)))))).
% 2.96/1.53 thf(fact_174_range__snd, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ A)))))).
% 2.96/1.53 thf(fact_175_fst__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F3:(C > A),G:(D > B),X:((product_prod @ C) @ D)]: (((((product_fst @ A) @ B) @ (((((((product_map_prod @ C) @ A) @ D) @ B) @ F3) @ G) @ X)) = (F3 @ (((product_fst @ C) @ D) @ X)))))).
% 2.96/1.53 thf(fact_176_snd__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F3:(C > B),G:(D > A),X:((product_prod @ C) @ D)]: (((((product_snd @ B) @ A) @ (((((((product_map_prod @ C) @ B) @ D) @ A) @ F3) @ G) @ X)) = (G @ (((product_snd @ C) @ D) @ X)))))).
% 2.96/1.53 thf(fact_177_apfst__eq__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F3:(C > A),X:((product_prod @ C) @ B),G:(C > A)]: ((((((((product_apfst @ C) @ A) @ B) @ F3) @ X) = (((((product_apfst @ C) @ A) @ B) @ G) @ X)) = ((F3 @ (((product_fst @ C) @ B) @ X)) = (G @ (((product_fst @ C) @ B) @ X))))))).
% 2.96/1.53 thf(fact_178_fst__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,F3:(C > A),X:((product_prod @ C) @ B)]: (((((product_fst @ A) @ B) @ (((((product_apfst @ C) @ A) @ B) @ F3) @ X)) = (F3 @ (((product_fst @ C) @ B) @ X)))))).
% 2.96/1.53 thf(fact_179_snd__apfst, axiom, (! [B:$tType,A:$tType,C:$tType,F3:(C > B),X:((product_prod @ C) @ A)]: (((((product_snd @ B) @ A) @ (((((product_apfst @ C) @ B) @ A) @ F3) @ X)) = (((product_snd @ C) @ A) @ X))))).
% 2.96/1.53 thf(fact_180_fst__apsnd, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(C > B),X:((product_prod @ A) @ C)]: (((((product_fst @ A) @ B) @ (((((product_apsnd @ C) @ B) @ A) @ F3) @ X)) = (((product_fst @ A) @ C) @ X))))).
% 2.96/1.53 thf(fact_181_apsnd__eq__conv, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(C > B),X:((product_prod @ A) @ C),G:(C > B)]: ((((((((product_apsnd @ C) @ B) @ A) @ F3) @ X) = (((((product_apsnd @ C) @ B) @ A) @ G) @ X)) = ((F3 @ (((product_snd @ A) @ C) @ X)) = (G @ (((product_snd @ A) @ C) @ X))))))).
% 2.96/1.53 thf(fact_182_snd__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F3:(C > A),X:((product_prod @ B) @ C)]: (((((product_snd @ B) @ A) @ (((((product_apsnd @ C) @ A) @ B) @ F3) @ X)) = (F3 @ (((product_snd @ B) @ C) @ X)))))).
% 2.96/1.53 thf(fact_183_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 2.96/1.53 thf(fact_184_snd__swap, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X)) = (((product_fst @ A) @ B) @ X))))).
% 2.96/1.53 thf(fact_185_fst__swap, axiom, (! [A:$tType,B:$tType,X:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X)) = (((product_snd @ B) @ A) @ X))))).
% 2.96/1.53 thf(fact_186_apsnd__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F3:(C > B),G:(D > A),X:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F3) @ (((((product_apfst @ D) @ A) @ C) @ G) @ X)) = ((((product_Pair @ A) @ B) @ (G @ (((product_fst @ D) @ C) @ X))) @ (F3 @ (((product_snd @ D) @ C) @ X))))))).
% 2.96/1.53 thf(fact_187_apfst__apsnd, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,F3:(C > A),G:(D > B),X:((product_prod @ C) @ D)]: (((((((product_apfst @ C) @ A) @ B) @ F3) @ (((((product_apsnd @ D) @ B) @ C) @ G) @ X)) = ((((product_Pair @ A) @ B) @ (F3 @ (((product_fst @ C) @ D) @ X))) @ (G @ (((product_snd @ C) @ D) @ X))))))).
% 2.96/1.53 thf(fact_188_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P)) @ (((product_fst @ A) @ B) @ P)))))))).
% 2.96/1.53 thf(fact_189_fst__eqD, axiom, (! [B:$tType,A:$tType,X:A,Y:B,A2:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X) @ Y)) = A2) => (X = A2))))).
% 2.96/1.53 thf(fact_190_snd__eqD, axiom, (! [B:$tType,A:$tType,X:B,Y:A,A2:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = A2) => (Y = A2))))).
% 2.96/1.53 thf(fact_191_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X23:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X23)) = X1)))).
% 2.96/1.53 thf(fact_192_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X23:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X23)) = X23)))).
% 2.96/1.53 thf(fact_193_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 2.96/1.53 thf(fact_194_surjective__pairing, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: ((T2 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T2)) @ (((product_snd @ A) @ B) @ T2)))))).
% 2.96/1.53 thf(fact_195_total__onI, axiom, (! [A:$tType,A3:(set @ A),R:(set @ ((product_prod @ A) @ A))]: (((! [X4:A,Y2:A]: (((((member @ A) @ X4) @ A3) => ((((member @ A) @ Y2) @ A3) => ((X4 != Y2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y2)) @ R) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ X4)) @ R))))))) => (((total_on @ A) @ A3) @ R))))).
% 2.96/1.53 thf(fact_196_total__on__def, axiom, (! [A:$tType]: (((total_on @ A) = (^ [A5:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: ((! [X2:A]: (((((member @ A) @ X2) @ A5) => (! [Y4:A]: (((((member @ A) @ Y4) @ A5) => ((X2 != Y4) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ R3) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X2)) @ R3))))))))))))))).
% 2.96/1.53 thf(fact_197_prod__eqI, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),Q:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P2) = (((product_fst @ A) @ B) @ Q)) => (((((product_snd @ A) @ B) @ P2) = (((product_snd @ A) @ B) @ Q)) => (P2 = Q)))))).
% 2.96/1.53 thf(fact_198_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))).
% 2.96/1.53 thf(fact_199_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y5:((product_prod @ A) @ B),Z2:((product_prod @ A) @ B)]: ((Y5 = Z2))) = (^ [S:((product_prod @ A) @ B),T3:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S) = (((product_fst @ A) @ B) @ T3)) & ((((product_snd @ A) @ B) @ S) = (((product_snd @ A) @ B) @ T3))))))))).
% 2.96/1.53 thf(fact_200_exI__realizer, axiom, (! [B:$tType,A:$tType,P3:(A > (B > $o)),Y:A,X:B]: ((((P3 @ Y) @ X) => ((P3 @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))))))).
% 2.96/1.53 thf(fact_201_conjI__realizer, axiom, (! [A:$tType,B:$tType,P3:(A > $o),P2:A,Q2:(B > $o),Q:B]: (((P3 @ P2) => ((Q2 @ Q) => ((P3 @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q))) & (Q2 @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q))))))))).
% 2.96/1.53 thf(fact_202_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P3:(A > (B > $o)),X:A,Y:B,A2:((product_prod @ A) @ B)]: ((((P3 @ X) @ Y) => ((A2 = ((((product_Pair @ A) @ B) @ X) @ Y)) => ((P3 @ (((product_fst @ A) @ B) @ A2)) @ (((product_snd @ A) @ B) @ A2))))))).
% 2.96/1.53 thf(fact_203_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B2:A,P2:((product_prod @ B) @ A)]: (((B2 = (((product_snd @ B) @ A) @ P2)) = (? [A7:B]: ((P2 = ((((product_Pair @ B) @ A) @ A7) @ B2)))))))).
% 2.96/1.53 thf(fact_204_sndI, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),Y:A,Z:B]: (((X = ((((product_Pair @ A) @ B) @ Y) @ Z)) => ((((product_snd @ A) @ B) @ X) = Z))))).
% 2.96/1.53 thf(fact_205_fstI, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),Y:A,Z:B]: (((X = ((((product_Pair @ A) @ B) @ Y) @ Z)) => ((((product_fst @ A) @ B) @ X) = Y))))).
% 2.96/1.53 thf(fact_206_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A2:A,P2:((product_prod @ A) @ B)]: (((A2 = (((product_fst @ A) @ B) @ P2)) = (? [B5:B]: ((P2 = ((((product_Pair @ A) @ B) @ A2) @ B5)))))))).
% 2.96/1.53 thf(fact_207_total__lex__prod, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ B) @ B))]: (((((total_on @ A) @ (top_top @ (set @ A))) @ R) => ((((total_on @ B) @ (top_top @ (set @ B))) @ S3) => (((total_on @ ((product_prod @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) @ ((((lex_prod @ A) @ B) @ R) @ S3))))))).
% 2.96/1.53 thf(fact_208_sndOp__def, axiom, (! [B:$tType,A:$tType,C:$tType]: (((((bNF_sndOp @ C) @ A) @ B) = (^ [P5:(C > (A > $o)),Q4:(A > (B > $o)),Ac:((product_prod @ C) @ B)]: (((((product_Pair @ A) @ B) @ (((((((bNF_pick_middlep @ C) @ A) @ B) @ P5) @ Q4) @ (((product_fst @ C) @ B) @ Ac)) @ (((product_snd @ C) @ B) @ Ac))) @ (((product_snd @ C) @ B) @ Ac)))))))).
% 2.96/1.53 thf(fact_209_in__lex__prod, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A6:A,B4:B,R:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Pair @ A) @ B) @ A6) @ B4))) @ ((((lex_prod @ A) @ B) @ R) @ S3)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A6)) @ R) | ((A2 = A6) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B2) @ B4)) @ S3))))))).
% 2.96/1.53 thf(fact_210_fstOp__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((bNF_fstOp @ A) @ B) @ C) = (^ [P5:(A > (B > $o)),Q4:(B > (C > $o)),Ac:((product_prod @ A) @ C)]: (((((product_Pair @ A) @ B) @ (((product_fst @ A) @ C) @ Ac)) @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P5) @ Q4) @ (((product_fst @ A) @ C) @ Ac)) @ (((product_snd @ A) @ C) @ Ac))))))))).
% 2.96/1.53 thf(fact_211_wf__map__prod__image, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),F3:(A > B)]: ((((wf @ A) @ R) => (((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((wf @ B) @ ((((image @ ((product_prod @ A) @ A)) @ ((product_prod @ B) @ B)) @ ((((((product_map_prod @ A) @ B) @ A) @ B) @ F3) @ F3)) @ R))))))).
% 2.96/1.53 thf(fact_212_wf__induct__rule, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),P3:(A > $o),A2:A]: ((((wf @ A) @ R) => ((! [X4:A]: (((! [Y6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ X4)) @ R) => (P3 @ Y6)))) => (P3 @ X4)))) => (P3 @ A2)))))).
% 2.96/1.53 thf(fact_213_wf__eq__minimal, axiom, (! [A:$tType]: (((wf @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [Q4:(set @ A)]: (((? [X2:A]: ((((member @ A) @ X2) @ Q4))) => (? [X2:A]: (((((member @ A) @ X2) @ Q4) & (! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X2)) @ R3) => (~ ((((member @ A) @ Y4) @ Q4)))))))))))))))))).
% 2.96/1.53 thf(fact_214_wf__not__refl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A2:A]: ((((wf @ A) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R))))))).
% 2.96/1.53 thf(fact_215_wf__not__sym, axiom, (! [A:$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)))))))).
% 2.96/1.53 thf(fact_216_wf__irrefl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A2:A]: ((((wf @ A) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R))))))).
% 2.96/1.53 thf(fact_217_wf__induct, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),P3:(A > $o),A2:A]: ((((wf @ A) @ R) => ((! [X4:A]: (((! [Y6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ X4)) @ R) => (P3 @ Y6)))) => (P3 @ X4)))) => (P3 @ A2)))))).
% 2.96/1.53 thf(fact_218_wf__asym, axiom, (! [A:$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)))))))).
% 2.96/1.53 thf(fact_219_wfUNIVI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((! [P6:(A > $o),X4:A]: (((! [Xa3:A]: (((! [Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ Xa3)) @ R) => (P6 @ Y2)))) => (P6 @ Xa3)))) => (P6 @ X4)))) => ((wf @ A) @ R))))).
% 2.96/1.53 thf(fact_220_wfI__min, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((! [X4:A,Q5:(set @ A)]: (((((member @ A) @ X4) @ Q5) => (? [Xa3:A]: (((((member @ A) @ Xa3) @ Q5) & (! [Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ Xa3)) @ R2) => (~ ((((member @ A) @ Y2) @ Q5)))))))))))) => ((wf @ A) @ R2))))).
% 2.96/1.53 thf(fact_221_wfE__min, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),X:A,Q2:(set @ A)]: ((((wf @ A) @ R2) => ((((member @ A) @ X) @ Q2) => (~ ((! [Z3:A]: (((((member @ A) @ Z3) @ Q2) => (~ ((! [Y6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ Z3)) @ R2) => (~ ((((member @ A) @ Y6) @ Q2)))))))))))))))))).
% 2.96/1.53 thf(fact_222_wf__def, axiom, (! [A:$tType]: (((wf @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [P5:(A > $o)]: (((! [X2:A]: (((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X2)) @ R3) => (P5 @ Y4)))) => (P5 @ X2)))) => (! [X6:A]: ((P5 @ X6)))))))))))).
% 2.96/1.53 thf(fact_223_dependent__wf__choice, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ A)),P3:((A > B) > (A > (B > $o)))]: ((((wf @ A) @ R2) => ((! [F7:(A > B),G3:(A > B),X4:A,R4:B]: (((! [Z4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z4) @ X4)) @ R2) => ((F7 @ Z4) = (G3 @ Z4))))) => ((((P3 @ F7) @ X4) @ R4) = (((P3 @ G3) @ X4) @ R4))))) => ((! [X4:A,F7:(A > B)]: (((! [Y6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ X4)) @ R2) => (((P3 @ F7) @ Y6) @ (F7 @ Y6))))) => (? [X_1:B]: ((((P3 @ F7) @ X4) @ X_1)))))) => (? [F7:(A > B)]: ((! [X5:A]: ((((P3 @ F7) @ X5) @ (F7 @ X5)))))))))))).
% 2.96/1.53 thf(fact_224_same__fstI, axiom, (! [B:$tType,A:$tType,P3:(A > $o),X:A,Y7:B,Y:B,R2:(A > (set @ ((product_prod @ B) @ B)))]: (((P3 @ X) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y7) @ Y)) @ (R2 @ X)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y7)) @ ((((product_Pair @ A) @ B) @ X) @ Y))) @ ((((same_fst @ A) @ B) @ P3) @ R2))))))).
% 2.96/1.53 thf(fact_225_image2__eqI, axiom, (! [A:$tType,C:$tType,B:$tType,B2:A,F3:(B > A),X:B,C3:C,G:(B > C),A3:(set @ B)]: (((B2 = (F3 @ 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) @ F3) @ G)))))))).
% 2.96/1.53 thf(fact_226_the__inv__into__into, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),X:B,B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ A3) => ((((member @ B) @ X) @ ((((image @ A) @ B) @ F3) @ A3)) => ((((ord_less_eq @ (set @ A)) @ A3) @ B6) => (((member @ A) @ (((((the_inv_into @ A) @ B) @ A3) @ F3) @ X)) @ B6))))))).
% 2.96/1.53 thf(fact_227_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.96/1.53 thf(fact_228_subsetI, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A3) => (((member @ A) @ X4) @ B6)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B6))))).
% 2.96/1.53 thf(fact_229_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ A3) => (A3 = B6)))))).
% 2.96/1.53 thf(fact_230_inj__on__image__eq__iff, axiom, (! [B:$tType,A:$tType,F3:(A > B),C4:(set @ A),A3:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ C4) => ((((ord_less_eq @ (set @ A)) @ A3) @ C4) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => ((((((image @ A) @ B) @ F3) @ A3) = ((((image @ A) @ B) @ F3) @ B6)) = (A3 = B6)))))))).
% 2.96/1.53 thf(fact_231_inj__on__image__mem__iff, axiom, (! [B:$tType,A:$tType,F3:(A > B),B6:(set @ A),A2:A,A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ B6) => ((((member @ A) @ A2) @ B6) => ((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((member @ B) @ (F3 @ A2)) @ ((((image @ A) @ B) @ F3) @ A3)) = (((member @ A) @ A2) @ A3)))))))).
% 2.96/1.53 thf(fact_232_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) => (A2 = (top_top @ A))))))))).
% 2.96/1.53 thf(fact_233_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) = (A2 = (top_top @ A))))))))).
% 2.96/1.53 thf(fact_234_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ (top_top @ A)))))))).
% 2.96/1.53 thf(fact_235_subset__UNIV, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ (top_top @ (set @ A)))))).
% 2.96/1.53 thf(fact_236_inj__on__subset, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ A3) => ((((ord_less_eq @ (set @ A)) @ B6) @ A3) => ((((inj_on @ A) @ B) @ F3) @ B6)))))).
% 2.96/1.53 thf(fact_237_subset__inj__on, axiom, (! [B:$tType,A:$tType,F3:(A > B),B6:(set @ A),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ B6) => ((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((inj_on @ A) @ B) @ F3) @ A3)))))).
