%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT150^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n018.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 08:44:56 AM UTC 2026
% Result : Theorem 5.35s 2.11s
% Output : Refutation 5.35s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : DAT150^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.09 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n018.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Tue Feb 24 06:09:01 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.38 ---- Original DTF file ---
% 0.16/0.38 thf(spec,logic,$$dhol).
% 0.16/0.38 %------------------------------------------------------------------------------
% 0.16/0.38 % File : DAT150^1 : TPTP v9.2.1. Released v7.0.0.
% 0.16/0.38 % Domain : Data Structures
% 0.16/0.38 % Problem : Coinductive stream 331
% 0.16/0.38 % Version : [Bla16] axioms : Especial.
% 0.16/0.38 % English :
% 0.16/0.38
% 0.16/0.38 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.16/0.38 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.16/0.38 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.16/0.38 % Source : [Bla16]
% 0.16/0.38 % Names : coinductive_stream__331.p [Bla16]
% 0.16/0.38
% 0.16/0.38 % Status : Theorem
% 0.16/0.38 % Rating : 0.33 v9.1.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.16/0.38 % Syntax : Number of formulae : 381 ( 142 unt; 68 typ; 0 def)
% 0.16/0.38 % Number of atoms : 839 ( 246 equ; 0 cnn)
% 0.16/0.38 % Maximal formula atoms : 13 ( 2 avg)
% 0.16/0.38 % Number of connectives : 3986 ( 56 ~; 1 |; 31 &;3476 @)
% 0.16/0.38 % ( 0 <=>; 422 =>; 0 <=; 0 <~>)
% 0.16/0.38 % Maximal formula depth : 19 ( 8 avg)
% 0.16/0.38 % Number of types : 3 ( 2 usr)
% 0.16/0.38 % Number of type conns : 299 ( 299 >; 0 *; 0 +; 0 <<)
% 0.16/0.38 % Number of symbols : 68 ( 66 usr; 1 con; 0-6 aty)
% 0.16/0.38 % Number of variables : 1113 ( 40 ^; 980 !; 25 ?;1113 :)
% 0.16/0.38 % ( 68 !>; 0 ?*; 0 @-; 0 @+)
% 0.16/0.38 % SPC : TH1_THM_EQU_NAR
% 0.16/0.38
% 0.16/0.38 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.16/0.38 % 2016-07-13 15:14:17.934
% 0.16/0.38 %------------------------------------------------------------------------------
% 0.16/0.38 %----Could-be-implicit typings (6)
% 0.16/0.38 thf(ty_t_Coinductive__List_Ollist,type,
% 0.16/0.38 coinductive_llist: $tType > $tType ).
% 0.16/0.38
% 0.16/0.38 thf(ty_t_Stream_Ostream,type,
% 0.16/0.38 stream: $tType > $tType ).
% 0.16/0.38
% 0.16/0.38 thf(ty_t_Set_Oset,type,
% 0.16/0.38 set: $tType > $tType ).
% 0.16/0.38
% 0.16/0.38 thf(ty_t_Nat_Onat,type,
% 0.16/0.38 nat: $tType ).
% 0.16/0.38
% 0.16/0.38 thf(ty_t_itself,type,
% 0.16/0.38 itself: $tType > $tType ).
% 0.16/0.38
% 0.16/0.38 thf(ty_tf_a,type,
% 0.16/0.38 a: $tType ).
% 0.16/0.38
% 0.16/0.38 %----Explicit typings (62)
% 0.16/0.38 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.16/0.38 finite_finite:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_HOL_Otype,type,
% 0.16/0.38 type:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Groups_Ozero,type,
% 0.16/0.38 zero:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Orderings_Obot,type,
% 0.16/0.38 bot:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Orderings_Oord,type,
% 0.16/0.38 ord:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Orderings_Otop,type,
% 0.16/0.38 top:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Orderings_Oorder,type,
% 0.16/0.38 order:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Orderings_Ono__bot,type,
% 0.16/0.38 no_bot:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Groups_Omonoid__add,type,
% 0.16/0.38 monoid_add:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Orderings_Olinorder,type,
% 0.16/0.38 linorder:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Orderings_Opreorder,type,
% 0.16/0.38 preorder:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Countable_Ocountable,type,
% 0.16/0.38 countable:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.16/0.38 order_bot:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Orderings_Oorder__top,type,
% 0.16/0.38 order_top:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Complete__Lattices_OSup,type,
% 0.16/0.38 complete_Sup:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.16/0.38 linordered_semidom:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Complete__Partial__Order_Occpo,type,
% 0.16/0.38 comple1141879883l_ccpo:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.16/0.38 ordere779506340up_add:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.16/0.38 comple187826305attice:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.16/0.38 canoni770627133id_add:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice,type,
% 0.16/0.38 counta840220525attice:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
% 0.16/0.38 condit378418413attice:
% 0.16/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Coinductive__List_Ogen__lset,type,
% 0.16/0.38 coinductive_gen_lset:
% 0.16/0.38 !>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Coinductive__List_Oinf__llist,type,
% 0.16/0.38 coindu68654304_llist:
% 0.16/0.38 !>[A: $tType] : ( ( nat > A ) > ( coinductive_llist @ A ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.16/0.38 coinductive_lfinite:
% 0.16/0.38 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Coinductive__List_Ollist_Olmap,type,
% 0.16/0.38 coinductive_lmap:
% 0.16/0.38 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( coinductive_llist @ A ) > ( coinductive_llist @ Aa ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 0.16/0.38 coinductive_lset:
% 0.16/0.38 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Coinductive__List_Ollist_Opred__llist,type,
% 0.16/0.38 coindu543516966_llist:
% 0.16/0.38 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Coinductive__List_Olmember,type,
% 0.16/0.38 coinductive_lmember:
% 0.16/0.38 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Coinductive__List_Olnth,type,
% 0.16/0.38 coinductive_lnth:
% 0.16/0.38 !>[A: $tType] : ( ( coinductive_llist @ A ) > nat > A ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Coinductive__List_Omonoid__add__class_Ollistsum,type,
% 0.16/0.38 coindu780009021istsum:
% 0.16/0.38 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ollist__of__stream,type,
% 0.16/0.38 coindu1724414836stream:
% 0.16/0.38 !>[A: $tType] : ( ( stream @ A ) > ( coinductive_llist @ A ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Oof__seq,type,
% 0.16/0.38 coindu176146587of_seq:
% 0.16/0.38 !>[A: $tType] : ( ( nat > A ) > ( stream @ A ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Ocr__stream,type,
% 0.16/0.38 coindu1183105481stream:
% 0.16/0.38 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__of__llist,type,
% 0.16/0.38 coindu2010755910_llist:
% 0.16/0.38 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 0.16/0.38 complete_Sup_Sup:
% 0.16/0.38 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Complete__Partial__Order2_Ocompact,type,
% 0.16/0.38 comple2143767107ompact:
% 0.16/0.38 !>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > A > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
% 0.16/0.38 condit1810911227_above:
% 0.16/0.38 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Countable_Ofrom__nat,type,
% 0.16/0.38 from_nat:
% 0.16/0.38 !>[A: $tType] : ( nat > A ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Fun_Oswap,type,
% 0.16/0.38 swap:
% 0.16/0.38 !>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.16/0.38 zero_zero:
% 0.16/0.38 !>[A: $tType] : A ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.16/0.38 bot_bot:
% 0.16/0.38 !>[A: $tType] : A ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.16/0.38 ord_less_eq:
% 0.16/0.38 !>[A: $tType] : ( A > A > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.16/0.38 top_top:
% 0.16/0.38 !>[A: $tType] : A ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Pure_Otype,type,
% 0.16/0.38 type2:
% 0.16/0.38 !>[A: $tType] : ( itself @ A ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Set_OBall,type,
% 0.16/0.38 ball:
% 0.16/0.38 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Set_OCollect,type,
% 0.16/0.38 collect:
% 0.16/0.38 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Set_OPow,type,
% 0.16/0.38 pow:
% 0.16/0.38 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Set_Obind,type,
% 0.16/0.38 bind:
% 0.16/0.38 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Set_Oimage,type,
% 0.16/0.38 image:
% 0.16/0.38 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Set_Othe__elem,type,
% 0.16/0.38 the_elem:
% 0.16/0.38 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Stream_Osmap2,type,
% 0.16/0.38 smap2:
% 0.16/0.38 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( stream @ A ) > ( stream @ B ) > ( stream @ C ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Stream_Osmember,type,
% 0.16/0.38 smember:
% 0.16/0.38 !>[A: $tType] : ( A > ( stream @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Stream_Osmerge,type,
% 0.16/0.38 smerge:
% 0.16/0.38 !>[A: $tType] : ( ( stream @ ( stream @ A ) ) > ( stream @ A ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Stream_Osnth,type,
% 0.16/0.38 snth:
% 0.16/0.38 !>[A: $tType] : ( ( stream @ A ) > nat > A ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Stream_Ostream_Opred__stream,type,
% 0.16/0.38 pred_stream:
% 0.16/0.38 !>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Stream_Ostream_Osmap,type,
% 0.16/0.38 smap:
% 0.16/0.38 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( stream @ A ) > ( stream @ Aa ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Stream_Ostream_Osset,type,
% 0.16/0.38 sset:
% 0.16/0.38 !>[A: $tType] : ( ( stream @ A ) > ( set @ A ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Stream_Ostream__all,type,
% 0.16/0.38 stream_all:
% 0.16/0.38 !>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_Stream_Ostreams,type,
% 0.16/0.38 streams:
% 0.16/0.38 !>[A: $tType] : ( ( set @ A ) > ( set @ ( stream @ A ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_c_member,type,
% 0.16/0.38 member:
% 0.16/0.38 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.16/0.38
% 0.16/0.38 thf(sy_v_f,type,
% 0.16/0.38 f: nat > a ).
% 0.16/0.38
% 0.16/0.38 %----Relevant facts (255)
% 0.16/0.38 thf(fact_0_stream__all__def,axiom,
% 0.16/0.38 ! [A: $tType] :
% 0.16/0.38 ( ( stream_all @ A )
% 0.16/0.38 = ( ^ [P: A > $o,S: stream @ A] :
% 0.16/0.38 ! [P2: nat] : ( P @ ( snth @ A @ S @ P2 ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % stream_all_def
% 0.16/0.38 thf(fact_1_snth__smap2,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,S1: stream @ B,S2: stream @ C,N: nat] :
% 0.16/0.38 ( ( snth @ A @ ( smap2 @ B @ C @ A @ F @ S1 @ S2 ) @ N )
% 0.16/0.38 = ( F @ ( snth @ B @ S1 @ N ) @ ( snth @ C @ S2 @ N ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % snth_smap2
% 0.16/0.38 thf(fact_2_smap2__alt,axiom,
% 0.16/0.38 ! [A: $tType,B: $tType,C: $tType,F: B > C > A,S1: stream @ B,S2: stream @ C,S3: stream @ A] :
% 0.16/0.38 ( ( ( smap2 @ B @ C @ A @ F @ S1 @ S2 )
% 0.16/0.38 = S3 )
% 0.16/0.38 = ( ! [N2: nat] :
% 0.16/0.38 ( ( F @ ( snth @ B @ S1 @ N2 ) @ ( snth @ C @ S2 @ N2 ) )
% 0.16/0.38 = ( snth @ A @ S3 @ N2 ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % smap2_alt
% 0.16/0.38 thf(fact_3_of__seq_Oabs__eq,axiom,
% 0.16/0.38 ! [A: $tType] :
% 0.16/0.38 ( ( coindu176146587of_seq @ A )
% 0.16/0.38 = ( ^ [X: nat > A] : ( coindu2010755910_llist @ A @ ( coindu68654304_llist @ A @ X ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % of_seq.abs_eq
% 0.16/0.38 thf(fact_4_of__seq_Orep__eq,axiom,
% 0.16/0.38 ! [A: $tType,X2: nat > A] :
% 0.16/0.38 ( ( coindu1724414836stream @ A @ ( coindu176146587of_seq @ A @ X2 ) )
% 0.16/0.38 = ( coindu68654304_llist @ A @ X2 ) ) ).
% 0.16/0.38
% 0.16/0.38 % of_seq.rep_eq
% 0.16/0.38 thf(fact_5_snth__in,axiom,
% 0.16/0.38 ! [A: $tType,S3: stream @ A,X3: set @ A,N: nat] :
% 0.16/0.38 ( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ X3 ) )
% 0.16/0.38 => ( member @ A @ ( snth @ A @ S3 @ N ) @ X3 ) ) ).
% 0.16/0.38
% 0.16/0.38 % snth_in
% 0.16/0.38 thf(fact_6_streams__iff__snth,axiom,
% 0.16/0.38 ! [A: $tType,S3: stream @ A,X3: set @ A] :
% 0.16/0.38 ( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ X3 ) )
% 0.16/0.38 = ( ! [N2: nat] : ( member @ A @ ( snth @ A @ S3 @ N2 ) @ X3 ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % streams_iff_snth
% 0.16/0.38 thf(fact_7_snth__smap,axiom,
% 0.16/0.38 ! [A: $tType,B: $tType,F: B > A,S3: stream @ B,N: nat] :
% 0.16/0.38 ( ( snth @ A @ ( smap @ B @ A @ F @ S3 ) @ N )
% 0.16/0.38 = ( F @ ( snth @ B @ S3 @ N ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % snth_smap
% 0.16/0.38 thf(fact_8_snth__sset,axiom,
% 0.16/0.38 ! [A: $tType,S3: stream @ A,N: nat] : ( member @ A @ ( snth @ A @ S3 @ N ) @ ( sset @ A @ S3 ) ) ).
% 0.16/0.38
% 0.16/0.38 % snth_sset
% 0.16/0.38 thf(fact_9_lnth__list__of__stream,axiom,
% 0.16/0.38 ! [A: $tType,Xs: stream @ A] :
% 0.16/0.38 ( ( coinductive_lnth @ A @ ( coindu1724414836stream @ A @ Xs ) )
% 0.16/0.38 = ( snth @ A @ Xs ) ) ).
% 0.16/0.38
% 0.16/0.38 % lnth_list_of_stream
% 0.16/0.38 thf(fact_10_smap__alt,axiom,
% 0.16/0.38 ! [A: $tType,B: $tType,F: B > A,S3: stream @ B,S4: stream @ A] :
% 0.16/0.38 ( ( ( smap @ B @ A @ F @ S3 )
% 0.16/0.38 = S4 )
% 0.16/0.38 = ( ! [N2: nat] :
% 0.16/0.38 ( ( F @ ( snth @ B @ S3 @ N2 ) )
% 0.16/0.38 = ( snth @ A @ S4 @ N2 ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % smap_alt
% 0.16/0.38 thf(fact_11_snth__sset__smerge,axiom,
% 0.16/0.38 ! [A: $tType,Ss: stream @ ( stream @ A ),N: nat,M: nat] : ( member @ A @ ( snth @ A @ ( snth @ ( stream @ A ) @ Ss @ N ) @ M ) @ ( sset @ A @ ( smerge @ A @ Ss ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % snth_sset_smerge
% 0.16/0.38 thf(fact_12_stream_ORep__inverse,axiom,
% 0.16/0.38 ! [A: $tType,X2: stream @ A] :
% 0.16/0.38 ( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ X2 ) )
% 0.16/0.38 = X2 ) ).
% 0.16/0.38
% 0.16/0.38 % stream.Rep_inverse
% 0.16/0.38 thf(fact_13_stream__of__llist__llist__of__stream,axiom,
% 0.16/0.38 ! [A: $tType,Xs: stream @ A] :
% 0.16/0.38 ( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ Xs ) )
% 0.16/0.38 = Xs ) ).
% 0.16/0.38
% 0.16/0.38 % stream_of_llist_llist_of_stream
% 0.16/0.38 thf(fact_14_smap__streams,axiom,
% 0.16/0.38 ! [A: $tType,B: $tType,S3: stream @ A,A2: set @ A,F: A > B,B2: set @ B] :
% 0.16/0.38 ( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) )
% 0.16/0.38 => ( ! [X4: A] :
% 0.16/0.38 ( ( member @ A @ X4 @ A2 )
% 0.16/0.38 => ( member @ B @ ( F @ X4 ) @ B2 ) )
% 0.16/0.38 => ( member @ ( stream @ B ) @ ( smap @ A @ B @ F @ S3 ) @ ( streams @ B @ B2 ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % smap_streams
% 0.16/0.38 thf(fact_15_stream_Omap__cong,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType,X2: stream @ A,Ya: stream @ A,F: A > B,G: A > B] :
% 0.16/0.38 ( ( X2 = Ya )
% 0.16/0.38 => ( ! [Z: A] :
% 0.16/0.38 ( ( member @ A @ Z @ ( sset @ A @ Ya ) )
% 0.16/0.38 => ( ( F @ Z )
% 0.16/0.38 = ( G @ Z ) ) )
% 0.16/0.38 => ( ( smap @ A @ B @ F @ X2 )
% 0.16/0.38 = ( smap @ A @ B @ G @ Ya ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % stream.map_cong
% 0.16/0.38 thf(fact_16_stream_Omap__cong0,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType,X2: stream @ A,F: A > B,G: A > B] :
% 0.16/0.38 ( ! [Z: A] :
% 0.16/0.38 ( ( member @ A @ Z @ ( sset @ A @ X2 ) )
% 0.16/0.38 => ( ( F @ Z )
% 0.16/0.38 = ( G @ Z ) ) )
% 0.16/0.38 => ( ( smap @ A @ B @ F @ X2 )
% 0.16/0.38 = ( smap @ A @ B @ G @ X2 ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % stream.map_cong0
% 0.16/0.38 thf(fact_17_stream_Oinj__map__strong,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType,X2: stream @ A,Xa: stream @ A,F: A > B,Fa: A > B] :
% 0.16/0.38 ( ! [Z: A,Za: A] :
% 0.16/0.38 ( ( member @ A @ Z @ ( sset @ A @ X2 ) )
% 0.16/0.38 => ( ( member @ A @ Za @ ( sset @ A @ Xa ) )
% 0.16/0.38 => ( ( ( F @ Z )
% 0.16/0.38 = ( Fa @ Za ) )
% 0.16/0.38 => ( Z = Za ) ) ) )
% 0.16/0.38 => ( ( ( smap @ A @ B @ F @ X2 )
% 0.16/0.38 = ( smap @ A @ B @ Fa @ Xa ) )
% 0.16/0.38 => ( X2 = Xa ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % stream.inj_map_strong
% 0.16/0.38 thf(fact_18_stream_ORep__inject,axiom,
% 0.16/0.38 ! [A: $tType,X2: stream @ A,Y: stream @ A] :
% 0.16/0.38 ( ( ( coindu1724414836stream @ A @ X2 )
% 0.16/0.38 = ( coindu1724414836stream @ A @ Y ) )
% 0.16/0.38 = ( X2 = Y ) ) ).
% 0.16/0.38
% 0.16/0.38 % stream.Rep_inject
% 0.16/0.38 thf(fact_19_lnth__inf__llist,axiom,
% 0.16/0.38 ! [A: $tType,F: nat > A,N: nat] :
% 0.16/0.38 ( ( coinductive_lnth @ A @ ( coindu68654304_llist @ A @ F ) @ N )
% 0.16/0.38 = ( F @ N ) ) ).
% 0.16/0.38
% 0.16/0.38 % lnth_inf_llist
% 0.16/0.38 thf(fact_20_inf__llist__inj,axiom,
% 0.16/0.38 ! [A: $tType,F: nat > A,G: nat > A] :
% 0.16/0.38 ( ( ( coindu68654304_llist @ A @ F )
% 0.16/0.38 = ( coindu68654304_llist @ A @ G ) )
% 0.16/0.38 = ( F = G ) ) ).
% 0.16/0.38
% 0.16/0.38 % inf_llist_inj
% 0.16/0.38 thf(fact_21_Stream_Osmember__def,axiom,
% 0.16/0.38 ! [A: $tType] :
% 0.16/0.38 ( ( smember @ A )
% 0.16/0.38 = ( ^ [X: A,S: stream @ A] : ( member @ A @ X @ ( sset @ A @ S ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % Stream.smember_def
% 0.16/0.38 thf(fact_22_stream__all__iff,axiom,
% 0.16/0.38 ! [A: $tType] :
% 0.16/0.38 ( ( stream_all @ A )
% 0.16/0.38 = ( ^ [P: A > $o,S: stream @ A] :
% 0.16/0.38 ! [X: A] :
% 0.16/0.38 ( ( member @ A @ X @ ( sset @ A @ S ) )
% 0.16/0.38 => ( P @ X ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % stream_all_iff
% 0.16/0.38 thf(fact_23_llist__of__stream__stream__of__llist,axiom,
% 0.16/0.38 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.16/0.38 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.16/0.38 => ( ( coindu1724414836stream @ A @ ( coindu2010755910_llist @ A @ Xs ) )
% 0.16/0.38 = Xs ) ) ).
% 0.16/0.38
% 0.16/0.38 % llist_of_stream_stream_of_llist
% 0.16/0.38 thf(fact_24_lset__llist__of__stream,axiom,
% 0.16/0.38 ! [A: $tType,Xs: stream @ A] :
% 0.16/0.38 ( ( coinductive_lset @ A @ ( coindu1724414836stream @ A @ Xs ) )
% 0.16/0.38 = ( sset @ A @ Xs ) ) ).
% 0.16/0.38
% 0.16/0.38 % lset_llist_of_stream
% 0.16/0.38 thf(fact_25_sset__streams,axiom,
% 0.16/0.38 ! [A: $tType,S3: stream @ A,A2: set @ A] :
% 0.16/0.38 ( ( ord_less_eq @ ( set @ A ) @ ( sset @ A @ S3 ) @ A2 )
% 0.16/0.38 => ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % sset_streams
% 0.16/0.38 thf(fact_26_streams__sset,axiom,
% 0.16/0.38 ! [A: $tType,S3: stream @ A,A2: set @ A] :
% 0.16/0.38 ( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) )
% 0.16/0.38 => ( ord_less_eq @ ( set @ A ) @ ( sset @ A @ S3 ) @ A2 ) ) ).
% 0.16/0.38
% 0.16/0.38 % streams_sset
% 0.16/0.38 thf(fact_27_streams__iff__sset,axiom,
% 0.16/0.38 ! [A: $tType,S3: stream @ A,A2: set @ A] :
% 0.16/0.38 ( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) )
% 0.16/0.38 = ( ord_less_eq @ ( set @ A ) @ ( sset @ A @ S3 ) @ A2 ) ) ).
% 0.16/0.38
% 0.16/0.38 % streams_iff_sset
% 0.16/0.38 thf(fact_28_lmap__llist__of__stream,axiom,
% 0.16/0.38 ! [A: $tType,B: $tType,F: B > A,Xs: stream @ B] :
% 0.16/0.38 ( ( coinductive_lmap @ B @ A @ F @ ( coindu1724414836stream @ B @ Xs ) )
% 0.16/0.38 = ( coindu1724414836stream @ A @ ( smap @ B @ A @ F @ Xs ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % lmap_llist_of_stream
% 0.16/0.38 thf(fact_29_stream_Oset__map,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType,F: A > B,V: stream @ A] :
% 0.16/0.38 ( ( sset @ B @ ( smap @ A @ B @ F @ V ) )
% 0.16/0.38 = ( image @ A @ B @ F @ ( sset @ A @ V ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % stream.set_map
% 0.16/0.38 thf(fact_30_lfinite__lmap,axiom,
% 0.16/0.38 ! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B] :
% 0.16/0.38 ( ( coinductive_lfinite @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
% 0.16/0.38 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 0.16/0.38
% 0.16/0.38 % lfinite_lmap
% 0.16/0.38 thf(fact_31_llist_Oset__map,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType,F: A > B,V: coinductive_llist @ A] :
% 0.16/0.38 ( ( coinductive_lset @ B @ ( coinductive_lmap @ A @ B @ F @ V ) )
% 0.16/0.38 = ( image @ A @ B @ F @ ( coinductive_lset @ A @ V ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % llist.set_map
% 0.16/0.38 thf(fact_32_inf__llist__lnth,axiom,
% 0.16/0.38 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.16/0.38 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.16/0.38 => ( ( coindu68654304_llist @ A @ ( coinductive_lnth @ A @ Xs ) )
% 0.16/0.38 = Xs ) ) ).
% 0.16/0.38
% 0.16/0.38 % inf_llist_lnth
% 0.16/0.38 thf(fact_33_sset__stream__of__llist,axiom,
% 0.16/0.38 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.16/0.38 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.16/0.38 => ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.16/0.38 => ( ( sset @ A @ ( coindu2010755910_llist @ A @ Xs ) )
% 0.16/0.38 = ( coinductive_lset @ A @ Xs ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % sset_stream_of_llist
% 0.16/0.38 thf(fact_34_smap__stream__of__llist,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType,Xs: coinductive_llist @ A,F: A > B] :
% 0.16/0.38 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.16/0.38 => ( ( smap @ A @ B @ F @ ( coindu2010755910_llist @ A @ Xs ) )
% 0.16/0.38 = ( coindu2010755910_llist @ B @ ( coinductive_lmap @ A @ B @ F @ Xs ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % smap_stream_of_llist
% 0.16/0.38 thf(fact_35_llist_Oinj__map__strong,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType,X2: coinductive_llist @ A,Xa: coinductive_llist @ A,F: A > B,Fa: A > B] :
% 0.16/0.38 ( ! [Z: A,Za: A] :
% 0.16/0.38 ( ( member @ A @ Z @ ( coinductive_lset @ A @ X2 ) )
% 0.16/0.38 => ( ( member @ A @ Za @ ( coinductive_lset @ A @ Xa ) )
% 0.16/0.38 => ( ( ( F @ Z )
% 0.16/0.38 = ( Fa @ Za ) )
% 0.16/0.38 => ( Z = Za ) ) ) )
% 0.16/0.38 => ( ( ( coinductive_lmap @ A @ B @ F @ X2 )
% 0.16/0.38 = ( coinductive_lmap @ A @ B @ Fa @ Xa ) )
% 0.16/0.38 => ( X2 = Xa ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % llist.inj_map_strong
% 0.16/0.38 thf(fact_36_llist_Omap__cong0,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType,X2: coinductive_llist @ A,F: A > B,G: A > B] :
% 0.16/0.38 ( ! [Z: A] :
% 0.16/0.38 ( ( member @ A @ Z @ ( coinductive_lset @ A @ X2 ) )
% 0.16/0.38 => ( ( F @ Z )
% 0.16/0.38 = ( G @ Z ) ) )
% 0.16/0.38 => ( ( coinductive_lmap @ A @ B @ F @ X2 )
% 0.16/0.38 = ( coinductive_lmap @ A @ B @ G @ X2 ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % llist.map_cong0
% 0.16/0.38 thf(fact_37_llist_Omap__cong,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,F: A > B,G: A > B] :
% 0.16/0.38 ( ( X2 = Ya )
% 0.16/0.38 => ( ! [Z: A] :
% 0.16/0.38 ( ( member @ A @ Z @ ( coinductive_lset @ A @ Ya ) )
% 0.16/0.38 => ( ( F @ Z )
% 0.16/0.38 = ( G @ Z ) ) )
% 0.16/0.38 => ( ( coinductive_lmap @ A @ B @ F @ X2 )
% 0.16/0.38 = ( coinductive_lmap @ A @ B @ G @ Ya ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % llist.map_cong
% 0.16/0.38 thf(fact_38_wlog__linorder__le,axiom,
% 0.16/0.38 ! [A: $tType] :
% 0.16/0.38 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.38 => ! [P3: A > A > $o,B3: A,A3: A] :
% 0.16/0.38 ( ! [A4: A,B4: A] :
% 0.16/0.38 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.16/0.38 => ( P3 @ A4 @ B4 ) )
% 0.16/0.38 => ( ( ( P3 @ B3 @ A3 )
% 0.16/0.38 => ( P3 @ A3 @ B3 ) )
% 0.16/0.38 => ( P3 @ A3 @ B3 ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % wlog_linorder_le
% 0.16/0.38 thf(fact_39_streams__mono2,axiom,
% 0.16/0.38 ! [A: $tType,S5: set @ A,T: set @ A] :
% 0.16/0.38 ( ( ord_less_eq @ ( set @ A ) @ S5 @ T )
% 0.16/0.38 => ( ord_less_eq @ ( set @ ( stream @ A ) ) @ ( streams @ A @ S5 ) @ ( streams @ A @ T ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % streams_mono2
% 0.16/0.38 thf(fact_40_lfinite__inf__llist,axiom,
% 0.16/0.38 ! [A: $tType,F: nat > A] :
% 0.16/0.38 ~ ( coinductive_lfinite @ A @ ( coindu68654304_llist @ A @ F ) ) ).
% 0.16/0.38
% 0.16/0.38 % lfinite_inf_llist
% 0.16/0.38 thf(fact_41_lfinite__llist__of__stream,axiom,
% 0.16/0.38 ! [A: $tType,Xs: stream @ A] :
% 0.16/0.38 ~ ( coinductive_lfinite @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% 0.16/0.38
% 0.16/0.38 % lfinite_llist_of_stream
% 0.16/0.38 thf(fact_42_streams__mono,axiom,
% 0.16/0.38 ! [A: $tType,S3: stream @ A,A2: set @ A,B2: set @ A] :
% 0.16/0.38 ( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) )
% 0.16/0.38 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.38 => ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ B2 ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % streams_mono
% 0.16/0.38 thf(fact_43_not__False__in__image__Ball,axiom,
% 0.16/0.38 ! [A: $tType,P3: A > $o,A2: set @ A] :
% 0.16/0.38 ( ( ~ ( member @ $o @ $false @ ( image @ A @ $o @ P3 @ A2 ) ) )
% 0.16/0.38 = ( ! [X: A] :
% 0.16/0.38 ( ( member @ A @ X @ A2 )
% 0.16/0.38 => ( P3 @ X ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % not_False_in_image_Ball
% 0.16/0.38 thf(fact_44_subset__antisym,axiom,
% 0.16/0.38 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.16/0.38 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.38 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.16/0.38 => ( A2 = B2 ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % subset_antisym
% 0.16/0.38 thf(fact_45_mem__Collect__eq,axiom,
% 0.16/0.38 ! [A: $tType,A3: A,P3: A > $o] :
% 0.16/0.38 ( ( member @ A @ A3 @ ( collect @ A @ P3 ) )
% 0.16/0.38 = ( P3 @ A3 ) ) ).
% 0.16/0.38
% 0.16/0.38 % mem_Collect_eq
% 0.16/0.38 thf(fact_46_Collect__mem__eq,axiom,
% 0.16/0.38 ! [A: $tType,A2: set @ A] :
% 0.16/0.38 ( ( collect @ A
% 0.16/0.38 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 0.16/0.38 = A2 ) ).
% 0.16/0.38
% 0.16/0.38 % Collect_mem_eq
% 0.16/0.38 thf(fact_47_Collect__cong,axiom,
% 0.16/0.38 ! [A: $tType,P3: A > $o,Q: A > $o] :
% 0.16/0.38 ( ! [X4: A] :
% 0.16/0.38 ( ( P3 @ X4 )
% 0.16/0.38 = ( Q @ X4 ) )
% 0.16/0.38 => ( ( collect @ A @ P3 )
% 0.16/0.38 = ( collect @ A @ Q ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % Collect_cong
% 0.16/0.38 thf(fact_48_ext,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.16/0.38 ( ! [X4: A] :
% 0.16/0.38 ( ( F @ X4 )
% 0.16/0.38 = ( G @ X4 ) )
% 0.16/0.38 => ( F = G ) ) ).
