%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT147^1 : 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 : n010.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 2.88s 1.65s
% Output : Refutation 2.88s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : DAT147^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n010.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 06:08:48 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.28/0.45 ---- Original DTF file ---
% 0.28/0.46 thf(spec,logic,$$dhol).
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % File : DAT147^1 : TPTP v9.2.1. Released v7.0.0.
% 0.28/0.46 % Domain : Data Structures
% 0.28/0.46 % Problem : Coinductive stream 192
% 0.28/0.46 % Version : [Bla16] axioms : Especial.
% 0.28/0.46 % English :
% 0.28/0.46
% 0.28/0.46 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.28/0.46 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.28/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.28/0.46 % Source : [Bla16]
% 0.28/0.46 % Names : coinductive_stream__192.p [Bla16]
% 0.28/0.46
% 0.28/0.46 % Status : Theorem
% 0.28/0.46 % Rating : 0.33 v9.1.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.28/0.46 % Syntax : Number of formulae : 336 ( 133 unt; 54 typ; 0 def)
% 0.28/0.46 % Number of atoms : 636 ( 270 equ; 0 cnn)
% 0.28/0.46 % Maximal formula atoms : 8 ( 2 avg)
% 0.28/0.46 % Number of connectives : 3790 ( 112 ~; 17 |; 50 &;3350 @)
% 0.28/0.46 % ( 0 <=>; 261 =>; 0 <=; 0 <~>)
% 0.28/0.46 % Maximal formula depth : 19 ( 7 avg)
% 0.28/0.46 % Number of types : 2 ( 1 usr)
% 0.28/0.46 % Number of type conns : 211 ( 211 >; 0 *; 0 +; 0 <<)
% 0.28/0.46 % Number of symbols : 56 ( 53 usr; 5 con; 0-6 aty)
% 0.28/0.46 % Number of variables : 984 ( 48 ^; 862 !; 25 ?; 984 :)
% 0.28/0.46 % ( 49 !>; 0 ?*; 0 @-; 0 @+)
% 0.28/0.46 % SPC : TH1_THM_EQU_NAR
% 0.28/0.46
% 0.28/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.28/0.46 % 2016-07-13 15:13:31.164
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 %----Could-be-implicit typings (5)
% 0.28/0.46 thf(ty_t_Coinductive__List_Ollist,type,
% 0.28/0.46 coinductive_llist: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Stream_Ostream,type,
% 0.28/0.46 stream: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Set_Oset,type,
% 0.28/0.46 set: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_itself,type,
% 0.28/0.46 itself: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_tf_a,type,
% 0.28/0.46 a: $tType ).
% 0.28/0.46
% 0.28/0.46 %----Explicit typings (49)
% 0.28/0.46 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.28/0.46 bounded_lattice:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_HOL_Otype,type,
% 0.28/0.46 type:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Obot,type,
% 0.28/0.46 bot:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Lattices_Olattice,type,
% 0.28/0.46 lattice:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Lattices_Odistrib__lattice,type,
% 0.28/0.46 distrib_lattice:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 0.28/0.46 semilattice_inf:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.28/0.46 semilattice_sup:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 0.28/0.46 bounded_lattice_bot:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 0.28/0.46 bounde1808546759up_bot:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Ogen__lset,type,
% 0.28/0.46 coinductive_gen_lset:
% 0.28/0.46 !>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Olappend,type,
% 0.28/0.46 coinductive_lappend:
% 0.28/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Olconcat,type,
% 0.28/0.46 coinductive_lconcat:
% 0.28/0.46 !>[A: $tType] : ( ( coinductive_llist @ ( coinductive_llist @ A ) ) > ( coinductive_llist @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Oldistinct,type,
% 0.28/0.46 coindu351974385stinct:
% 0.28/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_OldropWhile,type,
% 0.28/0.46 coindu218763757pWhile:
% 0.28/0.46 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Olfilter,type,
% 0.28/0.46 coinductive_lfilter:
% 0.28/0.46 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.28/0.46 coinductive_lfinite:
% 0.28/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Ollast,type,
% 0.28/0.46 coinductive_llast:
% 0.28/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 0.28/0.46 coinductive_LCons:
% 0.28/0.46 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 0.28/0.46 coinductive_LNil:
% 0.28/0.46 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Ollist_Ocase__llist,type,
% 0.28/0.46 coindu1381640503_llist:
% 0.28/0.46 !>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Ollist_Olhd,type,
% 0.28/0.46 coinductive_lhd:
% 0.28/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 0.28/0.46 coinductive_lnull:
% 0.28/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 0.28/0.46 coinductive_lset:
% 0.28/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Ollist_Oltl,type,
% 0.28/0.46 coinductive_ltl:
% 0.28/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Olmember,type,
% 0.28/0.46 coinductive_lmember:
% 0.28/0.46 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Olsetp,type,
% 0.28/0.46 coinductive_lsetp:
% 0.28/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > A > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Olstrict__prefix,type,
% 0.28/0.46 coindu1478340336prefix:
% 0.28/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_OltakeWhile,type,
% 0.28/0.46 coindu501562517eWhile:
% 0.28/0.46 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 0.28/0.46 coindu1441602521_llist:
% 0.28/0.46 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ollist__of__stream,type,
% 0.28/0.46 coindu1724414836stream:
% 0.28/0.46 !>[A: $tType] : ( ( stream @ A ) > ( coinductive_llist @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Ocr__stream,type,
% 0.28/0.46 coindu1183105481stream:
% 0.28/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__of__llist,type,
% 0.28/0.46 coindu2010755910_llist:
% 0.28/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ounfold__stream,type,
% 0.28/0.46 coindu139217191stream:
% 0.28/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > A ) > A > ( stream @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_If,type,
% 0.28/0.46 if:
% 0.28/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.28/0.46 inf_inf:
% 0.28/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.28/0.46 sup_sup:
% 0.28/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.28/0.46 bot_bot:
% 0.28/0.46 !>[A: $tType] : A ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Pure_Otype,type,
% 0.28/0.46 type2:
% 0.28/0.46 !>[A: $tType] : ( itself @ A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set_OCollect,type,
% 0.28/0.46 collect:
% 0.28/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set_Odisjnt,type,
% 0.28/0.46 disjnt:
% 0.28/0.46 !>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set_Ois__empty,type,
% 0.28/0.46 is_empty:
% 0.28/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Stream_Osinterleave,type,
% 0.28/0.46 sinterleave:
% 0.28/0.46 !>[A: $tType] : ( ( stream @ A ) > ( stream @ A ) > ( stream @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Stream_Osmember,type,
% 0.28/0.46 smember:
% 0.28/0.46 !>[A: $tType] : ( A > ( stream @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Stream_Ostream_Oshd,type,
% 0.28/0.46 shd:
% 0.28/0.46 !>[A: $tType] : ( ( stream @ A ) > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Stream_Ostream_Osset,type,
% 0.28/0.46 sset:
% 0.28/0.46 !>[A: $tType] : ( ( stream @ A ) > ( set @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_member,type,
% 0.28/0.46 member:
% 0.28/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_x____,type,
% 0.28/0.46 x: a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_xs,type,
% 0.28/0.46 xs: stream @ a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_xsa____,type,
% 0.28/0.46 xsa: stream @ a ).
% 0.28/0.46
% 0.28/0.46 %----Relevant facts (254)
% 0.28/0.46 thf(fact_0__092_060open_062_092_060not_062_Alnull_A_Illist__of__stream_Axs_J_A_092_060Longrightarrow_062_Alhd_A_Illist__of__stream_Axs_J_A_092_060in_062_Alset_A_Illist__of__stream_Axs_J_092_060close_062,axiom,
% 0.28/0.46 ( ~ ( coinductive_lnull @ a @ ( coindu1724414836stream @ a @ xsa ) )
% 0.28/0.46 => ( member @ a @ ( coinductive_lhd @ a @ ( coindu1724414836stream @ a @ xsa ) ) @ ( coinductive_lset @ a @ ( coindu1724414836stream @ a @ xsa ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>\<not> lnull (llist_of_stream xs) \<Longrightarrow> lhd (llist_of_stream xs) \<in> lset (llist_of_stream xs)\<close>
% 0.28/0.46 thf(fact_1_stream_ORep__inject,axiom,
% 0.28/0.46 ! [A: $tType,X: stream @ A,Y: stream @ A] :
% 0.28/0.46 ( ( ( coindu1724414836stream @ A @ X )
% 0.28/0.46 = ( coindu1724414836stream @ A @ Y ) )
% 0.28/0.46 = ( X = Y ) ) ).
% 0.28/0.46
% 0.28/0.46 % stream.Rep_inject
% 0.28/0.46 thf(fact_2_lhd__llist__of__stream,axiom,
% 0.28/0.46 ! [A: $tType,Xs: stream @ A] :
% 0.28/0.46 ( ( coinductive_lhd @ A @ ( coindu1724414836stream @ A @ Xs ) )
% 0.28/0.46 = ( shd @ A @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % lhd_llist_of_stream
% 0.28/0.46 thf(fact_3_lnull__llist__of__stream,axiom,
% 0.28/0.46 ! [A: $tType,Xs: stream @ A] :
% 0.28/0.46 ~ ( coinductive_lnull @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % lnull_llist_of_stream
% 0.28/0.46 thf(fact_4_stream_ORep__inverse,axiom,
% 0.28/0.46 ! [A: $tType,X: stream @ A] :
% 0.28/0.46 ( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ X ) )
% 0.28/0.46 = X ) ).
% 0.28/0.46
% 0.28/0.46 % stream.Rep_inverse
% 0.28/0.46 thf(fact_5_stream__of__llist__llist__of__stream,axiom,
% 0.28/0.46 ! [A: $tType,Xs: stream @ A] :
% 0.28/0.46 ( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ Xs ) )
% 0.28/0.46 = Xs ) ).
% 0.28/0.46
% 0.28/0.46 % stream_of_llist_llist_of_stream
% 0.28/0.46 thf(fact_6_unfold__stream_Osimps_I1_J,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,G1: A > B,G2: A > A,A2: A] :
% 0.28/0.46 ( ( shd @ B @ ( coindu139217191stream @ A @ B @ G1 @ G2 @ A2 ) )
% 0.28/0.46 = ( G1 @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % unfold_stream.simps(1)
% 0.28/0.46 thf(fact_7_shd__stream__of__llist,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( shd @ A @ ( coindu2010755910_llist @ A @ Xs ) )
% 0.28/0.46 = ( coinductive_lhd @ A @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % shd_stream_of_llist
% 0.28/0.46 thf(fact_8_lset__lmember,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 = ( coinductive_lmember @ A @ X @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % lset_lmember
% 0.28/0.46 thf(fact_9_lfinite__llist__of__stream,axiom,
% 0.28/0.46 ! [A: $tType,Xs: stream @ A] :
% 0.28/0.46 ~ ( coinductive_lfinite @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfinite_llist_of_stream
% 0.28/0.46 thf(fact_10__092_060open_062x_A_092_060in_062_Asset_Axs_092_060close_062,axiom,
% 0.28/0.46 member @ a @ x @ ( sset @ a @ xs ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>x \<in> sset xs\<close>
% 0.28/0.46 thf(fact_11_llist_Oset__sel_I1_J,axiom,
% 0.28/0.46 ! [A: $tType,A2: coinductive_llist @ A] :
% 0.28/0.46 ( ~ ( coinductive_lnull @ A @ A2 )
% 0.28/0.46 => ( member @ A @ ( coinductive_lhd @ A @ A2 ) @ ( coinductive_lset @ A @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.set_sel(1)
% 0.28/0.46 thf(fact_12_sinterleave_Osimps_I1_J,axiom,
% 0.28/0.46 ! [A: $tType,S1: stream @ A,S2: stream @ A] :
% 0.28/0.46 ( ( shd @ A @ ( sinterleave @ A @ S1 @ S2 ) )
% 0.28/0.46 = ( shd @ A @ S1 ) ) ).
% 0.28/0.46
% 0.28/0.46 % sinterleave.simps(1)
% 0.28/0.46 thf(fact_13_Coinductive__List_Olset__into__lsetp,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ( coinductive_lsetp @ A @ Xs @ X ) ) ).
% 0.28/0.46
% 0.28/0.46 % Coinductive_List.lset_into_lsetp
% 0.28/0.46 thf(fact_14_llist__of__stream__stream__of__llist,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 => ( ( coindu1724414836stream @ A @ ( coindu2010755910_llist @ A @ Xs ) )
% 0.28/0.46 = Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist_of_stream_stream_of_llist
% 0.28/0.46 thf(fact_15_lzip_Oexhaust,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.28/0.46 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 | ( coinductive_lnull @ B @ Ys ) )
% 0.28/0.46 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lzip.exhaust
% 0.28/0.46 thf(fact_16_lappend_Oexhaust,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ~ ( coinductive_lnull @ A @ Ys ) )
% 0.28/0.46 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lappend.exhaust
% 0.28/0.46 thf(fact_17_lnull__imp__lfinite,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % lnull_imp_lfinite
% 0.28/0.46 thf(fact_18_ltakeWhile_Oexhaust,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.28/0.46 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 | ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
% 0.28/0.46 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltakeWhile.exhaust
% 0.28/0.46 thf(fact_19_shd__sset,axiom,
% 0.28/0.46 ! [A: $tType,A2: stream @ A] : ( member @ A @ ( shd @ A @ A2 ) @ ( sset @ A @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % shd_sset
% 0.28/0.46 thf(fact_20_lsetp__into__lset,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 0.28/0.46 ( ( coinductive_lsetp @ A @ Xs @ X )
% 0.28/0.46 => ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lsetp_into_lset
% 0.28/0.46 thf(fact_21_cr__streamI,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 => ( coindu1183105481stream @ A @ Xs @ ( coindu2010755910_llist @ A @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % cr_streamI
% 0.28/0.46 thf(fact_22_Stream_Osmember__def,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( smember @ A )
% 0.28/0.46 = ( ^ [X2: A,S: stream @ A] : ( member @ A @ X2 @ ( sset @ A @ S ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Stream.smember_def
% 0.28/0.46 thf(fact_23_llist__set__induct,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ( ! [Xs2: coinductive_llist @ A] :
% 0.28/0.46 ( ~ ( coinductive_lnull @ A @ Xs2 )
% 0.28/0.46 => ( P @ ( coinductive_lhd @ A @ Xs2 ) @ Xs2 ) )
% 0.28/0.46 => ( ! [Xs2: coinductive_llist @ A,Y2: A] :
% 0.28/0.46 ( ~ ( coinductive_lnull @ A @ Xs2 )
% 0.28/0.46 => ( ( member @ A @ Y2 @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
% 0.28/0.46 => ( ( P @ Y2 @ ( coinductive_ltl @ A @ Xs2 ) )
% 0.28/0.46 => ( P @ Y2 @ Xs2 ) ) ) )
% 0.28/0.46 => ( P @ X @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist_set_induct
% 0.28/0.46 thf(fact_24_lset__eq__empty,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( ( coinductive_lset @ A @ Xs )
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.46 = ( coinductive_lnull @ A @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % lset_eq_empty
% 0.28/0.46 thf(fact_25_lnull__ldropWhile,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lnull @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 0.28/0.46 = ( ! [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ( P @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lnull_ldropWhile
% 0.28/0.46 thf(fact_26_lhd__lconcat,axiom,
% 0.28/0.46 ! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A )] :
% 0.28/0.46 ( ~ ( coinductive_lnull @ ( coinductive_llist @ A ) @ Xss )
% 0.28/0.46 => ( ~ ( coinductive_lnull @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) )
% 0.28/0.46 => ( ( coinductive_lhd @ A @ ( coinductive_lconcat @ A @ Xss ) )
% 0.28/0.46 = ( coinductive_lhd @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lhd_lconcat
% 0.28/0.46 thf(fact_27_sset__sinterleave,axiom,
% 0.28/0.46 ! [A: $tType,S1: stream @ A,S2: stream @ A] :
% 0.28/0.46 ( ( sset @ A @ ( sinterleave @ A @ S1 @ S2 ) )
% 0.28/0.46 = ( sup_sup @ ( set @ A ) @ ( sset @ A @ S1 ) @ ( sset @ A @ S2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sset_sinterleave
% 0.28/0.46 thf(fact_28_ltakeWhile_Odisc__iff_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
% 0.28/0.46 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 & ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltakeWhile.disc_iff(2)
% 0.28/0.46 thf(fact_29_ltakeWhile_Odisc__iff_I1_J,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 0.28/0.46 = ( ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 | ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltakeWhile.disc_iff(1)
% 0.28/0.46 thf(fact_30_lnull__ltakeWhile,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 0.28/0.46 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lnull_ltakeWhile
% 0.28/0.46 thf(fact_31_lnull__lfilter,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lnull @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 0.28/0.46 = ( ! [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ~ ( P @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lnull_lfilter
% 0.28/0.46 thf(fact_32_lfilter__idem,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 0.28/0.46 = ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfilter_idem
% 0.28/0.46 thf(fact_33_lfinite__ltl,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lfinite @ A @ ( coinductive_ltl @ A @ Xs ) )
% 0.28/0.46 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfinite_ltl
% 0.28/0.46 thf(fact_34_ltakeWhile_Osimps_I4_J,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.28/0.46 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( ( P @ ( coinductive_lhd @ A @ Xs ) )
% 0.28/0.46 => ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 0.28/0.46 = ( coindu501562517eWhile @ A @ P @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltakeWhile.simps(4)
% 0.28/0.46 thf(fact_35_lnull__ltlI,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lnull_ltlI
% 0.28/0.46 thf(fact_36_in__lset__ltlD,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) )
% 0.28/0.46 => ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % in_lset_ltlD
% 0.28/0.46 thf(fact_37_lfilter__cong,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 0.28/0.46 ( ( Xs = Ys )
% 0.28/0.46 => ( ! [X3: A] :
% 0.28/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Ys ) )
% 0.28/0.46 => ( ( P @ X3 )
% 0.28/0.46 = ( Q @ X3 ) ) )
% 0.28/0.46 => ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.28/0.46 = ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfilter_cong
% 0.28/0.46 thf(fact_38_lfilter__id__conv,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.28/0.46 = Xs )
% 0.28/0.46 = ( ! [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ( P @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfilter_id_conv
% 0.28/0.46 thf(fact_39_lfinite__lfilterI,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.28/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 => ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfinite_lfilterI
% 0.28/0.46 thf(fact_40_ltakeWhile__all,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.28/0.46 ( ! [X3: A] :
% 0.28/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ( P @ X3 ) )
% 0.28/0.46 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 0.28/0.46 = Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltakeWhile_all
% 0.28/0.46 thf(fact_41_ltakeWhile__cong,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 0.28/0.46 ( ( Xs = Ys )
% 0.28/0.46 => ( ! [X3: A] :
% 0.28/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Ys ) )
% 0.28/0.46 => ( ( P @ X3 )
% 0.28/0.46 = ( Q @ X3 ) ) )
% 0.28/0.46 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 0.28/0.46 = ( coindu501562517eWhile @ A @ Q @ Ys ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltakeWhile_cong
% 0.28/0.46 thf(fact_42_lset__ltakeWhileD,axiom,
% 0.28/0.46 ! [A: $tType,X: A,P: A > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
% 0.28/0.46 => ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 & ( P @ X ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lset_ltakeWhileD
% 0.28/0.46 thf(fact_43_mem__Collect__eq,axiom,
% 0.28/0.46 ! [A: $tType,A2: A,P: A > $o] :
% 0.28/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.28/0.46 = ( P @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % mem_Collect_eq
% 0.28/0.46 thf(fact_44_Collect__mem__eq,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A] :
% 0.28/0.46 ( ( collect @ A
% 0.28/0.46 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.28/0.46 = A3 ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_mem_eq
% 0.28/0.46 thf(fact_45_Collect__cong,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.46 ( ! [X3: A] :
% 0.28/0.46 ( ( P @ X3 )
% 0.28/0.46 = ( Q @ X3 ) )
% 0.28/0.46 => ( ( collect @ A @ P )
% 0.28/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_cong
% 0.28/0.46 thf(fact_46_ext,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.28/0.46 ( ! [X3: A] :
% 0.28/0.46 ( ( F @ X3 )
% 0.28/0.46 = ( G @ X3 ) )
% 0.28/0.46 => ( F = G ) ) ).
% 0.28/0.46
% 0.28/0.46 % ext
% 0.28/0.46 thf(fact_47_ldropWhile__cong,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 0.28/0.46 ( ( Xs = Ys )
% 0.28/0.46 => ( ! [X3: A] :
% 0.28/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Ys ) )
% 0.28/0.46 => ( ( P @ X3 )
% 0.28/0.46 = ( Q @ X3 ) ) )
% 0.28/0.46 => ( ( coindu218763757pWhile @ A @ P @ Xs )
% 0.28/0.46 = ( coindu218763757pWhile @ A @ Q @ Ys ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ldropWhile_cong
% 0.28/0.46 thf(fact_48_in__lset__ldropWhileD,axiom,
% 0.28/0.46 ! [A: $tType,X: A,P: A > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) )
% 0.28/0.46 => ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % in_lset_ldropWhileD
% 0.28/0.46 thf(fact_49_sset__neq__empty,axiom,
% 0.28/0.46 ! [A: $tType,Xs: stream @ A] :
% 0.28/0.46 ( ( sset @ A @ Xs )
% 0.28/0.46 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sset_neq_empty
% 0.28/0.46 thf(fact_50_llist_Oset__sel_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,A2: coinductive_llist @ A,X: A] :
% 0.28/0.46 ( ~ ( coinductive_lnull @ A @ A2 )
% 0.28/0.46 => ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ A2 ) ) )
% 0.28/0.46 => ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.set_sel(2)
% 0.28/0.46 thf(fact_51_lfinite__induct,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.28/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 => ( ! [Xs2: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lnull @ A @ Xs2 )
% 0.28/0.46 => ( P @ Xs2 ) )
% 0.28/0.46 => ( ! [Xs2: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lfinite @ A @ Xs2 )
% 0.28/0.46 => ( ~ ( coinductive_lnull @ A @ Xs2 )
% 0.28/0.46 => ( ( P @ ( coinductive_ltl @ A @ Xs2 ) )
% 0.28/0.46 => ( P @ Xs2 ) ) ) )
% 0.28/0.46 => ( P @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfinite_induct
% 0.28/0.46 thf(fact_52_llist_Ocoinduct__strong,axiom,
% 0.28/0.46 ! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
% 0.28/0.46 ( ( R @ Llist @ Llist2 )
% 0.28/0.46 => ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
% 0.28/0.46 ( ( R @ Llist3 @ Llist4 )
% 0.28/0.46 => ( ( ( coinductive_lnull @ A @ Llist3 )
% 0.28/0.46 = ( coinductive_lnull @ A @ Llist4 ) )
% 0.28/0.46 & ( ~ ( coinductive_lnull @ A @ Llist3 )
% 0.28/0.46 => ( ~ ( coinductive_lnull @ A @ Llist4 )
% 0.28/0.46 => ( ( ( coinductive_lhd @ A @ Llist3 )
% 0.28/0.46 = ( coinductive_lhd @ A @ Llist4 ) )
% 0.28/0.46 & ( ( R @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) )
% 0.28/0.46 | ( ( coinductive_ltl @ A @ Llist3 )
% 0.28/0.46 = ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) ) )
% 0.28/0.46 => ( Llist = Llist2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.coinduct_strong
% 0.28/0.46 thf(fact_53_llist_Ocoinduct,axiom,
% 0.28/0.46 ! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
% 0.28/0.46 ( ( R @ Llist @ Llist2 )
% 0.28/0.46 => ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
% 0.28/0.46 ( ( R @ Llist3 @ Llist4 )
% 0.28/0.46 => ( ( ( coinductive_lnull @ A @ Llist3 )
% 0.28/0.46 = ( coinductive_lnull @ A @ Llist4 ) )
% 0.28/0.46 & ( ~ ( coinductive_lnull @ A @ Llist3 )
% 0.28/0.46 => ( ~ ( coinductive_lnull @ A @ Llist4 )
% 0.28/0.46 => ( ( ( coinductive_lhd @ A @ Llist3 )
% 0.28/0.46 = ( coinductive_lhd @ A @ Llist4 ) )
% 0.28/0.46 & ( R @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) )
% 0.28/0.46 => ( Llist = Llist2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.coinduct
% 0.28/0.46 thf(fact_54_llist_Oexpand,axiom,
% 0.28/0.46 ! [A: $tType,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
% 0.28/0.46 ( ( ( coinductive_lnull @ A @ Llist )
% 0.28/0.46 = ( coinductive_lnull @ A @ Llist2 ) )
% 0.28/0.46 => ( ( ~ ( coinductive_lnull @ A @ Llist )
% 0.28/0.46 => ( ~ ( coinductive_lnull @ A @ Llist2 )
% 0.28/0.46 => ( ( ( coinductive_lhd @ A @ Llist )
% 0.28/0.46 = ( coinductive_lhd @ A @ Llist2 ) )
% 0.28/0.46 & ( ( coinductive_ltl @ A @ Llist )
% 0.28/0.46 = ( coinductive_ltl @ A @ Llist2 ) ) ) ) )
% 0.28/0.46 => ( Llist = Llist2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.expand
% 0.28/0.46 thf(fact_55_lhd__ltakeWhile,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.28/0.46 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( ( P @ ( coinductive_lhd @ A @ Xs ) )
% 0.28/0.46 => ( ( coinductive_lhd @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 0.28/0.46 = ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lhd_ltakeWhile
% 0.28/0.46 thf(fact_56_ltakeWhile_Odisc_I1_J,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.28/0.46 ( ( ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 | ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
% 0.28/0.46 => ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltakeWhile.disc(1)
% 0.28/0.46 thf(fact_57_ltakeWhile_Odisc_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.28/0.46 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( ( P @ ( coinductive_lhd @ A @ Xs ) )
% 0.28/0.46 => ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltakeWhile.disc(2)
% 0.28/0.46 thf(fact_58_lfinite__ltakeWhile,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lfinite @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 0.28/0.46 = ( ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 | ? [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 & ~ ( P @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfinite_ltakeWhile
% 0.28/0.46 thf(fact_59_stream__of__llist__def,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( coindu2010755910_llist @ A )
% 0.28/0.46 = ( coindu139217191stream @ ( coinductive_llist @ A ) @ A @ ( coinductive_lhd @ A ) @ ( coinductive_ltl @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % stream_of_llist_def
% 0.28/0.46 thf(fact_60_lset__lnull,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_lset @ A @ Xs )
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lset_lnull
% 0.28/0.46 thf(fact_61_lfinite__ldropWhile,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lfinite @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 0.28/0.46 = ( ? [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 & ~ ( P @ X2 ) )
% 0.28/0.46 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfinite_ldropWhile
% 0.28/0.46 thf(fact_62_lhd__ldropWhile,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.28/0.46 ( ? [X4: A] :
% 0.28/0.46 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 & ~ ( P @ X4 ) )
% 0.28/0.46 => ~ ( P @ ( coinductive_lhd @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lhd_ldropWhile
% 0.28/0.46 thf(fact_63_lhd__ldropWhile__in__lset,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.28/0.46 ( ? [X4: A] :
% 0.28/0.46 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 & ~ ( P @ X4 ) )
% 0.28/0.46 => ( member @ A @ ( coinductive_lhd @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lhd_ldropWhile_in_lset
% 0.28/0.46 thf(fact_64_Un__empty,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.46 = ( ( A3
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.46 & ( B2
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Un_empty
% 0.28/0.46 thf(fact_65_sup__bot_Oright__neutral,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_bot.right_neutral
% 0.28/0.46 thf(fact_66_sup__bot_Oleft__neutral,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_bot.left_neutral
% 0.28/0.46 thf(fact_67_sup__eq__bot__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A] :
% 0.28/0.46 ( ( ( sup_sup @ A @ X @ Y )
% 0.28/0.46 = ( bot_bot @ A ) )
% 0.28/0.46 = ( ( X
% 0.28/0.46 = ( bot_bot @ A ) )
% 0.28/0.46 & ( Y
% 0.28/0.46 = ( bot_bot @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_eq_bot_iff
% 0.28/0.46 thf(fact_68_bot__eq__sup__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A] :
% 0.28/0.46 ( ( ( bot_bot @ A )
% 0.28/0.46 = ( sup_sup @ A @ X @ Y ) )
% 0.28/0.46 = ( ( X
% 0.28/0.46 = ( bot_bot @ A ) )
% 0.28/0.46 & ( Y
% 0.28/0.46 = ( bot_bot @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot_eq_sup_iff
% 0.28/0.46 thf(fact_69_Un__iff,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
% 0.28/0.46 = ( ( member @ A @ C @ A3 )
% 0.28/0.46 | ( member @ A @ C @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Un_iff
% 0.28/0.46 thf(fact_70_UnCI,axiom,
% 0.28/0.46 ! [A: $tType,C: A,B2: set @ A,A3: set @ A] :
% 0.28/0.46 ( ( ~ ( member @ A @ C @ B2 )
% 0.28/0.46 => ( member @ A @ C @ A3 ) )
% 0.28/0.46 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UnCI
% 0.28/0.46 thf(fact_71_sup_Oright__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A2: A,B3: A] :
% 0.28/0.46 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ B3 )
% 0.28/0.46 = ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup.right_idem
% 0.28/0.46 thf(fact_72_sup__left__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A] :
% 0.28/0.46 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.28/0.46 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_left_idem
% 0.28/0.46 thf(fact_73_empty__iff,axiom,
% 0.28/0.46 ! [A: $tType,C: A] :
% 0.28/0.46 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % empty_iff
% 0.28/0.46 thf(fact_74_all__not__in__conv,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A] :
% 0.28/0.46 ( ( ! [X2: A] :
% 0.28/0.46 ~ ( member @ A @ X2 @ A3 ) )
% 0.28/0.46 = ( A3
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % all_not_in_conv
% 0.28/0.46 thf(fact_75_Collect__empty__eq,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o] :
% 0.28/0.46 ( ( ( collect @ A @ P )
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.46 = ( ! [X2: A] :
% 0.28/0.46 ~ ( P @ X2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_empty_eq
% 0.28/0.46 thf(fact_76_empty__Collect__eq,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o] :
% 0.28/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.28/0.46 = ( collect @ A @ P ) )
% 0.28/0.46 = ( ! [X2: A] :
% 0.28/0.46 ~ ( P @ X2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % empty_Collect_eq
% 0.28/0.46 thf(fact_77_sup__apply,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.28/0.46 => ( ( sup_sup @ ( A > B ) )
% 0.28/0.46 = ( ^ [F2: A > B,G3: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_apply
% 0.28/0.46 thf(fact_78_sup_Oidem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( sup_sup @ A @ A2 @ A2 )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup.idem
% 0.28/0.46 thf(fact_79_sup__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A] :
% 0.28/0.46 ( ( sup_sup @ A @ X @ X )
% 0.28/0.46 = X ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_idem
% 0.28/0.46 thf(fact_80_sup_Oleft__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A2: A,B3: A] :
% 0.28/0.46 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B3 ) )
% 0.28/0.46 = ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup.left_idem
% 0.28/0.46 thf(fact_81_bot__set__def,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.28/0.46 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot_set_def
% 0.28/0.46 thf(fact_82_emptyE,axiom,
% 0.28/0.46 ! [A: $tType,A2: A] :
% 0.28/0.46 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % emptyE
% 0.28/0.46 thf(fact_83_equals0D,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,A2: A] :
% 0.28/0.46 ( ( A3
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.46 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % equals0D
% 0.28/0.46 thf(fact_84_equals0I,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A] :
% 0.28/0.46 ( ! [Y2: A] :
% 0.28/0.46 ~ ( member @ A @ Y2 @ A3 )
% 0.28/0.46 => ( A3
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % equals0I
% 0.28/0.46 thf(fact_85_ex__in__conv,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A] :
% 0.28/0.46 ( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.28/0.46 = ( A3
% 0.28/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ex_in_conv
% 0.28/0.46 thf(fact_86_inf__sup__aci_I8_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A] :
% 0.28/0.46 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.28/0.46 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_aci(8)
% 0.28/0.46 thf(fact_87_inf__sup__aci_I7_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A,Z: A] :
% 0.28/0.46 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.28/0.46 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_aci(7)
% 0.28/0.46 thf(fact_88_inf__sup__aci_I6_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A,Z: A] :
% 0.28/0.46 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 0.28/0.46 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_aci(6)
% 0.28/0.46 thf(fact_89_inf__sup__aci_I5_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ( ( sup_sup @ A )
% 0.28/0.46 = ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_aci(5)
% 0.28/0.46 thf(fact_90_sup__fun__def,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.28/0.46 => ( ( sup_sup @ ( A > B ) )
% 0.28/0.46 = ( ^ [F2: A > B,G3: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_fun_def
% 0.28/0.46 thf(fact_91_sup_Oassoc,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A2: A,B3: A,C: A] :
% 0.28/0.46 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ C )
% 0.28/0.46 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup.assoc
% 0.28/0.46 thf(fact_92_sup__assoc,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A,Z: A] :
% 0.28/0.46 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 0.28/0.46 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_assoc
% 0.28/0.46 thf(fact_93_sup_Ocommute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ( ( sup_sup @ A )
% 0.28/0.46 = ( ^ [A4: A,B4: A] : ( sup_sup @ A @ B4 @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup.commute
% 0.28/0.46 thf(fact_94_sup__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ( ( sup_sup @ A )
% 0.28/0.46 = ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_commute
% 0.28/0.46 thf(fact_95_sup_Oleft__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [B3: A,A2: A,C: A] :
% 0.28/0.46 ( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A2 @ C ) )
% 0.28/0.46 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup.left_commute
% 0.28/0.46 thf(fact_96_sup__left__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A,Z: A] :
% 0.28/0.46 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.28/0.46 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_left_commute
% 0.28/0.46 thf(fact_97_UnE,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
% 0.28/0.46 => ( ~ ( member @ A @ C @ A3 )
% 0.28/0.46 => ( member @ A @ C @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UnE
% 0.28/0.46 thf(fact_98_UnI1,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ A3 )
% 0.28/0.46 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UnI1
% 0.28/0.46 thf(fact_99_UnI2,axiom,
% 0.28/0.46 ! [A: $tType,C: A,B2: set @ A,A3: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ B2 )
% 0.28/0.46 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UnI2
% 0.28/0.46 thf(fact_100_bex__Un,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,P: A > $o] :
% 0.28/0.46 ( ( ? [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
% 0.28/0.46 & ( P @ X2 ) ) )
% 0.28/0.46 = ( ? [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ A3 )
% 0.28/0.46 & ( P @ X2 ) )
% 0.28/0.46 | ? [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ B2 )
% 0.28/0.46 & ( P @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bex_Un
% 0.28/0.46 thf(fact_101_ball__Un,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,P: A > $o] :
% 0.28/0.46 ( ( ! [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
% 0.28/0.46 => ( P @ X2 ) ) )
% 0.28/0.46 = ( ! [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ A3 )
% 0.28/0.46 => ( P @ X2 ) )
% 0.28/0.46 & ! [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ B2 )
% 0.28/0.46 => ( P @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ball_Un
% 0.28/0.46 thf(fact_102_Un__assoc,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 0.28/0.46 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ C2 )
% 0.28/0.46 = ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Un_assoc
% 0.28/0.46 thf(fact_103_Un__absorb,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A] :
% 0.28/0.46 ( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
% 0.28/0.46 = A3 ) ).