% 2.96/1.53 thf(fact_238_in__mono, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((member @ A) @ X) @ A3) => (((member @ A) @ X) @ B6)))))).
% 2.96/1.53 thf(fact_239_subsetD, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((member @ A) @ C3) @ A3) => (((member @ A) @ C3) @ B6)))))).
% 2.96/1.53 thf(fact_240_equalityE, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((A3 = B6) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A3)))))))))).
% 2.96/1.53 thf(fact_241_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B7:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A5) => (((member @ A) @ X2) @ B7)))))))))).
% 2.96/1.53 thf(fact_242_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((A3 = B6) => (((ord_less_eq @ (set @ A)) @ A3) @ B6))))).
% 2.96/1.53 thf(fact_243_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A)]: (((A3 = B6) => (((ord_less_eq @ (set @ A)) @ B6) @ A3))))).
% 2.96/1.53 thf(fact_244_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B7:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A5) => (((member @ A) @ T3) @ B7)))))))))).
% 2.96/1.53 thf(fact_245_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 2.96/1.53 thf(fact_246_Collect__mono, axiom, (! [A:$tType,P3:(A > $o),Q2:(A > $o)]: (((! [X4:A]: (((P3 @ X4) => (Q2 @ X4)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P3)) @ ((collect @ A) @ Q2)))))).
% 2.96/1.53 thf(fact_247_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => (((ord_less_eq @ (set @ A)) @ A3) @ C4)))))).
% 2.96/1.53 thf(fact_248_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G) => (((ord_less_eq @ B) @ (F3 @ X)) @ (G @ X))))))))).
% 2.96/1.53 thf(fact_249_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G) => (((ord_less_eq @ B) @ (F3 @ X)) @ (G @ X))))))))).
% 2.96/1.53 thf(fact_250_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G:(A > B)]: (((! [X4:A]: ((((ord_less_eq @ B) @ (F3 @ X4)) @ (G @ X4)))) => (((ord_less_eq @ (A > B)) @ F3) @ G)))))))).
% 2.96/1.53 thf(fact_251_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z2:(set @ A)]: ((Y5 = Z2))) = (^ [A5:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A5)))))))).
% 2.96/1.53 thf(fact_252_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F:(A > B),G4:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G4 @ X2))))))))))).
% 2.96/1.53 thf(fact_253_Collect__mono__iff, axiom, (! [A:$tType,P3:(A > $o),Q2:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P3)) @ ((collect @ A) @ Q2)) = (! [X2:A]: (((P3 @ X2) => (Q2 @ X2)))))))).
% 2.96/1.53 thf(fact_254_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F3:(B > A),B2:B,C3:B]: (((((ord_less_eq @ A) @ A2) @ (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C3) => ((! [X4:B,Y2:B]: (((((ord_less_eq @ B) @ X4) @ Y2) => (((ord_less_eq @ A) @ (F3 @ X4)) @ (F3 @ Y2))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C3))))))))))).
% 2.96/1.53 thf(fact_255_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F3:(A > C),C3:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F3 @ B2)) @ C3) => ((! [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) => (((ord_less_eq @ C) @ (F3 @ X4)) @ (F3 @ Y2))))) => (((ord_less_eq @ C) @ (F3 @ A2)) @ C3)))))))))).
% 2.96/1.53 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A8:$tType,A9:$tType]: (((order_top @ A9) => (order_top @ (A8 > A9)))))).
% 2.96/1.53 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 2.96/1.53 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 2.96/1.53 thf(tcon_fun___Orderings_Otop, axiom, (! [A8:$tType,A9:$tType]: (((top @ A9) => (top @ (A8 > A9)))))).
% 2.96/1.53 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 2.96/1.53 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, (preorder @ nat)).
% 2.96/1.53 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, (order @ nat)).
% 2.96/1.53 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, (ord @ nat)).
% 2.96/1.53 thf(tcon_Set_Oset___Orderings_Oorder__top_4, axiom, (! [A8:$tType]: ((order_top @ (set @ A8))))).
% 2.96/1.53 thf(tcon_Set_Oset___Orderings_Opreorder_5, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 2.96/1.53 thf(tcon_Set_Oset___Orderings_Oorder_6, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 2.96/1.53 thf(tcon_Set_Oset___Orderings_Otop_7, axiom, (! [A8:$tType]: ((top @ (set @ A8))))).
% 2.96/1.53 thf(tcon_Set_Oset___Orderings_Oord_8, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 2.96/1.53 thf(tcon_HOL_Obool___Orderings_Oorder__top_9, axiom, (order_top @ $o)).
% 2.96/1.53 thf(tcon_HOL_Obool___Orderings_Opreorder_10, axiom, (preorder @ $o)).
% 2.96/1.53 thf(tcon_HOL_Obool___Orderings_Oorder_11, axiom, (order @ $o)).
% 2.96/1.53 thf(tcon_HOL_Obool___Orderings_Otop_12, axiom, (top @ $o)).
% 2.96/1.53 thf(tcon_HOL_Obool___Orderings_Oord_13, axiom, (ord @ $o)).
% 2.96/1.53 thf(help_If_3_1_T, axiom, (! [P3:$o]: (((P3 = $true) | (P3 = $false))))).
% 2.96/1.53 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.96/1.53 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.96/1.53 thf(conj_0, conjecture, (((paraco876059933e_eval @ ((paraco753463838ge_int @ f) @ i)) @ p2) != (paraco2040174112le_Det @ $true))).
% 2.96/1.53 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.xGjE5LYYhO/DTF2DTF24296.p
% 2.96/1.53 ------------------------
% 2.96/1.56 ---- Cleaned THF ---
% 2.96/1.56 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv,type,
% 2.96/1.56 paraco415392788lle_tv: $tType ).
% 2.96/1.56
% 2.96/1.56 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm,type,
% 2.96/1.56 paraco414474393lle_fm: $tType ).
% 2.96/1.56
% 2.96/1.56 thf(ty_t_Product__Type_Oprod,type,
% 2.96/1.56 product_prod: $tType > $tType > $tType ).
% 2.96/1.56
% 2.96/1.56 thf(ty_t_String_Ochar,type,
% 2.96/1.56 char: $tType ).
% 2.96/1.56
% 2.96/1.56 thf(ty_t_List_Olist,type,
% 2.96/1.56 list: $tType > $tType ).
% 2.96/1.56
% 2.96/1.56 thf(ty_t_Set_Oset,type,
% 2.96/1.56 set: $tType > $tType ).
% 2.96/1.56
% 2.96/1.56 thf(ty_t_Nat_Onat,type,
% 2.96/1.56 nat: $tType ).
% 2.96/1.56
% 2.96/1.56 thf(sy_cl_HOL_Otype,type,
% 2.96/1.56 type:
% 2.96/1.56 !>[A: $tType] : $o ).
% 2.96/1.56
% 2.96/1.56 thf(sy_cl_Orderings_Oord,type,
% 2.96/1.56 ord:
% 2.96/1.56 !>[A: $tType] : $o ).
% 2.96/1.56
% 2.96/1.56 thf(sy_cl_Orderings_Otop,type,
% 2.96/1.56 top:
% 2.96/1.56 !>[A: $tType] : $o ).
% 2.96/1.56
% 2.96/1.56 thf(sy_cl_Orderings_Oorder,type,
% 2.96/1.56 order:
% 2.96/1.56 !>[A: $tType] : $o ).
% 2.96/1.56
% 2.96/1.56 thf(sy_cl_Orderings_Opreorder,type,
% 2.96/1.56 preorder:
% 2.96/1.56 !>[A: $tType] : $o ).
% 2.96/1.56
% 2.96/1.56 thf(sy_cl_Orderings_Oorder__top,type,
% 2.96/1.56 order_top:
% 2.96/1.56 !>[A: $tType] : $o ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_BNF__Def_OfstOp,type,
% 2.96/1.56 bNF_fstOp:
% 2.96/1.56 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_BNF__Def_Opick__middlep,type,
% 2.96/1.56 bNF_pick_middlep:
% 2.96/1.56 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_BNF__Def_OsndOp,type,
% 2.96/1.56 bNF_sndOp:
% 2.96/1.56 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 2.96/1.56 bNF_Greatest_image2:
% 2.96/1.56 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Fun_Oinj__on,type,
% 2.96/1.56 inj_on:
% 2.96/1.56 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Fun_Othe__inv__into,type,
% 2.96/1.56 the_inv_into:
% 2.96/1.56 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_If,type,
% 2.96/1.56 if:
% 2.96/1.56 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.96/1.56 ord_less_eq:
% 2.96/1.56 !>[A: $tType] : ( A > A > $o ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.96/1.56 top_top:
% 2.96/1.56 !>[A: $tType] : A ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__int,type,
% 2.96/1.56 paraco753463838ge_int: ( nat > nat ) > ( ( list @ char ) > paraco415392788lle_tv ) > ( list @ char ) > paraco415392788lle_tv ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv,type,
% 2.96/1.56 paraco1920534163nge_tv: ( nat > nat ) > paraco415392788lle_tv > paraco415392788lle_tv ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv__rel,type,
% 2.96/1.56 paraco2077297190tv_rel: ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) > ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) > $o ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval,type,
% 2.96/1.56 paraco876059933e_eval: ( ( list @ char ) > paraco415392788lle_tv ) > paraco414474393lle_fm > paraco415392788lle_tv ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H,type,
% 2.96/1.56 paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql,type,
% 2.96/1.56 paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H,type,
% 2.96/1.56 paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H,type,
% 2.96/1.56 paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro,type,
% 2.96/1.56 paraco27778325le_Pro: ( list @ char ) > paraco414474393lle_fm ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth,type,
% 2.96/1.56 paraco251304083_Truth: paraco414474393lle_fm ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_Ocase__fm,type,
% 2.96/1.56 paraco1246693743ase_fm:
% 2.96/1.56 !>[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 ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet,type,
% 2.96/1.56 paraco2040174112le_Det: $o > paraco415392788lle_tv ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet,type,
% 2.96/1.56 paraco676387099_Indet: nat > paraco415392788lle_tv ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Ocase__tv,type,
% 2.96/1.56 paraco490622181ase_tv:
% 2.96/1.56 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv,type,
% 2.96/1.56 paraco152590079rec_tv:
% 2.96/1.56 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid,type,
% 2.96/1.56 paraco769098683_valid: paraco414474393lle_fm > $o ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid__in,type,
% 2.96/1.56 paraco2086025920lid_in: ( set @ nat ) > paraco414474393lle_fm > $o ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Product__Type_OPair,type,
% 2.96/1.56 product_Pair:
% 2.96/1.56 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Product__Type_Oapfst,type,
% 2.96/1.56 product_apfst:
% 2.96/1.56 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Product__Type_Oapsnd,type,
% 2.96/1.56 product_apsnd:
% 2.96/1.56 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Product__Type_Obool_Ocase__bool,type,
% 2.96/1.56 product_case_bool:
% 2.96/1.56 !>[A: $tType] : ( A > A > $o > A ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Product__Type_Ocurry,type,
% 2.96/1.56 product_curry:
% 2.96/1.56 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.96/1.56 produc2004651681e_prod:
% 2.96/1.56 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Product__Type_Omap__prod,type,
% 2.96/1.56 product_map_prod:
% 2.96/1.56 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.96/1.56 product_rec_prod:
% 2.96/1.56 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 2.96/1.56 product_fst:
% 2.96/1.56 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 2.96/1.56 product_snd:
% 2.96/1.56 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.96/1.56 product_swap:
% 2.96/1.56 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Relation_Oinv__image,type,
% 2.96/1.56 inv_image:
% 2.96/1.56 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Relation_Oinv__imagep,type,
% 2.96/1.56 inv_imagep:
% 2.96/1.56 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Relation_Ototal__on,type,
% 2.96/1.56 total_on:
% 2.96/1.56 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Set_OCollect,type,
% 2.96/1.56 collect:
% 2.96/1.56 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Set_Oimage,type,
% 2.96/1.56 image:
% 2.96/1.56 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Wellfounded_Oaccp,type,
% 2.96/1.56 accp:
% 2.96/1.56 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Wellfounded_Olex__prod,type,
% 2.96/1.56 lex_prod:
% 2.96/1.56 !>[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 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Wellfounded_Owf,type,
% 2.96/1.56 wf:
% 2.96/1.56 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_Wfrec_Osame__fst,type,
% 2.96/1.56 same_fst:
% 2.96/1.56 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_c_member,type,
% 2.96/1.56 member:
% 2.96/1.56 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.96/1.56
% 2.96/1.56 thf(sy_v_f,type,
% 2.96/1.56 f: nat > nat ).
% 2.96/1.56
% 2.96/1.56 thf(sy_v_i,type,
% 2.96/1.56 i: ( list @ char ) > paraco415392788lle_tv ).
% 2.96/1.56
% 2.96/1.56 thf(sy_v_p1____,type,
% 2.96/1.56 p1: paraco414474393lle_fm ).
% 2.96/1.56
% 2.96/1.56 thf(sy_v_p2____,type,
% 2.96/1.56 p2: paraco414474393lle_fm ).