% 0.16/0.38
% 0.16/0.38 % ext
% 0.16/0.38 thf(fact_49_subsetI,axiom,
% 0.16/0.38 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.16/0.38 ( ! [X4: A] :
% 0.16/0.38 ( ( member @ A @ X4 @ A2 )
% 0.16/0.38 => ( member @ A @ X4 @ B2 ) )
% 0.16/0.38 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.16/0.38
% 0.16/0.38 % subsetI
% 0.16/0.38 thf(fact_50_image__eqI,axiom,
% 0.16/0.38 ! [A: $tType,B: $tType,B3: A,F: B > A,X2: B,A2: set @ B] :
% 0.16/0.38 ( ( B3
% 0.16/0.38 = ( F @ X2 ) )
% 0.16/0.38 => ( ( member @ B @ X2 @ A2 )
% 0.16/0.38 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % image_eqI
% 0.16/0.38 thf(fact_51_order__refl,axiom,
% 0.16/0.38 ! [A: $tType] :
% 0.16/0.38 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.38 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.16/0.38
% 0.16/0.38 % order_refl
% 0.16/0.38 thf(fact_52_Ball__Collect,axiom,
% 0.16/0.38 ! [A: $tType] :
% 0.16/0.38 ( ( ball @ A )
% 0.16/0.38 = ( ^ [A5: set @ A,P: A > $o] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % Ball_Collect
% 0.16/0.38 thf(fact_53_subset__image__iff,axiom,
% 0.16/0.38 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 0.16/0.38 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 0.16/0.38 = ( ? [AA: set @ B] :
% 0.16/0.38 ( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
% 0.16/0.38 & ( B2
% 0.16/0.38 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % subset_image_iff
% 0.16/0.38 thf(fact_54_image__subset__iff,axiom,
% 0.16/0.38 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 0.16/0.38 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 0.16/0.38 = ( ! [X: B] :
% 0.16/0.38 ( ( member @ B @ X @ A2 )
% 0.16/0.38 => ( member @ A @ ( F @ X ) @ B2 ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % image_subset_iff
% 0.16/0.38 thf(fact_55_subset__imageE,axiom,
% 0.16/0.38 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 0.16/0.38 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 0.16/0.38 => ~ ! [C2: set @ B] :
% 0.16/0.38 ( ( ord_less_eq @ ( set @ B ) @ C2 @ A2 )
% 0.16/0.38 => ( B2
% 0.16/0.38 != ( image @ B @ A @ F @ C2 ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % subset_imageE
% 0.16/0.38 thf(fact_56_le__funD,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType] :
% 0.16/0.38 ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.38 => ! [F: A > B,G: A > B,X2: A] :
% 0.16/0.38 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.16/0.38 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % le_funD
% 0.16/0.38 thf(fact_57_le__funE,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType] :
% 0.16/0.38 ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.38 => ! [F: A > B,G: A > B,X2: A] :
% 0.16/0.38 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.16/0.38 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % le_funE
% 0.16/0.38 thf(fact_58_le__funI,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType] :
% 0.16/0.38 ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.38 => ! [F: A > B,G: A > B] :
% 0.16/0.38 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.16/0.38 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % le_funI
% 0.16/0.38 thf(fact_59_le__fun__def,axiom,
% 0.16/0.38 ! [B: $tType,A: $tType] :
% 0.16/0.38 ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.38 => ( ( ord_less_eq @ ( A > B ) )
% 0.16/0.38 = ( ^ [F2: A > B,G2: A > B] :
% 0.16/0.38 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % le_fun_def
% 0.16/0.38 thf(fact_60_order__subst1,axiom,
% 0.16/0.38 ! [A: $tType,B: $tType] :
% 0.16/0.38 ( ( ( order @ B @ ( type2 @ B ) )
% 0.16/0.38 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.38 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.16/0.38 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.16/0.38 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 0.16/0.38 => ( ! [X4: B,Y2: B] :
% 0.16/0.38 ( ( ord_less_eq @ B @ X4 @ Y2 )
% 0.16/0.38 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.16/0.38 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % order_subst1
% 0.16/0.38 thf(fact_61_order__subst2,axiom,
% 0.16/0.38 ! [A: $tType,C: $tType] :
% 0.16/0.38 ( ( ( order @ C @ ( type2 @ C ) )
% 0.16/0.38 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.38 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 0.16/0.38 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.16/0.38 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 0.16/0.38 => ( ! [X4: A,Y2: A] :
% 0.16/0.38 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.16/0.38 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.16/0.38 => ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % order_subst2
% 0.16/0.38 thf(fact_62_ord__eq__le__subst,axiom,
% 0.16/0.38 ! [A: $tType,B: $tType] :
% 0.16/0.38 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.38 & ( ord @ A @ ( type2 @ A ) ) )
% 0.16/0.38 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.16/0.38 ( ( A3
% 0.16/0.38 = ( F @ B3 ) )
% 0.16/0.38 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 0.16/0.38 => ( ! [X4: B,Y2: B] :
% 0.16/0.38 ( ( ord_less_eq @ B @ X4 @ Y2 )
% 0.16/0.38 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.16/0.38 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % ord_eq_le_subst
% 0.16/0.38 thf(fact_63_ord__le__eq__subst,axiom,
% 0.16/0.38 ! [A: $tType,B: $tType] :
% 0.16/0.38 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.38 & ( ord @ A @ ( type2 @ A ) ) )
% 0.16/0.38 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 0.16/0.38 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.16/0.38 => ( ( ( F @ B3 )
% 0.16/0.38 = C3 )
% 0.16/0.38 => ( ! [X4: A,Y2: A] :
% 0.16/0.38 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.16/0.38 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.16/0.38 => ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % ord_le_eq_subst
% 0.16/0.38 thf(fact_64_eq__iff,axiom,
% 0.16/0.38 ! [A: $tType] :
% 0.16/0.38 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.38 => ( ( ^ [Y3: A,Z2: A] : ( Y3 = Z2 ) )
% 0.16/0.38 = ( ^ [X: A,Y4: A] :
% 0.16/0.38 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.16/0.38 & ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % eq_iff
% 0.16/0.38 thf(fact_65_antisym,axiom,
% 0.16/0.38 ! [A: $tType] :
% 0.16/0.38 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.38 => ! [X2: A,Y: A] :
% 0.16/0.38 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.16/0.38 => ( ( ord_less_eq @ A @ Y @ X2 )
% 0.16/0.38 => ( X2 = Y ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % antisym
% 0.16/0.38 thf(fact_66_linear,axiom,
% 0.16/0.38 ! [A: $tType] :
% 0.16/0.38 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.38 => ! [X2: A,Y: A] :
% 0.16/0.38 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.16/0.38 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.16/0.38
% 0.16/0.38 % linear
% 0.16/0.38 thf(fact_67_eq__refl,axiom,
% 0.16/0.38 ! [A: $tType] :
% 0.16/0.39 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: A,Y: A] :
% 0.16/0.39 ( ( X2 = Y )
% 0.16/0.39 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % eq_refl
% 0.16/0.39 thf(fact_68_le__cases,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: A,Y: A] :
% 0.16/0.39 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 0.16/0.39 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % le_cases
% 0.16/0.39 thf(fact_69_order_Otrans,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A3: A,B3: A,C3: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.16/0.39 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.16/0.39 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % order.trans
% 0.16/0.39 thf(fact_70_le__cases3,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: A,Y: A,Z3: A] :
% 0.16/0.39 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 0.16/0.39 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 0.16/0.39 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 0.16/0.39 => ~ ( ord_less_eq @ A @ X2 @ Z3 ) )
% 0.16/0.39 => ( ( ( ord_less_eq @ A @ X2 @ Z3 )
% 0.16/0.39 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 0.16/0.39 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 0.16/0.39 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 0.16/0.39 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.16/0.39 => ~ ( ord_less_eq @ A @ Z3 @ X2 ) )
% 0.16/0.39 => ~ ( ( ord_less_eq @ A @ Z3 @ X2 )
% 0.16/0.39 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % le_cases3
% 0.16/0.39 thf(fact_71_antisym__conv,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [Y: A,X2: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.16/0.39 => ( ( ord_less_eq @ A @ X2 @ Y )
% 0.16/0.39 = ( X2 = Y ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % antisym_conv
% 0.16/0.39 thf(fact_72_ord__eq__le__trans,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( ord @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A3: A,B3: A,C3: A] :
% 0.16/0.39 ( ( A3 = B3 )
% 0.16/0.39 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.16/0.39 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % ord_eq_le_trans
% 0.16/0.39 thf(fact_73_ord__le__eq__trans,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( ord @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A3: A,B3: A,C3: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.16/0.39 => ( ( B3 = C3 )
% 0.16/0.39 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % ord_le_eq_trans
% 0.16/0.39 thf(fact_74_order__class_Oorder_Oantisym,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A3: A,B3: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.16/0.39 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.16/0.39 => ( A3 = B3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % order_class.order.antisym
% 0.16/0.39 thf(fact_75_order__trans,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: A,Y: A,Z3: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.16/0.39 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.16/0.39 => ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % order_trans
% 0.16/0.39 thf(fact_76_dual__order_Orefl,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.16/0.39
% 0.16/0.39 % dual_order.refl
% 0.16/0.39 thf(fact_77_linorder__wlog,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [P3: A > A > $o,A3: A,B3: A] :
% 0.16/0.39 ( ! [A4: A,B4: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.16/0.39 => ( P3 @ A4 @ B4 ) )
% 0.16/0.39 => ( ! [A4: A,B4: A] :
% 0.16/0.39 ( ( P3 @ B4 @ A4 )
% 0.16/0.39 => ( P3 @ A4 @ B4 ) )
% 0.16/0.39 => ( P3 @ A3 @ B3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % linorder_wlog
% 0.16/0.39 thf(fact_78_dual__order_Otrans,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [B3: A,A3: A,C3: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.16/0.39 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 0.16/0.39 => ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % dual_order.trans
% 0.16/0.39 thf(fact_79_dual__order_Oantisym,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [B3: A,A3: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.16/0.39 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.16/0.39 => ( A3 = B3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % dual_order.antisym
% 0.16/0.39 thf(fact_80_imageI,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,X2: A,A2: set @ A,F: A > B] :
% 0.16/0.39 ( ( member @ A @ X2 @ A2 )
% 0.16/0.39 => ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % imageI
% 0.16/0.39 thf(fact_81_image__iff,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,Z3: A,F: B > A,A2: set @ B] :
% 0.16/0.39 ( ( member @ A @ Z3 @ ( image @ B @ A @ F @ A2 ) )
% 0.16/0.39 = ( ? [X: B] :
% 0.16/0.39 ( ( member @ B @ X @ A2 )
% 0.16/0.39 & ( Z3
% 0.16/0.39 = ( F @ X ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % image_iff
% 0.16/0.39 thf(fact_82_bex__imageD,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P3: A > $o] :
% 0.16/0.39 ( ? [X5: A] :
% 0.16/0.39 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A2 ) )
% 0.16/0.39 & ( P3 @ X5 ) )
% 0.16/0.39 => ? [X4: B] :
% 0.16/0.39 ( ( member @ B @ X4 @ A2 )
% 0.16/0.39 & ( P3 @ ( F @ X4 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % bex_imageD
% 0.16/0.39 thf(fact_83_image__cong,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,M2: set @ A,N3: set @ A,F: A > B,G: A > B] :
% 0.16/0.39 ( ( M2 = N3 )
% 0.16/0.39 => ( ! [X4: A] :
% 0.16/0.39 ( ( member @ A @ X4 @ N3 )
% 0.16/0.39 => ( ( F @ X4 )
% 0.16/0.39 = ( G @ X4 ) ) )
% 0.16/0.39 => ( ( image @ A @ B @ F @ M2 )
% 0.16/0.39 = ( image @ A @ B @ G @ N3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % image_cong
% 0.16/0.39 thf(fact_84_ball__imageD,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P3: A > $o] :
% 0.16/0.39 ( ! [X4: A] :
% 0.16/0.39 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A2 ) )
% 0.16/0.39 => ( P3 @ X4 ) )
% 0.16/0.39 => ! [X5: B] :
% 0.16/0.39 ( ( member @ B @ X5 @ A2 )
% 0.16/0.39 => ( P3 @ ( F @ X5 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % ball_imageD
% 0.16/0.39 thf(fact_85_rev__image__eqI,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,X2: A,A2: set @ A,B3: B,F: A > B] :
% 0.16/0.39 ( ( member @ A @ X2 @ A2 )
% 0.16/0.39 => ( ( B3
% 0.16/0.39 = ( F @ X2 ) )
% 0.16/0.39 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % rev_image_eqI
% 0.16/0.39 thf(fact_86_Inf_OINF__cong,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C4: B > A,D: B > A,Inf: ( set @ A ) > A] :
% 0.16/0.39 ( ( A2 = B2 )
% 0.16/0.39 => ( ! [X4: B] :
% 0.16/0.39 ( ( member @ B @ X4 @ B2 )
% 0.16/0.39 => ( ( C4 @ X4 )
% 0.16/0.39 = ( D @ X4 ) ) )
% 0.16/0.39 => ( ( Inf @ ( image @ B @ A @ C4 @ A2 ) )
% 0.16/0.39 = ( Inf @ ( image @ B @ A @ D @ B2 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Inf.INF_cong
% 0.16/0.39 thf(fact_87_Sup_OSUP__cong,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C4: B > A,D: B > A,Sup: ( set @ A ) > A] :
% 0.16/0.39 ( ( A2 = B2 )
% 0.16/0.39 => ( ! [X4: B] :
% 0.16/0.39 ( ( member @ B @ X4 @ B2 )
% 0.16/0.39 => ( ( C4 @ X4 )
% 0.16/0.39 = ( D @ X4 ) ) )
% 0.16/0.39 => ( ( Sup @ ( image @ B @ A @ C4 @ A2 ) )
% 0.16/0.39 = ( Sup @ ( image @ B @ A @ D @ B2 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup.SUP_cong
% 0.16/0.39 thf(fact_88_set__mp,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( ( member @ A @ X2 @ A2 )
% 0.16/0.39 => ( member @ A @ X2 @ B2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % set_mp
% 0.16/0.39 thf(fact_89_in__mono,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( ( member @ A @ X2 @ A2 )
% 0.16/0.39 => ( member @ A @ X2 @ B2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % in_mono
% 0.16/0.39 thf(fact_90_subsetD,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( ( member @ A @ C3 @ A2 )
% 0.16/0.39 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % subsetD
% 0.16/0.39 thf(fact_91_subsetCE,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( ( member @ A @ C3 @ A2 )
% 0.16/0.39 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % subsetCE
% 0.16/0.39 thf(fact_92_equalityE,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.16/0.39 ( ( A2 = B2 )
% 0.16/0.39 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % equalityE
% 0.16/0.39 thf(fact_93_subset__eq,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) )
% 0.16/0.39 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.16/0.39 ! [X: A] :
% 0.16/0.39 ( ( member @ A @ X @ A5 )
% 0.16/0.39 => ( member @ A @ X @ B5 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % subset_eq
% 0.16/0.39 thf(fact_94_equalityD1,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.16/0.39 ( ( A2 = B2 )
% 0.16/0.39 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.16/0.39
% 0.16/0.39 % equalityD1
% 0.16/0.39 thf(fact_95_equalityD2,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.16/0.39 ( ( A2 = B2 )
% 0.16/0.39 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.16/0.39
% 0.16/0.39 % equalityD2
% 0.16/0.39 thf(fact_96_set__rev__mp,axiom,
% 0.16/0.39 ! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
% 0.16/0.39 ( ( member @ A @ X2 @ A2 )
% 0.16/0.39 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( member @ A @ X2 @ B2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % set_rev_mp
% 0.16/0.39 thf(fact_97_subset__iff,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) )
% 0.16/0.39 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.16/0.39 ! [T2: A] :
% 0.16/0.39 ( ( member @ A @ T2 @ A5 )
% 0.16/0.39 => ( member @ A @ T2 @ B5 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % subset_iff
% 0.16/0.39 thf(fact_98_rev__subsetD,axiom,
% 0.16/0.39 ! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
% 0.16/0.39 ( ( member @ A @ C3 @ A2 )
% 0.16/0.39 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % rev_subsetD
% 0.16/0.39 thf(fact_99_subset__refl,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.16/0.39
% 0.16/0.39 % subset_refl
% 0.16/0.39 thf(fact_100_Collect__mono,axiom,
% 0.16/0.39 ! [A: $tType,P3: A > $o,Q: A > $o] :
% 0.16/0.39 ( ! [X4: A] :
% 0.16/0.39 ( ( P3 @ X4 )
% 0.16/0.39 => ( Q @ X4 ) )
% 0.16/0.39 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Collect_mono
% 0.16/0.39 thf(fact_101_subset__trans,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
% 0.16/0.39 => ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % subset_trans
% 0.16/0.39 thf(fact_102_set__eq__subset,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( ^ [Y3: set @ A,Z2: set @ A] : ( Y3 = Z2 ) )
% 0.16/0.39 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.16/0.39 & ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % set_eq_subset
% 0.16/0.39 thf(fact_103_contra__subsetD,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( ~ ( member @ A @ C3 @ B2 )
% 0.16/0.39 => ~ ( member @ A @ C3 @ A2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % contra_subsetD
% 0.16/0.39 thf(fact_104_Collect__mono__iff,axiom,
% 0.16/0.39 ! [A: $tType,P3: A > $o,Q: A > $o] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) )
% 0.16/0.39 = ( ! [X: A] :
% 0.16/0.39 ( ( P3 @ X )
% 0.16/0.39 => ( Q @ X ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Collect_mono_iff
% 0.16/0.39 thf(fact_105_Ball__def,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( ball @ A )
% 0.16/0.39 = ( ^ [A5: set @ A,P: A > $o] :
% 0.16/0.39 ! [X: A] :
% 0.16/0.39 ( ( member @ A @ X @ A5 )
% 0.16/0.39 => ( P @ X ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Ball_def
% 0.16/0.39 thf(fact_106_image__mono,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % image_mono
% 0.16/0.39 thf(fact_107_image__subsetI,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
% 0.16/0.39 ( ! [X4: A] :
% 0.16/0.39 ( ( member @ A @ X4 @ A2 )
% 0.16/0.39 => ( member @ B @ ( F @ X4 ) @ B2 ) )
% 0.16/0.39 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% 0.16/0.39
% 0.16/0.39 % image_subsetI
% 0.16/0.39 thf(fact_108_cr__streamI,axiom,
% 0.16/0.39 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.16/0.39 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.16/0.39 => ( coindu1183105481stream @ A @ Xs @ ( coindu2010755910_llist @ A @ Xs ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cr_streamI
% 0.16/0.39 thf(fact_109_llistsum__mono,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType] :
% 0.16/0.39 ( ( ( monoid_add @ B @ ( type2 @ B ) )
% 0.16/0.39 & ( ordere779506340up_add @ B @ ( type2 @ B ) ) )
% 0.16/0.39 => ! [Xs: coinductive_llist @ A,F: A > B,G: A > B] :
% 0.16/0.39 ( ! [X4: A] :
% 0.16/0.39 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 0.16/0.39 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.16/0.39 => ( ord_less_eq @ B @ ( coindu780009021istsum @ B @ ( coinductive_lmap @ A @ B @ F @ Xs ) ) @ ( coindu780009021istsum @ B @ ( coinductive_lmap @ A @ B @ G @ Xs ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % llistsum_mono
% 0.16/0.39 thf(fact_110_lset__lmember,axiom,
% 0.16/0.39 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
% 0.16/0.39 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.16/0.39 = ( coinductive_lmember @ A @ X2 @ Xs ) ) ).
% 0.16/0.39
% 0.16/0.39 % lset_lmember
% 0.16/0.39 thf(fact_111_ball__reg,axiom,
% 0.16/0.39 ! [A: $tType,R: set @ A,P3: A > $o,Q: A > $o] :
% 0.16/0.39 ( ! [X4: A] :
% 0.16/0.39 ( ( member @ A @ X4 @ R )
% 0.16/0.39 => ( ( P3 @ X4 )
% 0.16/0.39 => ( Q @ X4 ) ) )
% 0.16/0.39 => ( ! [X4: A] :
% 0.16/0.39 ( ( member @ A @ X4 @ R )
% 0.16/0.39 => ( P3 @ X4 ) )
% 0.16/0.39 => ! [X5: A] :
% 0.16/0.39 ( ( member @ A @ X5 @ R )
% 0.16/0.39 => ( Q @ X5 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % ball_reg
% 0.16/0.39 thf(fact_112_stream_Opred__set,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( pred_stream @ A )
% 0.16/0.39 = ( ^ [P: A > $o,X: stream @ A] :
% 0.16/0.39 ! [Y4: A] :
% 0.16/0.39 ( ( member @ A @ Y4 @ ( sset @ A @ X ) )
% 0.16/0.39 => ( P @ Y4 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % stream.pred_set
% 0.16/0.39 thf(fact_113_lfp_Oleq__refl,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.16/0.39
% 0.16/0.39 % lfp.leq_refl
% 0.16/0.39 thf(fact_114_gfp_Oleq__trans,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [Y: A,X2: A,Z3: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.16/0.39 => ( ( ord_less_eq @ A @ Z3 @ Y )
% 0.16/0.39 => ( ord_less_eq @ A @ Z3 @ X2 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % gfp.leq_trans
% 0.16/0.39 thf(fact_115_lfp_Oleq__trans,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: A,Y: A,Z3: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.16/0.39 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.16/0.39 => ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % lfp.leq_trans
% 0.16/0.39 thf(fact_116_stream_Opred__mono__strong,axiom,
% 0.16/0.39 ! [A: $tType,P3: A > $o,X2: stream @ A,Pa: A > $o] :
% 0.16/0.39 ( ( pred_stream @ A @ P3 @ X2 )
% 0.16/0.39 => ( ! [Z: A] :
% 0.16/0.39 ( ( member @ A @ Z @ ( sset @ A @ X2 ) )
% 0.16/0.39 => ( ( P3 @ Z )
% 0.16/0.39 => ( Pa @ Z ) ) )
% 0.16/0.39 => ( pred_stream @ A @ Pa @ X2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % stream.pred_mono_strong
% 0.16/0.39 thf(fact_117_stream_Opred__cong,axiom,
% 0.16/0.39 ! [A: $tType,X2: stream @ A,Ya: stream @ A,P3: A > $o,Pa: A > $o] :
% 0.16/0.39 ( ( X2 = Ya )
% 0.16/0.39 => ( ! [Z: A] :
% 0.16/0.39 ( ( member @ A @ Z @ ( sset @ A @ Ya ) )
% 0.16/0.39 => ( ( P3 @ Z )
% 0.16/0.39 = ( Pa @ Z ) ) )
% 0.16/0.39 => ( ( pred_stream @ A @ P3 @ X2 )
% 0.16/0.39 = ( pred_stream @ A @ Pa @ Ya ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % stream.pred_cong
% 0.16/0.39 thf(fact_118_lfp_Oleq__antisym,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: A,Y: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.16/0.39 => ( ( ord_less_eq @ A @ Y @ X2 )
% 0.16/0.39 => ( X2 = Y ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % lfp.leq_antisym
% 0.16/0.39 thf(fact_119_gfp_Oleq__antisym,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [Y: A,X2: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.16/0.39 => ( ( ord_less_eq @ A @ X2 @ Y )
% 0.16/0.39 => ( X2 = Y ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % gfp.leq_antisym
% 0.16/0.39 thf(fact_120_llistsum__inf,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( monoid_add @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [Xs: coinductive_llist @ A] :
% 0.16/0.39 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.16/0.39 => ( ( coindu780009021istsum @ A @ Xs )
% 0.16/0.39 = ( zero_zero @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % llistsum_inf
% 0.16/0.39 thf(fact_121_lset__inf__llist,axiom,
% 0.16/0.39 ! [A: $tType,F: nat > A] :
% 0.16/0.39 ( ( coinductive_lset @ A @ ( coindu68654304_llist @ A @ F ) )
% 0.16/0.39 = ( image @ nat @ A @ F @ ( top_top @ ( set @ nat ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % lset_inf_llist
% 0.16/0.39 thf(fact_122_sset__smerge,axiom,
% 0.16/0.39 ! [A: $tType,Ss: stream @ ( stream @ A )] :
% 0.16/0.39 ( ( sset @ A @ ( smerge @ A @ Ss ) )
% 0.16/0.39 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( stream @ A ) @ ( set @ A ) @ ( sset @ A ) @ ( sset @ ( stream @ A ) @ Ss ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % sset_smerge
% 0.16/0.39 thf(fact_123_llist_Opred__set,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( coindu543516966_llist @ A )
% 0.16/0.39 = ( ^ [P: A > $o,X: coinductive_llist @ A] :
% 0.16/0.39 ! [Y4: A] :
% 0.16/0.39 ( ( member @ A @ Y4 @ ( coinductive_lset @ A @ X ) )
% 0.16/0.39 => ( P @ Y4 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % llist.pred_set
% 0.16/0.39 thf(fact_124_top__apply,axiom,
% 0.16/0.39 ! [C: $tType,D2: $tType] :
% 0.16/0.39 ( ( top @ C @ ( type2 @ C ) )
% 0.16/0.39 => ( ( top_top @ ( D2 > C ) )
% 0.16/0.39 = ( ^ [X: D2] : ( top_top @ C ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % top_apply
% 0.16/0.39 thf(fact_125_UNIV__I,axiom,
% 0.16/0.39 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % UNIV_I
% 0.16/0.39 thf(fact_126_Union__iff,axiom,
% 0.16/0.39 ! [A: $tType,A2: A,C4: set @ ( set @ A )] :
% 0.16/0.39 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
% 0.16/0.39 = ( ? [X: set @ A] :
% 0.16/0.39 ( ( member @ ( set @ A ) @ X @ C4 )
% 0.16/0.39 & ( member @ A @ A2 @ X ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Union_iff
% 0.16/0.39 thf(fact_127_UnionI,axiom,
% 0.16/0.39 ! [A: $tType,X3: set @ A,C4: set @ ( set @ A ),A2: A] :
% 0.16/0.39 ( ( member @ ( set @ A ) @ X3 @ C4 )
% 0.16/0.39 => ( ( member @ A @ A2 @ X3 )
% 0.16/0.39 => ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % UnionI
% 0.16/0.39 thf(fact_128_UN__ball__bex__simps_I1_J,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ ( set @ A ),P3: A > $o] :
% 0.16/0.39 ( ( ! [X: A] :
% 0.16/0.39 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
% 0.16/0.39 => ( P3 @ X ) ) )
% 0.16/0.39 = ( ! [X: set @ A] :
% 0.16/0.39 ( ( member @ ( set @ A ) @ X @ A2 )
% 0.16/0.39 => ! [Y4: A] :
% 0.16/0.39 ( ( member @ A @ Y4 @ X )
% 0.16/0.39 => ( P3 @ Y4 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % UN_ball_bex_simps(1)
% 0.16/0.39 thf(fact_129_UN__ball__bex__simps_I3_J,axiom,
% 0.16/0.39 ! [D2: $tType,A2: set @ ( set @ D2 ),P3: D2 > $o] :
% 0.16/0.39 ( ( ? [X: D2] :
% 0.16/0.39 ( ( member @ D2 @ X @ ( complete_Sup_Sup @ ( set @ D2 ) @ A2 ) )
% 0.16/0.39 & ( P3 @ X ) ) )
% 0.16/0.39 = ( ? [X: set @ D2] :
% 0.16/0.39 ( ( member @ ( set @ D2 ) @ X @ A2 )
% 0.16/0.39 & ? [Y4: D2] :
% 0.16/0.39 ( ( member @ D2 @ Y4 @ X )
% 0.16/0.39 & ( P3 @ Y4 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % UN_ball_bex_simps(3)
% 0.16/0.39 thf(fact_130_UN__ball__bex__simps_I4_J,axiom,
% 0.16/0.39 ! [F3: $tType,E: $tType,B2: E > ( set @ F3 ),A2: set @ E,P3: F3 > $o] :
% 0.16/0.39 ( ( ? [X: F3] :
% 0.16/0.39 ( ( member @ F3 @ X @ ( complete_Sup_Sup @ ( set @ F3 ) @ ( image @ E @ ( set @ F3 ) @ B2 @ A2 ) ) )
% 0.16/0.39 & ( P3 @ X ) ) )
% 0.16/0.39 = ( ? [X: E] :
% 0.16/0.39 ( ( member @ E @ X @ A2 )
% 0.16/0.39 & ? [Y4: F3] :
% 0.16/0.39 ( ( member @ F3 @ Y4 @ ( B2 @ X ) )
% 0.16/0.39 & ( P3 @ Y4 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % UN_ball_bex_simps(4)
% 0.16/0.39 thf(fact_131_UN__ball__bex__simps_I2_J,axiom,
% 0.16/0.39 ! [C: $tType,B: $tType,B2: B > ( set @ C ),A2: set @ B,P3: C > $o] :
% 0.16/0.39 ( ( ! [X: C] :
% 0.16/0.39 ( ( member @ C @ X @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ B2 @ A2 ) ) )
% 0.16/0.39 => ( P3 @ X ) ) )
% 0.16/0.39 = ( ! [X: B] :
% 0.16/0.39 ( ( member @ B @ X @ A2 )
% 0.16/0.39 => ! [Y4: C] :
% 0.16/0.39 ( ( member @ C @ Y4 @ ( B2 @ X ) )
% 0.16/0.39 => ( P3 @ Y4 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % UN_ball_bex_simps(2)
% 0.16/0.39 thf(fact_132_bex__UN,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P3: A > $o] :
% 0.16/0.39 ( ( ? [X: A] :
% 0.16/0.39 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
% 0.16/0.39 & ( P3 @ X ) ) )
% 0.16/0.39 = ( ? [X: B] :
% 0.16/0.39 ( ( member @ B @ X @ A2 )
% 0.16/0.39 & ? [Y4: A] :
% 0.16/0.39 ( ( member @ A @ Y4 @ ( B2 @ X ) )
% 0.16/0.39 & ( P3 @ Y4 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % bex_UN
% 0.16/0.39 thf(fact_133_ball__UN,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P3: A > $o] :
% 0.16/0.39 ( ( ! [X: A] :
% 0.16/0.39 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
% 0.16/0.39 => ( P3 @ X ) ) )
% 0.16/0.39 = ( ! [X: B] :
% 0.16/0.39 ( ( member @ B @ X @ A2 )
% 0.16/0.39 => ! [Y4: A] :
% 0.16/0.39 ( ( member @ A @ Y4 @ ( B2 @ X ) )
% 0.16/0.39 => ( P3 @ Y4 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % ball_UN
% 0.16/0.39 thf(fact_134_streams__UNIV,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( streams @ A @ ( top_top @ ( set @ A ) ) )
% 0.16/0.39 = ( top_top @ ( set @ ( stream @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % streams_UNIV
% 0.16/0.39 thf(fact_135_Sup__UNIV,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
% 0.16/0.39 = ( top_top @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup_UNIV
% 0.16/0.39 thf(fact_136_Sup__eqI,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ A,X2: A] :
% 0.16/0.39 ( ! [Y2: A] :
% 0.16/0.39 ( ( member @ A @ Y2 @ A2 )
% 0.16/0.39 => ( ord_less_eq @ A @ Y2 @ X2 ) )
% 0.16/0.39 => ( ! [Y2: A] :
% 0.16/0.39 ( ! [Z4: A] :
% 0.16/0.39 ( ( member @ A @ Z4 @ A2 )
% 0.16/0.39 => ( ord_less_eq @ A @ Z4 @ Y2 ) )
% 0.16/0.39 => ( ord_less_eq @ A @ X2 @ Y2 ) )
% 0.16/0.39 => ( ( complete_Sup_Sup @ A @ A2 )
% 0.16/0.39 = X2 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup_eqI
% 0.16/0.39 thf(fact_137_complete__lattice__class_OSup__mono,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ A,B2: set @ A] :
% 0.16/0.39 ( ! [A4: A] :
% 0.16/0.39 ( ( member @ A @ A4 @ A2 )
% 0.16/0.39 => ? [X5: A] :
% 0.16/0.39 ( ( member @ A @ X5 @ B2 )
% 0.16/0.39 & ( ord_less_eq @ A @ A4 @ X5 ) ) )
% 0.16/0.39 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % complete_lattice_class.Sup_mono
% 0.16/0.39 thf(fact_138_Sup__least,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ A,Z3: A] :
% 0.16/0.39 ( ! [X4: A] :
% 0.16/0.39 ( ( member @ A @ X4 @ A2 )
% 0.16/0.39 => ( ord_less_eq @ A @ X4 @ Z3 ) )
% 0.16/0.39 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ Z3 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup_least
% 0.16/0.39 thf(fact_139_Sup__upper,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: A,A2: set @ A] :
% 0.16/0.39 ( ( member @ A @ X2 @ A2 )
% 0.16/0.39 => ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup_upper
% 0.16/0.39 thf(fact_140_Sup__le__iff,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ A,B3: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ B3 )
% 0.16/0.39 = ( ! [X: A] :
% 0.16/0.39 ( ( member @ A @ X @ A2 )
% 0.16/0.39 => ( ord_less_eq @ A @ X @ B3 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup_le_iff
% 0.16/0.39 thf(fact_141_Sup__upper2,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [U: A,A2: set @ A,V: A] :
% 0.16/0.39 ( ( member @ A @ U @ A2 )
% 0.16/0.39 => ( ( ord_less_eq @ A @ V @ U )
% 0.16/0.39 => ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup_upper2
% 0.16/0.39 thf(fact_142_top__greatest,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( order_top @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % top_greatest
% 0.16/0.39 thf(fact_143_top_Oextremum__unique,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( order_top @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A3: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.16/0.39 = ( A3
% 0.16/0.39 = ( top_top @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % top.extremum_unique
% 0.16/0.39 thf(fact_144_top_Oextremum__uniqueI,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( order_top @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A3: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.16/0.39 => ( A3
% 0.16/0.39 = ( top_top @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % top.extremum_uniqueI
% 0.16/0.39 thf(fact_145_SUP__cong,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType] :
% 0.16/0.39 ( ( complete_Sup @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ B,B2: set @ B,C4: B > A,D: B > A] :
% 0.16/0.39 ( ( A2 = B2 )
% 0.16/0.39 => ( ! [X4: B] :
% 0.16/0.39 ( ( member @ B @ X4 @ B2 )
% 0.16/0.39 => ( ( C4 @ X4 )
% 0.16/0.39 = ( D @ X4 ) ) )
% 0.16/0.39 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ C4 @ A2 ) )
% 0.16/0.39 = ( complete_Sup_Sup @ A @ ( image @ B @ A @ D @ B2 ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % SUP_cong
% 0.16/0.39 thf(fact_146_Union__upper,axiom,
% 0.16/0.39 ! [A: $tType,B2: set @ A,A2: set @ ( set @ A )] :
% 0.16/0.39 ( ( member @ ( set @ A ) @ B2 @ A2 )
% 0.16/0.39 => ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Union_upper
% 0.16/0.39 thf(fact_147_Union__least,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ ( set @ A ),C4: set @ A] :
% 0.16/0.39 ( ! [X6: set @ A] :
% 0.16/0.39 ( ( member @ ( set @ A ) @ X6 @ A2 )
% 0.16/0.39 => ( ord_less_eq @ ( set @ A ) @ X6 @ C4 ) )
% 0.16/0.39 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ C4 ) ) ).
% 0.16/0.39
% 0.16/0.39 % Union_least
% 0.16/0.39 thf(fact_148_Union__mono,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ B2 )
% 0.16/0.39 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Union_mono
% 0.16/0.39 thf(fact_149_range__eqI,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,B3: A,F: B > A,X2: B] :
% 0.16/0.39 ( ( B3
% 0.16/0.39 = ( F @ X2 ) )
% 0.16/0.39 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % range_eqI
% 0.16/0.39 thf(fact_150_rangeI,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,F: B > A,X2: B] : ( member @ A @ ( F @ X2 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % rangeI
% 0.16/0.39 thf(fact_151_subset__UNIV,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % subset_UNIV
% 0.16/0.39 thf(fact_152_Union__UNIV,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 0.16/0.39 = ( top_top @ ( set @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Union_UNIV
% 0.16/0.39 thf(fact_153_UnionE,axiom,
% 0.16/0.39 ! [A: $tType,A2: A,C4: set @ ( set @ A )] :
% 0.16/0.39 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
% 0.16/0.39 => ~ ! [X6: set @ A] :
% 0.16/0.39 ( ( member @ A @ A2 @ X6 )
% 0.16/0.39 => ~ ( member @ ( set @ A ) @ X6 @ C4 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % UnionE
% 0.16/0.39 thf(fact_154_UNIV__witness,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % UNIV_witness
% 0.16/0.39 thf(fact_155_UNIV__eq__I,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A] :
% 0.16/0.39 ( ! [X4: A] : ( member @ A @ X4 @ A2 )
% 0.16/0.39 => ( ( top_top @ ( set @ A ) )
% 0.16/0.39 = A2 ) ) ).
% 0.16/0.39
% 0.16/0.39 % UNIV_eq_I
% 0.16/0.39 thf(fact_156_llist_Opred__cong,axiom,
% 0.16/0.39 ! [A: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,P3: A > $o,Pa: A > $o] :
% 0.16/0.39 ( ( X2 = Ya )
% 0.16/0.39 => ( ! [Z: A] :
% 0.16/0.39 ( ( member @ A @ Z @ ( coinductive_lset @ A @ Ya ) )
% 0.16/0.39 => ( ( P3 @ Z )
% 0.16/0.39 = ( Pa @ Z ) ) )
% 0.16/0.39 => ( ( coindu543516966_llist @ A @ P3 @ X2 )
% 0.16/0.39 = ( coindu543516966_llist @ A @ Pa @ Ya ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % llist.pred_cong
% 0.16/0.39 thf(fact_157_llist_Opred__mono__strong,axiom,
% 0.16/0.39 ! [A: $tType,P3: A > $o,X2: coinductive_llist @ A,Pa: A > $o] :
% 0.16/0.39 ( ( coindu543516966_llist @ A @ P3 @ X2 )
% 0.16/0.39 => ( ! [Z: A] :
% 0.16/0.39 ( ( member @ A @ Z @ ( coinductive_lset @ A @ X2 ) )
% 0.16/0.39 => ( ( P3 @ Z )
% 0.16/0.39 => ( Pa @ Z ) ) )
% 0.16/0.39 => ( coindu543516966_llist @ A @ Pa @ X2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % llist.pred_mono_strong
% 0.16/0.39 thf(fact_158_SUP__eq,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,C: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ B,B2: set @ C,F: B > A,G: C > A] :
% 0.16/0.39 ( ! [I: B] :
% 0.16/0.39 ( ( member @ B @ I @ A2 )
% 0.16/0.39 => ? [X5: C] :
% 0.16/0.39 ( ( member @ C @ X5 @ B2 )
% 0.16/0.39 & ( ord_less_eq @ A @ ( F @ I ) @ ( G @ X5 ) ) ) )
% 0.16/0.39 => ( ! [J: C] :
% 0.16/0.39 ( ( member @ C @ J @ B2 )
% 0.16/0.39 => ? [X5: B] :
% 0.16/0.39 ( ( member @ B @ X5 @ A2 )
% 0.16/0.39 & ( ord_less_eq @ A @ ( G @ J ) @ ( F @ X5 ) ) ) )
% 0.16/0.39 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) )
% 0.16/0.39 = ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % SUP_eq
% 0.16/0.39 thf(fact_159_Sup__subset__mono,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ A,B2: set @ A] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup_subset_mono
% 0.16/0.39 thf(fact_160_sset__range,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( sset @ A )
% 0.16/0.39 = ( ^ [S: stream @ A] : ( image @ nat @ A @ ( snth @ A @ S ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % sset_range
% 0.16/0.39 thf(fact_161_llistsum__infllist,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( monoid_add @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [F: nat > A] :
% 0.16/0.39 ( ( coindu780009021istsum @ A @ ( coindu68654304_llist @ A @ F ) )
% 0.16/0.39 = ( zero_zero @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % llistsum_infllist
% 0.16/0.39 thf(fact_162_le__zero__eq,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [N: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.16/0.39 = ( N
% 0.16/0.39 = ( zero_zero @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % le_zero_eq
% 0.16/0.39 thf(fact_163_iso__tuple__UNIV__I,axiom,
% 0.16/0.39 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % iso_tuple_UNIV_I
% 0.16/0.39 thf(fact_164_surj__def,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,F: B > A] :
% 0.16/0.39 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.16/0.39 = ( top_top @ ( set @ A ) ) )
% 0.16/0.39 = ( ! [Y4: A] :
% 0.16/0.39 ? [X: B] :
% 0.16/0.39 ( Y4
% 0.16/0.39 = ( F @ X ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % surj_def
% 0.16/0.39 thf(fact_165_top__set__def,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( top_top @ ( set @ A ) )
% 0.16/0.39 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % top_set_def
% 0.16/0.39 thf(fact_166_zero__reorient,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( zero @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: A] :
% 0.16/0.39 ( ( ( zero_zero @ A )
% 0.16/0.39 = X2 )
% 0.16/0.39 = ( X2
% 0.16/0.39 = ( zero_zero @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % zero_reorient
% 0.16/0.39 thf(fact_167_zero__le,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% 0.16/0.39
% 0.16/0.39 % zero_le
% 0.16/0.39 thf(fact_168_surjD,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.16/0.39 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.16/0.39 = ( top_top @ ( set @ A ) ) )
% 0.16/0.39 => ? [X4: B] :
% 0.16/0.39 ( Y
% 0.16/0.39 = ( F @ X4 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % surjD
% 0.16/0.39 thf(fact_169_surjE,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.16/0.39 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.16/0.39 = ( top_top @ ( set @ A ) ) )
% 0.16/0.39 => ~ ! [X4: B] :
% 0.16/0.39 ( Y
% 0.16/0.39 != ( F @ X4 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % surjE
% 0.16/0.39 thf(fact_170_surjI,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 0.16/0.39 ( ! [X4: A] :
% 0.16/0.39 ( ( G @ ( F @ X4 ) )
% 0.16/0.39 = X4 )
% 0.16/0.39 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 0.16/0.39 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % surjI
% 0.16/0.39 thf(fact_171_cSup__eq,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 & ( no_bot @ A @ ( type2 @ A ) ) )
% 0.16/0.39 => ! [X3: set @ A,A3: A] :
% 0.16/0.39 ( ! [X4: A] :
% 0.16/0.39 ( ( member @ A @ X4 @ X3 )
% 0.16/0.39 => ( ord_less_eq @ A @ X4 @ A3 ) )
% 0.16/0.39 => ( ! [Y2: A] :
% 0.16/0.39 ( ! [X5: A] :
% 0.16/0.39 ( ( member @ A @ X5 @ X3 )
% 0.16/0.39 => ( ord_less_eq @ A @ X5 @ Y2 ) )
% 0.16/0.39 => ( ord_less_eq @ A @ A3 @ Y2 ) )
% 0.16/0.39 => ( ( complete_Sup_Sup @ A @ X3 )
% 0.16/0.39 = A3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSup_eq
% 0.16/0.39 thf(fact_172_cSup__eq__maximum,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [Z3: A,X3: set @ A] :
% 0.16/0.39 ( ( member @ A @ Z3 @ X3 )
% 0.16/0.39 => ( ! [X4: A] :
% 0.16/0.39 ( ( member @ A @ X4 @ X3 )
% 0.16/0.39 => ( ord_less_eq @ A @ X4 @ Z3 ) )
% 0.16/0.39 => ( ( complete_Sup_Sup @ A @ X3 )
% 0.16/0.39 = Z3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSup_eq_maximum
% 0.16/0.39 thf(fact_173_le__numeral__extra_I3_J,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.39 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % le_numeral_extra(3)
% 0.16/0.39 thf(fact_174_top1I,axiom,
% 0.16/0.39 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 0.16/0.39
% 0.16/0.39 % top1I
% 0.16/0.39 thf(fact_175_top__empty__eq,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( top_top @ ( A > $o ) )
% 0.16/0.39 = ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % top_empty_eq
% 0.16/0.39 thf(fact_176_top__conj_I2_J,axiom,
% 0.16/0.39 ! [A: $tType,P3: $o,X2: A] :
% 0.16/0.39 ( ( P3
% 0.16/0.39 & ( top_top @ ( A > $o ) @ X2 ) )
% 0.16/0.39 = P3 ) ).
% 0.16/0.39
% 0.16/0.39 % top_conj(2)
% 0.16/0.39 thf(fact_177_top__conj_I1_J,axiom,
% 0.16/0.39 ! [A: $tType,X2: A,P3: $o] :
% 0.16/0.39 ( ( ( top_top @ ( A > $o ) @ X2 )
% 0.16/0.39 & P3 )
% 0.16/0.39 = P3 ) ).
% 0.16/0.39
% 0.16/0.39 % top_conj(1)
% 0.16/0.39 thf(fact_178_Sup__SUP__eq,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( complete_Sup_Sup @ ( A > $o ) )
% 0.16/0.39 = ( ^ [S6: set @ ( A > $o ),X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S6 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup_SUP_eq
% 0.16/0.39 thf(fact_179_surj__from__nat,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( countable @ A @ ( type2 @ A ) )
% 0.16/0.39 => ( ( image @ nat @ A @ ( from_nat @ A ) @ ( top_top @ ( set @ nat ) ) )
% 0.16/0.39 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % surj_from_nat
% 0.16/0.39 thf(fact_180_member__bind,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,X2: A,P3: set @ B,F: B > ( set @ A )] :
% 0.16/0.39 ( ( member @ A @ X2 @ ( bind @ B @ A @ P3 @ F ) )
% 0.16/0.39 = ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F @ P3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % member_bind
% 0.16/0.39 thf(fact_181_image__Pow__mono,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 0.16/0.39 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) ) @ ( pow @ A @ B2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % image_Pow_mono
% 0.16/0.39 thf(fact_182_Sup1__I,axiom,
% 0.16/0.39 ! [A: $tType,P3: A > $o,A2: set @ ( A > $o ),A3: A] :
% 0.16/0.39 ( ( member @ ( A > $o ) @ P3 @ A2 )
% 0.16/0.39 => ( ( P3 @ A3 )
% 0.16/0.39 => ( complete_Sup_Sup @ ( A > $o ) @ A2 @ A3 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup1_I
% 0.16/0.39 thf(fact_183_Pow__iff,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.16/0.39 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 0.16/0.39 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.16/0.39
% 0.16/0.39 % Pow_iff
% 0.16/0.39 thf(fact_184_PowI,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % PowI
% 0.16/0.39 thf(fact_185_Union__Pow__eq,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A] :
% 0.16/0.39 ( ( complete_Sup_Sup @ ( set @ A ) @ ( pow @ A @ A2 ) )
% 0.16/0.39 = A2 ) ).
% 0.16/0.39
% 0.16/0.39 % Union_Pow_eq
% 0.16/0.39 thf(fact_186_Pow__UNIV,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( pow @ A @ ( top_top @ ( set @ A ) ) )
% 0.16/0.39 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Pow_UNIV
% 0.16/0.39 thf(fact_187_image__Pow__surj,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 0.16/0.39 ( ( ( image @ B @ A @ F @ A2 )
% 0.16/0.39 = B2 )
% 0.16/0.39 => ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) )
% 0.16/0.39 = ( pow @ A @ B2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % image_Pow_surj
% 0.16/0.39 thf(fact_188_Sup1__E,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ ( A > $o ),A3: A] :
% 0.16/0.39 ( ( complete_Sup_Sup @ ( A > $o ) @ A2 @ A3 )
% 0.16/0.39 => ~ ! [P4: A > $o] :
% 0.16/0.39 ( ( member @ ( A > $o ) @ P4 @ A2 )
% 0.16/0.39 => ~ ( P4 @ A3 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup1_E
% 0.16/0.39 thf(fact_189_Cantors__paradox,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A] :
% 0.16/0.39 ~ ? [X5: A > ( set @ A )] :
% 0.16/0.39 ( ( image @ A @ ( set @ A ) @ X5 @ A2 )
% 0.16/0.39 = ( pow @ A @ A2 ) ) ).