% 0.28/0.46
% 0.28/0.46 % Un_absorb
% 0.28/0.46 thf(fact_104_Un__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( sup_sup @ ( set @ A ) )
% 0.28/0.46 = ( ^ [A5: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Un_commute
% 0.28/0.46 thf(fact_105_Un__left__absorb,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
% 0.28/0.46 = ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % Un_left_absorb
% 0.28/0.46 thf(fact_106_Un__left__commute,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 0.28/0.46 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
% 0.28/0.46 = ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Un_left_commute
% 0.28/0.46 thf(fact_107_sup__bot__left,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A] :
% 0.28/0.46 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
% 0.28/0.46 = X ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_bot_left
% 0.28/0.46 thf(fact_108_sup__bot__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A] :
% 0.28/0.46 ( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
% 0.28/0.46 = X ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_bot_right
% 0.28/0.46 thf(fact_109_Un__empty__left,axiom,
% 0.28/0.46 ! [A: $tType,B2: set @ A] :
% 0.28/0.46 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
% 0.28/0.46 = B2 ) ).
% 0.28/0.46
% 0.28/0.46 % Un_empty_left
% 0.28/0.46 thf(fact_110_Un__empty__right,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A] :
% 0.28/0.46 ( ( sup_sup @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.46 = A3 ) ).
% 0.28/0.46
% 0.28/0.46 % Un_empty_right
% 0.28/0.46 thf(fact_111_bot__apply,axiom,
% 0.28/0.46 ! [C3: $tType,D: $tType] :
% 0.28/0.46 ( ( bot @ C3 @ ( type2 @ C3 ) )
% 0.28/0.46 => ( ( bot_bot @ ( D > C3 ) )
% 0.28/0.46 = ( ^ [X2: D] : ( bot_bot @ C3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot_apply
% 0.28/0.46 thf(fact_112_ltl__lconcat,axiom,
% 0.28/0.46 ! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A )] :
% 0.28/0.46 ( ~ ( coinductive_lnull @ ( coinductive_llist @ A ) @ Xss )
% 0.28/0.46 => ( ~ ( coinductive_lnull @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) )
% 0.28/0.46 => ( ( coinductive_ltl @ A @ ( coinductive_lconcat @ A @ Xss ) )
% 0.28/0.46 = ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) ) @ ( coinductive_lconcat @ A @ ( coinductive_ltl @ ( coinductive_llist @ A ) @ Xss ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltl_lconcat
% 0.28/0.46 thf(fact_113_gen__lset__def,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( coinductive_gen_lset @ A )
% 0.28/0.46 = ( ^ [A5: set @ A,Xs3: coinductive_llist @ A] : ( sup_sup @ ( set @ A ) @ A5 @ ( coinductive_lset @ A @ Xs3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % gen_lset_def
% 0.28/0.46 thf(fact_114_lset__code,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( coinductive_lset @ A )
% 0.28/0.46 = ( coinductive_gen_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lset_code
% 0.28/0.46 thf(fact_115_unfold__llist__id,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coindu1441602521_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_lnull @ A ) @ ( coinductive_lhd @ A ) @ ( coinductive_ltl @ A ) @ Xs )
% 0.28/0.46 = Xs ) ).
% 0.28/0.46
% 0.28/0.46 % unfold_llist_id
% 0.28/0.46 thf(fact_116_Set_Ois__empty__def,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( is_empty @ A )
% 0.28/0.46 = ( ^ [A5: set @ A] :
% 0.28/0.46 ( A5
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Set.is_empty_def
% 0.28/0.46 thf(fact_117_lnull__lappend,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.28/0.46 = ( ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 & ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lnull_lappend
% 0.28/0.46 thf(fact_118_lappend_Odisc__iff_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 0.28/0.46 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lappend.disc_iff(2)
% 0.28/0.46 thf(fact_119_lfinite__lappend,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.28/0.46 = ( ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 & ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfinite_lappend
% 0.28/0.46 thf(fact_120_lconcat__lappend,axiom,
% 0.28/0.46 ! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A ),Yss: coinductive_llist @ ( coinductive_llist @ A )] :
% 0.28/0.46 ( ( coinductive_lfinite @ ( coinductive_llist @ A ) @ Xss )
% 0.28/0.46 => ( ( coinductive_lconcat @ A @ ( coinductive_lappend @ ( coinductive_llist @ A ) @ Xss @ Yss ) )
% 0.28/0.46 = ( coinductive_lappend @ A @ ( coinductive_lconcat @ A @ Xss ) @ ( coinductive_lconcat @ A @ Yss ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lconcat_lappend
% 0.28/0.46 thf(fact_121_unfold__llist_Odisc__iff_I1_J,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A2: A] :
% 0.28/0.46 ( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
% 0.28/0.46 = ( P2 @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % unfold_llist.disc_iff(1)
% 0.28/0.46 thf(fact_122_unfold__llist_Odisc__iff_I2_J,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A2: A] :
% 0.28/0.46 ( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) )
% 0.28/0.46 = ( ~ ( P2 @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % unfold_llist.disc_iff(2)
% 0.28/0.46 thf(fact_123_ltl__lappend,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.28/0.46 = ( coinductive_ltl @ A @ Ys ) ) )
% 0.28/0.46 & ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.28/0.46 = ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ Xs ) @ Ys ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltl_lappend
% 0.28/0.46 thf(fact_124_lhd__lappend,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.28/0.46 = ( coinductive_lhd @ A @ Ys ) ) )
% 0.28/0.46 & ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.28/0.46 = ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lhd_lappend
% 0.28/0.46 thf(fact_125_lfilter__lappend__lfinite,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.28/0.46 = ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfilter_lappend_lfinite
% 0.28/0.46 thf(fact_126_lappend__ltakeWhile__ldropWhile,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lappend @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 0.28/0.46 = Xs ) ).
% 0.28/0.46
% 0.28/0.46 % lappend_ltakeWhile_ldropWhile
% 0.28/0.46 thf(fact_127_lset__lappend__lfinite,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.28/0.46 = ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lset_lappend_lfinite
% 0.28/0.46 thf(fact_128_lappend__assoc,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 0.28/0.46 = ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lappend_assoc
% 0.28/0.46 thf(fact_129_lappend__lnull2,axiom,
% 0.28/0.46 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lnull @ A @ Ys )
% 0.28/0.46 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.28/0.46 = Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % lappend_lnull2
% 0.28/0.46 thf(fact_130_lappend__lnull1,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.28/0.46 = Ys ) ) ).
% 0.28/0.46
% 0.28/0.46 % lappend_lnull1
% 0.28/0.46 thf(fact_131_lappend_Odisc_I1_J,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_lnull @ A @ Ys )
% 0.28/0.46 => ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lappend.disc(1)
% 0.28/0.46 thf(fact_132_lappend_Odisc_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 | ~ ( coinductive_lnull @ A @ Ys ) )
% 0.28/0.46 => ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lappend.disc(2)
% 0.28/0.46 thf(fact_133_lappend__inf,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.28/0.46 = Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % lappend_inf
% 0.28/0.46 thf(fact_134_unfold__llist_Odisc_I1_J,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 0.28/0.46 ( ( P2 @ A2 )
% 0.28/0.46 => ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % unfold_llist.disc(1)
% 0.28/0.46 thf(fact_135_unfold__llist_Odisc_I2_J,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 0.28/0.46 ( ~ ( P2 @ A2 )
% 0.28/0.46 => ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % unfold_llist.disc(2)
% 0.28/0.46 thf(fact_136_unfold__llist_Osimps_I4_J,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 0.28/0.46 ( ~ ( P2 @ A2 )
% 0.28/0.46 => ( ( coinductive_ltl @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
% 0.28/0.46 = ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % unfold_llist.simps(4)
% 0.28/0.46 thf(fact_137_unfold__llist_Osimps_I3_J,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 0.28/0.46 ( ~ ( P2 @ A2 )
% 0.28/0.46 => ( ( coinductive_lhd @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
% 0.28/0.46 = ( G21 @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % unfold_llist.simps(3)
% 0.28/0.46 thf(fact_138_lappend__ltl,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ Xs ) @ Ys )
% 0.28/0.46 = ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lappend_ltl
% 0.28/0.46 thf(fact_139_in__lset__lappend__iff,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 0.28/0.46 = ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 | ( ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 & ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % in_lset_lappend_iff
% 0.28/0.46 thf(fact_140_ltakeWhile__lappend1,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ( ~ ( P @ X )
% 0.28/0.46 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.28/0.46 = ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltakeWhile_lappend1
% 0.28/0.46 thf(fact_141_lfilter__eq__lappend__lfiniteD,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.28/0.46 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.28/0.46 = ( coinductive_lappend @ A @ Ys @ Zs ) )
% 0.28/0.46 => ( ( coinductive_lfinite @ A @ Ys )
% 0.28/0.46 => ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 0.28/0.46 ( ( Xs
% 0.28/0.46 = ( coinductive_lappend @ A @ Us @ Vs ) )
% 0.28/0.46 & ( coinductive_lfinite @ A @ Us )
% 0.28/0.46 & ( Ys
% 0.28/0.46 = ( coinductive_lfilter @ A @ P @ Us ) )
% 0.28/0.46 & ( Zs
% 0.28/0.46 = ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfilter_eq_lappend_lfiniteD
% 0.28/0.46 thf(fact_142_lset__lappend__conv,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.28/0.46 = ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) )
% 0.28/0.46 & ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.28/0.46 = ( coinductive_lset @ A @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lset_lappend_conv
% 0.28/0.46 thf(fact_143_bot__fun__def,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( bot @ B @ ( type2 @ B ) )
% 0.28/0.46 => ( ( bot_bot @ ( A > B ) )
% 0.28/0.46 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot_fun_def
% 0.28/0.46 thf(fact_144_bot__empty__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( bot_bot @ ( A > $o ) )
% 0.28/0.46 = ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot_empty_eq
% 0.28/0.46 thf(fact_145_Collect__empty__eq__bot,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o] :
% 0.28/0.46 ( ( ( collect @ A @ P )
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.46 = ( P
% 0.28/0.46 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_empty_eq_bot
% 0.28/0.46 thf(fact_146_lstrict__prefix__lappend__conv,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.28/0.46 = ( ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 & ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lstrict_prefix_lappend_conv
% 0.28/0.46 thf(fact_147_ldistinct__coinduct,axiom,
% 0.28/0.46 ! [A: $tType,X5: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( X5 @ Xs )
% 0.28/0.46 => ( ! [Xs2: coinductive_llist @ A] :
% 0.28/0.46 ( ( X5 @ Xs2 )
% 0.28/0.46 => ( ~ ( coinductive_lnull @ A @ Xs2 )
% 0.28/0.46 => ( ~ ( member @ A @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
% 0.28/0.46 & ( ( X5 @ ( coinductive_ltl @ A @ Xs2 ) )
% 0.28/0.46 | ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs2 ) ) ) ) ) )
% 0.28/0.46 => ( coindu351974385stinct @ A @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ldistinct_coinduct
% 0.28/0.46 thf(fact_148_llist__less__induct,axiom,
% 0.28/0.46 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ! [Xs2: coinductive_llist @ A] :
% 0.28/0.46 ( ! [Ys2: coinductive_llist @ A] :
% 0.28/0.46 ( ( coindu1478340336prefix @ A @ Ys2 @ Xs2 )
% 0.28/0.46 => ( P @ Ys2 ) )
% 0.28/0.46 => ( P @ Xs2 ) )
% 0.28/0.46 => ( P @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist_less_induct
% 0.28/0.46 thf(fact_149_ldistinct__ltlI,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coindu351974385stinct @ A @ Xs )
% 0.28/0.46 => ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ldistinct_ltlI
% 0.28/0.46 thf(fact_150_ldistinct__lfilterI,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.28/0.46 ( ( coindu351974385stinct @ A @ Xs )
% 0.28/0.46 => ( coindu351974385stinct @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ldistinct_lfilterI
% 0.28/0.46 thf(fact_151_lstrict__prefix__lfinite1,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( coindu1478340336prefix @ A @ Xs @ Ys )
% 0.28/0.46 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % lstrict_prefix_lfinite1
% 0.28/0.46 thf(fact_152_ldistinct__lhdD,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coindu351974385stinct @ A @ Xs )
% 0.28/0.46 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ~ ( member @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ldistinct_lhdD
% 0.28/0.46 thf(fact_153_ldistinct__lappend,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( coindu351974385stinct @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.28/0.46 = ( ( coindu351974385stinct @ A @ Xs )
% 0.28/0.46 & ( ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 => ( ( coindu351974385stinct @ A @ Ys )
% 0.28/0.46 & ( ( inf_inf @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) )
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ldistinct_lappend
% 0.28/0.46 thf(fact_154_ldistinct__lconcat,axiom,
% 0.28/0.46 ! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A )] :
% 0.28/0.46 ( ( coindu351974385stinct @ ( coinductive_llist @ A ) @ Xss )
% 0.28/0.46 => ( ! [Ys3: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ ( coinductive_llist @ A ) @ Ys3 @ ( coinductive_lset @ ( coinductive_llist @ A ) @ Xss ) )
% 0.28/0.46 => ( coindu351974385stinct @ A @ Ys3 ) )
% 0.28/0.46 => ( ! [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ ( coinductive_llist @ A ) @ Ys3 @ ( coinductive_lset @ ( coinductive_llist @ A ) @ Xss ) )
% 0.28/0.46 => ( ( member @ ( coinductive_llist @ A ) @ Zs2 @ ( coinductive_lset @ ( coinductive_llist @ A ) @ Xss ) )
% 0.28/0.46 => ( ( Ys3 != Zs2 )
% 0.28/0.46 => ( ( inf_inf @ ( set @ A ) @ ( coinductive_lset @ A @ Ys3 ) @ ( coinductive_lset @ A @ Zs2 ) )
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) )
% 0.28/0.46 => ( coindu351974385stinct @ A @ ( coinductive_lconcat @ A @ Xss ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ldistinct_lconcat
% 0.28/0.46 thf(fact_155_llist_Ocase__eq__if,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType] :
% 0.28/0.46 ( ( coindu1381640503_llist @ B @ A )
% 0.28/0.46 = ( ^ [F1: B,F22: A > ( coinductive_llist @ A ) > B,Llist5: coinductive_llist @ A] : ( if @ B @ ( coinductive_lnull @ A @ Llist5 ) @ F1 @ ( F22 @ ( coinductive_lhd @ A @ Llist5 ) @ ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.case_eq_if
% 0.28/0.46 thf(fact_156_inf__apply,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 0.28/0.46 => ( ( inf_inf @ ( A > B ) )
% 0.28/0.46 = ( ^ [F2: A > B,G3: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_apply
% 0.28/0.46 thf(fact_157_inf_Oidem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( inf_inf @ A @ A2 @ A2 )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.idem
% 0.28/0.46 thf(fact_158_inf__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X @ X )
% 0.28/0.46 = X ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_idem
% 0.28/0.46 thf(fact_159_inf_Oleft__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A2: A,B3: A] :
% 0.28/0.46 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B3 ) )
% 0.28/0.46 = ( inf_inf @ A @ A2 @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.left_idem
% 0.28/0.46 thf(fact_160_inf__left__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 0.28/0.46 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_left_idem
% 0.28/0.46 thf(fact_161_inf_Oright__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A2: A,B3: A] :
% 0.28/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B3 ) @ B3 )
% 0.28/0.46 = ( inf_inf @ A @ A2 @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.right_idem
% 0.28/0.46 thf(fact_162_inf__right__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A] :
% 0.28/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
% 0.28/0.46 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_right_idem
% 0.28/0.46 thf(fact_163_IntI,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ A3 )
% 0.28/0.46 => ( ( member @ A @ C @ B2 )
% 0.28/0.46 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % IntI
% 0.28/0.46 thf(fact_164_Int__iff,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
% 0.28/0.46 = ( ( member @ A @ C @ A3 )
% 0.28/0.46 & ( member @ A @ C @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_iff
% 0.28/0.46 thf(fact_165_inf__bot__left,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A] :
% 0.28/0.46 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
% 0.28/0.46 = ( bot_bot @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_bot_left
% 0.28/0.46 thf(fact_166_inf__bot__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
% 0.28/0.46 = ( bot_bot @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_bot_right
% 0.28/0.46 thf(fact_167_inf__sup__absorb,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.28/0.46 = X ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_absorb
% 0.28/0.46 thf(fact_168_sup__inf__absorb,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A] :
% 0.28/0.46 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 0.28/0.46 = X ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_inf_absorb
% 0.28/0.46 thf(fact_169_inf__sup__aci_I4_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 0.28/0.46 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_aci(4)
% 0.28/0.46 thf(fact_170_inf__sup__aci_I3_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A,Z: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.28/0.46 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_aci(3)
% 0.28/0.46 thf(fact_171_inf__sup__aci_I2_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A,Z: A] :
% 0.28/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 0.28/0.46 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_aci(2)
% 0.28/0.46 thf(fact_172_inf__sup__aci_I1_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ( ( inf_inf @ A )
% 0.28/0.46 = ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_aci(1)
% 0.28/0.46 thf(fact_173_IntE,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
% 0.28/0.46 => ~ ( ( member @ A @ C @ A3 )
% 0.28/0.46 => ~ ( member @ A @ C @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % IntE
% 0.28/0.46 thf(fact_174_IntD1,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
% 0.28/0.46 => ( member @ A @ C @ A3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % IntD1
% 0.28/0.46 thf(fact_175_IntD2,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
% 0.28/0.46 => ( member @ A @ C @ B2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % IntD2
% 0.28/0.46 thf(fact_176_Int__assoc,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ C2 )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_assoc
% 0.28/0.46 thf(fact_177_Int__absorb,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
% 0.28/0.46 = A3 ) ).
% 0.28/0.46
% 0.28/0.46 % Int_absorb
% 0.28/0.46 thf(fact_178_Int__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) )
% 0.28/0.46 = ( ^ [A5: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_commute
% 0.28/0.46 thf(fact_179_Int__left__absorb,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_left_absorb
% 0.28/0.46 thf(fact_180_inf__fun__def,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 0.28/0.46 => ( ( inf_inf @ ( A > B ) )
% 0.28/0.46 = ( ^ [F2: A > B,G3: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_fun_def
% 0.28/0.46 thf(fact_181_Int__left__commute,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_left_commute
% 0.28/0.46 thf(fact_182_inf_Oassoc,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A2: A,B3: A,C: A] :
% 0.28/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B3 ) @ C )
% 0.28/0.46 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.assoc
% 0.28/0.46 thf(fact_183_inf__assoc,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A,Z: A] :
% 0.28/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 0.28/0.46 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_assoc
% 0.28/0.46 thf(fact_184_inf_Ocommute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.28/0.46 => ( ( inf_inf @ A )
% 0.28/0.46 = ( ^ [A4: A,B4: A] : ( inf_inf @ A @ B4 @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.commute
% 0.28/0.46 thf(fact_185_inf__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.28/0.46 => ( ( inf_inf @ A )
% 0.28/0.46 = ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_commute
% 0.28/0.46 thf(fact_186_inf_Oleft__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [B3: A,A2: A,C: A] :
% 0.28/0.46 ( ( inf_inf @ A @ B3 @ ( inf_inf @ A @ A2 @ C ) )
% 0.28/0.46 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.left_commute
% 0.28/0.46 thf(fact_187_inf__left__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A,Z: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.28/0.46 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_left_commute
% 0.28/0.46 thf(fact_188_Un__Int__distrib2,axiom,
% 0.28/0.46 ! [A: $tType,B2: set @ A,C2: set @ A,A3: set @ A] :
% 0.28/0.46 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) @ A3 )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ A3 ) @ ( sup_sup @ ( set @ A ) @ C2 @ A3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Un_Int_distrib2
% 0.28/0.46 thf(fact_189_Int__Un__distrib2,axiom,
% 0.28/0.46 ! [A: $tType,B2: set @ A,C2: set @ A,A3: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) @ A3 )
% 0.28/0.46 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C2 @ A3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_Un_distrib2
% 0.28/0.46 thf(fact_190_Un__Int__distrib,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 0.28/0.46 ( ( sup_sup @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ ( sup_sup @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Un_Int_distrib
% 0.28/0.46 thf(fact_191_Int__Un__distrib,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
% 0.28/0.46 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ ( inf_inf @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_Un_distrib
% 0.28/0.46 thf(fact_192_Un__Int__crazy,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 0.28/0.46 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) @ ( inf_inf @ ( set @ A ) @ C2 @ A3 ) )
% 0.28/0.46 = ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) @ ( sup_sup @ ( set @ A ) @ C2 @ A3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Un_Int_crazy
% 0.28/0.46 thf(fact_193_disjoint__iff__not__equal,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.46 = ( ! [X2: A] :
% 0.28/0.46 ( ( member @ A @ X2 @ A3 )
% 0.28/0.46 => ! [Y3: A] :
% 0.28/0.46 ( ( member @ A @ Y3 @ B2 )
% 0.28/0.46 => ( X2 != Y3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % disjoint_iff_not_equal
% 0.28/0.46 thf(fact_194_Int__empty__right,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_empty_right
% 0.28/0.46 thf(fact_195_Int__empty__left,axiom,
% 0.28/0.46 ! [A: $tType,B2: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_empty_left
% 0.28/0.46 thf(fact_196_Int__emptyI,axiom,
% 0.28/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.28/0.46 ( ! [X3: A] :
% 0.28/0.46 ( ( member @ A @ X3 @ A3 )
% 0.28/0.46 => ~ ( member @ A @ X3 @ B2 ) )
% 0.28/0.46 => ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Int_emptyI
% 0.28/0.46 thf(fact_197_sup__inf__distrib2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [Y: A,Z: A,X: A] :
% 0.28/0.46 ( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z ) @ X )
% 0.28/0.46 = ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z @ X ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_inf_distrib2
% 0.28/0.46 thf(fact_198_sup__inf__distrib1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A,Z: A] :
% 0.28/0.46 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.28/0.46 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_inf_distrib1
% 0.28/0.46 thf(fact_199_inf__sup__distrib2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [Y: A,Z: A,X: A] :
% 0.28/0.46 ( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z ) @ X )
% 0.28/0.46 = ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z @ X ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_distrib2
% 0.28/0.46 thf(fact_200_inf__sup__distrib1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A,Z: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.28/0.46 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_distrib1
% 0.28/0.46 thf(fact_201_distrib__imp2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A,Z: A] :
% 0.28/0.46 ( ! [X3: A,Y2: A,Z2: A] :
% 0.28/0.46 ( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z2 ) )
% 0.28/0.46 = ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ ( sup_sup @ A @ X3 @ Z2 ) ) )
% 0.28/0.46 => ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.28/0.46 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % distrib_imp2
% 0.28/0.46 thf(fact_202_distrib__imp1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X: A,Y: A,Z: A] :
% 0.28/0.46 ( ! [X3: A,Y2: A,Z2: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) )
% 0.28/0.46 = ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ ( inf_inf @ A @ X3 @ Z2 ) ) )
% 0.28/0.46 => ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.28/0.46 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % distrib_imp1
% 0.28/0.46 thf(fact_203_disjnt__def,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( disjnt @ A )
% 0.28/0.46 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.28/0.46 ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
% 0.28/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % disjnt_def
% 0.28/0.46 thf(fact_204_lfilter__eq__LConsD,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( ( coinductive_lfilter @ A @ P @ Ys )
% 0.28/0.46 = ( coinductive_LCons @ A @ X @ Xs ) )
% 0.28/0.46 => ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 0.28/0.46 ( ( Ys
% 0.28/0.46 = ( coinductive_lappend @ A @ Us @ ( coinductive_LCons @ A @ X @ Vs ) ) )
% 0.28/0.46 & ( coinductive_lfinite @ A @ Us )
% 0.28/0.46 & ! [X4: A] :
% 0.28/0.46 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Us ) )
% 0.28/0.46 => ~ ( P @ X4 ) )
% 0.28/0.46 & ( P @ X )
% 0.28/0.46 & ( Xs
% 0.28/0.46 = ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfilter_eq_LConsD
% 0.28/0.46 thf(fact_205_ltakeWhile_Octr_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.28/0.46 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.28/0.46 => ( ( P @ ( coinductive_lhd @ A @ Xs ) )
% 0.28/0.46 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 0.28/0.46 = ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coindu501562517eWhile @ A @ P @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltakeWhile.ctr(2)
% 0.28/0.46 thf(fact_206_llist_Oinject,axiom,
% 0.28/0.46 ! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 0.28/0.46 ( ( ( coinductive_LCons @ A @ X21 @ X22 )
% 0.28/0.46 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 0.28/0.46 = ( ( X21 = Y21 )
% 0.28/0.46 & ( X22 = Y22 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.inject
% 0.28/0.46 thf(fact_207_lfinite__LCons,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.28/0.46 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfinite_LCons
% 0.28/0.46 thf(fact_208_lfinite__code_I2_J,axiom,
% 0.28/0.46 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 0.28/0.46 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
% 0.28/0.46 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfinite_code(2)
% 0.28/0.46 thf(fact_209_lappend__code_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,Xa: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X ) @ Ys )
% 0.28/0.46 = ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lappend_code(2)
% 0.28/0.46 thf(fact_210_lfilter__LCons,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( ( P @ X )
% 0.28/0.46 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.28/0.46 = ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
% 0.28/0.46 & ( ~ ( P @ X )
% 0.28/0.46 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.28/0.46 = ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfilter_LCons
% 0.28/0.46 thf(fact_211_ldropWhile__LCons,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( ( P @ X )
% 0.28/0.46 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.28/0.46 = ( coindu218763757pWhile @ A @ P @ Xs ) ) )
% 0.28/0.46 & ( ~ ( P @ X )
% 0.28/0.46 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.28/0.46 = ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ldropWhile_LCons
% 0.28/0.46 thf(fact_212_lconcat__LCons,axiom,
% 0.28/0.46 ! [B: $tType,Xs: coinductive_llist @ B,Xss: coinductive_llist @ ( coinductive_llist @ B )] :
% 0.28/0.46 ( ( coinductive_lconcat @ B @ ( coinductive_LCons @ ( coinductive_llist @ B ) @ Xs @ Xss ) )
% 0.28/0.46 = ( coinductive_lappend @ B @ Xs @ ( coinductive_lconcat @ B @ Xss ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lconcat_LCons
% 0.28/0.46 thf(fact_213_unfold__llist__eq__LCons,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B3: B,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B3 )
% 0.28/0.46 = ( coinductive_LCons @ A @ X @ Xs ) )
% 0.28/0.46 = ( ~ ( IS_LNIL @ B3 )
% 0.28/0.46 & ( X
% 0.28/0.46 = ( LHD @ B3 ) )
% 0.28/0.46 & ( Xs
% 0.28/0.46 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % unfold_llist_eq_LCons
% 0.28/0.46 thf(fact_214_lstrict__prefix__code_I4_J,axiom,
% 0.28/0.46 ! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
% 0.28/0.46 ( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
% 0.28/0.46 = ( ( X = Y )
% 0.28/0.46 & ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lstrict_prefix_code(4)
% 0.28/0.46 thf(fact_215_ldistinct__LCons,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.28/0.46 = ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 & ( coindu351974385stinct @ A @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ldistinct_LCons
% 0.28/0.46 thf(fact_216_lhd__LCons__ltl,axiom,
% 0.28/0.46 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.28/0.46 ( ~ ( coinductive_lnull @ A @ Llist )
% 0.28/0.46 => ( ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) )
% 0.28/0.46 = Llist ) ) ).