% 2.96/1.56
% 2.96/1.56 thf(fact_0_a_H_H_H,axiom,
% 2.96/1.56 ( ( paraco876059933e_eval @ i @ p2 )
% 2.96/1.56 != ( paraco2040174112le_Det @ $true ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_1_a_H_H,axiom,
% 2.96/1.56 ( ( paraco876059933e_eval @ i @ p1 )
% 2.96/1.56 != ( paraco2040174112le_Det @ $true ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_2_a_H_H_H_H,axiom,
% 2.96/1.56 ( ( paraco876059933e_eval @ i @ p1 )
% 2.96/1.56 != ( paraco2040174112le_Det @ $false ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_3_b_H_H,axiom,
% 2.96/1.56 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
% 2.96/1.56 != ( paraco2040174112le_Det @ $true ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_4_a,axiom,
% 2.96/1.56 ( ( paraco876059933e_eval @ i @ p2 )
% 2.96/1.56 = ( paraco2040174112le_Det @ $false ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_5_a_H,axiom,
% 2.96/1.56 ( ( paraco876059933e_eval @ i @ p1 )
% 2.96/1.56 != ( paraco876059933e_eval @ i @ p2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_6_b,axiom,
% 2.96/1.56 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 )
% 2.96/1.56 = ( paraco2040174112le_Det @ $false ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_7_b_H,axiom,
% 2.96/1.56 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
% 2.96/1.56 != ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_8_tv_Oinject_I1_J,axiom,
% 2.96/1.56 ! [X1: $o,Y1: $o] :
% 2.96/1.56 ( ( ( paraco2040174112le_Det @ X1 )
% 2.96/1.56 = ( paraco2040174112le_Det @ Y1 ) )
% 2.96/1.56 = ( X1 = Y1 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_9_ih2,axiom,
% 2.96/1.56 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 )
% 2.96/1.56 = ( paraco1920534163nge_tv @ f @ ( paraco876059933e_eval @ i @ p2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_10_ih1,axiom,
% 2.96/1.56 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p1 )
% 2.96/1.56 = ( paraco1920534163nge_tv @ f @ ( paraco876059933e_eval @ i @ p1 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_11__092_060open_062change__tv_Af_A_Ieval_Ai_Ap2_J_A_061_ADet_AFalse_092_060close_062,axiom,
% 2.96/1.56 ( ( paraco1920534163nge_tv @ f @ ( paraco876059933e_eval @ i @ p2 ) )
% 2.96/1.56 = ( paraco2040174112le_Det @ $false ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_12_valid__def,axiom,
% 2.96/1.56 ( paraco769098683_valid
% 2.96/1.56 = ( ^ [P: paraco414474393lle_fm] :
% 2.96/1.56 ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 2.96/1.56 ( ( paraco876059933e_eval @ I @ P )
% 2.96/1.56 = ( paraco2040174112le_Det @ $true ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_13_yes,axiom,
% 2.96/1.56 ( ( paraco876059933e_eval @ i @ ( paraco1628874225le_Eql @ p1 @ p2 ) )
% 2.96/1.56 = ( paraco876059933e_eval @ i @ ( paraco329115265le_Neg @ p1 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_14_eval_Osimps_I2_J,axiom,
% 2.96/1.56 ! [I2: ( list @ char ) > paraco415392788lle_tv] :
% 2.96/1.56 ( ( paraco876059933e_eval @ I2 @ paraco251304083_Truth )
% 2.96/1.56 = ( paraco2040174112le_Det @ $true ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_15_tv_Osimps_I7_J,axiom,
% 2.96/1.56 ! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
% 2.96/1.56 ( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
% 2.96/1.56 = ( F1 @ X1 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_16_eval_Osimps_I5_J,axiom,
% 2.96/1.56 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.96/1.56 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 = ( paraco876059933e_eval @ I2 @ Q ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
% 2.96/1.56 = ( paraco2040174112le_Det @ $true ) ) )
% 2.96/1.56 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 != ( paraco876059933e_eval @ I2 @ Q ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
% 2.96/1.56 = ( paraco2040174112le_Det @ $false ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_17_eval__equality__simplify,axiom,
% 2.96/1.56 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P2 @ Q ) )
% 2.96/1.56 = ( paraco2040174112le_Det
% 2.96/1.56 @ ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 = ( paraco876059933e_eval @ I2 @ Q ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_18_change__int__def,axiom,
% 2.96/1.56 ( paraco753463838ge_int
% 2.96/1.56 = ( ^ [F: nat > nat,I: ( list @ char ) > paraco415392788lle_tv,S: list @ char] : ( paraco1920534163nge_tv @ F @ ( I @ S ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_19_eval_Osimps_I4_J,axiom,
% 2.96/1.56 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.96/1.56 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 = ( paraco876059933e_eval @ I2 @ Q ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 2.96/1.56 = ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 2.96/1.56 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 != ( paraco876059933e_eval @ I2 @ Q ) )
% 2.96/1.56 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 = ( paraco2040174112le_Det @ $true ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 2.96/1.56 = ( paraco876059933e_eval @ I2 @ Q ) ) )
% 2.96/1.56 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 != ( paraco2040174112le_Det @ $true ) )
% 2.96/1.56 => ( ( ( ( paraco876059933e_eval @ I2 @ Q )
% 2.96/1.56 = ( paraco2040174112le_Det @ $true ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 2.96/1.56 = ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 2.96/1.56 & ( ( ( paraco876059933e_eval @ I2 @ Q )
% 2.96/1.56 != ( paraco2040174112le_Det @ $true ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 2.96/1.56 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_20_tv_Osimps_I5_J,axiom,
% 2.96/1.56 ! [A: $tType,F1: $o > A,F2: nat > A,X1: $o] :
% 2.96/1.56 ( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco2040174112le_Det @ X1 ) )
% 2.96/1.56 = ( F1 @ X1 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_21_eval__negation,axiom,
% 2.96/1.56 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm] :
% 2.96/1.56 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 = ( paraco2040174112le_Det @ $false ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 2.96/1.56 = ( paraco2040174112le_Det @ $true ) ) )
% 2.96/1.56 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 != ( paraco2040174112le_Det @ $false ) )
% 2.96/1.56 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 = ( paraco2040174112le_Det @ $true ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 2.96/1.56 = ( paraco2040174112le_Det @ $false ) ) )
% 2.96/1.56 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 != ( paraco2040174112le_Det @ $true ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 2.96/1.56 = ( paraco876059933e_eval @ I2 @ P2 ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_22_assms,axiom,
% 2.96/1.56 inj_on @ nat @ nat @ f @ ( top_top @ ( set @ nat ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_23_change__tv_Osimps_I1_J,axiom,
% 2.96/1.56 ! [F3: nat > nat,B2: $o] :
% 2.96/1.56 ( ( paraco1920534163nge_tv @ F3 @ ( paraco2040174112le_Det @ B2 ) )
% 2.96/1.56 = ( paraco2040174112le_Det @ B2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_24_fm_Oinject_I2_J,axiom,
% 2.96/1.56 ! [X3: paraco414474393lle_fm,Y3: paraco414474393lle_fm] :
% 2.96/1.56 ( ( ( paraco329115265le_Neg @ X3 )
% 2.96/1.56 = ( paraco329115265le_Neg @ Y3 ) )
% 2.96/1.56 = ( X3 = Y3 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_25_fm_Oinject_I3_J,axiom,
% 2.96/1.56 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,Y41: paraco414474393lle_fm,Y42: paraco414474393lle_fm] :
% 2.96/1.56 ( ( ( paraco2100061555le_Con @ X41 @ X42 )
% 2.96/1.56 = ( paraco2100061555le_Con @ Y41 @ Y42 ) )
% 2.96/1.56 = ( ( X41 = Y41 )
% 2.96/1.56 & ( X42 = Y42 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_26_fm_Oinject_I5_J,axiom,
% 2.96/1.56 ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm,Y61: paraco414474393lle_fm,Y62: paraco414474393lle_fm] :
% 2.96/1.56 ( ( ( paraco1628874225le_Eql @ X61 @ X62 )
% 2.96/1.56 = ( paraco1628874225le_Eql @ Y61 @ Y62 ) )
% 2.96/1.56 = ( ( X61 = Y61 )
% 2.96/1.56 & ( X62 = Y62 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_27_fm_Oinject_I4_J,axiom,
% 2.96/1.56 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,Y51: paraco414474393lle_fm,Y52: paraco414474393lle_fm] :
% 2.96/1.56 ( ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 2.96/1.56 = ( paraco2084319816le_Eql @ Y51 @ Y52 ) )
% 2.96/1.56 = ( ( X51 = Y51 )
% 2.96/1.56 & ( X52 = Y52 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_28_fm_Odistinct_I29_J,axiom,
% 2.96/1.56 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 2.96/1.56 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_29_fm_Odistinct_I27_J,axiom,
% 2.96/1.56 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 2.96/1.56 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_30_fm_Odistinct_I25_J,axiom,
% 2.96/1.56 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 2.96/1.56 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_31_fm_Odistinct_I23_J,axiom,
% 2.96/1.56 ! [X3: paraco414474393lle_fm,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco329115265le_Neg @ X3 )
% 2.96/1.56 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_32_fm_Odistinct_I21_J,axiom,
% 2.96/1.56 ! [X3: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco329115265le_Neg @ X3 )
% 2.96/1.56 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_33_fm_Odistinct_I19_J,axiom,
% 2.96/1.56 ! [X3: paraco414474393lle_fm,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco329115265le_Neg @ X3 )
% 2.96/1.56 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_34_fm_Odistinct_I17_J,axiom,
% 2.96/1.56 ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.96/1.56 ( paraco251304083_Truth
% 2.96/1.56 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_35_fm_Odistinct_I15_J,axiom,
% 2.96/1.56 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.96/1.56 ( paraco251304083_Truth
% 2.96/1.56 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_36_fm_Odistinct_I13_J,axiom,
% 2.96/1.56 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.96/1.56 ( paraco251304083_Truth
% 2.96/1.56 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_37_fm_Odistinct_I11_J,axiom,
% 2.96/1.56 ! [X3: paraco414474393lle_fm] :
% 2.96/1.56 ( paraco251304083_Truth
% 2.96/1.56 != ( paraco329115265le_Neg @ X3 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_38_conjunction,axiom,
% 2.96/1.56 ! [P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco769098683_valid @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 2.96/1.56 = ( ( paraco769098683_valid @ P2 )
% 2.96/1.56 & ( paraco769098683_valid @ Q ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_39_eval__equality,axiom,
% 2.96/1.56 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.96/1.56 ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 = ( paraco876059933e_eval @ I2 @ Q ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.96/1.56 = ( paraco2040174112le_Det @ $true ) ) )
% 2.96/1.56 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 != ( paraco876059933e_eval @ I2 @ Q ) )
% 2.96/1.56 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 = ( paraco2040174112le_Det @ $true ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.96/1.56 = ( paraco876059933e_eval @ I2 @ Q ) ) )
% 2.96/1.56 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 != ( paraco2040174112le_Det @ $true ) )
% 2.96/1.56 => ( ( ( ( paraco876059933e_eval @ I2 @ Q )
% 2.96/1.56 = ( paraco2040174112le_Det @ $true ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.96/1.56 = ( paraco876059933e_eval @ I2 @ P2 ) ) )
% 2.96/1.56 & ( ( ( paraco876059933e_eval @ I2 @ Q )
% 2.96/1.56 != ( paraco2040174112le_Det @ $true ) )
% 2.96/1.56 => ( ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 = ( paraco2040174112le_Det @ $false ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.96/1.56 = ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ Q ) ) ) )
% 2.96/1.56 & ( ( ( paraco876059933e_eval @ I2 @ P2 )
% 2.96/1.56 != ( paraco2040174112le_Det @ $false ) )
% 2.96/1.56 => ( ( ( ( paraco876059933e_eval @ I2 @ Q )
% 2.96/1.56 = ( paraco2040174112le_Det @ $false ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.96/1.56 = ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) ) ) )
% 2.96/1.56 & ( ( ( paraco876059933e_eval @ I2 @ Q )
% 2.96/1.56 != ( paraco2040174112le_Det @ $false ) )
% 2.96/1.56 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P2 @ Q ) )
% 2.96/1.56 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_40_double__negation,axiom,
% 2.96/1.56 ( paraco876059933e_eval
% 2.96/1.56 = ( ^ [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] : ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ ( paraco329115265le_Neg @ P ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_41_UNIV__I,axiom,
% 2.96/1.56 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_42_iso__tuple__UNIV__I,axiom,
% 2.96/1.56 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_43_top__apply,axiom,
% 2.96/1.56 ! [C: $tType,D: $tType] :
% 2.96/1.56 ( ( top @ C )
% 2.96/1.56 => ( ( top_top @ ( D > C ) )
% 2.96/1.56 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_44_injD,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,X: A,Y: A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 2.96/1.56 => ( ( ( F3 @ X )
% 2.96/1.56 = ( F3 @ Y ) )
% 2.96/1.56 => ( X = Y ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_45_mem__Collect__eq,axiom,
% 2.96/1.56 ! [A: $tType,A2: A,P3: A > $o] :
% 2.96/1.56 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 2.96/1.56 = ( P3 @ A2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_46_Collect__mem__eq,axiom,
% 2.96/1.56 ! [A: $tType,A3: set @ A] :
% 2.96/1.56 ( ( collect @ A
% 2.96/1.56 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 2.96/1.56 = A3 ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_47_Collect__cong,axiom,
% 2.96/1.56 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 2.96/1.56 ( ! [X4: A] :
% 2.96/1.56 ( ( P3 @ X4 )
% 2.96/1.56 = ( Q2 @ X4 ) )
% 2.96/1.56 => ( ( collect @ A @ P3 )
% 2.96/1.56 = ( collect @ A @ Q2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_48_ext,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 2.96/1.56 ( ! [X4: A] :
% 2.96/1.56 ( ( F3 @ X4 )
% 2.96/1.56 = ( G @ X4 ) )
% 2.96/1.56 => ( F3 = G ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_49_injI,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B] :
% 2.96/1.56 ( ! [X4: A,Y2: A] :
% 2.96/1.56 ( ( ( F3 @ X4 )
% 2.96/1.56 = ( F3 @ Y2 ) )
% 2.96/1.56 => ( X4 = Y2 ) )
% 2.96/1.56 => ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_50_inj__eq,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,X: A,Y: A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 2.96/1.56 => ( ( ( F3 @ X )
% 2.96/1.56 = ( F3 @ Y ) )
% 2.96/1.56 = ( X = Y ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_51_inj__def,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 2.96/1.56 = ( ! [X2: A,Y4: A] :
% 2.96/1.56 ( ( ( F3 @ X2 )
% 2.96/1.56 = ( F3 @ Y4 ) )
% 2.96/1.56 => ( X2 = Y4 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_52_fm_Oinduct,axiom,
% 2.96/1.56 ! [P3: paraco414474393lle_fm > $o,Fm: paraco414474393lle_fm] :
% 2.96/1.56 ( ! [X4: list @ char] : ( P3 @ ( paraco27778325le_Pro @ X4 ) )
% 2.96/1.56 => ( ( P3 @ paraco251304083_Truth )
% 2.96/1.56 => ( ! [X4: paraco414474393lle_fm] :
% 2.96/1.56 ( ( P3 @ X4 )
% 2.96/1.56 => ( P3 @ ( paraco329115265le_Neg @ X4 ) ) )
% 2.96/1.56 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 2.96/1.56 ( ( P3 @ X1a )
% 2.96/1.56 => ( ( P3 @ X22 )
% 2.96/1.56 => ( P3 @ ( paraco2100061555le_Con @ X1a @ X22 ) ) ) )
% 2.96/1.56 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 2.96/1.56 ( ( P3 @ X1a )
% 2.96/1.56 => ( ( P3 @ X22 )
% 2.96/1.56 => ( P3 @ ( paraco2084319816le_Eql @ X1a @ X22 ) ) ) )
% 2.96/1.56 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 2.96/1.56 ( ( P3 @ X1a )
% 2.96/1.56 => ( ( P3 @ X22 )
% 2.96/1.56 => ( P3 @ ( paraco1628874225le_Eql @ X1a @ X22 ) ) ) )
% 2.96/1.56 => ( P3 @ Fm ) ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_53_fm_Oexhaust,axiom,
% 2.96/1.56 ! [Y: paraco414474393lle_fm] :
% 2.96/1.56 ( ! [X12: list @ char] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( paraco27778325le_Pro @ X12 ) )
% 2.96/1.56 => ( ( Y != paraco251304083_Truth )
% 2.96/1.56 => ( ! [X32: paraco414474393lle_fm] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( paraco329115265le_Neg @ X32 ) )
% 2.96/1.56 => ( ! [X412: paraco414474393lle_fm,X422: paraco414474393lle_fm] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( paraco2100061555le_Con @ X412 @ X422 ) )
% 2.96/1.56 => ( ! [X512: paraco414474393lle_fm,X522: paraco414474393lle_fm] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( paraco2084319816le_Eql @ X512 @ X522 ) )
% 2.96/1.56 => ~ ! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_54_props_Ocases,axiom,
% 2.96/1.56 ! [X: paraco414474393lle_fm] :
% 2.96/1.56 ( ( X != paraco251304083_Truth )
% 2.96/1.56 => ( ! [S2: list @ char] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( paraco27778325le_Pro @ S2 ) )
% 2.96/1.56 => ( ! [P4: paraco414474393lle_fm] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( paraco329115265le_Neg @ P4 ) )
% 2.96/1.56 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( paraco2100061555le_Con @ P4 @ Q3 ) )
% 2.96/1.56 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( paraco2084319816le_Eql @ P4 @ Q3 ) )
% 2.96/1.56 => ~ ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_55_fm_Oinject_I1_J,axiom,
% 2.96/1.56 ! [X1: list @ char,Y1: list @ char] :
% 2.96/1.56 ( ( ( paraco27778325le_Pro @ X1 )
% 2.96/1.56 = ( paraco27778325le_Pro @ Y1 ) )
% 2.96/1.56 = ( X1 = Y1 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_56_top__set__def,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ( ( top_top @ ( set @ A ) )