% 0.16/0.39
% 0.16/0.39 % Cantors_paradox
% 0.16/0.39 thf(fact_190_PowD,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.16/0.39 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 0.16/0.39 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.16/0.39
% 0.16/0.39 % PowD
% 0.16/0.39 thf(fact_191_Pow__top,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% 0.16/0.39
% 0.16/0.39 % Pow_top
% 0.16/0.39 thf(fact_192_Pow__mono,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Pow_mono
% 0.16/0.39 thf(fact_193_subset__Pow__Union,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % subset_Pow_Union
% 0.16/0.39 thf(fact_194_bind__UNION,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType] :
% 0.16/0.39 ( ( bind @ B @ A )
% 0.16/0.39 = ( ^ [A5: set @ B,F2: B > ( set @ A )] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ A5 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % bind_UNION
% 0.16/0.39 thf(fact_195_surj__swap__iff,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,A3: B,B3: B,F: B > A] :
% 0.16/0.39 ( ( ( image @ B @ A @ ( swap @ B @ A @ A3 @ B3 @ F ) @ ( top_top @ ( set @ B ) ) )
% 0.16/0.39 = ( top_top @ ( set @ A ) ) )
% 0.16/0.39 = ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.16/0.39 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % surj_swap_iff
% 0.16/0.39 thf(fact_196_cSUP__least,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ B,F: B > A,M2: A] :
% 0.16/0.39 ( ( A2
% 0.16/0.39 != ( bot_bot @ ( set @ B ) ) )
% 0.16/0.39 => ( ! [X4: B] :
% 0.16/0.39 ( ( member @ B @ X4 @ A2 )
% 0.16/0.39 => ( ord_less_eq @ A @ ( F @ X4 ) @ M2 ) )
% 0.16/0.39 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ M2 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSUP_least
% 0.16/0.39 thf(fact_197_SUP__eq__iff,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [I2: set @ B,C3: A,F: B > A] :
% 0.16/0.39 ( ( I2
% 0.16/0.39 != ( bot_bot @ ( set @ B ) ) )
% 0.16/0.39 => ( ! [I: B] :
% 0.16/0.39 ( ( member @ B @ I @ I2 )
% 0.16/0.39 => ( ord_less_eq @ A @ C3 @ ( F @ I ) ) )
% 0.16/0.39 => ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ I2 ) )
% 0.16/0.39 = C3 )
% 0.16/0.39 = ( ! [X: B] :
% 0.16/0.39 ( ( member @ B @ X @ I2 )
% 0.16/0.39 => ( ( F @ X )
% 0.16/0.39 = C3 ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % SUP_eq_iff
% 0.16/0.39 thf(fact_198_bot__apply,axiom,
% 0.16/0.39 ! [C: $tType,D2: $tType] :
% 0.16/0.39 ( ( bot @ C @ ( type2 @ C ) )
% 0.16/0.39 => ( ( bot_bot @ ( D2 > C ) )
% 0.16/0.39 = ( ^ [X: D2] : ( bot_bot @ C ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % bot_apply
% 0.16/0.39 thf(fact_199_empty__iff,axiom,
% 0.16/0.39 ! [A: $tType,C3: A] :
% 0.16/0.39 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % empty_iff
% 0.16/0.39 thf(fact_200_all__not__in__conv,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A] :
% 0.16/0.39 ( ( ! [X: A] :
% 0.16/0.39 ~ ( member @ A @ X @ A2 ) )
% 0.16/0.39 = ( A2
% 0.16/0.39 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % all_not_in_conv
% 0.16/0.39 thf(fact_201_Collect__empty__eq,axiom,
% 0.16/0.39 ! [A: $tType,P3: A > $o] :
% 0.16/0.39 ( ( ( collect @ A @ P3 )
% 0.16/0.39 = ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 = ( ! [X: A] :
% 0.16/0.39 ~ ( P3 @ X ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Collect_empty_eq
% 0.16/0.39 thf(fact_202_empty__Collect__eq,axiom,
% 0.16/0.39 ! [A: $tType,P3: A > $o] :
% 0.16/0.39 ( ( ( bot_bot @ ( set @ A ) )
% 0.16/0.39 = ( collect @ A @ P3 ) )
% 0.16/0.39 = ( ! [X: A] :
% 0.16/0.39 ~ ( P3 @ X ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % empty_Collect_eq
% 0.16/0.39 thf(fact_203_image__is__empty,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
% 0.16/0.39 ( ( ( image @ B @ A @ F @ A2 )
% 0.16/0.39 = ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 = ( A2
% 0.16/0.39 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % image_is_empty
% 0.16/0.39 thf(fact_204_empty__is__image,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
% 0.16/0.39 ( ( ( bot_bot @ ( set @ A ) )
% 0.16/0.39 = ( image @ B @ A @ F @ A2 ) )
% 0.16/0.39 = ( A2
% 0.16/0.39 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % empty_is_image
% 0.16/0.39 thf(fact_205_image__empty,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,F: B > A] :
% 0.16/0.39 ( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
% 0.16/0.39 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % image_empty
% 0.16/0.39 thf(fact_206_Sup__bot__conv_I1_J,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ A] :
% 0.16/0.39 ( ( ( complete_Sup_Sup @ A @ A2 )
% 0.16/0.39 = ( bot_bot @ A ) )
% 0.16/0.39 = ( ! [X: A] :
% 0.16/0.39 ( ( member @ A @ X @ A2 )
% 0.16/0.39 => ( X
% 0.16/0.39 = ( bot_bot @ A ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup_bot_conv(1)
% 0.16/0.39 thf(fact_207_Sup__bot__conv_I2_J,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ A] :
% 0.16/0.39 ( ( ( bot_bot @ A )
% 0.16/0.39 = ( complete_Sup_Sup @ A @ A2 ) )
% 0.16/0.39 = ( ! [X: A] :
% 0.16/0.39 ( ( member @ A @ X @ A2 )
% 0.16/0.39 => ( X
% 0.16/0.39 = ( bot_bot @ A ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup_bot_conv(2)
% 0.16/0.39 thf(fact_208_subset__empty,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A] :
% 0.16/0.39 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 = ( A2
% 0.16/0.39 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % subset_empty
% 0.16/0.39 thf(fact_209_empty__subsetI,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 0.16/0.39
% 0.16/0.39 % empty_subsetI
% 0.16/0.39 thf(fact_210_ball__empty,axiom,
% 0.16/0.39 ! [A: $tType,P3: A > $o,X5: A] :
% 0.16/0.39 ( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 => ( P3 @ X5 ) ) ).
% 0.16/0.39
% 0.16/0.39 % ball_empty
% 0.16/0.39 thf(fact_211_swap__image__eq,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,A3: A,A2: set @ A,B3: A,F: A > B] :
% 0.16/0.39 ( ( member @ A @ A3 @ A2 )
% 0.16/0.39 => ( ( member @ A @ B3 @ A2 )
% 0.16/0.39 => ( ( image @ A @ B @ ( swap @ A @ B @ A3 @ B3 @ F ) @ A2 )
% 0.16/0.39 = ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % swap_image_eq
% 0.16/0.39 thf(fact_212_empty__bind,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,F: B > ( set @ A )] :
% 0.16/0.39 ( ( bind @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F )
% 0.16/0.39 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % empty_bind
% 0.16/0.39 thf(fact_213_Sup__empty,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 = ( bot_bot @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Sup_empty
% 0.16/0.39 thf(fact_214_empty__Union__conv,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ ( set @ A )] :
% 0.16/0.39 ( ( ( bot_bot @ ( set @ A ) )
% 0.16/0.39 = ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
% 0.16/0.39 = ( ! [X: set @ A] :
% 0.16/0.39 ( ( member @ ( set @ A ) @ X @ A2 )
% 0.16/0.39 => ( X
% 0.16/0.39 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % empty_Union_conv
% 0.16/0.39 thf(fact_215_Union__empty__conv,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ ( set @ A )] :
% 0.16/0.39 ( ( ( complete_Sup_Sup @ ( set @ A ) @ A2 )
% 0.16/0.39 = ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 = ( ! [X: set @ A] :
% 0.16/0.39 ( ( member @ ( set @ A ) @ X @ A2 )
% 0.16/0.39 => ( X
% 0.16/0.39 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Union_empty_conv
% 0.16/0.39 thf(fact_216_Union__empty,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 0.16/0.39 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Union_empty
% 0.16/0.39 thf(fact_217_Pow__not__empty,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A] :
% 0.16/0.39 ( ( pow @ A @ A2 )
% 0.16/0.39 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % Pow_not_empty
% 0.16/0.39 thf(fact_218_Pow__bottom,axiom,
% 0.16/0.39 ! [A: $tType,B2: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B2 ) ) ).
% 0.16/0.39
% 0.16/0.39 % Pow_bottom
% 0.16/0.39 thf(fact_219_empty__not__UNIV,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( bot_bot @ ( set @ A ) )
% 0.16/0.39 != ( top_top @ ( set @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % empty_not_UNIV
% 0.16/0.39 thf(fact_220_sset__neq__empty,axiom,
% 0.16/0.39 ! [A: $tType,Xs: stream @ A] :
% 0.16/0.39 ( ( sset @ A @ Xs )
% 0.16/0.39 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % sset_neq_empty
% 0.16/0.39 thf(fact_221_emptyE,axiom,
% 0.16/0.39 ! [A: $tType,A3: A] :
% 0.16/0.39 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % emptyE
% 0.16/0.39 thf(fact_222_equals0D,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A,A3: A] :
% 0.16/0.39 ( ( A2
% 0.16/0.39 = ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 0.16/0.39
% 0.16/0.39 % equals0D
% 0.16/0.39 thf(fact_223_equals0I,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A] :
% 0.16/0.39 ( ! [Y2: A] :
% 0.16/0.39 ~ ( member @ A @ Y2 @ A2 )
% 0.16/0.39 => ( A2
% 0.16/0.39 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % equals0I
% 0.16/0.39 thf(fact_224_ex__in__conv,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A] :
% 0.16/0.39 ( ( ? [X: A] : ( member @ A @ X @ A2 ) )
% 0.16/0.39 = ( A2
% 0.16/0.39 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % ex_in_conv
% 0.16/0.39 thf(fact_225_bot__fun__def,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType] :
% 0.16/0.39 ( ( bot @ B @ ( type2 @ B ) )
% 0.16/0.39 => ( ( bot_bot @ ( A > B ) )
% 0.16/0.39 = ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % bot_fun_def
% 0.16/0.39 thf(fact_226_streams__empty,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( streams @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 = ( bot_bot @ ( set @ ( stream @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % streams_empty
% 0.16/0.39 thf(fact_227_bot_Oextremum,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 0.16/0.39
% 0.16/0.39 % bot.extremum
% 0.16/0.39 thf(fact_228_bot_Oextremum__unique,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A3: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.16/0.39 = ( A3
% 0.16/0.39 = ( bot_bot @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % bot.extremum_unique
% 0.16/0.39 thf(fact_229_bot_Oextremum__uniqueI,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A3: A] :
% 0.16/0.39 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.16/0.39 => ( A3
% 0.16/0.39 = ( bot_bot @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % bot.extremum_uniqueI
% 0.16/0.39 thf(fact_230_less__eq__Sup,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ A,U: A] :
% 0.16/0.39 ( ! [V2: A] :
% 0.16/0.39 ( ( member @ A @ V2 @ A2 )
% 0.16/0.39 => ( ord_less_eq @ A @ U @ V2 ) )
% 0.16/0.39 => ( ( A2
% 0.16/0.39 != ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % less_eq_Sup
% 0.16/0.39 thf(fact_231_cSup__least,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X3: set @ A,Z3: A] :
% 0.16/0.39 ( ( X3
% 0.16/0.39 != ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 => ( ! [X4: A] :
% 0.16/0.39 ( ( member @ A @ X4 @ X3 )
% 0.16/0.39 => ( ord_less_eq @ A @ X4 @ Z3 ) )
% 0.16/0.39 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X3 ) @ Z3 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSup_least
% 0.16/0.39 thf(fact_232_cSup__eq__non__empty,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X3: set @ A,A3: A] :
% 0.16/0.39 ( ( X3
% 0.16/0.39 != ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 => ( ! [X4: A] :
% 0.16/0.39 ( ( member @ A @ X4 @ X3 )
% 0.16/0.39 => ( ord_less_eq @ A @ X4 @ A3 ) )
% 0.16/0.39 => ( ! [Y2: A] :
% 0.16/0.39 ( ! [X5: A] :
% 0.16/0.39 ( ( member @ A @ X5 @ X3 )
% 0.16/0.39 => ( ord_less_eq @ A @ X5 @ Y2 ) )
% 0.16/0.39 => ( ord_less_eq @ A @ A3 @ Y2 ) )
% 0.16/0.39 => ( ( complete_Sup_Sup @ A @ X3 )
% 0.16/0.39 = A3 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSup_eq_non_empty
% 0.16/0.39 thf(fact_233_SUP__eq__const,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [I2: set @ B,F: B > A,X2: A] :
% 0.16/0.39 ( ( I2
% 0.16/0.39 != ( bot_bot @ ( set @ B ) ) )
% 0.16/0.39 => ( ! [I: B] :
% 0.16/0.39 ( ( member @ B @ I @ I2 )
% 0.16/0.39 => ( ( F @ I )
% 0.16/0.39 = X2 ) )
% 0.16/0.39 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ I2 ) )
% 0.16/0.39 = X2 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % SUP_eq_const
% 0.16/0.39 thf(fact_234_surj__imp__surj__swap,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,F: B > A,A3: B,B3: B] :
% 0.16/0.39 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.16/0.39 = ( top_top @ ( set @ A ) ) )
% 0.16/0.39 => ( ( image @ B @ A @ ( swap @ B @ A @ A3 @ B3 @ F ) @ ( top_top @ ( set @ B ) ) )
% 0.16/0.39 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % surj_imp_surj_swap
% 0.16/0.39 thf(fact_235_ccSup__empty,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( counta840220525attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 = ( bot_bot @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % ccSup_empty
% 0.16/0.39 thf(fact_236_subset__emptyI,axiom,
% 0.16/0.39 ! [A: $tType,A2: set @ A] :
% 0.16/0.39 ( ! [X4: A] :
% 0.16/0.39 ~ ( member @ A @ X4 @ A2 )
% 0.16/0.39 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % subset_emptyI
% 0.16/0.39 thf(fact_237_bot__set__def,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( bot_bot @ ( set @ A ) )
% 0.16/0.39 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % bot_set_def
% 0.16/0.39 thf(fact_238_lset__code,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( coinductive_lset @ A )
% 0.16/0.39 = ( coinductive_gen_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % lset_code
% 0.16/0.39 thf(fact_239_the__elem__image__unique,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,X2: A] :
% 0.16/0.39 ( ( A2
% 0.16/0.39 != ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 => ( ! [Y2: A] :
% 0.16/0.39 ( ( member @ A @ Y2 @ A2 )
% 0.16/0.39 => ( ( F @ Y2 )
% 0.16/0.39 = ( F @ X2 ) ) )
% 0.16/0.39 => ( ( the_elem @ B @ ( image @ A @ B @ F @ A2 ) )
% 0.16/0.39 = ( F @ X2 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % the_elem_image_unique
% 0.16/0.39 thf(fact_240_cSUP__subset__mono,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ B,G: B > A,B2: set @ B,F: B > A] :
% 0.16/0.39 ( ( A2
% 0.16/0.39 != ( bot_bot @ ( set @ B ) ) )
% 0.16/0.39 => ( ( condit1810911227_above @ A @ ( image @ B @ A @ G @ B2 ) )
% 0.16/0.39 => ( ( ord_less_eq @ ( set @ B ) @ A2 @ B2 )
% 0.16/0.39 => ( ! [X4: B] :
% 0.16/0.39 ( ( member @ B @ X4 @ B2 )
% 0.16/0.39 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.16/0.39 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B2 ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSUP_subset_mono
% 0.16/0.39 thf(fact_241_compact__bot,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( comple1141879883l_ccpo @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: A] :
% 0.16/0.39 ( ( X2
% 0.16/0.39 = ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) )
% 0.16/0.39 => ( comple2143767107ompact @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ X2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % compact_bot
% 0.16/0.39 thf(fact_242_bdd__aboveI,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ A,M2: A] :
% 0.16/0.39 ( ! [X4: A] :
% 0.16/0.39 ( ( member @ A @ X4 @ A2 )
% 0.16/0.39 => ( ord_less_eq @ A @ X4 @ M2 ) )
% 0.16/0.39 => ( condit1810911227_above @ A @ A2 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % bdd_aboveI
% 0.16/0.39 thf(fact_243_cSup__le__iff,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [S5: set @ A,A3: A] :
% 0.16/0.39 ( ( S5
% 0.16/0.39 != ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 => ( ( condit1810911227_above @ A @ S5 )
% 0.16/0.39 => ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S5 ) @ A3 )
% 0.16/0.39 = ( ! [X: A] :
% 0.16/0.39 ( ( member @ A @ X @ S5 )
% 0.16/0.39 => ( ord_less_eq @ A @ X @ A3 ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSup_le_iff
% 0.16/0.39 thf(fact_244_cSup__mono,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [B2: set @ A,A2: set @ A] :
% 0.16/0.39 ( ( B2
% 0.16/0.39 != ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 => ( ( condit1810911227_above @ A @ A2 )
% 0.16/0.39 => ( ! [B4: A] :
% 0.16/0.39 ( ( member @ A @ B4 @ B2 )
% 0.16/0.39 => ? [X5: A] :
% 0.16/0.39 ( ( member @ A @ X5 @ A2 )
% 0.16/0.39 & ( ord_less_eq @ A @ B4 @ X5 ) ) )
% 0.16/0.39 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B2 ) @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSup_mono
% 0.16/0.39 thf(fact_245_bdd__above__def,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.39 => ( ( condit1810911227_above @ A )
% 0.16/0.39 = ( ^ [A5: set @ A] :
% 0.16/0.39 ? [M3: A] :
% 0.16/0.39 ! [X: A] :
% 0.16/0.39 ( ( member @ A @ X @ A5 )
% 0.16/0.39 => ( ord_less_eq @ A @ X @ M3 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % bdd_above_def
% 0.16/0.39 thf(fact_246_bdd__above__mono,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [B2: set @ A,A2: set @ A] :
% 0.16/0.39 ( ( condit1810911227_above @ A @ B2 )
% 0.16/0.39 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( condit1810911227_above @ A @ A2 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % bdd_above_mono
% 0.16/0.39 thf(fact_247_cSup__upper2,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: A,X3: set @ A,Y: A] :
% 0.16/0.39 ( ( member @ A @ X2 @ X3 )
% 0.16/0.39 => ( ( ord_less_eq @ A @ Y @ X2 )
% 0.16/0.39 => ( ( condit1810911227_above @ A @ X3 )
% 0.16/0.39 => ( ord_less_eq @ A @ Y @ ( complete_Sup_Sup @ A @ X3 ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSup_upper2
% 0.16/0.39 thf(fact_248_cSup__upper,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: A,X3: set @ A] :
% 0.16/0.39 ( ( member @ A @ X2 @ X3 )
% 0.16/0.39 => ( ( condit1810911227_above @ A @ X3 )
% 0.16/0.39 => ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ X3 ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSup_upper
% 0.16/0.39 thf(fact_249_bdd__aboveI2,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType] :
% 0.16/0.39 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ B,F: B > A,M2: A] :
% 0.16/0.39 ( ! [X4: B] :
% 0.16/0.39 ( ( member @ B @ X4 @ A2 )
% 0.16/0.39 => ( ord_less_eq @ A @ ( F @ X4 ) @ M2 ) )
% 0.16/0.39 => ( condit1810911227_above @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % bdd_aboveI2
% 0.16/0.39 thf(fact_250_cSUP__upper2,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [F: B > A,A2: set @ B,X2: B,U: A] :
% 0.16/0.39 ( ( condit1810911227_above @ A @ ( image @ B @ A @ F @ A2 ) )
% 0.16/0.39 => ( ( member @ B @ X2 @ A2 )
% 0.16/0.39 => ( ( ord_less_eq @ A @ U @ ( F @ X2 ) )
% 0.16/0.39 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSUP_upper2
% 0.16/0.39 thf(fact_251_cSUP__upper,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [X2: B,A2: set @ B,F: B > A] :
% 0.16/0.39 ( ( member @ B @ X2 @ A2 )
% 0.16/0.39 => ( ( condit1810911227_above @ A @ ( image @ B @ A @ F @ A2 ) )
% 0.16/0.39 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSUP_upper
% 0.16/0.39 thf(fact_252_cSUP__le__iff,axiom,
% 0.16/0.39 ! [A: $tType,B: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ B,F: B > A,U: A] :
% 0.16/0.39 ( ( A2
% 0.16/0.39 != ( bot_bot @ ( set @ B ) ) )
% 0.16/0.39 => ( ( condit1810911227_above @ A @ ( image @ B @ A @ F @ A2 ) )
% 0.16/0.39 => ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ U )
% 0.16/0.39 = ( ! [X: B] :
% 0.16/0.39 ( ( member @ B @ X @ A2 )
% 0.16/0.39 => ( ord_less_eq @ A @ ( F @ X ) @ U ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSUP_le_iff
% 0.16/0.39 thf(fact_253_cSUP__mono,axiom,
% 0.16/0.39 ! [B: $tType,A: $tType,C: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ B,G: C > A,B2: set @ C,F: B > A] :
% 0.16/0.39 ( ( A2
% 0.16/0.39 != ( bot_bot @ ( set @ B ) ) )
% 0.16/0.39 => ( ( condit1810911227_above @ A @ ( image @ C @ A @ G @ B2 ) )
% 0.16/0.39 => ( ! [N4: B] :
% 0.16/0.39 ( ( member @ B @ N4 @ A2 )
% 0.16/0.39 => ? [X5: C] :
% 0.16/0.39 ( ( member @ C @ X5 @ B2 )
% 0.16/0.39 & ( ord_less_eq @ A @ ( F @ N4 ) @ ( G @ X5 ) ) ) )
% 0.16/0.39 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSUP_mono
% 0.16/0.39 thf(fact_254_cSup__subset__mono,axiom,
% 0.16/0.39 ! [A: $tType] :
% 0.16/0.39 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.16/0.39 => ! [A2: set @ A,B2: set @ A] :
% 0.16/0.39 ( ( A2
% 0.16/0.39 != ( bot_bot @ ( set @ A ) ) )
% 0.16/0.39 => ( ( condit1810911227_above @ A @ B2 )
% 0.16/0.39 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.16/0.39 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 % cSup_subset_mono
% 0.16/0.39
% 0.16/0.39 %----Type constructors (57)
% 0.16/0.39 thf(tcon_HOL_Obool___Finite__Set_Ofinite,axiom,
% 0.16/0.39 finite_finite @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
% 0.16/0.39 ! [A6: $tType] :
% 0.16/0.39 ( ( finite_finite @ A6 @ ( type2 @ A6 ) )
% 0.16/0.39 => ( finite_finite @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Finite__Set_Ofinite_2,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( ( finite_finite @ A6 @ ( type2 @ A6 ) )
% 0.16/0.39 & ( finite_finite @ A7 @ ( type2 @ A7 ) ) )
% 0.16/0.39 => ( finite_finite @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
% 0.16/0.39 => ( condit378418413attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( counta840220525attice @ A7 @ ( type2 @ A7 ) )
% 0.16/0.39 => ( counta840220525attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
% 0.16/0.39 => ( comple187826305attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
% 0.16/0.39 => ( comple1141879883l_ccpo @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Complete__Lattices_OSup,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( complete_Sup @ A7 @ ( type2 @ A7 ) )
% 0.16/0.39 => ( complete_Sup @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( order_top @ A7 @ ( type2 @ A7 ) )
% 0.16/0.39 => ( order_top @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( order_bot @ A7 @ ( type2 @ A7 ) )
% 0.16/0.39 => ( order_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Countable_Ocountable,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( ( finite_finite @ A6 @ ( type2 @ A6 ) )
% 0.16/0.39 & ( countable @ A7 @ ( type2 @ A7 ) ) )
% 0.16/0.39 => ( countable @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 0.16/0.39 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( order @ A7 @ ( type2 @ A7 ) )
% 0.16/0.39 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Orderings_Otop,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( top @ A7 @ ( type2 @ A7 ) )
% 0.16/0.39 => ( top @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Orderings_Oord,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 0.16/0.39 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_fun___Orderings_Obot,axiom,
% 0.16/0.39 ! [A6: $tType,A7: $tType] :
% 0.16/0.39 ( ( bot @ A7 @ ( type2 @ A7 ) )
% 0.16/0.39 => ( bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_3,axiom,
% 0.16/0.39 condit378418413attice @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.16/0.39 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.16/0.39 ordere779506340up_add @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 0.16/0.39 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Complete__Lattices_OSup_4,axiom,
% 0.16/0.39 complete_Sup @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Orderings_Oorder__bot_5,axiom,
% 0.16/0.39 order_bot @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Countable_Ocountable_6,axiom,
% 0.16/0.39 countable @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Orderings_Opreorder_7,axiom,
% 0.16/0.39 preorder @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.16/0.39 linorder @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 0.16/0.39 monoid_add @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Orderings_Oorder_8,axiom,
% 0.16/0.39 order @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Orderings_Oord_9,axiom,
% 0.16/0.39 ord @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Orderings_Obot_10,axiom,
% 0.16/0.39 bot @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.16/0.39 zero @ nat @ ( type2 @ nat ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom,
% 0.16/0.39 ! [A6: $tType] : ( condit378418413attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_12,axiom,
% 0.16/0.39 ! [A6: $tType] : ( counta840220525attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_13,axiom,
% 0.16/0.39 ! [A6: $tType] : ( comple187826305attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_14,axiom,
% 0.16/0.39 ! [A6: $tType] : ( comple1141879883l_ccpo @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Complete__Lattices_OSup_15,axiom,
% 0.16/0.39 ! [A6: $tType] : ( complete_Sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Orderings_Oorder__top_16,axiom,
% 0.16/0.39 ! [A6: $tType] : ( order_top @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Orderings_Oorder__bot_17,axiom,
% 0.16/0.39 ! [A6: $tType] : ( order_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Countable_Ocountable_18,axiom,
% 0.16/0.39 ! [A6: $tType] :
% 0.16/0.39 ( ( finite_finite @ A6 @ ( type2 @ A6 ) )
% 0.16/0.39 => ( countable @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Orderings_Opreorder_19,axiom,
% 0.16/0.39 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Orderings_Oorder_20,axiom,
% 0.16/0.39 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Orderings_Otop_21,axiom,
% 0.16/0.39 ! [A6: $tType] : ( top @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Orderings_Oord_22,axiom,
% 0.16/0.39 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_Set_Oset___Orderings_Obot_23,axiom,
% 0.16/0.39 ! [A6: $tType] : ( bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_24,axiom,
% 0.16/0.39 condit378418413attice @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_25,axiom,
% 0.16/0.39 counta840220525attice @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_26,axiom,
% 0.16/0.39 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_27,axiom,
% 0.16/0.39 comple1141879883l_ccpo @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Complete__Lattices_OSup_28,axiom,
% 0.16/0.39 complete_Sup @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Orderings_Oorder__top_29,axiom,
% 0.16/0.39 order_top @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Orderings_Oorder__bot_30,axiom,
% 0.16/0.39 order_bot @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Countable_Ocountable_31,axiom,
% 0.16/0.39 countable @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Orderings_Opreorder_32,axiom,
% 0.16/0.39 preorder @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Orderings_Olinorder_33,axiom,
% 0.16/0.39 linorder @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Orderings_Oorder_34,axiom,
% 0.16/0.39 order @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Orderings_Otop_35,axiom,
% 0.16/0.39 top @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Orderings_Oord_36,axiom,
% 0.16/0.39 ord @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 thf(tcon_HOL_Obool___Orderings_Obot_37,axiom,
% 0.16/0.39 bot @ $o @ ( type2 @ $o ) ).
% 0.16/0.39
% 0.16/0.39 %----Conjectures (1)
% 0.16/0.39 thf(conj_0,conjecture,
% 0.16/0.39 ( ( snth @ a @ ( coindu176146587of_seq @ a @ f ) )
% 0.16/0.39 = f ) ).
% 0.16/0.39
% 0.16/0.39 %------------------------------------------------------------------------------
% 0.16/0.39 ------------------------
% 3.07/1.62 ---- Embedded in THF ---
% 3.07/1.62 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.07/1.62 %%% Generated on Tue Feb 24 06:09:02 EST 2026
% 3.07/1.62 %%% using '$$dhol' embedding, version 1.3.0.
% 3.07/1.62 %%% Logic specification used:
% 3.07/1.62 %%% thf(spec, logic, $$dhol).
% 3.07/1.62
% 3.07/1.62 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.dF9X7yUCOB/DTF2DTF3754.p
% 3.07/1.62 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 3.07/1.62 thf(ty_t_Stream_Ostream, type, stream: ($tType > $tType)).
% 3.07/1.62 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.07/1.62 thf(ty_t_Nat_Onat, type, nat: $tType).
% 3.07/1.62 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 3.07/1.62 thf(ty_tf_a, type, a: $tType).
% 3.07/1.62 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Groups_Omonoid__add, type, monoid_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Countable_Ocountable, type, countable: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Complete__Lattices_OSup, type, complete_Sup: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Complete__Partial__Order_Occpo, type, comple1141879883l_ccpo: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice, type, counta840220525attice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice, type, condit378418413attice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.07/1.62 thf(sy_c_Coinductive__List_Ogen__lset, type, coinductive_gen_lset: (!> [A:$tType]: (((set @ A) > ((coinductive_llist @ A) > (set @ A)))))).
% 3.07/1.62 thf(sy_c_Coinductive__List_Oinf__llist, type, coindu68654304_llist: (!> [A:$tType]: (((nat > A) > (coinductive_llist @ A))))).
% 3.07/1.62 thf(sy_c_Coinductive__List_Olfinite, type, coinductive_lfinite: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 3.07/1.62 thf(sy_c_Coinductive__List_Ollist_Olmap, type, coinductive_lmap: (!> [A:$tType,Aa:$tType]: (((A > Aa) > ((coinductive_llist @ A) > (coinductive_llist @ Aa)))))).
% 3.07/1.62 thf(sy_c_Coinductive__List_Ollist_Olset, type, coinductive_lset: (!> [A:$tType]: (((coinductive_llist @ A) > (set @ A))))).
% 3.07/1.62 thf(sy_c_Coinductive__List_Ollist_Opred__llist, type, coindu543516966_llist: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > $o))))).
% 3.07/1.62 thf(sy_c_Coinductive__List_Olmember, type, coinductive_lmember: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > $o))))).
% 3.07/1.62 thf(sy_c_Coinductive__List_Olnth, type, coinductive_lnth: (!> [A:$tType]: (((coinductive_llist @ A) > (nat > A))))).
% 3.07/1.62 thf(sy_c_Coinductive__List_Omonoid__add__class_Ollistsum, type, coindu780009021istsum: (!> [A:$tType]: (((coinductive_llist @ A) > A)))).
% 3.07/1.62 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ollist__of__stream, type, coindu1724414836stream: (!> [A:$tType]: (((stream @ A) > (coinductive_llist @ A))))).
% 3.07/1.62 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Oof__seq, type, coindu176146587of_seq: (!> [A:$tType]: (((nat > A) > (stream @ A))))).
% 3.07/1.62 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Ocr__stream, type, coindu1183105481stream: (!> [A:$tType]: (((coinductive_llist @ A) > ((stream @ A) > $o))))).
% 3.07/1.62 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__of__llist, type, coindu2010755910_llist: (!> [A:$tType]: (((coinductive_llist @ A) > (stream @ A))))).
% 3.07/1.62 thf(sy_c_Complete__Lattices_OSup__class_OSup, type, complete_Sup_Sup: (!> [A:$tType]: (((set @ A) > A)))).
% 3.07/1.62 thf(sy_c_Complete__Partial__Order2_Ocompact, type, comple2143767107ompact: (!> [A:$tType]: ((((set @ A) > A) > ((A > (A > $o)) > (A > $o)))))).
% 3.07/1.62 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above, type, condit1810911227_above: (!> [A:$tType]: (((set @ A) > $o)))).
% 3.07/1.62 thf(sy_c_Countable_Ofrom__nat, type, from_nat: (!> [A:$tType]: ((nat > A)))).
% 3.07/1.62 thf(sy_c_Fun_Oswap, type, swap: (!> [A:$tType,B:$tType]: ((A > (A > ((A > B) > (A > B))))))).
% 3.07/1.62 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 3.07/1.62 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 3.07/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.07/1.62 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 3.07/1.62 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 3.07/1.62 thf(sy_c_Set_OBall, type, ball: (!> [A:$tType]: (((set @ A) > ((A > $o) > $o))))).
% 3.07/1.62 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.07/1.62 thf(sy_c_Set_OPow, type, pow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 3.07/1.62 thf(sy_c_Set_Obind, type, bind: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ B)))))).
% 3.07/1.62 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 3.07/1.62 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 3.07/1.62 thf(sy_c_Stream_Osmap2, type, smap2: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > ((stream @ A) > ((stream @ B) > (stream @ C))))))).
% 3.07/1.62 thf(sy_c_Stream_Osmember, type, smember: (!> [A:$tType]: ((A > ((stream @ A) > $o))))).
% 3.07/1.62 thf(sy_c_Stream_Osmerge, type, smerge: (!> [A:$tType]: (((stream @ (stream @ A)) > (stream @ A))))).
% 3.07/1.62 thf(sy_c_Stream_Osnth, type, snth: (!> [A:$tType]: (((stream @ A) > (nat > A))))).
% 3.07/1.62 thf(sy_c_Stream_Ostream_Opred__stream, type, pred_stream: (!> [A:$tType]: (((A > $o) > ((stream @ A) > $o))))).
% 3.07/1.62 thf(sy_c_Stream_Ostream_Osmap, type, smap: (!> [A:$tType,Aa:$tType]: (((A > Aa) > ((stream @ A) > (stream @ Aa)))))).
% 3.07/1.62 thf(sy_c_Stream_Ostream_Osset, type, sset: (!> [A:$tType]: (((stream @ A) > (set @ A))))).
% 3.07/1.62 thf(sy_c_Stream_Ostream__all, type, stream_all: (!> [A:$tType]: (((A > $o) > ((stream @ A) > $o))))).
% 3.07/1.62 thf(sy_c_Stream_Ostreams, type, streams: (!> [A:$tType]: (((set @ A) > (set @ (stream @ A)))))).
% 3.07/1.62 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.07/1.62 thf(sy_v_f, type, f: (nat > a)).
% 3.07/1.62 thf(fact_0_stream__all__def, axiom, (! [A:$tType]: (((stream_all @ A) = (^ [P:(A > $o),S:(stream @ A)]: ((! [P2:nat]: ((P @ (((snth @ A) @ S) @ P2)))))))))).
% 3.07/1.62 thf(fact_1_snth__smap2, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),S1:(stream @ B),S2:(stream @ C),N:nat]: (((((snth @ A) @ ((((((smap2 @ B) @ C) @ A) @ F) @ S1) @ S2)) @ N) = ((F @ (((snth @ B) @ S1) @ N)) @ (((snth @ C) @ S2) @ N)))))).
% 3.07/1.62 thf(fact_2_smap2__alt, axiom, (! [A:$tType,B:$tType,C:$tType,F:(B > (C > A)),S1:(stream @ B),S2:(stream @ C),S3:(stream @ A)]: (((((((((smap2 @ B) @ C) @ A) @ F) @ S1) @ S2) = S3) = (! [N2:nat]: ((((F @ (((snth @ B) @ S1) @ N2)) @ (((snth @ C) @ S2) @ N2)) = (((snth @ A) @ S3) @ N2)))))))).
% 3.07/1.62 thf(fact_3_of__seq_Oabs__eq, axiom, (! [A:$tType]: (((coindu176146587of_seq @ A) = (^ [X:(nat > A)]: (((coindu2010755910_llist @ A) @ ((coindu68654304_llist @ A) @ X)))))))).
% 3.07/1.62 thf(fact_4_of__seq_Orep__eq, axiom, (! [A:$tType,X2:(nat > A)]: ((((coindu1724414836stream @ A) @ ((coindu176146587of_seq @ A) @ X2)) = ((coindu68654304_llist @ A) @ X2))))).
% 3.07/1.62 thf(fact_5_snth__in, axiom, (! [A:$tType,S3:(stream @ A),X3:(set @ A),N:nat]: (((((member @ (stream @ A)) @ S3) @ ((streams @ A) @ X3)) => (((member @ A) @ (((snth @ A) @ S3) @ N)) @ X3))))).
% 3.07/1.62 thf(fact_6_streams__iff__snth, axiom, (! [A:$tType,S3:(stream @ A),X3:(set @ A)]: (((((member @ (stream @ A)) @ S3) @ ((streams @ A) @ X3)) = (! [N2:nat]: ((((member @ A) @ (((snth @ A) @ S3) @ N2)) @ X3))))))).
% 3.07/1.62 thf(fact_7_snth__smap, axiom, (! [A:$tType,B:$tType,F:(B > A),S3:(stream @ B),N:nat]: (((((snth @ A) @ ((((smap @ B) @ A) @ F) @ S3)) @ N) = (F @ (((snth @ B) @ S3) @ N)))))).
% 3.07/1.62 thf(fact_8_snth__sset, axiom, (! [A:$tType,S3:(stream @ A),N:nat]: ((((member @ A) @ (((snth @ A) @ S3) @ N)) @ ((sset @ A) @ S3))))).
% 3.07/1.62 thf(fact_9_lnth__list__of__stream, axiom, (! [A:$tType,Xs:(stream @ A)]: ((((coinductive_lnth @ A) @ ((coindu1724414836stream @ A) @ Xs)) = ((snth @ A) @ Xs))))).
% 3.07/1.62 thf(fact_10_smap__alt, axiom, (! [A:$tType,B:$tType,F:(B > A),S3:(stream @ B),S4:(stream @ A)]: (((((((smap @ B) @ A) @ F) @ S3) = S4) = (! [N2:nat]: (((F @ (((snth @ B) @ S3) @ N2)) = (((snth @ A) @ S4) @ N2)))))))).
% 3.07/1.62 thf(fact_11_snth__sset__smerge, axiom, (! [A:$tType,Ss:(stream @ (stream @ A)),N:nat,M:nat]: ((((member @ A) @ (((snth @ A) @ (((snth @ (stream @ A)) @ Ss) @ N)) @ M)) @ ((sset @ A) @ ((smerge @ A) @ Ss)))))).
% 3.07/1.62 thf(fact_12_stream_ORep__inverse, axiom, (! [A:$tType,X2:(stream @ A)]: ((((coindu2010755910_llist @ A) @ ((coindu1724414836stream @ A) @ X2)) = X2)))).
% 3.07/1.62 thf(fact_13_stream__of__llist__llist__of__stream, axiom, (! [A:$tType,Xs:(stream @ A)]: ((((coindu2010755910_llist @ A) @ ((coindu1724414836stream @ A) @ Xs)) = Xs)))).
% 3.07/1.62 thf(fact_14_smap__streams, axiom, (! [A:$tType,B:$tType,S3:(stream @ A),A2:(set @ A),F:(A > B),B2:(set @ B)]: (((((member @ (stream @ A)) @ S3) @ ((streams @ A) @ A2)) => ((! [X4:A]: (((((member @ A) @ X4) @ A2) => (((member @ B) @ (F @ X4)) @ B2)))) => (((member @ (stream @ B)) @ ((((smap @ A) @ B) @ F) @ S3)) @ ((streams @ B) @ B2))))))).
% 3.07/1.62 thf(fact_15_stream_Omap__cong, axiom, (! [B:$tType,A:$tType,X2:(stream @ A),Ya:(stream @ A),F:(A > B),G:(A > B)]: (((X2 = Ya) => ((! [Z:A]: (((((member @ A) @ Z) @ ((sset @ A) @ Ya)) => ((F @ Z) = (G @ Z))))) => (((((smap @ A) @ B) @ F) @ X2) = ((((smap @ A) @ B) @ G) @ Ya))))))).
% 3.07/1.62 thf(fact_16_stream_Omap__cong0, axiom, (! [B:$tType,A:$tType,X2:(stream @ A),F:(A > B),G:(A > B)]: (((! [Z:A]: (((((member @ A) @ Z) @ ((sset @ A) @ X2)) => ((F @ Z) = (G @ Z))))) => (((((smap @ A) @ B) @ F) @ X2) = ((((smap @ A) @ B) @ G) @ X2)))))).
% 3.07/1.62 thf(fact_17_stream_Oinj__map__strong, axiom, (! [B:$tType,A:$tType,X2:(stream @ A),Xa:(stream @ A),F:(A > B),Fa:(A > B)]: (((! [Z:A,Za:A]: (((((member @ A) @ Z) @ ((sset @ A) @ X2)) => ((((member @ A) @ Za) @ ((sset @ A) @ Xa)) => (((F @ Z) = (Fa @ Za)) => (Z = Za)))))) => ((((((smap @ A) @ B) @ F) @ X2) = ((((smap @ A) @ B) @ Fa) @ Xa)) => (X2 = Xa)))))).