% 0.28/0.46
% 0.28/0.46 % lhd_LCons_ltl
% 0.28/0.46 thf(fact_217_llist_Osimps_I5_J,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,F12: B,F23: A > ( coinductive_llist @ A ) > B,X21: A,X22: coinductive_llist @ A] :
% 0.28/0.46 ( ( coindu1381640503_llist @ B @ A @ F12 @ F23 @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 0.28/0.46 = ( F23 @ X21 @ X22 ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.simps(5)
% 0.28/0.46 thf(fact_218_llist_Oset__induct,axiom,
% 0.28/0.46 ! [A: $tType,X: A,A2: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) )
% 0.28/0.46 => ( ! [Z1: A,Z22: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) )
% 0.28/0.46 => ( ! [Z1: A,Z22: coinductive_llist @ A,Xa2: A] :
% 0.28/0.46 ( ( member @ A @ Xa2 @ ( coinductive_lset @ A @ Z22 ) )
% 0.28/0.46 => ( ( P @ Xa2 @ Z22 )
% 0.28/0.46 => ( P @ Xa2 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) ) ) )
% 0.28/0.46 => ( P @ X @ A2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.set_induct
% 0.28/0.46 thf(fact_219_llist_Oset__cases,axiom,
% 0.28/0.46 ! [A: $tType,E: A,A2: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ A @ E @ ( coinductive_lset @ A @ A2 ) )
% 0.28/0.46 => ( ! [Z22: coinductive_llist @ A] :
% 0.28/0.46 ( A2
% 0.28/0.46 != ( coinductive_LCons @ A @ E @ Z22 ) )
% 0.28/0.46 => ~ ! [Z1: A,Z22: coinductive_llist @ A] :
% 0.28/0.46 ( ( A2
% 0.28/0.46 = ( coinductive_LCons @ A @ Z1 @ Z22 ) )
% 0.28/0.46 => ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z22 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.set_cases
% 0.28/0.46 thf(fact_220_lset__induct_H,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
% 0.28/0.46 => ( ! [X6: A,Xs2: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
% 0.28/0.46 => ( ( P @ Xs2 )
% 0.28/0.46 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) )
% 0.28/0.46 => ( P @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lset_induct'
% 0.28/0.46 thf(fact_221_lset__induct,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
% 0.28/0.46 => ( ! [X6: A,Xs2: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
% 0.28/0.46 => ( ( X != X6 )
% 0.28/0.46 => ( ( P @ Xs2 )
% 0.28/0.46 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) ) )
% 0.28/0.46 => ( P @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lset_induct
% 0.28/0.46 thf(fact_222_lset__cases,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ( ! [Xs4: coinductive_llist @ A] :
% 0.28/0.46 ( Xs
% 0.28/0.46 != ( coinductive_LCons @ A @ X @ Xs4 ) )
% 0.28/0.46 => ~ ! [X6: A,Xs4: coinductive_llist @ A] :
% 0.28/0.46 ( ( Xs
% 0.28/0.46 = ( coinductive_LCons @ A @ X6 @ Xs4 ) )
% 0.28/0.46 => ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs4 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lset_cases
% 0.28/0.46 thf(fact_223_llist_Oset__intros_I1_J,axiom,
% 0.28/0.46 ! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.set_intros(1)
% 0.28/0.46 thf(fact_224_llist_Oset__intros_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
% 0.28/0.46 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.set_intros(2)
% 0.28/0.46 thf(fact_225_lset__intros_I1_J,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lset_intros(1)
% 0.28/0.46 thf(fact_226_lset__intros_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,X7: A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X7 @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lset_intros(2)
% 0.28/0.46 thf(fact_227_lmember__code_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.28/0.46 = ( ( X = Y )
% 0.28/0.46 | ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lmember_code(2)
% 0.28/0.46 thf(fact_228_not__lnull__conv,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( ~ ( coinductive_lnull @ A @ Xs ) )
% 0.28/0.46 = ( ? [X2: A,Xs5: coinductive_llist @ A] :
% 0.28/0.46 ( Xs
% 0.28/0.46 = ( coinductive_LCons @ A @ X2 @ Xs5 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_lnull_conv
% 0.28/0.46 thf(fact_229_llist_OdiscI_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
% 0.28/0.46 ( ( Llist
% 0.28/0.46 = ( coinductive_LCons @ A @ X21 @ X22 ) )
% 0.28/0.46 => ~ ( coinductive_lnull @ A @ Llist ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.discI(2)
% 0.28/0.46 thf(fact_230_llist_Odisc_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.28/0.46 ~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 0.28/0.46
% 0.28/0.46 % llist.disc(2)
% 0.28/0.46 thf(fact_231_ltl__simps_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_ltl @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 0.28/0.46 = X22 ) ).
% 0.28/0.46
% 0.28/0.46 % ltl_simps(2)
% 0.28/0.46 thf(fact_232_lhd__LCons,axiom,
% 0.28/0.46 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lhd @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 0.28/0.46 = X21 ) ).
% 0.28/0.46
% 0.28/0.46 % lhd_LCons
% 0.28/0.46 thf(fact_233_lfilter__LCons__seek,axiom,
% 0.28/0.46 ! [A: $tType,P2: A > $o,X: A,L: coinductive_llist @ A] :
% 0.28/0.46 ( ~ ( P2 @ X )
% 0.28/0.46 => ( ( coinductive_lfilter @ A @ P2 @ ( coinductive_LCons @ A @ X @ L ) )
% 0.28/0.46 = ( coinductive_lfilter @ A @ P2 @ L ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfilter_LCons_seek
% 0.28/0.46 thf(fact_234_lfilter__LCons__found,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( P @ X )
% 0.28/0.46 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.28/0.46 = ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfilter_LCons_found
% 0.28/0.46 thf(fact_235_disjnt__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( disjnt @ A )
% 0.28/0.46 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.28/0.46 ! [X2: A] :
% 0.28/0.46 ~ ( ( member @ A @ X2 @ A5 )
% 0.28/0.46 & ( member @ A @ X2 @ B5 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % disjnt_iff
% 0.28/0.46 thf(fact_236_lsetp_Oinducts,axiom,
% 0.28/0.46 ! [A: $tType,X1: coinductive_llist @ A,X23: A,P: ( coinductive_llist @ A ) > A > $o] :
% 0.28/0.46 ( ( coinductive_lsetp @ A @ X1 @ X23 )
% 0.28/0.46 => ( ! [X3: A,Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X3 @ Xs2 ) @ X3 )
% 0.28/0.46 => ( ! [Xs2: coinductive_llist @ A,X3: A,X6: A] :
% 0.28/0.46 ( ( coinductive_lsetp @ A @ Xs2 @ X3 )
% 0.28/0.46 => ( ( P @ Xs2 @ X3 )
% 0.28/0.46 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) @ X3 ) ) )
% 0.28/0.46 => ( P @ X1 @ X23 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lsetp.inducts
% 0.28/0.46 thf(fact_237_lsetp_Osimps,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( coinductive_lsetp @ A )
% 0.28/0.46 = ( ^ [A12: coinductive_llist @ A,A23: A] :
% 0.28/0.46 ( ? [X2: A,Xs3: coinductive_llist @ A] :
% 0.28/0.46 ( ( A12
% 0.28/0.46 = ( coinductive_LCons @ A @ X2 @ Xs3 ) )
% 0.28/0.46 & ( A23 = X2 ) )
% 0.28/0.46 | ? [Xs3: coinductive_llist @ A,X2: A,X8: A] :
% 0.28/0.46 ( ( A12
% 0.28/0.46 = ( coinductive_LCons @ A @ X8 @ Xs3 ) )
% 0.28/0.46 & ( A23 = X2 )
% 0.28/0.46 & ( coinductive_lsetp @ A @ Xs3 @ X2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lsetp.simps
% 0.28/0.46 thf(fact_238_lsetp_Ocases,axiom,
% 0.28/0.46 ! [A: $tType,A1: coinductive_llist @ A,A22: A] :
% 0.28/0.46 ( ( coinductive_lsetp @ A @ A1 @ A22 )
% 0.28/0.46 => ( ! [X3: A] :
% 0.28/0.46 ( ? [Xs2: coinductive_llist @ A] :
% 0.28/0.46 ( A1
% 0.28/0.46 = ( coinductive_LCons @ A @ X3 @ Xs2 ) )
% 0.28/0.46 => ( A22 != X3 ) )
% 0.28/0.46 => ~ ! [Xs2: coinductive_llist @ A] :
% 0.28/0.46 ( ? [X6: A] :
% 0.28/0.46 ( A1
% 0.28/0.46 = ( coinductive_LCons @ A @ X6 @ Xs2 ) )
% 0.28/0.46 => ~ ( coinductive_lsetp @ A @ Xs2 @ A22 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lsetp.cases
% 0.28/0.46 thf(fact_239_lsetp_Ointros_I1_J,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( coinductive_lsetp @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ X ) ).
% 0.28/0.46
% 0.28/0.46 % lsetp.intros(1)
% 0.28/0.46 thf(fact_240_lsetp_Ointros_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,X: A,X7: A] :
% 0.28/0.46 ( ( coinductive_lsetp @ A @ Xs @ X )
% 0.28/0.46 => ( coinductive_lsetp @ A @ ( coinductive_LCons @ A @ X7 @ Xs ) @ X ) ) ).
% 0.28/0.46
% 0.28/0.46 % lsetp.intros(2)
% 0.28/0.46 thf(fact_241_lfinite__LConsI,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 0.28/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfinite_LConsI
% 0.28/0.46 thf(fact_242_unfold__llist_Octr_I2_J,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 0.28/0.46 ( ~ ( P2 @ A2 )
% 0.28/0.46 => ( ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
% 0.28/0.46 = ( coinductive_LCons @ B @ ( G21 @ A2 ) @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % unfold_llist.ctr(2)
% 0.28/0.46 thf(fact_243_ldistinct_OLCons,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ( ( coindu351974385stinct @ A @ Xs )
% 0.28/0.46 => ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ldistinct.LCons
% 0.28/0.46 thf(fact_244_split__llist,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ? [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 0.28/0.46 ( ( Xs
% 0.28/0.46 = ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
% 0.28/0.46 & ( coinductive_lfinite @ A @ Ys3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % split_llist
% 0.28/0.46 thf(fact_245_split__llist__first,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.28/0.46 => ? [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 0.28/0.46 ( ( Xs
% 0.28/0.46 = ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
% 0.28/0.46 & ( coinductive_lfinite @ A @ Ys3 )
% 0.28/0.46 & ~ ( member @ A @ X @ ( coinductive_lset @ A @ Ys3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % split_llist_first
% 0.28/0.46 thf(fact_246_llast__lappend__LCons,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.28/0.46 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
% 0.28/0.46 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % llast_lappend_LCons
% 0.28/0.46 thf(fact_247_ltakeWhile_Ocode,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( coindu501562517eWhile @ A )
% 0.28/0.46 = ( ^ [P3: A > $o,Xs3: coinductive_llist @ A] :
% 0.28/0.46 ( if @ ( coinductive_llist @ A )
% 0.28/0.46 @ ( ( coinductive_lnull @ A @ Xs3 )
% 0.28/0.46 | ~ ( P3 @ ( coinductive_lhd @ A @ Xs3 ) ) )
% 0.28/0.46 @ ( coinductive_LNil @ A )
% 0.28/0.46 @ ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs3 ) @ ( coindu501562517eWhile @ A @ P3 @ ( coinductive_ltl @ A @ Xs3 ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltakeWhile.code
% 0.28/0.46 thf(fact_248_lfinite__code_I1_J,axiom,
% 0.28/0.46 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfinite_code(1)
% 0.28/0.46 thf(fact_249_lappend__LNil2,axiom,
% 0.28/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
% 0.28/0.46 = Xs ) ).
% 0.28/0.46
% 0.28/0.46 % lappend_LNil2
% 0.28/0.46 thf(fact_250_lappend__code_I1_J,axiom,
% 0.28/0.46 ! [A: $tType,Ys: coinductive_llist @ A] :
% 0.28/0.46 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
% 0.28/0.46 = Ys ) ).
% 0.28/0.46
% 0.28/0.46 % lappend_code(1)
% 0.28/0.46 thf(fact_251_lfilter__LNil,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o] :
% 0.28/0.46 ( ( coinductive_lfilter @ A @ P @ ( coinductive_LNil @ A ) )
% 0.28/0.46 = ( coinductive_LNil @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % lfilter_LNil
% 0.28/0.46 thf(fact_252_ltakeWhile__LNil,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o] :
% 0.28/0.46 ( ( coindu501562517eWhile @ A @ P @ ( coinductive_LNil @ A ) )
% 0.28/0.46 = ( coinductive_LNil @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % ltakeWhile_LNil
% 0.28/0.46 thf(fact_253_lconcat__LNil,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( coinductive_lconcat @ A @ ( coinductive_LNil @ ( coinductive_llist @ A ) ) )
% 0.28/0.46 = ( coinductive_LNil @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % lconcat_LNil
% 0.28/0.46
% 0.28/0.46 %----Type constructors (24)
% 0.28/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.28/0.46 bounded_lattice @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.28/0.46 ! [A6: $tType] : ( bounded_lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.28/0.46 ! [A6: $tType,A7: $tType] :
% 0.28/0.46 ( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
% 0.28/0.46 => ( bounded_lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 0.28/0.46 ! [A6: $tType,A7: $tType] :
% 0.28/0.46 ( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
% 0.28/0.46 => ( bounde1808546759up_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 0.28/0.46 ! [A6: $tType,A7: $tType] :
% 0.28/0.46 ( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
% 0.28/0.46 => ( bounded_lattice_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.28/0.46 ! [A6: $tType,A7: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A7 @ ( type2 @ A7 ) )
% 0.28/0.46 => ( semilattice_sup @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.28/0.46 ! [A6: $tType,A7: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A7 @ ( type2 @ A7 ) )
% 0.28/0.46 => ( semilattice_inf @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
% 0.28/0.46 ! [A6: $tType,A7: $tType] :
% 0.28/0.46 ( ( distrib_lattice @ A7 @ ( type2 @ A7 ) )
% 0.28/0.46 => ( distrib_lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.28/0.46 ! [A6: $tType,A7: $tType] :
% 0.28/0.46 ( ( lattice @ A7 @ ( type2 @ A7 ) )
% 0.28/0.46 => ( lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Orderings_Obot,axiom,
% 0.28/0.46 ! [A6: $tType,A7: $tType] :
% 0.28/0.46 ( ( bot @ A7 @ ( type2 @ A7 ) )
% 0.28/0.46 => ( bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3,axiom,
% 0.28/0.46 ! [A6: $tType] : ( bounde1808546759up_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4,axiom,
% 0.28/0.46 ! [A6: $tType] : ( bounded_lattice_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5,axiom,
% 0.28/0.46 ! [A6: $tType] : ( semilattice_sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_6,axiom,
% 0.28/0.46 ! [A6: $tType] : ( semilattice_inf @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_7,axiom,
% 0.28/0.46 ! [A6: $tType] : ( distrib_lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Lattices_Olattice_8,axiom,
% 0.28/0.46 ! [A6: $tType] : ( lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
% 0.28/0.46 ! [A6: $tType] : ( bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_10,axiom,
% 0.28/0.46 bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_11,axiom,
% 0.28/0.46 bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_12,axiom,
% 0.28/0.46 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_13,axiom,
% 0.28/0.46 semilattice_inf @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_14,axiom,
% 0.28/0.46 distrib_lattice @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Lattices_Olattice_15,axiom,
% 0.28/0.46 lattice @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
% 0.28/0.46 bot @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 %----Helper facts (3)
% 0.28/0.46 thf(help_If_3_1_T,axiom,
% 0.28/0.46 ! [P: $o] :
% 0.28/0.46 ( ( P = $true )
% 0.28/0.46 | ( P = $false ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(help_If_2_1_T,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Y: A] :
% 0.28/0.46 ( ( if @ A @ $false @ X @ Y )
% 0.28/0.46 = Y ) ).
% 0.28/0.46
% 0.28/0.46 thf(help_If_1_1_T,axiom,
% 0.28/0.46 ! [A: $tType,X: A,Y: A] :
% 0.28/0.46 ( ( if @ A @ $true @ X @ Y )
% 0.28/0.46 = X ) ).
% 0.28/0.46
% 0.28/0.46 %----Conjectures (1)
% 0.28/0.46 thf(conj_0,conjecture,
% 0.28/0.46 member @ a @ ( shd @ a @ xsa ) @ ( coinductive_lset @ a @ ( coindu1724414836stream @ a @ xsa ) ) ).
% 0.28/0.46
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 ------------------------
% 2.80/1.51 ---- Embedded in THF ---
% 2.80/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.80/1.52 %%% Generated on Tue Feb 24 06:08:49 EST 2026
% 2.80/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.80/1.52 %%% Logic specification used:
% 2.80/1.52 %%% thf(spec, logic, $$dhol).
% 2.80/1.52
% 2.80/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.61YHUtgMPz/DTF2DTF26799.p
% 2.80/1.52 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 2.80/1.52 thf(ty_t_Stream_Ostream, type, stream: ($tType > $tType)).
% 2.80/1.52 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.80/1.52 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.80/1.52 thf(ty_tf_a, type, a: $tType).
% 2.80/1.52 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.80/1.52 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.80/1.52 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.80/1.52 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.80/1.52 thf(sy_cl_Lattices_Odistrib__lattice, type, distrib_lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.80/1.52 thf(sy_cl_Lattices_Osemilattice__inf, type, semilattice_inf: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.80/1.52 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.80/1.52 thf(sy_cl_Lattices_Obounded__lattice__bot, type, bounded_lattice_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.80/1.52 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot, type, bounde1808546759up_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Ogen__lset, type, coinductive_gen_lset: (!> [A:$tType]: (((set @ A) > ((coinductive_llist @ A) > (set @ A)))))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Olappend, type, coinductive_lappend: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Olconcat, type, coinductive_lconcat: (!> [A:$tType]: (((coinductive_llist @ (coinductive_llist @ A)) > (coinductive_llist @ A))))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Oldistinct, type, coindu351974385stinct: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.80/1.52 thf(sy_c_Coinductive__List_OldropWhile, type, coindu218763757pWhile: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Olfilter, type, coinductive_lfilter: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Olfinite, type, coinductive_lfinite: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Ollast, type, coinductive_llast: (!> [A:$tType]: (((coinductive_llist @ A) > A)))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Ollist_OLCons, type, coinductive_LCons: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Ollist_OLNil, type, coinductive_LNil: (!> [A:$tType]: ((coinductive_llist @ A)))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Ollist_Ocase__llist, type, coindu1381640503_llist: (!> [B:$tType,A:$tType]: ((B > ((A > ((coinductive_llist @ A) > B)) > ((coinductive_llist @ A) > B)))))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Ollist_Olhd, type, coinductive_lhd: (!> [A:$tType]: (((coinductive_llist @ A) > A)))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Ollist_Olnull, type, coinductive_lnull: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Ollist_Olset, type, coinductive_lset: (!> [A:$tType]: (((coinductive_llist @ A) > (set @ A))))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Ollist_Oltl, type, coinductive_ltl: (!> [A:$tType]: (((coinductive_llist @ A) > (coinductive_llist @ A))))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Olmember, type, coinductive_lmember: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > $o))))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Olsetp, type, coinductive_lsetp: (!> [A:$tType]: (((coinductive_llist @ A) > (A > $o))))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Olstrict__prefix, type, coindu1478340336prefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 2.80/1.52 thf(sy_c_Coinductive__List_OltakeWhile, type, coindu501562517eWhile: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.80/1.52 thf(sy_c_Coinductive__List_Ounfold__llist, type, coindu1441602521_llist: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > B) > ((A > A) > (A > (coinductive_llist @ B)))))))).
% 2.80/1.52 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ollist__of__stream, type, coindu1724414836stream: (!> [A:$tType]: (((stream @ A) > (coinductive_llist @ A))))).
% 2.80/1.52 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Ocr__stream, type, coindu1183105481stream: (!> [A:$tType]: (((coinductive_llist @ A) > ((stream @ A) > $o))))).
% 2.80/1.52 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__of__llist, type, coindu2010755910_llist: (!> [A:$tType]: (((coinductive_llist @ A) > (stream @ A))))).
% 2.80/1.52 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ounfold__stream, type, coindu139217191stream: (!> [A:$tType,B:$tType]: (((A > B) > ((A > A) > (A > (stream @ B))))))).
% 2.80/1.52 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.80/1.52 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 2.80/1.52 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.80/1.52 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.80/1.52 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.80/1.52 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.80/1.52 thf(sy_c_Set_Odisjnt, type, disjnt: (!> [A:$tType]: (((set @ A) > ((set @ A) > $o))))).
% 2.80/1.52 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.80/1.52 thf(sy_c_Stream_Osinterleave, type, sinterleave: (!> [A:$tType]: (((stream @ A) > ((stream @ A) > (stream @ A)))))).
% 2.80/1.52 thf(sy_c_Stream_Osmember, type, smember: (!> [A:$tType]: ((A > ((stream @ A) > $o))))).
% 2.80/1.52 thf(sy_c_Stream_Ostream_Oshd, type, shd: (!> [A:$tType]: (((stream @ A) > A)))).
% 2.80/1.52 thf(sy_c_Stream_Ostream_Osset, type, sset: (!> [A:$tType]: (((stream @ A) > (set @ A))))).
% 2.80/1.52 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.80/1.52 thf(sy_v_x____, type, x: a).
% 2.80/1.52 thf(sy_v_xs, type, xs: (stream @ a)).
% 2.80/1.52 thf(sy_v_xsa____, type, xsa: (stream @ a)).
% 2.80/1.52 thf(fact_0__092_060open_062_092_060not_062_Alnull_A_Illist__of__stream_Axs_J_A_092_060Longrightarrow_062_Alhd_A_Illist__of__stream_Axs_J_A_092_060in_062_Alset_A_Illist__of__stream_Axs_J_092_060close_062, axiom, ((~ (((coinductive_lnull @ a) @ ((coindu1724414836stream @ a) @ xsa)))) => (((member @ a) @ ((coinductive_lhd @ a) @ ((coindu1724414836stream @ a) @ xsa))) @ ((coinductive_lset @ a) @ ((coindu1724414836stream @ a) @ xsa))))).
% 2.80/1.52 thf(fact_1_stream_ORep__inject, axiom, (! [A:$tType,X:(stream @ A),Y:(stream @ A)]: (((((coindu1724414836stream @ A) @ X) = ((coindu1724414836stream @ A) @ Y)) = (X = Y))))).
% 2.80/1.52 thf(fact_2_lhd__llist__of__stream, axiom, (! [A:$tType,Xs:(stream @ A)]: ((((coinductive_lhd @ A) @ ((coindu1724414836stream @ A) @ Xs)) = ((shd @ A) @ Xs))))).
% 2.80/1.52 thf(fact_3_lnull__llist__of__stream, axiom, (! [A:$tType,Xs:(stream @ A)]: ((~ (((coinductive_lnull @ A) @ ((coindu1724414836stream @ A) @ Xs))))))).
% 2.80/1.52 thf(fact_4_stream_ORep__inverse, axiom, (! [A:$tType,X:(stream @ A)]: ((((coindu2010755910_llist @ A) @ ((coindu1724414836stream @ A) @ X)) = X)))).
% 2.80/1.52 thf(fact_5_stream__of__llist__llist__of__stream, axiom, (! [A:$tType,Xs:(stream @ A)]: ((((coindu2010755910_llist @ A) @ ((coindu1724414836stream @ A) @ Xs)) = Xs)))).
% 2.80/1.52 thf(fact_6_unfold__stream_Osimps_I1_J, axiom, (! [B:$tType,A:$tType,G1:(A > B),G2:(A > A),A2:A]: ((((shd @ B) @ (((((coindu139217191stream @ A) @ B) @ G1) @ G2) @ A2)) = (G1 @ A2))))).
% 2.80/1.52 thf(fact_7_shd__stream__of__llist, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((shd @ A) @ ((coindu2010755910_llist @ A) @ Xs)) = ((coinductive_lhd @ A) @ Xs))))).
% 2.80/1.52 thf(fact_8_lset__lmember, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) = (((coinductive_lmember @ A) @ X) @ Xs))))).
% 2.80/1.52 thf(fact_9_lfinite__llist__of__stream, axiom, (! [A:$tType,Xs:(stream @ A)]: ((~ (((coinductive_lfinite @ A) @ ((coindu1724414836stream @ A) @ Xs))))))).
% 2.80/1.52 thf(fact_10__092_060open_062x_A_092_060in_062_Asset_Axs_092_060close_062, axiom, (((member @ a) @ x) @ ((sset @ a) @ xs))).
% 2.80/1.52 thf(fact_11_llist_Oset__sel_I1_J, axiom, (! [A:$tType,A2:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ A2))) => (((member @ A) @ ((coinductive_lhd @ A) @ A2)) @ ((coinductive_lset @ A) @ A2)))))).
% 2.80/1.52 thf(fact_12_sinterleave_Osimps_I1_J, axiom, (! [A:$tType,S1:(stream @ A),S2:(stream @ A)]: ((((shd @ A) @ (((sinterleave @ A) @ S1) @ S2)) = ((shd @ A) @ S1))))).
% 2.80/1.52 thf(fact_13_Coinductive__List_Olset__into__lsetp, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (((coinductive_lsetp @ A) @ Xs) @ X))))).
% 2.80/1.52 thf(fact_14_llist__of__stream__stream__of__llist, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => (((coindu1724414836stream @ A) @ ((coindu2010755910_llist @ A) @ Xs)) = Xs))))).