% 2.96/1.56 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_57_eval_Osimps_I1_J,axiom,
% 2.96/1.56 ! [I2: ( list @ char ) > paraco415392788lle_tv,S3: list @ char] :
% 2.96/1.56 ( ( paraco876059933e_eval @ I2 @ ( paraco27778325le_Pro @ S3 ) )
% 2.96/1.56 = ( I2 @ S3 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_58_fm_Odistinct_I3_J,axiom,
% 2.96/1.56 ! [X1: list @ char,X3: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco27778325le_Pro @ X1 )
% 2.96/1.56 != ( paraco329115265le_Neg @ X3 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_59_fm_Odistinct_I5_J,axiom,
% 2.96/1.56 ! [X1: list @ char,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco27778325le_Pro @ X1 )
% 2.96/1.56 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_60_fm_Odistinct_I9_J,axiom,
% 2.96/1.56 ! [X1: list @ char,X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco27778325le_Pro @ X1 )
% 2.96/1.56 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_61_fm_Odistinct_I7_J,axiom,
% 2.96/1.56 ! [X1: list @ char,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco27778325le_Pro @ X1 )
% 2.96/1.56 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_62_fm_Odistinct_I1_J,axiom,
% 2.96/1.56 ! [X1: list @ char] :
% 2.96/1.56 ( ( paraco27778325le_Pro @ X1 )
% 2.96/1.56 != paraco251304083_Truth ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_63_change__tv__injection,axiom,
% 2.96/1.56 ! [F3: nat > nat] :
% 2.96/1.56 ( ( inj_on @ nat @ nat @ F3 @ ( top_top @ ( set @ nat ) ) )
% 2.96/1.56 => ( inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ ( paraco1920534163nge_tv @ F3 ) @ ( top_top @ ( set @ paraco415392788lle_tv ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_64_UNIV__witness,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_65_UNIV__eq__I,axiom,
% 2.96/1.56 ! [A: $tType,A3: set @ A] :
% 2.96/1.56 ( ! [X4: A] : ( member @ A @ X4 @ A3 )
% 2.96/1.56 => ( ( top_top @ ( set @ A ) )
% 2.96/1.56 = A3 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_66_inj__on__inverseI,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,A3: set @ A,G: B > A,F3: A > B] :
% 2.96/1.56 ( ! [X4: A] :
% 2.96/1.56 ( ( member @ A @ X4 @ A3 )
% 2.96/1.56 => ( ( G @ ( F3 @ X4 ) )
% 2.96/1.56 = X4 ) )
% 2.96/1.56 => ( inj_on @ A @ B @ F3 @ A3 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_67_inj__on__contraD,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X: A,Y: A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ A3 )
% 2.96/1.56 => ( ( X != Y )
% 2.96/1.56 => ( ( member @ A @ X @ A3 )
% 2.96/1.56 => ( ( member @ A @ Y @ A3 )
% 2.96/1.56 => ( ( F3 @ X )
% 2.96/1.56 != ( F3 @ Y ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_68_inj__on__eq__iff,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X: A,Y: A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ A3 )
% 2.96/1.56 => ( ( member @ A @ X @ A3 )
% 2.96/1.56 => ( ( member @ A @ Y @ A3 )
% 2.96/1.56 => ( ( ( F3 @ X )
% 2.96/1.56 = ( F3 @ Y ) )
% 2.96/1.56 = ( X = Y ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_69_inj__on__cong,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,A3: set @ A,F3: A > B,G: A > B] :
% 2.96/1.56 ( ! [A4: A] :
% 2.96/1.56 ( ( member @ A @ A4 @ A3 )
% 2.96/1.56 => ( ( F3 @ A4 )
% 2.96/1.56 = ( G @ A4 ) ) )
% 2.96/1.56 => ( ( inj_on @ A @ B @ F3 @ A3 )
% 2.96/1.56 = ( inj_on @ A @ B @ G @ A3 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_70_inj__on__def,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType] :
% 2.96/1.56 ( ( inj_on @ A @ B )
% 2.96/1.56 = ( ^ [F: A > B,A5: set @ A] :
% 2.96/1.56 ! [X2: A] :
% 2.96/1.56 ( ( member @ A @ X2 @ A5 )
% 2.96/1.56 => ! [Y4: A] :
% 2.96/1.56 ( ( member @ A @ Y4 @ A5 )
% 2.96/1.56 => ( ( ( F @ X2 )
% 2.96/1.56 = ( F @ Y4 ) )
% 2.96/1.56 => ( X2 = Y4 ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_71_inj__onI,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,A3: set @ A,F3: A > B] :
% 2.96/1.56 ( ! [X4: A,Y2: A] :
% 2.96/1.56 ( ( member @ A @ X4 @ A3 )
% 2.96/1.56 => ( ( member @ A @ Y2 @ A3 )
% 2.96/1.56 => ( ( ( F3 @ X4 )
% 2.96/1.56 = ( F3 @ Y2 ) )
% 2.96/1.56 => ( X4 = Y2 ) ) ) )
% 2.96/1.56 => ( inj_on @ A @ B @ F3 @ A3 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_72_inj__onD,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X: A,Y: A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ A3 )
% 2.96/1.56 => ( ( ( F3 @ X )
% 2.96/1.56 = ( F3 @ Y ) )
% 2.96/1.56 => ( ( member @ A @ X @ A3 )
% 2.96/1.56 => ( ( member @ A @ Y @ A3 )
% 2.96/1.56 => ( X = Y ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_73_props_Oinduct,axiom,
% 2.96/1.56 ! [P3: paraco414474393lle_fm > $o,A0: paraco414474393lle_fm] :
% 2.96/1.56 ( ( P3 @ paraco251304083_Truth )
% 2.96/1.56 => ( ! [S2: list @ char] : ( P3 @ ( paraco27778325le_Pro @ S2 ) )
% 2.96/1.56 => ( ! [P4: paraco414474393lle_fm] :
% 2.96/1.56 ( ( P3 @ P4 )
% 2.96/1.56 => ( P3 @ ( paraco329115265le_Neg @ P4 ) ) )
% 2.96/1.56 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.96/1.56 ( ( P3 @ P4 )
% 2.96/1.56 => ( ( P3 @ Q3 )
% 2.96/1.56 => ( P3 @ ( paraco2100061555le_Con @ P4 @ Q3 ) ) ) )
% 2.96/1.56 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.96/1.56 ( ( P3 @ P4 )
% 2.96/1.56 => ( ( P3 @ Q3 )
% 2.96/1.56 => ( P3 @ ( paraco2084319816le_Eql @ P4 @ Q3 ) ) ) )
% 2.96/1.56 => ( ! [P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.96/1.56 ( ( P3 @ P4 )
% 2.96/1.56 => ( ( P3 @ Q3 )
% 2.96/1.56 => ( P3 @ ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) )
% 2.96/1.56 => ( P3 @ A0 ) ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_74_eval_Ocases,axiom,
% 2.96/1.56 ! [X: product_prod @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm] :
% 2.96/1.56 ( ! [I3: ( list @ char ) > paraco415392788lle_tv,S2: list @ char] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco27778325le_Pro @ S2 ) ) )
% 2.96/1.56 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ paraco251304083_Truth ) )
% 2.96/1.56 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco329115265le_Neg @ P4 ) ) )
% 2.96/1.56 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2100061555le_Con @ P4 @ Q3 ) ) )
% 2.96/1.56 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2084319816le_Eql @ P4 @ Q3 ) ) )
% 2.96/1.56 => ~ ! [I3: ( list @ char ) > paraco415392788lle_tv,P4: paraco414474393lle_fm,Q3: paraco414474393lle_fm] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco1628874225le_Eql @ P4 @ Q3 ) ) ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_75_top__empty__eq,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ( ( top_top @ ( A > $o ) )
% 2.96/1.56 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_76_change__tv_Oelims,axiom,
% 2.96/1.56 ! [X: nat > nat,Xa: paraco415392788lle_tv,Y: paraco415392788lle_tv] :
% 2.96/1.56 ( ( ( paraco1920534163nge_tv @ X @ Xa )
% 2.96/1.56 = Y )
% 2.96/1.56 => ( ! [B3: $o] :
% 2.96/1.56 ( ( Xa
% 2.96/1.56 = ( paraco2040174112le_Det @ B3 ) )
% 2.96/1.56 => ( Y
% 2.96/1.56 != ( paraco2040174112le_Det @ B3 ) ) )
% 2.96/1.56 => ~ ! [N: nat] :
% 2.96/1.56 ( ( Xa
% 2.96/1.56 = ( paraco676387099_Indet @ N ) )
% 2.96/1.56 => ( Y
% 2.96/1.56 != ( paraco676387099_Indet @ ( X @ N ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_77_valid__valid__in,axiom,
% 2.96/1.56 ! [P2: paraco414474393lle_fm,U: set @ nat] :
% 2.96/1.56 ( ( paraco769098683_valid @ P2 )
% 2.96/1.56 => ( paraco2086025920lid_in @ U @ P2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_78_transfer,axiom,
% 2.96/1.56 ! [U: set @ nat,P2: paraco414474393lle_fm] :
% 2.96/1.56 ( ~ ( paraco2086025920lid_in @ U @ P2 )
% 2.96/1.56 => ~ ( paraco769098683_valid @ P2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_79_the__inv__f__f,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,X: A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 2.96/1.56 => ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ ( F3 @ X ) )
% 2.96/1.56 = X ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_80_conjunction__in,axiom,
% 2.96/1.56 ! [U: set @ nat,P2: paraco414474393lle_fm,Q: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco2086025920lid_in @ U @ ( paraco2100061555le_Con @ P2 @ Q ) )
% 2.96/1.56 = ( ( paraco2086025920lid_in @ U @ P2 )
% 2.96/1.56 & ( paraco2086025920lid_in @ U @ Q ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_81_fm_Osimps_I37_J,axiom,
% 2.96/1.56 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F32: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A] :
% 2.96/1.56 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F32 @ F4 @ F5 @ F6 @ paraco251304083_Truth )
% 2.96/1.56 = F2 ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_82_tv_Oinject_I2_J,axiom,
% 2.96/1.56 ! [X23: nat,Y22: nat] :
% 2.96/1.56 ( ( ( paraco676387099_Indet @ X23 )
% 2.96/1.56 = ( paraco676387099_Indet @ Y22 ) )
% 2.96/1.56 = ( X23 = Y22 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_83_top1I,axiom,
% 2.96/1.56 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_84_string__tv_Oinduct,axiom,
% 2.96/1.56 ! [P3: paraco415392788lle_tv > $o,A0: paraco415392788lle_tv] :
% 2.96/1.56 ( ( P3 @ ( paraco2040174112le_Det @ $true ) )
% 2.96/1.56 => ( ( P3 @ ( paraco2040174112le_Det @ $false ) )
% 2.96/1.56 => ( ! [N: nat] : ( P3 @ ( paraco676387099_Indet @ N ) )
% 2.96/1.56 => ( P3 @ A0 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_85_change__tv_Oinduct,axiom,
% 2.96/1.56 ! [P3: ( nat > nat ) > paraco415392788lle_tv > $o,A0: nat > nat,A1: paraco415392788lle_tv] :
% 2.96/1.56 ( ! [F7: nat > nat,B3: $o] : ( P3 @ F7 @ ( paraco2040174112le_Det @ B3 ) )
% 2.96/1.56 => ( ! [F7: nat > nat,N: nat] : ( P3 @ F7 @ ( paraco676387099_Indet @ N ) )
% 2.96/1.56 => ( P3 @ A0 @ A1 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_86_string__tv_Ocases,axiom,
% 2.96/1.56 ! [X: paraco415392788lle_tv] :
% 2.96/1.56 ( ( X
% 2.96/1.56 != ( paraco2040174112le_Det @ $true ) )
% 2.96/1.56 => ( ( X
% 2.96/1.56 != ( paraco2040174112le_Det @ $false ) )
% 2.96/1.56 => ~ ! [N: nat] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( paraco676387099_Indet @ N ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_87_tv_Oexhaust,axiom,
% 2.96/1.56 ! [Y: paraco415392788lle_tv] :
% 2.96/1.56 ( ! [X12: $o] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( paraco2040174112le_Det @ X12 ) )
% 2.96/1.56 => ~ ! [X22: nat] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( paraco676387099_Indet @ X22 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_88_tv_Oinduct,axiom,
% 2.96/1.56 ! [P3: paraco415392788lle_tv > $o,Tv: paraco415392788lle_tv] :
% 2.96/1.56 ( ! [X4: $o] : ( P3 @ ( paraco2040174112le_Det @ X4 ) )
% 2.96/1.56 => ( ! [X4: nat] : ( P3 @ ( paraco676387099_Indet @ X4 ) )
% 2.96/1.56 => ( P3 @ Tv ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_89_tv_Odistinct_I1_J,axiom,
% 2.96/1.56 ! [X1: $o,X23: nat] :
% 2.96/1.56 ( ( paraco2040174112le_Det @ X1 )
% 2.96/1.56 != ( paraco676387099_Indet @ X23 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_90_the__inv__into__f__f,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X: A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ A3 )
% 2.96/1.56 => ( ( member @ A @ X @ A3 )
% 2.96/1.56 => ( ( the_inv_into @ A @ B @ A3 @ F3 @ ( F3 @ X ) )
% 2.96/1.56 = X ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_91_the__inv__into__f__eq,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X: A,Y: B] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ A3 )
% 2.96/1.56 => ( ( ( F3 @ X )
% 2.96/1.56 = Y )
% 2.96/1.56 => ( ( member @ A @ X @ A3 )
% 2.96/1.56 => ( ( the_inv_into @ A @ B @ A3 @ F3 @ Y )
% 2.96/1.56 = X ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_92_tv_Osimps_I6_J,axiom,
% 2.96/1.56 ! [A: $tType,F1: $o > A,F2: nat > A,X23: nat] :
% 2.96/1.56 ( ( paraco490622181ase_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X23 ) )
% 2.96/1.56 = ( F2 @ X23 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_93_change__tv_Osimps_I2_J,axiom,
% 2.96/1.56 ! [F3: nat > nat,N2: nat] :
% 2.96/1.56 ( ( paraco1920534163nge_tv @ F3 @ ( paraco676387099_Indet @ N2 ) )
% 2.96/1.56 = ( paraco676387099_Indet @ ( F3 @ N2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_94_fm_Osimps_I36_J,axiom,
% 2.96/1.56 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F32: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X1: list @ char] :
% 2.96/1.56 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F32 @ F4 @ F5 @ F6 @ ( paraco27778325le_Pro @ X1 ) )
% 2.96/1.56 = ( F1 @ X1 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_95_tv_Osimps_I8_J,axiom,
% 2.96/1.56 ! [A: $tType,F1: $o > A,F2: nat > A,X23: nat] :
% 2.96/1.56 ( ( paraco152590079rec_tv @ A @ F1 @ F2 @ ( paraco676387099_Indet @ X23 ) )
% 2.96/1.56 = ( F2 @ X23 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_96_fm_Osimps_I38_J,axiom,
% 2.96/1.56 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F32: paraco414474393lle_fm > A,F4: paraco414474393lle_fm > paraco414474393lle_fm > A,F5: paraco414474393lle_fm > paraco414474393lle_fm > A,F6: paraco414474393lle_fm > paraco414474393lle_fm > A,X3: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F32 @ F4 @ F5 @ F6 @ ( paraco329115265le_Neg @ X3 ) )
% 2.96/1.56 = ( F32 @ X3 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_97_fm_Osimps_I39_J,axiom,
% 2.96/1.56 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F32: 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] :
% 2.96/1.56 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F32 @ F4 @ F5 @ F6 @ ( paraco2100061555le_Con @ X41 @ X42 ) )
% 2.96/1.56 = ( F4 @ X41 @ X42 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_98_fm_Osimps_I41_J,axiom,
% 2.96/1.56 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F32: 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] :
% 2.96/1.56 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F32 @ F4 @ F5 @ F6 @ ( paraco1628874225le_Eql @ X61 @ X62 ) )
% 2.96/1.56 = ( F6 @ X61 @ X62 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_99_fm_Osimps_I40_J,axiom,
% 2.96/1.56 ! [A: $tType,F1: ( list @ char ) > A,F2: A,F32: 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] :
% 2.96/1.56 ( ( paraco1246693743ase_fm @ A @ F1 @ F2 @ F32 @ F4 @ F5 @ F6 @ ( paraco2084319816le_Eql @ X51 @ X52 ) )
% 2.96/1.56 = ( F5 @ X51 @ X52 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_100_prod_Oinject,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,X1: A,X23: B,Y1: A,Y22: B] :
% 2.96/1.56 ( ( ( product_Pair @ A @ B @ X1 @ X23 )
% 2.96/1.56 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 2.96/1.56 = ( ( X1 = Y1 )
% 2.96/1.56 & ( X23 = Y22 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_101_old_Oprod_Oinject,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
% 2.96/1.56 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.96/1.56 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 2.96/1.56 = ( ( A2 = A6 )
% 2.96/1.56 & ( B2 = B4 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_102_eval_Osimps_I3_J,axiom,
% 2.96/1.56 ! [I2: ( list @ char ) > paraco415392788lle_tv,P2: paraco414474393lle_fm] :
% 2.96/1.56 ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P2 ) )
% 2.96/1.56 = ( paraco490622181ase_tv @ paraco415392788lle_tv @ ( product_case_bool @ paraco415392788lle_tv @ ( paraco2040174112le_Det @ $false ) @ ( paraco2040174112le_Det @ $true ) ) @ paraco676387099_Indet @ ( paraco876059933e_eval @ I2 @ P2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_103_top__conj_I2_J,axiom,
% 2.96/1.56 ! [A: $tType,P3: $o,X: A] :
% 2.96/1.56 ( ( P3
% 2.96/1.56 & ( top_top @ ( A > $o ) @ X ) )
% 2.96/1.56 = P3 ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_104_top__conj_I1_J,axiom,
% 2.96/1.56 ! [A: $tType,X: A,P3: $o] :
% 2.96/1.56 ( ( ( top_top @ ( A > $o ) @ X )
% 2.96/1.56 & P3 )
% 2.96/1.56 = P3 ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_105_surj__pair,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.96/1.56 ? [X4: A,Y2: B] :
% 2.96/1.56 ( P2
% 2.96/1.56 = ( product_Pair @ A @ B @ X4 @ Y2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_106_bool_Osplit__sel__asm,axiom,
% 2.96/1.56 ! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
% 2.96/1.56 ( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 2.96/1.56 = ( ~ ( ( Bool
% 2.96/1.56 & ~ ( P3 @ F1 ) )
% 2.96/1.56 | ( ~ Bool
% 2.96/1.56 & ~ ( P3 @ F2 ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_107_bool_Ocase__distrib,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,H: A > B,F1: A,F2: A,Bool: $o] :
% 2.96/1.56 ( ( H @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 2.96/1.56 = ( product_case_bool @ B @ ( H @ F1 ) @ ( H @ F2 ) @ Bool ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_108_bool_Ocase__eq__if,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ( ( product_case_bool @ A )
% 2.96/1.56 = ( ^ [F12: A,F22: A,Bool2: $o] : ( if @ A @ Bool2 @ F12 @ F22 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_109_bool_Osplit__sel,axiom,
% 2.96/1.56 ! [A: $tType,P3: A > $o,F1: A,F2: A,Bool: $o] :
% 2.96/1.56 ( ( P3 @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 2.96/1.56 = ( ( Bool
% 2.96/1.56 => ( P3 @ F1 ) )
% 2.96/1.56 & ( ~ Bool
% 2.96/1.56 => ( P3 @ F2 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_110_old_Obool_Osimps_I3_J,axiom,
% 2.96/1.56 ! [A: $tType,F1: A,F2: A] :
% 2.96/1.56 ( ( product_case_bool @ A @ F1 @ F2 @ $true )
% 2.96/1.56 = F1 ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_111_old_Obool_Osimps_I4_J,axiom,
% 2.96/1.56 ! [A: $tType,F1: A,F2: A] :
% 2.96/1.56 ( ( product_case_bool @ A @ F1 @ F2 @ $false )
% 2.96/1.56 = F2 ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_112_change__tv_Ocases,axiom,
% 2.96/1.56 ! [X: product_prod @ ( nat > nat ) @ paraco415392788lle_tv] :
% 2.96/1.56 ( ! [F7: nat > nat,B3: $o] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ F7 @ ( paraco2040174112le_Det @ B3 ) ) )
% 2.96/1.56 => ~ ! [F7: nat > nat,N: nat] :