% 3.07/1.62 thf(fact_18_stream_ORep__inject, axiom, (! [A:$tType,X2:(stream @ A),Y:(stream @ A)]: (((((coindu1724414836stream @ A) @ X2) = ((coindu1724414836stream @ A) @ Y)) = (X2 = Y))))).
% 3.07/1.62 thf(fact_19_lnth__inf__llist, axiom, (! [A:$tType,F:(nat > A),N:nat]: (((((coinductive_lnth @ A) @ ((coindu68654304_llist @ A) @ F)) @ N) = (F @ N))))).
% 3.07/1.62 thf(fact_20_inf__llist__inj, axiom, (! [A:$tType,F:(nat > A),G:(nat > A)]: (((((coindu68654304_llist @ A) @ F) = ((coindu68654304_llist @ A) @ G)) = (F = G))))).
% 3.07/1.62 thf(fact_21_Stream_Osmember__def, axiom, (! [A:$tType]: (((smember @ A) = (^ [X:A,S:(stream @ A)]: ((((member @ A) @ X) @ ((sset @ A) @ S)))))))).
% 3.07/1.62 thf(fact_22_stream__all__iff, axiom, (! [A:$tType]: (((stream_all @ A) = (^ [P:(A > $o),S:(stream @ A)]: ((! [X:A]: (((((member @ A) @ X) @ ((sset @ A) @ S)) => (P @ X)))))))))).
% 3.07/1.62 thf(fact_23_llist__of__stream__stream__of__llist, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => (((coindu1724414836stream @ A) @ ((coindu2010755910_llist @ A) @ Xs)) = Xs))))).
% 3.07/1.62 thf(fact_24_lset__llist__of__stream, axiom, (! [A:$tType,Xs:(stream @ A)]: ((((coinductive_lset @ A) @ ((coindu1724414836stream @ A) @ Xs)) = ((sset @ A) @ Xs))))).
% 3.07/1.62 thf(fact_25_sset__streams, axiom, (! [A:$tType,S3:(stream @ A),A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((sset @ A) @ S3)) @ A2) => (((member @ (stream @ A)) @ S3) @ ((streams @ A) @ A2)))))).
% 3.07/1.62 thf(fact_26_streams__sset, axiom, (! [A:$tType,S3:(stream @ A),A2:(set @ A)]: (((((member @ (stream @ A)) @ S3) @ ((streams @ A) @ A2)) => (((ord_less_eq @ (set @ A)) @ ((sset @ A) @ S3)) @ A2))))).
% 3.07/1.62 thf(fact_27_streams__iff__sset, axiom, (! [A:$tType,S3:(stream @ A),A2:(set @ A)]: (((((member @ (stream @ A)) @ S3) @ ((streams @ A) @ A2)) = (((ord_less_eq @ (set @ A)) @ ((sset @ A) @ S3)) @ A2))))).
% 3.07/1.62 thf(fact_28_lmap__llist__of__stream, axiom, (! [A:$tType,B:$tType,F:(B > A),Xs:(stream @ B)]: ((((((coinductive_lmap @ B) @ A) @ F) @ ((coindu1724414836stream @ B) @ Xs)) = ((coindu1724414836stream @ A) @ ((((smap @ B) @ A) @ F) @ Xs)))))).
% 3.07/1.62 thf(fact_29_stream_Oset__map, axiom, (! [B:$tType,A:$tType,F:(A > B),V:(stream @ A)]: ((((sset @ B) @ ((((smap @ A) @ B) @ F) @ V)) = ((((image @ A) @ B) @ F) @ ((sset @ A) @ V)))))).
% 3.07/1.62 thf(fact_30_lfinite__lmap, axiom, (! [A:$tType,B:$tType,F:(B > A),Xs:(coinductive_llist @ B)]: ((((coinductive_lfinite @ A) @ ((((coinductive_lmap @ B) @ A) @ F) @ Xs)) = ((coinductive_lfinite @ B) @ Xs))))).
% 3.07/1.62 thf(fact_31_llist_Oset__map, axiom, (! [B:$tType,A:$tType,F:(A > B),V:(coinductive_llist @ A)]: ((((coinductive_lset @ B) @ ((((coinductive_lmap @ A) @ B) @ F) @ V)) = ((((image @ A) @ B) @ F) @ ((coinductive_lset @ A) @ V)))))).
% 3.07/1.62 thf(fact_32_inf__llist__lnth, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => (((coindu68654304_llist @ A) @ ((coinductive_lnth @ A) @ Xs)) = Xs))))).
% 3.07/1.62 thf(fact_33_sset__stream__of__llist, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((~ (((coinductive_lfinite @ A) @ Xs))) => (((sset @ A) @ ((coindu2010755910_llist @ A) @ Xs)) = ((coinductive_lset @ A) @ Xs))))))).
% 3.07/1.62 thf(fact_34_smap__stream__of__llist, axiom, (! [B:$tType,A:$tType,Xs:(coinductive_llist @ A),F:(A > B)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => (((((smap @ A) @ B) @ F) @ ((coindu2010755910_llist @ A) @ Xs)) = ((coindu2010755910_llist @ B) @ ((((coinductive_lmap @ A) @ B) @ F) @ Xs))))))).
% 3.07/1.62 thf(fact_35_llist_Oinj__map__strong, axiom, (! [B:$tType,A:$tType,X2:(coinductive_llist @ A),Xa:(coinductive_llist @ A),F:(A > B),Fa:(A > B)]: (((! [Z:A,Za:A]: (((((member @ A) @ Z) @ ((coinductive_lset @ A) @ X2)) => ((((member @ A) @ Za) @ ((coinductive_lset @ A) @ Xa)) => (((F @ Z) = (Fa @ Za)) => (Z = Za)))))) => ((((((coinductive_lmap @ A) @ B) @ F) @ X2) = ((((coinductive_lmap @ A) @ B) @ Fa) @ Xa)) => (X2 = Xa)))))).
% 3.07/1.62 thf(fact_36_llist_Omap__cong0, axiom, (! [B:$tType,A:$tType,X2:(coinductive_llist @ A),F:(A > B),G:(A > B)]: (((! [Z:A]: (((((member @ A) @ Z) @ ((coinductive_lset @ A) @ X2)) => ((F @ Z) = (G @ Z))))) => (((((coinductive_lmap @ A) @ B) @ F) @ X2) = ((((coinductive_lmap @ A) @ B) @ G) @ X2)))))).
% 3.07/1.62 thf(fact_37_llist_Omap__cong, axiom, (! [B:$tType,A:$tType,X2:(coinductive_llist @ A),Ya:(coinductive_llist @ A),F:(A > B),G:(A > B)]: (((X2 = Ya) => ((! [Z:A]: (((((member @ A) @ Z) @ ((coinductive_lset @ A) @ Ya)) => ((F @ Z) = (G @ Z))))) => (((((coinductive_lmap @ A) @ B) @ F) @ X2) = ((((coinductive_lmap @ A) @ B) @ G) @ Ya))))))).
% 3.07/1.62 thf(fact_38_wlog__linorder__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P3:(A > (A > $o)),B3:A,A3:A]: (((! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((P3 @ A4) @ B4)))) => ((((P3 @ B3) @ A3) => ((P3 @ A3) @ B3)) => ((P3 @ A3) @ B3))))))))).
% 3.07/1.62 thf(fact_39_streams__mono2, axiom, (! [A:$tType,S5:(set @ A),T:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S5) @ T) => (((ord_less_eq @ (set @ (stream @ A))) @ ((streams @ A) @ S5)) @ ((streams @ A) @ T)))))).
% 3.07/1.62 thf(fact_40_lfinite__inf__llist, axiom, (! [A:$tType,F:(nat > A)]: ((~ (((coinductive_lfinite @ A) @ ((coindu68654304_llist @ A) @ F))))))).
% 3.07/1.62 thf(fact_41_lfinite__llist__of__stream, axiom, (! [A:$tType,Xs:(stream @ A)]: ((~ (((coinductive_lfinite @ A) @ ((coindu1724414836stream @ A) @ Xs))))))).
% 3.07/1.62 thf(fact_42_streams__mono, axiom, (! [A:$tType,S3:(stream @ A),A2:(set @ A),B2:(set @ A)]: (((((member @ (stream @ A)) @ S3) @ ((streams @ A) @ A2)) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ (stream @ A)) @ S3) @ ((streams @ A) @ B2))))))).
% 3.07/1.62 thf(fact_43_not__False__in__image__Ball, axiom, (! [A:$tType,P3:(A > $o),A2:(set @ A)]: (((~ ((((member @ $o) @ $false) @ ((((image @ A) @ $o) @ P3) @ A2)))) = (! [X:A]: (((((member @ A) @ X) @ A2) => (P3 @ X)))))))).
% 3.07/1.62 thf(fact_44_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (A2 = B2)))))).
% 3.07/1.62 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P3:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P3)) = (P3 @ A3))))).
% 3.07/1.62 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A2)))) = A2)))).
% 3.07/1.62 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P3:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P3 @ X4) = (Q @ X4)))) => (((collect @ A) @ P3) = ((collect @ A) @ Q)))))).
% 3.07/1.62 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 3.07/1.62 thf(fact_49_subsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A2) => (((member @ A) @ X4) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 3.07/1.62 thf(fact_50_image__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X2:B,A2:(set @ B)]: (((B3 = (F @ X2)) => ((((member @ B) @ X2) @ A2) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ A2))))))).
% 3.07/1.62 thf(fact_51_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 3.07/1.62 thf(fact_52_Ball__Collect, axiom, (! [A:$tType]: (((ball @ A) = (^ [A5:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ A5) @ ((collect @ A) @ P)))))))).
% 3.07/1.62 thf(fact_53_subset__image__iff, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F) @ A2)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A2) & (B2 = ((((image @ B) @ A) @ F) @ AA))))))))).
% 3.07/1.62 thf(fact_54_image__subset__iff, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A2)) @ B2) = (! [X:B]: (((((member @ B) @ X) @ A2) => (((member @ A) @ (F @ X)) @ B2)))))))).
% 3.07/1.62 thf(fact_55_subset__imageE, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F) @ A2)) => (~ ((! [C2:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C2) @ A2) => (B2 != ((((image @ B) @ A) @ F) @ C2))))))))))).
% 3.07/1.62 thf(fact_56_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 3.07/1.62 thf(fact_57_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 3.07/1.62 thf(fact_58_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X4:A]: ((((ord_less_eq @ B) @ (F @ X4)) @ (G @ X4)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 3.07/1.62 thf(fact_59_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ (G2 @ X))))))))))).
% 3.07/1.62 thf(fact_60_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C3) => ((! [X4:B,Y2:B]: (((((ord_less_eq @ B) @ X4) @ Y2) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A3) @ (F @ C3))))))))))).
% 3.07/1.62 thf(fact_61_order__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F:(A > C),C3:C]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C3) => ((! [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) => (((ord_less_eq @ C) @ (F @ X4)) @ (F @ Y2))))) => (((ord_less_eq @ C) @ (F @ A3)) @ C3)))))))))).
% 3.07/1.62 thf(fact_62_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((A3 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C3) => ((! [X4:B,Y2:B]: (((((ord_less_eq @ B) @ X4) @ Y2) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A3) @ (F @ C3))))))))))).
% 3.07/1.62 thf(fact_63_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F:(A > B),C3:B]: (((((ord_less_eq @ A) @ A3) @ B3) => (((F @ B3) = C3) => ((! [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) => (((ord_less_eq @ B) @ (F @ X4)) @ (F @ Y2))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C3)))))))))).
% 3.07/1.62 thf(fact_64_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y3:A,Z2:A]: ((Y3 = Z2))) = (^ [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X))))))))).
% 3.07/1.62 thf(fact_65_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ X2) => (X2 = Y))))))))).
% 3.07/1.62 thf(fact_66_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) | (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 3.07/1.62 thf(fact_67_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X2:A,Y:A]: (((X2 = Y) => (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 3.07/1.62 thf(fact_68_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X2:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y))) => (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 3.07/1.62 thf(fact_69_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 3.07/1.62 thf(fact_70_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X2:A,Y:A,Z3:A]: ((((((ord_less_eq @ A) @ X2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z3)))) => (((((ord_less_eq @ A) @ Y) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z3)))) => (((((ord_less_eq @ A) @ X2) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ Y)))) => (((((ord_less_eq @ A) @ Z3) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X2)))) => (((((ord_less_eq @ A) @ Y) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z3) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y))))))))))))))))).
% 3.07/1.62 thf(fact_71_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y) = (X2 = Y))))))))).
% 3.07/1.62 thf(fact_72_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C3:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 3.07/1.62 thf(fact_73_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((B3 = C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 3.07/1.62 thf(fact_74_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ A3) => (A3 = B3))))))))).
% 3.07/1.62 thf(fact_75_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X2:A,Y:A,Z3:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z3) => (((ord_less_eq @ A) @ X2) @ Z3))))))))).
% 3.07/1.62 thf(fact_76_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 3.07/1.62 thf(fact_77_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P3:(A > (A > $o)),A3:A,B3:A]: (((! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((P3 @ A4) @ B4)))) => ((! [A4:A,B4:A]: ((((P3 @ B4) @ A4) => ((P3 @ A4) @ B4)))) => ((P3 @ A3) @ B3))))))))).
% 3.07/1.62 thf(fact_78_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C3) @ B3) => (((ord_less_eq @ A) @ C3) @ A3))))))))).
% 3.07/1.62 thf(fact_79_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ A3) @ B3) => (A3 = B3))))))))).
% 3.07/1.62 thf(fact_80_imageI, axiom, (! [B:$tType,A:$tType,X2:A,A2:(set @ A),F:(A > B)]: (((((member @ A) @ X2) @ A2) => (((member @ B) @ (F @ X2)) @ ((((image @ A) @ B) @ F) @ A2)))))).
% 3.07/1.62 thf(fact_81_image__iff, axiom, (! [A:$tType,B:$tType,Z3:A,F:(B > A),A2:(set @ B)]: (((((member @ A) @ Z3) @ ((((image @ B) @ A) @ F) @ A2)) = (? [X:B]: (((((member @ B) @ X) @ A2) & (Z3 = (F @ X))))))))).
% 3.07/1.62 thf(fact_82_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P3:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F) @ A2)) & (P3 @ X5)))) => (? [X4:B]: (((((member @ B) @ X4) @ A2) & (P3 @ (F @ X4))))))))).
% 3.07/1.62 thf(fact_83_image__cong, axiom, (! [B:$tType,A:$tType,M2:(set @ A),N3:(set @ A),F:(A > B),G:(A > B)]: (((M2 = N3) => ((! [X4:A]: (((((member @ A) @ X4) @ N3) => ((F @ X4) = (G @ X4))))) => (((((image @ A) @ B) @ F) @ M2) = ((((image @ A) @ B) @ G) @ N3))))))).
% 3.07/1.62 thf(fact_84_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P3:(A > $o)]: (((! [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F) @ A2)) => (P3 @ X4)))) => (! [X5:B]: (((((member @ B) @ X5) @ A2) => (P3 @ (F @ X5))))))))).
% 3.07/1.62 thf(fact_85_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X2:A,A2:(set @ A),B3:B,F:(A > B)]: (((((member @ A) @ X2) @ A2) => ((B3 = (F @ X2)) => (((member @ B) @ B3) @ ((((image @ A) @ B) @ F) @ A2))))))).
% 3.07/1.62 thf(fact_86_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A2:(set @ B),B2:(set @ B),C4:(B > A),D:(B > A),Inf:((set @ A) > A)]: (((A2 = B2) => ((! [X4:B]: (((((member @ B) @ X4) @ B2) => ((C4 @ X4) = (D @ X4))))) => ((Inf @ ((((image @ B) @ A) @ C4) @ A2)) = (Inf @ ((((image @ B) @ A) @ D) @ B2)))))))).
% 3.07/1.62 thf(fact_87_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A2:(set @ B),B2:(set @ B),C4:(B > A),D:(B > A),Sup:((set @ A) > A)]: (((A2 = B2) => ((! [X4:B]: (((((member @ B) @ X4) @ B2) => ((C4 @ X4) = (D @ X4))))) => ((Sup @ ((((image @ B) @ A) @ C4) @ A2)) = (Sup @ ((((image @ B) @ A) @ D) @ B2)))))))).
% 3.07/1.62 thf(fact_88_set__mp, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X2) @ A2) => (((member @ A) @ X2) @ B2)))))).
% 3.07/1.62 thf(fact_89_in__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X2) @ A2) => (((member @ A) @ X2) @ B2)))))).
% 3.07/1.62 thf(fact_90_subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C3) @ A2) => (((member @ A) @ C3) @ B2)))))).
% 3.07/1.62 thf(fact_91_subsetCE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C3) @ A2) => (((member @ A) @ C3) @ B2)))))).
% 3.07/1.62 thf(fact_92_equalityE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))))).
% 3.07/1.62 thf(fact_93_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ A5) => (((member @ A) @ X) @ B5)))))))))).
% 3.07/1.62 thf(fact_94_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 3.07/1.62 thf(fact_95_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 3.07/1.62 thf(fact_96_set__rev__mp, axiom, (! [A:$tType,X2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ X2) @ A2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ A) @ X2) @ B2)))))).
% 3.07/1.62 thf(fact_97_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A5) => (((member @ A) @ T2) @ B5)))))))))).
% 3.07/1.62 thf(fact_98_rev__subsetD, axiom, (! [A:$tType,C3:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ A2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ A) @ C3) @ B2)))))).
% 3.07/1.62 thf(fact_99_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 3.07/1.62 thf(fact_100_Collect__mono, axiom, (! [A:$tType,P3:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P3 @ X4) => (Q @ X4)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P3)) @ ((collect @ A) @ Q)))))).
% 3.07/1.62 thf(fact_101_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C4) => (((ord_less_eq @ (set @ A)) @ A2) @ C4)))))).
% 3.07/1.62 thf(fact_102_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y3:(set @ A),Z2:(set @ A)]: ((Y3 = Z2))) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) & (((ord_less_eq @ (set @ A)) @ B5) @ A5)))))))).
% 3.07/1.62 thf(fact_103_contra__subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((~ ((((member @ A) @ C3) @ B2))) => (~ ((((member @ A) @ C3) @ A2)))))))).
% 3.07/1.62 thf(fact_104_Collect__mono__iff, axiom, (! [A:$tType,P3:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P3)) @ ((collect @ A) @ Q)) = (! [X:A]: (((P3 @ X) => (Q @ X)))))))).
% 3.07/1.62 thf(fact_105_Ball__def, axiom, (! [A:$tType]: (((ball @ A) = (^ [A5:(set @ A),P:(A > $o)]: ((! [X:A]: (((((member @ A) @ X) @ A5) => (P @ X)))))))))).
% 3.07/1.62 thf(fact_106_image__mono, axiom, (! [B:$tType,A:$tType,A2:(set @ A),B2:(set @ A),F:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ ((((image @ A) @ B) @ F) @ B2)))))).
% 3.07/1.62 thf(fact_107_image__subsetI, axiom, (! [A:$tType,B:$tType,A2:(set @ A),F:(A > B),B2:(set @ B)]: (((! [X4:A]: (((((member @ A) @ X4) @ A2) => (((member @ B) @ (F @ X4)) @ B2)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ B2))))).
% 3.07/1.62 thf(fact_108_cr__streamI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => (((coindu1183105481stream @ A) @ Xs) @ ((coindu2010755910_llist @ A) @ Xs)))))).
% 3.07/1.62 thf(fact_109_llistsum__mono, axiom, (! [B:$tType,A:$tType]: (((((monoid_add @ B) @ (type2 @ B)) & ((ordere779506340up_add @ B) @ (type2 @ B))) => (! [Xs:(coinductive_llist @ A),F:(A > B),G:(A > B)]: (((! [X4:A]: (((((member @ A) @ X4) @ ((coinductive_lset @ A) @ Xs)) => (((ord_less_eq @ B) @ (F @ X4)) @ (G @ X4))))) => (((ord_less_eq @ B) @ ((coindu780009021istsum @ B) @ ((((coinductive_lmap @ A) @ B) @ F) @ Xs))) @ ((coindu780009021istsum @ B) @ ((((coinductive_lmap @ A) @ B) @ G) @ Xs)))))))))).
% 3.07/1.62 thf(fact_110_lset__lmember, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) = (((coinductive_lmember @ A) @ X2) @ Xs))))).
% 3.07/1.62 thf(fact_111_ball__reg, axiom, (! [A:$tType,R:(set @ A),P3:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((((member @ A) @ X4) @ R) => ((P3 @ X4) => (Q @ X4))))) => ((! [X4:A]: (((((member @ A) @ X4) @ R) => (P3 @ X4)))) => (! [X5:A]: (((((member @ A) @ X5) @ R) => (Q @ X5))))))))).
% 3.07/1.62 thf(fact_112_stream_Opred__set, axiom, (! [A:$tType]: (((pred_stream @ A) = (^ [P:(A > $o),X:(stream @ A)]: ((! [Y4:A]: (((((member @ A) @ Y4) @ ((sset @ A) @ X)) => (P @ Y4)))))))))).
% 3.07/1.62 thf(fact_113_lfp_Oleq__refl, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 3.07/1.62 thf(fact_114_gfp_Oleq__trans, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [Y:A,X2:A,Z3:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ Z3) @ Y) => (((ord_less_eq @ A) @ Z3) @ X2))))))))).
% 3.07/1.62 thf(fact_115_lfp_Oleq__trans, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X2:A,Y:A,Z3:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z3) => (((ord_less_eq @ A) @ X2) @ Z3))))))))).
% 3.07/1.62 thf(fact_116_stream_Opred__mono__strong, axiom, (! [A:$tType,P3:(A > $o),X2:(stream @ A),Pa:(A > $o)]: (((((pred_stream @ A) @ P3) @ X2) => ((! [Z:A]: (((((member @ A) @ Z) @ ((sset @ A) @ X2)) => ((P3 @ Z) => (Pa @ Z))))) => (((pred_stream @ A) @ Pa) @ X2)))))).
% 3.07/1.62 thf(fact_117_stream_Opred__cong, axiom, (! [A:$tType,X2:(stream @ A),Ya:(stream @ A),P3:(A > $o),Pa:(A > $o)]: (((X2 = Ya) => ((! [Z:A]: (((((member @ A) @ Z) @ ((sset @ A) @ Ya)) => ((P3 @ Z) = (Pa @ Z))))) => ((((pred_stream @ A) @ P3) @ X2) = (((pred_stream @ A) @ Pa) @ Ya))))))).
% 3.07/1.62 thf(fact_118_lfp_Oleq__antisym, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ X2) => (X2 = Y))))))))).
% 3.07/1.62 thf(fact_119_gfp_Oleq__antisym, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y) => (X2 = Y))))))))).
% 3.07/1.62 thf(fact_120_llistsum__inf, axiom, (! [A:$tType]: ((((monoid_add @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => (((coindu780009021istsum @ A) @ Xs) = (zero_zero @ A))))))))).
% 3.07/1.62 thf(fact_121_lset__inf__llist, axiom, (! [A:$tType,F:(nat > A)]: ((((coinductive_lset @ A) @ ((coindu68654304_llist @ A) @ F)) = ((((image @ nat) @ A) @ F) @ (top_top @ (set @ nat))))))).
% 3.07/1.62 thf(fact_122_sset__smerge, axiom, (! [A:$tType,Ss:(stream @ (stream @ A))]: ((((sset @ A) @ ((smerge @ A) @ Ss)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ (stream @ A)) @ (set @ A)) @ (sset @ A)) @ ((sset @ (stream @ A)) @ Ss))))))).
% 3.07/1.62 thf(fact_123_llist_Opred__set, axiom, (! [A:$tType]: (((coindu543516966_llist @ A) = (^ [P:(A > $o),X:(coinductive_llist @ A)]: ((! [Y4:A]: (((((member @ A) @ Y4) @ ((coinductive_lset @ A) @ X)) => (P @ Y4)))))))))).
% 3.07/1.62 thf(fact_124_top__apply, axiom, (! [C:$tType,D2:$tType]: ((((top @ C) @ (type2 @ C)) => ((top_top @ (D2 > C)) = (^ [X:D2]: ((top_top @ C)))))))).
% 3.07/1.62 thf(fact_125_UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 3.07/1.62 thf(fact_126_Union__iff, axiom, (! [A:$tType,A2:A,C4:(set @ (set @ A))]: (((((member @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ C4)) = (? [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ C4) & (((member @ A) @ A2) @ X)))))))).
% 3.07/1.62 thf(fact_127_UnionI, axiom, (! [A:$tType,X3:(set @ A),C4:(set @ (set @ A)),A2:A]: (((((member @ (set @ A)) @ X3) @ C4) => ((((member @ A) @ A2) @ X3) => (((member @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ C4))))))).
% 3.07/1.62 thf(fact_128_UN__ball__bex__simps_I1_J, axiom, (! [A:$tType,A2:(set @ (set @ A)),P3:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) => (P3 @ X)))) = (! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ A2) => (! [Y4:A]: (((((member @ A) @ Y4) @ X) => (P3 @ Y4))))))))))).
% 3.07/1.62 thf(fact_129_UN__ball__bex__simps_I3_J, axiom, (! [D2:$tType,A2:(set @ (set @ D2)),P3:(D2 > $o)]: (((? [X:D2]: (((((member @ D2) @ X) @ ((complete_Sup_Sup @ (set @ D2)) @ A2)) & (P3 @ X)))) = (? [X:(set @ D2)]: (((((member @ (set @ D2)) @ X) @ A2) & (? [Y4:D2]: (((((member @ D2) @ Y4) @ X) & (P3 @ Y4))))))))))).
% 3.07/1.62 thf(fact_130_UN__ball__bex__simps_I4_J, axiom, (! [F3:$tType,E:$tType,B2:(E > (set @ F3)),A2:(set @ E),P3:(F3 > $o)]: (((? [X:F3]: (((((member @ F3) @ X) @ ((complete_Sup_Sup @ (set @ F3)) @ ((((image @ E) @ (set @ F3)) @ B2) @ A2))) & (P3 @ X)))) = (? [X:E]: (((((member @ E) @ X) @ A2) & (? [Y4:F3]: (((((member @ F3) @ Y4) @ (B2 @ X)) & (P3 @ Y4))))))))))).
% 3.07/1.62 thf(fact_131_UN__ball__bex__simps_I2_J, axiom, (! [C:$tType,B:$tType,B2:(B > (set @ C)),A2:(set @ B),P3:(C > $o)]: (((! [X:C]: (((((member @ C) @ X) @ ((complete_Sup_Sup @ (set @ C)) @ ((((image @ B) @ (set @ C)) @ B2) @ A2))) => (P3 @ X)))) = (! [X:B]: (((((member @ B) @ X) @ A2) => (! [Y4:C]: (((((member @ C) @ Y4) @ (B2 @ X)) => (P3 @ Y4))))))))))).
% 3.07/1.62 thf(fact_132_bex__UN, axiom, (! [A:$tType,B:$tType,B2:(B > (set @ A)),A2:(set @ B),P3:(A > $o)]: (((? [X:A]: (((((member @ A) @ X) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B2) @ A2))) & (P3 @ X)))) = (? [X:B]: (((((member @ B) @ X) @ A2) & (? [Y4:A]: (((((member @ A) @ Y4) @ (B2 @ X)) & (P3 @ Y4))))))))))).
% 3.07/1.62 thf(fact_133_ball__UN, axiom, (! [A:$tType,B:$tType,B2:(B > (set @ A)),A2:(set @ B),P3:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B2) @ A2))) => (P3 @ X)))) = (! [X:B]: (((((member @ B) @ X) @ A2) => (! [Y4:A]: (((((member @ A) @ Y4) @ (B2 @ X)) => (P3 @ Y4))))))))))).
% 3.07/1.62 thf(fact_134_streams__UNIV, axiom, (! [A:$tType]: ((((streams @ A) @ (top_top @ (set @ A))) = (top_top @ (set @ (stream @ A))))))).
% 3.07/1.62 thf(fact_135_Sup__UNIV, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (((complete_Sup_Sup @ A) @ (top_top @ (set @ A))) = (top_top @ A)))))).
% 3.07/1.62 thf(fact_136_Sup__eqI, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A),X2:A]: (((! [Y2:A]: (((((member @ A) @ Y2) @ A2) => (((ord_less_eq @ A) @ Y2) @ X2)))) => ((! [Y2:A]: (((! [Z4:A]: (((((member @ A) @ Z4) @ A2) => (((ord_less_eq @ A) @ Z4) @ Y2)))) => (((ord_less_eq @ A) @ X2) @ Y2)))) => (((complete_Sup_Sup @ A) @ A2) = X2))))))))).
% 3.07/1.62 thf(fact_137_complete__lattice__class_OSup__mono, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A),B2:(set @ A)]: (((! [A4:A]: (((((member @ A) @ A4) @ A2) => (? [X5:A]: (((((member @ A) @ X5) @ B2) & (((ord_less_eq @ A) @ A4) @ X5))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ ((complete_Sup_Sup @ A) @ B2))))))))).
% 3.07/1.62 thf(fact_138_Sup__least, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A),Z3:A]: (((! [X4:A]: (((((member @ A) @ X4) @ A2) => (((ord_less_eq @ A) @ X4) @ Z3)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ Z3)))))))).
% 3.07/1.62 thf(fact_139_Sup__upper, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X2:A,A2:(set @ A)]: (((((member @ A) @ X2) @ A2) => (((ord_less_eq @ A) @ X2) @ ((complete_Sup_Sup @ A) @ A2))))))))).
% 3.07/1.62 thf(fact_140_Sup__le__iff, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A),B3:A]: (((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ B3) = (! [X:A]: (((((member @ A) @ X) @ A2) => (((ord_less_eq @ A) @ X) @ B3))))))))))).
% 3.07/1.62 thf(fact_141_Sup__upper2, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [U:A,A2:(set @ A),V:A]: (((((member @ A) @ U) @ A2) => ((((ord_less_eq @ A) @ V) @ U) => (((ord_less_eq @ A) @ V) @ ((complete_Sup_Sup @ A) @ A2)))))))))).
% 3.07/1.62 thf(fact_142_top__greatest, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ (top_top @ A)))))))).
% 3.07/1.62 thf(fact_143_top_Oextremum__unique, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) = (A3 = (top_top @ A))))))))).
% 3.07/1.62 thf(fact_144_top_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) => (A3 = (top_top @ A))))))))).
% 3.07/1.62 thf(fact_145_SUP__cong, axiom, (! [A:$tType,B:$tType]: ((((complete_Sup @ A) @ (type2 @ A)) => (! [A2:(set @ B),B2:(set @ B),C4:(B > A),D:(B > A)]: (((A2 = B2) => ((! [X4:B]: (((((member @ B) @ X4) @ B2) => ((C4 @ X4) = (D @ X4))))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ C4) @ A2)) = ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ D) @ B2))))))))))).
% 3.07/1.62 thf(fact_146_Union__upper, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ (set @ A))]: (((((member @ (set @ A)) @ B2) @ A2) => (((ord_less_eq @ (set @ A)) @ B2) @ ((complete_Sup_Sup @ (set @ A)) @ A2)))))).
% 3.07/1.62 thf(fact_147_Union__least, axiom, (! [A:$tType,A2:(set @ (set @ A)),C4:(set @ A)]: (((! [X6:(set @ A)]: (((((member @ (set @ A)) @ X6) @ A2) => (((ord_less_eq @ (set @ A)) @ X6) @ C4)))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) @ C4))))).
% 3.07/1.62 thf(fact_148_Union__mono, axiom, (! [A:$tType,A2:(set @ (set @ A)),B2:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) @ ((complete_Sup_Sup @ (set @ A)) @ B2)))))).
% 3.07/1.62 thf(fact_149_range__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X2:B]: (((B3 = (F @ X2)) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 3.07/1.62 thf(fact_150_rangeI, axiom, (! [A:$tType,B:$tType,F:(B > A),X2:B]: ((((member @ A) @ (F @ X2)) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B))))))).
% 3.07/1.62 thf(fact_151_subset__UNIV, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (top_top @ (set @ A)))))).
% 3.07/1.62 thf(fact_152_Union__UNIV, axiom, (! [A:$tType]: ((((complete_Sup_Sup @ (set @ A)) @ (top_top @ (set @ (set @ A)))) = (top_top @ (set @ A)))))).
% 3.07/1.62 thf(fact_153_UnionE, axiom, (! [A:$tType,A2:A,C4:(set @ (set @ A))]: (((((member @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ C4)) => (~ ((! [X6:(set @ A)]: (((((member @ A) @ A2) @ X6) => (~ ((((member @ (set @ A)) @ X6) @ C4)))))))))))).
% 3.07/1.62 thf(fact_154_UNIV__witness, axiom, (! [A:$tType]: ((? [X4:A]: ((((member @ A) @ X4) @ (top_top @ (set @ A)))))))).
% 3.07/1.62 thf(fact_155_UNIV__eq__I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X4:A]: ((((member @ A) @ X4) @ A2))) => ((top_top @ (set @ A)) = A2))))).
% 3.07/1.62 thf(fact_156_llist_Opred__cong, axiom, (! [A:$tType,X2:(coinductive_llist @ A),Ya:(coinductive_llist @ A),P3:(A > $o),Pa:(A > $o)]: (((X2 = Ya) => ((! [Z:A]: (((((member @ A) @ Z) @ ((coinductive_lset @ A) @ Ya)) => ((P3 @ Z) = (Pa @ Z))))) => ((((coindu543516966_llist @ A) @ P3) @ X2) = (((coindu543516966_llist @ A) @ Pa) @ Ya))))))).
% 3.07/1.62 thf(fact_157_llist_Opred__mono__strong, axiom, (! [A:$tType,P3:(A > $o),X2:(coinductive_llist @ A),Pa:(A > $o)]: (((((coindu543516966_llist @ A) @ P3) @ X2) => ((! [Z:A]: (((((member @ A) @ Z) @ ((coinductive_lset @ A) @ X2)) => ((P3 @ Z) => (Pa @ Z))))) => (((coindu543516966_llist @ A) @ Pa) @ X2)))))).
% 3.07/1.62 thf(fact_158_SUP__eq, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ B),B2:(set @ C),F:(B > A),G:(C > A)]: (((! [I:B]: (((((member @ B) @ I) @ A2) => (? [X5:C]: (((((member @ C) @ X5) @ B2) & (((ord_less_eq @ A) @ (F @ I)) @ (G @ X5)))))))) => ((! [J:C]: (((((member @ C) @ J) @ B2) => (? [X5:B]: (((((member @ B) @ X5) @ A2) & (((ord_less_eq @ A) @ (G @ J)) @ (F @ X5)))))))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2)) = ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ G) @ B2))))))))))).
% 3.07/1.62 thf(fact_159_Sup__subset__mono, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ ((complete_Sup_Sup @ A) @ B2))))))))).
% 3.07/1.62 thf(fact_160_sset__range, axiom, (! [A:$tType]: (((sset @ A) = (^ [S:(stream @ A)]: (((((image @ nat) @ A) @ ((snth @ A) @ S)) @ (top_top @ (set @ nat))))))))).
% 3.07/1.62 thf(fact_161_llistsum__infllist, axiom, (! [A:$tType]: ((((monoid_add @ A) @ (type2 @ A)) => (! [F:(nat > A)]: ((((coindu780009021istsum @ A) @ ((coindu68654304_llist @ A) @ F)) = (zero_zero @ A)))))))).
% 3.07/1.62 thf(fact_162_le__zero__eq, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 3.07/1.62 thf(fact_163_iso__tuple__UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 3.07/1.62 thf(fact_164_surj__def, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y4:A]: ((? [X:B]: ((Y4 = (F @ X)))))))))).
% 3.07/1.62 thf(fact_165_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 3.07/1.62 thf(fact_166_zero__reorient, axiom, (! [A:$tType]: ((((zero @ A) @ (type2 @ A)) => (! [X2:A]: ((((zero_zero @ A) = X2) = (X2 = (zero_zero @ A))))))))).
% 3.07/1.62 thf(fact_167_zero__le, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [X2:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X2))))))).
% 3.07/1.62 thf(fact_168_surjD, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X4:B]: ((Y = (F @ X4)))))))).
% 3.07/1.62 thf(fact_169_surjE, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X4:B]: ((Y != (F @ X4)))))))))).
% 3.07/1.62 thf(fact_170_surjI, axiom, (! [B:$tType,A:$tType,G:(B > A),F:(A > B)]: (((! [X4:A]: (((G @ (F @ X4)) = X4))) => (((((image @ B) @ A) @ G) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 3.07/1.62 thf(fact_171_cSup__eq, axiom, (! [A:$tType]: (((((condit378418413attice @ A) @ (type2 @ A)) & ((no_bot @ A) @ (type2 @ A))) => (! [X3:(set @ A),A3:A]: (((! [X4:A]: (((((member @ A) @ X4) @ X3) => (((ord_less_eq @ A) @ X4) @ A3)))) => ((! [Y2:A]: (((! [X5:A]: (((((member @ A) @ X5) @ X3) => (((ord_less_eq @ A) @ X5) @ Y2)))) => (((ord_less_eq @ A) @ A3) @ Y2)))) => (((complete_Sup_Sup @ A) @ X3) = A3))))))))).
% 3.07/1.62 thf(fact_172_cSup__eq__maximum, axiom, (! [A:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [Z3:A,X3:(set @ A)]: (((((member @ A) @ Z3) @ X3) => ((! [X4:A]: (((((member @ A) @ X4) @ X3) => (((ord_less_eq @ A) @ X4) @ Z3)))) => (((complete_Sup_Sup @ A) @ X3) = Z3))))))))).
% 3.07/1.62 thf(fact_173_le__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 3.07/1.62 thf(fact_174_top1I, axiom, (! [A:$tType,X2:A]: (((top_top @ (A > $o)) @ X2)))).
% 3.07/1.62 thf(fact_175_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A))))))))).
% 3.07/1.62 thf(fact_176_top__conj_I2_J, axiom, (! [A:$tType,P3:$o,X2:A]: (((P3 & ((top_top @ (A > $o)) @ X2)) = P3)))).
% 3.07/1.62 thf(fact_177_top__conj_I1_J, axiom, (! [A:$tType,X2:A,P3:$o]: (((((top_top @ (A > $o)) @ X2) & P3) = P3)))).
% 3.07/1.62 thf(fact_178_Sup__SUP__eq, axiom, (! [A:$tType]: (((complete_Sup_Sup @ (A > $o)) = (^ [S6:(set @ (A > $o)),X:A]: ((((member @ A) @ X) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ (A > $o)) @ (set @ A)) @ (collect @ A)) @ S6))))))))).
% 3.07/1.62 thf(fact_179_surj__from__nat, axiom, (! [A:$tType]: ((((countable @ A) @ (type2 @ A)) => (((((image @ nat) @ A) @ (from_nat @ A)) @ (top_top @ (set @ nat))) = (top_top @ (set @ A))))))).
% 3.07/1.62 thf(fact_180_member__bind, axiom, (! [A:$tType,B:$tType,X2:A,P3:(set @ B),F:(B > (set @ A))]: (((((member @ A) @ X2) @ ((((bind @ B) @ A) @ P3) @ F)) = (((member @ A) @ X2) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ F) @ P3))))))).
% 3.07/1.62 thf(fact_181_image__Pow__mono, axiom, (! [B:$tType,A:$tType,F:(B > A),A2:(set @ B),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A2)) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F)) @ ((pow @ B) @ A2))) @ ((pow @ A) @ B2)))))).
% 3.07/1.62 thf(fact_182_Sup1__I, axiom, (! [A:$tType,P3:(A > $o),A2:(set @ (A > $o)),A3:A]: (((((member @ (A > $o)) @ P3) @ A2) => ((P3 @ A3) => (((complete_Sup_Sup @ (A > $o)) @ A2) @ A3)))))).
% 3.07/1.62 thf(fact_183_Pow__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)) = (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 3.07/1.62 thf(fact_184_PowI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)))))).
% 3.07/1.62 thf(fact_185_Union__Pow__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((complete_Sup_Sup @ (set @ A)) @ ((pow @ A) @ A2)) = A2)))).