% 2.80/1.52 thf(fact_15_lzip_Oexhaust, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((~ ((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ B) @ Ys)))) => (~ (((~ (((coinductive_lnull @ A) @ Xs))) => ((coinductive_lnull @ B) @ Ys)))))))).
% 2.80/1.52 thf(fact_16_lappend_Oexhaust, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Xs) => (~ (((coinductive_lnull @ A) @ Ys)))) => ((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Ys)))))))).
% 2.80/1.52 thf(fact_17_lnull__imp__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((coinductive_lfinite @ A) @ Xs))))).
% 2.80/1.52 thf(fact_18_ltakeWhile_Oexhaust, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((~ ((((coinductive_lnull @ A) @ Xs) | (~ ((P @ ((coinductive_lhd @ A) @ Xs))))))) => (~ (((~ (((coinductive_lnull @ A) @ Xs))) => (~ ((P @ ((coinductive_lhd @ A) @ Xs))))))))))).
% 2.80/1.52 thf(fact_19_shd__sset, axiom, (! [A:$tType,A2:(stream @ A)]: ((((member @ A) @ ((shd @ A) @ A2)) @ ((sset @ A) @ A2))))).
% 2.80/1.52 thf(fact_20_lsetp__into__lset, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),X:A]: (((((coinductive_lsetp @ A) @ Xs) @ X) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)))))).
% 2.80/1.52 thf(fact_21_cr__streamI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => (((coindu1183105481stream @ A) @ Xs) @ ((coindu2010755910_llist @ A) @ Xs)))))).
% 2.80/1.52 thf(fact_22_Stream_Osmember__def, axiom, (! [A:$tType]: (((smember @ A) = (^ [X2:A,S:(stream @ A)]: ((((member @ A) @ X2) @ ((sset @ A) @ S)))))))).
% 2.80/1.52 thf(fact_23_llist__set__induct, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),P:(A > ((coinductive_llist @ A) > $o))]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs2:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ Xs2))) => ((P @ ((coinductive_lhd @ A) @ Xs2)) @ Xs2)))) => ((! [Xs2:(coinductive_llist @ A),Y2:A]: (((~ (((coinductive_lnull @ A) @ Xs2))) => ((((member @ A) @ Y2) @ ((coinductive_lset @ A) @ ((coinductive_ltl @ A) @ Xs2))) => (((P @ Y2) @ ((coinductive_ltl @ A) @ Xs2)) => ((P @ Y2) @ Xs2)))))) => ((P @ X) @ Xs))))))).
% 2.80/1.52 thf(fact_24_lset__eq__empty, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_lset @ A) @ Xs) = (bot_bot @ (set @ A))) = ((coinductive_lnull @ A) @ Xs))))).
% 2.80/1.52 thf(fact_25_lnull__ldropWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coindu218763757pWhile @ A) @ P) @ Xs)) = (! [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => (P @ X2)))))))).
% 2.80/1.52 thf(fact_26_lhd__lconcat, axiom, (! [A:$tType,Xss:(coinductive_llist @ (coinductive_llist @ A))]: (((~ (((coinductive_lnull @ (coinductive_llist @ A)) @ Xss))) => ((~ (((coinductive_lnull @ A) @ ((coinductive_lhd @ (coinductive_llist @ A)) @ Xss)))) => (((coinductive_lhd @ A) @ ((coinductive_lconcat @ A) @ Xss)) = ((coinductive_lhd @ A) @ ((coinductive_lhd @ (coinductive_llist @ A)) @ Xss)))))))).
% 2.80/1.52 thf(fact_27_sset__sinterleave, axiom, (! [A:$tType,S1:(stream @ A),S2:(stream @ A)]: ((((sset @ A) @ (((sinterleave @ A) @ S1) @ S2)) = (((sup_sup @ (set @ A)) @ ((sset @ A) @ S1)) @ ((sset @ A) @ S2)))))).
% 2.80/1.52 thf(fact_28_ltakeWhile_Odisc__iff_I2_J, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)))) = ((~ (((coinductive_lnull @ A) @ Xs))) & (P @ ((coinductive_lhd @ A) @ Xs))))))).
% 2.80/1.52 thf(fact_29_ltakeWhile_Odisc__iff_I1_J, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) = (((coinductive_lnull @ A) @ Xs) | (~ ((P @ ((coinductive_lhd @ A) @ Xs))))))))).
% 2.80/1.52 thf(fact_30_lnull__ltakeWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) = ((~ (((coinductive_lnull @ A) @ Xs))) => (~ ((P @ ((coinductive_lhd @ A) @ Xs))))))))).
% 2.80/1.52 thf(fact_31_lnull__lfilter, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)) = (! [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => (~ ((P @ X2)))))))))).
% 2.80/1.52 thf(fact_32_lfilter__idem, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((coinductive_lfilter @ A) @ P) @ (((coinductive_lfilter @ A) @ P) @ Xs)) = (((coinductive_lfilter @ A) @ P) @ Xs))))).
% 2.80/1.52 thf(fact_33_lfinite__ltl, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ ((coinductive_ltl @ A) @ Xs)) = ((coinductive_lfinite @ A) @ Xs))))).
% 2.80/1.52 thf(fact_34_ltakeWhile_Osimps_I4_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((P @ ((coinductive_lhd @ A) @ Xs)) => (((coinductive_ltl @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) = (((coindu501562517eWhile @ A) @ P) @ ((coinductive_ltl @ A) @ Xs)))))))).
% 2.80/1.52 thf(fact_35_lnull__ltlI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((coinductive_lnull @ A) @ ((coinductive_ltl @ A) @ Xs)))))).
% 2.80/1.52 thf(fact_36_in__lset__ltlD, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ ((coinductive_ltl @ A) @ Xs))) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)))))).
% 2.80/1.52 thf(fact_37_lfilter__cong, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Ys)) => ((P @ X3) = (Q @ X3))))) => ((((coinductive_lfilter @ A) @ P) @ Xs) = (((coinductive_lfilter @ A) @ Q) @ Ys))))))).
% 2.80/1.52 thf(fact_38_lfilter__id__conv, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Xs) = Xs) = (! [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => (P @ X2)))))))).
% 2.80/1.52 thf(fact_39_lfinite__lfilterI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: ((((coinductive_lfinite @ A) @ Xs) => ((coinductive_lfinite @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)))))).
% 2.80/1.52 thf(fact_40_ltakeWhile__all, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => (P @ X3)))) => ((((coindu501562517eWhile @ A) @ P) @ Xs) = Xs))))).
% 2.80/1.52 thf(fact_41_ltakeWhile__cong, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Ys)) => ((P @ X3) = (Q @ X3))))) => ((((coindu501562517eWhile @ A) @ P) @ Xs) = (((coindu501562517eWhile @ A) @ Q) @ Ys))))))).
% 2.80/1.52 thf(fact_42_lset__ltakeWhileD, axiom, (! [A:$tType,X:A,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs))) => ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) & (P @ X)))))).
% 2.80/1.52 thf(fact_43_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.80/1.52 thf(fact_44_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A3)))) = A3)))).
% 2.80/1.52 thf(fact_45_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.80/1.52 thf(fact_46_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.80/1.52 thf(fact_47_ldropWhile__cong, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Ys)) => ((P @ X3) = (Q @ X3))))) => ((((coindu218763757pWhile @ A) @ P) @ Xs) = (((coindu218763757pWhile @ A) @ Q) @ Ys))))))).
% 2.80/1.52 thf(fact_48_in__lset__ldropWhileD, axiom, (! [A:$tType,X:A,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coindu218763757pWhile @ A) @ P) @ Xs))) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)))))).
% 2.80/1.52 thf(fact_49_sset__neq__empty, axiom, (! [A:$tType,Xs:(stream @ A)]: ((((sset @ A) @ Xs) != (bot_bot @ (set @ A)))))).
% 2.80/1.52 thf(fact_50_llist_Oset__sel_I2_J, axiom, (! [A:$tType,A2:(coinductive_llist @ A),X:A]: (((~ (((coinductive_lnull @ A) @ A2))) => ((((member @ A) @ X) @ ((coinductive_lset @ A) @ ((coinductive_ltl @ A) @ A2))) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ A2))))))).
% 2.80/1.52 thf(fact_51_lfinite__induct, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: ((((coinductive_lfinite @ A) @ Xs) => ((! [Xs2:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs2) => (P @ Xs2)))) => ((! [Xs2:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs2) => ((~ (((coinductive_lnull @ A) @ Xs2))) => ((P @ ((coinductive_ltl @ A) @ Xs2)) => (P @ Xs2)))))) => (P @ Xs))))))).
% 2.80/1.52 thf(fact_52_llist_Ocoinduct__strong, axiom, (! [A:$tType,R:((coinductive_llist @ A) > ((coinductive_llist @ A) > $o)),Llist:(coinductive_llist @ A),Llist2:(coinductive_llist @ A)]: ((((R @ Llist) @ Llist2) => ((! [Llist3:(coinductive_llist @ A),Llist4:(coinductive_llist @ A)]: ((((R @ Llist3) @ Llist4) => ((((coinductive_lnull @ A) @ Llist3) = ((coinductive_lnull @ A) @ Llist4)) & ((~ (((coinductive_lnull @ A) @ Llist3))) => ((~ (((coinductive_lnull @ A) @ Llist4))) => ((((coinductive_lhd @ A) @ Llist3) = ((coinductive_lhd @ A) @ Llist4)) & (((R @ ((coinductive_ltl @ A) @ Llist3)) @ ((coinductive_ltl @ A) @ Llist4)) | (((coinductive_ltl @ A) @ Llist3) = ((coinductive_ltl @ A) @ Llist4)))))))))) => (Llist = Llist2)))))).
% 2.80/1.52 thf(fact_53_llist_Ocoinduct, axiom, (! [A:$tType,R:((coinductive_llist @ A) > ((coinductive_llist @ A) > $o)),Llist:(coinductive_llist @ A),Llist2:(coinductive_llist @ A)]: ((((R @ Llist) @ Llist2) => ((! [Llist3:(coinductive_llist @ A),Llist4:(coinductive_llist @ A)]: ((((R @ Llist3) @ Llist4) => ((((coinductive_lnull @ A) @ Llist3) = ((coinductive_lnull @ A) @ Llist4)) & ((~ (((coinductive_lnull @ A) @ Llist3))) => ((~ (((coinductive_lnull @ A) @ Llist4))) => ((((coinductive_lhd @ A) @ Llist3) = ((coinductive_lhd @ A) @ Llist4)) & ((R @ ((coinductive_ltl @ A) @ Llist3)) @ ((coinductive_ltl @ A) @ Llist4))))))))) => (Llist = Llist2)))))).
% 2.80/1.52 thf(fact_54_llist_Oexpand, axiom, (! [A:$tType,Llist:(coinductive_llist @ A),Llist2:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Llist) = ((coinductive_lnull @ A) @ Llist2)) => (((~ (((coinductive_lnull @ A) @ Llist))) => ((~ (((coinductive_lnull @ A) @ Llist2))) => ((((coinductive_lhd @ A) @ Llist) = ((coinductive_lhd @ A) @ Llist2)) & (((coinductive_ltl @ A) @ Llist) = ((coinductive_ltl @ A) @ Llist2))))) => (Llist = Llist2)))))).
% 2.80/1.52 thf(fact_55_lhd__ltakeWhile, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((P @ ((coinductive_lhd @ A) @ Xs)) => (((coinductive_lhd @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) = ((coinductive_lhd @ A) @ Xs))))))).
% 2.80/1.52 thf(fact_56_ltakeWhile_Odisc_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((((coinductive_lnull @ A) @ Xs) | (~ ((P @ ((coinductive_lhd @ A) @ Xs))))) => ((coinductive_lnull @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)))))).
% 2.80/1.52 thf(fact_57_ltakeWhile_Odisc_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((P @ ((coinductive_lhd @ A) @ Xs)) => (~ (((coinductive_lnull @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs))))))))).
% 2.80/1.52 thf(fact_58_lfinite__ltakeWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) = (((coinductive_lfinite @ A) @ Xs) | (? [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P @ X2))))))))))).
% 2.80/1.52 thf(fact_59_stream__of__llist__def, axiom, (! [A:$tType]: (((coindu2010755910_llist @ A) = ((((coindu139217191stream @ (coinductive_llist @ A)) @ A) @ (coinductive_lhd @ A)) @ (coinductive_ltl @ A)))))).
% 2.80/1.52 thf(fact_60_lset__lnull, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => (((coinductive_lset @ A) @ Xs) = (bot_bot @ (set @ A))))))).
% 2.80/1.52 thf(fact_61_lfinite__ldropWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coindu218763757pWhile @ A) @ P) @ Xs)) = ((? [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P @ X2)))))) => ((coinductive_lfinite @ A) @ Xs)))))).
% 2.80/1.52 thf(fact_62_lhd__ldropWhile, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P @ X4)))))) => (~ ((P @ ((coinductive_lhd @ A) @ (((coindu218763757pWhile @ A) @ P) @ Xs))))))))).
% 2.80/1.52 thf(fact_63_lhd__ldropWhile__in__lset, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P @ X4)))))) => (((member @ A) @ ((coinductive_lhd @ A) @ (((coindu218763757pWhile @ A) @ P) @ Xs))) @ ((coinductive_lset @ A) @ Xs)))))).
% 2.80/1.52 thf(fact_64_Un__empty, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: ((((((sup_sup @ (set @ A)) @ A3) @ B2) = (bot_bot @ (set @ A))) = ((A3 = (bot_bot @ (set @ A))) & (B2 = (bot_bot @ (set @ A)))))))).
% 2.80/1.52 thf(fact_65_sup__bot_Oright__neutral, axiom, (! [A:$tType]: ((((bounde1808546759up_bot @ A) @ (type2 @ A)) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ (bot_bot @ A)) = A2))))))).
% 2.80/1.52 thf(fact_66_sup__bot_Oleft__neutral, axiom, (! [A:$tType]: ((((bounde1808546759up_bot @ A) @ (type2 @ A)) => (! [A2:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ A2) = A2))))))).
% 2.80/1.52 thf(fact_67_sup__eq__bot__iff, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((((sup_sup @ A) @ X) @ Y) = (bot_bot @ A)) = ((X = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 2.80/1.52 thf(fact_68_bot__eq__sup__iff, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ X) @ Y)) = ((X = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 2.80/1.52 thf(fact_69_Un__iff, axiom, (! [A:$tType,C:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A3) @ B2)) = ((((member @ A) @ C) @ A3) | (((member @ A) @ C) @ B2)))))).
% 2.80/1.52 thf(fact_70_UnCI, axiom, (! [A:$tType,C:A,B2:(set @ A),A3:(set @ A)]: ((((~ ((((member @ A) @ C) @ B2))) => (((member @ A) @ C) @ A3)) => (((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A3) @ B2)))))).
% 2.80/1.52 thf(fact_71_sup_Oright__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B3)) @ B3) = (((sup_sup @ A) @ A2) @ B3)))))))).
% 2.80/1.52 thf(fact_72_sup__left__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 2.80/1.52 thf(fact_73_empty__iff, axiom, (! [A:$tType,C:A]: ((~ ((((member @ A) @ C) @ (bot_bot @ (set @ A)))))))).
% 2.80/1.52 thf(fact_74_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X2:A]: ((~ ((((member @ A) @ X2) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.80/1.52 thf(fact_75_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.80/1.52 thf(fact_76_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.80/1.52 thf(fact_77_sup__apply, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G3:(A > B),X2:A]: ((((sup_sup @ B) @ (F2 @ X2)) @ (G3 @ X2))))))))).
% 2.80/1.52 thf(fact_78_sup_Oidem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ A2) = A2))))))).
% 2.80/1.52 thf(fact_79_sup__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ X) @ X) = X))))))).
% 2.80/1.52 thf(fact_80_sup_Oleft__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B3)) = (((sup_sup @ A) @ A2) @ B3)))))))).
% 2.80/1.52 thf(fact_81_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.80/1.52 thf(fact_82_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.80/1.52 thf(fact_83_equals0D, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A3))))))).
% 2.80/1.52 thf(fact_84_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y2:A]: ((~ ((((member @ A) @ Y2) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))).
% 2.80/1.52 thf(fact_85_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X2:A]: ((((member @ A) @ X2) @ A3))) = (A3 != (bot_bot @ (set @ A))))))).
% 2.80/1.52 thf(fact_86_inf__sup__aci_I8_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 2.80/1.52 thf(fact_87_inf__sup__aci_I7_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.80/1.52 thf(fact_88_inf__sup__aci_I6_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 2.80/1.52 thf(fact_89_inf__sup__aci_I5_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X2:A,Y3:A]: ((((sup_sup @ A) @ Y3) @ X2)))))))).
% 2.80/1.52 thf(fact_90_sup__fun__def, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G3:(A > B),X2:A]: ((((sup_sup @ B) @ (F2 @ X2)) @ (G3 @ X2))))))))).
% 2.80/1.52 thf(fact_91_sup_Oassoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B3)) @ C) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B3) @ C))))))))).
% 2.80/1.52 thf(fact_92_sup__assoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 2.80/1.52 thf(fact_93_sup_Ocommute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [A4:A,B4:A]: ((((sup_sup @ A) @ B4) @ A4)))))))).
% 2.80/1.52 thf(fact_94_sup__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X2:A,Y3:A]: ((((sup_sup @ A) @ Y3) @ X2)))))))).
% 2.80/1.52 thf(fact_95_sup_Oleft__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,A2:A,C:A]: (((((sup_sup @ A) @ B3) @ (((sup_sup @ A) @ A2) @ C)) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B3) @ C))))))))).
% 2.80/1.52 thf(fact_96_sup__left__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.80/1.52 thf(fact_97_UnE, axiom, (! [A:$tType,C:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A3) @ B2)) => ((~ ((((member @ A) @ C) @ A3))) => (((member @ A) @ C) @ B2)))))).
% 2.80/1.52 thf(fact_98_UnI1, axiom, (! [A:$tType,C:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A3) @ B2)))))).
% 2.80/1.52 thf(fact_99_UnI2, axiom, (! [A:$tType,C:A,B2:(set @ A),A3:(set @ A)]: (((((member @ A) @ C) @ B2) => (((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A3) @ B2)))))).
% 2.80/1.52 thf(fact_100_bex__Un, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A3) @ B2)) & (P @ X2)))) = ((? [X2:A]: (((((member @ A) @ X2) @ A3) & (P @ X2)))) | (? [X2:A]: (((((member @ A) @ X2) @ B2) & (P @ X2))))))))).
% 2.80/1.52 thf(fact_101_ball__Un, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A3) @ B2)) => (P @ X2)))) = ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (P @ X2)))) & (! [X2:A]: (((((member @ A) @ X2) @ B2) => (P @ X2))))))))).
% 2.80/1.52 thf(fact_102_Un__assoc, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B2)) @ C2) = (((sup_sup @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ B2) @ C2)))))).
% 2.80/1.52 thf(fact_103_Un__absorb, axiom, (! [A:$tType,A3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ A3) = A3)))).
% 2.80/1.52 thf(fact_104_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((((sup_sup @ (set @ A)) @ B5) @ A5))))))).
% 2.80/1.52 thf(fact_105_Un__left__absorb, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ A3) @ B2)) = (((sup_sup @ (set @ A)) @ A3) @ B2))))).
% 2.80/1.52 thf(fact_106_Un__left__commute, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ B2) @ C2)) = (((sup_sup @ (set @ A)) @ B2) @ (((sup_sup @ (set @ A)) @ A3) @ C2)))))).
% 2.80/1.52 thf(fact_107_sup__bot__left, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ X) = X))))))).
% 2.80/1.52 thf(fact_108_sup__bot__right, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ X) @ (bot_bot @ A)) = X))))))).
% 2.80/1.52 thf(fact_109_Un__empty__left, axiom, (! [A:$tType,B2:(set @ A)]: (((((sup_sup @ (set @ A)) @ (bot_bot @ (set @ A))) @ B2) = B2)))).
% 2.80/1.52 thf(fact_110_Un__empty__right, axiom, (! [A:$tType,A3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = A3)))).
% 2.80/1.52 thf(fact_111_bot__apply, axiom, (! [C3:$tType,D:$tType]: ((((bot @ C3) @ (type2 @ C3)) => ((bot_bot @ (D > C3)) = (^ [X2:D]: ((bot_bot @ C3)))))))).
% 2.80/1.52 thf(fact_112_ltl__lconcat, axiom, (! [A:$tType,Xss:(coinductive_llist @ (coinductive_llist @ A))]: (((~ (((coinductive_lnull @ (coinductive_llist @ A)) @ Xss))) => ((~ (((coinductive_lnull @ A) @ ((coinductive_lhd @ (coinductive_llist @ A)) @ Xss)))) => (((coinductive_ltl @ A) @ ((coinductive_lconcat @ A) @ Xss)) = (((coinductive_lappend @ A) @ ((coinductive_ltl @ A) @ ((coinductive_lhd @ (coinductive_llist @ A)) @ Xss))) @ ((coinductive_lconcat @ A) @ ((coinductive_ltl @ (coinductive_llist @ A)) @ Xss))))))))).
% 2.80/1.52 thf(fact_113_gen__lset__def, axiom, (! [A:$tType]: (((coinductive_gen_lset @ A) = (^ [A5:(set @ A),Xs3:(coinductive_llist @ A)]: ((((sup_sup @ (set @ A)) @ A5) @ ((coinductive_lset @ A) @ Xs3)))))))).
% 2.80/1.52 thf(fact_114_lset__code, axiom, (! [A:$tType]: (((coinductive_lset @ A) = ((coinductive_gen_lset @ A) @ (bot_bot @ (set @ A))))))).
% 2.80/1.52 thf(fact_115_unfold__llist__id, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((((((coindu1441602521_llist @ (coinductive_llist @ A)) @ A) @ (coinductive_lnull @ A)) @ (coinductive_lhd @ A)) @ (coinductive_ltl @ A)) @ Xs) = Xs)))).
% 2.80/1.52 thf(fact_116_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A5:(set @ A)]: ((A5 = (bot_bot @ (set @ A))))))))).
% 2.80/1.52 thf(fact_117_lnull__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lnull @ A) @ Xs) & ((coinductive_lnull @ A) @ Ys)))))).
% 2.80/1.52 thf(fact_118_lappend_Odisc__iff_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)))) = ((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Ys)))))))).
% 2.80/1.52 thf(fact_119_lfinite__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lfinite @ A) @ Xs) & ((coinductive_lfinite @ A) @ Ys)))))).
% 2.80/1.52 thf(fact_120_lconcat__lappend, axiom, (! [A:$tType,Xss:(coinductive_llist @ (coinductive_llist @ A)),Yss:(coinductive_llist @ (coinductive_llist @ A))]: ((((coinductive_lfinite @ (coinductive_llist @ A)) @ Xss) => (((coinductive_lconcat @ A) @ (((coinductive_lappend @ (coinductive_llist @ A)) @ Xss) @ Yss)) = (((coinductive_lappend @ A) @ ((coinductive_lconcat @ A) @ Xss)) @ ((coinductive_lconcat @ A) @ Yss))))))).
% 2.80/1.52 thf(fact_121_unfold__llist_Odisc__iff_I1_J, axiom, (! [B:$tType,A:$tType,P2:(A > $o),G21:(A > B),G22:(A > A),A2:A]: ((((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P2) @ G21) @ G22) @ A2)) = (P2 @ A2))))).
% 2.80/1.52 thf(fact_122_unfold__llist_Odisc__iff_I2_J, axiom, (! [B:$tType,A:$tType,P2:(A > $o),G21:(A > B),G22:(A > A),A2:A]: (((~ (((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P2) @ G21) @ G22) @ A2)))) = (~ ((P2 @ A2))))))).
% 2.80/1.52 thf(fact_123_ltl__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Xs) => (((coinductive_ltl @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((coinductive_ltl @ A) @ Ys))) & ((~ (((coinductive_lnull @ A) @ Xs))) => (((coinductive_ltl @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lappend @ A) @ ((coinductive_ltl @ A) @ Xs)) @ Ys))))))).
% 2.80/1.52 thf(fact_124_lhd__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Xs) => (((coinductive_lhd @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((coinductive_lhd @ A) @ Ys))) & ((~ (((coinductive_lnull @ A) @ Xs))) => (((coinductive_lhd @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((coinductive_lhd @ A) @ Xs))))))).
% 2.80/1.52 thf(fact_125_lfilter__lappend__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lappend @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)) @ (((coinductive_lfilter @ A) @ P) @ Ys))))))).
% 2.80/1.52 thf(fact_126_lappend__ltakeWhile__ldropWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) @ (((coindu218763757pWhile @ A) @ P) @ Xs)) = Xs)))).
% 2.80/1.52 thf(fact_127_lset__lappend__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs) => (((coinductive_lset @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((sup_sup @ (set @ A)) @ ((coinductive_lset @ A) @ Xs)) @ ((coinductive_lset @ A) @ Ys))))))).
% 2.80/1.52 thf(fact_128_lappend__assoc, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) @ Zs) = (((coinductive_lappend @ A) @ Xs) @ (((coinductive_lappend @ A) @ Ys) @ Zs)))))).
% 2.80/1.52 thf(fact_129_lappend__lnull2, axiom, (! [A:$tType,Ys:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Ys) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = Xs))))).
% 2.80/1.52 thf(fact_130_lappend__lnull1, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = Ys))))).
% 2.80/1.52 thf(fact_131_lappend_Odisc_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => (((coinductive_lnull @ A) @ Ys) => ((coinductive_lnull @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys))))))).
% 2.80/1.52 thf(fact_132_lappend_Odisc_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Ys)))) => (~ (((coinductive_lnull @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)))))))).
% 2.80/1.52 thf(fact_133_lappend__inf, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = Xs))))).
% 2.80/1.52 thf(fact_134_unfold__llist_Odisc_I1_J, axiom, (! [B:$tType,A:$tType,P2:(A > $o),A2:A,G21:(A > B),G22:(A > A)]: (((P2 @ A2) => ((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P2) @ G21) @ G22) @ A2)))))).
% 2.80/1.52 thf(fact_135_unfold__llist_Odisc_I2_J, axiom, (! [B:$tType,A:$tType,P2:(A > $o),A2:A,G21:(A > B),G22:(A > A)]: (((~ ((P2 @ A2))) => (~ (((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P2) @ G21) @ G22) @ A2)))))))).
% 2.80/1.52 thf(fact_136_unfold__llist_Osimps_I4_J, axiom, (! [B:$tType,A:$tType,P2:(A > $o),A2:A,G21:(A > B),G22:(A > A)]: (((~ ((P2 @ A2))) => (((coinductive_ltl @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P2) @ G21) @ G22) @ A2)) = ((((((coindu1441602521_llist @ A) @ B) @ P2) @ G21) @ G22) @ (G22 @ A2))))))).
% 2.80/1.52 thf(fact_137_unfold__llist_Osimps_I3_J, axiom, (! [B:$tType,A:$tType,P2:(A > $o),A2:A,G21:(A > B),G22:(A > A)]: (((~ ((P2 @ A2))) => (((coinductive_lhd @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P2) @ G21) @ G22) @ A2)) = (G21 @ A2)))))).
% 2.80/1.52 thf(fact_138_lappend__ltl, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((((coinductive_lappend @ A) @ ((coinductive_ltl @ A) @ Xs)) @ Ys) = ((coinductive_ltl @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys))))))).
% 2.80/1.52 thf(fact_139_in__lset__lappend__iff, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys))) = ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) | (((coinductive_lfinite @ A) @ Xs) & (((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys)))))))).
% 2.80/1.52 thf(fact_140_ltakeWhile__lappend1, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),P:(A > $o),Ys:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((~ ((P @ X))) => ((((coindu501562517eWhile @ A) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coindu501562517eWhile @ A) @ P) @ Xs))))))).
% 2.80/1.52 thf(fact_141_lfilter__eq__lappend__lfiniteD, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Xs) = (((coinductive_lappend @ A) @ Ys) @ Zs)) => (((coinductive_lfinite @ A) @ Ys) => (? [Us:(coinductive_llist @ A),Vs:(coinductive_llist @ A)]: (((Xs = (((coinductive_lappend @ A) @ Us) @ Vs)) & (((coinductive_lfinite @ A) @ Us) & ((Ys = (((coinductive_lfilter @ A) @ P) @ Us)) & (Zs = (((coinductive_lfilter @ A) @ P) @ Vs)))))))))))).
% 2.80/1.52 thf(fact_142_lset__lappend__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lfinite @ A) @ Xs) => (((coinductive_lset @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((sup_sup @ (set @ A)) @ ((coinductive_lset @ A) @ Xs)) @ ((coinductive_lset @ A) @ Ys)))) & ((~ (((coinductive_lfinite @ A) @ Xs))) => (((coinductive_lset @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((coinductive_lset @ A) @ Xs))))))).
% 2.80/1.52 thf(fact_143_bot__fun__def, axiom, (! [B:$tType,A:$tType]: ((((bot @ B) @ (type2 @ B)) => ((bot_bot @ (A > B)) = (^ [X2:A]: ((bot_bot @ B)))))))).
% 2.80/1.52 thf(fact_144_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (bot_bot @ (set @ A))))))))).
% 2.80/1.52 thf(fact_145_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.80/1.52 thf(fact_146_lstrict__prefix__lappend__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Xs) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lfinite @ A) @ Xs) & (~ (((coinductive_lnull @ A) @ Ys)))))))).