% 2.96/1.56 ( X
% 2.96/1.56 != ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ F7 @ ( paraco676387099_Indet @ N ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_113_old_Oprod_Oinducts,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.96/1.56 ( ! [A4: A,B3: B] : ( P3 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
% 2.96/1.56 => ( P3 @ Prod ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_114_old_Oprod_Oexhaust,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.96/1.56 ~ ! [A4: A,B3: B] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( product_Pair @ A @ B @ A4 @ B3 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_115_prod__induct7,axiom,
% 2.96/1.56 ! [G2: $tType,F8: $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 @ F8 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) ) )] :
% 2.96/1.56 ( ! [A4: A,B3: B,C2: C,D2: D,E2: E,F7: F8,G3: G2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F8 @ G2 ) @ E2 @ ( product_Pair @ F8 @ G2 @ F7 @ G3 ) ) ) ) ) ) )
% 2.96/1.56 => ( P3 @ X ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_116_prod__induct6,axiom,
% 2.96/1.56 ! [F8: $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 @ F8 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) ) )] :
% 2.96/1.56 ( ! [A4: A,B3: B,C2: C,D2: D,E2: E,F7: F8] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F8 ) @ D2 @ ( product_Pair @ E @ F8 @ E2 @ F7 ) ) ) ) ) )
% 2.96/1.56 => ( P3 @ X ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_117_prod__induct5,axiom,
% 2.96/1.56 ! [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 ) ) )] :
% 2.96/1.56 ( ! [A4: A,B3: B,C2: C,D2: D,E2: E] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( 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 ) ) ) ) )
% 2.96/1.56 => ( P3 @ X ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_118_prod__induct4,axiom,
% 2.96/1.56 ! [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 ) )] :
% 2.96/1.56 ( ! [A4: A,B3: B,C2: C,D2: D] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 2.96/1.56 => ( P3 @ X ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_119_prod__induct3,axiom,
% 2.96/1.56 ! [C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.96/1.56 ( ! [A4: A,B3: B,C2: C] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B3 @ C2 ) ) )
% 2.96/1.56 => ( P3 @ X ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_120_prod__cases7,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F8: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) ) )] :
% 2.96/1.56 ~ ! [A4: A,B3: B,C2: C,D2: D,E2: E,F7: F8,G3: G2] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F8 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F8 @ G2 ) @ E2 @ ( product_Pair @ F8 @ G2 @ F7 @ G3 ) ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_121_prod__cases6,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F8: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) ) )] :
% 2.96/1.56 ~ ! [A4: A,B3: B,C2: C,D2: D,E2: E,F7: F8] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F8 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F8 ) @ D2 @ ( product_Pair @ E @ F8 @ E2 @ F7 ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_122_prod__cases5,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.96/1.56 ~ ! [A4: A,B3: B,C2: C,D2: D,E2: E] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( 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 ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_123_prod__cases4,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.96/1.56 ~ ! [A4: A,B3: B,C2: C,D2: D] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_124_prod__cases3,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.96/1.56 ~ ! [A4: A,B3: B,C2: C] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B3 @ C2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_125_Pair__inject,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
% 2.96/1.56 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.96/1.56 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 2.96/1.56 => ~ ( ( A2 = A6 )
% 2.96/1.56 => ( B2 != B4 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_126_prod__cases,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.96/1.56 ( ! [A4: A,B3: B] : ( P3 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
% 2.96/1.56 => ( P3 @ P2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_127_old_Oprod_Orec,axiom,
% 2.96/1.56 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.96/1.56 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.96/1.56 = ( F1 @ A2 @ B2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_128_internal__case__prod__conv,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 2.96/1.56 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 2.96/1.56 = ( C3 @ A2 @ B2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_129_inj__apsnd,axiom,
% 2.96/1.56 ! [A: $tType,C: $tType,B: $tType,F3: B > C] :
% 2.96/1.56 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.96/1.56 = ( inj_on @ B @ C @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_130_inj__apfst,axiom,
% 2.96/1.56 ! [B: $tType,C: $tType,A: $tType,F3: A > C] :
% 2.96/1.56 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.96/1.56 = ( inj_on @ A @ C @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_131_ssubst__Pair__rhs,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,R: A,S3: B,R2: set @ ( product_prod @ A @ B ),S4: B] :
% 2.96/1.56 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R2 )
% 2.96/1.56 => ( ( S4 = S3 )
% 2.96/1.56 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S4 ) @ R2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_132_apfst__conv,axiom,
% 2.96/1.56 ! [C: $tType,A: $tType,B: $tType,F3: C > A,X: C,Y: B] :
% 2.96/1.56 ( ( product_apfst @ C @ A @ B @ F3 @ ( product_Pair @ C @ B @ X @ Y ) )
% 2.96/1.56 = ( product_Pair @ A @ B @ ( F3 @ X ) @ Y ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_133_apsnd__conv,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,C: $tType,F3: C > B,X: A,Y: C] :
% 2.96/1.56 ( ( product_apsnd @ C @ B @ A @ F3 @ ( product_Pair @ A @ C @ X @ Y ) )
% 2.96/1.56 = ( product_Pair @ A @ B @ X @ ( F3 @ Y ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_134_apsnd__apfst__commute,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: C > B,G: D > A,P2: product_prod @ D @ C] :
% 2.96/1.56 ( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apfst @ D @ A @ C @ G @ P2 ) )
% 2.96/1.56 = ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F3 @ P2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_135_change__tv_Opelims,axiom,
% 2.96/1.56 ! [X: nat > nat,Xa: paraco415392788lle_tv,Y: paraco415392788lle_tv] :
% 2.96/1.56 ( ( ( paraco1920534163nge_tv @ X @ Xa )
% 2.96/1.56 = Y )
% 2.96/1.56 => ( ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ Xa ) )
% 2.96/1.56 => ( ! [B3: $o] :
% 2.96/1.56 ( ( Xa
% 2.96/1.56 = ( paraco2040174112le_Det @ B3 ) )
% 2.96/1.56 => ( ( Y
% 2.96/1.56 = ( paraco2040174112le_Det @ B3 ) )
% 2.96/1.56 => ~ ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ ( paraco2040174112le_Det @ B3 ) ) ) ) )
% 2.96/1.56 => ~ ! [N: nat] :
% 2.96/1.56 ( ( Xa
% 2.96/1.56 = ( paraco676387099_Indet @ N ) )
% 2.96/1.56 => ( ( Y
% 2.96/1.56 = ( paraco676387099_Indet @ ( X @ N ) ) )
% 2.96/1.56 => ~ ( accp @ ( product_prod @ ( nat > nat ) @ paraco415392788lle_tv ) @ paraco2077297190tv_rel @ ( product_Pair @ ( nat > nat ) @ paraco415392788lle_tv @ X @ ( paraco676387099_Indet @ N ) ) ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_136_prod_Oinj__map,axiom,
% 2.96/1.56 ! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F2: B > D] :
% 2.96/1.56 ( ( inj_on @ A @ C @ F1 @ ( top_top @ ( set @ A ) ) )
% 2.96/1.56 => ( ( inj_on @ B @ D @ F2 @ ( top_top @ ( set @ B ) ) )
% 2.96/1.56 => ( 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 ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_137_curry__conv,axiom,
% 2.96/1.56 ! [A: $tType,C: $tType,B: $tType] :
% 2.96/1.56 ( ( product_curry @ B @ C @ A )
% 2.96/1.56 = ( ^ [F: ( product_prod @ B @ C ) > A,A7: B,B5: C] : ( F @ ( product_Pair @ B @ C @ A7 @ B5 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_138_in__inv__imagep,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType] :
% 2.96/1.56 ( ( inv_imagep @ A @ B )
% 2.96/1.56 = ( ^ [R3: A > A > $o,F: B > A,X2: B,Y4: B] : ( R3 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_139_curryI,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.96/1.56 ( ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.96/1.56 => ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_140_map__prod__simp,axiom,
% 2.96/1.56 ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G: D > B,A2: C,B2: D] :
% 2.96/1.56 ( ( product_map_prod @ C @ A @ D @ B @ F3 @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 2.96/1.56 = ( product_Pair @ A @ B @ ( F3 @ A2 ) @ ( G @ B2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_141_curryD,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.96/1.56 ( ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 )
% 2.96/1.56 => ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_142_curryE,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.96/1.56 ( ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 )
% 2.96/1.56 => ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_143_the__inv__into__onto,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ A3 )
% 2.96/1.56 => ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A3 @ F3 ) @ ( image @ A @ B @ F3 @ A3 ) )
% 2.96/1.56 = A3 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_144_in__inv__image,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ B @ B ),F3: A > B] :
% 2.96/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R @ F3 ) )
% 2.96/1.56 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) @ R ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_145_map__prod__imageI,axiom,
% 2.96/1.56 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),F3: A > C,G: B > D] :
% 2.96/1.56 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 2.96/1.56 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F3 @ A2 ) @ ( G @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G ) @ R2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_146_inj__swap,axiom,
% 2.96/1.56 ! [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 ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_147_image__eqI,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,B2: A,F3: B > A,X: B,A3: set @ B] :
% 2.96/1.56 ( ( B2
% 2.96/1.56 = ( F3 @ X ) )
% 2.96/1.56 => ( ( member @ B @ X @ A3 )
% 2.96/1.56 => ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_148_swap__swap,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 2.96/1.56 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 2.96/1.56 = P2 ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_149_swap__simp,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,X: B,Y: A] :
% 2.96/1.56 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 2.96/1.56 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_150_pair__in__swap__image,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,Y: A,X: B,A3: set @ ( product_prod @ B @ A )] :
% 2.96/1.56 ( ( 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 ) )
% 2.96/1.56 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A3 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_151_surj__swap,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType] :
% 2.96/1.56 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.96/1.56 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_152_map__prod__surj,axiom,
% 2.96/1.56 ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: A > B,G: C > D] :
% 2.96/1.56 ( ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 2.96/1.56 = ( top_top @ ( set @ B ) ) )
% 2.96/1.56 => ( ( ( image @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
% 2.96/1.56 = ( top_top @ ( set @ D ) ) )
% 2.96/1.56 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 2.96/1.56 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_153_rev__image__eqI,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,X: A,A3: set @ A,B2: B,F3: A > B] :
% 2.96/1.56 ( ( member @ A @ X @ A3 )
% 2.96/1.56 => ( ( B2
% 2.96/1.56 = ( F3 @ X ) )
% 2.96/1.56 => ( member @ B @ B2 @ ( image @ A @ B @ F3 @ A3 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_154_ball__imageD,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P3: A > $o] :
% 2.96/1.56 ( ! [X4: A] :
% 2.96/1.56 ( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A3 ) )
% 2.96/1.56 => ( P3 @ X4 ) )
% 2.96/1.56 => ! [X5: B] :
% 2.96/1.56 ( ( member @ B @ X5 @ A3 )
% 2.96/1.56 => ( P3 @ ( F3 @ X5 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_155_image__cong,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,M: set @ A,N3: set @ A,F3: A > B,G: A > B] :
% 2.96/1.56 ( ( M = N3 )
% 2.96/1.56 => ( ! [X4: A] :
% 2.96/1.56 ( ( member @ A @ X4 @ N3 )
% 2.96/1.56 => ( ( F3 @ X4 )
% 2.96/1.56 = ( G @ X4 ) ) )
% 2.96/1.56 => ( ( image @ A @ B @ F3 @ M )
% 2.96/1.56 = ( image @ A @ B @ G @ N3 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_156_bex__imageD,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P3: A > $o] :
% 2.96/1.56 ( ? [X5: A] :
% 2.96/1.56 ( ( member @ A @ X5 @ ( image @ B @ A @ F3 @ A3 ) )
% 2.96/1.56 & ( P3 @ X5 ) )
% 2.96/1.56 => ? [X4: B] :
% 2.96/1.56 ( ( member @ B @ X4 @ A3 )
% 2.96/1.56 & ( P3 @ ( F3 @ X4 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_157_image__iff,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,Z: A,F3: B > A,A3: set @ B] :
% 2.96/1.56 ( ( member @ A @ Z @ ( image @ B @ A @ F3 @ A3 ) )
% 2.96/1.56 = ( ? [X2: B] :
% 2.96/1.56 ( ( member @ B @ X2 @ A3 )
% 2.96/1.56 & ( Z
% 2.96/1.56 = ( F3 @ X2 ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_158_imageI,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,X: A,A3: set @ A,F3: A > B] :
% 2.96/1.56 ( ( member @ A @ X @ A3 )
% 2.96/1.56 => ( member @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A3 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_159_inj__on__image__iff,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,A3: set @ A,G: A > B,F3: A > A] :
% 2.96/1.56 ( ! [X4: A] :
% 2.96/1.56 ( ( member @ A @ X4 @ A3 )
% 2.96/1.56 => ! [Xa2: A] :
% 2.96/1.56 ( ( member @ A @ Xa2 @ A3 )
% 2.96/1.56 => ( ( ( G @ ( F3 @ X4 ) )
% 2.96/1.56 = ( G @ ( F3 @ Xa2 ) ) )
% 2.96/1.56 = ( ( G @ X4 )
% 2.96/1.56 = ( G @ Xa2 ) ) ) ) )
% 2.96/1.56 => ( ( inj_on @ A @ A @ F3 @ A3 )
% 2.96/1.56 => ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F3 @ A3 ) )
% 2.96/1.56 = ( inj_on @ A @ B @ G @ A3 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_160_surjD,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,F3: B > A,Y: A] :
% 2.96/1.56 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 2.96/1.56 = ( top_top @ ( set @ A ) ) )
% 2.96/1.56 => ? [X4: B] :
% 2.96/1.56 ( Y
% 2.96/1.56 = ( F3 @ X4 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_161_surjE,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,F3: B > A,Y: A] :
% 2.96/1.56 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 2.96/1.56 = ( top_top @ ( set @ A ) ) )
% 2.96/1.56 => ~ ! [X4: B] :
% 2.96/1.56 ( Y
% 2.96/1.56 != ( F3 @ X4 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_162_surjI,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,G: B > A,F3: A > B] :
% 2.96/1.56 ( ! [X4: A] :
% 2.96/1.56 ( ( G @ ( F3 @ X4 ) )
% 2.96/1.56 = X4 )
% 2.96/1.56 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 2.96/1.56 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_163_rangeI,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,F3: B > A,X: B] : ( member @ A @ ( F3 @ X ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_164_surj__def,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: B > A] :
% 2.96/1.56 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 2.96/1.56 = ( top_top @ ( set @ A ) ) )
% 2.96/1.56 = ( ! [Y4: A] :
% 2.96/1.56 ? [X2: B] :
% 2.96/1.56 ( Y4
% 2.96/1.56 = ( F3 @ X2 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_165_range__eqI,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,B2: A,F3: B > A,X: B] :
% 2.96/1.56 ( ( B2
% 2.96/1.56 = ( F3 @ X ) )
% 2.96/1.56 => ( member @ A @ B2 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_166_range__ex1__eq,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,B2: B] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 2.96/1.56 => ( ( member @ B @ B2 @ ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
% 2.96/1.56 = ( ? [X2: A] :
% 2.96/1.56 ( ( B2
% 2.96/1.56 = ( F3 @ X2 ) )
% 2.96/1.56 & ! [Y4: A] :
% 2.96/1.56 ( ( B2
% 2.96/1.56 = ( F3 @ Y4 ) )
% 2.96/1.56 => ( Y4 = X2 ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_167_inj__image__eq__iff,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B6: set @ A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 2.96/1.56 => ( ( ( image @ A @ B @ F3 @ A3 )
% 2.96/1.56 = ( image @ A @ B @ F3 @ B6 ) )
% 2.96/1.56 = ( A3 = B6 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_168_inj__image__mem__iff,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,A2: A,A3: set @ A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 2.96/1.56 => ( ( member @ B @ ( F3 @ A2 ) @ ( image @ A @ B @ F3 @ A3 ) )
% 2.96/1.56 = ( member @ A @ A2 @ A3 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_169_prod__fun__imageE,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F3: C > A,G: D > B,R2: set @ ( product_prod @ C @ D )] :
% 2.96/1.56 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G ) @ R2 ) )
% 2.96/1.56 => ~ ! [X4: C,Y2: D] :
% 2.96/1.56 ( ( C3
% 2.96/1.56 = ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G @ Y2 ) ) )