% 3.07/1.62 thf(fact_186_Pow__UNIV, axiom, (! [A:$tType]: ((((pow @ A) @ (top_top @ (set @ A))) = (top_top @ (set @ (set @ A))))))).
% 3.07/1.62 thf(fact_187_image__Pow__surj, axiom, (! [B:$tType,A:$tType,F:(B > A),A2:(set @ B),B2:(set @ A)]: (((((((image @ B) @ A) @ F) @ A2) = B2) => (((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F)) @ ((pow @ B) @ A2)) = ((pow @ A) @ B2)))))).
% 3.07/1.62 thf(fact_188_Sup1__E, axiom, (! [A:$tType,A2:(set @ (A > $o)),A3:A]: (((((complete_Sup_Sup @ (A > $o)) @ A2) @ A3) => (~ ((! [P4:(A > $o)]: (((((member @ (A > $o)) @ P4) @ A2) => (~ ((P4 @ A3)))))))))))).
% 3.07/1.62 thf(fact_189_Cantors__paradox, axiom, (! [A:$tType,A2:(set @ A)]: ((~ ((? [X5:(A > (set @ A))]: ((((((image @ A) @ (set @ A)) @ X5) @ A2) = ((pow @ A) @ A2))))))))).
% 3.07/1.62 thf(fact_190_PowD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 3.07/1.62 thf(fact_191_Pow__top, axiom, (! [A:$tType,A2:(set @ A)]: ((((member @ (set @ A)) @ A2) @ ((pow @ A) @ A2))))).
% 3.07/1.62 thf(fact_192_Pow__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((pow @ A) @ A2)) @ ((pow @ A) @ B2)))))).
% 3.07/1.62 thf(fact_193_subset__Pow__Union, axiom, (! [A:$tType,A2:(set @ (set @ A))]: ((((ord_less_eq @ (set @ (set @ A))) @ A2) @ ((pow @ A) @ ((complete_Sup_Sup @ (set @ A)) @ A2)))))).
% 3.07/1.62 thf(fact_194_bind__UNION, axiom, (! [A:$tType,B:$tType]: ((((bind @ B) @ A) = (^ [A5:(set @ B),F2:(B > (set @ A))]: (((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ F2) @ A5)))))))).
% 3.07/1.62 thf(fact_195_surj__swap__iff, axiom, (! [B:$tType,A:$tType,A3:B,B3:B,F:(B > A)]: (((((((image @ B) @ A) @ (((((swap @ B) @ A) @ A3) @ B3) @ F)) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 3.07/1.62 thf(fact_196_cSUP__least, axiom, (! [B:$tType,A:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [A2:(set @ B),F:(B > A),M2:A]: (((A2 != (bot_bot @ (set @ B))) => ((! [X4:B]: (((((member @ B) @ X4) @ A2) => (((ord_less_eq @ A) @ (F @ X4)) @ M2)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2))) @ M2))))))))).
% 3.07/1.62 thf(fact_197_SUP__eq__iff, axiom, (! [A:$tType,B:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [I2:(set @ B),C3:A,F:(B > A)]: (((I2 != (bot_bot @ (set @ B))) => ((! [I:B]: (((((member @ B) @ I) @ I2) => (((ord_less_eq @ A) @ C3) @ (F @ I))))) => ((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ I2)) = C3) = (! [X:B]: (((((member @ B) @ X) @ I2) => ((F @ X) = C3))))))))))))).
% 3.07/1.62 thf(fact_198_bot__apply, axiom, (! [C:$tType,D2:$tType]: ((((bot @ C) @ (type2 @ C)) => ((bot_bot @ (D2 > C)) = (^ [X:D2]: ((bot_bot @ C)))))))).
% 3.07/1.62 thf(fact_199_empty__iff, axiom, (! [A:$tType,C3:A]: ((~ ((((member @ A) @ C3) @ (bot_bot @ (set @ A)))))))).
% 3.07/1.62 thf(fact_200_all__not__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X:A]: ((~ ((((member @ A) @ X) @ A2))))) = (A2 = (bot_bot @ (set @ A))))))).
% 3.07/1.62 thf(fact_201_Collect__empty__eq, axiom, (! [A:$tType,P3:(A > $o)]: (((((collect @ A) @ P3) = (bot_bot @ (set @ A))) = (! [X:A]: ((~ ((P3 @ X))))))))).
% 3.07/1.62 thf(fact_202_empty__Collect__eq, axiom, (! [A:$tType,P3:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P3)) = (! [X:A]: ((~ ((P3 @ X))))))))).
% 3.07/1.62 thf(fact_203_image__is__empty, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B)]: (((((((image @ B) @ A) @ F) @ A2) = (bot_bot @ (set @ A))) = (A2 = (bot_bot @ (set @ B))))))).
% 3.07/1.62 thf(fact_204_empty__is__image, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B)]: ((((bot_bot @ (set @ A)) = ((((image @ B) @ A) @ F) @ A2)) = (A2 = (bot_bot @ (set @ B))))))).
% 3.07/1.62 thf(fact_205_image__empty, axiom, (! [B:$tType,A:$tType,F:(B > A)]: ((((((image @ B) @ A) @ F) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))).
% 3.07/1.62 thf(fact_206_Sup__bot__conv_I1_J, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A)]: (((((complete_Sup_Sup @ A) @ A2) = (bot_bot @ A)) = (! [X:A]: (((((member @ A) @ X) @ A2) => (X = (bot_bot @ A)))))))))))).
% 3.07/1.62 thf(fact_207_Sup__bot__conv_I2_J, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A)]: ((((bot_bot @ A) = ((complete_Sup_Sup @ A) @ A2)) = (! [X:A]: (((((member @ A) @ X) @ A2) => (X = (bot_bot @ A)))))))))))).
% 3.07/1.62 thf(fact_208_subset__empty, axiom, (! [A:$tType,A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))) = (A2 = (bot_bot @ (set @ A))))))).
% 3.07/1.62 thf(fact_209_empty__subsetI, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A2)))).
% 3.07/1.62 thf(fact_210_ball__empty, axiom, (! [A:$tType,P3:(A > $o),X5:A]: (((((member @ A) @ X5) @ (bot_bot @ (set @ A))) => (P3 @ X5))))).
% 3.07/1.62 thf(fact_211_swap__image__eq, axiom, (! [B:$tType,A:$tType,A3:A,A2:(set @ A),B3:A,F:(A > B)]: (((((member @ A) @ A3) @ A2) => ((((member @ A) @ B3) @ A2) => (((((image @ A) @ B) @ (((((swap @ A) @ B) @ A3) @ B3) @ F)) @ A2) = ((((image @ A) @ B) @ F) @ A2))))))).
% 3.07/1.62 thf(fact_212_empty__bind, axiom, (! [B:$tType,A:$tType,F:(B > (set @ A))]: ((((((bind @ B) @ A) @ (bot_bot @ (set @ B))) @ F) = (bot_bot @ (set @ A)))))).
% 3.07/1.62 thf(fact_213_Sup__empty, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (((complete_Sup_Sup @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ A)))))).
% 3.07/1.62 thf(fact_214_empty__Union__conv, axiom, (! [A:$tType,A2:(set @ (set @ A))]: ((((bot_bot @ (set @ A)) = ((complete_Sup_Sup @ (set @ A)) @ A2)) = (! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ A2) => (X = (bot_bot @ (set @ A)))))))))).
% 3.07/1.62 thf(fact_215_Union__empty__conv, axiom, (! [A:$tType,A2:(set @ (set @ A))]: (((((complete_Sup_Sup @ (set @ A)) @ A2) = (bot_bot @ (set @ A))) = (! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ A2) => (X = (bot_bot @ (set @ A)))))))))).
% 3.07/1.62 thf(fact_216_Union__empty, axiom, (! [A:$tType]: ((((complete_Sup_Sup @ (set @ A)) @ (bot_bot @ (set @ (set @ A)))) = (bot_bot @ (set @ A)))))).
% 3.07/1.62 thf(fact_217_Pow__not__empty, axiom, (! [A:$tType,A2:(set @ A)]: ((((pow @ A) @ A2) != (bot_bot @ (set @ (set @ A))))))).
% 3.07/1.62 thf(fact_218_Pow__bottom, axiom, (! [A:$tType,B2:(set @ A)]: ((((member @ (set @ A)) @ (bot_bot @ (set @ A))) @ ((pow @ A) @ B2))))).
% 3.07/1.62 thf(fact_219_empty__not__UNIV, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) != (top_top @ (set @ A)))))).
% 3.07/1.62 thf(fact_220_sset__neq__empty, axiom, (! [A:$tType,Xs:(stream @ A)]: ((((sset @ A) @ Xs) != (bot_bot @ (set @ A)))))).
% 3.07/1.62 thf(fact_221_emptyE, axiom, (! [A:$tType,A3:A]: ((~ ((((member @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 3.07/1.62 thf(fact_222_equals0D, axiom, (! [A:$tType,A2:(set @ A),A3:A]: (((A2 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A3) @ A2))))))).
% 3.07/1.62 thf(fact_223_equals0I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [Y2:A]: ((~ ((((member @ A) @ Y2) @ A2))))) => (A2 = (bot_bot @ (set @ A))))))).
% 3.07/1.62 thf(fact_224_ex__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((? [X:A]: ((((member @ A) @ X) @ A2))) = (A2 != (bot_bot @ (set @ A))))))).
% 3.07/1.62 thf(fact_225_bot__fun__def, axiom, (! [B:$tType,A:$tType]: ((((bot @ B) @ (type2 @ B)) => ((bot_bot @ (A > B)) = (^ [X:A]: ((bot_bot @ B)))))))).
% 3.07/1.62 thf(fact_226_streams__empty, axiom, (! [A:$tType]: ((((streams @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ (stream @ A))))))).
% 3.07/1.62 thf(fact_227_bot_Oextremum, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A3))))))).
% 3.07/1.62 thf(fact_228_bot_Oextremum__unique, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) = (A3 = (bot_bot @ A))))))))).
% 3.07/1.62 thf(fact_229_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) => (A3 = (bot_bot @ A))))))))).
% 3.07/1.62 thf(fact_230_less__eq__Sup, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A),U:A]: (((! [V2:A]: (((((member @ A) @ V2) @ A2) => (((ord_less_eq @ A) @ U) @ V2)))) => ((A2 != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ U) @ ((complete_Sup_Sup @ A) @ A2)))))))))).
% 3.07/1.62 thf(fact_231_cSup__least, axiom, (! [A:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [X3:(set @ A),Z3:A]: (((X3 != (bot_bot @ (set @ A))) => ((! [X4:A]: (((((member @ A) @ X4) @ X3) => (((ord_less_eq @ A) @ X4) @ Z3)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ X3)) @ Z3))))))))).
% 3.07/1.62 thf(fact_232_cSup__eq__non__empty, axiom, (! [A:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [X3:(set @ A),A3:A]: (((X3 != (bot_bot @ (set @ A))) => ((! [X4:A]: (((((member @ A) @ X4) @ X3) => (((ord_less_eq @ A) @ X4) @ A3)))) => ((! [Y2:A]: (((! [X5:A]: (((((member @ A) @ X5) @ X3) => (((ord_less_eq @ A) @ X5) @ Y2)))) => (((ord_less_eq @ A) @ A3) @ Y2)))) => (((complete_Sup_Sup @ A) @ X3) = A3)))))))))).
% 3.07/1.62 thf(fact_233_SUP__eq__const, axiom, (! [B:$tType,A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [I2:(set @ B),F:(B > A),X2:A]: (((I2 != (bot_bot @ (set @ B))) => ((! [I:B]: (((((member @ B) @ I) @ I2) => ((F @ I) = X2)))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ I2)) = X2))))))))).
% 3.07/1.62 thf(fact_234_surj__imp__surj__swap, axiom, (! [B:$tType,A:$tType,F:(B > A),A3:B,B3:B]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((image @ B) @ A) @ (((((swap @ B) @ A) @ A3) @ B3) @ F)) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 3.07/1.62 thf(fact_235_ccSup__empty, axiom, (! [A:$tType]: ((((counta840220525attice @ A) @ (type2 @ A)) => (((complete_Sup_Sup @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ A)))))).
% 3.07/1.62 thf(fact_236_subset__emptyI, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A2))))) => (((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))))))).
% 3.07/1.62 thf(fact_237_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 3.07/1.62 thf(fact_238_lset__code, axiom, (! [A:$tType]: (((coinductive_lset @ A) = ((coinductive_gen_lset @ A) @ (bot_bot @ (set @ A))))))).
% 3.07/1.62 thf(fact_239_the__elem__image__unique, axiom, (! [B:$tType,A:$tType,A2:(set @ A),F:(A > B),X2:A]: (((A2 != (bot_bot @ (set @ A))) => ((! [Y2:A]: (((((member @ A) @ Y2) @ A2) => ((F @ Y2) = (F @ X2))))) => (((the_elem @ B) @ ((((image @ A) @ B) @ F) @ A2)) = (F @ X2))))))).
% 3.07/1.62 thf(fact_240_cSUP__subset__mono, axiom, (! [A:$tType,B:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [A2:(set @ B),G:(B > A),B2:(set @ B),F:(B > A)]: (((A2 != (bot_bot @ (set @ B))) => (((condit1810911227_above @ A) @ ((((image @ B) @ A) @ G) @ B2)) => ((((ord_less_eq @ (set @ B)) @ A2) @ B2) => ((! [X4:B]: (((((member @ B) @ X4) @ B2) => (((ord_less_eq @ A) @ (F @ X4)) @ (G @ X4))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2))) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ G) @ B2))))))))))))).
% 3.07/1.62 thf(fact_241_compact__bot, axiom, (! [A:$tType]: ((((comple1141879883l_ccpo @ A) @ (type2 @ A)) => (! [X2:A]: (((X2 = ((complete_Sup_Sup @ A) @ (bot_bot @ (set @ A)))) => ((((comple2143767107ompact @ A) @ (complete_Sup_Sup @ A)) @ (ord_less_eq @ A)) @ X2)))))))).
% 3.07/1.62 thf(fact_242_bdd__aboveI, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:(set @ A),M2:A]: (((! [X4:A]: (((((member @ A) @ X4) @ A2) => (((ord_less_eq @ A) @ X4) @ M2)))) => ((condit1810911227_above @ A) @ A2)))))))).
% 3.07/1.62 thf(fact_243_cSup__le__iff, axiom, (! [A:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [S5:(set @ A),A3:A]: (((S5 != (bot_bot @ (set @ A))) => (((condit1810911227_above @ A) @ S5) => ((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ S5)) @ A3) = (! [X:A]: (((((member @ A) @ X) @ S5) => (((ord_less_eq @ A) @ X) @ A3))))))))))))).
% 3.07/1.62 thf(fact_244_cSup__mono, axiom, (! [A:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [B2:(set @ A),A2:(set @ A)]: (((B2 != (bot_bot @ (set @ A))) => (((condit1810911227_above @ A) @ A2) => ((! [B4:A]: (((((member @ A) @ B4) @ B2) => (? [X5:A]: (((((member @ A) @ X5) @ A2) & (((ord_less_eq @ A) @ B4) @ X5))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ B2)) @ ((complete_Sup_Sup @ A) @ A2))))))))))).
% 3.07/1.62 thf(fact_245_bdd__above__def, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((condit1810911227_above @ A) = (^ [A5:(set @ A)]: ((? [M3:A]: ((! [X:A]: (((((member @ A) @ X) @ A5) => (((ord_less_eq @ A) @ X) @ M3))))))))))))).
% 3.07/1.62 thf(fact_246_bdd__above__mono, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [B2:(set @ A),A2:(set @ A)]: ((((condit1810911227_above @ A) @ B2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((condit1810911227_above @ A) @ A2))))))))).
% 3.07/1.62 thf(fact_247_cSup__upper2, axiom, (! [A:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [X2:A,X3:(set @ A),Y:A]: (((((member @ A) @ X2) @ X3) => ((((ord_less_eq @ A) @ Y) @ X2) => (((condit1810911227_above @ A) @ X3) => (((ord_less_eq @ A) @ Y) @ ((complete_Sup_Sup @ A) @ X3))))))))))).
% 3.07/1.62 thf(fact_248_cSup__upper, axiom, (! [A:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [X2:A,X3:(set @ A)]: (((((member @ A) @ X2) @ X3) => (((condit1810911227_above @ A) @ X3) => (((ord_less_eq @ A) @ X2) @ ((complete_Sup_Sup @ A) @ X3)))))))))).
% 3.07/1.62 thf(fact_249_bdd__aboveI2, axiom, (! [A:$tType,B:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:(set @ B),F:(B > A),M2:A]: (((! [X4:B]: (((((member @ B) @ X4) @ A2) => (((ord_less_eq @ A) @ (F @ X4)) @ M2)))) => ((condit1810911227_above @ A) @ ((((image @ B) @ A) @ F) @ A2))))))))).
% 3.07/1.62 thf(fact_250_cSUP__upper2, axiom, (! [A:$tType,B:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [F:(B > A),A2:(set @ B),X2:B,U:A]: ((((condit1810911227_above @ A) @ ((((image @ B) @ A) @ F) @ A2)) => ((((member @ B) @ X2) @ A2) => ((((ord_less_eq @ A) @ U) @ (F @ X2)) => (((ord_less_eq @ A) @ U) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2)))))))))))).
% 3.07/1.62 thf(fact_251_cSUP__upper, axiom, (! [A:$tType,B:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [X2:B,A2:(set @ B),F:(B > A)]: (((((member @ B) @ X2) @ A2) => (((condit1810911227_above @ A) @ ((((image @ B) @ A) @ F) @ A2)) => (((ord_less_eq @ A) @ (F @ X2)) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2))))))))))).
% 3.07/1.62 thf(fact_252_cSUP__le__iff, axiom, (! [A:$tType,B:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [A2:(set @ B),F:(B > A),U:A]: (((A2 != (bot_bot @ (set @ B))) => (((condit1810911227_above @ A) @ ((((image @ B) @ A) @ F) @ A2)) => ((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2))) @ U) = (! [X:B]: (((((member @ B) @ X) @ A2) => (((ord_less_eq @ A) @ (F @ X)) @ U))))))))))))).
% 3.07/1.62 thf(fact_253_cSUP__mono, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [A2:(set @ B),G:(C > A),B2:(set @ C),F:(B > A)]: (((A2 != (bot_bot @ (set @ B))) => (((condit1810911227_above @ A) @ ((((image @ C) @ A) @ G) @ B2)) => ((! [N4:B]: (((((member @ B) @ N4) @ A2) => (? [X5:C]: (((((member @ C) @ X5) @ B2) & (((ord_less_eq @ A) @ (F @ N4)) @ (G @ X5)))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2))) @ ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ G) @ B2)))))))))))).
% 3.07/1.62 thf(fact_254_cSup__subset__mono, axiom, (! [A:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [A2:(set @ A),B2:(set @ A)]: (((A2 != (bot_bot @ (set @ A))) => (((condit1810911227_above @ A) @ B2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ ((complete_Sup_Sup @ A) @ B2))))))))))).
% 3.07/1.62 thf(tcon_HOL_Obool___Finite__Set_Ofinite, axiom, ((finite_finite @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_Set_Oset___Finite__Set_Ofinite_1, axiom, (! [A6:$tType]: ((((finite_finite @ A6) @ (type2 @ A6)) => ((finite_finite @ (set @ A6)) @ (type2 @ (set @ A6))))))).
% 3.07/1.62 thf(tcon_fun___Finite__Set_Ofinite_2, axiom, (! [A6:$tType,A7:$tType]: (((((finite_finite @ A6) @ (type2 @ A6)) & ((finite_finite @ A7) @ (type2 @ A7))) => ((finite_finite @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice, axiom, (! [A6:$tType,A7:$tType]: ((((comple187826305attice @ A7) @ (type2 @ A7)) => ((condit378418413attice @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice, axiom, (! [A6:$tType,A7:$tType]: ((((counta840220525attice @ A7) @ (type2 @ A7)) => ((counta840220525attice @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A6:$tType,A7:$tType]: ((((comple187826305attice @ A7) @ (type2 @ A7)) => ((comple187826305attice @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_fun___Complete__Partial__Order_Occpo, axiom, (! [A6:$tType,A7:$tType]: ((((comple187826305attice @ A7) @ (type2 @ A7)) => ((comple1141879883l_ccpo @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_fun___Complete__Lattices_OSup, axiom, (! [A6:$tType,A7:$tType]: ((((complete_Sup @ A7) @ (type2 @ A7)) => ((complete_Sup @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A6:$tType,A7:$tType]: ((((order_top @ A7) @ (type2 @ A7)) => ((order_top @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A6:$tType,A7:$tType]: ((((order_bot @ A7) @ (type2 @ A7)) => ((order_bot @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_fun___Countable_Ocountable, axiom, (! [A6:$tType,A7:$tType]: (((((finite_finite @ A6) @ (type2 @ A6)) & ((countable @ A7) @ (type2 @ A7))) => ((countable @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A6:$tType,A7:$tType]: ((((preorder @ A7) @ (type2 @ A7)) => ((preorder @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: ((((order @ A7) @ (type2 @ A7)) => ((order @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_fun___Orderings_Otop, axiom, (! [A6:$tType,A7:$tType]: ((((top @ A7) @ (type2 @ A7)) => ((top @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_fun___Orderings_Oord, axiom, (! [A6:$tType,A7:$tType]: ((((ord @ A7) @ (type2 @ A7)) => ((ord @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_fun___Orderings_Obot, axiom, (! [A6:$tType,A7:$tType]: ((((bot @ A7) @ (type2 @ A7)) => ((bot @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.07/1.62 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_3, axiom, ((condit378418413attice @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, axiom, ((ordere779506340up_add @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Nat_Onat___Rings_Olinordered__semidom, axiom, ((linordered_semidom @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Nat_Onat___Complete__Lattices_OSup_4, axiom, ((complete_Sup @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Nat_Onat___Orderings_Oorder__bot_5, axiom, ((order_bot @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Nat_Onat___Countable_Ocountable_6, axiom, ((countable @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Nat_Onat___Orderings_Opreorder_7, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Nat_Onat___Groups_Omonoid__add, axiom, ((monoid_add @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Nat_Onat___Orderings_Oorder_8, axiom, ((order @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Nat_Onat___Orderings_Oord_9, axiom, ((ord @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Nat_Onat___Orderings_Obot_10, axiom, ((bot @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Nat_Onat___Groups_Ozero, axiom, ((zero @ nat) @ (type2 @ nat))).
% 3.07/1.62 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11, axiom, (! [A6:$tType]: (((condit378418413attice @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.07/1.62 thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_12, axiom, (! [A6:$tType]: (((counta840220525attice @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.07/1.62 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_13, axiom, (! [A6:$tType]: (((comple187826305attice @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.07/1.62 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_14, axiom, (! [A6:$tType]: (((comple1141879883l_ccpo @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.07/1.62 thf(tcon_Set_Oset___Complete__Lattices_OSup_15, axiom, (! [A6:$tType]: (((complete_Sup @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.07/1.62 thf(tcon_Set_Oset___Orderings_Oorder__top_16, axiom, (! [A6:$tType]: (((order_top @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.07/1.62 thf(tcon_Set_Oset___Orderings_Oorder__bot_17, axiom, (! [A6:$tType]: (((order_bot @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.07/1.62 thf(tcon_Set_Oset___Countable_Ocountable_18, axiom, (! [A6:$tType]: ((((finite_finite @ A6) @ (type2 @ A6)) => ((countable @ (set @ A6)) @ (type2 @ (set @ A6))))))).
% 3.07/1.62 thf(tcon_Set_Oset___Orderings_Opreorder_19, axiom, (! [A6:$tType]: (((preorder @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.07/1.62 thf(tcon_Set_Oset___Orderings_Oorder_20, axiom, (! [A6:$tType]: (((order @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.07/1.62 thf(tcon_Set_Oset___Orderings_Otop_21, axiom, (! [A6:$tType]: (((top @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.07/1.62 thf(tcon_Set_Oset___Orderings_Oord_22, axiom, (! [A6:$tType]: (((ord @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.07/1.62 thf(tcon_Set_Oset___Orderings_Obot_23, axiom, (! [A6:$tType]: (((bot @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.07/1.62 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_24, axiom, ((condit378418413attice @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_25, axiom, ((counta840220525attice @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_26, axiom, ((comple187826305attice @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_27, axiom, ((comple1141879883l_ccpo @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_HOL_Obool___Complete__Lattices_OSup_28, axiom, ((complete_Sup @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_HOL_Obool___Orderings_Oorder__top_29, axiom, ((order_top @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_HOL_Obool___Orderings_Oorder__bot_30, axiom, ((order_bot @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_HOL_Obool___Countable_Ocountable_31, axiom, ((countable @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_HOL_Obool___Orderings_Opreorder_32, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_HOL_Obool___Orderings_Olinorder_33, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_HOL_Obool___Orderings_Oorder_34, axiom, ((order @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_HOL_Obool___Orderings_Otop_35, axiom, ((top @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_HOL_Obool___Orderings_Oord_36, axiom, ((ord @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(tcon_HOL_Obool___Orderings_Obot_37, axiom, ((bot @ $o) @ (type2 @ $o))).
% 3.07/1.62 thf(conj_0, conjecture, (((snth @ a) @ ((coindu176146587of_seq @ a) @ f)) = f)).
% 3.07/1.62 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.dF9X7yUCOB/DTF2DTF3754.p
% 3.07/1.62 ------------------------
% 3.07/1.67 ---- Cleaned THF ---
% 3.07/1.67 thf(ty_t_Coinductive__List_Ollist,type,
% 3.07/1.67 coinductive_llist: $tType > $tType ).
% 3.07/1.67
% 3.07/1.67 thf(ty_t_Stream_Ostream,type,
% 3.07/1.67 stream: $tType > $tType ).
% 3.07/1.67
% 3.07/1.67 thf(ty_t_Set_Oset,type,
% 3.07/1.67 set: $tType > $tType ).
% 3.07/1.67
% 3.07/1.67 thf(ty_t_Nat_Onat,type,
% 3.07/1.67 nat: $tType ).
% 3.07/1.67
% 3.07/1.67 thf(ty_t_itself,type,
% 3.07/1.67 itself: $tType > $tType ).
% 3.07/1.67
% 3.07/1.67 thf(ty_tf_a,type,
% 3.07/1.67 a: $tType ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Finite__Set_Ofinite,type,
% 3.07/1.67 finite_finite:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_HOL_Otype,type,
% 3.07/1.67 type:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Groups_Ozero,type,
% 3.07/1.67 zero:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Orderings_Obot,type,
% 3.07/1.67 bot:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Orderings_Oord,type,
% 3.07/1.67 ord:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Orderings_Otop,type,
% 3.07/1.67 top:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Orderings_Oorder,type,
% 3.07/1.67 order:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Orderings_Ono__bot,type,
% 3.07/1.67 no_bot:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Groups_Omonoid__add,type,
% 3.07/1.67 monoid_add:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Orderings_Olinorder,type,
% 3.07/1.67 linorder:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Orderings_Opreorder,type,
% 3.07/1.67 preorder:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Countable_Ocountable,type,
% 3.07/1.67 countable:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Orderings_Oorder__bot,type,
% 3.07/1.67 order_bot:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Orderings_Oorder__top,type,
% 3.07/1.67 order_top:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Complete__Lattices_OSup,type,
% 3.07/1.67 complete_Sup:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Rings_Olinordered__semidom,type,
% 3.07/1.67 linordered_semidom:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Complete__Partial__Order_Occpo,type,
% 3.07/1.67 comple1141879883l_ccpo:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 3.07/1.67 ordere779506340up_add:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 3.07/1.67 comple187826305attice:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 3.07/1.67 canoni770627133id_add:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice,type,
% 3.07/1.67 counta840220525attice:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
% 3.07/1.67 condit378418413attice:
% 3.07/1.67 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Coinductive__List_Ogen__lset,type,
% 3.07/1.67 coinductive_gen_lset:
% 3.07/1.67 !>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Coinductive__List_Oinf__llist,type,
% 3.07/1.67 coindu68654304_llist:
% 3.07/1.67 !>[A: $tType] : ( ( nat > A ) > ( coinductive_llist @ A ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Coinductive__List_Olfinite,type,
% 3.07/1.67 coinductive_lfinite:
% 3.07/1.67 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Coinductive__List_Ollist_Olmap,type,
% 3.07/1.67 coinductive_lmap:
% 3.07/1.67 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( coinductive_llist @ A ) > ( coinductive_llist @ Aa ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 3.07/1.67 coinductive_lset:
% 3.07/1.67 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Coinductive__List_Ollist_Opred__llist,type,
% 3.07/1.67 coindu543516966_llist:
% 3.07/1.67 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Coinductive__List_Olmember,type,
% 3.07/1.67 coinductive_lmember:
% 3.07/1.67 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Coinductive__List_Olnth,type,
% 3.07/1.67 coinductive_lnth:
% 3.07/1.67 !>[A: $tType] : ( ( coinductive_llist @ A ) > nat > A ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Coinductive__List_Omonoid__add__class_Ollistsum,type,
% 3.07/1.67 coindu780009021istsum:
% 3.07/1.67 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ollist__of__stream,type,
% 3.07/1.67 coindu1724414836stream:
% 3.07/1.67 !>[A: $tType] : ( ( stream @ A ) > ( coinductive_llist @ A ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Oof__seq,type,
% 3.07/1.67 coindu176146587of_seq:
% 3.07/1.67 !>[A: $tType] : ( ( nat > A ) > ( stream @ A ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Ocr__stream,type,
% 3.07/1.67 coindu1183105481stream:
% 3.07/1.67 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__of__llist,type,
% 3.07/1.67 coindu2010755910_llist:
% 3.07/1.67 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 3.07/1.67 complete_Sup_Sup:
% 3.07/1.67 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Complete__Partial__Order2_Ocompact,type,
% 3.07/1.67 comple2143767107ompact:
% 3.07/1.67 !>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > A > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
% 3.07/1.67 condit1810911227_above:
% 3.07/1.67 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Countable_Ofrom__nat,type,
% 3.07/1.67 from_nat:
% 3.07/1.67 !>[A: $tType] : ( nat > A ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Fun_Oswap,type,
% 3.07/1.67 swap:
% 3.07/1.67 !>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 3.07/1.67 zero_zero:
% 3.07/1.67 !>[A: $tType] : A ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Orderings_Obot__class_Obot,type,
% 3.07/1.67 bot_bot:
% 3.07/1.67 !>[A: $tType] : A ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.07/1.67 ord_less_eq:
% 3.07/1.67 !>[A: $tType] : ( A > A > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Orderings_Otop__class_Otop,type,
% 3.07/1.67 top_top:
% 3.07/1.67 !>[A: $tType] : A ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Pure_Otype,type,
% 3.07/1.67 type2:
% 3.07/1.67 !>[A: $tType] : ( itself @ A ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Set_OBall,type,
% 3.07/1.67 ball:
% 3.07/1.67 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Set_OCollect,type,
% 3.07/1.67 collect:
% 3.07/1.67 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Set_OPow,type,
% 3.07/1.67 pow:
% 3.07/1.67 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Set_Obind,type,
% 3.07/1.67 bind:
% 3.07/1.67 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Set_Oimage,type,
% 3.07/1.67 image:
% 3.07/1.67 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Set_Othe__elem,type,
% 3.07/1.67 the_elem:
% 3.07/1.67 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Stream_Osmap2,type,
% 3.07/1.67 smap2:
% 3.07/1.67 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( stream @ A ) > ( stream @ B ) > ( stream @ C ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Stream_Osmember,type,
% 3.07/1.67 smember:
% 3.07/1.67 !>[A: $tType] : ( A > ( stream @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Stream_Osmerge,type,
% 3.07/1.67 smerge:
% 3.07/1.67 !>[A: $tType] : ( ( stream @ ( stream @ A ) ) > ( stream @ A ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Stream_Osnth,type,
% 3.07/1.67 snth:
% 3.07/1.67 !>[A: $tType] : ( ( stream @ A ) > nat > A ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Stream_Ostream_Opred__stream,type,
% 3.07/1.67 pred_stream:
% 3.07/1.67 !>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Stream_Ostream_Osmap,type,
% 3.07/1.67 smap:
% 3.07/1.67 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( stream @ A ) > ( stream @ Aa ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Stream_Ostream_Osset,type,
% 3.07/1.67 sset:
% 3.07/1.67 !>[A: $tType] : ( ( stream @ A ) > ( set @ A ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Stream_Ostream__all,type,
% 3.07/1.67 stream_all:
% 3.07/1.67 !>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_Stream_Ostreams,type,
% 3.07/1.67 streams:
% 3.07/1.67 !>[A: $tType] : ( ( set @ A ) > ( set @ ( stream @ A ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_c_member,type,
% 3.07/1.67 member:
% 3.07/1.67 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.07/1.67
% 3.07/1.67 thf(sy_v_f,type,
% 3.07/1.67 f: nat > a ).
% 3.07/1.67
% 3.07/1.67 thf(fact_0_stream__all__def,axiom,
% 3.07/1.67 ! [A: $tType] :
% 3.07/1.67 ( ( stream_all @ A )
% 3.07/1.67 = ( ^ [P: A > $o,S: stream @ A] :
% 3.07/1.67 ! [P2: nat] : ( P @ ( snth @ A @ S @ P2 ) ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_1_snth__smap2,axiom,
% 3.07/1.67 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,S1: stream @ B,S2: stream @ C,N: nat] :
% 3.07/1.67 ( ( snth @ A @ ( smap2 @ B @ C @ A @ F @ S1 @ S2 ) @ N )
% 3.07/1.67 = ( F @ ( snth @ B @ S1 @ N ) @ ( snth @ C @ S2 @ N ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_2_smap2__alt,axiom,
% 3.07/1.67 ! [A: $tType,B: $tType,C: $tType,F: B > C > A,S1: stream @ B,S2: stream @ C,S3: stream @ A] :
% 3.07/1.67 ( ( ( smap2 @ B @ C @ A @ F @ S1 @ S2 )
% 3.07/1.67 = S3 )
% 3.07/1.67 = ( ! [N2: nat] :
% 3.07/1.67 ( ( F @ ( snth @ B @ S1 @ N2 ) @ ( snth @ C @ S2 @ N2 ) )
% 3.07/1.67 = ( snth @ A @ S3 @ N2 ) ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_3_of__seq_Oabs__eq,axiom,
% 3.07/1.67 ! [A: $tType] :
% 3.07/1.67 ( ( coindu176146587of_seq @ A )
% 3.07/1.67 = ( ^ [X: nat > A] : ( coindu2010755910_llist @ A @ ( coindu68654304_llist @ A @ X ) ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_4_of__seq_Orep__eq,axiom,
% 3.07/1.67 ! [A: $tType,X2: nat > A] :
% 3.07/1.67 ( ( coindu1724414836stream @ A @ ( coindu176146587of_seq @ A @ X2 ) )
% 3.07/1.67 = ( coindu68654304_llist @ A @ X2 ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_5_snth__in,axiom,
% 3.07/1.67 ! [A: $tType,S3: stream @ A,X3: set @ A,N: nat] :
% 3.07/1.67 ( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ X3 ) )
% 3.07/1.67 => ( member @ A @ ( snth @ A @ S3 @ N ) @ X3 ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_6_streams__iff__snth,axiom,
% 3.07/1.67 ! [A: $tType,S3: stream @ A,X3: set @ A] :
% 3.07/1.67 ( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ X3 ) )
% 3.07/1.67 = ( ! [N2: nat] : ( member @ A @ ( snth @ A @ S3 @ N2 ) @ X3 ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_7_snth__smap,axiom,
% 3.07/1.67 ! [A: $tType,B: $tType,F: B > A,S3: stream @ B,N: nat] :
% 3.07/1.67 ( ( snth @ A @ ( smap @ B @ A @ F @ S3 ) @ N )
% 3.07/1.67 = ( F @ ( snth @ B @ S3 @ N ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_8_snth__sset,axiom,
% 3.07/1.67 ! [A: $tType,S3: stream @ A,N: nat] : ( member @ A @ ( snth @ A @ S3 @ N ) @ ( sset @ A @ S3 ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_9_lnth__list__of__stream,axiom,
% 3.07/1.67 ! [A: $tType,Xs: stream @ A] :
% 3.07/1.67 ( ( coinductive_lnth @ A @ ( coindu1724414836stream @ A @ Xs ) )
% 3.07/1.67 = ( snth @ A @ Xs ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_10_smap__alt,axiom,
% 3.07/1.67 ! [A: $tType,B: $tType,F: B > A,S3: stream @ B,S4: stream @ A] :
% 3.07/1.67 ( ( ( smap @ B @ A @ F @ S3 )
% 3.07/1.67 = S4 )
% 3.07/1.67 = ( ! [N2: nat] :
% 3.07/1.67 ( ( F @ ( snth @ B @ S3 @ N2 ) )
% 3.07/1.67 = ( snth @ A @ S4 @ N2 ) ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_11_snth__sset__smerge,axiom,
% 3.07/1.67 ! [A: $tType,Ss: stream @ ( stream @ A ),N: nat,M: nat] : ( member @ A @ ( snth @ A @ ( snth @ ( stream @ A ) @ Ss @ N ) @ M ) @ ( sset @ A @ ( smerge @ A @ Ss ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_12_stream_ORep__inverse,axiom,
% 3.07/1.67 ! [A: $tType,X2: stream @ A] :
% 3.07/1.67 ( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ X2 ) )
% 3.07/1.67 = X2 ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_13_stream__of__llist__llist__of__stream,axiom,
% 3.07/1.67 ! [A: $tType,Xs: stream @ A] :
% 3.07/1.67 ( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ Xs ) )
% 3.07/1.67 = Xs ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_14_smap__streams,axiom,
% 3.07/1.67 ! [A: $tType,B: $tType,S3: stream @ A,A2: set @ A,F: A > B,B2: set @ B] :
% 3.07/1.67 ( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) )
% 3.07/1.67 => ( ! [X4: A] :
% 3.07/1.67 ( ( member @ A @ X4 @ A2 )
% 3.07/1.67 => ( member @ B @ ( F @ X4 ) @ B2 ) )
% 3.07/1.67 => ( member @ ( stream @ B ) @ ( smap @ A @ B @ F @ S3 ) @ ( streams @ B @ B2 ) ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_15_stream_Omap__cong,axiom,
% 3.07/1.67 ! [B: $tType,A: $tType,X2: stream @ A,Ya: stream @ A,F: A > B,G: A > B] :
% 3.07/1.67 ( ( X2 = Ya )
% 3.07/1.67 => ( ! [Z: A] :
% 3.07/1.67 ( ( member @ A @ Z @ ( sset @ A @ Ya ) )
% 3.07/1.67 => ( ( F @ Z )
% 3.07/1.67 = ( G @ Z ) ) )
% 3.07/1.67 => ( ( smap @ A @ B @ F @ X2 )
% 3.07/1.67 = ( smap @ A @ B @ G @ Ya ) ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_16_stream_Omap__cong0,axiom,
% 3.07/1.67 ! [B: $tType,A: $tType,X2: stream @ A,F: A > B,G: A > B] :
% 3.07/1.67 ( ! [Z: A] :
% 3.07/1.67 ( ( member @ A @ Z @ ( sset @ A @ X2 ) )
% 3.07/1.67 => ( ( F @ Z )
% 3.07/1.67 = ( G @ Z ) ) )
% 3.07/1.67 => ( ( smap @ A @ B @ F @ X2 )
% 3.07/1.67 = ( smap @ A @ B @ G @ X2 ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_17_stream_Oinj__map__strong,axiom,
% 3.07/1.67 ! [B: $tType,A: $tType,X2: stream @ A,Xa: stream @ A,F: A > B,Fa: A > B] :
% 3.07/1.67 ( ! [Z: A,Za: A] :
% 3.07/1.67 ( ( member @ A @ Z @ ( sset @ A @ X2 ) )
% 3.07/1.67 => ( ( member @ A @ Za @ ( sset @ A @ Xa ) )
% 3.07/1.67 => ( ( ( F @ Z )
% 3.07/1.67 = ( Fa @ Za ) )
% 3.07/1.67 => ( Z = Za ) ) ) )
% 3.07/1.67 => ( ( ( smap @ A @ B @ F @ X2 )
% 3.07/1.67 = ( smap @ A @ B @ Fa @ Xa ) )
% 3.07/1.67 => ( X2 = Xa ) ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_18_stream_ORep__inject,axiom,
% 3.07/1.67 ! [A: $tType,X2: stream @ A,Y: stream @ A] :
% 3.07/1.67 ( ( ( coindu1724414836stream @ A @ X2 )
% 3.07/1.67 = ( coindu1724414836stream @ A @ Y ) )
% 3.07/1.67 = ( X2 = Y ) ) ).