% 2.80/1.52 thf(fact_147_ldistinct__coinduct, axiom, (! [A:$tType,X5:((coinductive_llist @ A) > $o),Xs:(coinductive_llist @ A)]: (((X5 @ Xs) => ((! [Xs2:(coinductive_llist @ A)]: (((X5 @ Xs2) => ((~ (((coinductive_lnull @ A) @ Xs2))) => ((~ ((((member @ A) @ ((coinductive_lhd @ A) @ Xs2)) @ ((coinductive_lset @ A) @ ((coinductive_ltl @ A) @ Xs2))))) & ((X5 @ ((coinductive_ltl @ A) @ Xs2)) | ((coindu351974385stinct @ A) @ ((coinductive_ltl @ A) @ Xs2)))))))) => ((coindu351974385stinct @ A) @ Xs)))))).
% 2.80/1.52 thf(fact_148_llist__less__induct, axiom, (! [A:$tType,P:((coinductive_llist @ A) > $o),Xs:(coinductive_llist @ A)]: (((! [Xs2:(coinductive_llist @ A)]: (((! [Ys2:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Ys2) @ Xs2) => (P @ Ys2)))) => (P @ Xs2)))) => (P @ Xs))))).
% 2.80/1.52 thf(fact_149_ldistinct__ltlI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coindu351974385stinct @ A) @ Xs) => ((coindu351974385stinct @ A) @ ((coinductive_ltl @ A) @ Xs)))))).
% 2.80/1.52 thf(fact_150_ldistinct__lfilterI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: ((((coindu351974385stinct @ A) @ Xs) => ((coindu351974385stinct @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)))))).
% 2.80/1.52 thf(fact_151_lstrict__prefix__lfinite1, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Xs) @ Ys) => ((coinductive_lfinite @ A) @ Xs))))).
% 2.80/1.52 thf(fact_152_ldistinct__lhdD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coindu351974385stinct @ A) @ Xs) => ((~ (((coinductive_lnull @ A) @ Xs))) => (~ ((((member @ A) @ ((coinductive_lhd @ A) @ Xs)) @ ((coinductive_lset @ A) @ ((coinductive_ltl @ A) @ Xs)))))))))).
% 2.80/1.52 thf(fact_153_ldistinct__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coindu351974385stinct @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coindu351974385stinct @ A) @ Xs) & (((coinductive_lfinite @ A) @ Xs) => (((coindu351974385stinct @ A) @ Ys) & ((((inf_inf @ (set @ A)) @ ((coinductive_lset @ A) @ Xs)) @ ((coinductive_lset @ A) @ Ys)) = (bot_bot @ (set @ A)))))))))).
% 2.80/1.52 thf(fact_154_ldistinct__lconcat, axiom, (! [A:$tType,Xss:(coinductive_llist @ (coinductive_llist @ A))]: ((((coindu351974385stinct @ (coinductive_llist @ A)) @ Xss) => ((! [Ys3:(coinductive_llist @ A)]: (((((member @ (coinductive_llist @ A)) @ Ys3) @ ((coinductive_lset @ (coinductive_llist @ A)) @ Xss)) => ((coindu351974385stinct @ A) @ Ys3)))) => ((! [Ys3:(coinductive_llist @ A),Zs2:(coinductive_llist @ A)]: (((((member @ (coinductive_llist @ A)) @ Ys3) @ ((coinductive_lset @ (coinductive_llist @ A)) @ Xss)) => ((((member @ (coinductive_llist @ A)) @ Zs2) @ ((coinductive_lset @ (coinductive_llist @ A)) @ Xss)) => ((Ys3 != Zs2) => ((((inf_inf @ (set @ A)) @ ((coinductive_lset @ A) @ Ys3)) @ ((coinductive_lset @ A) @ Zs2)) = (bot_bot @ (set @ A)))))))) => ((coindu351974385stinct @ A) @ ((coinductive_lconcat @ A) @ Xss)))))))).
% 2.80/1.52 thf(fact_155_llist_Ocase__eq__if, axiom, (! [A:$tType,B:$tType]: ((((coindu1381640503_llist @ B) @ A) = (^ [F1:B,F22:(A > ((coinductive_llist @ A) > B)),Llist5:(coinductive_llist @ A)]: (((((if @ B) @ ((coinductive_lnull @ A) @ Llist5)) @ F1) @ ((F22 @ ((coinductive_lhd @ A) @ Llist5)) @ ((coinductive_ltl @ A) @ Llist5))))))))).
% 2.80/1.52 thf(fact_156_inf__apply, axiom, (! [B:$tType,A:$tType]: ((((semilattice_inf @ B) @ (type2 @ B)) => ((inf_inf @ (A > B)) = (^ [F2:(A > B),G3:(A > B),X2:A]: ((((inf_inf @ B) @ (F2 @ X2)) @ (G3 @ X2))))))))).
% 2.80/1.52 thf(fact_157_inf_Oidem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A]: (((((inf_inf @ A) @ A2) @ A2) = A2))))))).
% 2.80/1.52 thf(fact_158_inf__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A]: (((((inf_inf @ A) @ X) @ X) = X))))))).
% 2.80/1.52 thf(fact_159_inf_Oleft__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ A2) @ B3)) = (((inf_inf @ A) @ A2) @ B3)))))))).
% 2.80/1.52 thf(fact_160_inf__left__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y)) = (((inf_inf @ A) @ X) @ Y)))))))).
% 2.80/1.52 thf(fact_161_inf_Oright__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B3)) @ B3) = (((inf_inf @ A) @ A2) @ B3)))))))).
% 2.80/1.52 thf(fact_162_inf__right__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Y) = (((inf_inf @ A) @ X) @ Y)))))))).
% 2.80/1.52 thf(fact_163_IntI, axiom, (! [A:$tType,C:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ A3) => ((((member @ A) @ C) @ B2) => (((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B2))))))).
% 2.80/1.52 thf(fact_164_Int__iff, axiom, (! [A:$tType,C:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B2)) = ((((member @ A) @ C) @ A3) & (((member @ A) @ C) @ B2)))))).
% 2.80/1.52 thf(fact_165_inf__bot__left, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A]: (((((inf_inf @ A) @ (bot_bot @ A)) @ X) = (bot_bot @ A)))))))).
% 2.80/1.52 thf(fact_166_inf__bot__right, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A]: (((((inf_inf @ A) @ X) @ (bot_bot @ A)) = (bot_bot @ A)))))))).
% 2.80/1.52 thf(fact_167_inf__sup__absorb, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = X))))))).
% 2.80/1.52 thf(fact_168_sup__inf__absorb, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ X) @ Y)) = X))))))).
% 2.80/1.52 thf(fact_169_inf__sup__aci_I4_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y)) = (((inf_inf @ A) @ X) @ Y)))))))).
% 2.80/1.52 thf(fact_170_inf__sup__aci_I3_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ Y) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 2.80/1.52 thf(fact_171_inf__sup__aci_I2_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Z) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z))))))))).
% 2.80/1.52 thf(fact_172_inf__sup__aci_I1_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => ((inf_inf @ A) = (^ [X2:A,Y3:A]: ((((inf_inf @ A) @ Y3) @ X2)))))))).
% 2.80/1.52 thf(fact_173_IntE, axiom, (! [A:$tType,C:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B2)) => (~ (((((member @ A) @ C) @ A3) => (~ ((((member @ A) @ C) @ B2)))))))))).
% 2.80/1.52 thf(fact_174_IntD1, axiom, (! [A:$tType,C:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B2)) => (((member @ A) @ C) @ A3))))).
% 2.80/1.52 thf(fact_175_IntD2, axiom, (! [A:$tType,C:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B2)) => (((member @ A) @ C) @ B2))))).
% 2.80/1.52 thf(fact_176_Int__assoc, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B2)) @ C2) = (((inf_inf @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ B2) @ C2)))))).
% 2.80/1.52 thf(fact_177_Int__absorb, axiom, (! [A:$tType,A3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ A3) = A3)))).
% 2.80/1.52 thf(fact_178_Int__commute, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((((inf_inf @ (set @ A)) @ B5) @ A5))))))).
% 2.80/1.52 thf(fact_179_Int__left__absorb, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ A3) @ B2)) = (((inf_inf @ (set @ A)) @ A3) @ B2))))).
% 2.80/1.52 thf(fact_180_inf__fun__def, axiom, (! [B:$tType,A:$tType]: ((((semilattice_inf @ B) @ (type2 @ B)) => ((inf_inf @ (A > B)) = (^ [F2:(A > B),G3:(A > B),X2:A]: ((((inf_inf @ B) @ (F2 @ X2)) @ (G3 @ X2))))))))).
% 2.80/1.52 thf(fact_181_Int__left__commute, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ B2) @ C2)) = (((inf_inf @ (set @ A)) @ B2) @ (((inf_inf @ (set @ A)) @ A3) @ C2)))))).
% 2.80/1.52 thf(fact_182_inf_Oassoc, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B3)) @ C) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B3) @ C))))))))).
% 2.80/1.52 thf(fact_183_inf__assoc, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Z) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z))))))))).
% 2.80/1.52 thf(fact_184_inf_Ocommute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => ((inf_inf @ A) = (^ [A4:A,B4:A]: ((((inf_inf @ A) @ B4) @ A4)))))))).
% 2.80/1.52 thf(fact_185_inf__commute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => ((inf_inf @ A) = (^ [X2:A,Y3:A]: ((((inf_inf @ A) @ Y3) @ X2)))))))).
% 2.80/1.52 thf(fact_186_inf_Oleft__commute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [B3:A,A2:A,C:A]: (((((inf_inf @ A) @ B3) @ (((inf_inf @ A) @ A2) @ C)) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B3) @ C))))))))).
% 2.80/1.52 thf(fact_187_inf__left__commute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ Y) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 2.80/1.52 thf(fact_188_Un__Int__distrib2, axiom, (! [A:$tType,B2:(set @ A),C2:(set @ A),A3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ B2) @ C2)) @ A3) = (((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ B2) @ A3)) @ (((sup_sup @ (set @ A)) @ C2) @ A3)))))).
% 2.80/1.52 thf(fact_189_Int__Un__distrib2, axiom, (! [A:$tType,B2:(set @ A),C2:(set @ A),A3:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ B2) @ C2)) @ A3) = (((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ B2) @ A3)) @ (((inf_inf @ (set @ A)) @ C2) @ A3)))))).
% 2.80/1.52 thf(fact_190_Un__Int__distrib, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ B2) @ C2)) = (((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B2)) @ (((sup_sup @ (set @ A)) @ A3) @ C2)))))).
% 2.80/1.52 thf(fact_191_Int__Un__distrib, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ B2) @ C2)) = (((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B2)) @ (((inf_inf @ (set @ A)) @ A3) @ C2)))))).
% 2.80/1.52 thf(fact_192_Un__Int__crazy, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B2)) @ (((inf_inf @ (set @ A)) @ B2) @ C2))) @ (((inf_inf @ (set @ A)) @ C2) @ A3)) = (((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B2)) @ (((sup_sup @ (set @ A)) @ B2) @ C2))) @ (((sup_sup @ (set @ A)) @ C2) @ A3)))))).
% 2.80/1.52 thf(fact_193_disjoint__iff__not__equal, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A3) @ B2) = (bot_bot @ (set @ A))) = (! [X2:A]: (((((member @ A) @ X2) @ A3) => (! [Y3:A]: (((((member @ A) @ Y3) @ B2) => (X2 != Y3))))))))))).
% 2.80/1.52 thf(fact_194_Int__empty__right, axiom, (! [A:$tType,A3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ A)))))).
% 2.80/1.52 thf(fact_195_Int__empty__left, axiom, (! [A:$tType,B2:(set @ A)]: (((((inf_inf @ (set @ A)) @ (bot_bot @ (set @ A))) @ B2) = (bot_bot @ (set @ A)))))).
% 2.80/1.52 thf(fact_196_Int__emptyI, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A3) => (~ ((((member @ A) @ X3) @ B2)))))) => ((((inf_inf @ (set @ A)) @ A3) @ B2) = (bot_bot @ (set @ A))))))).
% 2.80/1.52 thf(fact_197_sup__inf__distrib2, axiom, (! [A:$tType]: ((((distrib_lattice @ A) @ (type2 @ A)) => (! [Y:A,Z:A,X:A]: (((((sup_sup @ A) @ (((inf_inf @ A) @ Y) @ Z)) @ X) = (((inf_inf @ A) @ (((sup_sup @ A) @ Y) @ X)) @ (((sup_sup @ A) @ Z) @ X))))))))).
% 2.80/1.52 thf(fact_198_sup__inf__distrib1, axiom, (! [A:$tType]: ((((distrib_lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y)) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.80/1.52 thf(fact_199_inf__sup__distrib2, axiom, (! [A:$tType]: ((((distrib_lattice @ A) @ (type2 @ A)) => (! [Y:A,Z:A,X:A]: (((((inf_inf @ A) @ (((sup_sup @ A) @ Y) @ Z)) @ X) = (((sup_sup @ A) @ (((inf_inf @ A) @ Y) @ X)) @ (((inf_inf @ A) @ Z) @ X))))))))).
% 2.80/1.52 thf(fact_200_inf__sup__distrib1, axiom, (! [A:$tType]: ((((distrib_lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y)) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 2.80/1.52 thf(fact_201_distrib__imp2, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((! [X3:A,Y2:A,Z2:A]: (((((sup_sup @ A) @ X3) @ (((inf_inf @ A) @ Y2) @ Z2)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X3) @ Y2)) @ (((sup_sup @ A) @ X3) @ Z2))))) => ((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y)) @ (((inf_inf @ A) @ X) @ Z)))))))))).
% 2.80/1.52 thf(fact_202_distrib__imp1, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((! [X3:A,Y2:A,Z2:A]: (((((inf_inf @ A) @ X3) @ (((sup_sup @ A) @ Y2) @ Z2)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X3) @ Y2)) @ (((inf_inf @ A) @ X3) @ Z2))))) => ((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y)) @ (((sup_sup @ A) @ X) @ Z)))))))))).
% 2.80/1.52 thf(fact_203_disjnt__def, axiom, (! [A:$tType]: (((disjnt @ A) = (^ [A5:(set @ A),B5:(set @ A)]: (((((inf_inf @ (set @ A)) @ A5) @ B5) = (bot_bot @ (set @ A))))))))).
% 2.80/1.52 thf(fact_204_lfilter__eq__LConsD, axiom, (! [A:$tType,P:(A > $o),Ys:(coinductive_llist @ A),X:A,Xs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Ys) = (((coinductive_LCons @ A) @ X) @ Xs)) => (? [Us:(coinductive_llist @ A),Vs:(coinductive_llist @ A)]: (((Ys = (((coinductive_lappend @ A) @ Us) @ (((coinductive_LCons @ A) @ X) @ Vs))) & (((coinductive_lfinite @ A) @ Us) & ((! [X4:A]: (((((member @ A) @ X4) @ ((coinductive_lset @ A) @ Us)) => (~ ((P @ X4)))))) & ((P @ X) & (Xs = (((coinductive_lfilter @ A) @ P) @ Vs)))))))))))).
% 2.80/1.52 thf(fact_205_ltakeWhile_Octr_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((P @ ((coinductive_lhd @ A) @ Xs)) => ((((coindu501562517eWhile @ A) @ P) @ Xs) = (((coinductive_LCons @ A) @ ((coinductive_lhd @ A) @ Xs)) @ (((coindu501562517eWhile @ A) @ P) @ ((coinductive_ltl @ A) @ Xs))))))))).
% 2.80/1.52 thf(fact_206_llist_Oinject, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A),Y21:A,Y22:(coinductive_llist @ A)]: ((((((coinductive_LCons @ A) @ X21) @ X22) = (((coinductive_LCons @ A) @ Y21) @ Y22)) = ((X21 = Y21) & (X22 = Y22)))))).
% 2.80/1.52 thf(fact_207_lfinite__LCons, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = ((coinductive_lfinite @ A) @ Xs))))).
% 2.80/1.52 thf(fact_208_lfinite__code_I2_J, axiom, (! [B:$tType,X:B,Xs:(coinductive_llist @ B)]: ((((coinductive_lfinite @ B) @ (((coinductive_LCons @ B) @ X) @ Xs)) = ((coinductive_lfinite @ B) @ Xs))))).
% 2.80/1.52 thf(fact_209_lappend__code_I2_J, axiom, (! [A:$tType,Xa:A,X:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coinductive_LCons @ A) @ Xa) @ X)) @ Ys) = (((coinductive_LCons @ A) @ Xa) @ (((coinductive_lappend @ A) @ X) @ Ys)))))).
% 2.80/1.52 thf(fact_210_lfilter__LCons, axiom, (! [A:$tType,P:(A > $o),X:A,Xs:(coinductive_llist @ A)]: ((((P @ X) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_LCons @ A) @ X) @ (((coinductive_lfilter @ A) @ P) @ Xs)))) & ((~ ((P @ X))) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_lfilter @ A) @ P) @ Xs))))))).
% 2.80/1.52 thf(fact_211_ldropWhile__LCons, axiom, (! [A:$tType,P:(A > $o),X:A,Xs:(coinductive_llist @ A)]: ((((P @ X) => ((((coindu218763757pWhile @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coindu218763757pWhile @ A) @ P) @ Xs))) & ((~ ((P @ X))) => ((((coindu218763757pWhile @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_LCons @ A) @ X) @ Xs))))))).
% 2.80/1.52 thf(fact_212_lconcat__LCons, axiom, (! [B:$tType,Xs:(coinductive_llist @ B),Xss:(coinductive_llist @ (coinductive_llist @ B))]: ((((coinductive_lconcat @ B) @ (((coinductive_LCons @ (coinductive_llist @ B)) @ Xs) @ Xss)) = (((coinductive_lappend @ B) @ Xs) @ ((coinductive_lconcat @ B) @ Xss)))))).
% 2.80/1.52 thf(fact_213_unfold__llist__eq__LCons, axiom, (! [A:$tType,B:$tType,IS_LNIL:(B > $o),LHD:(B > A),LTL:(B > B),B3:B,X:A,Xs:(coinductive_llist @ A)]: (((((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ B3) = (((coinductive_LCons @ A) @ X) @ Xs)) = ((~ ((IS_LNIL @ B3))) & ((X = (LHD @ B3)) & (Xs = ((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ (LTL @ B3))))))))).
% 2.80/1.52 thf(fact_214_lstrict__prefix__code_I4_J, axiom, (! [B:$tType,X:B,Xs:(coinductive_llist @ B),Y:B,Ys:(coinductive_llist @ B)]: (((((coindu1478340336prefix @ B) @ (((coinductive_LCons @ B) @ X) @ Xs)) @ (((coinductive_LCons @ B) @ Y) @ Ys)) = ((X = Y) & (((coindu1478340336prefix @ B) @ Xs) @ Ys)))))).
% 2.80/1.52 thf(fact_215_ldistinct__LCons, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((((coindu351974385stinct @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = ((~ ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)))) & ((coindu351974385stinct @ A) @ Xs)))))).
% 2.80/1.52 thf(fact_216_lhd__LCons__ltl, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ Llist))) => ((((coinductive_LCons @ A) @ ((coinductive_lhd @ A) @ Llist)) @ ((coinductive_ltl @ A) @ Llist)) = Llist))))).
% 2.80/1.52 thf(fact_217_llist_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,F12:B,F23:(A > ((coinductive_llist @ A) > B)),X21:A,X22:(coinductive_llist @ A)]: (((((((coindu1381640503_llist @ B) @ A) @ F12) @ F23) @ (((coinductive_LCons @ A) @ X21) @ X22)) = ((F23 @ X21) @ X22))))).
% 2.80/1.52 thf(fact_218_llist_Oset__induct, axiom, (! [A:$tType,X:A,A2:(coinductive_llist @ A),P:(A > ((coinductive_llist @ A) > $o))]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ A2)) => ((! [Z1:A,Z22:(coinductive_llist @ A)]: (((P @ Z1) @ (((coinductive_LCons @ A) @ Z1) @ Z22)))) => ((! [Z1:A,Z22:(coinductive_llist @ A),Xa2:A]: (((((member @ A) @ Xa2) @ ((coinductive_lset @ A) @ Z22)) => (((P @ Xa2) @ Z22) => ((P @ Xa2) @ (((coinductive_LCons @ A) @ Z1) @ Z22)))))) => ((P @ X) @ A2))))))).
% 2.80/1.52 thf(fact_219_llist_Oset__cases, axiom, (! [A:$tType,E:A,A2:(coinductive_llist @ A)]: (((((member @ A) @ E) @ ((coinductive_lset @ A) @ A2)) => ((! [Z22:(coinductive_llist @ A)]: ((A2 != (((coinductive_LCons @ A) @ E) @ Z22)))) => (~ ((! [Z1:A,Z22:(coinductive_llist @ A)]: (((A2 = (((coinductive_LCons @ A) @ Z1) @ Z22)) => (~ ((((member @ A) @ E) @ ((coinductive_lset @ A) @ Z22)))))))))))))).
% 2.80/1.52 thf(fact_220_lset__induct_H, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs2:(coinductive_llist @ A)]: ((P @ (((coinductive_LCons @ A) @ X) @ Xs2)))) => ((! [X6:A,Xs2:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs2)) => ((P @ Xs2) => (P @ (((coinductive_LCons @ A) @ X6) @ Xs2)))))) => (P @ Xs))))))).
% 2.80/1.52 thf(fact_221_lset__induct, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs2:(coinductive_llist @ A)]: ((P @ (((coinductive_LCons @ A) @ X) @ Xs2)))) => ((! [X6:A,Xs2:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs2)) => ((X != X6) => ((P @ Xs2) => (P @ (((coinductive_LCons @ A) @ X6) @ Xs2))))))) => (P @ Xs))))))).
% 2.80/1.52 thf(fact_222_lset__cases, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs4:(coinductive_llist @ A)]: ((Xs != (((coinductive_LCons @ A) @ X) @ Xs4)))) => (~ ((! [X6:A,Xs4:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ X6) @ Xs4)) => (~ ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs4)))))))))))))).
% 2.80/1.52 thf(fact_223_llist_Oset__intros_I1_J, axiom, (! [A:$tType,A1:A,A22:(coinductive_llist @ A)]: ((((member @ A) @ A1) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ A1) @ A22)))))).
% 2.80/1.52 thf(fact_224_llist_Oset__intros_I2_J, axiom, (! [A:$tType,X:A,A22:(coinductive_llist @ A),A1:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ A22)) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ A1) @ A22))))))).
% 2.80/1.52 thf(fact_225_lset__intros_I1_J, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)))))).
% 2.80/1.52 thf(fact_226_lset__intros_I2_J, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),X7:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X7) @ Xs))))))).
% 2.80/1.52 thf(fact_227_lmember__code_I2_J, axiom, (! [A:$tType,X:A,Y:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lmember @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Ys)) = ((X = Y) | (((coinductive_lmember @ A) @ X) @ Ys)))))).
% 2.80/1.52 thf(fact_228_not__lnull__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ Xs))) = (? [X2:A,Xs5:(coinductive_llist @ A)]: ((Xs = (((coinductive_LCons @ A) @ X2) @ Xs5)))))))).
% 2.80/1.52 thf(fact_229_llist_OdiscI_I2_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A),X21:A,X22:(coinductive_llist @ A)]: (((Llist = (((coinductive_LCons @ A) @ X21) @ X22)) => (~ (((coinductive_lnull @ A) @ Llist))))))).
% 2.80/1.52 thf(fact_230_llist_Odisc_I2_J, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: ((~ (((coinductive_lnull @ A) @ (((coinductive_LCons @ A) @ X21) @ X22))))))).
% 2.80/1.52 thf(fact_231_ltl__simps_I2_J, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: ((((coinductive_ltl @ A) @ (((coinductive_LCons @ A) @ X21) @ X22)) = X22)))).
% 2.80/1.52 thf(fact_232_lhd__LCons, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: ((((coinductive_lhd @ A) @ (((coinductive_LCons @ A) @ X21) @ X22)) = X21)))).
% 2.80/1.52 thf(fact_233_lfilter__LCons__seek, axiom, (! [A:$tType,P2:(A > $o),X:A,L:(coinductive_llist @ A)]: (((~ ((P2 @ X))) => ((((coinductive_lfilter @ A) @ P2) @ (((coinductive_LCons @ A) @ X) @ L)) = (((coinductive_lfilter @ A) @ P2) @ L)))))).
% 2.80/1.52 thf(fact_234_lfilter__LCons__found, axiom, (! [A:$tType,P:(A > $o),X:A,Xs:(coinductive_llist @ A)]: (((P @ X) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_LCons @ A) @ X) @ (((coinductive_lfilter @ A) @ P) @ Xs))))))).
% 2.80/1.52 thf(fact_235_disjnt__iff, axiom, (! [A:$tType]: (((disjnt @ A) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [X2:A]: ((~ (((((member @ A) @ X2) @ A5) & (((member @ A) @ X2) @ B5)))))))))))).
% 2.80/1.52 thf(fact_236_lsetp_Oinducts, axiom, (! [A:$tType,X1:(coinductive_llist @ A),X23:A,P:((coinductive_llist @ A) > (A > $o))]: (((((coinductive_lsetp @ A) @ X1) @ X23) => ((! [X3:A,Xs2:(coinductive_llist @ A)]: (((P @ (((coinductive_LCons @ A) @ X3) @ Xs2)) @ X3))) => ((! [Xs2:(coinductive_llist @ A),X3:A,X6:A]: (((((coinductive_lsetp @ A) @ Xs2) @ X3) => (((P @ Xs2) @ X3) => ((P @ (((coinductive_LCons @ A) @ X6) @ Xs2)) @ X3))))) => ((P @ X1) @ X23))))))).
% 2.80/1.52 thf(fact_237_lsetp_Osimps, axiom, (! [A:$tType]: (((coinductive_lsetp @ A) = (^ [A12:(coinductive_llist @ A),A23:A]: (((? [X2:A,Xs3:(coinductive_llist @ A)]: (((A12 = (((coinductive_LCons @ A) @ X2) @ Xs3)) & (A23 = X2)))) | (? [Xs3:(coinductive_llist @ A),X2:A,X8:A]: (((A12 = (((coinductive_LCons @ A) @ X8) @ Xs3)) & ((A23 = X2) & (((coinductive_lsetp @ A) @ Xs3) @ X2)))))))))))).
% 2.80/1.52 thf(fact_238_lsetp_Ocases, axiom, (! [A:$tType,A1:(coinductive_llist @ A),A22:A]: (((((coinductive_lsetp @ A) @ A1) @ A22) => ((! [X3:A]: (((? [Xs2:(coinductive_llist @ A)]: ((A1 = (((coinductive_LCons @ A) @ X3) @ Xs2)))) => (A22 != X3)))) => (~ ((! [Xs2:(coinductive_llist @ A)]: (((? [X6:A]: ((A1 = (((coinductive_LCons @ A) @ X6) @ Xs2)))) => (~ ((((coinductive_lsetp @ A) @ Xs2) @ A22))))))))))))).
% 2.80/1.52 thf(fact_239_lsetp_Ointros_I1_J, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((((coinductive_lsetp @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) @ X)))).
% 2.80/1.52 thf(fact_240_lsetp_Ointros_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),X:A,X7:A]: (((((coinductive_lsetp @ A) @ Xs) @ X) => (((coinductive_lsetp @ A) @ (((coinductive_LCons @ A) @ X7) @ Xs)) @ X))))).
% 2.80/1.52 thf(fact_241_lfinite__LConsI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),X:A]: ((((coinductive_lfinite @ A) @ Xs) => ((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)))))).
% 2.80/1.52 thf(fact_242_unfold__llist_Octr_I2_J, axiom, (! [B:$tType,A:$tType,P2:(A > $o),A2:A,G21:(A > B),G22:(A > A)]: (((~ ((P2 @ A2))) => (((((((coindu1441602521_llist @ A) @ B) @ P2) @ G21) @ G22) @ A2) = (((coinductive_LCons @ B) @ (G21 @ A2)) @ ((((((coindu1441602521_llist @ A) @ B) @ P2) @ G21) @ G22) @ (G22 @ A2)))))))).
% 2.80/1.52 thf(fact_243_ldistinct_OLCons, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((~ ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)))) => (((coindu351974385stinct @ A) @ Xs) => ((coindu351974385stinct @ A) @ (((coinductive_LCons @ A) @ X) @ Xs))))))).
% 2.80/1.52 thf(fact_244_split__llist, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (? [Ys3:(coinductive_llist @ A),Zs2:(coinductive_llist @ A)]: (((Xs = (((coinductive_lappend @ A) @ Ys3) @ (((coinductive_LCons @ A) @ X) @ Zs2))) & ((coinductive_lfinite @ A) @ Ys3)))))))).
% 2.80/1.52 thf(fact_245_split__llist__first, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (? [Ys3:(coinductive_llist @ A),Zs2:(coinductive_llist @ A)]: (((Xs = (((coinductive_lappend @ A) @ Ys3) @ (((coinductive_LCons @ A) @ X) @ Zs2))) & (((coinductive_lfinite @ A) @ Ys3) & (~ ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys3)))))))))))).
% 2.80/1.52 thf(fact_246_llast__lappend__LCons, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Y:A,Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs) => (((coinductive_llast @ A) @ (((coinductive_lappend @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y) @ Ys))) = ((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ Y) @ Ys))))))).
% 2.80/1.52 thf(fact_247_ltakeWhile_Ocode, axiom, (! [A:$tType]: (((coindu501562517eWhile @ A) = (^ [P3:(A > $o),Xs3:(coinductive_llist @ A)]: (((((if @ (coinductive_llist @ A)) @ (((coinductive_lnull @ A) @ Xs3) | (~ ((P3 @ ((coinductive_lhd @ A) @ Xs3)))))) @ (coinductive_LNil @ A)) @ (((coinductive_LCons @ A) @ ((coinductive_lhd @ A) @ Xs3)) @ (((coindu501562517eWhile @ A) @ P3) @ ((coinductive_ltl @ A) @ Xs3)))))))))).