% 2.96/1.56 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X4 @ Y2 ) @ R2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_170_f__the__inv__into__f,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,F3: A > B,A3: set @ A,Y: B] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ A3 )
% 2.96/1.56 => ( ( member @ B @ Y @ ( image @ A @ B @ F3 @ A3 ) )
% 2.96/1.56 => ( ( F3 @ ( the_inv_into @ A @ B @ A3 @ F3 @ Y ) )
% 2.96/1.56 = Y ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_171_inj__on__the__inv__into,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ A3 )
% 2.96/1.56 => ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A3 @ F3 ) @ ( image @ A @ B @ F3 @ A3 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_172_total__inv__image,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,R: set @ ( product_prod @ B @ B )] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 2.96/1.56 => ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ R )
% 2.96/1.56 => ( total_on @ A @ ( top_top @ ( set @ A ) ) @ ( inv_image @ B @ A @ R @ F3 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_173_range__fst,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType] :
% 2.96/1.56 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.96/1.56 = ( top_top @ ( set @ A ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_174_range__snd,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType] :
% 2.96/1.56 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.96/1.56 = ( top_top @ ( set @ A ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_175_fst__map__prod,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > A,G: D > B,X: product_prod @ C @ D] :
% 2.96/1.56 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G @ X ) )
% 2.96/1.56 = ( F3 @ ( product_fst @ C @ D @ X ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_176_snd__map__prod,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > B,G: D > A,X: product_prod @ C @ D] :
% 2.96/1.56 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F3 @ G @ X ) )
% 2.96/1.56 = ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_177_apfst__eq__conv,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,C: $tType,F3: C > A,X: product_prod @ C @ B,G: C > A] :
% 2.96/1.56 ( ( ( product_apfst @ C @ A @ B @ F3 @ X )
% 2.96/1.56 = ( product_apfst @ C @ A @ B @ G @ X ) )
% 2.96/1.56 = ( ( F3 @ ( product_fst @ C @ B @ X ) )
% 2.96/1.56 = ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_178_fst__apfst,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,C: $tType,F3: C > A,X: product_prod @ C @ B] :
% 2.96/1.56 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F3 @ X ) )
% 2.96/1.56 = ( F3 @ ( product_fst @ C @ B @ X ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_179_snd__apfst,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,C: $tType,F3: C > B,X: product_prod @ C @ A] :
% 2.96/1.56 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F3 @ X ) )
% 2.96/1.56 = ( product_snd @ C @ A @ X ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_180_fst__apsnd,axiom,
% 2.96/1.56 ! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C] :
% 2.96/1.56 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F3 @ X ) )
% 2.96/1.56 = ( product_fst @ A @ C @ X ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_181_apsnd__eq__conv,axiom,
% 2.96/1.56 ! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C,G: C > B] :
% 2.96/1.56 ( ( ( product_apsnd @ C @ B @ A @ F3 @ X )
% 2.96/1.56 = ( product_apsnd @ C @ B @ A @ G @ X ) )
% 2.96/1.56 = ( ( F3 @ ( product_snd @ A @ C @ X ) )
% 2.96/1.56 = ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_182_snd__apsnd,axiom,
% 2.96/1.56 ! [A: $tType,C: $tType,B: $tType,F3: C > A,X: product_prod @ B @ C] :
% 2.96/1.56 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F3 @ X ) )
% 2.96/1.56 = ( F3 @ ( product_snd @ B @ C @ X ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_183_prod_Ocollapse,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.96/1.56 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 2.96/1.56 = Prod ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_184_snd__swap,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 2.96/1.56 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 2.96/1.56 = ( product_fst @ A @ B @ X ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_185_fst__swap,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 2.96/1.56 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 2.96/1.56 = ( product_snd @ B @ A @ X ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_186_apsnd__apfst,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: C > B,G: D > A,X: product_prod @ D @ C] :
% 2.96/1.56 ( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apfst @ D @ A @ C @ G @ X ) )
% 2.96/1.56 = ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F3 @ ( product_snd @ D @ C @ X ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_187_apfst__apsnd,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,D: $tType,C: $tType,F3: C > A,G: D > B,X: product_prod @ C @ D] :
% 2.96/1.56 ( ( product_apfst @ C @ A @ B @ F3 @ ( product_apsnd @ D @ B @ C @ G @ X ) )
% 2.96/1.56 = ( product_Pair @ A @ B @ ( F3 @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_188_prod_Oswap__def,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType] :
% 2.96/1.56 ( ( product_swap @ A @ B )
% 2.96/1.56 = ( ^ [P: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P ) @ ( product_fst @ A @ B @ P ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_189_fst__eqD,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
% 2.96/1.56 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 2.96/1.56 = A2 )
% 2.96/1.56 => ( X = A2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_190_snd__eqD,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
% 2.96/1.56 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 2.96/1.56 = A2 )
% 2.96/1.56 => ( Y = A2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_191_fst__conv,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,X1: A,X23: B] :
% 2.96/1.56 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X23 ) )
% 2.96/1.56 = X1 ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_192_snd__conv,axiom,
% 2.96/1.56 ! [Aa: $tType,A: $tType,X1: Aa,X23: A] :
% 2.96/1.56 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X23 ) )
% 2.96/1.56 = X23 ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_193_prod_Oexhaust__sel,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.96/1.56 ( Prod
% 2.96/1.56 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_194_surjective__pairing,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 2.96/1.56 ( T2
% 2.96/1.56 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_195_total__onI,axiom,
% 2.96/1.56 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A )] :
% 2.96/1.56 ( ! [X4: A,Y2: A] :
% 2.96/1.56 ( ( member @ A @ X4 @ A3 )
% 2.96/1.56 => ( ( member @ A @ Y2 @ A3 )
% 2.96/1.56 => ( ( X4 != Y2 )
% 2.96/1.56 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y2 ) @ R )
% 2.96/1.56 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X4 ) @ R ) ) ) ) )
% 2.96/1.56 => ( total_on @ A @ A3 @ R ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_196_total__on__def,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ( ( total_on @ A )
% 2.96/1.56 = ( ^ [A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 2.96/1.56 ! [X2: A] :
% 2.96/1.56 ( ( member @ A @ X2 @ A5 )
% 2.96/1.56 => ! [Y4: A] :
% 2.96/1.56 ( ( member @ A @ Y4 @ A5 )
% 2.96/1.56 => ( ( X2 != Y4 )
% 2.96/1.56 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R3 )
% 2.96/1.56 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R3 ) ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_197_prod__eqI,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q: product_prod @ A @ B] :
% 2.96/1.56 ( ( ( product_fst @ A @ B @ P2 )
% 2.96/1.56 = ( product_fst @ A @ B @ Q ) )
% 2.96/1.56 => ( ( ( product_snd @ A @ B @ P2 )
% 2.96/1.56 = ( product_snd @ A @ B @ Q ) )
% 2.96/1.56 => ( P2 = Q ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_198_prod_Oexpand,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 2.96/1.56 ( ( ( ( product_fst @ A @ B @ Prod )
% 2.96/1.56 = ( product_fst @ A @ B @ Prod2 ) )
% 2.96/1.56 & ( ( product_snd @ A @ B @ Prod )
% 2.96/1.56 = ( product_snd @ A @ B @ Prod2 ) ) )
% 2.96/1.56 => ( Prod = Prod2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_199_prod__eq__iff,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType] :
% 2.96/1.56 ( ( ^ [Y5: product_prod @ A @ B,Z2: product_prod @ A @ B] : ( Y5 = Z2 ) )
% 2.96/1.56 = ( ^ [S: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 2.96/1.56 ( ( ( product_fst @ A @ B @ S )
% 2.96/1.56 = ( product_fst @ A @ B @ T3 ) )
% 2.96/1.56 & ( ( product_snd @ A @ B @ S )
% 2.96/1.56 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_200_exI__realizer,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,P3: A > B > $o,Y: A,X: B] :
% 2.96/1.56 ( ( P3 @ Y @ X )
% 2.96/1.56 => ( P3 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_201_conjI__realizer,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,P3: A > $o,P2: A,Q2: B > $o,Q: B] :
% 2.96/1.56 ( ( P3 @ P2 )
% 2.96/1.56 => ( ( Q2 @ Q )
% 2.96/1.56 => ( ( P3 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) )
% 2.96/1.56 & ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_202_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,P3: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
% 2.96/1.56 ( ( P3 @ X @ Y )
% 2.96/1.56 => ( ( A2
% 2.96/1.56 = ( product_Pair @ A @ B @ X @ Y ) )
% 2.96/1.56 => ( P3 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_203_eq__snd__iff,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 2.96/1.56 ( ( B2
% 2.96/1.56 = ( product_snd @ B @ A @ P2 ) )
% 2.96/1.56 = ( ? [A7: B] :
% 2.96/1.56 ( P2
% 2.96/1.56 = ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_204_sndI,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
% 2.96/1.56 ( ( X
% 2.96/1.56 = ( product_Pair @ A @ B @ Y @ Z ) )
% 2.96/1.56 => ( ( product_snd @ A @ B @ X )
% 2.96/1.56 = Z ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_205_fstI,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
% 2.96/1.56 ( ( X
% 2.96/1.56 = ( product_Pair @ A @ B @ Y @ Z ) )
% 2.96/1.56 => ( ( product_fst @ A @ B @ X )
% 2.96/1.56 = Y ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_206_eq__fst__iff,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 2.96/1.56 ( ( A2
% 2.96/1.56 = ( product_fst @ A @ B @ P2 ) )
% 2.96/1.56 = ( ? [B5: B] :
% 2.96/1.56 ( P2
% 2.96/1.56 = ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_207_total__lex__prod,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
% 2.96/1.56 ( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R )
% 2.96/1.56 => ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ S3 )
% 2.96/1.56 => ( total_on @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( lex_prod @ A @ B @ R @ S3 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_208_sndOp__def,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,C: $tType] :
% 2.96/1.56 ( ( bNF_sndOp @ C @ A @ B )
% 2.96/1.56 = ( ^ [P5: C > A > $o,Q4: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P5 @ Q4 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_209_in__lex__prod,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
% 2.96/1.56 ( ( 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 ) )
% 2.96/1.56 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A6 ) @ R )
% 2.96/1.56 | ( ( A2 = A6 )
% 2.96/1.56 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B4 ) @ S3 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_210_fstOp__def,axiom,
% 2.96/1.56 ! [C: $tType,B: $tType,A: $tType] :
% 2.96/1.56 ( ( bNF_fstOp @ A @ B @ C )
% 2.96/1.56 = ( ^ [P5: A > B > $o,Q4: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P5 @ Q4 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_211_wf__map__prod__image,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F3: A > B] :
% 2.96/1.56 ( ( wf @ A @ R )
% 2.96/1.56 => ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 2.96/1.56 => ( wf @ B @ ( image @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( product_map_prod @ A @ B @ A @ B @ F3 @ F3 ) @ R ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_212_wf__induct__rule,axiom,
% 2.96/1.56 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P3: A > $o,A2: A] :
% 2.96/1.56 ( ( wf @ A @ R )
% 2.96/1.56 => ( ! [X4: A] :
% 2.96/1.56 ( ! [Y6: A] :
% 2.96/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X4 ) @ R )
% 2.96/1.56 => ( P3 @ Y6 ) )
% 2.96/1.56 => ( P3 @ X4 ) )
% 2.96/1.56 => ( P3 @ A2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_213_wf__eq__minimal,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ( ( wf @ A )
% 2.96/1.56 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 2.96/1.56 ! [Q4: set @ A] :
% 2.96/1.56 ( ? [X2: A] : ( member @ A @ X2 @ Q4 )
% 2.96/1.56 => ? [X2: A] :
% 2.96/1.56 ( ( member @ A @ X2 @ Q4 )
% 2.96/1.56 & ! [Y4: A] :
% 2.96/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R3 )
% 2.96/1.56 => ~ ( member @ A @ Y4 @ Q4 ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_214_wf__not__refl,axiom,
% 2.96/1.56 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
% 2.96/1.56 ( ( wf @ A @ R )
% 2.96/1.56 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_215_wf__not__sym,axiom,
% 2.96/1.56 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X: A] :
% 2.96/1.56 ( ( wf @ A @ R )
% 2.96/1.56 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X ) @ R )
% 2.96/1.56 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ R ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_216_wf__irrefl,axiom,
% 2.96/1.56 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
% 2.96/1.56 ( ( wf @ A @ R )
% 2.96/1.56 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_217_wf__induct,axiom,
% 2.96/1.56 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P3: A > $o,A2: A] :
% 2.96/1.56 ( ( wf @ A @ R )
% 2.96/1.56 => ( ! [X4: A] :
% 2.96/1.56 ( ! [Y6: A] :
% 2.96/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X4 ) @ R )
% 2.96/1.56 => ( P3 @ Y6 ) )
% 2.96/1.56 => ( P3 @ X4 ) )
% 2.96/1.56 => ( P3 @ A2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_218_wf__asym,axiom,
% 2.96/1.56 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X: A] :
% 2.96/1.56 ( ( wf @ A @ R )
% 2.96/1.56 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X ) @ R )
% 2.96/1.56 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ R ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_219_wfUNIVI,axiom,
% 2.96/1.56 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 2.96/1.56 ( ! [P6: A > $o,X4: A] :
% 2.96/1.56 ( ! [Xa3: A] :
% 2.96/1.56 ( ! [Y2: A] :
% 2.96/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa3 ) @ R )
% 2.96/1.56 => ( P6 @ Y2 ) )
% 2.96/1.56 => ( P6 @ Xa3 ) )
% 2.96/1.56 => ( P6 @ X4 ) )
% 2.96/1.56 => ( wf @ A @ R ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_220_wfI__min,axiom,
% 2.96/1.56 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 2.96/1.56 ( ! [X4: A,Q5: set @ A] :
% 2.96/1.56 ( ( member @ A @ X4 @ Q5 )
% 2.96/1.56 => ? [Xa3: A] :
% 2.96/1.56 ( ( member @ A @ Xa3 @ Q5 )
% 2.96/1.56 & ! [Y2: A] :
% 2.96/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa3 ) @ R2 )
% 2.96/1.56 => ~ ( member @ A @ Y2 @ Q5 ) ) ) )
% 2.96/1.56 => ( wf @ A @ R2 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_221_wfE__min,axiom,
% 2.96/1.56 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Q2: set @ A] :
% 2.96/1.56 ( ( wf @ A @ R2 )
% 2.96/1.56 => ( ( member @ A @ X @ Q2 )
% 2.96/1.56 => ~ ! [Z3: A] :
% 2.96/1.56 ( ( member @ A @ Z3 @ Q2 )
% 2.96/1.56 => ~ ! [Y6: A] :
% 2.96/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R2 )
% 2.96/1.56 => ~ ( member @ A @ Y6 @ Q2 ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_222_wf__def,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ( ( wf @ A )
% 2.96/1.56 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 2.96/1.56 ! [P5: A > $o] :
% 2.96/1.56 ( ! [X2: A] :
% 2.96/1.56 ( ! [Y4: A] :
% 2.96/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R3 )
% 2.96/1.56 => ( P5 @ Y4 ) )
% 2.96/1.56 => ( P5 @ X2 ) )
% 2.96/1.56 => ! [X6: A] : ( P5 @ X6 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_223_dependent__wf__choice,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P3: ( A > B ) > A > B > $o] :
% 2.96/1.56 ( ( wf @ A @ R2 )
% 2.96/1.56 => ( ! [F7: A > B,G3: A > B,X4: A,R4: B] :
% 2.96/1.56 ( ! [Z4: A] :
% 2.96/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X4 ) @ R2 )
% 2.96/1.56 => ( ( F7 @ Z4 )
% 2.96/1.56 = ( G3 @ Z4 ) ) )
% 2.96/1.56 => ( ( P3 @ F7 @ X4 @ R4 )
% 2.96/1.56 = ( P3 @ G3 @ X4 @ R4 ) ) )
% 2.96/1.56 => ( ! [X4: A,F7: A > B] :
% 2.96/1.56 ( ! [Y6: A] :
% 2.96/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X4 ) @ R2 )
% 2.96/1.56 => ( P3 @ F7 @ Y6 @ ( F7 @ Y6 ) ) )
% 2.96/1.56 => ? [X_1: B] : ( P3 @ F7 @ X4 @ X_1 ) )
% 2.96/1.56 => ? [F7: A > B] :
% 2.96/1.56 ! [X5: A] : ( P3 @ F7 @ X5 @ ( F7 @ X5 ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_224_same__fstI,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,P3: A > $o,X: A,Y7: B,Y: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 2.96/1.56 ( ( P3 @ X )
% 2.96/1.56 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R2 @ X ) )
% 2.96/1.56 => ( 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 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_225_image2__eqI,axiom,
% 2.96/1.56 ! [A: $tType,C: $tType,B: $tType,B2: A,F3: B > A,X: B,C3: C,G: B > C,A3: set @ B] :
% 2.96/1.56 ( ( B2
% 2.96/1.56 = ( F3 @ X ) )
% 2.96/1.56 => ( ( C3
% 2.96/1.56 = ( G @ X ) )
% 2.96/1.56 => ( ( member @ B @ X @ A3 )