% 3.07/1.67
% 3.07/1.67 thf(fact_19_lnth__inf__llist,axiom,
% 3.07/1.68 ! [A: $tType,F: nat > A,N: nat] :
% 3.07/1.68 ( ( coinductive_lnth @ A @ ( coindu68654304_llist @ A @ F ) @ N )
% 3.07/1.68 = ( F @ N ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_20_inf__llist__inj,axiom,
% 3.07/1.68 ! [A: $tType,F: nat > A,G: nat > A] :
% 3.07/1.68 ( ( ( coindu68654304_llist @ A @ F )
% 3.07/1.68 = ( coindu68654304_llist @ A @ G ) )
% 3.07/1.68 = ( F = G ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_21_Stream_Osmember__def,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( smember @ A )
% 3.07/1.68 = ( ^ [X: A,S: stream @ A] : ( member @ A @ X @ ( sset @ A @ S ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_22_stream__all__iff,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( stream_all @ A )
% 3.07/1.68 = ( ^ [P: A > $o,S: stream @ A] :
% 3.07/1.68 ! [X: A] :
% 3.07/1.68 ( ( member @ A @ X @ ( sset @ A @ S ) )
% 3.07/1.68 => ( P @ X ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_23_llist__of__stream__stream__of__llist,axiom,
% 3.07/1.68 ! [A: $tType,Xs: coinductive_llist @ A] :
% 3.07/1.68 ( ~ ( coinductive_lfinite @ A @ Xs )
% 3.07/1.68 => ( ( coindu1724414836stream @ A @ ( coindu2010755910_llist @ A @ Xs ) )
% 3.07/1.68 = Xs ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_24_lset__llist__of__stream,axiom,
% 3.07/1.68 ! [A: $tType,Xs: stream @ A] :
% 3.07/1.68 ( ( coinductive_lset @ A @ ( coindu1724414836stream @ A @ Xs ) )
% 3.07/1.68 = ( sset @ A @ Xs ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_25_sset__streams,axiom,
% 3.07/1.68 ! [A: $tType,S3: stream @ A,A2: set @ A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ ( sset @ A @ S3 ) @ A2 )
% 3.07/1.68 => ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_26_streams__sset,axiom,
% 3.07/1.68 ! [A: $tType,S3: stream @ A,A2: set @ A] :
% 3.07/1.68 ( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) )
% 3.07/1.68 => ( ord_less_eq @ ( set @ A ) @ ( sset @ A @ S3 ) @ A2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_27_streams__iff__sset,axiom,
% 3.07/1.68 ! [A: $tType,S3: stream @ A,A2: set @ A] :
% 3.07/1.68 ( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) )
% 3.07/1.68 = ( ord_less_eq @ ( set @ A ) @ ( sset @ A @ S3 ) @ A2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_28_lmap__llist__of__stream,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,F: B > A,Xs: stream @ B] :
% 3.07/1.68 ( ( coinductive_lmap @ B @ A @ F @ ( coindu1724414836stream @ B @ Xs ) )
% 3.07/1.68 = ( coindu1724414836stream @ A @ ( smap @ B @ A @ F @ Xs ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_29_stream_Oset__map,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,F: A > B,V: stream @ A] :
% 3.07/1.68 ( ( sset @ B @ ( smap @ A @ B @ F @ V ) )
% 3.07/1.68 = ( image @ A @ B @ F @ ( sset @ A @ V ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_30_lfinite__lmap,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B] :
% 3.07/1.68 ( ( coinductive_lfinite @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
% 3.07/1.68 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_31_llist_Oset__map,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,F: A > B,V: coinductive_llist @ A] :
% 3.07/1.68 ( ( coinductive_lset @ B @ ( coinductive_lmap @ A @ B @ F @ V ) )
% 3.07/1.68 = ( image @ A @ B @ F @ ( coinductive_lset @ A @ V ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_32_inf__llist__lnth,axiom,
% 3.07/1.68 ! [A: $tType,Xs: coinductive_llist @ A] :
% 3.07/1.68 ( ~ ( coinductive_lfinite @ A @ Xs )
% 3.07/1.68 => ( ( coindu68654304_llist @ A @ ( coinductive_lnth @ A @ Xs ) )
% 3.07/1.68 = Xs ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_33_sset__stream__of__llist,axiom,
% 3.07/1.68 ! [A: $tType,Xs: coinductive_llist @ A] :
% 3.07/1.68 ( ~ ( coinductive_lfinite @ A @ Xs )
% 3.07/1.68 => ( ~ ( coinductive_lfinite @ A @ Xs )
% 3.07/1.68 => ( ( sset @ A @ ( coindu2010755910_llist @ A @ Xs ) )
% 3.07/1.68 = ( coinductive_lset @ A @ Xs ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_34_smap__stream__of__llist,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,Xs: coinductive_llist @ A,F: A > B] :
% 3.07/1.68 ( ~ ( coinductive_lfinite @ A @ Xs )
% 3.07/1.68 => ( ( smap @ A @ B @ F @ ( coindu2010755910_llist @ A @ Xs ) )
% 3.07/1.68 = ( coindu2010755910_llist @ B @ ( coinductive_lmap @ A @ B @ F @ Xs ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_35_llist_Oinj__map__strong,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,X2: coinductive_llist @ A,Xa: coinductive_llist @ A,F: A > B,Fa: A > B] :
% 3.07/1.68 ( ! [Z: A,Za: A] :
% 3.07/1.68 ( ( member @ A @ Z @ ( coinductive_lset @ A @ X2 ) )
% 3.07/1.68 => ( ( member @ A @ Za @ ( coinductive_lset @ A @ Xa ) )
% 3.07/1.68 => ( ( ( F @ Z )
% 3.07/1.68 = ( Fa @ Za ) )
% 3.07/1.68 => ( Z = Za ) ) ) )
% 3.07/1.68 => ( ( ( coinductive_lmap @ A @ B @ F @ X2 )
% 3.07/1.68 = ( coinductive_lmap @ A @ B @ Fa @ Xa ) )
% 3.07/1.68 => ( X2 = Xa ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_36_llist_Omap__cong0,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,X2: coinductive_llist @ A,F: A > B,G: A > B] :
% 3.07/1.68 ( ! [Z: A] :
% 3.07/1.68 ( ( member @ A @ Z @ ( coinductive_lset @ A @ X2 ) )
% 3.07/1.68 => ( ( F @ Z )
% 3.07/1.68 = ( G @ Z ) ) )
% 3.07/1.68 => ( ( coinductive_lmap @ A @ B @ F @ X2 )
% 3.07/1.68 = ( coinductive_lmap @ A @ B @ G @ X2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_37_llist_Omap__cong,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,F: A > B,G: A > B] :
% 3.07/1.68 ( ( X2 = Ya )
% 3.07/1.68 => ( ! [Z: A] :
% 3.07/1.68 ( ( member @ A @ Z @ ( coinductive_lset @ A @ Ya ) )
% 3.07/1.68 => ( ( F @ Z )
% 3.07/1.68 = ( G @ Z ) ) )
% 3.07/1.68 => ( ( coinductive_lmap @ A @ B @ F @ X2 )
% 3.07/1.68 = ( coinductive_lmap @ A @ B @ G @ Ya ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_38_wlog__linorder__le,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( linorder @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [P3: A > A > $o,B3: A,A3: A] :
% 3.07/1.68 ( ! [A4: A,B4: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.07/1.68 => ( P3 @ A4 @ B4 ) )
% 3.07/1.68 => ( ( ( P3 @ B3 @ A3 )
% 3.07/1.68 => ( P3 @ A3 @ B3 ) )
% 3.07/1.68 => ( P3 @ A3 @ B3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_39_streams__mono2,axiom,
% 3.07/1.68 ! [A: $tType,S5: set @ A,T: set @ A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ S5 @ T )
% 3.07/1.68 => ( ord_less_eq @ ( set @ ( stream @ A ) ) @ ( streams @ A @ S5 ) @ ( streams @ A @ T ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_40_lfinite__inf__llist,axiom,
% 3.07/1.68 ! [A: $tType,F: nat > A] :
% 3.07/1.68 ~ ( coinductive_lfinite @ A @ ( coindu68654304_llist @ A @ F ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_41_lfinite__llist__of__stream,axiom,
% 3.07/1.68 ! [A: $tType,Xs: stream @ A] :
% 3.07/1.68 ~ ( coinductive_lfinite @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_42_streams__mono,axiom,
% 3.07/1.68 ! [A: $tType,S3: stream @ A,A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) )
% 3.07/1.68 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ B2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_43_not__False__in__image__Ball,axiom,
% 3.07/1.68 ! [A: $tType,P3: A > $o,A2: set @ A] :
% 3.07/1.68 ( ( ~ ( member @ $o @ $false @ ( image @ A @ $o @ P3 @ A2 ) ) )
% 3.07/1.68 = ( ! [X: A] :
% 3.07/1.68 ( ( member @ A @ X @ A2 )
% 3.07/1.68 => ( P3 @ X ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_44_subset__antisym,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 3.07/1.68 => ( A2 = B2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_45_mem__Collect__eq,axiom,
% 3.07/1.68 ! [A: $tType,A3: A,P3: A > $o] :
% 3.07/1.68 ( ( member @ A @ A3 @ ( collect @ A @ P3 ) )
% 3.07/1.68 = ( P3 @ A3 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_46_Collect__mem__eq,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] :
% 3.07/1.68 ( ( collect @ A
% 3.07/1.68 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 3.07/1.68 = A2 ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_47_Collect__cong,axiom,
% 3.07/1.68 ! [A: $tType,P3: A > $o,Q: A > $o] :
% 3.07/1.68 ( ! [X4: A] :
% 3.07/1.68 ( ( P3 @ X4 )
% 3.07/1.68 = ( Q @ X4 ) )
% 3.07/1.68 => ( ( collect @ A @ P3 )
% 3.07/1.68 = ( collect @ A @ Q ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_48_ext,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.07/1.68 ( ! [X4: A] :
% 3.07/1.68 ( ( F @ X4 )
% 3.07/1.68 = ( G @ X4 ) )
% 3.07/1.68 => ( F = G ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_49_subsetI,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ! [X4: A] :
% 3.07/1.68 ( ( member @ A @ X4 @ A2 )
% 3.07/1.68 => ( member @ A @ X4 @ B2 ) )
% 3.07/1.68 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_50_image__eqI,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,B3: A,F: B > A,X2: B,A2: set @ B] :
% 3.07/1.68 ( ( B3
% 3.07/1.68 = ( F @ X2 ) )
% 3.07/1.68 => ( ( member @ B @ X2 @ A2 )
% 3.07/1.68 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_51_order__refl,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( preorder @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_52_Ball__Collect,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( ball @ A )
% 3.07/1.68 = ( ^ [A5: set @ A,P: A > $o] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_53_subset__image__iff,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 3.07/1.68 = ( ? [AA: set @ B] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
% 3.07/1.68 & ( B2
% 3.07/1.68 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_54_image__subset__iff,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 3.07/1.68 = ( ! [X: B] :
% 3.07/1.68 ( ( member @ B @ X @ A2 )
% 3.07/1.68 => ( member @ A @ ( F @ X ) @ B2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_55_subset__imageE,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 3.07/1.68 => ~ ! [C2: set @ B] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ B ) @ C2 @ A2 )
% 3.07/1.68 => ( B2
% 3.07/1.68 != ( image @ B @ A @ F @ C2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_56_le__funD,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType] :
% 3.07/1.68 ( ( ord @ B @ ( type2 @ B ) )
% 3.07/1.68 => ! [F: A > B,G: A > B,X2: A] :
% 3.07/1.68 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.07/1.68 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_57_le__funE,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType] :
% 3.07/1.68 ( ( ord @ B @ ( type2 @ B ) )
% 3.07/1.68 => ! [F: A > B,G: A > B,X2: A] :
% 3.07/1.68 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.07/1.68 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_58_le__funI,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType] :
% 3.07/1.68 ( ( ord @ B @ ( type2 @ B ) )
% 3.07/1.68 => ! [F: A > B,G: A > B] :
% 3.07/1.68 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
% 3.07/1.68 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_59_le__fun__def,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType] :
% 3.07/1.68 ( ( ord @ B @ ( type2 @ B ) )
% 3.07/1.68 => ( ( ord_less_eq @ ( A > B ) )
% 3.07/1.68 = ( ^ [F2: A > B,G2: A > B] :
% 3.07/1.68 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_60_order__subst1,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType] :
% 3.07/1.68 ( ( ( order @ B @ ( type2 @ B ) )
% 3.07/1.68 & ( order @ A @ ( type2 @ A ) ) )
% 3.07/1.68 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 3.07/1.68 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 3.07/1.68 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 3.07/1.68 => ( ! [X4: B,Y2: B] :
% 3.07/1.68 ( ( ord_less_eq @ B @ X4 @ Y2 )
% 3.07/1.68 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 3.07/1.68 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_61_order__subst2,axiom,
% 3.07/1.68 ! [A: $tType,C: $tType] :
% 3.07/1.68 ( ( ( order @ C @ ( type2 @ C ) )
% 3.07/1.68 & ( order @ A @ ( type2 @ A ) ) )
% 3.07/1.68 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 3.07/1.68 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.07/1.68 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 3.07/1.68 => ( ! [X4: A,Y2: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 3.07/1.68 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 3.07/1.68 => ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_62_ord__eq__le__subst,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType] :
% 3.07/1.68 ( ( ( ord @ B @ ( type2 @ B ) )
% 3.07/1.68 & ( ord @ A @ ( type2 @ A ) ) )
% 3.07/1.68 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 3.07/1.68 ( ( A3
% 3.07/1.68 = ( F @ B3 ) )
% 3.07/1.68 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 3.07/1.68 => ( ! [X4: B,Y2: B] :
% 3.07/1.68 ( ( ord_less_eq @ B @ X4 @ Y2 )
% 3.07/1.68 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 3.07/1.68 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_63_ord__le__eq__subst,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType] :
% 3.07/1.68 ( ( ( ord @ B @ ( type2 @ B ) )
% 3.07/1.68 & ( ord @ A @ ( type2 @ A ) ) )
% 3.07/1.68 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 3.07/1.68 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.07/1.68 => ( ( ( F @ B3 )
% 3.07/1.68 = C3 )
% 3.07/1.68 => ( ! [X4: A,Y2: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 3.07/1.68 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 3.07/1.68 => ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_64_eq__iff,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order @ A @ ( type2 @ A ) )
% 3.07/1.68 => ( ( ^ [Y3: A,Z2: A] : ( Y3 = Z2 ) )
% 3.07/1.68 = ( ^ [X: A,Y4: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ X @ Y4 )
% 3.07/1.68 & ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_65_antisym,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A,Y: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.07/1.68 => ( ( ord_less_eq @ A @ Y @ X2 )
% 3.07/1.68 => ( X2 = Y ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_66_linear,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( linorder @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A,Y: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.07/1.68 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_67_eq__refl,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( preorder @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A,Y: A] :
% 3.07/1.68 ( ( X2 = Y )
% 3.07/1.68 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_68_le__cases,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( linorder @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A,Y: A] :
% 3.07/1.68 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 3.07/1.68 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_69_order_Otrans,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A3: A,B3: A,C3: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.07/1.68 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 3.07/1.68 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_70_le__cases3,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( linorder @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A,Y: A,Z3: A] :
% 3.07/1.68 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 3.07/1.68 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 3.07/1.68 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 3.07/1.68 => ~ ( ord_less_eq @ A @ X2 @ Z3 ) )
% 3.07/1.68 => ( ( ( ord_less_eq @ A @ X2 @ Z3 )
% 3.07/1.68 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 3.07/1.68 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 3.07/1.68 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 3.07/1.68 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 3.07/1.68 => ~ ( ord_less_eq @ A @ Z3 @ X2 ) )
% 3.07/1.68 => ~ ( ( ord_less_eq @ A @ Z3 @ X2 )
% 3.07/1.68 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_71_antisym__conv,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [Y: A,X2: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ Y @ X2 )
% 3.07/1.68 => ( ( ord_less_eq @ A @ X2 @ Y )
% 3.07/1.68 = ( X2 = Y ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_72_ord__eq__le__trans,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( ord @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A3: A,B3: A,C3: A] :
% 3.07/1.68 ( ( A3 = B3 )
% 3.07/1.68 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 3.07/1.68 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_73_ord__le__eq__trans,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( ord @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A3: A,B3: A,C3: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.07/1.68 => ( ( B3 = C3 )
% 3.07/1.68 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_74_order__class_Oorder_Oantisym,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A3: A,B3: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.07/1.68 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.07/1.68 => ( A3 = B3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_75_order__trans,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( preorder @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A,Y: A,Z3: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.07/1.68 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 3.07/1.68 => ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_76_dual__order_Orefl,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_77_linorder__wlog,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( linorder @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [P3: A > A > $o,A3: A,B3: A] :
% 3.07/1.68 ( ! [A4: A,B4: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.07/1.68 => ( P3 @ A4 @ B4 ) )
% 3.07/1.68 => ( ! [A4: A,B4: A] :
% 3.07/1.68 ( ( P3 @ B4 @ A4 )
% 3.07/1.68 => ( P3 @ A4 @ B4 ) )
% 3.07/1.68 => ( P3 @ A3 @ B3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_78_dual__order_Otrans,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [B3: A,A3: A,C3: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.07/1.68 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 3.07/1.68 => ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_79_dual__order_Oantisym,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [B3: A,A3: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.07/1.68 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.07/1.68 => ( A3 = B3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_80_imageI,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,X2: A,A2: set @ A,F: A > B] :
% 3.07/1.68 ( ( member @ A @ X2 @ A2 )
% 3.07/1.68 => ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_81_image__iff,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,Z3: A,F: B > A,A2: set @ B] :
% 3.07/1.68 ( ( member @ A @ Z3 @ ( image @ B @ A @ F @ A2 ) )
% 3.07/1.68 = ( ? [X: B] :
% 3.07/1.68 ( ( member @ B @ X @ A2 )
% 3.07/1.68 & ( Z3
% 3.07/1.68 = ( F @ X ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_82_bex__imageD,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P3: A > $o] :
% 3.07/1.68 ( ? [X5: A] :
% 3.07/1.68 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A2 ) )
% 3.07/1.68 & ( P3 @ X5 ) )
% 3.07/1.68 => ? [X4: B] :
% 3.07/1.68 ( ( member @ B @ X4 @ A2 )
% 3.07/1.68 & ( P3 @ ( F @ X4 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_83_image__cong,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,M2: set @ A,N3: set @ A,F: A > B,G: A > B] :
% 3.07/1.68 ( ( M2 = N3 )
% 3.07/1.68 => ( ! [X4: A] :
% 3.07/1.68 ( ( member @ A @ X4 @ N3 )
% 3.07/1.68 => ( ( F @ X4 )
% 3.07/1.68 = ( G @ X4 ) ) )
% 3.07/1.68 => ( ( image @ A @ B @ F @ M2 )
% 3.07/1.68 = ( image @ A @ B @ G @ N3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_84_ball__imageD,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P3: A > $o] :
% 3.07/1.68 ( ! [X4: A] :
% 3.07/1.68 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A2 ) )
% 3.07/1.68 => ( P3 @ X4 ) )
% 3.07/1.68 => ! [X5: B] :
% 3.07/1.68 ( ( member @ B @ X5 @ A2 )
% 3.07/1.68 => ( P3 @ ( F @ X5 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_85_rev__image__eqI,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,X2: A,A2: set @ A,B3: B,F: A > B] :
% 3.07/1.68 ( ( member @ A @ X2 @ A2 )
% 3.07/1.68 => ( ( B3
% 3.07/1.68 = ( F @ X2 ) )
% 3.07/1.68 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_86_Inf_OINF__cong,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C4: B > A,D: B > A,Inf: ( set @ A ) > A] :
% 3.07/1.68 ( ( A2 = B2 )
% 3.07/1.68 => ( ! [X4: B] :
% 3.07/1.68 ( ( member @ B @ X4 @ B2 )
% 3.07/1.68 => ( ( C4 @ X4 )
% 3.07/1.68 = ( D @ X4 ) ) )
% 3.07/1.68 => ( ( Inf @ ( image @ B @ A @ C4 @ A2 ) )
% 3.07/1.68 = ( Inf @ ( image @ B @ A @ D @ B2 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_87_Sup_OSUP__cong,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C4: B > A,D: B > A,Sup: ( set @ A ) > A] :
% 3.07/1.68 ( ( A2 = B2 )
% 3.07/1.68 => ( ! [X4: B] :
% 3.07/1.68 ( ( member @ B @ X4 @ B2 )
% 3.07/1.68 => ( ( C4 @ X4 )
% 3.07/1.68 = ( D @ X4 ) ) )
% 3.07/1.68 => ( ( Sup @ ( image @ B @ A @ C4 @ A2 ) )
% 3.07/1.68 = ( Sup @ ( image @ B @ A @ D @ B2 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_88_set__mp,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( ( member @ A @ X2 @ A2 )
% 3.07/1.68 => ( member @ A @ X2 @ B2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_89_in__mono,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( ( member @ A @ X2 @ A2 )
% 3.07/1.68 => ( member @ A @ X2 @ B2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_90_subsetD,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( ( member @ A @ C3 @ A2 )
% 3.07/1.68 => ( member @ A @ C3 @ B2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_91_subsetCE,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( ( member @ A @ C3 @ A2 )
% 3.07/1.68 => ( member @ A @ C3 @ B2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_92_equalityE,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ( A2 = B2 )
% 3.07/1.68 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_93_subset__eq,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) )
% 3.07/1.68 = ( ^ [A5: set @ A,B5: set @ A] :
% 3.07/1.68 ! [X: A] :
% 3.07/1.68 ( ( member @ A @ X @ A5 )
% 3.07/1.68 => ( member @ A @ X @ B5 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_94_equalityD1,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ( A2 = B2 )
% 3.07/1.68 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_95_equalityD2,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ( A2 = B2 )
% 3.07/1.68 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_96_set__rev__mp,axiom,
% 3.07/1.68 ! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ( member @ A @ X2 @ A2 )
% 3.07/1.68 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( member @ A @ X2 @ B2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_97_subset__iff,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) )
% 3.07/1.68 = ( ^ [A5: set @ A,B5: set @ A] :
% 3.07/1.68 ! [T2: A] :
% 3.07/1.68 ( ( member @ A @ T2 @ A5 )
% 3.07/1.68 => ( member @ A @ T2 @ B5 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_98_rev__subsetD,axiom,
% 3.07/1.68 ! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ( member @ A @ C3 @ A2 )
% 3.07/1.68 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( member @ A @ C3 @ B2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_99_subset__refl,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_100_Collect__mono,axiom,
% 3.07/1.68 ! [A: $tType,P3: A > $o,Q: A > $o] :
% 3.07/1.68 ( ! [X4: A] :
% 3.07/1.68 ( ( P3 @ X4 )
% 3.07/1.68 => ( Q @ X4 ) )
% 3.07/1.68 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_101_subset__trans,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
% 3.07/1.68 => ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_102_set__eq__subset,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( ^ [Y3: set @ A,Z2: set @ A] : ( Y3 = Z2 ) )
% 3.07/1.68 = ( ^ [A5: set @ A,B5: set @ A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 3.07/1.68 & ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_103_contra__subsetD,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( ~ ( member @ A @ C3 @ B2 )
% 3.07/1.68 => ~ ( member @ A @ C3 @ A2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_104_Collect__mono__iff,axiom,
% 3.07/1.68 ! [A: $tType,P3: A > $o,Q: A > $o] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) )
% 3.07/1.68 = ( ! [X: A] :
% 3.07/1.68 ( ( P3 @ X )
% 3.07/1.68 => ( Q @ X ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_105_Ball__def,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( ball @ A )
% 3.07/1.68 = ( ^ [A5: set @ A,P: A > $o] :
% 3.07/1.68 ! [X: A] :
% 3.07/1.68 ( ( member @ A @ X @ A5 )
% 3.07/1.68 => ( P @ X ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_106_image__mono,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_107_image__subsetI,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
% 3.07/1.68 ( ! [X4: A] :
% 3.07/1.68 ( ( member @ A @ X4 @ A2 )
% 3.07/1.68 => ( member @ B @ ( F @ X4 ) @ B2 ) )
% 3.07/1.68 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_108_cr__streamI,axiom,
% 3.07/1.68 ! [A: $tType,Xs: coinductive_llist @ A] :
% 3.07/1.68 ( ~ ( coinductive_lfinite @ A @ Xs )
% 3.07/1.68 => ( coindu1183105481stream @ A @ Xs @ ( coindu2010755910_llist @ A @ Xs ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_109_llistsum__mono,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType] :
% 3.07/1.68 ( ( ( monoid_add @ B @ ( type2 @ B ) )
% 3.07/1.68 & ( ordere779506340up_add @ B @ ( type2 @ B ) ) )
% 3.07/1.68 => ! [Xs: coinductive_llist @ A,F: A > B,G: A > B] :
% 3.07/1.68 ( ! [X4: A] :
% 3.07/1.68 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 3.07/1.68 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 3.07/1.68 => ( ord_less_eq @ B @ ( coindu780009021istsum @ B @ ( coinductive_lmap @ A @ B @ F @ Xs ) ) @ ( coindu780009021istsum @ B @ ( coinductive_lmap @ A @ B @ G @ Xs ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_110_lset__lmember,axiom,
% 3.07/1.68 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
% 3.07/1.68 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 3.07/1.68 = ( coinductive_lmember @ A @ X2 @ Xs ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_111_ball__reg,axiom,
% 3.07/1.68 ! [A: $tType,R: set @ A,P3: A > $o,Q: A > $o] :
% 3.07/1.68 ( ! [X4: A] :
% 3.07/1.68 ( ( member @ A @ X4 @ R )
% 3.07/1.68 => ( ( P3 @ X4 )
% 3.07/1.68 => ( Q @ X4 ) ) )
% 3.07/1.68 => ( ! [X4: A] :
% 3.07/1.68 ( ( member @ A @ X4 @ R )
% 3.07/1.68 => ( P3 @ X4 ) )
% 3.07/1.68 => ! [X5: A] :
% 3.07/1.68 ( ( member @ A @ X5 @ R )
% 3.07/1.68 => ( Q @ X5 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_112_stream_Opred__set,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( pred_stream @ A )
% 3.07/1.68 = ( ^ [P: A > $o,X: stream @ A] :
% 3.07/1.68 ! [Y4: A] :
% 3.07/1.68 ( ( member @ A @ Y4 @ ( sset @ A @ X ) )
% 3.07/1.68 => ( P @ Y4 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_113_lfp_Oleq__refl,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_114_gfp_Oleq__trans,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [Y: A,X2: A,Z3: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ Y @ X2 )
% 3.07/1.68 => ( ( ord_less_eq @ A @ Z3 @ Y )
% 3.07/1.68 => ( ord_less_eq @ A @ Z3 @ X2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_115_lfp_Oleq__trans,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A,Y: A,Z3: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.07/1.68 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 3.07/1.68 => ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_116_stream_Opred__mono__strong,axiom,
% 3.07/1.68 ! [A: $tType,P3: A > $o,X2: stream @ A,Pa: A > $o] :
% 3.07/1.68 ( ( pred_stream @ A @ P3 @ X2 )
% 3.07/1.68 => ( ! [Z: A] :
% 3.07/1.68 ( ( member @ A @ Z @ ( sset @ A @ X2 ) )
% 3.07/1.68 => ( ( P3 @ Z )
% 3.07/1.68 => ( Pa @ Z ) ) )
% 3.07/1.68 => ( pred_stream @ A @ Pa @ X2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_117_stream_Opred__cong,axiom,
% 3.07/1.68 ! [A: $tType,X2: stream @ A,Ya: stream @ A,P3: A > $o,Pa: A > $o] :
% 3.07/1.68 ( ( X2 = Ya )
% 3.07/1.68 => ( ! [Z: A] :
% 3.07/1.68 ( ( member @ A @ Z @ ( sset @ A @ Ya ) )
% 3.07/1.68 => ( ( P3 @ Z )
% 3.07/1.68 = ( Pa @ Z ) ) )
% 3.07/1.68 => ( ( pred_stream @ A @ P3 @ X2 )
% 3.07/1.68 = ( pred_stream @ A @ Pa @ Ya ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_118_lfp_Oleq__antisym,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A,Y: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.07/1.68 => ( ( ord_less_eq @ A @ Y @ X2 )
% 3.07/1.68 => ( X2 = Y ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_119_gfp_Oleq__antisym,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [Y: A,X2: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ Y @ X2 )
% 3.07/1.68 => ( ( ord_less_eq @ A @ X2 @ Y )
% 3.07/1.68 => ( X2 = Y ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_120_llistsum__inf,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( monoid_add @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [Xs: coinductive_llist @ A] :
% 3.07/1.68 ( ~ ( coinductive_lfinite @ A @ Xs )
% 3.07/1.68 => ( ( coindu780009021istsum @ A @ Xs )
% 3.07/1.68 = ( zero_zero @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_121_lset__inf__llist,axiom,
% 3.07/1.68 ! [A: $tType,F: nat > A] :
% 3.07/1.68 ( ( coinductive_lset @ A @ ( coindu68654304_llist @ A @ F ) )
% 3.07/1.68 = ( image @ nat @ A @ F @ ( top_top @ ( set @ nat ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_122_sset__smerge,axiom,
% 3.07/1.68 ! [A: $tType,Ss: stream @ ( stream @ A )] :
% 3.07/1.68 ( ( sset @ A @ ( smerge @ A @ Ss ) )
% 3.07/1.68 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( stream @ A ) @ ( set @ A ) @ ( sset @ A ) @ ( sset @ ( stream @ A ) @ Ss ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_123_llist_Opred__set,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( coindu543516966_llist @ A )
% 3.07/1.68 = ( ^ [P: A > $o,X: coinductive_llist @ A] :
% 3.07/1.68 ! [Y4: A] :
% 3.07/1.68 ( ( member @ A @ Y4 @ ( coinductive_lset @ A @ X ) )
% 3.07/1.68 => ( P @ Y4 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_124_top__apply,axiom,
% 3.07/1.68 ! [C: $tType,D2: $tType] :
% 3.07/1.68 ( ( top @ C @ ( type2 @ C ) )
% 3.07/1.68 => ( ( top_top @ ( D2 > C ) )
% 3.07/1.68 = ( ^ [X: D2] : ( top_top @ C ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_125_UNIV__I,axiom,
% 3.07/1.68 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_126_Union__iff,axiom,
% 3.07/1.68 ! [A: $tType,A2: A,C4: set @ ( set @ A )] :
% 3.07/1.68 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
% 3.07/1.68 = ( ? [X: set @ A] :
% 3.07/1.68 ( ( member @ ( set @ A ) @ X @ C4 )
% 3.07/1.68 & ( member @ A @ A2 @ X ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_127_UnionI,axiom,
% 3.07/1.68 ! [A: $tType,X3: set @ A,C4: set @ ( set @ A ),A2: A] :
% 3.07/1.68 ( ( member @ ( set @ A ) @ X3 @ C4 )
% 3.07/1.68 => ( ( member @ A @ A2 @ X3 )
% 3.07/1.68 => ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_128_UN__ball__bex__simps_I1_J,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ ( set @ A ),P3: A > $o] :
% 3.07/1.68 ( ( ! [X: A] :
% 3.07/1.68 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
% 3.07/1.68 => ( P3 @ X ) ) )
% 3.07/1.68 = ( ! [X: set @ A] :
% 3.07/1.68 ( ( member @ ( set @ A ) @ X @ A2 )
% 3.07/1.68 => ! [Y4: A] :
% 3.07/1.68 ( ( member @ A @ Y4 @ X )
% 3.07/1.68 => ( P3 @ Y4 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_129_UN__ball__bex__simps_I3_J,axiom,
% 3.07/1.68 ! [D2: $tType,A2: set @ ( set @ D2 ),P3: D2 > $o] :
% 3.07/1.68 ( ( ? [X: D2] :
% 3.07/1.68 ( ( member @ D2 @ X @ ( complete_Sup_Sup @ ( set @ D2 ) @ A2 ) )
% 3.07/1.68 & ( P3 @ X ) ) )
% 3.07/1.68 = ( ? [X: set @ D2] :
% 3.07/1.68 ( ( member @ ( set @ D2 ) @ X @ A2 )
% 3.07/1.68 & ? [Y4: D2] :
% 3.07/1.68 ( ( member @ D2 @ Y4 @ X )
% 3.07/1.68 & ( P3 @ Y4 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_130_UN__ball__bex__simps_I4_J,axiom,
% 3.07/1.68 ! [F3: $tType,E: $tType,B2: E > ( set @ F3 ),A2: set @ E,P3: F3 > $o] :
% 3.07/1.68 ( ( ? [X: F3] :
% 3.07/1.68 ( ( member @ F3 @ X @ ( complete_Sup_Sup @ ( set @ F3 ) @ ( image @ E @ ( set @ F3 ) @ B2 @ A2 ) ) )
% 3.07/1.68 & ( P3 @ X ) ) )
% 3.07/1.68 = ( ? [X: E] :
% 3.07/1.68 ( ( member @ E @ X @ A2 )
% 3.07/1.68 & ? [Y4: F3] :
% 3.07/1.68 ( ( member @ F3 @ Y4 @ ( B2 @ X ) )
% 3.07/1.68 & ( P3 @ Y4 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_131_UN__ball__bex__simps_I2_J,axiom,
% 3.07/1.68 ! [C: $tType,B: $tType,B2: B > ( set @ C ),A2: set @ B,P3: C > $o] :
% 3.07/1.68 ( ( ! [X: C] :
% 3.07/1.68 ( ( member @ C @ X @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ B2 @ A2 ) ) )
% 3.07/1.68 => ( P3 @ X ) ) )
% 3.07/1.68 = ( ! [X: B] :
% 3.07/1.68 ( ( member @ B @ X @ A2 )
% 3.07/1.68 => ! [Y4: C] :
% 3.07/1.68 ( ( member @ C @ Y4 @ ( B2 @ X ) )
% 3.07/1.68 => ( P3 @ Y4 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_132_bex__UN,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P3: A > $o] :
% 3.07/1.68 ( ( ? [X: A] :
% 3.07/1.68 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
% 3.07/1.68 & ( P3 @ X ) ) )
% 3.07/1.68 = ( ? [X: B] :
% 3.07/1.68 ( ( member @ B @ X @ A2 )
% 3.07/1.68 & ? [Y4: A] :
% 3.07/1.68 ( ( member @ A @ Y4 @ ( B2 @ X ) )
% 3.07/1.68 & ( P3 @ Y4 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_133_ball__UN,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P3: A > $o] :
% 3.07/1.68 ( ( ! [X: A] :
% 3.07/1.68 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
% 3.07/1.68 => ( P3 @ X ) ) )
% 3.07/1.68 = ( ! [X: B] :
% 3.07/1.68 ( ( member @ B @ X @ A2 )
% 3.07/1.68 => ! [Y4: A] :
% 3.07/1.68 ( ( member @ A @ Y4 @ ( B2 @ X ) )
% 3.07/1.68 => ( P3 @ Y4 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_134_streams__UNIV,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( streams @ A @ ( top_top @ ( set @ A ) ) )
% 3.07/1.68 = ( top_top @ ( set @ ( stream @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_135_Sup__UNIV,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
% 3.07/1.68 = ( top_top @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_136_Sup__eqI,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ A,X2: A] :
% 3.07/1.68 ( ! [Y2: A] :
% 3.07/1.68 ( ( member @ A @ Y2 @ A2 )
% 3.07/1.68 => ( ord_less_eq @ A @ Y2 @ X2 ) )
% 3.07/1.68 => ( ! [Y2: A] :
% 3.07/1.68 ( ! [Z4: A] :
% 3.07/1.68 ( ( member @ A @ Z4 @ A2 )
% 3.07/1.68 => ( ord_less_eq @ A @ Z4 @ Y2 ) )
% 3.07/1.68 => ( ord_less_eq @ A @ X2 @ Y2 ) )
% 3.07/1.68 => ( ( complete_Sup_Sup @ A @ A2 )
% 3.07/1.68 = X2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_137_complete__lattice__class_OSup__mono,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ! [A4: A] :
% 3.07/1.68 ( ( member @ A @ A4 @ A2 )
% 3.07/1.68 => ? [X5: A] :
% 3.07/1.68 ( ( member @ A @ X5 @ B2 )
% 3.07/1.68 & ( ord_less_eq @ A @ A4 @ X5 ) ) )
% 3.07/1.68 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_138_Sup__least,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ A,Z3: A] :
% 3.07/1.68 ( ! [X4: A] :
% 3.07/1.68 ( ( member @ A @ X4 @ A2 )
% 3.07/1.68 => ( ord_less_eq @ A @ X4 @ Z3 ) )
% 3.07/1.68 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ Z3 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_139_Sup__upper,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A,A2: set @ A] :
% 3.07/1.68 ( ( member @ A @ X2 @ A2 )
% 3.07/1.68 => ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_140_Sup__le__iff,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ A,B3: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ B3 )
% 3.07/1.68 = ( ! [X: A] :
% 3.07/1.68 ( ( member @ A @ X @ A2 )
% 3.07/1.68 => ( ord_less_eq @ A @ X @ B3 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_141_Sup__upper2,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [U: A,A2: set @ A,V: A] :
% 3.07/1.68 ( ( member @ A @ U @ A2 )
% 3.07/1.68 => ( ( ord_less_eq @ A @ V @ U )
% 3.07/1.68 => ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_142_top__greatest,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order_top @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_143_top_Oextremum__unique,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order_top @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A3: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 3.07/1.68 = ( A3
% 3.07/1.68 = ( top_top @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_144_top_Oextremum__uniqueI,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order_top @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A3: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 3.07/1.68 => ( A3
% 3.07/1.68 = ( top_top @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_145_SUP__cong,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType] :
% 3.07/1.68 ( ( complete_Sup @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ B,B2: set @ B,C4: B > A,D: B > A] :
% 3.07/1.68 ( ( A2 = B2 )
% 3.07/1.68 => ( ! [X4: B] :
% 3.07/1.68 ( ( member @ B @ X4 @ B2 )
% 3.07/1.68 => ( ( C4 @ X4 )
% 3.07/1.68 = ( D @ X4 ) ) )
% 3.07/1.68 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ C4 @ A2 ) )
% 3.07/1.68 = ( complete_Sup_Sup @ A @ ( image @ B @ A @ D @ B2 ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_146_Union__upper,axiom,
% 3.07/1.68 ! [A: $tType,B2: set @ A,A2: set @ ( set @ A )] :
% 3.07/1.68 ( ( member @ ( set @ A ) @ B2 @ A2 )
% 3.07/1.68 => ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_147_Union__least,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ ( set @ A ),C4: set @ A] :
% 3.07/1.68 ( ! [X6: set @ A] :
% 3.07/1.68 ( ( member @ ( set @ A ) @ X6 @ A2 )
% 3.07/1.68 => ( ord_less_eq @ ( set @ A ) @ X6 @ C4 ) )
% 3.07/1.68 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ C4 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_148_Union__mono,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ B2 )
% 3.07/1.68 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_149_range__eqI,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,B3: A,F: B > A,X2: B] :
% 3.07/1.68 ( ( B3
% 3.07/1.68 = ( F @ X2 ) )
% 3.07/1.68 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_150_rangeI,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,F: B > A,X2: B] : ( member @ A @ ( F @ X2 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_151_subset__UNIV,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_152_Union__UNIV,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 3.07/1.68 = ( top_top @ ( set @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_153_UnionE,axiom,
% 3.07/1.68 ! [A: $tType,A2: A,C4: set @ ( set @ A )] :
% 3.07/1.68 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
% 3.07/1.68 => ~ ! [X6: set @ A] :
% 3.07/1.68 ( ( member @ A @ A2 @ X6 )
% 3.07/1.68 => ~ ( member @ ( set @ A ) @ X6 @ C4 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_154_UNIV__witness,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_155_UNIV__eq__I,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] :
% 3.07/1.68 ( ! [X4: A] : ( member @ A @ X4 @ A2 )
% 3.07/1.68 => ( ( top_top @ ( set @ A ) )
% 3.07/1.68 = A2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_156_llist_Opred__cong,axiom,
% 3.07/1.68 ! [A: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,P3: A > $o,Pa: A > $o] :
% 3.07/1.68 ( ( X2 = Ya )
% 3.07/1.68 => ( ! [Z: A] :
% 3.07/1.68 ( ( member @ A @ Z @ ( coinductive_lset @ A @ Ya ) )
% 3.07/1.68 => ( ( P3 @ Z )
% 3.07/1.68 = ( Pa @ Z ) ) )
% 3.07/1.68 => ( ( coindu543516966_llist @ A @ P3 @ X2 )