% 2.80/1.52 thf(fact_248_lfinite__code_I1_J, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) @ (coinductive_LNil @ A))))).
% 2.80/1.52 thf(fact_249_lappend__LNil2, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ Xs) @ (coinductive_LNil @ A)) = Xs)))).
% 2.80/1.52 thf(fact_250_lappend__code_I1_J, axiom, (! [A:$tType,Ys:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (coinductive_LNil @ A)) @ Ys) = Ys)))).
% 2.80/1.52 thf(fact_251_lfilter__LNil, axiom, (! [A:$tType,P:(A > $o)]: (((((coinductive_lfilter @ A) @ P) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.80/1.52 thf(fact_252_ltakeWhile__LNil, axiom, (! [A:$tType,P:(A > $o)]: (((((coindu501562517eWhile @ A) @ P) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.80/1.52 thf(fact_253_lconcat__LNil, axiom, (! [A:$tType]: ((((coinductive_lconcat @ A) @ (coinductive_LNil @ (coinductive_llist @ A))) = (coinductive_LNil @ A))))).
% 2.80/1.52 thf(tcon_HOL_Obool___Lattices_Obounded__lattice, axiom, ((bounded_lattice @ $o) @ (type2 @ $o))).
% 2.80/1.52 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1, axiom, (! [A6:$tType]: (((bounded_lattice @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.80/1.52 thf(tcon_fun___Lattices_Obounded__lattice_2, axiom, (! [A6:$tType,A7:$tType]: ((((bounded_lattice @ A7) @ (type2 @ A7)) => ((bounded_lattice @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.80/1.52 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot, axiom, (! [A6:$tType,A7:$tType]: ((((bounded_lattice @ A7) @ (type2 @ A7)) => ((bounde1808546759up_bot @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.80/1.52 thf(tcon_fun___Lattices_Obounded__lattice__bot, axiom, (! [A6:$tType,A7:$tType]: ((((bounded_lattice @ A7) @ (type2 @ A7)) => ((bounded_lattice_bot @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.80/1.52 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A6:$tType,A7:$tType]: ((((semilattice_sup @ A7) @ (type2 @ A7)) => ((semilattice_sup @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.80/1.52 thf(tcon_fun___Lattices_Osemilattice__inf, axiom, (! [A6:$tType,A7:$tType]: ((((semilattice_inf @ A7) @ (type2 @ A7)) => ((semilattice_inf @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.80/1.52 thf(tcon_fun___Lattices_Odistrib__lattice, axiom, (! [A6:$tType,A7:$tType]: ((((distrib_lattice @ A7) @ (type2 @ A7)) => ((distrib_lattice @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.80/1.52 thf(tcon_fun___Lattices_Olattice, axiom, (! [A6:$tType,A7:$tType]: ((((lattice @ A7) @ (type2 @ A7)) => ((lattice @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.80/1.52 thf(tcon_fun___Orderings_Obot, axiom, (! [A6:$tType,A7:$tType]: ((((bot @ A7) @ (type2 @ A7)) => ((bot @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.80/1.52 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3, axiom, (! [A6:$tType]: (((bounde1808546759up_bot @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.80/1.52 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4, axiom, (! [A6:$tType]: (((bounded_lattice_bot @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.80/1.52 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5, axiom, (! [A6:$tType]: (((semilattice_sup @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.80/1.52 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_6, axiom, (! [A6:$tType]: (((semilattice_inf @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.80/1.52 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_7, axiom, (! [A6:$tType]: (((distrib_lattice @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.80/1.52 thf(tcon_Set_Oset___Lattices_Olattice_8, axiom, (! [A6:$tType]: (((lattice @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.80/1.52 thf(tcon_Set_Oset___Orderings_Obot_9, axiom, (! [A6:$tType]: (((bot @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.80/1.52 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_10, axiom, ((bounde1808546759up_bot @ $o) @ (type2 @ $o))).
% 2.80/1.52 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_11, axiom, ((bounded_lattice_bot @ $o) @ (type2 @ $o))).
% 2.80/1.52 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_12, axiom, ((semilattice_sup @ $o) @ (type2 @ $o))).
% 2.80/1.52 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_13, axiom, ((semilattice_inf @ $o) @ (type2 @ $o))).
% 2.80/1.52 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_14, axiom, ((distrib_lattice @ $o) @ (type2 @ $o))).
% 2.80/1.52 thf(tcon_HOL_Obool___Lattices_Olattice_15, axiom, ((lattice @ $o) @ (type2 @ $o))).
% 2.80/1.52 thf(tcon_HOL_Obool___Orderings_Obot_16, axiom, ((bot @ $o) @ (type2 @ $o))).
% 2.80/1.52 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.80/1.52 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.80/1.52 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.80/1.52 thf(conj_0, conjecture, (((member @ a) @ ((shd @ a) @ xsa)) @ ((coinductive_lset @ a) @ ((coindu1724414836stream @ a) @ xsa)))).
% 2.80/1.52 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.61YHUtgMPz/DTF2DTF26799.p
% 2.80/1.52 ------------------------
% 2.80/1.56 ---- Cleaned THF ---
% 2.80/1.56 thf(ty_t_Coinductive__List_Ollist,type,
% 2.80/1.56 coinductive_llist: $tType > $tType ).
% 2.80/1.56
% 2.80/1.56 thf(ty_t_Stream_Ostream,type,
% 2.80/1.56 stream: $tType > $tType ).
% 2.80/1.56
% 2.80/1.56 thf(ty_t_Set_Oset,type,
% 2.80/1.56 set: $tType > $tType ).
% 2.80/1.56
% 2.80/1.56 thf(ty_t_itself,type,
% 2.80/1.56 itself: $tType > $tType ).
% 2.80/1.56
% 2.80/1.56 thf(ty_tf_a,type,
% 2.80/1.56 a: $tType ).
% 2.80/1.56
% 2.80/1.56 thf(sy_cl_Lattices_Obounded__lattice,type,
% 2.80/1.56 bounded_lattice:
% 2.80/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_cl_HOL_Otype,type,
% 2.80/1.56 type:
% 2.80/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_cl_Orderings_Obot,type,
% 2.80/1.56 bot:
% 2.80/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_cl_Lattices_Olattice,type,
% 2.80/1.56 lattice:
% 2.80/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_cl_Lattices_Odistrib__lattice,type,
% 2.80/1.56 distrib_lattice:
% 2.80/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 2.80/1.56 semilattice_inf:
% 2.80/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.80/1.56 semilattice_sup:
% 2.80/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 2.80/1.56 bounded_lattice_bot:
% 2.80/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 2.80/1.56 bounde1808546759up_bot:
% 2.80/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Ogen__lset,type,
% 2.80/1.56 coinductive_gen_lset:
% 2.80/1.56 !>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Olappend,type,
% 2.80/1.56 coinductive_lappend:
% 2.80/1.56 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Olconcat,type,
% 2.80/1.56 coinductive_lconcat:
% 2.80/1.56 !>[A: $tType] : ( ( coinductive_llist @ ( coinductive_llist @ A ) ) > ( coinductive_llist @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Oldistinct,type,
% 2.80/1.56 coindu351974385stinct:
% 2.80/1.56 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_OldropWhile,type,
% 2.80/1.56 coindu218763757pWhile:
% 2.80/1.56 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Olfilter,type,
% 2.80/1.56 coinductive_lfilter:
% 2.80/1.56 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Olfinite,type,
% 2.80/1.56 coinductive_lfinite:
% 2.80/1.56 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Ollast,type,
% 2.80/1.56 coinductive_llast:
% 2.80/1.56 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 2.80/1.56 coinductive_LCons:
% 2.80/1.56 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 2.80/1.56 coinductive_LNil:
% 2.80/1.56 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Ollist_Ocase__llist,type,
% 2.80/1.56 coindu1381640503_llist:
% 2.80/1.56 !>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Ollist_Olhd,type,
% 2.80/1.56 coinductive_lhd:
% 2.80/1.56 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 2.80/1.56 coinductive_lnull:
% 2.80/1.56 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 2.80/1.56 coinductive_lset:
% 2.80/1.56 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Ollist_Oltl,type,
% 2.80/1.56 coinductive_ltl:
% 2.80/1.56 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Olmember,type,
% 2.80/1.56 coinductive_lmember:
% 2.80/1.56 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Olsetp,type,
% 2.80/1.56 coinductive_lsetp:
% 2.80/1.56 !>[A: $tType] : ( ( coinductive_llist @ A ) > A > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Olstrict__prefix,type,
% 2.80/1.56 coindu1478340336prefix:
% 2.80/1.56 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_OltakeWhile,type,
% 2.80/1.56 coindu501562517eWhile:
% 2.80/1.56 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 2.80/1.56 coindu1441602521_llist:
% 2.80/1.56 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ollist__of__stream,type,
% 2.80/1.56 coindu1724414836stream:
% 2.80/1.56 !>[A: $tType] : ( ( stream @ A ) > ( coinductive_llist @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Ocr__stream,type,
% 2.80/1.56 coindu1183105481stream:
% 2.80/1.56 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__of__llist,type,
% 2.80/1.56 coindu2010755910_llist:
% 2.80/1.56 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ounfold__stream,type,
% 2.80/1.56 coindu139217191stream:
% 2.80/1.56 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > A ) > A > ( stream @ B ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_If,type,
% 2.80/1.56 if:
% 2.80/1.56 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 2.80/1.56 inf_inf:
% 2.80/1.56 !>[A: $tType] : ( A > A > A ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.80/1.56 sup_sup:
% 2.80/1.56 !>[A: $tType] : ( A > A > A ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.80/1.56 bot_bot:
% 2.80/1.56 !>[A: $tType] : A ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Pure_Otype,type,
% 2.80/1.56 type2:
% 2.80/1.56 !>[A: $tType] : ( itself @ A ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Set_OCollect,type,
% 2.80/1.56 collect:
% 2.80/1.56 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Set_Odisjnt,type,
% 2.80/1.56 disjnt:
% 2.80/1.56 !>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Set_Ois__empty,type,
% 2.80/1.56 is_empty:
% 2.80/1.56 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Stream_Osinterleave,type,
% 2.80/1.56 sinterleave:
% 2.80/1.56 !>[A: $tType] : ( ( stream @ A ) > ( stream @ A ) > ( stream @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Stream_Osmember,type,
% 2.80/1.56 smember:
% 2.80/1.56 !>[A: $tType] : ( A > ( stream @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Stream_Ostream_Oshd,type,
% 2.80/1.56 shd:
% 2.80/1.56 !>[A: $tType] : ( ( stream @ A ) > A ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_Stream_Ostream_Osset,type,
% 2.80/1.56 sset:
% 2.80/1.56 !>[A: $tType] : ( ( stream @ A ) > ( set @ A ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_c_member,type,
% 2.80/1.56 member:
% 2.80/1.56 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.80/1.56
% 2.80/1.56 thf(sy_v_x____,type,
% 2.80/1.56 x: a ).
% 2.80/1.56
% 2.80/1.56 thf(sy_v_xs,type,
% 2.80/1.56 xs: stream @ a ).
% 2.80/1.56
% 2.80/1.56 thf(sy_v_xsa____,type,
% 2.80/1.56 xsa: stream @ a ).
% 2.80/1.56
% 2.80/1.56 thf(fact_0__092_060open_062_092_060not_062_Alnull_A_Illist__of__stream_Axs_J_A_092_060Longrightarrow_062_Alhd_A_Illist__of__stream_Axs_J_A_092_060in_062_Alset_A_Illist__of__stream_Axs_J_092_060close_062,axiom,
% 2.80/1.56 ( ~ ( coinductive_lnull @ a @ ( coindu1724414836stream @ a @ xsa ) )
% 2.80/1.56 => ( member @ a @ ( coinductive_lhd @ a @ ( coindu1724414836stream @ a @ xsa ) ) @ ( coinductive_lset @ a @ ( coindu1724414836stream @ a @ xsa ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_1_stream_ORep__inject,axiom,
% 2.80/1.56 ! [A: $tType,X: stream @ A,Y: stream @ A] :
% 2.80/1.56 ( ( ( coindu1724414836stream @ A @ X )
% 2.80/1.56 = ( coindu1724414836stream @ A @ Y ) )
% 2.80/1.56 = ( X = Y ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_2_lhd__llist__of__stream,axiom,
% 2.80/1.56 ! [A: $tType,Xs: stream @ A] :
% 2.80/1.56 ( ( coinductive_lhd @ A @ ( coindu1724414836stream @ A @ Xs ) )
% 2.80/1.56 = ( shd @ A @ Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_3_lnull__llist__of__stream,axiom,
% 2.80/1.56 ! [A: $tType,Xs: stream @ A] :
% 2.80/1.56 ~ ( coinductive_lnull @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_4_stream_ORep__inverse,axiom,
% 2.80/1.56 ! [A: $tType,X: stream @ A] :
% 2.80/1.56 ( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ X ) )
% 2.80/1.56 = X ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_5_stream__of__llist__llist__of__stream,axiom,
% 2.80/1.56 ! [A: $tType,Xs: stream @ A] :
% 2.80/1.56 ( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ Xs ) )
% 2.80/1.56 = Xs ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_6_unfold__stream_Osimps_I1_J,axiom,
% 2.80/1.56 ! [B: $tType,A: $tType,G1: A > B,G2: A > A,A2: A] :
% 2.80/1.56 ( ( shd @ B @ ( coindu139217191stream @ A @ B @ G1 @ G2 @ A2 ) )
% 2.80/1.56 = ( G1 @ A2 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_7_shd__stream__of__llist,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( shd @ A @ ( coindu2010755910_llist @ A @ Xs ) )
% 2.80/1.56 = ( coinductive_lhd @ A @ Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_8_lset__lmember,axiom,
% 2.80/1.56 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 = ( coinductive_lmember @ A @ X @ Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_9_lfinite__llist__of__stream,axiom,
% 2.80/1.56 ! [A: $tType,Xs: stream @ A] :
% 2.80/1.56 ~ ( coinductive_lfinite @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_10__092_060open_062x_A_092_060in_062_Asset_Axs_092_060close_062,axiom,
% 2.80/1.56 member @ a @ x @ ( sset @ a @ xs ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_11_llist_Oset__sel_I1_J,axiom,
% 2.80/1.56 ! [A: $tType,A2: coinductive_llist @ A] :
% 2.80/1.56 ( ~ ( coinductive_lnull @ A @ A2 )
% 2.80/1.56 => ( member @ A @ ( coinductive_lhd @ A @ A2 ) @ ( coinductive_lset @ A @ A2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_12_sinterleave_Osimps_I1_J,axiom,
% 2.80/1.56 ! [A: $tType,S1: stream @ A,S2: stream @ A] :
% 2.80/1.56 ( ( shd @ A @ ( sinterleave @ A @ S1 @ S2 ) )
% 2.80/1.56 = ( shd @ A @ S1 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_13_Coinductive__List_Olset__into__lsetp,axiom,
% 2.80/1.56 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 => ( coinductive_lsetp @ A @ Xs @ X ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_14_llist__of__stream__stream__of__llist,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 => ( ( coindu1724414836stream @ A @ ( coindu2010755910_llist @ A @ Xs ) )
% 2.80/1.56 = Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_15_lzip_Oexhaust,axiom,
% 2.80/1.56 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.80/1.56 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 | ( coinductive_lnull @ B @ Ys ) )
% 2.80/1.56 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_16_lappend_Oexhaust,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ~ ( coinductive_lnull @ A @ Ys ) )
% 2.80/1.56 => ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_17_lnull__imp__lfinite,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_18_ltakeWhile_Oexhaust,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.80/1.56 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 | ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
% 2.80/1.56 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_19_shd__sset,axiom,
% 2.80/1.56 ! [A: $tType,A2: stream @ A] : ( member @ A @ ( shd @ A @ A2 ) @ ( sset @ A @ A2 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_20_lsetp__into__lset,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 2.80/1.56 ( ( coinductive_lsetp @ A @ Xs @ X )
% 2.80/1.56 => ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_21_cr__streamI,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 => ( coindu1183105481stream @ A @ Xs @ ( coindu2010755910_llist @ A @ Xs ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_22_Stream_Osmember__def,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( smember @ A )
% 2.80/1.56 = ( ^ [X2: A,S: stream @ A] : ( member @ A @ X2 @ ( sset @ A @ S ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_23_llist__set__induct,axiom,
% 2.80/1.56 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
% 2.80/1.56 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 => ( ! [Xs2: coinductive_llist @ A] :
% 2.80/1.56 ( ~ ( coinductive_lnull @ A @ Xs2 )
% 2.80/1.56 => ( P @ ( coinductive_lhd @ A @ Xs2 ) @ Xs2 ) )
% 2.80/1.56 => ( ! [Xs2: coinductive_llist @ A,Y2: A] :
% 2.80/1.56 ( ~ ( coinductive_lnull @ A @ Xs2 )
% 2.80/1.56 => ( ( member @ A @ Y2 @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
% 2.80/1.56 => ( ( P @ Y2 @ ( coinductive_ltl @ A @ Xs2 ) )
% 2.80/1.56 => ( P @ Y2 @ Xs2 ) ) ) )
% 2.80/1.56 => ( P @ X @ Xs ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_24_lset__eq__empty,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( ( coinductive_lset @ A @ Xs )
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.80/1.56 = ( coinductive_lnull @ A @ Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_25_lnull__ldropWhile,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lnull @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 2.80/1.56 = ( ! [X2: A] :
% 2.80/1.56 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 => ( P @ X2 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_26_lhd__lconcat,axiom,
% 2.80/1.56 ! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A )] :
% 2.80/1.56 ( ~ ( coinductive_lnull @ ( coinductive_llist @ A ) @ Xss )
% 2.80/1.56 => ( ~ ( coinductive_lnull @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) )
% 2.80/1.56 => ( ( coinductive_lhd @ A @ ( coinductive_lconcat @ A @ Xss ) )
% 2.80/1.56 = ( coinductive_lhd @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_27_sset__sinterleave,axiom,
% 2.80/1.56 ! [A: $tType,S1: stream @ A,S2: stream @ A] :
% 2.80/1.56 ( ( sset @ A @ ( sinterleave @ A @ S1 @ S2 ) )
% 2.80/1.56 = ( sup_sup @ ( set @ A ) @ ( sset @ A @ S1 ) @ ( sset @ A @ S2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_28_ltakeWhile_Odisc__iff_I2_J,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
% 2.80/1.56 = ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 & ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_29_ltakeWhile_Odisc__iff_I1_J,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 2.80/1.56 = ( ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 | ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_30_lnull__ltakeWhile,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 2.80/1.56 = ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_31_lnull__lfilter,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lnull @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 2.80/1.56 = ( ! [X2: A] :
% 2.80/1.56 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 => ~ ( P @ X2 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_32_lfilter__idem,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 2.80/1.56 = ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_33_lfinite__ltl,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lfinite @ A @ ( coinductive_ltl @ A @ Xs ) )
% 2.80/1.56 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_34_ltakeWhile_Osimps_I4_J,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.80/1.56 ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( ( P @ ( coinductive_lhd @ A @ Xs ) )
% 2.80/1.56 => ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 2.80/1.56 = ( coindu501562517eWhile @ A @ P @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_35_lnull__ltlI,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_36_in__lset__ltlD,axiom,
% 2.80/1.56 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) )
% 2.80/1.56 => ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_37_lfilter__cong,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 2.80/1.56 ( ( Xs = Ys )
% 2.80/1.56 => ( ! [X3: A] :
% 2.80/1.56 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Ys ) )
% 2.80/1.56 => ( ( P @ X3 )
% 2.80/1.56 = ( Q @ X3 ) ) )
% 2.80/1.56 => ( ( coinductive_lfilter @ A @ P @ Xs )
% 2.80/1.56 = ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_38_lfilter__id__conv,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 2.80/1.56 = Xs )
% 2.80/1.56 = ( ! [X2: A] :
% 2.80/1.56 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 => ( P @ X2 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_39_lfinite__lfilterI,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.80/1.56 ( ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 => ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_40_ltakeWhile__all,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.80/1.56 ( ! [X3: A] :
% 2.80/1.56 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 => ( P @ X3 ) )
% 2.80/1.56 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 2.80/1.56 = Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_41_ltakeWhile__cong,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 2.80/1.56 ( ( Xs = Ys )
% 2.80/1.56 => ( ! [X3: A] :
% 2.80/1.56 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Ys ) )
% 2.80/1.56 => ( ( P @ X3 )
% 2.80/1.56 = ( Q @ X3 ) ) )
% 2.80/1.56 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 2.80/1.56 = ( coindu501562517eWhile @ A @ Q @ Ys ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_42_lset__ltakeWhileD,axiom,
% 2.80/1.56 ! [A: $tType,X: A,P: A > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
% 2.80/1.56 => ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 & ( P @ X ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_43_mem__Collect__eq,axiom,
% 2.80/1.56 ! [A: $tType,A2: A,P: A > $o] :
% 2.80/1.56 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.80/1.56 = ( P @ A2 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_44_Collect__mem__eq,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A] :
% 2.80/1.56 ( ( collect @ A
% 2.80/1.56 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 2.80/1.56 = A3 ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_45_Collect__cong,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.80/1.56 ( ! [X3: A] :
% 2.80/1.56 ( ( P @ X3 )
% 2.80/1.56 = ( Q @ X3 ) )
% 2.80/1.56 => ( ( collect @ A @ P )
% 2.80/1.56 = ( collect @ A @ Q ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_46_ext,axiom,
% 2.80/1.56 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.80/1.56 ( ! [X3: A] :
% 2.80/1.56 ( ( F @ X3 )
% 2.80/1.56 = ( G @ X3 ) )
% 2.80/1.56 => ( F = G ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_47_ldropWhile__cong,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 2.80/1.56 ( ( Xs = Ys )
% 2.80/1.56 => ( ! [X3: A] :
% 2.80/1.56 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Ys ) )
% 2.80/1.56 => ( ( P @ X3 )
% 2.80/1.56 = ( Q @ X3 ) ) )
% 2.80/1.56 => ( ( coindu218763757pWhile @ A @ P @ Xs )
% 2.80/1.56 = ( coindu218763757pWhile @ A @ Q @ Ys ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_48_in__lset__ldropWhileD,axiom,
% 2.80/1.56 ! [A: $tType,X: A,P: A > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) )
% 2.80/1.56 => ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_49_sset__neq__empty,axiom,
% 2.80/1.56 ! [A: $tType,Xs: stream @ A] :
% 2.80/1.56 ( ( sset @ A @ Xs )
% 2.80/1.56 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_50_llist_Oset__sel_I2_J,axiom,
% 2.80/1.56 ! [A: $tType,A2: coinductive_llist @ A,X: A] :
% 2.80/1.56 ( ~ ( coinductive_lnull @ A @ A2 )
% 2.80/1.56 => ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ A2 ) ) )
% 2.80/1.56 => ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_51_lfinite__induct,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 2.80/1.56 ( ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 => ( ! [Xs2: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lnull @ A @ Xs2 )
% 2.80/1.56 => ( P @ Xs2 ) )
% 2.80/1.56 => ( ! [Xs2: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lfinite @ A @ Xs2 )
% 2.80/1.56 => ( ~ ( coinductive_lnull @ A @ Xs2 )
% 2.80/1.56 => ( ( P @ ( coinductive_ltl @ A @ Xs2 ) )
% 2.80/1.56 => ( P @ Xs2 ) ) ) )
% 2.80/1.56 => ( P @ Xs ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_52_llist_Ocoinduct__strong,axiom,
% 2.80/1.56 ! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
% 2.80/1.56 ( ( R @ Llist @ Llist2 )
% 2.80/1.56 => ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
% 2.80/1.56 ( ( R @ Llist3 @ Llist4 )
% 2.80/1.56 => ( ( ( coinductive_lnull @ A @ Llist3 )
% 2.80/1.56 = ( coinductive_lnull @ A @ Llist4 ) )
% 2.80/1.56 & ( ~ ( coinductive_lnull @ A @ Llist3 )
% 2.80/1.56 => ( ~ ( coinductive_lnull @ A @ Llist4 )
% 2.80/1.56 => ( ( ( coinductive_lhd @ A @ Llist3 )
% 2.80/1.56 = ( coinductive_lhd @ A @ Llist4 ) )
% 2.80/1.56 & ( ( R @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) )
% 2.80/1.56 | ( ( coinductive_ltl @ A @ Llist3 )
% 2.80/1.56 = ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) ) )
% 2.80/1.56 => ( Llist = Llist2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_53_llist_Ocoinduct,axiom,
% 2.80/1.56 ! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
% 2.80/1.56 ( ( R @ Llist @ Llist2 )
% 2.80/1.56 => ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
% 2.80/1.56 ( ( R @ Llist3 @ Llist4 )
% 2.80/1.56 => ( ( ( coinductive_lnull @ A @ Llist3 )
% 2.80/1.56 = ( coinductive_lnull @ A @ Llist4 ) )
% 2.80/1.56 & ( ~ ( coinductive_lnull @ A @ Llist3 )
% 2.80/1.56 => ( ~ ( coinductive_lnull @ A @ Llist4 )
% 2.80/1.56 => ( ( ( coinductive_lhd @ A @ Llist3 )
% 2.80/1.56 = ( coinductive_lhd @ A @ Llist4 ) )
% 2.80/1.56 & ( R @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) )
% 2.80/1.56 => ( Llist = Llist2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_54_llist_Oexpand,axiom,
% 2.80/1.56 ! [A: $tType,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
% 2.80/1.56 ( ( ( coinductive_lnull @ A @ Llist )
% 2.80/1.56 = ( coinductive_lnull @ A @ Llist2 ) )
% 2.80/1.56 => ( ( ~ ( coinductive_lnull @ A @ Llist )
% 2.80/1.56 => ( ~ ( coinductive_lnull @ A @ Llist2 )
% 2.80/1.56 => ( ( ( coinductive_lhd @ A @ Llist )
% 2.80/1.56 = ( coinductive_lhd @ A @ Llist2 ) )
% 2.80/1.56 & ( ( coinductive_ltl @ A @ Llist )
% 2.80/1.56 = ( coinductive_ltl @ A @ Llist2 ) ) ) ) )
% 2.80/1.56 => ( Llist = Llist2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_55_lhd__ltakeWhile,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.80/1.56 ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( ( P @ ( coinductive_lhd @ A @ Xs ) )
% 2.80/1.56 => ( ( coinductive_lhd @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 2.80/1.56 = ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_56_ltakeWhile_Odisc_I1_J,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.80/1.56 ( ( ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 | ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
% 2.80/1.56 => ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_57_ltakeWhile_Odisc_I2_J,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.80/1.56 ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( ( P @ ( coinductive_lhd @ A @ Xs ) )
% 2.80/1.56 => ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_58_lfinite__ltakeWhile,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lfinite @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 2.80/1.56 = ( ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 | ? [X2: A] :
% 2.80/1.56 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 & ~ ( P @ X2 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_59_stream__of__llist__def,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( coindu2010755910_llist @ A )
% 2.80/1.56 = ( coindu139217191stream @ ( coinductive_llist @ A ) @ A @ ( coinductive_lhd @ A ) @ ( coinductive_ltl @ A ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_60_lset__lnull,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( ( coinductive_lset @ A @ Xs )
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_61_lfinite__ldropWhile,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lfinite @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 2.80/1.56 = ( ? [X2: A] :
% 2.80/1.56 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 & ~ ( P @ X2 ) )
% 2.80/1.56 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_62_lhd__ldropWhile,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.80/1.56 ( ? [X4: A] :
% 2.80/1.56 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 & ~ ( P @ X4 ) )
% 2.80/1.56 => ~ ( P @ ( coinductive_lhd @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_63_lhd__ldropWhile__in__lset,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.80/1.56 ( ? [X4: A] :
% 2.80/1.56 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 & ~ ( P @ X4 ) )
% 2.80/1.56 => ( member @ A @ ( coinductive_lhd @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_64_Un__empty,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.80/1.56 ( ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.80/1.56 = ( ( A3
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.80/1.56 & ( B2
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_65_sup__bot_Oright__neutral,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [A2: A] :
% 2.80/1.56 ( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
% 2.80/1.56 = A2 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_66_sup__bot_Oleft__neutral,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [A2: A] :
% 2.80/1.56 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
% 2.80/1.56 = A2 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_67_sup__eq__bot__iff,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A] :
% 2.80/1.56 ( ( ( sup_sup @ A @ X @ Y )
% 2.80/1.56 = ( bot_bot @ A ) )
% 2.80/1.56 = ( ( X
% 2.80/1.56 = ( bot_bot @ A ) )
% 2.80/1.56 & ( Y
% 2.80/1.56 = ( bot_bot @ A ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_68_bot__eq__sup__iff,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A] :
% 2.80/1.56 ( ( ( bot_bot @ A )
% 2.80/1.56 = ( sup_sup @ A @ X @ Y ) )
% 2.80/1.56 = ( ( X
% 2.80/1.56 = ( bot_bot @ A ) )
% 2.80/1.56 & ( Y
% 2.80/1.56 = ( bot_bot @ A ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_69_Un__iff,axiom,
% 2.80/1.56 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 2.80/1.56 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
% 2.80/1.56 = ( ( member @ A @ C @ A3 )
% 2.80/1.56 | ( member @ A @ C @ B2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_70_UnCI,axiom,
% 2.80/1.56 ! [A: $tType,C: A,B2: set @ A,A3: set @ A] :
% 2.80/1.56 ( ( ~ ( member @ A @ C @ B2 )
% 2.80/1.56 => ( member @ A @ C @ A3 ) )
% 2.80/1.56 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_71_sup_Oright__idem,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [A2: A,B3: A] :
% 2.80/1.56 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ B3 )
% 2.80/1.56 = ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_72_sup__left__idem,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A] :
% 2.80/1.56 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.80/1.56 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_73_empty__iff,axiom,
% 2.80/1.56 ! [A: $tType,C: A] :
% 2.80/1.56 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_74_all__not__in__conv,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A] :
% 2.80/1.56 ( ( ! [X2: A] :
% 2.80/1.56 ~ ( member @ A @ X2 @ A3 ) )
% 2.80/1.56 = ( A3
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_75_Collect__empty__eq,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o] :