% 2.96/1.56 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F3 @ G ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_226_the__inv__into__into,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X: B,B6: set @ A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ A3 )
% 2.96/1.56 => ( ( member @ B @ X @ ( image @ A @ B @ F3 @ A3 ) )
% 2.96/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 2.96/1.56 => ( member @ A @ ( the_inv_into @ A @ B @ A3 @ F3 @ X ) @ B6 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_227_order__refl,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ( ( preorder @ A )
% 2.96/1.56 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_228_subsetI,axiom,
% 2.96/1.56 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 2.96/1.56 ( ! [X4: A] :
% 2.96/1.56 ( ( member @ A @ X4 @ A3 )
% 2.96/1.56 => ( member @ A @ X4 @ B6 ) )
% 2.96/1.56 => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_229_subset__antisym,axiom,
% 2.96/1.56 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 2.96/1.56 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 2.96/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
% 2.96/1.56 => ( A3 = B6 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_230_inj__on__image__eq__iff,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,C4: set @ A,A3: set @ A,B6: set @ A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ C4 )
% 2.96/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ C4 )
% 2.96/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 2.96/1.56 => ( ( ( image @ A @ B @ F3 @ A3 )
% 2.96/1.56 = ( image @ A @ B @ F3 @ B6 ) )
% 2.96/1.56 = ( A3 = B6 ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_231_inj__on__image__mem__iff,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,B6: set @ A,A2: A,A3: set @ A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ B6 )
% 2.96/1.56 => ( ( member @ A @ A2 @ B6 )
% 2.96/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 2.96/1.56 => ( ( member @ B @ ( F3 @ A2 ) @ ( image @ A @ B @ F3 @ A3 ) )
% 2.96/1.56 = ( member @ A @ A2 @ A3 ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_232_top_Oextremum__uniqueI,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ( ( order_top @ A )
% 2.96/1.56 => ! [A2: A] :
% 2.96/1.56 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.96/1.56 => ( A2
% 2.96/1.56 = ( top_top @ A ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_233_top_Oextremum__unique,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ( ( order_top @ A )
% 2.96/1.56 => ! [A2: A] :
% 2.96/1.56 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.96/1.56 = ( A2
% 2.96/1.56 = ( top_top @ A ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_234_top__greatest,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ( ( order_top @ A )
% 2.96/1.56 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_235_subset__UNIV,axiom,
% 2.96/1.56 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_236_inj__on__subset,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B6: set @ A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ A3 )
% 2.96/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
% 2.96/1.56 => ( inj_on @ A @ B @ F3 @ B6 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_237_subset__inj__on,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType,F3: A > B,B6: set @ A,A3: set @ A] :
% 2.96/1.56 ( ( inj_on @ A @ B @ F3 @ B6 )
% 2.96/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 2.96/1.56 => ( inj_on @ A @ B @ F3 @ A3 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_238_in__mono,axiom,
% 2.96/1.56 ! [A: $tType,A3: set @ A,B6: set @ A,X: A] :
% 2.96/1.56 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 2.96/1.56 => ( ( member @ A @ X @ A3 )
% 2.96/1.56 => ( member @ A @ X @ B6 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_239_subsetD,axiom,
% 2.96/1.56 ! [A: $tType,A3: set @ A,B6: set @ A,C3: A] :
% 2.96/1.56 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 2.96/1.56 => ( ( member @ A @ C3 @ A3 )
% 2.96/1.56 => ( member @ A @ C3 @ B6 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_240_equalityE,axiom,
% 2.96/1.56 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 2.96/1.56 ( ( A3 = B6 )
% 2.96/1.56 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 2.96/1.56 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_241_subset__eq,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ( ( ord_less_eq @ ( set @ A ) )
% 2.96/1.56 = ( ^ [A5: set @ A,B7: set @ A] :
% 2.96/1.56 ! [X2: A] :
% 2.96/1.56 ( ( member @ A @ X2 @ A5 )
% 2.96/1.56 => ( member @ A @ X2 @ B7 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_242_equalityD1,axiom,
% 2.96/1.56 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 2.96/1.56 ( ( A3 = B6 )
% 2.96/1.56 => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_243_equalityD2,axiom,
% 2.96/1.56 ! [A: $tType,A3: set @ A,B6: set @ A] :
% 2.96/1.56 ( ( A3 = B6 )
% 2.96/1.56 => ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_244_subset__iff,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ( ( ord_less_eq @ ( set @ A ) )
% 2.96/1.56 = ( ^ [A5: set @ A,B7: set @ A] :
% 2.96/1.56 ! [T3: A] :
% 2.96/1.56 ( ( member @ A @ T3 @ A5 )
% 2.96/1.56 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_245_subset__refl,axiom,
% 2.96/1.56 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_246_Collect__mono,axiom,
% 2.96/1.56 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 2.96/1.56 ( ! [X4: A] :
% 2.96/1.56 ( ( P3 @ X4 )
% 2.96/1.56 => ( Q2 @ X4 ) )
% 2.96/1.56 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q2 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_247_subset__trans,axiom,
% 2.96/1.56 ! [A: $tType,A3: set @ A,B6: set @ A,C4: set @ A] :
% 2.96/1.56 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 2.96/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 2.96/1.56 => ( ord_less_eq @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_248_le__funD,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType] :
% 2.96/1.56 ( ( ord @ B )
% 2.96/1.56 => ! [F3: A > B,G: A > B,X: A] :
% 2.96/1.56 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 2.96/1.56 => ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_249_le__funE,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType] :
% 2.96/1.56 ( ( ord @ B )
% 2.96/1.56 => ! [F3: A > B,G: A > B,X: A] :
% 2.96/1.56 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 2.96/1.56 => ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_250_le__funI,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType] :
% 2.96/1.56 ( ( ord @ B )
% 2.96/1.56 => ! [F3: A > B,G: A > B] :
% 2.96/1.56 ( ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
% 2.96/1.56 => ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_251_set__eq__subset,axiom,
% 2.96/1.56 ! [A: $tType] :
% 2.96/1.56 ( ( ^ [Y5: set @ A,Z2: set @ A] : ( Y5 = Z2 ) )
% 2.96/1.56 = ( ^ [A5: set @ A,B7: set @ A] :
% 2.96/1.56 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B7 )
% 2.96/1.56 & ( ord_less_eq @ ( set @ A ) @ B7 @ A5 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_252_le__fun__def,axiom,
% 2.96/1.56 ! [B: $tType,A: $tType] :
% 2.96/1.56 ( ( ord @ B )
% 2.96/1.56 => ( ( ord_less_eq @ ( A > B ) )
% 2.96/1.56 = ( ^ [F: A > B,G4: A > B] :
% 2.96/1.56 ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G4 @ X2 ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_253_Collect__mono__iff,axiom,
% 2.96/1.56 ! [A: $tType,P3: A > $o,Q2: A > $o] :
% 2.96/1.56 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q2 ) )
% 2.96/1.56 = ( ! [X2: A] :
% 2.96/1.56 ( ( P3 @ X2 )
% 2.96/1.56 => ( Q2 @ X2 ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_254_order__subst1,axiom,
% 2.96/1.56 ! [A: $tType,B: $tType] :
% 2.96/1.56 ( ( ( order @ B )
% 2.96/1.56 & ( order @ A ) )
% 2.96/1.56 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 2.96/1.56 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 2.96/1.56 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 2.96/1.56 => ( ! [X4: B,Y2: B] :
% 2.96/1.56 ( ( ord_less_eq @ B @ X4 @ Y2 )
% 2.96/1.56 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y2 ) ) )
% 2.96/1.56 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(fact_255_order__subst2,axiom,
% 2.96/1.56 ! [A: $tType,C: $tType] :
% 2.96/1.56 ( ( ( order @ C )
% 2.96/1.56 & ( order @ A ) )
% 2.96/1.56 => ! [A2: A,B2: A,F3: A > C,C3: C] :
% 2.96/1.56 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.96/1.56 => ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
% 2.96/1.56 => ( ! [X4: A,Y2: A] :
% 2.96/1.56 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.96/1.56 => ( ord_less_eq @ C @ ( F3 @ X4 ) @ ( F3 @ Y2 ) ) )
% 2.96/1.56 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.96/1.56 ! [A8: $tType,A9: $tType] :
% 2.96/1.56 ( ( order_top @ A9 )
% 2.96/1.56 => ( order_top @ ( A8 > A9 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.96/1.56 ! [A8: $tType,A9: $tType] :
% 2.96/1.56 ( ( preorder @ A9 )
% 2.96/1.56 => ( preorder @ ( A8 > A9 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.96/1.56 ! [A8: $tType,A9: $tType] :
% 2.96/1.56 ( ( order @ A9 )
% 2.96/1.56 => ( order @ ( A8 > A9 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_fun___Orderings_Otop,axiom,
% 2.96/1.56 ! [A8: $tType,A9: $tType] :
% 2.96/1.56 ( ( top @ A9 )
% 2.96/1.56 => ( top @ ( A8 > A9 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_fun___Orderings_Oord,axiom,
% 2.96/1.56 ! [A8: $tType,A9: $tType] :
% 2.96/1.56 ( ( ord @ A9 )
% 2.96/1.56 => ( ord @ ( A8 > A9 ) ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.96/1.56 preorder @ nat ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.96/1.56 order @ nat ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.96/1.56 ord @ nat ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_Set_Oset___Orderings_Oorder__top_4,axiom,
% 2.96/1.56 ! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 2.96/1.56 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 2.96/1.56 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
% 2.96/1.56 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 2.96/1.56 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_HOL_Obool___Orderings_Oorder__top_9,axiom,
% 2.96/1.56 order_top @ $o ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 2.96/1.56 preorder @ $o ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 2.96/1.56 order @ $o ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_HOL_Obool___Orderings_Otop_12,axiom,
% 2.96/1.56 top @ $o ).
% 2.96/1.56
% 2.96/1.56 thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
% 2.96/1.56 ord @ $o ).
% 2.96/1.56
% 2.96/1.56 thf(help_If_3_1_T,axiom,
% 2.96/1.56 ! [P3: $o] :
% 2.96/1.56 ( ( P3 = $true )
% 2.96/1.56 | ( P3 = $false ) ) ).
% 2.96/1.56
% 2.96/1.56 thf(help_If_2_1_T,axiom,
% 2.96/1.56 ! [A: $tType,X: A,Y: A] :
% 2.96/1.56 ( ( if @ A @ $false @ X @ Y )
% 2.96/1.56 = Y ) ).
% 2.96/1.56
% 2.96/1.56 thf(help_If_1_1_T,axiom,
% 2.96/1.56 ! [A: $tType,X: A,Y: A] :
% 2.96/1.56 ( ( if @ A @ $true @ X @ Y )
% 2.96/1.56 = X ) ).
% 2.96/1.56
% 2.96/1.56 thf(conj_0,conjecture,
% 2.96/1.56 ( ( paraco876059933e_eval @ ( paraco753463838ge_int @ f @ i ) @ p2 )
% 2.96/1.56 != ( paraco2040174112le_Det @ $true ) ) ).
% 2.96/1.56 ------------------------
% 2.96/1.59 % (24582)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24296 for (2999ds/183Mi)
% 2.96/1.59 % (24583)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_24296 for (2999ds/4Mi)
% 2.96/1.59 % (24584)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_24296 for (2999ds/27Mi)
% 2.96/1.59 % (24583)Instruction limit reached!
% 2.96/1.59 % (24583)------------------------------
% 2.96/1.59 % (24583)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.59 % (24583)Termination reason: Unknown
% 2.96/1.59 % (24583)Termination phase: shuffling
% 2.96/1.59
% 2.96/1.59 % (24583)Memory used [KB]: 1407
% 2.96/1.59 % (24583)Time elapsed: 0.003 s
% 2.96/1.59 % (24583)Instructions burned: 4 (million)
% 2.96/1.59 % (24583)------------------------------
% 2.96/1.59 % (24583)------------------------------
% 3.04/1.60 % (24586)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_24296 for (2999ds/2Mi)
% 3.04/1.60 % (24587)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_24296 for (2999ds/275Mi)
% 3.04/1.60 % (24588)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24296 for (2999ds/18Mi)
% 3.04/1.60 % (24585)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24296 for (2999ds/2Mi)
% 3.04/1.60 % (24585)Instruction limit reached!
% 3.04/1.60 % (24585)------------------------------
% 3.04/1.60 % (24585)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.60 % (24585)Termination reason: Unknown
% 3.04/1.60 % (24585)Termination phase: shuffling
% 3.04/1.60
% 3.04/1.60 % (24585)Memory used [KB]: 1407
% 3.04/1.60 % (24585)Time elapsed: 0.003 s
% 3.04/1.60 % (24585)Instructions burned: 2 (million)
% 3.04/1.60 % (24585)------------------------------
% 3.04/1.60 % (24585)------------------------------
% 3.04/1.60 % (24586)Instruction limit reached!
% 3.04/1.60 % (24586)------------------------------
% 3.04/1.60 % (24586)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.60 % (24586)Termination reason: Unknown
% 3.04/1.60 % (24586)Termination phase: shuffling
% 3.04/1.60
% 3.04/1.60 % (24586)Memory used [KB]: 1407
% 3.04/1.60 % (24586)Time elapsed: 0.004 s
% 3.04/1.60 % (24586)Instructions burned: 3 (million)
% 3.04/1.60 % (24586)------------------------------
% 3.04/1.60 % (24586)------------------------------
% 3.04/1.60 % (24584)Instruction limit reached!
% 3.04/1.60 % (24584)------------------------------
% 3.04/1.60 % (24584)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.60 % (24584)Termination reason: Unknown
% 3.04/1.60 % (24584)Termination phase: Property scanning
% 3.04/1.60
% 3.04/1.60 % (24584)Memory used [KB]: 1918
% 3.04/1.60 % (24584)Time elapsed: 0.011 s
% 3.04/1.60 % (24584)Instructions burned: 30 (million)
% 3.04/1.60 % (24584)------------------------------
% 3.04/1.60 % (24584)------------------------------
% 3.04/1.60 % (24589)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24296 for (2999ds/3Mi)
% 3.04/1.61 % (24589)Instruction limit reached!
% 3.04/1.61 % (24589)------------------------------
% 3.04/1.61 % (24589)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.61 % (24589)Termination reason: Unknown
% 3.04/1.61 % (24589)Termination phase: shuffling
% 3.04/1.61
% 3.04/1.61 % (24589)Memory used [KB]: 1535
% 3.04/1.61 % (24589)Time elapsed: 0.003 s
% 3.04/1.61 % (24589)Instructions burned: 5 (million)
% 3.04/1.61 % (24589)------------------------------
% 3.04/1.61 % (24589)------------------------------
% 3.04/1.61 % (24592)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_24296 for (2999ds/3Mi)
% 3.04/1.61 % (24588)Instruction limit reached!
% 3.04/1.61 % (24588)------------------------------
% 3.04/1.61 % (24588)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.61 % (24588)Termination reason: Unknown
% 3.04/1.61 % (24588)Termination phase: shuffling
% 3.04/1.61
% 3.04/1.61 % (24588)Memory used [KB]: 1791
% 3.04/1.61 % (24588)Time elapsed: 0.014 s
% 3.04/1.61 % (24588)Instructions burned: 19 (million)
% 3.04/1.61 % (24588)------------------------------
% 3.04/1.61 % (24588)------------------------------
% 3.04/1.61 % (24592)Instruction limit reached!
% 3.04/1.61 % (24592)------------------------------
% 3.04/1.61 % (24592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.61 % (24592)Termination reason: Unknown
% 3.04/1.61 % (24592)Termination phase: shuffling
% 3.04/1.61
% 3.04/1.61 % (24592)Memory used [KB]: 1407
% 3.04/1.61 % (24592)Time elapsed: 0.002 s
% 3.04/1.61 % (24592)Instructions burned: 3 (million)
% 3.04/1.61 % (24592)------------------------------
% 3.04/1.61 % (24592)------------------------------
% 3.04/1.62 % (24593)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_24296 for (2999ds/1041Mi)
% 3.04/1.62 % (24590)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_24296 for (2999ds/37Mi)
% 3.04/1.62 % (24591)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_24296 for (2999ds/15Mi)
% 3.04/1.62 % (24595)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_24296 for (2999ds/16Mi)
% 3.04/1.63 % (24594)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_24296 for (2999ds/7Mi)
% 3.04/1.63 % (24595)Instruction limit reached!
% 3.04/1.63 % (24595)------------------------------
% 3.04/1.63 % (24595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.63 % (24595)Termination reason: Unknown
% 3.04/1.63 % (24595)Termination phase: shuffling
% 3.04/1.63
% 3.04/1.63 % (24595)Memory used [KB]: 1791
% 3.04/1.63 % (24595)Time elapsed: 0.007 s
% 3.04/1.63 % (24595)Instructions burned: 18 (million)
% 3.04/1.63 % (24595)------------------------------
% 3.04/1.63 % (24595)------------------------------
% 3.04/1.63 % (24591)Instruction limit reached!
% 3.04/1.63 % (24591)------------------------------
% 3.04/1.63 % (24591)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.63 % (24591)Termination reason: Unknown
% 3.04/1.63 % (24591)Termination phase: shuffling
% 3.04/1.63
% 3.04/1.63 % (24591)Memory used [KB]: 1791
% 3.04/1.63 % (24591)Time elapsed: 0.012 s
% 3.04/1.63 % (24591)Instructions burned: 16 (million)
% 3.04/1.63 % (24591)------------------------------
% 3.04/1.63 % (24591)------------------------------
% 3.04/1.63 % (24587)First to succeed.
% 3.04/1.63 % (24594)Instruction limit reached!
% 3.04/1.63 % (24594)------------------------------
% 3.04/1.63 % (24594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.63 % (24594)Termination reason: Unknown
% 3.04/1.63 % (24594)Termination phase: shuffling
% 3.04/1.63
% 3.04/1.63 % (24594)Memory used [KB]: 1663
% 3.04/1.63 % (24594)Time elapsed: 0.006 s
% 3.04/1.63 % (24594)Instructions burned: 9 (million)
% 3.04/1.63 % (24594)------------------------------
% 3.04/1.63 % (24594)------------------------------
% 3.04/1.63 % (24590)Instruction limit reached!