% 3.07/1.68 = ( coindu543516966_llist @ A @ Pa @ Ya ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_157_llist_Opred__mono__strong,axiom,
% 3.07/1.68 ! [A: $tType,P3: A > $o,X2: coinductive_llist @ A,Pa: A > $o] :
% 3.07/1.68 ( ( coindu543516966_llist @ A @ P3 @ X2 )
% 3.07/1.68 => ( ! [Z: A] :
% 3.07/1.68 ( ( member @ A @ Z @ ( coinductive_lset @ A @ X2 ) )
% 3.07/1.68 => ( ( P3 @ Z )
% 3.07/1.68 => ( Pa @ Z ) ) )
% 3.07/1.68 => ( coindu543516966_llist @ A @ Pa @ X2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_158_SUP__eq,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,C: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ B,B2: set @ C,F: B > A,G: C > A] :
% 3.07/1.68 ( ! [I: B] :
% 3.07/1.68 ( ( member @ B @ I @ A2 )
% 3.07/1.68 => ? [X5: C] :
% 3.07/1.68 ( ( member @ C @ X5 @ B2 )
% 3.07/1.68 & ( ord_less_eq @ A @ ( F @ I ) @ ( G @ X5 ) ) ) )
% 3.07/1.68 => ( ! [J: C] :
% 3.07/1.68 ( ( member @ C @ J @ B2 )
% 3.07/1.68 => ? [X5: B] :
% 3.07/1.68 ( ( member @ B @ X5 @ A2 )
% 3.07/1.68 & ( ord_less_eq @ A @ ( G @ J ) @ ( F @ X5 ) ) ) )
% 3.07/1.68 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) )
% 3.07/1.68 = ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_159_Sup__subset__mono,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_160_sset__range,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( sset @ A )
% 3.07/1.68 = ( ^ [S: stream @ A] : ( image @ nat @ A @ ( snth @ A @ S ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_161_llistsum__infllist,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( monoid_add @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [F: nat > A] :
% 3.07/1.68 ( ( coindu780009021istsum @ A @ ( coindu68654304_llist @ A @ F ) )
% 3.07/1.68 = ( zero_zero @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_162_le__zero__eq,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [N: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 3.07/1.68 = ( N
% 3.07/1.68 = ( zero_zero @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_163_iso__tuple__UNIV__I,axiom,
% 3.07/1.68 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_164_surj__def,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,F: B > A] :
% 3.07/1.68 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 3.07/1.68 = ( top_top @ ( set @ A ) ) )
% 3.07/1.68 = ( ! [Y4: A] :
% 3.07/1.68 ? [X: B] :
% 3.07/1.68 ( Y4
% 3.07/1.68 = ( F @ X ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_165_top__set__def,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( top_top @ ( set @ A ) )
% 3.07/1.68 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_166_zero__reorient,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( zero @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A] :
% 3.07/1.68 ( ( ( zero_zero @ A )
% 3.07/1.68 = X2 )
% 3.07/1.68 = ( X2
% 3.07/1.68 = ( zero_zero @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_167_zero__le,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_168_surjD,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 3.07/1.68 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 3.07/1.68 = ( top_top @ ( set @ A ) ) )
% 3.07/1.68 => ? [X4: B] :
% 3.07/1.68 ( Y
% 3.07/1.68 = ( F @ X4 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_169_surjE,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 3.07/1.68 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 3.07/1.68 = ( top_top @ ( set @ A ) ) )
% 3.07/1.68 => ~ ! [X4: B] :
% 3.07/1.68 ( Y
% 3.07/1.68 != ( F @ X4 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_170_surjI,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 3.07/1.68 ( ! [X4: A] :
% 3.07/1.68 ( ( G @ ( F @ X4 ) )
% 3.07/1.68 = X4 )
% 3.07/1.68 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 3.07/1.68 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_171_cSup__eq,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 & ( no_bot @ A @ ( type2 @ A ) ) )
% 3.07/1.68 => ! [X3: set @ A,A3: A] :
% 3.07/1.68 ( ! [X4: A] :
% 3.07/1.68 ( ( member @ A @ X4 @ X3 )
% 3.07/1.68 => ( ord_less_eq @ A @ X4 @ A3 ) )
% 3.07/1.68 => ( ! [Y2: A] :
% 3.07/1.68 ( ! [X5: A] :
% 3.07/1.68 ( ( member @ A @ X5 @ X3 )
% 3.07/1.68 => ( ord_less_eq @ A @ X5 @ Y2 ) )
% 3.07/1.68 => ( ord_less_eq @ A @ A3 @ Y2 ) )
% 3.07/1.68 => ( ( complete_Sup_Sup @ A @ X3 )
% 3.07/1.68 = A3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_172_cSup__eq__maximum,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [Z3: A,X3: set @ A] :
% 3.07/1.68 ( ( member @ A @ Z3 @ X3 )
% 3.07/1.68 => ( ! [X4: A] :
% 3.07/1.68 ( ( member @ A @ X4 @ X3 )
% 3.07/1.68 => ( ord_less_eq @ A @ X4 @ Z3 ) )
% 3.07/1.68 => ( ( complete_Sup_Sup @ A @ X3 )
% 3.07/1.68 = Z3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_173_le__numeral__extra_I3_J,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 3.07/1.68 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_174_top1I,axiom,
% 3.07/1.68 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_175_top__empty__eq,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( top_top @ ( A > $o ) )
% 3.07/1.68 = ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_176_top__conj_I2_J,axiom,
% 3.07/1.68 ! [A: $tType,P3: $o,X2: A] :
% 3.07/1.68 ( ( P3
% 3.07/1.68 & ( top_top @ ( A > $o ) @ X2 ) )
% 3.07/1.68 = P3 ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_177_top__conj_I1_J,axiom,
% 3.07/1.68 ! [A: $tType,X2: A,P3: $o] :
% 3.07/1.68 ( ( ( top_top @ ( A > $o ) @ X2 )
% 3.07/1.68 & P3 )
% 3.07/1.68 = P3 ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_178_Sup__SUP__eq,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( complete_Sup_Sup @ ( A > $o ) )
% 3.07/1.68 = ( ^ [S6: set @ ( A > $o ),X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S6 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_179_surj__from__nat,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( countable @ A @ ( type2 @ A ) )
% 3.07/1.68 => ( ( image @ nat @ A @ ( from_nat @ A ) @ ( top_top @ ( set @ nat ) ) )
% 3.07/1.68 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_180_member__bind,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,X2: A,P3: set @ B,F: B > ( set @ A )] :
% 3.07/1.68 ( ( member @ A @ X2 @ ( bind @ B @ A @ P3 @ F ) )
% 3.07/1.68 = ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F @ P3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_181_image__Pow__mono,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 3.07/1.68 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) ) @ ( pow @ A @ B2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_182_Sup1__I,axiom,
% 3.07/1.68 ! [A: $tType,P3: A > $o,A2: set @ ( A > $o ),A3: A] :
% 3.07/1.68 ( ( member @ ( A > $o ) @ P3 @ A2 )
% 3.07/1.68 => ( ( P3 @ A3 )
% 3.07/1.68 => ( complete_Sup_Sup @ ( A > $o ) @ A2 @ A3 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_183_Pow__iff,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 3.07/1.68 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_184_PowI,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_185_Union__Pow__eq,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] :
% 3.07/1.68 ( ( complete_Sup_Sup @ ( set @ A ) @ ( pow @ A @ A2 ) )
% 3.07/1.68 = A2 ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_186_Pow__UNIV,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( pow @ A @ ( top_top @ ( set @ A ) ) )
% 3.07/1.68 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_187_image__Pow__surj,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 3.07/1.68 ( ( ( image @ B @ A @ F @ A2 )
% 3.07/1.68 = B2 )
% 3.07/1.68 => ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) )
% 3.07/1.68 = ( pow @ A @ B2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_188_Sup1__E,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ ( A > $o ),A3: A] :
% 3.07/1.68 ( ( complete_Sup_Sup @ ( A > $o ) @ A2 @ A3 )
% 3.07/1.68 => ~ ! [P4: A > $o] :
% 3.07/1.68 ( ( member @ ( A > $o ) @ P4 @ A2 )
% 3.07/1.68 => ~ ( P4 @ A3 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_189_Cantors__paradox,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] :
% 3.07/1.68 ~ ? [X5: A > ( set @ A )] :
% 3.07/1.68 ( ( image @ A @ ( set @ A ) @ X5 @ A2 )
% 3.07/1.68 = ( pow @ A @ A2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_190_PowD,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 3.07/1.68 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_191_Pow__top,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_192_Pow__mono,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_193_subset__Pow__Union,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_194_bind__UNION,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType] :
% 3.07/1.68 ( ( bind @ B @ A )
% 3.07/1.68 = ( ^ [A5: set @ B,F2: B > ( set @ A )] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ A5 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_195_surj__swap__iff,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,A3: B,B3: B,F: B > A] :
% 3.07/1.68 ( ( ( image @ B @ A @ ( swap @ B @ A @ A3 @ B3 @ F ) @ ( top_top @ ( set @ B ) ) )
% 3.07/1.68 = ( top_top @ ( set @ A ) ) )
% 3.07/1.68 = ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 3.07/1.68 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_196_cSUP__least,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ B,F: B > A,M2: A] :
% 3.07/1.68 ( ( A2
% 3.07/1.68 != ( bot_bot @ ( set @ B ) ) )
% 3.07/1.68 => ( ! [X4: B] :
% 3.07/1.68 ( ( member @ B @ X4 @ A2 )
% 3.07/1.68 => ( ord_less_eq @ A @ ( F @ X4 ) @ M2 ) )
% 3.07/1.68 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ M2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_197_SUP__eq__iff,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [I2: set @ B,C3: A,F: B > A] :
% 3.07/1.68 ( ( I2
% 3.07/1.68 != ( bot_bot @ ( set @ B ) ) )
% 3.07/1.68 => ( ! [I: B] :
% 3.07/1.68 ( ( member @ B @ I @ I2 )
% 3.07/1.68 => ( ord_less_eq @ A @ C3 @ ( F @ I ) ) )
% 3.07/1.68 => ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ I2 ) )
% 3.07/1.68 = C3 )
% 3.07/1.68 = ( ! [X: B] :
% 3.07/1.68 ( ( member @ B @ X @ I2 )
% 3.07/1.68 => ( ( F @ X )
% 3.07/1.68 = C3 ) ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_198_bot__apply,axiom,
% 3.07/1.68 ! [C: $tType,D2: $tType] :
% 3.07/1.68 ( ( bot @ C @ ( type2 @ C ) )
% 3.07/1.68 => ( ( bot_bot @ ( D2 > C ) )
% 3.07/1.68 = ( ^ [X: D2] : ( bot_bot @ C ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_199_empty__iff,axiom,
% 3.07/1.68 ! [A: $tType,C3: A] :
% 3.07/1.68 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_200_all__not__in__conv,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] :
% 3.07/1.68 ( ( ! [X: A] :
% 3.07/1.68 ~ ( member @ A @ X @ A2 ) )
% 3.07/1.68 = ( A2
% 3.07/1.68 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_201_Collect__empty__eq,axiom,
% 3.07/1.68 ! [A: $tType,P3: A > $o] :
% 3.07/1.68 ( ( ( collect @ A @ P3 )
% 3.07/1.68 = ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 = ( ! [X: A] :
% 3.07/1.68 ~ ( P3 @ X ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_202_empty__Collect__eq,axiom,
% 3.07/1.68 ! [A: $tType,P3: A > $o] :
% 3.07/1.68 ( ( ( bot_bot @ ( set @ A ) )
% 3.07/1.68 = ( collect @ A @ P3 ) )
% 3.07/1.68 = ( ! [X: A] :
% 3.07/1.68 ~ ( P3 @ X ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_203_image__is__empty,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
% 3.07/1.68 ( ( ( image @ B @ A @ F @ A2 )
% 3.07/1.68 = ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 = ( A2
% 3.07/1.68 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_204_empty__is__image,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
% 3.07/1.68 ( ( ( bot_bot @ ( set @ A ) )
% 3.07/1.68 = ( image @ B @ A @ F @ A2 ) )
% 3.07/1.68 = ( A2
% 3.07/1.68 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_205_image__empty,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,F: B > A] :
% 3.07/1.68 ( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
% 3.07/1.68 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_206_Sup__bot__conv_I1_J,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ A] :
% 3.07/1.68 ( ( ( complete_Sup_Sup @ A @ A2 )
% 3.07/1.68 = ( bot_bot @ A ) )
% 3.07/1.68 = ( ! [X: A] :
% 3.07/1.68 ( ( member @ A @ X @ A2 )
% 3.07/1.68 => ( X
% 3.07/1.68 = ( bot_bot @ A ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_207_Sup__bot__conv_I2_J,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ A] :
% 3.07/1.68 ( ( ( bot_bot @ A )
% 3.07/1.68 = ( complete_Sup_Sup @ A @ A2 ) )
% 3.07/1.68 = ( ! [X: A] :
% 3.07/1.68 ( ( member @ A @ X @ A2 )
% 3.07/1.68 => ( X
% 3.07/1.68 = ( bot_bot @ A ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_208_subset__empty,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] :
% 3.07/1.68 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 = ( A2
% 3.07/1.68 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_209_empty__subsetI,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_210_ball__empty,axiom,
% 3.07/1.68 ! [A: $tType,P3: A > $o,X5: A] :
% 3.07/1.68 ( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 => ( P3 @ X5 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_211_swap__image__eq,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,A3: A,A2: set @ A,B3: A,F: A > B] :
% 3.07/1.68 ( ( member @ A @ A3 @ A2 )
% 3.07/1.68 => ( ( member @ A @ B3 @ A2 )
% 3.07/1.68 => ( ( image @ A @ B @ ( swap @ A @ B @ A3 @ B3 @ F ) @ A2 )
% 3.07/1.68 = ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_212_empty__bind,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,F: B > ( set @ A )] :
% 3.07/1.68 ( ( bind @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F )
% 3.07/1.68 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_213_Sup__empty,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 = ( bot_bot @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_214_empty__Union__conv,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ ( set @ A )] :
% 3.07/1.68 ( ( ( bot_bot @ ( set @ A ) )
% 3.07/1.68 = ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
% 3.07/1.68 = ( ! [X: set @ A] :
% 3.07/1.68 ( ( member @ ( set @ A ) @ X @ A2 )
% 3.07/1.68 => ( X
% 3.07/1.68 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_215_Union__empty__conv,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ ( set @ A )] :
% 3.07/1.68 ( ( ( complete_Sup_Sup @ ( set @ A ) @ A2 )
% 3.07/1.68 = ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 = ( ! [X: set @ A] :
% 3.07/1.68 ( ( member @ ( set @ A ) @ X @ A2 )
% 3.07/1.68 => ( X
% 3.07/1.68 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_216_Union__empty,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 3.07/1.68 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_217_Pow__not__empty,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] :
% 3.07/1.68 ( ( pow @ A @ A2 )
% 3.07/1.68 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_218_Pow__bottom,axiom,
% 3.07/1.68 ! [A: $tType,B2: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_219_empty__not__UNIV,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( bot_bot @ ( set @ A ) )
% 3.07/1.68 != ( top_top @ ( set @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_220_sset__neq__empty,axiom,
% 3.07/1.68 ! [A: $tType,Xs: stream @ A] :
% 3.07/1.68 ( ( sset @ A @ Xs )
% 3.07/1.68 != ( bot_bot @ ( set @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_221_emptyE,axiom,
% 3.07/1.68 ! [A: $tType,A3: A] :
% 3.07/1.68 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_222_equals0D,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A,A3: A] :
% 3.07/1.68 ( ( A2
% 3.07/1.68 = ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_223_equals0I,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] :
% 3.07/1.68 ( ! [Y2: A] :
% 3.07/1.68 ~ ( member @ A @ Y2 @ A2 )
% 3.07/1.68 => ( A2
% 3.07/1.68 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_224_ex__in__conv,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] :
% 3.07/1.68 ( ( ? [X: A] : ( member @ A @ X @ A2 ) )
% 3.07/1.68 = ( A2
% 3.07/1.68 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_225_bot__fun__def,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType] :
% 3.07/1.68 ( ( bot @ B @ ( type2 @ B ) )
% 3.07/1.68 => ( ( bot_bot @ ( A > B ) )
% 3.07/1.68 = ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_226_streams__empty,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( streams @ A @ ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 = ( bot_bot @ ( set @ ( stream @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_227_bot_Oextremum,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order_bot @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_228_bot_Oextremum__unique,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order_bot @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A3: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 3.07/1.68 = ( A3
% 3.07/1.68 = ( bot_bot @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_229_bot_Oextremum__uniqueI,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( order_bot @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A3: A] :
% 3.07/1.68 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 3.07/1.68 => ( A3
% 3.07/1.68 = ( bot_bot @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_230_less__eq__Sup,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ A,U: A] :
% 3.07/1.68 ( ! [V2: A] :
% 3.07/1.68 ( ( member @ A @ V2 @ A2 )
% 3.07/1.68 => ( ord_less_eq @ A @ U @ V2 ) )
% 3.07/1.68 => ( ( A2
% 3.07/1.68 != ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_231_cSup__least,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X3: set @ A,Z3: A] :
% 3.07/1.68 ( ( X3
% 3.07/1.68 != ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 => ( ! [X4: A] :
% 3.07/1.68 ( ( member @ A @ X4 @ X3 )
% 3.07/1.68 => ( ord_less_eq @ A @ X4 @ Z3 ) )
% 3.07/1.68 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X3 ) @ Z3 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_232_cSup__eq__non__empty,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X3: set @ A,A3: A] :
% 3.07/1.68 ( ( X3
% 3.07/1.68 != ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 => ( ! [X4: A] :
% 3.07/1.68 ( ( member @ A @ X4 @ X3 )
% 3.07/1.68 => ( ord_less_eq @ A @ X4 @ A3 ) )
% 3.07/1.68 => ( ! [Y2: A] :
% 3.07/1.68 ( ! [X5: A] :
% 3.07/1.68 ( ( member @ A @ X5 @ X3 )
% 3.07/1.68 => ( ord_less_eq @ A @ X5 @ Y2 ) )
% 3.07/1.68 => ( ord_less_eq @ A @ A3 @ Y2 ) )
% 3.07/1.68 => ( ( complete_Sup_Sup @ A @ X3 )
% 3.07/1.68 = A3 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_233_SUP__eq__const,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [I2: set @ B,F: B > A,X2: A] :
% 3.07/1.68 ( ( I2
% 3.07/1.68 != ( bot_bot @ ( set @ B ) ) )
% 3.07/1.68 => ( ! [I: B] :
% 3.07/1.68 ( ( member @ B @ I @ I2 )
% 3.07/1.68 => ( ( F @ I )
% 3.07/1.68 = X2 ) )
% 3.07/1.68 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ I2 ) )
% 3.07/1.68 = X2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_234_surj__imp__surj__swap,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,F: B > A,A3: B,B3: B] :
% 3.07/1.68 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 3.07/1.68 = ( top_top @ ( set @ A ) ) )
% 3.07/1.68 => ( ( image @ B @ A @ ( swap @ B @ A @ A3 @ B3 @ F ) @ ( top_top @ ( set @ B ) ) )
% 3.07/1.68 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_235_ccSup__empty,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( counta840220525attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 = ( bot_bot @ A ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_236_subset__emptyI,axiom,
% 3.07/1.68 ! [A: $tType,A2: set @ A] :
% 3.07/1.68 ( ! [X4: A] :
% 3.07/1.68 ~ ( member @ A @ X4 @ A2 )
% 3.07/1.68 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_237_bot__set__def,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( bot_bot @ ( set @ A ) )
% 3.07/1.68 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_238_lset__code,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( coinductive_lset @ A )
% 3.07/1.68 = ( coinductive_gen_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_239_the__elem__image__unique,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,X2: A] :
% 3.07/1.68 ( ( A2
% 3.07/1.68 != ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 => ( ! [Y2: A] :
% 3.07/1.68 ( ( member @ A @ Y2 @ A2 )
% 3.07/1.68 => ( ( F @ Y2 )
% 3.07/1.68 = ( F @ X2 ) ) )
% 3.07/1.68 => ( ( the_elem @ B @ ( image @ A @ B @ F @ A2 ) )
% 3.07/1.68 = ( F @ X2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_240_cSUP__subset__mono,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ B,G: B > A,B2: set @ B,F: B > A] :
% 3.07/1.68 ( ( A2
% 3.07/1.68 != ( bot_bot @ ( set @ B ) ) )
% 3.07/1.68 => ( ( condit1810911227_above @ A @ ( image @ B @ A @ G @ B2 ) )
% 3.07/1.68 => ( ( ord_less_eq @ ( set @ B ) @ A2 @ B2 )
% 3.07/1.68 => ( ! [X4: B] :
% 3.07/1.68 ( ( member @ B @ X4 @ B2 )
% 3.07/1.68 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 3.07/1.68 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B2 ) ) ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_241_compact__bot,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( comple1141879883l_ccpo @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A] :
% 3.07/1.68 ( ( X2
% 3.07/1.68 = ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) )
% 3.07/1.68 => ( comple2143767107ompact @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ X2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_242_bdd__aboveI,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( preorder @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ A,M2: A] :
% 3.07/1.68 ( ! [X4: A] :
% 3.07/1.68 ( ( member @ A @ X4 @ A2 )
% 3.07/1.68 => ( ord_less_eq @ A @ X4 @ M2 ) )
% 3.07/1.68 => ( condit1810911227_above @ A @ A2 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_243_cSup__le__iff,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [S5: set @ A,A3: A] :
% 3.07/1.68 ( ( S5
% 3.07/1.68 != ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 => ( ( condit1810911227_above @ A @ S5 )
% 3.07/1.68 => ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S5 ) @ A3 )
% 3.07/1.68 = ( ! [X: A] :
% 3.07/1.68 ( ( member @ A @ X @ S5 )
% 3.07/1.68 => ( ord_less_eq @ A @ X @ A3 ) ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_244_cSup__mono,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [B2: set @ A,A2: set @ A] :
% 3.07/1.68 ( ( B2
% 3.07/1.68 != ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 => ( ( condit1810911227_above @ A @ A2 )
% 3.07/1.68 => ( ! [B4: A] :
% 3.07/1.68 ( ( member @ A @ B4 @ B2 )
% 3.07/1.68 => ? [X5: A] :
% 3.07/1.68 ( ( member @ A @ X5 @ A2 )
% 3.07/1.68 & ( ord_less_eq @ A @ B4 @ X5 ) ) )
% 3.07/1.68 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B2 ) @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_245_bdd__above__def,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( preorder @ A @ ( type2 @ A ) )
% 3.07/1.68 => ( ( condit1810911227_above @ A )
% 3.07/1.68 = ( ^ [A5: set @ A] :
% 3.07/1.68 ? [M3: A] :
% 3.07/1.68 ! [X: A] :
% 3.07/1.68 ( ( member @ A @ X @ A5 )
% 3.07/1.68 => ( ord_less_eq @ A @ X @ M3 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_246_bdd__above__mono,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( preorder @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [B2: set @ A,A2: set @ A] :
% 3.07/1.68 ( ( condit1810911227_above @ A @ B2 )
% 3.07/1.68 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( condit1810911227_above @ A @ A2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_247_cSup__upper2,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A,X3: set @ A,Y: A] :
% 3.07/1.68 ( ( member @ A @ X2 @ X3 )
% 3.07/1.68 => ( ( ord_less_eq @ A @ Y @ X2 )
% 3.07/1.68 => ( ( condit1810911227_above @ A @ X3 )
% 3.07/1.68 => ( ord_less_eq @ A @ Y @ ( complete_Sup_Sup @ A @ X3 ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_248_cSup__upper,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: A,X3: set @ A] :
% 3.07/1.68 ( ( member @ A @ X2 @ X3 )
% 3.07/1.68 => ( ( condit1810911227_above @ A @ X3 )
% 3.07/1.68 => ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ X3 ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_249_bdd__aboveI2,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType] :
% 3.07/1.68 ( ( preorder @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ B,F: B > A,M2: A] :
% 3.07/1.68 ( ! [X4: B] :
% 3.07/1.68 ( ( member @ B @ X4 @ A2 )
% 3.07/1.68 => ( ord_less_eq @ A @ ( F @ X4 ) @ M2 ) )
% 3.07/1.68 => ( condit1810911227_above @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_250_cSUP__upper2,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [F: B > A,A2: set @ B,X2: B,U: A] :
% 3.07/1.68 ( ( condit1810911227_above @ A @ ( image @ B @ A @ F @ A2 ) )
% 3.07/1.68 => ( ( member @ B @ X2 @ A2 )
% 3.07/1.68 => ( ( ord_less_eq @ A @ U @ ( F @ X2 ) )
% 3.07/1.68 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_251_cSUP__upper,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [X2: B,A2: set @ B,F: B > A] :
% 3.07/1.68 ( ( member @ B @ X2 @ A2 )
% 3.07/1.68 => ( ( condit1810911227_above @ A @ ( image @ B @ A @ F @ A2 ) )
% 3.07/1.68 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_252_cSUP__le__iff,axiom,
% 3.07/1.68 ! [A: $tType,B: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ B,F: B > A,U: A] :
% 3.07/1.68 ( ( A2
% 3.07/1.68 != ( bot_bot @ ( set @ B ) ) )
% 3.07/1.68 => ( ( condit1810911227_above @ A @ ( image @ B @ A @ F @ A2 ) )
% 3.07/1.68 => ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ U )
% 3.07/1.68 = ( ! [X: B] :
% 3.07/1.68 ( ( member @ B @ X @ A2 )
% 3.07/1.68 => ( ord_less_eq @ A @ ( F @ X ) @ U ) ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_253_cSUP__mono,axiom,
% 3.07/1.68 ! [B: $tType,A: $tType,C: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ B,G: C > A,B2: set @ C,F: B > A] :
% 3.07/1.68 ( ( A2
% 3.07/1.68 != ( bot_bot @ ( set @ B ) ) )
% 3.07/1.68 => ( ( condit1810911227_above @ A @ ( image @ C @ A @ G @ B2 ) )
% 3.07/1.68 => ( ! [N4: B] :
% 3.07/1.68 ( ( member @ B @ N4 @ A2 )
% 3.07/1.68 => ? [X5: C] :
% 3.07/1.68 ( ( member @ C @ X5 @ B2 )
% 3.07/1.68 & ( ord_less_eq @ A @ ( F @ N4 ) @ ( G @ X5 ) ) ) )
% 3.07/1.68 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(fact_254_cSup__subset__mono,axiom,
% 3.07/1.68 ! [A: $tType] :
% 3.07/1.68 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 3.07/1.68 => ! [A2: set @ A,B2: set @ A] :
% 3.07/1.68 ( ( A2
% 3.07/1.68 != ( bot_bot @ ( set @ A ) ) )
% 3.07/1.68 => ( ( condit1810911227_above @ A @ B2 )
% 3.07/1.68 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.07/1.68 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Finite__Set_Ofinite,axiom,
% 3.07/1.68 finite_finite @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
% 3.07/1.68 ! [A6: $tType] :
% 3.07/1.68 ( ( finite_finite @ A6 @ ( type2 @ A6 ) )
% 3.07/1.68 => ( finite_finite @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Finite__Set_Ofinite_2,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( ( finite_finite @ A6 @ ( type2 @ A6 ) )
% 3.07/1.68 & ( finite_finite @ A7 @ ( type2 @ A7 ) ) )
% 3.07/1.68 => ( finite_finite @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
% 3.07/1.68 => ( condit378418413attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( counta840220525attice @ A7 @ ( type2 @ A7 ) )
% 3.07/1.68 => ( counta840220525attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
% 3.07/1.68 => ( comple187826305attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
% 3.07/1.68 => ( comple1141879883l_ccpo @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Complete__Lattices_OSup,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( complete_Sup @ A7 @ ( type2 @ A7 ) )
% 3.07/1.68 => ( complete_Sup @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( order_top @ A7 @ ( type2 @ A7 ) )
% 3.07/1.68 => ( order_top @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( order_bot @ A7 @ ( type2 @ A7 ) )
% 3.07/1.68 => ( order_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Countable_Ocountable,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( ( finite_finite @ A6 @ ( type2 @ A6 ) )
% 3.07/1.68 & ( countable @ A7 @ ( type2 @ A7 ) ) )
% 3.07/1.68 => ( countable @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 3.07/1.68 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( order @ A7 @ ( type2 @ A7 ) )
% 3.07/1.68 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Orderings_Otop,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( top @ A7 @ ( type2 @ A7 ) )
% 3.07/1.68 => ( top @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Orderings_Oord,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 3.07/1.68 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_fun___Orderings_Obot,axiom,
% 3.07/1.68 ! [A6: $tType,A7: $tType] :
% 3.07/1.68 ( ( bot @ A7 @ ( type2 @ A7 ) )
% 3.07/1.68 => ( bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_3,axiom,
% 3.07/1.68 condit378418413attice @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 3.07/1.68 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 3.07/1.68 ordere779506340up_add @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 3.07/1.68 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Complete__Lattices_OSup_4,axiom,
% 3.07/1.68 complete_Sup @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Orderings_Oorder__bot_5,axiom,
% 3.07/1.68 order_bot @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Countable_Ocountable_6,axiom,
% 3.07/1.68 countable @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Orderings_Opreorder_7,axiom,
% 3.07/1.68 preorder @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 3.07/1.68 linorder @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 3.07/1.68 monoid_add @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Orderings_Oorder_8,axiom,
% 3.07/1.68 order @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Orderings_Oord_9,axiom,
% 3.07/1.68 ord @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Orderings_Obot_10,axiom,
% 3.07/1.68 bot @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 3.07/1.68 zero @ nat @ ( type2 @ nat ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom,
% 3.07/1.68 ! [A6: $tType] : ( condit378418413attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_12,axiom,
% 3.07/1.68 ! [A6: $tType] : ( counta840220525attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_13,axiom,
% 3.07/1.68 ! [A6: $tType] : ( comple187826305attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_14,axiom,
% 3.07/1.68 ! [A6: $tType] : ( comple1141879883l_ccpo @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Complete__Lattices_OSup_15,axiom,
% 3.07/1.68 ! [A6: $tType] : ( complete_Sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Orderings_Oorder__top_16,axiom,
% 3.07/1.68 ! [A6: $tType] : ( order_top @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Orderings_Oorder__bot_17,axiom,
% 3.07/1.68 ! [A6: $tType] : ( order_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Countable_Ocountable_18,axiom,
% 3.07/1.68 ! [A6: $tType] :
% 3.07/1.68 ( ( finite_finite @ A6 @ ( type2 @ A6 ) )
% 3.07/1.68 => ( countable @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Orderings_Opreorder_19,axiom,
% 3.07/1.68 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Orderings_Oorder_20,axiom,
% 3.07/1.68 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Orderings_Otop_21,axiom,
% 3.07/1.68 ! [A6: $tType] : ( top @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Orderings_Oord_22,axiom,
% 3.07/1.68 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_Set_Oset___Orderings_Obot_23,axiom,
% 3.07/1.68 ! [A6: $tType] : ( bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_24,axiom,
% 3.07/1.68 condit378418413attice @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_25,axiom,
% 3.07/1.68 counta840220525attice @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_26,axiom,
% 3.07/1.68 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_27,axiom,
% 3.07/1.68 comple1141879883l_ccpo @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Complete__Lattices_OSup_28,axiom,
% 3.07/1.68 complete_Sup @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Orderings_Oorder__top_29,axiom,
% 3.07/1.68 order_top @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Orderings_Oorder__bot_30,axiom,
% 3.07/1.68 order_bot @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Countable_Ocountable_31,axiom,
% 3.07/1.68 countable @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Orderings_Opreorder_32,axiom,
% 3.07/1.68 preorder @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Orderings_Olinorder_33,axiom,
% 3.07/1.68 linorder @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Orderings_Oorder_34,axiom,
% 3.07/1.68 order @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Orderings_Otop_35,axiom,
% 3.07/1.68 top @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Orderings_Oord_36,axiom,
% 3.07/1.68 ord @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(tcon_HOL_Obool___Orderings_Obot_37,axiom,
% 3.07/1.68 bot @ $o @ ( type2 @ $o ) ).
% 3.07/1.68
% 3.07/1.68 thf(conj_0,conjecture,
% 3.07/1.68 ( ( snth @ a @ ( coindu176146587of_seq @ a @ f ) )
% 3.07/1.68 = f ) ).
% 3.07/1.68 ------------------------
% 3.15/1.71 % (3858)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_3754 for (2999ds/275Mi)
% 3.15/1.71 % (3853)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/183Mi)
% 3.15/1.71 % (3855)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_3754 for (2999ds/27Mi)
% 3.15/1.71 % (3854)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_3754 for (2999ds/4Mi)
% 3.15/1.71 % (3856)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/2Mi)
% 3.15/1.71 % (3857)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_3754 for (2999ds/2Mi)
% 3.15/1.71 % (3859)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/18Mi)
% 3.15/1.71 % (3856)Instruction limit reached!
% 3.15/1.71 % (3856)------------------------------
% 3.15/1.71 % (3856)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.71 % (3856)Termination reason: Unknown
% 3.15/1.71 % (3857)Instruction limit reached!
% 3.15/1.71 % (3857)------------------------------
% 3.15/1.71 % (3857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.71 % (3857)Termination reason: Unknown
% 3.15/1.71 % (3857)Termination phase: shuffling
% 3.15/1.71
% 3.15/1.71 % (3857)Memory used [KB]: 1407
% 3.15/1.71 % (3857)Time elapsed: 0.003 s
% 3.15/1.71 % (3857)Instructions burned: 2 (million)
% 3.15/1.71 % (3857)------------------------------
% 3.15/1.71 % (3857)------------------------------
% 3.15/1.71 % (3856)Termination phase: shuffling
% 3.15/1.71
% 3.15/1.71 % (3856)Memory used [KB]: 1407
% 3.15/1.71 % (3856)Time elapsed: 0.003 s
% 3.15/1.71 % (3856)Instructions burned: 2 (million)
% 3.15/1.71 % (3856)------------------------------
% 3.15/1.71 % (3856)------------------------------
% 3.15/1.72 % (3854)Instruction limit reached!
% 3.15/1.72 % (3854)------------------------------
% 3.15/1.72 % (3854)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.72 % (3854)Termination reason: Unknown
% 3.15/1.72 % (3854)Termination phase: shuffling
% 3.15/1.72
% 3.15/1.72 % (3854)Memory used [KB]: 1407
% 3.15/1.72 % (3854)Time elapsed: 0.004 s
% 3.15/1.72 % (3854)Instructions burned: 4 (million)
% 3.15/1.72 % (3854)------------------------------
% 3.15/1.72 % (3854)------------------------------
% 3.15/1.72 % (3859)Instruction limit reached!
% 3.15/1.72 % (3859)------------------------------
% 3.15/1.72 % (3859)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.72 % (3859)Termination reason: Unknown
% 3.15/1.72 % (3859)Termination phase: shuffling
% 3.15/1.72
% 3.15/1.72 % (3859)Memory used [KB]: 1791
% 3.15/1.72 % (3859)Time elapsed: 0.012 s
% 3.15/1.72 % (3859)Instructions burned: 19 (million)
% 3.15/1.72 % (3859)------------------------------
% 3.15/1.72 % (3859)------------------------------
% 3.15/1.73 % (3855)Instruction limit reached!
% 3.15/1.73 % (3855)------------------------------
% 3.15/1.73 % (3855)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.73 % (3855)Termination reason: Unknown
% 3.15/1.73 % (3855)Termination phase: Property scanning
% 3.15/1.73
% 3.15/1.73 % (3855)Memory used [KB]: 1918
% 3.15/1.73 % (3855)Time elapsed: 0.016 s
% 3.15/1.73 % (3855)Instructions burned: 27 (million)
% 3.15/1.73 % (3855)------------------------------
% 3.15/1.73 % (3855)------------------------------
% 3.15/1.73 % (3860)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/3Mi)
% 3.15/1.73 % (3861)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_3754 for (2999ds/37Mi)
% 3.15/1.73 % (3862)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_3754 for (2999ds/15Mi)
% 3.15/1.73 % (3860)Instruction limit reached!
% 3.15/1.73 % (3860)------------------------------
% 3.15/1.73 % (3860)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.73 % (3860)Termination reason: Unknown
% 3.15/1.73 % (3860)Termination phase: shuffling
% 3.15/1.73
% 3.15/1.73 % (3860)Memory used [KB]: 1407
% 3.15/1.73 % (3860)Time elapsed: 0.003 s
% 3.15/1.73 % (3860)Instructions burned: 3 (million)
% 3.15/1.73 % (3860)------------------------------
% 3.15/1.73 % (3860)------------------------------
% 3.15/1.74 % (3862)Instruction limit reached!
% 3.15/1.74 % (3862)------------------------------
% 3.15/1.74 % (3862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.74 % (3862)Termination reason: Unknown
% 3.15/1.74 % (3862)Termination phase: shuffling
% 3.15/1.74
% 3.15/1.74 % (3862)Memory used [KB]: 1663
% 3.15/1.74 % (3862)Time elapsed: 0.009 s
% 3.15/1.74 % (3862)Instructions burned: 15 (million)
% 3.15/1.74 % (3862)------------------------------
% 3.15/1.74 % (3862)------------------------------
% 3.15/1.74 % (3863)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_3754 for (2999ds/3Mi)
% 3.15/1.74 % (3863)Instruction limit reached!
% 3.15/1.74 % (3863)------------------------------
% 3.15/1.74 % (3863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.74 % (3863)Termination reason: Unknown
% 3.15/1.74 % (3863)Termination phase: shuffling
% 3.15/1.74
% 3.15/1.74 % (3863)Memory used [KB]: 1407
% 3.15/1.74 % (3863)Time elapsed: 0.003 s
% 3.15/1.74 % (3863)Instructions burned: 3 (million)
% 3.15/1.74 % (3863)------------------------------
% 3.15/1.74 % (3863)------------------------------
% 3.15/1.74 % (3864)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_3754 for (2999ds/1041Mi)
% 3.15/1.74 % (3858)Refutation not found, incomplete strategy
% 3.15/1.74 % (3858)------------------------------
% 3.15/1.74 % (3858)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.74 % (3858)Termination reason: Refutation not found, incomplete strategy
% 3.15/1.74
% 3.15/1.74
% 3.15/1.74 % (3858)Memory used [KB]: 6652
% 3.15/1.74 % (3858)Time elapsed: 0.033 s
% 3.15/1.74 % (3858)Instructions burned: 62 (million)
% 3.15/1.74 % (3858)------------------------------
% 3.15/1.74 % (3858)------------------------------
% 3.15/1.75 % (3865)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/7Mi)
% 3.15/1.75 % (3861)Instruction limit reached!
% 3.15/1.75 % (3861)------------------------------
% 3.15/1.75 % (3861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.75 % (3861)Termination reason: Unknown
% 3.15/1.75 % (3861)Termination phase: shuffling
% 3.15/1.75
% 3.15/1.75 % (3861)Memory used [KB]: 2174
% 3.15/1.75 % (3861)Time elapsed: 0.019 s
% 3.15/1.75 % (3861)Instructions burned: 38 (million)
% 3.15/1.75 % (3861)------------------------------
% 3.15/1.75 % (3861)------------------------------
% 3.15/1.75 % (3865)Instruction limit reached!