% 2.80/1.56 ( ( ( collect @ A @ P )
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.80/1.56 = ( ! [X2: A] :
% 2.80/1.56 ~ ( P @ X2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_76_empty__Collect__eq,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o] :
% 2.80/1.56 ( ( ( bot_bot @ ( set @ A ) )
% 2.80/1.56 = ( collect @ A @ P ) )
% 2.80/1.56 = ( ! [X2: A] :
% 2.80/1.56 ~ ( P @ X2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_77_sup__apply,axiom,
% 2.80/1.56 ! [B: $tType,A: $tType] :
% 2.80/1.56 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 2.80/1.56 => ( ( sup_sup @ ( A > B ) )
% 2.80/1.56 = ( ^ [F2: A > B,G3: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_78_sup_Oidem,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [A2: A] :
% 2.80/1.56 ( ( sup_sup @ A @ A2 @ A2 )
% 2.80/1.56 = A2 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_79_sup__idem,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A] :
% 2.80/1.56 ( ( sup_sup @ A @ X @ X )
% 2.80/1.56 = X ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_80_sup_Oleft__idem,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [A2: A,B3: A] :
% 2.80/1.56 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B3 ) )
% 2.80/1.56 = ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_81_bot__set__def,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( bot_bot @ ( set @ A ) )
% 2.80/1.56 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_82_emptyE,axiom,
% 2.80/1.56 ! [A: $tType,A2: A] :
% 2.80/1.56 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_83_equals0D,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,A2: A] :
% 2.80/1.56 ( ( A3
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.80/1.56 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_84_equals0I,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A] :
% 2.80/1.56 ( ! [Y2: A] :
% 2.80/1.56 ~ ( member @ A @ Y2 @ A3 )
% 2.80/1.56 => ( A3
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_85_ex__in__conv,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A] :
% 2.80/1.56 ( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
% 2.80/1.56 = ( A3
% 2.80/1.56 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_86_inf__sup__aci_I8_J,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A] :
% 2.80/1.56 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.80/1.56 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_87_inf__sup__aci_I7_J,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A,Z: A] :
% 2.80/1.56 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.80/1.56 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_88_inf__sup__aci_I6_J,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A,Z: A] :
% 2.80/1.56 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 2.80/1.56 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_89_inf__sup__aci_I5_J,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.80/1.56 => ( ( sup_sup @ A )
% 2.80/1.56 = ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_90_sup__fun__def,axiom,
% 2.80/1.56 ! [B: $tType,A: $tType] :
% 2.80/1.56 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 2.80/1.56 => ( ( sup_sup @ ( A > B ) )
% 2.80/1.56 = ( ^ [F2: A > B,G3: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_91_sup_Oassoc,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [A2: A,B3: A,C: A] :
% 2.80/1.56 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ C )
% 2.80/1.56 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_92_sup__assoc,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A,Z: A] :
% 2.80/1.56 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 2.80/1.56 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_93_sup_Ocommute,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.80/1.56 => ( ( sup_sup @ A )
% 2.80/1.56 = ( ^ [A4: A,B4: A] : ( sup_sup @ A @ B4 @ A4 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_94_sup__commute,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.80/1.56 => ( ( sup_sup @ A )
% 2.80/1.56 = ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_95_sup_Oleft__commute,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [B3: A,A2: A,C: A] :
% 2.80/1.56 ( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A2 @ C ) )
% 2.80/1.56 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_96_sup__left__commute,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A,Z: A] :
% 2.80/1.56 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.80/1.56 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_97_UnE,axiom,
% 2.80/1.56 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 2.80/1.56 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
% 2.80/1.56 => ( ~ ( member @ A @ C @ A3 )
% 2.80/1.56 => ( member @ A @ C @ B2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_98_UnI1,axiom,
% 2.80/1.56 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 2.80/1.56 ( ( member @ A @ C @ A3 )
% 2.80/1.56 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_99_UnI2,axiom,
% 2.80/1.56 ! [A: $tType,C: A,B2: set @ A,A3: set @ A] :
% 2.80/1.56 ( ( member @ A @ C @ B2 )
% 2.80/1.56 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_100_bex__Un,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A,P: A > $o] :
% 2.80/1.56 ( ( ? [X2: A] :
% 2.80/1.56 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
% 2.80/1.56 & ( P @ X2 ) ) )
% 2.80/1.56 = ( ? [X2: A] :
% 2.80/1.56 ( ( member @ A @ X2 @ A3 )
% 2.80/1.56 & ( P @ X2 ) )
% 2.80/1.56 | ? [X2: A] :
% 2.80/1.56 ( ( member @ A @ X2 @ B2 )
% 2.80/1.56 & ( P @ X2 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_101_ball__Un,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A,P: A > $o] :
% 2.80/1.56 ( ( ! [X2: A] :
% 2.80/1.56 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
% 2.80/1.56 => ( P @ X2 ) ) )
% 2.80/1.56 = ( ! [X2: A] :
% 2.80/1.56 ( ( member @ A @ X2 @ A3 )
% 2.80/1.56 => ( P @ X2 ) )
% 2.80/1.56 & ! [X2: A] :
% 2.80/1.56 ( ( member @ A @ X2 @ B2 )
% 2.80/1.56 => ( P @ X2 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_102_Un__assoc,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 2.80/1.56 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ C2 )
% 2.80/1.56 = ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_103_Un__absorb,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A] :
% 2.80/1.56 ( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
% 2.80/1.56 = A3 ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_104_Un__commute,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( sup_sup @ ( set @ A ) )
% 2.80/1.56 = ( ^ [A5: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_105_Un__left__absorb,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.80/1.56 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
% 2.80/1.56 = ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_106_Un__left__commute,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 2.80/1.56 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
% 2.80/1.56 = ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_107_sup__bot__left,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A] :
% 2.80/1.56 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
% 2.80/1.56 = X ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_108_sup__bot__right,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A] :
% 2.80/1.56 ( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
% 2.80/1.56 = X ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_109_Un__empty__left,axiom,
% 2.80/1.56 ! [A: $tType,B2: set @ A] :
% 2.80/1.56 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
% 2.80/1.56 = B2 ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_110_Un__empty__right,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A] :
% 2.80/1.56 ( ( sup_sup @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.80/1.56 = A3 ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_111_bot__apply,axiom,
% 2.80/1.56 ! [C3: $tType,D: $tType] :
% 2.80/1.56 ( ( bot @ C3 @ ( type2 @ C3 ) )
% 2.80/1.56 => ( ( bot_bot @ ( D > C3 ) )
% 2.80/1.56 = ( ^ [X2: D] : ( bot_bot @ C3 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_112_ltl__lconcat,axiom,
% 2.80/1.56 ! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A )] :
% 2.80/1.56 ( ~ ( coinductive_lnull @ ( coinductive_llist @ A ) @ Xss )
% 2.80/1.56 => ( ~ ( coinductive_lnull @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) )
% 2.80/1.56 => ( ( coinductive_ltl @ A @ ( coinductive_lconcat @ A @ Xss ) )
% 2.80/1.56 = ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) ) @ ( coinductive_lconcat @ A @ ( coinductive_ltl @ ( coinductive_llist @ A ) @ Xss ) ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_113_gen__lset__def,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( coinductive_gen_lset @ A )
% 2.80/1.56 = ( ^ [A5: set @ A,Xs3: coinductive_llist @ A] : ( sup_sup @ ( set @ A ) @ A5 @ ( coinductive_lset @ A @ Xs3 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_114_lset__code,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( coinductive_lset @ A )
% 2.80/1.56 = ( coinductive_gen_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_115_unfold__llist__id,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coindu1441602521_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_lnull @ A ) @ ( coinductive_lhd @ A ) @ ( coinductive_ltl @ A ) @ Xs )
% 2.80/1.56 = Xs ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_116_Set_Ois__empty__def,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( is_empty @ A )
% 2.80/1.56 = ( ^ [A5: set @ A] :
% 2.80/1.56 ( A5
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_117_lnull__lappend,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.80/1.56 = ( ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 & ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_118_lappend_Odisc__iff_I2_J,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 2.80/1.56 = ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_119_lfinite__lappend,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.80/1.56 = ( ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 & ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_120_lconcat__lappend,axiom,
% 2.80/1.56 ! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A ),Yss: coinductive_llist @ ( coinductive_llist @ A )] :
% 2.80/1.56 ( ( coinductive_lfinite @ ( coinductive_llist @ A ) @ Xss )
% 2.80/1.56 => ( ( coinductive_lconcat @ A @ ( coinductive_lappend @ ( coinductive_llist @ A ) @ Xss @ Yss ) )
% 2.80/1.56 = ( coinductive_lappend @ A @ ( coinductive_lconcat @ A @ Xss ) @ ( coinductive_lconcat @ A @ Yss ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_121_unfold__llist_Odisc__iff_I1_J,axiom,
% 2.80/1.56 ! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A2: A] :
% 2.80/1.56 ( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
% 2.80/1.56 = ( P2 @ A2 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_122_unfold__llist_Odisc__iff_I2_J,axiom,
% 2.80/1.56 ! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A2: A] :
% 2.80/1.56 ( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) )
% 2.80/1.56 = ( ~ ( P2 @ A2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_123_ltl__lappend,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.80/1.56 = ( coinductive_ltl @ A @ Ys ) ) )
% 2.80/1.56 & ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.80/1.56 = ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ Xs ) @ Ys ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_124_lhd__lappend,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.80/1.56 = ( coinductive_lhd @ A @ Ys ) ) )
% 2.80/1.56 & ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.80/1.56 = ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_125_lfilter__lappend__lfinite,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.80/1.56 = ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_126_lappend__ltakeWhile__ldropWhile,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lappend @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 2.80/1.56 = Xs ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_127_lset__lappend__lfinite,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 => ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.80/1.56 = ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_128_lappend__assoc,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 2.80/1.56 = ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_129_lappend__lnull2,axiom,
% 2.80/1.56 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lnull @ A @ Ys )
% 2.80/1.56 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.80/1.56 = Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_130_lappend__lnull1,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.80/1.56 = Ys ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_131_lappend_Odisc_I1_J,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( ( coinductive_lnull @ A @ Ys )
% 2.80/1.56 => ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_132_lappend_Odisc_I2_J,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 | ~ ( coinductive_lnull @ A @ Ys ) )
% 2.80/1.56 => ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_133_lappend__inf,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.80/1.56 = Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_134_unfold__llist_Odisc_I1_J,axiom,
% 2.80/1.56 ! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 2.80/1.56 ( ( P2 @ A2 )
% 2.80/1.56 => ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_135_unfold__llist_Odisc_I2_J,axiom,
% 2.80/1.56 ! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 2.80/1.56 ( ~ ( P2 @ A2 )
% 2.80/1.56 => ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_136_unfold__llist_Osimps_I4_J,axiom,
% 2.80/1.56 ! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 2.80/1.56 ( ~ ( P2 @ A2 )
% 2.80/1.56 => ( ( coinductive_ltl @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
% 2.80/1.56 = ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_137_unfold__llist_Osimps_I3_J,axiom,
% 2.80/1.56 ! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 2.80/1.56 ( ~ ( P2 @ A2 )
% 2.80/1.56 => ( ( coinductive_lhd @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
% 2.80/1.56 = ( G21 @ A2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_138_lappend__ltl,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ( ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ Xs ) @ Ys )
% 2.80/1.56 = ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_139_in__lset__lappend__iff,axiom,
% 2.80/1.56 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 2.80/1.56 = ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 | ( ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 & ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_140_ltakeWhile__lappend1,axiom,
% 2.80/1.56 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.56 => ( ~ ( P @ X )
% 2.80/1.56 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.80/1.56 = ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_141_lfilter__eq__lappend__lfiniteD,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 2.80/1.56 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 2.80/1.56 = ( coinductive_lappend @ A @ Ys @ Zs ) )
% 2.80/1.56 => ( ( coinductive_lfinite @ A @ Ys )
% 2.80/1.56 => ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 2.80/1.56 ( ( Xs
% 2.80/1.56 = ( coinductive_lappend @ A @ Us @ Vs ) )
% 2.80/1.56 & ( coinductive_lfinite @ A @ Us )
% 2.80/1.56 & ( Ys
% 2.80/1.56 = ( coinductive_lfilter @ A @ P @ Us ) )
% 2.80/1.56 & ( Zs
% 2.80/1.56 = ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_142_lset__lappend__conv,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 => ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.80/1.56 = ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) )
% 2.80/1.56 & ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 => ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.80/1.56 = ( coinductive_lset @ A @ Xs ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_143_bot__fun__def,axiom,
% 2.80/1.56 ! [B: $tType,A: $tType] :
% 2.80/1.56 ( ( bot @ B @ ( type2 @ B ) )
% 2.80/1.56 => ( ( bot_bot @ ( A > B ) )
% 2.80/1.56 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_144_bot__empty__eq,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( bot_bot @ ( A > $o ) )
% 2.80/1.56 = ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_145_Collect__empty__eq__bot,axiom,
% 2.80/1.56 ! [A: $tType,P: A > $o] :
% 2.80/1.56 ( ( ( collect @ A @ P )
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.80/1.56 = ( P
% 2.80/1.56 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_146_lstrict__prefix__lappend__conv,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.80/1.56 = ( ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 & ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_147_ldistinct__coinduct,axiom,
% 2.80/1.56 ! [A: $tType,X5: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( X5 @ Xs )
% 2.80/1.56 => ( ! [Xs2: coinductive_llist @ A] :
% 2.80/1.56 ( ( X5 @ Xs2 )
% 2.80/1.56 => ( ~ ( coinductive_lnull @ A @ Xs2 )
% 2.80/1.56 => ( ~ ( member @ A @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
% 2.80/1.56 & ( ( X5 @ ( coinductive_ltl @ A @ Xs2 ) )
% 2.80/1.56 | ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs2 ) ) ) ) ) )
% 2.80/1.56 => ( coindu351974385stinct @ A @ Xs ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_148_llist__less__induct,axiom,
% 2.80/1.56 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ! [Xs2: coinductive_llist @ A] :
% 2.80/1.56 ( ! [Ys2: coinductive_llist @ A] :
% 2.80/1.56 ( ( coindu1478340336prefix @ A @ Ys2 @ Xs2 )
% 2.80/1.56 => ( P @ Ys2 ) )
% 2.80/1.56 => ( P @ Xs2 ) )
% 2.80/1.56 => ( P @ Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_149_ldistinct__ltlI,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coindu351974385stinct @ A @ Xs )
% 2.80/1.56 => ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_150_ldistinct__lfilterI,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.80/1.56 ( ( coindu351974385stinct @ A @ Xs )
% 2.80/1.56 => ( coindu351974385stinct @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_151_lstrict__prefix__lfinite1,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( coindu1478340336prefix @ A @ Xs @ Ys )
% 2.80/1.56 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_152_ldistinct__lhdD,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.80/1.56 ( ( coindu351974385stinct @ A @ Xs )
% 2.80/1.56 => ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.56 => ~ ( member @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_153_ldistinct__lappend,axiom,
% 2.80/1.56 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.56 ( ( coindu351974385stinct @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.80/1.56 = ( ( coindu351974385stinct @ A @ Xs )
% 2.80/1.56 & ( ( coinductive_lfinite @ A @ Xs )
% 2.80/1.56 => ( ( coindu351974385stinct @ A @ Ys )
% 2.80/1.56 & ( ( inf_inf @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) )
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_154_ldistinct__lconcat,axiom,
% 2.80/1.56 ! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A )] :
% 2.80/1.56 ( ( coindu351974385stinct @ ( coinductive_llist @ A ) @ Xss )
% 2.80/1.56 => ( ! [Ys3: coinductive_llist @ A] :
% 2.80/1.56 ( ( member @ ( coinductive_llist @ A ) @ Ys3 @ ( coinductive_lset @ ( coinductive_llist @ A ) @ Xss ) )
% 2.80/1.56 => ( coindu351974385stinct @ A @ Ys3 ) )
% 2.80/1.56 => ( ! [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 2.80/1.56 ( ( member @ ( coinductive_llist @ A ) @ Ys3 @ ( coinductive_lset @ ( coinductive_llist @ A ) @ Xss ) )
% 2.80/1.56 => ( ( member @ ( coinductive_llist @ A ) @ Zs2 @ ( coinductive_lset @ ( coinductive_llist @ A ) @ Xss ) )
% 2.80/1.56 => ( ( Ys3 != Zs2 )
% 2.80/1.56 => ( ( inf_inf @ ( set @ A ) @ ( coinductive_lset @ A @ Ys3 ) @ ( coinductive_lset @ A @ Zs2 ) )
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ) )
% 2.80/1.56 => ( coindu351974385stinct @ A @ ( coinductive_lconcat @ A @ Xss ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_155_llist_Ocase__eq__if,axiom,
% 2.80/1.56 ! [A: $tType,B: $tType] :
% 2.80/1.56 ( ( coindu1381640503_llist @ B @ A )
% 2.80/1.56 = ( ^ [F1: B,F22: A > ( coinductive_llist @ A ) > B,Llist5: coinductive_llist @ A] : ( if @ B @ ( coinductive_lnull @ A @ Llist5 ) @ F1 @ ( F22 @ ( coinductive_lhd @ A @ Llist5 ) @ ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_156_inf__apply,axiom,
% 2.80/1.56 ! [B: $tType,A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 2.80/1.56 => ( ( inf_inf @ ( A > B ) )
% 2.80/1.56 = ( ^ [F2: A > B,G3: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_157_inf_Oidem,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [A2: A] :
% 2.80/1.56 ( ( inf_inf @ A @ A2 @ A2 )
% 2.80/1.56 = A2 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_158_inf__idem,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A] :
% 2.80/1.56 ( ( inf_inf @ A @ X @ X )
% 2.80/1.56 = X ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_159_inf_Oleft__idem,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [A2: A,B3: A] :
% 2.80/1.56 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B3 ) )
% 2.80/1.56 = ( inf_inf @ A @ A2 @ B3 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_160_inf__left__idem,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A] :
% 2.80/1.56 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 2.80/1.56 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_161_inf_Oright__idem,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [A2: A,B3: A] :
% 2.80/1.56 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B3 ) @ B3 )
% 2.80/1.56 = ( inf_inf @ A @ A2 @ B3 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_162_inf__right__idem,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A] :
% 2.80/1.56 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
% 2.80/1.56 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_163_IntI,axiom,
% 2.80/1.56 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 2.80/1.56 ( ( member @ A @ C @ A3 )
% 2.80/1.56 => ( ( member @ A @ C @ B2 )
% 2.80/1.56 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_164_Int__iff,axiom,
% 2.80/1.56 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 2.80/1.56 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
% 2.80/1.56 = ( ( member @ A @ C @ A3 )
% 2.80/1.56 & ( member @ A @ C @ B2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_165_inf__bot__left,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A] :
% 2.80/1.56 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
% 2.80/1.56 = ( bot_bot @ A ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_166_inf__bot__right,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A] :
% 2.80/1.56 ( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
% 2.80/1.56 = ( bot_bot @ A ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_167_inf__sup__absorb,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A] :
% 2.80/1.56 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.80/1.56 = X ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_168_sup__inf__absorb,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A] :
% 2.80/1.56 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 2.80/1.56 = X ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_169_inf__sup__aci_I4_J,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A] :
% 2.80/1.56 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 2.80/1.56 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_170_inf__sup__aci_I3_J,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A,Z: A] :
% 2.80/1.56 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.80/1.56 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_171_inf__sup__aci_I2_J,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A,Z: A] :
% 2.80/1.56 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 2.80/1.56 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_172_inf__sup__aci_I1_J,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.80/1.56 => ( ( inf_inf @ A )
% 2.80/1.56 = ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_173_IntE,axiom,
% 2.80/1.56 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 2.80/1.56 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
% 2.80/1.56 => ~ ( ( member @ A @ C @ A3 )
% 2.80/1.56 => ~ ( member @ A @ C @ B2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_174_IntD1,axiom,
% 2.80/1.56 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 2.80/1.56 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
% 2.80/1.56 => ( member @ A @ C @ A3 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_175_IntD2,axiom,
% 2.80/1.56 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 2.80/1.56 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
% 2.80/1.56 => ( member @ A @ C @ B2 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_176_Int__assoc,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 2.80/1.56 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ C2 )
% 2.80/1.56 = ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_177_Int__absorb,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A] :
% 2.80/1.56 ( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
% 2.80/1.56 = A3 ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_178_Int__commute,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( inf_inf @ ( set @ A ) )
% 2.80/1.56 = ( ^ [A5: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_179_Int__left__absorb,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.80/1.56 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
% 2.80/1.56 = ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_180_inf__fun__def,axiom,
% 2.80/1.56 ! [B: $tType,A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 2.80/1.56 => ( ( inf_inf @ ( A > B ) )
% 2.80/1.56 = ( ^ [F2: A > B,G3: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_181_Int__left__commute,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 2.80/1.56 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
% 2.80/1.56 = ( inf_inf @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_182_inf_Oassoc,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [A2: A,B3: A,C: A] :
% 2.80/1.56 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B3 ) @ C )
% 2.80/1.56 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_183_inf__assoc,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A,Z: A] :
% 2.80/1.56 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 2.80/1.56 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_184_inf_Ocommute,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.80/1.56 => ( ( inf_inf @ A )
% 2.80/1.56 = ( ^ [A4: A,B4: A] : ( inf_inf @ A @ B4 @ A4 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_185_inf__commute,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.80/1.56 => ( ( inf_inf @ A )
% 2.80/1.56 = ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_186_inf_Oleft__commute,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [B3: A,A2: A,C: A] :
% 2.80/1.56 ( ( inf_inf @ A @ B3 @ ( inf_inf @ A @ A2 @ C ) )
% 2.80/1.56 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_187_inf__left__commute,axiom,
% 2.80/1.56 ! [A: $tType] :
% 2.80/1.56 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.80/1.56 => ! [X: A,Y: A,Z: A] :
% 2.80/1.56 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.80/1.56 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_188_Un__Int__distrib2,axiom,
% 2.80/1.56 ! [A: $tType,B2: set @ A,C2: set @ A,A3: set @ A] :
% 2.80/1.56 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) @ A3 )
% 2.80/1.56 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ A3 ) @ ( sup_sup @ ( set @ A ) @ C2 @ A3 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_189_Int__Un__distrib2,axiom,
% 2.80/1.56 ! [A: $tType,B2: set @ A,C2: set @ A,A3: set @ A] :
% 2.80/1.56 ( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) @ A3 )
% 2.80/1.56 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C2 @ A3 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_190_Un__Int__distrib,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 2.80/1.56 ( ( sup_sup @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
% 2.80/1.56 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ ( sup_sup @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_191_Int__Un__distrib,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 2.80/1.56 ( ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
% 2.80/1.56 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ ( inf_inf @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_192_Un__Int__crazy,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 2.80/1.56 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) @ ( inf_inf @ ( set @ A ) @ C2 @ A3 ) )
% 2.80/1.56 = ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) @ ( sup_sup @ ( set @ A ) @ C2 @ A3 ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_193_disjoint__iff__not__equal,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.80/1.56 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.80/1.56 = ( ! [X2: A] :
% 2.80/1.56 ( ( member @ A @ X2 @ A3 )
% 2.80/1.56 => ! [Y3: A] :
% 2.80/1.56 ( ( member @ A @ Y3 @ B2 )
% 2.80/1.56 => ( X2 != Y3 ) ) ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_194_Int__empty__right,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A] :
% 2.80/1.56 ( ( inf_inf @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_195_Int__empty__left,axiom,
% 2.80/1.56 ! [A: $tType,B2: set @ A] :
% 2.80/1.56 ( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
% 2.80/1.56 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.80/1.56
% 2.80/1.56 thf(fact_196_Int__emptyI,axiom,
% 2.80/1.56 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.80/1.56 ( ! [X3: A] :
% 2.80/1.56 ( ( member @ A @ X3 @ A3 )
% 2.80/1.57 => ~ ( member @ A @ X3 @ B2 ) )
% 2.80/1.57 => ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
% 2.80/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_197_sup__inf__distrib2,axiom,
% 2.80/1.57 ! [A: $tType] :
% 2.80/1.57 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 2.80/1.57 => ! [Y: A,Z: A,X: A] :
% 2.80/1.57 ( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z ) @ X )
% 2.80/1.57 = ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z @ X ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_198_sup__inf__distrib1,axiom,
% 2.80/1.57 ! [A: $tType] :
% 2.80/1.57 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 2.80/1.57 => ! [X: A,Y: A,Z: A] :
% 2.80/1.57 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.80/1.57 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_199_inf__sup__distrib2,axiom,
% 2.80/1.57 ! [A: $tType] :
% 2.80/1.57 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 2.80/1.57 => ! [Y: A,Z: A,X: A] :
% 2.80/1.57 ( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z ) @ X )
% 2.80/1.57 = ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z @ X ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_200_inf__sup__distrib1,axiom,
% 2.80/1.57 ! [A: $tType] :
% 2.80/1.57 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 2.80/1.57 => ! [X: A,Y: A,Z: A] :
% 2.80/1.57 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.80/1.57 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_201_distrib__imp2,axiom,
% 2.80/1.57 ! [A: $tType] :
% 2.80/1.57 ( ( lattice @ A @ ( type2 @ A ) )
% 2.80/1.57 => ! [X: A,Y: A,Z: A] :
% 2.80/1.57 ( ! [X3: A,Y2: A,Z2: A] :
% 2.80/1.57 ( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z2 ) )