% 3.04/1.63 % (24590)------------------------------
% 3.04/1.63 % (24590)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.63 % (24590)Termination reason: Unknown
% 3.04/1.63 % (24590)Termination phase: shuffling
% 3.04/1.63
% 3.04/1.63 % (24590)Memory used [KB]: 2174
% 3.04/1.63 % (24590)Time elapsed: 0.016 s
% 3.04/1.63 % (24590)Instructions burned: 38 (million)
% 3.04/1.63 % (24590)------------------------------
% 3.04/1.63 % (24590)------------------------------
% 3.04/1.64 % (24587)Refutation found. Thanks to Tanya!
% 3.04/1.64 % SZS status Theorem for DTF2THF_24296
% 3.04/1.64 % SZS output start Proof for DTF2THF_24296
% 3.04/1.64 thf(type_def_6, type, product_prod: ($tType * $tType) > $tType).
% 3.04/1.64 thf(type_def_7, type, set: ($tType) > $tType).
% 3.04/1.64 thf(type_def_8, type, nat: $tType).
% 3.04/1.64 thf(type_def_9, type, char: $tType).
% 3.04/1.64 thf(type_def_10, type, list: ($tType) > $tType).
% 3.04/1.64 thf(type_def_11, type, paraco415392788lle_tv: $tType).
% 3.04/1.64 thf(type_def_12, type, paraco414474393lle_fm: $tType).
% 3.04/1.64 thf(func_def_0, type, paraco415392788lle_tv: $tType).
% 3.04/1.64 thf(func_def_1, type, paraco414474393lle_fm: $tType).
% 3.04/1.64 thf(func_def_2, type, product_prod: ($tType * $tType) > $tType).
% 3.04/1.64 thf(func_def_3, type, char: $tType).
% 3.04/1.64 thf(func_def_4, type, list: ($tType) > $tType).
% 3.04/1.64 thf(func_def_5, type, set: ($tType) > $tType).
% 3.04/1.64 thf(func_def_6, type, nat: $tType).
% 3.04/1.64 thf(func_def_7, type, type: !>[X0: $tType]:($o)).
% 3.04/1.64 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 3.04/1.64 thf(func_def_9, type, top: !>[X0: $tType]:($o)).
% 3.04/1.64 thf(func_def_10, type, order: !>[X0: $tType]:($o)).
% 3.04/1.64 thf(func_def_11, type, preorder: !>[X0: $tType]:($o)).
% 3.04/1.64 thf(func_def_12, type, order_top: !>[X0: $tType]:($o)).
% 3.04/1.64 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)).
% 3.04/1.64 thf(func_def_14, type, bNF_pick_middlep: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > X1)).
% 3.04/1.64 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)).
% 3.04/1.64 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)).
% 3.04/1.64 thf(func_def_17, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 3.04/1.64 thf(func_def_18, type, the_inv_into: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X1 > X0)).
% 3.04/1.64 thf(func_def_19, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.04/1.64 thf(func_def_20, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.04/1.64 thf(func_def_21, type, top_top: !>[X0: $tType]:(X0)).
% 3.04/1.64 thf(func_def_22, type, paraco753463838ge_int: (nat > nat) > (list @ char > paraco415392788lle_tv) > list @ char > paraco415392788lle_tv).
% 3.04/1.64 thf(func_def_23, type, paraco1920534163nge_tv: (nat > nat) > paraco415392788lle_tv > paraco415392788lle_tv).
% 3.04/1.64 thf(func_def_24, type, paraco2077297190tv_rel: product_prod @ (nat > nat) @ paraco415392788lle_tv > product_prod @ (nat > nat) @ paraco415392788lle_tv > $o).
% 3.04/1.64 thf(func_def_25, type, paraco876059933e_eval: (list @ char > paraco415392788lle_tv) > paraco414474393lle_fm > paraco415392788lle_tv).
% 3.04/1.64 thf(func_def_26, type, paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm).
% 3.04/1.64 thf(func_def_27, type, paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm).
% 3.04/1.64 thf(func_def_28, type, paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm).
% 3.04/1.64 thf(func_def_29, type, paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm).
% 3.04/1.64 thf(func_def_30, type, paraco27778325le_Pro: list @ char > paraco414474393lle_fm).
% 3.04/1.64 thf(func_def_31, type, paraco251304083_Truth: paraco414474393lle_fm).
% 3.04/1.64 thf(func_def_32, 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)).
% 3.04/1.64 thf(func_def_33, type, paraco2040174112le_Det: $o > paraco415392788lle_tv).
% 3.04/1.64 thf(func_def_34, type, paraco676387099_Indet: nat > paraco415392788lle_tv).
% 3.04/1.64 thf(func_def_35, type, paraco490622181ase_tv: !>[X0: $tType]:(($o > X0) > (nat > X0) > paraco415392788lle_tv > X0)).
% 3.04/1.64 thf(func_def_36, type, paraco152590079rec_tv: !>[X0: $tType]:(($o > X0) > (nat > X0) > paraco415392788lle_tv > X0)).
% 3.04/1.64 thf(func_def_37, type, paraco769098683_valid: paraco414474393lle_fm > $o).
% 3.04/1.64 thf(func_def_38, type, paraco2086025920lid_in: set @ nat > paraco414474393lle_fm > $o).
% 3.04/1.64 thf(func_def_39, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.04/1.64 thf(func_def_40, type, product_apfst: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 3.04/1.64 thf(func_def_41, type, product_apsnd: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X2 @ X0 > product_prod @ X2 @ X1)).
% 3.04/1.64 thf(func_def_42, type, product_case_bool: !>[X0: $tType]:(X0 > X0 > $o > X0)).
% 3.04/1.64 thf(func_def_43, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 3.04/1.64 thf(func_def_44, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.04/1.64 thf(func_def_45, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 3.04/1.64 thf(func_def_46, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.04/1.64 thf(func_def_47, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.04/1.64 thf(func_def_48, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.04/1.64 thf(func_def_49, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 3.04/1.64 thf(func_def_50, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 3.04/1.64 thf(func_def_51, type, inv_imagep: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > X1 > X1 > $o)).
% 3.04/1.64 thf(func_def_52, type, total_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 3.04/1.64 thf(func_def_53, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.04/1.64 thf(func_def_54, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 3.04/1.64 thf(func_def_55, type, accp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > $o)).
% 3.04/1.64 thf(func_def_56, 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)).
% 3.04/1.64 thf(func_def_57, type, wf: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 3.04/1.64 thf(func_def_58, 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)).
% 3.04/1.64 thf(func_def_59, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.04/1.64 thf(func_def_60, type, f: nat > nat).
% 3.04/1.64 thf(func_def_61, type, i: list @ char > paraco415392788lle_tv).
% 3.04/1.64 thf(func_def_62, type, p1: paraco414474393lle_fm).
% 3.04/1.64 thf(func_def_63, type, p2: paraco414474393lle_fm).
% 3.04/1.64 thf(func_def_79, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.04/1.64 thf(func_def_80, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.04/1.64 thf(func_def_81, type, sK2: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.04/1.64 thf(func_def_82, type, sK3: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.04/1.64 thf(func_def_83, type, sK4: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.04/1.64 thf(func_def_84, type, sK5: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 3.04/1.64 thf(func_def_85, type, sK6: (paraco415392788lle_tv > $o) > nat).
% 3.04/1.64 thf(func_def_86, type, sK7: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X1)).
% 3.04/1.64 thf(func_def_87, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X1)).
% 3.04/1.64 thf(func_def_88, type, sK9: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > set @ X1 > X1)).
% 3.04/1.64 thf(func_def_89, type, sK10: paraco415392788lle_tv > paraco415392788lle_tv > $o).
% 3.04/1.64 thf(func_def_90, type, sK11: paraco415392788lle_tv > paraco415392788lle_tv > (nat > nat) > nat).
% 3.04/1.64 thf(func_def_91, type, sK12: !>[X0: $tType]:(X0)).
% 3.04/1.64 thf(func_def_92, type, sK13: !>[X0: $tType]:(set @ X0 > X0)).
% 3.04/1.64 thf(func_def_93, type, sK14: paraco415392788lle_tv > nat).
% 3.04/1.64 thf(func_def_95, type, ph16: !>[X0: $tType]:(X0)).
% 3.04/1.64 thf(f822,plain,(
% 3.04/1.64 $false),
% 3.04/1.64 inference(trivial_inequality_removal,[status(thm)],[f821])).
% 3.04/1.64 thf(f821,plain,(
% 3.04/1.64 ($true = $false)),
% 3.04/1.64 inference(equality_resolution,[status(thm)],[f816])).
% 3.04/1.64 thf(f816,plain,(
% 3.04/1.64 ( ! [X0 : $o] : (((paraco2040174112le_Det @ X0) != (paraco2040174112le_Det @ $false)) | ($true = X0)) )),
% 3.04/1.64 inference(superposition,[status(thm)],[f712,f813])).
% 3.04/1.64 thf(f813,plain,(
% 3.04/1.64 ((paraco2040174112le_Det @ $true) = (paraco2040174112le_Det @ $false))),
% 3.04/1.64 inference(forward_demodulation,[status(thm)],[f812,f811])).
% 3.04/1.64 thf(f811,plain,(
% 3.04/1.64 ((paraco876059933e_eval @ (^[Y0 : list @ char]: (paraco1920534163nge_tv @ f @ (i @ Y0))) @ p2) = (paraco2040174112le_Det @ $false))),
% 3.04/1.64 inference(beta_eta_normalization,[status(thm)],[f786])).
% 3.04/1.64 thf(f786,plain,(
% 3.04/1.64 ((paraco876059933e_eval @ ((^[Y0 : nat > nat]: ((^[Y1 : list @ char > paraco415392788lle_tv]: ((^[Y2 : list @ char]: (paraco1920534163nge_tv @ Y0 @ (Y1 @ Y2))))))) @ f @ i) @ p2) = (paraco2040174112le_Det @ $false))),
% 3.04/1.64 inference(definition_unfolding,[status(thm)],[f718,f764])).
% 3.04/1.64 thf(f764,plain,(
% 3.04/1.64 (paraco753463838ge_int = (^[Y0 : nat > nat]: ((^[Y1 : list @ char > paraco415392788lle_tv]: ((^[Y2 : list @ char]: (paraco1920534163nge_tv @ Y0 @ (Y1 @ Y2))))))))),
% 3.04/1.64 inference(cnf_transformation,[status(thm)],[f520])).
% 3.04/1.64 thf(f520,plain,(
% 3.04/1.64 (paraco753463838ge_int = (^[Y0 : nat > nat]: ((^[Y1 : list @ char > paraco415392788lle_tv]: ((^[Y2 : list @ char]: (paraco1920534163nge_tv @ Y0 @ (Y1 @ Y2))))))))),
% 3.04/1.64 inference(fool_elimination,[status(thm)],[f519])).
% 3.04/1.64 thf(f519,plain,(
% 3.04/1.64 (paraco753463838ge_int = (^[X0 : nat > nat, X1 : list @ char > paraco415392788lle_tv, X2 : list @ char] : (paraco1920534163nge_tv @ X0 @ (X1 @ X2))))),
% 3.04/1.64 inference(rectify,[status(thm)],[f19])).
% 3.04/1.64 thf(f19,axiom,(
% 3.04/1.64 (paraco753463838ge_int = (^[X14 : nat > nat, X8 : list @ char > paraco415392788lle_tv, X15 : list @ char] : (paraco1920534163nge_tv @ X14 @ (X8 @ X15))))),
% 3.04/1.64 file('/export/starexec/sandbox/tmp/tmp.xGjE5LYYhO/DTF2THF_24296.p',fact_18_change__int__def)).
% 3.04/1.64 thf(f718,plain,(
% 3.04/1.64 ((paraco876059933e_eval @ (paraco753463838ge_int @ f @ i) @ p2) = (paraco2040174112le_Det @ $false))),
% 3.04/1.64 inference(cnf_transformation,[status(thm)],[f557])).
% 3.04/1.64 thf(f557,plain,(
% 3.04/1.64 ((paraco876059933e_eval @ (paraco753463838ge_int @ f @ i) @ p2) = (paraco2040174112le_Det @ $false))),
% 3.04/1.64 inference(fool_elimination,[status(thm)],[f7])).
% 3.04/1.64 thf(f7,axiom,(
% 3.04/1.64 ((paraco2040174112le_Det @ $false) = (paraco876059933e_eval @ (paraco753463838ge_int @ f @ i) @ p2))),
% 3.04/1.64 file('/export/starexec/sandbox/tmp/tmp.xGjE5LYYhO/DTF2THF_24296.p',fact_6_b)).
% 3.04/1.64 thf(f812,plain,(
% 3.04/1.64 ((paraco876059933e_eval @ (^[Y0 : list @ char]: (paraco1920534163nge_tv @ f @ (i @ Y0))) @ p2) = (paraco2040174112le_Det @ $true))),
% 3.04/1.64 inference(beta_eta_normalization,[status(thm)],[f790])).
% 3.04/1.64 thf(f790,plain,(
% 3.04/1.64 ((paraco876059933e_eval @ ((^[Y0 : nat > nat]: ((^[Y1 : list @ char > paraco415392788lle_tv]: ((^[Y2 : list @ char]: (paraco1920534163nge_tv @ Y0 @ (Y1 @ Y2))))))) @ f @ i) @ p2) = (paraco2040174112le_Det @ $true))),
% 3.04/1.64 inference(definition_unfolding,[status(thm)],[f778,f764])).
% 3.04/1.64 thf(f778,plain,(
% 3.04/1.64 ((paraco876059933e_eval @ (paraco753463838ge_int @ f @ i) @ p2) = (paraco2040174112le_Det @ $true))),
% 3.04/1.64 inference(cnf_transformation,[status(thm)],[f643])).
% 3.04/1.64 thf(f643,plain,(
% 3.04/1.64 ((paraco876059933e_eval @ (paraco753463838ge_int @ f @ i) @ p2) = (paraco2040174112le_Det @ $true))),
% 3.04/1.64 inference(flattening,[status(thm)],[f296])).
% 3.04/1.64 thf(f296,plain,(
% 3.04/1.64 ~((paraco876059933e_eval @ (paraco753463838ge_int @ f @ i) @ p2) != (paraco2040174112le_Det @ $true))),
% 3.04/1.64 inference(fool_elimination,[status(thm)],[f279])).
% 3.04/1.64 thf(f279,negated_conjecture,(
% 3.04/1.64 ~((paraco2040174112le_Det @ $true) != (paraco876059933e_eval @ (paraco753463838ge_int @ f @ i) @ p2))),
% 3.04/1.64 inference(negated_conjecture,[status(cth)],[f278])).
% 3.04/1.64 thf(f278,conjecture,(
% 3.04/1.64 ((paraco2040174112le_Det @ $true) != (paraco876059933e_eval @ (paraco753463838ge_int @ f @ i) @ p2))),
% 3.04/1.64 file('/export/starexec/sandbox/tmp/tmp.xGjE5LYYhO/DTF2THF_24296.p',conj_0)).
% 3.04/1.64 thf(f712,plain,(
% 3.04/1.64 ( ! [X0 : $o,X1 : $o] : (((paraco2040174112le_Det @ X0) != (paraco2040174112le_Det @ X1)) | (X0 = X1)) )),
% 3.04/1.64 inference(cnf_transformation,[status(thm)],[f680])).
% 3.04/1.64 thf(f680,plain,(
% 3.04/1.64 ! [X0 : $o,X1 : $o] : ((((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1)) | (X0 != X1)) & ((X0 = X1) | ((paraco2040174112le_Det @ X0) != (paraco2040174112le_Det @ X1))))),
% 3.04/1.64 inference(nnf_transformation,[status(thm)],[f480])).
% 3.04/1.64 thf(f480,plain,(
% 3.04/1.64 ! [X0 : $o,X1 : $o] : (((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1)) <=> (X0 = X1))),
% 3.04/1.64 inference(fool_elimination,[status(thm)],[f479])).
% 3.04/1.64 thf(f479,plain,(
% 3.04/1.64 ! [X0 : $o,X1 : $o] : ((X0 = X1) = ((paraco2040174112le_Det @ X0) = (paraco2040174112le_Det @ X1)))),
% 3.04/1.64 inference(rectify,[status(thm)],[f9])).
% 3.04/1.64 thf(f9,axiom,(
% 3.04/1.64 ! [X5 : $o,X6 : $o] : ((X5 = X6) = ((paraco2040174112le_Det @ X5) = (paraco2040174112le_Det @ X6)))),
% 3.04/1.64 file('/export/starexec/sandbox/tmp/tmp.xGjE5LYYhO/DTF2THF_24296.p',fact_8_tv_Oinject_I1_J)).
% 3.04/1.64 % SZS output end Proof for DTF2THF_24296
% 3.04/1.64 % (24587)------------------------------
% 3.04/1.64 % (24587)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.64 % (24587)Termination reason: Refutation
% 3.04/1.64
% 3.04/1.64 % (24587)Memory used [KB]: 6524
% 3.04/1.64 % (24587)Time elapsed: 0.034 s
% 3.04/1.64 % (24587)Instructions burned: 51 (million)
% 3.04/1.64 % (24587)------------------------------
% 3.04/1.64 % (24587)------------------------------
% 3.04/1.64 % (24581)Success in time 0.066 s
%------------------------------------------------------------------------------