% 3.15/1.75 % (3865)------------------------------
% 3.15/1.75 % (3865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.75 % (3865)Termination reason: Unknown
% 3.15/1.75 % (3865)Termination phase: shuffling
% 3.15/1.75
% 3.15/1.75 % (3865)Memory used [KB]: 1535
% 3.15/1.75 % (3865)Time elapsed: 0.005 s
% 3.15/1.75 % (3865)Instructions burned: 7 (million)
% 3.15/1.75 % (3865)------------------------------
% 3.15/1.75 % (3865)------------------------------
% 3.15/1.76 % (3866)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/16Mi)
% 3.15/1.76 % (3867)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/3Mi)
% 3.15/1.76 % (3867)Instruction limit reached!
% 3.15/1.76 % (3867)------------------------------
% 3.15/1.76 % (3867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.76 % (3867)Termination reason: Unknown
% 3.15/1.76 % (3867)Termination phase: shuffling
% 3.15/1.76
% 3.15/1.76 % (3867)Memory used [KB]: 1407
% 3.15/1.76 % (3867)Time elapsed: 0.003 s
% 3.15/1.76 % (3867)Instructions burned: 3 (million)
% 3.15/1.76 % (3867)------------------------------
% 3.15/1.76 % (3867)------------------------------
% 3.15/1.76 % (3868)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/3Mi)
% 3.15/1.76 % (3868)Instruction limit reached!
% 3.15/1.76 % (3868)------------------------------
% 3.15/1.76 % (3868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.76 % (3868)Termination reason: Unknown
% 3.15/1.76 % (3868)Termination phase: shuffling
% 3.15/1.76
% 3.15/1.76 % (3868)Memory used [KB]: 1407
% 3.15/1.76 % (3868)Time elapsed: 0.003 s
% 3.15/1.76 % (3868)Instructions burned: 4 (million)
% 3.15/1.76 % (3868)------------------------------
% 3.15/1.76 % (3868)------------------------------
% 3.15/1.76 % (3866)Instruction limit reached!
% 3.15/1.76 % (3866)------------------------------
% 3.15/1.76 % (3866)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.76 % (3866)Termination reason: Unknown
% 3.15/1.76 % (3866)Termination phase: shuffling
% 3.15/1.76
% 3.15/1.76 % (3866)Memory used [KB]: 1791
% 3.15/1.76 % (3866)Time elapsed: 0.010 s
% 3.15/1.76 % (3866)Instructions burned: 17 (million)
% 3.15/1.76 % (3866)------------------------------
% 3.15/1.76 % (3866)------------------------------
% 3.15/1.76 % (3869)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/7Mi)
% 3.15/1.77 % (3870)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/3Mi)
% 3.15/1.77 % (3869)Instruction limit reached!
% 3.15/1.77 % (3869)------------------------------
% 3.15/1.77 % (3869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.77 % (3869)Termination reason: Unknown
% 3.15/1.77 % (3869)Termination phase: shuffling
% 3.15/1.77
% 3.15/1.77 % (3869)Memory used [KB]: 1407
% 3.15/1.77 % (3869)Time elapsed: 0.005 s
% 3.15/1.77 % (3869)Instructions burned: 7 (million)
% 3.15/1.77 % (3869)------------------------------
% 3.15/1.77 % (3869)------------------------------
% 3.15/1.77 % (3870)Instruction limit reached!
% 3.15/1.77 % (3870)------------------------------
% 3.15/1.77 % (3870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.77 % (3870)Termination reason: Unknown
% 3.15/1.77 % (3870)Termination phase: shuffling
% 3.15/1.77
% 3.15/1.77 % (3870)Memory used [KB]: 1407
% 3.15/1.77 % (3870)Time elapsed: 0.003 s
% 3.15/1.77 % (3870)Instructions burned: 3 (million)
% 3.15/1.77 % (3870)------------------------------
% 3.15/1.77 % (3870)------------------------------
% 3.15/1.77 % (3871)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/4Mi)
% 3.15/1.78 % (3871)Instruction limit reached!
% 3.15/1.78 % (3871)------------------------------
% 3.15/1.78 % (3871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.78 % (3871)Termination reason: Unknown
% 3.15/1.78 % (3871)Termination phase: shuffling
% 3.15/1.78
% 3.15/1.78 % (3871)Memory used [KB]: 1407
% 3.15/1.78 % (3871)Time elapsed: 0.004 s
% 3.15/1.78 % (3871)Instructions burned: 5 (million)
% 3.15/1.78 % (3871)------------------------------
% 3.15/1.78 % (3871)------------------------------
% 3.15/1.78 % (3873)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/710Mi)
% 3.15/1.78 % (3872)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/18Mi)
% 3.15/1.78 % (3874)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/6Mi)
% 3.15/1.79 % (3875)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/902Mi)
% 3.15/1.79 % (3874)Instruction limit reached!
% 3.15/1.79 % (3874)------------------------------
% 3.15/1.79 % (3874)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.79 % (3874)Termination reason: Unknown
% 3.15/1.79 % (3874)Termination phase: shuffling
% 3.15/1.79
% 3.15/1.79 % (3874)Memory used [KB]: 1535
% 3.15/1.79 % (3874)Time elapsed: 0.005 s
% 3.15/1.79 % (3874)Instructions burned: 7 (million)
% 3.15/1.79 % (3874)------------------------------
% 3.15/1.79 % (3874)------------------------------
% 3.15/1.79 % (3872)Instruction limit reached!
% 3.15/1.79 % (3872)------------------------------
% 3.15/1.79 % (3872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.79 % (3872)Termination reason: Unknown
% 3.15/1.79 % (3872)Termination phase: shuffling
% 3.15/1.79
% 3.15/1.79 % (3872)Memory used [KB]: 1791
% 3.15/1.79 % (3872)Time elapsed: 0.011 s
% 3.15/1.79 % (3872)Instructions burned: 18 (million)
% 3.15/1.79 % (3872)------------------------------
% 3.15/1.79 % (3872)------------------------------
% 3.15/1.79 % (3876)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/21Mi)
% 3.15/1.80 % (3853)Instruction limit reached!
% 3.15/1.80 % (3853)------------------------------
% 3.15/1.80 % (3853)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.80 % (3853)Termination reason: Unknown
% 3.15/1.80 % (3853)Termination phase: Saturation
% 3.15/1.80
% 3.15/1.80 % (3853)Memory used [KB]: 7291
% 3.15/1.80 % (3853)Time elapsed: 0.090 s
% 3.15/1.80 % (3853)Instructions burned: 184 (million)
% 3.15/1.80 % (3853)------------------------------
% 3.15/1.80 % (3853)------------------------------
% 3.15/1.80 % (3877)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_3754 for (2999ds/5Mi)
% 3.15/1.80 % (3876)Instruction limit reached!
% 3.15/1.80 % (3876)------------------------------
% 3.15/1.80 % (3876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.80 % (3876)Termination reason: Unknown
% 3.15/1.80 % (3876)Termination phase: shuffling
% 3.15/1.80
% 3.15/1.80 % (3876)Memory used [KB]: 1791
% 3.15/1.80 % (3876)Time elapsed: 0.012 s
% 3.15/1.80 % (3876)Instructions burned: 21 (million)
% 3.15/1.80 % (3876)------------------------------
% 3.15/1.80 % (3876)------------------------------
% 3.15/1.80 % (3878)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_3754 for (2998ds/6Mi)
% 3.15/1.81 % (3877)Instruction limit reached!
% 3.15/1.81 % (3877)------------------------------
% 3.15/1.81 % (3877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.81 % (3877)Termination reason: Unknown
% 3.15/1.81 % (3877)Termination phase: shuffling
% 3.15/1.81
% 3.15/1.81 % (3877)Memory used [KB]: 1407
% 3.15/1.81 % (3877)Time elapsed: 0.004 s
% 3.15/1.81 % (3877)Instructions burned: 5 (million)
% 3.15/1.81 % (3877)------------------------------
% 3.15/1.81 % (3877)------------------------------
% 3.15/1.81 % (3878)Instruction limit reached!
% 3.15/1.81 % (3878)------------------------------
% 3.15/1.81 % (3878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.81 % (3878)Termination reason: Unknown
% 3.15/1.81 % (3878)Termination phase: shuffling
% 3.15/1.81
% 3.15/1.81 % (3878)Memory used [KB]: 1407
% 3.15/1.81 % (3878)Time elapsed: 0.004 s
% 3.15/1.81 % (3878)Instructions burned: 6 (million)
% 3.15/1.81 % (3878)------------------------------
% 3.15/1.81 % (3878)------------------------------
% 3.15/1.82 % (3879)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_3754 for (2998ds/377Mi)
% 3.15/1.82 % (3880)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_3754 for (2998ds/779Mi)
% 3.15/1.82 % (3881)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_3754 for (2998ds/19Mi)
% 3.15/1.82 % (3882)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_3754 for (2998ds/879Mi)
% 3.15/1.83 % (3881)Instruction limit reached!
% 3.15/1.83 % (3881)------------------------------
% 3.15/1.83 % (3881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.83 % (3881)Termination reason: Unknown
% 3.15/1.83 % (3881)Termination phase: shuffling
% 3.15/1.83
% 3.15/1.83 % (3881)Memory used [KB]: 1791
% 3.15/1.83 % (3881)Time elapsed: 0.010 s
% 3.15/1.83 % (3881)Instructions burned: 19 (million)
% 3.15/1.83 % (3881)------------------------------
% 3.15/1.83 % (3881)------------------------------
% 3.15/1.85 % (3883)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_3754 for (2998ds/17Mi)
% 3.15/1.85 % (3875)Refutation not found, incomplete strategy
% 3.15/1.85 % (3875)------------------------------
% 3.15/1.85 % (3875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.85 % (3875)Termination reason: Refutation not found, incomplete strategy
% 3.15/1.85
% 3.15/1.85
% 3.15/1.85 % (3875)Memory used [KB]: 7419
% 3.15/1.85 % (3875)Time elapsed: 0.066 s
% 3.15/1.85 % (3875)Instructions burned: 132 (million)
% 3.15/1.85 % (3875)------------------------------
% 3.15/1.85 % (3875)------------------------------
% 3.15/1.86 % (3883)Instruction limit reached!
% 3.15/1.86 % (3883)------------------------------
% 3.15/1.86 % (3883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.86 % (3883)Termination reason: Unknown
% 3.15/1.86 % (3883)Termination phase: shuffling
% 3.15/1.86
% 3.15/1.86 % (3883)Memory used [KB]: 1791
% 3.15/1.86 % (3883)Time elapsed: 0.010 s
% 3.15/1.86 % (3883)Instructions burned: 17 (million)
% 3.15/1.86 % (3883)------------------------------
% 3.15/1.86 % (3883)------------------------------
% 3.15/1.87 % (3884)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_3754 for (2998ds/3Mi)
% 3.15/1.87 % (3884)Instruction limit reached!
% 3.15/1.87 % (3884)------------------------------
% 3.15/1.87 % (3884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.87 % (3884)Termination reason: Unknown
% 3.15/1.87 % (3884)Termination phase: shuffling
% 3.15/1.87
% 3.15/1.87 % (3884)Memory used [KB]: 1407
% 3.15/1.87 % (3884)Time elapsed: 0.003 s
% 3.15/1.87 % (3884)Instructions burned: 3 (million)
% 3.15/1.87 % (3884)------------------------------
% 3.15/1.87 % (3884)------------------------------
% 3.15/1.87 % (3885)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_3754 for (2998ds/30Mi)
% 3.15/1.89 % (3886)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_3754 for (2998ds/127Mi)
% 3.15/1.89 % (3885)Instruction limit reached!
% 3.15/1.89 % (3885)------------------------------
% 3.15/1.89 % (3885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.89 % (3885)Termination reason: Unknown
% 3.15/1.89 % (3885)Termination phase: Property scanning
% 3.15/1.89
% 3.15/1.89 % (3885)Memory used [KB]: 1918
% 3.15/1.89 % (3885)Time elapsed: 0.017 s
% 3.15/1.89 % (3885)Instructions burned: 32 (million)
% 3.15/1.89 % (3885)------------------------------
% 3.15/1.89 % (3885)------------------------------
% 3.15/1.90 % (3887)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_3754 for (2998ds/100Mi)
% 3.15/1.95 % (3886)Instruction limit reached!
% 3.15/1.95 % (3886)------------------------------
% 3.15/1.95 % (3886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.95 % (3886)Termination reason: Unknown
% 3.15/1.95 % (3886)Termination phase: Saturation
% 3.15/1.95
% 3.15/1.95 % (3886)Memory used [KB]: 7036
% 3.15/1.95 % (3886)Time elapsed: 0.062 s
% 3.15/1.95 % (3886)Instructions burned: 127 (million)
% 3.15/1.95 % (3886)------------------------------
% 3.15/1.95 % (3886)------------------------------
% 3.15/1.95 % (3887)Instruction limit reached!
% 3.15/1.95 % (3887)------------------------------
% 3.15/1.95 % (3887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.95 % (3887)Termination reason: Unknown
% 3.15/1.95 % (3887)Termination phase: Property scanning
% 3.15/1.95
% 3.15/1.95 % (3887)Memory used [KB]: 2430
% 3.15/1.95 % (3887)Time elapsed: 0.047 s
% 3.15/1.95 % (3887)Instructions burned: 100 (million)
% 3.15/1.95 % (3887)------------------------------
% 3.15/1.95 % (3887)------------------------------
% 3.15/1.96 % (3888)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_3754 for (2997ds/3Mi)
% 3.15/1.96 % (3888)Instruction limit reached!
% 3.15/1.96 % (3888)------------------------------
% 3.15/1.96 % (3888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.96 % (3888)Termination reason: Unknown
% 3.15/1.96 % (3888)Termination phase: shuffling
% 3.15/1.96
% 3.15/1.96 % (3888)Memory used [KB]: 1407
% 3.15/1.96 % (3888)Time elapsed: 0.004 s
% 3.15/1.96 % (3888)Instructions burned: 5 (million)
% 3.15/1.96 % (3888)------------------------------
% 3.15/1.96 % (3888)------------------------------
% 3.15/1.96 % (3889)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_3754 for (2997ds/20Mi)
% 3.15/1.98 % (3889)Instruction limit reached!
% 3.15/1.98 % (3889)------------------------------
% 3.15/1.98 % (3889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.98 % (3889)Termination reason: Unknown
% 3.15/1.98 % (3889)Termination phase: shuffling
% 3.15/1.98
% 3.15/1.98 % (3889)Memory used [KB]: 1918
% 3.15/1.98 % (3889)Time elapsed: 0.012 s
% 3.15/1.98 % (3889)Instructions burned: 20 (million)
% 3.15/1.98 % (3889)------------------------------
% 3.15/1.98 % (3889)------------------------------
% 3.15/1.98 % (3891)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_3754 for (2997ds/86Mi)
% 3.15/1.99 % (3892)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_3754 for (2997ds/107Mi)
% 3.15/2.01 % (3879)Instruction limit reached!
% 3.15/2.01 % (3879)------------------------------
% 3.15/2.01 % (3879)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/2.01 % (3879)Termination reason: Unknown
% 3.15/2.01 % (3879)Termination phase: Saturation
% 3.15/2.01
% 3.15/2.01 % (3879)Memory used [KB]: 8315
% 3.15/2.01 % (3879)Time elapsed: 0.191 s
% 3.15/2.01 % (3879)Instructions burned: 378 (million)
% 3.15/2.01 % (3879)------------------------------
% 3.15/2.01 % (3879)------------------------------
% 3.15/2.02 % (3891)Instruction limit reached!
% 3.15/2.02 % (3891)------------------------------
% 3.15/2.02 % (3891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/2.02 % (3891)Termination reason: Unknown
% 3.15/2.02 % (3891)Termination phase: Property scanning
% 3.15/2.02
% 3.15/2.02 % (3891)Memory used [KB]: 2302
% 3.15/2.02 % (3891)Time elapsed: 0.043 s
% 3.15/2.02 % (3891)Instructions burned: 87 (million)
% 3.15/2.02 % (3891)------------------------------
% 3.15/2.02 % (3891)------------------------------
% 5.35/2.03 % (3893)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_3754 for (2996ds/39Mi)
% 5.35/2.04 % (3894)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_3754 for (2996ds/448Mi)
% 5.35/2.04 % (3893)Instruction limit reached!
% 5.35/2.04 % (3893)------------------------------
% 5.35/2.04 % (3893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.35/2.04 % (3893)Termination reason: Unknown
% 5.35/2.04 % (3893)Termination phase: shuffling
% 5.35/2.04
% 5.35/2.04 % (3893)Memory used [KB]: 2174
% 5.35/2.04 % (3893)Time elapsed: 0.020 s
% 5.35/2.04 % (3893)Instructions burned: 39 (million)
% 5.35/2.04 % (3893)------------------------------
% 5.35/2.04 % (3893)------------------------------
% 5.35/2.04 % (3892)Instruction limit reached!
% 5.35/2.04 % (3892)------------------------------
% 5.35/2.04 % (3892)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.35/2.04 % (3892)Termination reason: Unknown
% 5.35/2.04 % (3892)Termination phase: Saturation
% 5.35/2.04
% 5.35/2.04 % (3892)Memory used [KB]: 7036
% 5.35/2.04 % (3892)Time elapsed: 0.054 s
% 5.35/2.04 % (3892)Instructions burned: 108 (million)
% 5.35/2.04 % (3892)------------------------------
% 5.35/2.04 % (3892)------------------------------
% 5.35/2.06 % (3895)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_3754 for (2996ds/46Mi)
% 5.35/2.06 % (3896)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_3754 for (2996ds/98Mi)
% 5.35/2.08 % (3895)Instruction limit reached!
% 5.35/2.08 % (3895)------------------------------
% 5.35/2.08 % (3895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.35/2.08 % (3895)Termination reason: Unknown
% 5.35/2.08 % (3895)Termination phase: Preprocessing 3
% 5.35/2.08
% 5.35/2.08 % (3895)Memory used [KB]: 2046
% 5.35/2.08 % (3895)Time elapsed: 0.025 s
% 5.35/2.08 % (3895)Instructions burned: 47 (million)
% 5.35/2.08 % (3895)------------------------------
% 5.35/2.08 % (3895)------------------------------
% 5.35/2.10 % (3897)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_3754 for (2995ds/507Mi)
% 5.35/2.10 % (3896)Instruction limit reached!
% 5.35/2.10 % (3896)------------------------------
% 5.35/2.10 % (3896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.35/2.10 % (3896)Termination reason: Unknown
% 5.35/2.10 % (3896)Termination phase: Property scanning
% 5.35/2.10
% 5.35/2.10 % (3896)Memory used [KB]: 2430
% 5.35/2.10 % (3896)Time elapsed: 0.044 s
% 5.35/2.10 % (3896)Instructions burned: 100 (million)
% 5.35/2.10 % (3896)------------------------------
% 5.35/2.10 % (3896)------------------------------
% 5.35/2.11 % (3864)First to succeed.
% 5.35/2.11 % (3898)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_3754 for (2995ds/149Mi)
% 5.35/2.11 % (3864)Refutation found. Thanks to Tanya!
% 5.35/2.11 % SZS status Theorem for DTF2THF_3754
% 5.35/2.11 % SZS output start Proof for DTF2THF_3754
% 5.35/2.11 thf(type_def_5, type, itself: ($tType) > $tType).
% 5.35/2.11 thf(type_def_7, type, set: ($tType) > $tType).
% 5.35/2.11 thf(type_def_8, type, coinductive_llist: ($tType) > $tType).
% 5.35/2.11 thf(type_def_9, type, nat: $tType).
% 5.35/2.11 thf(type_def_10, type, stream: ($tType) > $tType).
% 5.35/2.11 thf(type_def_11, type, a: $tType).
% 5.35/2.11 thf(func_def_0, type, coinductive_llist: ($tType) > $tType).
% 5.35/2.11 thf(func_def_1, type, stream: ($tType) > $tType).
% 5.35/2.11 thf(func_def_2, type, set: ($tType) > $tType).
% 5.35/2.11 thf(func_def_3, type, nat: $tType).
% 5.35/2.11 thf(func_def_4, type, itself: ($tType) > $tType).
% 5.35/2.11 thf(func_def_5, type, a: $tType).
% 5.35/2.11 thf(func_def_6, type, finite_finite: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_7, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_8, type, zero: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_9, type, bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_10, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_11, type, top: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_12, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_13, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_14, type, monoid_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_15, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_16, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_17, type, countable: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_18, type, order_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_19, type, order_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_20, type, complete_Sup: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_21, type, linordered_semidom: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_22, type, comple1141879883l_ccpo: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_23, type, ordere779506340up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_24, type, comple187826305attice: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_25, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_26, type, counta840220525attice: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_27, type, condit378418413attice: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.35/2.11 thf(func_def_28, type, coinductive_gen_lset: !>[X0: $tType]:(set @ X0 > coinductive_llist @ X0 > set @ X0)).
% 5.35/2.11 thf(func_def_29, type, coindu68654304_llist: !>[X0: $tType]:((nat > X0) > coinductive_llist @ X0)).
% 5.35/2.11 thf(func_def_30, type, coinductive_lfinite: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 5.35/2.11 thf(func_def_31, type, coinductive_lmap: !>[X0: $tType, X1: $tType]:((X0 > X1) > coinductive_llist @ X0 > coinductive_llist @ X1)).
% 5.35/2.11 thf(func_def_32, type, coinductive_lset: !>[X0: $tType]:(coinductive_llist @ X0 > set @ X0)).
% 5.35/2.11 thf(func_def_33, type, coindu543516966_llist: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > $o)).
% 5.35/2.11 thf(func_def_34, type, coinductive_lmember: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > $o)).
% 5.35/2.11 thf(func_def_35, type, coinductive_lnth: !>[X0: $tType]:(coinductive_llist @ X0 > nat > X0)).
% 5.35/2.11 thf(func_def_36, type, coindu780009021istsum: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 5.35/2.11 thf(func_def_37, type, coindu1724414836stream: !>[X0: $tType]:(stream @ X0 > coinductive_llist @ X0)).
% 5.35/2.11 thf(func_def_38, type, coindu176146587of_seq: !>[X0: $tType]:((nat > X0) > stream @ X0)).
% 5.35/2.11 thf(func_def_39, type, coindu1183105481stream: !>[X0: $tType]:(coinductive_llist @ X0 > stream @ X0 > $o)).
% 5.35/2.11 thf(func_def_40, type, coindu2010755910_llist: !>[X0: $tType]:(coinductive_llist @ X0 > stream @ X0)).
% 5.35/2.11 thf(func_def_41, type, complete_Sup_Sup: !>[X0: $tType]:(set @ X0 > X0)).
% 5.35/2.11 thf(func_def_42, type, comple2143767107ompact: !>[X0: $tType]:((set @ X0 > X0) > (X0 > X0 > $o) > X0 > $o)).
% 5.35/2.11 thf(func_def_43, type, condit1810911227_above: !>[X0: $tType]:(set @ X0 > $o)).
% 5.35/2.11 thf(func_def_44, type, from_nat: !>[X0: $tType]:(nat > X0)).
% 5.35/2.11 thf(func_def_45, type, swap: !>[X0: $tType, X1: $tType]:(X0 > X0 > (X0 > X1) > X0 > X1)).
% 5.35/2.11 thf(func_def_46, type, zero_zero: !>[X0: $tType]:(X0)).
% 5.35/2.11 thf(func_def_47, type, bot_bot: !>[X0: $tType]:(X0)).
% 5.35/2.11 thf(func_def_48, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 5.35/2.11 thf(func_def_49, type, top_top: !>[X0: $tType]:(X0)).
% 5.35/2.11 thf(func_def_50, type, type2: !>[X0: $tType]:(itself @ X0)).
% 5.35/2.11 thf(func_def_51, type, ball: !>[X0: $tType]:(set @ X0 > (X0 > $o) > $o)).
% 5.35/2.11 thf(func_def_52, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 5.35/2.11 thf(func_def_53, type, pow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 5.35/2.11 thf(func_def_54, type, bind: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ X1)).
% 5.35/2.11 thf(func_def_55, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 5.35/2.11 thf(func_def_56, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 5.35/2.11 thf(func_def_57, type, smap2: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > stream @ X0 > stream @ X1 > stream @ X2)).
% 5.35/2.11 thf(func_def_58, type, smember: !>[X0: $tType]:(X0 > stream @ X0 > $o)).
% 5.35/2.11 thf(func_def_59, type, smerge: !>[X0: $tType]:(stream @ stream @ X0 > stream @ X0)).
% 5.35/2.11 thf(func_def_60, type, snth: !>[X0: $tType]:(stream @ X0 > nat > X0)).
% 5.35/2.11 thf(func_def_61, type, pred_stream: !>[X0: $tType]:((X0 > $o) > stream @ X0 > $o)).
% 5.35/2.11 thf(func_def_62, type, smap: !>[X0: $tType, X1: $tType]:((X0 > X1) > stream @ X0 > stream @ X1)).
% 5.35/2.11 thf(func_def_63, type, sset: !>[X0: $tType]:(stream @ X0 > set @ X0)).
% 5.35/2.11 thf(func_def_64, type, stream_all: !>[X0: $tType]:((X0 > $o) > stream @ X0 > $o)).
% 5.35/2.11 thf(func_def_65, type, streams: !>[X0: $tType]:(set @ X0 > set @ stream @ X0)).
% 5.35/2.11 thf(func_def_66, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 5.35/2.11 thf(func_def_67, type, f: nat > a).
% 5.35/2.11 thf(func_def_86, type, ph1: !>[X0: $tType]:(X0)).
% 5.35/2.11 thf(func_def_87, type, sK2: !>[X0: $tType]:(X0)).
% 5.35/2.11 thf(func_def_88, type, sK3: nat).
% 5.35/2.11 thf(func_def_89, type, sK4: !>[X0: $tType]:(X0)).
% 5.35/2.11 thf(func_def_90, type, sK5: nat).
% 5.35/2.11 thf(f2765,plain,(
% 5.35/2.11 $false),
% 5.35/2.11 inference(trivial_inequality_removal,[status(thm)],[f2764])).
% 5.35/2.11 thf(f2764,plain,(
% 5.35/2.11 ((f @ sK5) != (f @ sK5))),
% 5.35/2.11 inference(superposition,[status(thm)],[f2549,f2740])).
% 5.35/2.11 thf(f2740,plain,(
% 5.35/2.11 ( ! [X0 : $tType,X2 : nat > X0,X1 : nat] : (((X2 @ X1) = (coinductive_lnth @ X0 @ (coindu68654304_llist @ X0 @ X2) @ X1))) )),
% 5.35/2.11 inference(equality_proxy_clausification,[status(thm)],[f2737])).
% 5.35/2.11 thf(f2737,plain,(
% 5.35/2.11 ( ! [X0 : $tType,X2 : nat > X0,X1 : nat] : (($true = ((coinductive_lnth @ X0 @ (coindu68654304_llist @ X0 @ X2) @ X1) = (X2 @ X1)))) )),
% 5.35/2.11 inference(beta_eta_normalization,[status(thm)],[f2736])).
% 5.35/2.11 thf(f2736,plain,(
% 5.35/2.11 ( ! [X0 : $tType,X2 : nat > X0,X1 : nat] : (($true = ((^[Y0 : nat > X0]: ((coinductive_lnth @ X0 @ (coindu68654304_llist @ X0 @ Y0) @ X1) = (Y0 @ X1))) @ X2))) )),
% 5.35/2.11 inference(pi_clausification,[status(thm)],[f2733])).
% 5.35/2.11 thf(f2733,plain,(
% 5.35/2.11 ( ! [X0 : $tType,X1 : nat] : (($true = (!! @ (nat > X0) @ (^[Y0 : nat > X0]: ((coinductive_lnth @ X0 @ (coindu68654304_llist @ X0 @ Y0) @ X1) = (Y0 @ X1)))))) )),
% 5.35/2.11 inference(beta_eta_normalization,[status(thm)],[f2732])).
% 5.35/2.11 thf(f2732,plain,(
% 5.35/2.11 ( ! [X0 : $tType,X1 : nat] : (($true = ((^[Y0 : nat]: (!! @ (nat > X0) @ (^[Y1 : nat > X0]: ((coinductive_lnth @ X0 @ (coindu68654304_llist @ X0 @ Y1) @ Y0) = (Y1 @ Y0))))) @ X1))) )),
% 5.35/2.11 inference(pi_clausification,[status(thm)],[f1625])).
% 5.35/2.11 thf(f1625,plain,(
% 5.35/2.11 ( ! [X0 : $tType] : (($true = (!! @ nat @ (^[Y0 : nat]: (!! @ (nat > X0) @ (^[Y1 : nat > X0]: ((coinductive_lnth @ X0 @ (coindu68654304_llist @ X0 @ Y1) @ Y0) = (Y1 @ Y0)))))))) )),
% 5.35/2.11 inference(cnf_transformation,[status(thm)],[f1109])).
% 5.35/2.11 thf(f1109,plain,(
% 5.35/2.11 ! [X0 : $tType] : ($true = (!! @ nat @ (^[Y0 : nat]: (!! @ (nat > X0) @ (^[Y1 : nat > X0]: ((coinductive_lnth @ X0 @ (coindu68654304_llist @ X0 @ Y1) @ Y0) = (Y1 @ Y0)))))))),
% 5.35/2.11 inference(closure,[status(thm)],[f642])).
% 5.35/2.11 thf(f642,plain,(
% 5.35/2.11 ( ! [X0 : $tType] : (($true = (!! @ nat @ (^[Y0 : nat]: (!! @ (nat > X0) @ (^[Y1 : nat > X0]: ((coinductive_lnth @ X0 @ (coindu68654304_llist @ X0 @ Y1) @ Y0) = (Y1 @ Y0)))))))) )),
% 5.35/2.11 inference(fool_elimination,[status(thm)],[f641])).
% 5.35/2.11 thf(f641,plain,(
% 5.35/2.11 ! [X0 : $tType,X1 : nat > X0,X2 : nat] : ((coinductive_lnth @ X0 @ (coindu68654304_llist @ X0 @ X1) @ X2) = (X1 @ X2))),
% 5.35/2.11 inference(rectify,[status(thm)],[f20])).
% 5.35/2.11 thf(f20,axiom,(
% 5.35/2.11 ! [X0 : $tType,X7 : nat > X0,X10 : nat] : ((coinductive_lnth @ X0 @ (coindu68654304_llist @ X0 @ X7) @ X10) = (X7 @ X10))),
% 5.35/2.11 file('/export/starexec/sandbox/tmp/tmp.dF9X7yUCOB/DTF2THF_3754.p',fact_19_lnth__inf__llist)).
% 5.35/2.11 thf(f2549,plain,(
% 5.35/2.11 ((f @ sK5) != (coinductive_lnth @ a @ (coindu68654304_llist @ a @ f) @ sK5))),
% 5.35/2.11 inference(negative_extensionality,[status(thm)],[f2548])).
% 5.35/2.11 thf(f2548,plain,(
% 5.35/2.11 (f != (coinductive_lnth @ a @ (coindu68654304_llist @ a @ f)))),
% 5.35/2.11 inference(equality_proxy_clausification,[status(thm)],[f2539])).
% 5.35/2.11 thf(f2539,plain,(
% 5.35/2.11 ($false = (f = (coinductive_lnth @ a @ (coindu68654304_llist @ a @ f))))),
% 5.35/2.11 inference(backward_demodulation,[status(thm)],[f2008,f2535])).
% 5.35/2.11 thf(f2535,plain,(
% 5.35/2.11 ( ! [X0 : $tType,X1 : nat > X0] : (((coinductive_lnth @ X0 @ (coindu68654304_llist @ X0 @ X1)) = (snth @ X0 @ (coindu176146587of_seq @ X0 @ X1)))) )),
% 5.35/2.11 inference(superposition,[status(thm)],[f2514,f2530])).
% 5.35/2.11 thf(f2530,plain,(
% 5.35/2.11 ( ! [X0 : $tType,X1 : nat > X0] : (((coindu68654304_llist @ X0 @ X1) = (coindu1724414836stream @ X0 @ (coindu176146587of_seq @ X0 @ X1)))) )),
% 5.35/2.11 inference(equality_proxy_clausification,[status(thm)],[f2527])).
% 5.35/2.11 thf(f2527,plain,(
% 5.35/2.11 ( ! [X0 : $tType,X1 : nat > X0] : (($true = ((coindu68654304_llist @ X0 @ X1) = (coindu1724414836stream @ X0 @ (coindu176146587of_seq @ X0 @ X1))))) )),
% 5.35/2.11 inference(beta_eta_normalization,[status(thm)],[f2526])).
% 5.35/2.11 thf(f2526,plain,(
% 5.35/2.11 ( ! [X0 : $tType,X1 : nat > X0] : (($true = ((^[Y0 : nat > X0]: ((coindu68654304_llist @ X0 @ Y0) = (coindu1724414836stream @ X0 @ (coindu176146587of_seq @ X0 @ Y0)))) @ X1))) )),
% 5.35/2.11 inference(pi_clausification,[status(thm)],[f1398])).
% 5.35/2.11 thf(f1398,plain,(
% 5.35/2.11 ( ! [X0 : $tType] : (($true = (!! @ (nat > X0) @ (^[Y0 : nat > X0]: ((coindu68654304_llist @ X0 @ Y0) = (coindu1724414836stream @ X0 @ (coindu176146587of_seq @ X0 @ Y0))))))) )),
% 5.35/2.11 inference(cnf_transformation,[status(thm)],[f1195])).
% 5.35/2.11 thf(f1195,plain,(
% 5.35/2.11 ! [X0 : $tType] : ($true = (!! @ (nat > X0) @ (^[Y0 : nat > X0]: ((coindu68654304_llist @ X0 @ Y0) = (coindu1724414836stream @ X0 @ (coindu176146587of_seq @ X0 @ Y0))))))),
% 5.35/2.11 inference(closure,[status(thm)],[f792])).
% 5.35/2.11 thf(f792,plain,(
% 5.35/2.11 ( ! [X0 : $tType] : (($true = (!! @ (nat > X0) @ (^[Y0 : nat > X0]: ((coindu68654304_llist @ X0 @ Y0) = (coindu1724414836stream @ X0 @ (coindu176146587of_seq @ X0 @ Y0))))))) )),
% 5.35/2.11 inference(fool_elimination,[status(thm)],[f791])).
% 5.35/2.11 thf(f791,plain,(
% 5.35/2.11 ! [X0 : $tType,X1 : nat > X0] : ((coindu68654304_llist @ X0 @ X1) = (coindu1724414836stream @ X0 @ (coindu176146587of_seq @ X0 @ X1)))),
% 5.35/2.11 inference(rectify,[status(thm)],[f5])).
% 5.35/2.11 thf(f5,axiom,(
% 5.35/2.11 ! [X0 : $tType,X14 : nat > X0] : ((coindu1724414836stream @ X0 @ (coindu176146587of_seq @ X0 @ X14)) = (coindu68654304_llist @ X0 @ X14))),
% 5.35/2.11 file('/export/starexec/sandbox/tmp/tmp.dF9X7yUCOB/DTF2THF_3754.p',fact_4_of__seq_Orep__eq)).
% 5.35/2.11 thf(f2514,plain,(
% 5.35/2.11 ( ! [X0 : $tType,X1 : stream @ X0] : (((snth @ X0 @ X1) = (coinductive_lnth @ X0 @ (coindu1724414836stream @ X0 @ X1)))) )),
% 5.35/2.11 inference(equality_proxy_clausification,[status(thm)],[f2509])).
% 5.35/2.11 thf(f2509,plain,(
% 5.35/2.11 ( ! [X0 : $tType,X1 : stream @ X0] : (($true = ((snth @ X0 @ X1) = (coinductive_lnth @ X0 @ (coindu1724414836stream @ X0 @ X1))))) )),
% 5.35/2.11 inference(beta_eta_normalization,[status(thm)],[f2508])).
% 5.35/2.11 thf(f2508,plain,(
% 5.35/2.11 ( ! [X0 : $tType,X1 : stream @ X0] : (($true = ((^[Y0 : stream @ X0]: ((snth @ X0 @ Y0) = (coinductive_lnth @ X0 @ (coindu1724414836stream @ X0 @ Y0)))) @ X1))) )),
% 5.35/2.11 inference(pi_clausification,[status(thm)],[f1460])).
% 5.35/2.11 thf(f1460,plain,(
% 5.35/2.11 ( ! [X0 : $tType] : (($true = (!! @ stream @ X0 @ (^[Y0 : stream @ X0]: ((snth @ X0 @ Y0) = (coinductive_lnth @ X0 @ (coindu1724414836stream @ X0 @ Y0))))))) )),
% 5.35/2.11 inference(cnf_transformation,[status(thm)],[f1130])).
% 5.35/2.11 thf(f1130,plain,(
% 5.35/2.11 ! [X0 : $tType] : ($true = (!! @ stream @ X0 @ (^[Y0 : stream @ X0]: ((snth @ X0 @ Y0) = (coinductive_lnth @ X0 @ (coindu1724414836stream @ X0 @ Y0))))))),
% 5.35/2.11 inference(closure,[status(thm)],[f682])).
% 5.35/2.11 thf(f682,plain,(
% 5.35/2.11 ( ! [X0 : $tType] : (($true = (!! @ stream @ X0 @ (^[Y0 : stream @ X0]: ((snth @ X0 @ Y0) = (coinductive_lnth @ X0 @ (coindu1724414836stream @ X0 @ Y0))))))) )),
% 5.35/2.11 inference(fool_elimination,[status(thm)],[f681])).
% 5.35/2.11 thf(f681,plain,(
% 5.35/2.11 ! [X0 : $tType,X1 : stream @ X0] : ((snth @ X0 @ X1) = (coinductive_lnth @ X0 @ (coindu1724414836stream @ X0 @ X1)))),
% 5.35/2.11 inference(rectify,[status(thm)],[f10])).
% 5.35/2.11 thf(f10,axiom,(
% 5.35/2.11 ! [X0 : $tType,X16 : stream @ X0] : ((coinductive_lnth @ X0 @ (coindu1724414836stream @ X0 @ X16)) = (snth @ X0 @ X16))),
% 5.35/2.11 file('/export/starexec/sandbox/tmp/tmp.dF9X7yUCOB/DTF2THF_3754.p',fact_9_lnth__list__of__stream)).
% 5.35/2.11 thf(f2008,plain,(
% 5.35/2.11 ((f = (snth @ a @ (coindu176146587of_seq @ a @ f))) = $false)),
% 5.35/2.11 inference(not_proxy_clausification,[status(thm)],[f1520])).
% 5.35/2.11 thf(f1520,plain,(
% 5.35/2.11 ($true = (~ (f = (snth @ a @ (coindu176146587of_seq @ a @ f)))))),
% 5.35/2.11 inference(cnf_transformation,[status(thm)],[f494])).
% 5.35/2.11 thf(f494,plain,(
% 5.35/2.11 ($true = (~ (f = (snth @ a @ (coindu176146587of_seq @ a @ f)))))),
% 5.35/2.11 inference(fool_elimination,[status(thm)],[f314])).
% 5.35/2.11 thf(f314,negated_conjecture,(
% 5.35/2.11 ~(f = (snth @ a @ (coindu176146587of_seq @ a @ f)))),
% 5.35/2.11 inference(negated_conjecture,[status(cth)],[f313])).
% 5.35/2.11 thf(f313,conjecture,(
% 5.35/2.11 (f = (snth @ a @ (coindu176146587of_seq @ a @ f)))),
% 5.35/2.11 file('/export/starexec/sandbox/tmp/tmp.dF9X7yUCOB/DTF2THF_3754.p',conj_0)).
% 5.35/2.11 % SZS output end Proof for DTF2THF_3754
% 5.35/2.11 % (3864)------------------------------
% 5.35/2.11 % (3864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.35/2.11 % (3864)Termination reason: Refutation
% 5.35/2.11
% 5.35/2.11 % (3864)Memory used [KB]: 9594
% 5.35/2.11 % (3864)Time elapsed: 0.368 s
% 5.35/2.11 % (3864)Instructions burned: 741 (million)
% 5.35/2.11 % (3864)------------------------------
% 5.35/2.11 % (3864)------------------------------
% 5.35/2.11 % (3852)Success in time 0.429 s
%------------------------------------------------------------------------------