% 2.80/1.57 = ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ ( sup_sup @ A @ X3 @ Z2 ) ) )
% 2.80/1.57 => ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.80/1.57 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_202_distrib__imp1,axiom,
% 2.80/1.57 ! [A: $tType] :
% 2.80/1.57 ( ( lattice @ A @ ( type2 @ A ) )
% 2.80/1.57 => ! [X: A,Y: A,Z: A] :
% 2.80/1.57 ( ! [X3: A,Y2: A,Z2: A] :
% 2.80/1.57 ( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) )
% 2.80/1.57 = ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ ( inf_inf @ A @ X3 @ Z2 ) ) )
% 2.80/1.57 => ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.80/1.57 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_203_disjnt__def,axiom,
% 2.80/1.57 ! [A: $tType] :
% 2.80/1.57 ( ( disjnt @ A )
% 2.80/1.57 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.80/1.57 ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
% 2.80/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_204_lfilter__eq__LConsD,axiom,
% 2.80/1.57 ! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.57 ( ( ( coinductive_lfilter @ A @ P @ Ys )
% 2.80/1.57 = ( coinductive_LCons @ A @ X @ Xs ) )
% 2.80/1.57 => ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 2.80/1.57 ( ( Ys
% 2.80/1.57 = ( coinductive_lappend @ A @ Us @ ( coinductive_LCons @ A @ X @ Vs ) ) )
% 2.80/1.57 & ( coinductive_lfinite @ A @ Us )
% 2.80/1.57 & ! [X4: A] :
% 2.80/1.57 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Us ) )
% 2.80/1.57 => ~ ( P @ X4 ) )
% 2.80/1.57 & ( P @ X )
% 2.80/1.57 & ( Xs
% 2.80/1.57 = ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_205_ltakeWhile_Octr_I2_J,axiom,
% 2.80/1.57 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.80/1.57 ( ~ ( coinductive_lnull @ A @ Xs )
% 2.80/1.57 => ( ( P @ ( coinductive_lhd @ A @ Xs ) )
% 2.80/1.57 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 2.80/1.57 = ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coindu501562517eWhile @ A @ P @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_206_llist_Oinject,axiom,
% 2.80/1.57 ! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 2.80/1.57 ( ( ( coinductive_LCons @ A @ X21 @ X22 )
% 2.80/1.57 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 2.80/1.57 = ( ( X21 = Y21 )
% 2.80/1.57 & ( X22 = Y22 ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_207_lfinite__LCons,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.57 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.80/1.57 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_208_lfinite__code_I2_J,axiom,
% 2.80/1.57 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 2.80/1.57 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
% 2.80/1.57 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_209_lappend__code_I2_J,axiom,
% 2.80/1.57 ! [A: $tType,Xa: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.80/1.57 ( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X ) @ Ys )
% 2.80/1.57 = ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_210_lfilter__LCons,axiom,
% 2.80/1.57 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.57 ( ( ( P @ X )
% 2.80/1.57 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.80/1.57 = ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
% 2.80/1.57 & ( ~ ( P @ X )
% 2.80/1.57 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.80/1.57 = ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_211_ldropWhile__LCons,axiom,
% 2.80/1.57 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.57 ( ( ( P @ X )
% 2.80/1.57 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.80/1.57 = ( coindu218763757pWhile @ A @ P @ Xs ) ) )
% 2.80/1.57 & ( ~ ( P @ X )
% 2.80/1.57 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.80/1.57 = ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_212_lconcat__LCons,axiom,
% 2.80/1.57 ! [B: $tType,Xs: coinductive_llist @ B,Xss: coinductive_llist @ ( coinductive_llist @ B )] :
% 2.80/1.57 ( ( coinductive_lconcat @ B @ ( coinductive_LCons @ ( coinductive_llist @ B ) @ Xs @ Xss ) )
% 2.80/1.57 = ( coinductive_lappend @ B @ Xs @ ( coinductive_lconcat @ B @ Xss ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_213_unfold__llist__eq__LCons,axiom,
% 2.80/1.57 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B3: B,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.57 ( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B3 )
% 2.80/1.57 = ( coinductive_LCons @ A @ X @ Xs ) )
% 2.80/1.57 = ( ~ ( IS_LNIL @ B3 )
% 2.80/1.57 & ( X
% 2.80/1.57 = ( LHD @ B3 ) )
% 2.80/1.57 & ( Xs
% 2.80/1.57 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B3 ) ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_214_lstrict__prefix__code_I4_J,axiom,
% 2.80/1.57 ! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
% 2.80/1.57 ( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
% 2.80/1.57 = ( ( X = Y )
% 2.80/1.57 & ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_215_ldistinct__LCons,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.57 ( ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.80/1.57 = ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.57 & ( coindu351974385stinct @ A @ Xs ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_216_lhd__LCons__ltl,axiom,
% 2.80/1.57 ! [A: $tType,Llist: coinductive_llist @ A] :
% 2.80/1.57 ( ~ ( coinductive_lnull @ A @ Llist )
% 2.80/1.57 => ( ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) )
% 2.80/1.57 = Llist ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_217_llist_Osimps_I5_J,axiom,
% 2.80/1.57 ! [B: $tType,A: $tType,F12: B,F23: A > ( coinductive_llist @ A ) > B,X21: A,X22: coinductive_llist @ A] :
% 2.80/1.57 ( ( coindu1381640503_llist @ B @ A @ F12 @ F23 @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 2.80/1.57 = ( F23 @ X21 @ X22 ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_218_llist_Oset__induct,axiom,
% 2.80/1.57 ! [A: $tType,X: A,A2: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
% 2.80/1.57 ( ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) )
% 2.80/1.57 => ( ! [Z1: A,Z22: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) )
% 2.80/1.57 => ( ! [Z1: A,Z22: coinductive_llist @ A,Xa2: A] :
% 2.80/1.57 ( ( member @ A @ Xa2 @ ( coinductive_lset @ A @ Z22 ) )
% 2.80/1.57 => ( ( P @ Xa2 @ Z22 )
% 2.80/1.57 => ( P @ Xa2 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) ) ) )
% 2.80/1.57 => ( P @ X @ A2 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_219_llist_Oset__cases,axiom,
% 2.80/1.57 ! [A: $tType,E: A,A2: coinductive_llist @ A] :
% 2.80/1.57 ( ( member @ A @ E @ ( coinductive_lset @ A @ A2 ) )
% 2.80/1.57 => ( ! [Z22: coinductive_llist @ A] :
% 2.80/1.57 ( A2
% 2.80/1.57 != ( coinductive_LCons @ A @ E @ Z22 ) )
% 2.80/1.57 => ~ ! [Z1: A,Z22: coinductive_llist @ A] :
% 2.80/1.57 ( ( A2
% 2.80/1.57 = ( coinductive_LCons @ A @ Z1 @ Z22 ) )
% 2.80/1.57 => ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z22 ) ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_220_lset__induct_H,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 2.80/1.57 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.57 => ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
% 2.80/1.57 => ( ! [X6: A,Xs2: coinductive_llist @ A] :
% 2.80/1.57 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
% 2.80/1.57 => ( ( P @ Xs2 )
% 2.80/1.57 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) )
% 2.80/1.57 => ( P @ Xs ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_221_lset__induct,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 2.80/1.57 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.57 => ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
% 2.80/1.57 => ( ! [X6: A,Xs2: coinductive_llist @ A] :
% 2.80/1.57 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
% 2.80/1.57 => ( ( X != X6 )
% 2.80/1.57 => ( ( P @ Xs2 )
% 2.80/1.57 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) ) )
% 2.80/1.57 => ( P @ Xs ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_222_lset__cases,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.57 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.57 => ( ! [Xs4: coinductive_llist @ A] :
% 2.80/1.57 ( Xs
% 2.80/1.57 != ( coinductive_LCons @ A @ X @ Xs4 ) )
% 2.80/1.57 => ~ ! [X6: A,Xs4: coinductive_llist @ A] :
% 2.80/1.57 ( ( Xs
% 2.80/1.57 = ( coinductive_LCons @ A @ X6 @ Xs4 ) )
% 2.80/1.57 => ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs4 ) ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_223_llist_Oset__intros_I1_J,axiom,
% 2.80/1.57 ! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_224_llist_Oset__intros_I2_J,axiom,
% 2.80/1.57 ! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
% 2.80/1.57 ( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
% 2.80/1.57 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_225_lset__intros_I1_J,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_226_lset__intros_I2_J,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Xs: coinductive_llist @ A,X7: A] :
% 2.80/1.57 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.57 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X7 @ Xs ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_227_lmember__code_I2_J,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
% 2.80/1.57 ( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 2.80/1.57 = ( ( X = Y )
% 2.80/1.57 | ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_228_not__lnull__conv,axiom,
% 2.80/1.57 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.80/1.57 ( ( ~ ( coinductive_lnull @ A @ Xs ) )
% 2.80/1.57 = ( ? [X2: A,Xs5: coinductive_llist @ A] :
% 2.80/1.57 ( Xs
% 2.80/1.57 = ( coinductive_LCons @ A @ X2 @ Xs5 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_229_llist_OdiscI_I2_J,axiom,
% 2.80/1.57 ! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
% 2.80/1.57 ( ( Llist
% 2.80/1.57 = ( coinductive_LCons @ A @ X21 @ X22 ) )
% 2.80/1.57 => ~ ( coinductive_lnull @ A @ Llist ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_230_llist_Odisc_I2_J,axiom,
% 2.80/1.57 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 2.80/1.57 ~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_231_ltl__simps_I2_J,axiom,
% 2.80/1.57 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 2.80/1.57 ( ( coinductive_ltl @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 2.80/1.57 = X22 ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_232_lhd__LCons,axiom,
% 2.80/1.57 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 2.80/1.57 ( ( coinductive_lhd @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 2.80/1.57 = X21 ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_233_lfilter__LCons__seek,axiom,
% 2.80/1.57 ! [A: $tType,P2: A > $o,X: A,L: coinductive_llist @ A] :
% 2.80/1.57 ( ~ ( P2 @ X )
% 2.80/1.57 => ( ( coinductive_lfilter @ A @ P2 @ ( coinductive_LCons @ A @ X @ L ) )
% 2.80/1.57 = ( coinductive_lfilter @ A @ P2 @ L ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_234_lfilter__LCons__found,axiom,
% 2.80/1.57 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.57 ( ( P @ X )
% 2.80/1.57 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.80/1.57 = ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_235_disjnt__iff,axiom,
% 2.80/1.57 ! [A: $tType] :
% 2.80/1.57 ( ( disjnt @ A )
% 2.80/1.57 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.80/1.57 ! [X2: A] :
% 2.80/1.57 ~ ( ( member @ A @ X2 @ A5 )
% 2.80/1.57 & ( member @ A @ X2 @ B5 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_236_lsetp_Oinducts,axiom,
% 2.80/1.57 ! [A: $tType,X1: coinductive_llist @ A,X23: A,P: ( coinductive_llist @ A ) > A > $o] :
% 2.80/1.57 ( ( coinductive_lsetp @ A @ X1 @ X23 )
% 2.80/1.57 => ( ! [X3: A,Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X3 @ Xs2 ) @ X3 )
% 2.80/1.57 => ( ! [Xs2: coinductive_llist @ A,X3: A,X6: A] :
% 2.80/1.57 ( ( coinductive_lsetp @ A @ Xs2 @ X3 )
% 2.80/1.57 => ( ( P @ Xs2 @ X3 )
% 2.80/1.57 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) @ X3 ) ) )
% 2.80/1.57 => ( P @ X1 @ X23 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_237_lsetp_Osimps,axiom,
% 2.80/1.57 ! [A: $tType] :
% 2.80/1.57 ( ( coinductive_lsetp @ A )
% 2.80/1.57 = ( ^ [A12: coinductive_llist @ A,A23: A] :
% 2.80/1.57 ( ? [X2: A,Xs3: coinductive_llist @ A] :
% 2.80/1.57 ( ( A12
% 2.80/1.57 = ( coinductive_LCons @ A @ X2 @ Xs3 ) )
% 2.80/1.57 & ( A23 = X2 ) )
% 2.80/1.57 | ? [Xs3: coinductive_llist @ A,X2: A,X8: A] :
% 2.80/1.57 ( ( A12
% 2.80/1.57 = ( coinductive_LCons @ A @ X8 @ Xs3 ) )
% 2.80/1.57 & ( A23 = X2 )
% 2.80/1.57 & ( coinductive_lsetp @ A @ Xs3 @ X2 ) ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_238_lsetp_Ocases,axiom,
% 2.80/1.57 ! [A: $tType,A1: coinductive_llist @ A,A22: A] :
% 2.80/1.57 ( ( coinductive_lsetp @ A @ A1 @ A22 )
% 2.80/1.57 => ( ! [X3: A] :
% 2.80/1.57 ( ? [Xs2: coinductive_llist @ A] :
% 2.80/1.57 ( A1
% 2.80/1.57 = ( coinductive_LCons @ A @ X3 @ Xs2 ) )
% 2.80/1.57 => ( A22 != X3 ) )
% 2.80/1.57 => ~ ! [Xs2: coinductive_llist @ A] :
% 2.80/1.57 ( ? [X6: A] :
% 2.80/1.57 ( A1
% 2.80/1.57 = ( coinductive_LCons @ A @ X6 @ Xs2 ) )
% 2.80/1.57 => ~ ( coinductive_lsetp @ A @ Xs2 @ A22 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_239_lsetp_Ointros_I1_J,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( coinductive_lsetp @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ X ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_240_lsetp_Ointros_I2_J,axiom,
% 2.80/1.57 ! [A: $tType,Xs: coinductive_llist @ A,X: A,X7: A] :
% 2.80/1.57 ( ( coinductive_lsetp @ A @ Xs @ X )
% 2.80/1.57 => ( coinductive_lsetp @ A @ ( coinductive_LCons @ A @ X7 @ Xs ) @ X ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_241_lfinite__LConsI,axiom,
% 2.80/1.57 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 2.80/1.57 ( ( coinductive_lfinite @ A @ Xs )
% 2.80/1.57 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_242_unfold__llist_Octr_I2_J,axiom,
% 2.80/1.57 ! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 2.80/1.57 ( ~ ( P2 @ A2 )
% 2.80/1.57 => ( ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
% 2.80/1.57 = ( coinductive_LCons @ B @ ( G21 @ A2 ) @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_243_ldistinct_OLCons,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.57 ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.57 => ( ( coindu351974385stinct @ A @ Xs )
% 2.80/1.57 => ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_244_split__llist,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.57 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.57 => ? [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 2.80/1.57 ( ( Xs
% 2.80/1.57 = ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
% 2.80/1.57 & ( coinductive_lfinite @ A @ Ys3 ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_245_split__llist__first,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.80/1.57 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.80/1.57 => ? [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 2.80/1.57 ( ( Xs
% 2.80/1.57 = ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
% 2.80/1.57 & ( coinductive_lfinite @ A @ Ys3 )
% 2.80/1.57 & ~ ( member @ A @ X @ ( coinductive_lset @ A @ Ys3 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_246_llast__lappend__LCons,axiom,
% 2.80/1.57 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 2.80/1.57 ( ( coinductive_lfinite @ A @ Xs )
% 2.80/1.57 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
% 2.80/1.57 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_247_ltakeWhile_Ocode,axiom,
% 2.80/1.57 ! [A: $tType] :
% 2.80/1.57 ( ( coindu501562517eWhile @ A )
% 2.80/1.57 = ( ^ [P3: A > $o,Xs3: coinductive_llist @ A] :
% 2.80/1.57 ( if @ ( coinductive_llist @ A )
% 2.80/1.57 @ ( ( coinductive_lnull @ A @ Xs3 )
% 2.80/1.57 | ~ ( P3 @ ( coinductive_lhd @ A @ Xs3 ) ) )
% 2.80/1.57 @ ( coinductive_LNil @ A )
% 2.80/1.57 @ ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs3 ) @ ( coindu501562517eWhile @ A @ P3 @ ( coinductive_ltl @ A @ Xs3 ) ) ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_248_lfinite__code_I1_J,axiom,
% 2.80/1.57 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_249_lappend__LNil2,axiom,
% 2.80/1.57 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.80/1.57 ( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
% 2.80/1.57 = Xs ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_250_lappend__code_I1_J,axiom,
% 2.80/1.57 ! [A: $tType,Ys: coinductive_llist @ A] :
% 2.80/1.57 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
% 2.80/1.57 = Ys ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_251_lfilter__LNil,axiom,
% 2.80/1.57 ! [A: $tType,P: A > $o] :
% 2.80/1.57 ( ( coinductive_lfilter @ A @ P @ ( coinductive_LNil @ A ) )
% 2.80/1.57 = ( coinductive_LNil @ A ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_252_ltakeWhile__LNil,axiom,
% 2.80/1.57 ! [A: $tType,P: A > $o] :
% 2.80/1.57 ( ( coindu501562517eWhile @ A @ P @ ( coinductive_LNil @ A ) )
% 2.80/1.57 = ( coinductive_LNil @ A ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(fact_253_lconcat__LNil,axiom,
% 2.80/1.57 ! [A: $tType] :
% 2.80/1.57 ( ( coinductive_lconcat @ A @ ( coinductive_LNil @ ( coinductive_llist @ A ) ) )
% 2.80/1.57 = ( coinductive_LNil @ A ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 2.80/1.57 bounded_lattice @ $o @ ( type2 @ $o ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 2.80/1.57 ! [A6: $tType] : ( bounded_lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 2.80/1.57 ! [A6: $tType,A7: $tType] :
% 2.80/1.57 ( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
% 2.80/1.57 => ( bounded_lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 2.80/1.57 ! [A6: $tType,A7: $tType] :
% 2.80/1.57 ( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
% 2.80/1.57 => ( bounde1808546759up_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 2.80/1.57 ! [A6: $tType,A7: $tType] :
% 2.80/1.57 ( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
% 2.80/1.57 => ( bounded_lattice_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.80/1.57 ! [A6: $tType,A7: $tType] :
% 2.80/1.57 ( ( semilattice_sup @ A7 @ ( type2 @ A7 ) )
% 2.80/1.57 => ( semilattice_sup @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 2.80/1.57 ! [A6: $tType,A7: $tType] :
% 2.80/1.57 ( ( semilattice_inf @ A7 @ ( type2 @ A7 ) )
% 2.80/1.57 => ( semilattice_inf @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
% 2.80/1.57 ! [A6: $tType,A7: $tType] :
% 2.80/1.57 ( ( distrib_lattice @ A7 @ ( type2 @ A7 ) )
% 2.80/1.57 => ( distrib_lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.80/1.57 ! [A6: $tType,A7: $tType] :
% 2.80/1.57 ( ( lattice @ A7 @ ( type2 @ A7 ) )
% 2.80/1.57 => ( lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_fun___Orderings_Obot,axiom,
% 2.80/1.57 ! [A6: $tType,A7: $tType] :
% 2.80/1.57 ( ( bot @ A7 @ ( type2 @ A7 ) )
% 2.80/1.57 => ( bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3,axiom,
% 2.80/1.57 ! [A6: $tType] : ( bounde1808546759up_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4,axiom,
% 2.80/1.57 ! [A6: $tType] : ( bounded_lattice_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5,axiom,
% 2.80/1.57 ! [A6: $tType] : ( semilattice_sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_6,axiom,
% 2.80/1.57 ! [A6: $tType] : ( semilattice_inf @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_7,axiom,
% 2.80/1.57 ! [A6: $tType] : ( distrib_lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_Set_Oset___Lattices_Olattice_8,axiom,
% 2.80/1.57 ! [A6: $tType] : ( lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
% 2.80/1.57 ! [A6: $tType] : ( bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_10,axiom,
% 2.80/1.57 bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_11,axiom,
% 2.80/1.57 bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_12,axiom,
% 2.80/1.57 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_13,axiom,
% 2.80/1.57 semilattice_inf @ $o @ ( type2 @ $o ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_14,axiom,
% 2.80/1.57 distrib_lattice @ $o @ ( type2 @ $o ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_HOL_Obool___Lattices_Olattice_15,axiom,
% 2.80/1.57 lattice @ $o @ ( type2 @ $o ) ).
% 2.80/1.57
% 2.80/1.57 thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
% 2.80/1.57 bot @ $o @ ( type2 @ $o ) ).
% 2.80/1.57
% 2.80/1.57 thf(help_If_3_1_T,axiom,
% 2.80/1.57 ! [P: $o] :
% 2.80/1.57 ( ( P = $true )
% 2.80/1.57 | ( P = $false ) ) ).
% 2.80/1.57
% 2.80/1.57 thf(help_If_2_1_T,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Y: A] :
% 2.80/1.57 ( ( if @ A @ $false @ X @ Y )
% 2.80/1.57 = Y ) ).
% 2.80/1.57
% 2.80/1.57 thf(help_If_1_1_T,axiom,
% 2.80/1.57 ! [A: $tType,X: A,Y: A] :
% 2.80/1.57 ( ( if @ A @ $true @ X @ Y )
% 2.80/1.57 = X ) ).
% 2.80/1.57
% 2.80/1.57 thf(conj_0,conjecture,
% 2.80/1.57 member @ a @ ( shd @ a @ xsa ) @ ( coinductive_lset @ a @ ( coindu1724414836stream @ a @ xsa ) ) ).
% 2.80/1.57 ------------------------
% 2.80/1.59 % (27090)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_26799 for (2999ds/2Mi)
% 2.80/1.59 % (27087)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_26799 for (2999ds/4Mi)
% 2.80/1.59 % (27088)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_26799 for (2999ds/27Mi)
% 2.80/1.59 % (27089)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_26799 for (2999ds/2Mi)
% 2.80/1.59 % (27089)Instruction limit reached!
% 2.80/1.59 % (27089)------------------------------
% 2.80/1.59 % (27089)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.59 % (27089)Termination reason: Unknown
% 2.80/1.59 % (27089)Termination phase: shuffling
% 2.80/1.59
% 2.80/1.59 % (27089)Memory used [KB]: 1279
% 2.80/1.59 % (27089)Time elapsed: 0.002 s
% 2.80/1.59 % (27089)Instructions burned: 2 (million)
% 2.80/1.59 % (27089)------------------------------
% 2.80/1.59 % (27089)------------------------------
% 2.80/1.59 % (27090)Instruction limit reached!
% 2.80/1.59 % (27090)------------------------------
% 2.80/1.59 % (27090)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.59 % (27090)Termination reason: Unknown
% 2.80/1.59 % (27090)Termination phase: shuffling
% 2.80/1.59
% 2.80/1.59 % (27090)Memory used [KB]: 1407
% 2.80/1.59 % (27090)Time elapsed: 0.003 s
% 2.80/1.59 % (27090)Instructions burned: 4 (million)
% 2.80/1.59 % (27090)------------------------------
% 2.80/1.59 % (27090)------------------------------
% 2.80/1.59 % (27087)Instruction limit reached!
% 2.80/1.59 % (27087)------------------------------
% 2.80/1.59 % (27087)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.59 % (27087)Termination reason: Unknown
% 2.80/1.59 % (27087)Termination phase: shuffling
% 2.80/1.59
% 2.80/1.59 % (27087)Memory used [KB]: 1407
% 2.80/1.59 % (27087)Time elapsed: 0.003 s
% 2.80/1.59 % (27087)Instructions burned: 6 (million)
% 2.80/1.59 % (27087)------------------------------
% 2.80/1.59 % (27087)------------------------------
% 2.80/1.59 % (27086)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_26799 for (2999ds/183Mi)
% 2.80/1.60 % (27091)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_26799 for (2999ds/275Mi)
% 2.80/1.60 % (27092)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_26799 for (2999ds/18Mi)
% 2.88/1.60 % (27088)Instruction limit reached!
% 2.88/1.60 % (27088)------------------------------
% 2.88/1.60 % (27088)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.60 % (27088)Termination reason: Unknown
% 2.88/1.60 % (27088)Termination phase: Property scanning
% 2.88/1.60
% 2.88/1.60 % (27088)Memory used [KB]: 1663
% 2.88/1.60 % (27088)Time elapsed: 0.011 s
% 2.88/1.60 % (27088)Instructions burned: 29 (million)
% 2.88/1.60 % (27088)------------------------------
% 2.88/1.60 % (27088)------------------------------
% 2.88/1.60 % (27094)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_26799 for (2999ds/37Mi)
% 2.88/1.60 % (27093)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_26799 for (2999ds/3Mi)
% 2.88/1.60 % (27095)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_26799 for (2999ds/15Mi)
% 2.88/1.60 % (27093)Instruction limit reached!
% 2.88/1.60 % (27093)------------------------------
% 2.88/1.60 % (27093)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.60 % (27093)Termination reason: Unknown
% 2.88/1.60 % (27093)Termination phase: shuffling
% 2.88/1.60
% 2.88/1.60 % (27093)Memory used [KB]: 1407
% 2.88/1.60 % (27093)Time elapsed: 0.003 s
% 2.88/1.60 % (27093)Instructions burned: 3 (million)
% 2.88/1.60 % (27093)------------------------------
% 2.88/1.60 % (27093)------------------------------
% 2.88/1.61 % (27095)Instruction limit reached!
% 2.88/1.61 % (27095)------------------------------
% 2.88/1.61 % (27095)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.61 % (27095)Termination reason: Unknown
% 2.88/1.61 % (27095)Termination phase: shuffling
% 2.88/1.61
% 2.88/1.61 % (27095)Memory used [KB]: 1663
% 2.88/1.61 % (27095)Time elapsed: 0.008 s
% 2.88/1.61 % (27095)Instructions burned: 17 (million)
% 2.88/1.61 % (27095)------------------------------
% 2.88/1.61 % (27095)------------------------------
% 2.88/1.61 % (27092)Instruction limit reached!
% 2.88/1.61 % (27092)------------------------------
% 2.88/1.61 % (27092)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.61 % (27092)Termination reason: Unknown
% 2.88/1.61 % (27092)Termination phase: shuffling
% 2.88/1.61
% 2.88/1.61 % (27092)Memory used [KB]: 1663
% 2.88/1.61 % (27092)Time elapsed: 0.014 s
% 2.88/1.61 % (27092)Instructions burned: 18 (million)
% 2.88/1.61 % (27092)------------------------------
% 2.88/1.61 % (27092)------------------------------
% 2.88/1.61 % (27096)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_26799 for (2999ds/3Mi)
% 2.88/1.61 % (27096)Instruction limit reached!
% 2.88/1.61 % (27096)------------------------------
% 2.88/1.61 % (27096)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.61 % (27096)Termination reason: Unknown
% 2.88/1.61 % (27096)Termination phase: shuffling
% 2.88/1.61
% 2.88/1.61 % (27096)Memory used [KB]: 1407
% 2.88/1.61 % (27096)Time elapsed: 0.003 s
% 2.88/1.61 % (27096)Instructions burned: 3 (million)
% 2.88/1.61 % (27096)------------------------------
% 2.88/1.61 % (27096)------------------------------
% 2.88/1.61 % (27094)Instruction limit reached!
% 2.88/1.61 % (27094)------------------------------
% 2.88/1.61 % (27094)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.61 % (27094)Termination reason: Unknown
% 2.88/1.61 % (27094)Termination phase: Property scanning
% 2.88/1.61
% 2.88/1.61 % (27094)Memory used [KB]: 2046
% 2.88/1.61 % (27094)Time elapsed: 0.012 s
% 2.88/1.61 % (27094)Instructions burned: 38 (million)
% 2.88/1.61 % (27094)------------------------------
% 2.88/1.61 % (27094)------------------------------
% 2.88/1.61 % (27097)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_26799 for (2999ds/1041Mi)
% 2.88/1.62 % (27098)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_26799 for (2999ds/7Mi)
% 2.88/1.62 % (27091)Refutation not found, incomplete strategy
% 2.88/1.62 % (27091)------------------------------
% 2.88/1.62 % (27091)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.62 % (27091)Termination reason: Refutation not found, incomplete strategy
% 2.88/1.62
% 2.88/1.62
% 2.88/1.62 % (27091)Memory used [KB]: 6268
% 2.88/1.62 % (27091)Time elapsed: 0.025 s
% 2.88/1.62 % (27091)Instructions burned: 41 (million)
% 2.88/1.62 % (27091)------------------------------
% 2.88/1.62 % (27091)------------------------------
% 2.88/1.62 % (27101)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_26799 for (2999ds/3Mi)
% 2.88/1.62 % (27098)Instruction limit reached!
% 2.88/1.62 % (27098)------------------------------
% 2.88/1.62 % (27098)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.62 % (27098)Termination reason: Unknown
% 2.88/1.62 % (27098)Termination phase: shuffling
% 2.88/1.62
% 2.88/1.62 % (27098)Memory used [KB]: 1535
% 2.88/1.62 % (27098)Time elapsed: 0.005 s
% 2.88/1.62 % (27098)Instructions burned: 9 (million)
% 2.88/1.62 % (27098)------------------------------
% 2.88/1.62 % (27098)------------------------------
% 2.88/1.62 % (27100)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_26799 for (2999ds/3Mi)
% 2.88/1.62 % (27101)Instruction limit reached!
% 2.88/1.62 % (27101)------------------------------
% 2.88/1.62 % (27101)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.62 % (27101)Termination reason: Unknown
% 2.88/1.62 % (27101)Termination phase: shuffling
% 2.88/1.62
% 2.88/1.62 % (27101)Memory used [KB]: 1407
% 2.88/1.62 % (27101)Time elapsed: 0.002 s
% 2.88/1.62 % (27101)Instructions burned: 3 (million)
% 2.88/1.62 % (27101)------------------------------
% 2.88/1.62 % (27101)------------------------------
% 2.88/1.62 % (27100)Instruction limit reached!
% 2.88/1.62 % (27100)------------------------------
% 2.88/1.62 % (27100)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.62 % (27100)Termination reason: Unknown
% 2.88/1.62 % (27100)Termination phase: shuffling
% 2.88/1.62
% 2.88/1.62 % (27100)Memory used [KB]: 1407
% 2.88/1.62 % (27100)Time elapsed: 0.003 s
% 2.88/1.62 % (27100)Instructions burned: 3 (million)
% 2.88/1.62 % (27100)------------------------------
% 2.88/1.62 % (27100)------------------------------
% 2.88/1.62 % (27099)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_26799 for (2999ds/16Mi)
% 2.88/1.63 % (27104)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_26799 for (2999ds/4Mi)
% 2.88/1.63 % (27102)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_26799 for (2999ds/7Mi)
% 2.88/1.63 % (27105)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_26799 for (2999ds/18Mi)
% 2.88/1.63 % (27104)Instruction limit reached!
% 2.88/1.63 % (27104)------------------------------
% 2.88/1.63 % (27104)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.63 % (27104)Termination reason: Unknown
% 2.88/1.63 % (27104)Termination phase: shuffling
% 2.88/1.63
% 2.88/1.63 % (27104)Memory used [KB]: 1407
% 2.88/1.63 % (27104)Time elapsed: 0.003 s
% 2.88/1.63 % (27104)Instructions burned: 5 (million)
% 2.88/1.63 % (27104)------------------------------
% 2.88/1.63 % (27104)------------------------------
% 2.88/1.63 % (27102)Instruction limit reached!
% 2.88/1.63 % (27102)------------------------------
% 2.88/1.63 % (27102)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.63 % (27102)Termination reason: Unknown
% 2.88/1.63 % (27102)Termination phase: shuffling
% 2.88/1.63
% 2.88/1.63 % (27102)Memory used [KB]: 1535
% 2.88/1.63 % (27102)Time elapsed: 0.005 s
% 2.88/1.63 % (27102)Instructions burned: 9 (million)
% 2.88/1.63 % (27102)------------------------------
% 2.88/1.63 % (27102)------------------------------
% 2.88/1.63 % (27099)Instruction limit reached!
% 2.88/1.63 % (27099)------------------------------
% 2.88/1.63 % (27099)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.63 % (27099)Termination reason: Unknown
% 2.88/1.63 % (27099)Termination phase: shuffling
% 2.88/1.63
% 2.88/1.63 % (27099)Memory used [KB]: 1663
% 2.88/1.63 % (27099)Time elapsed: 0.012 s
% 2.88/1.63 % (27099)Instructions burned: 16 (million)
% 2.88/1.63 % (27099)------------------------------
% 2.88/1.63 % (27099)------------------------------
% 2.88/1.64 % (27103)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_26799 for (2999ds/3Mi)
% 2.88/1.64 % (27105)Instruction limit reached!
% 2.88/1.64 % (27105)------------------------------
% 2.88/1.64 % (27105)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.64 % (27105)Termination reason: Unknown
% 2.88/1.64 % (27105)Termination phase: shuffling
% 2.88/1.64
% 2.88/1.64 % (27105)Memory used [KB]: 1663
% 2.88/1.64 % (27105)Time elapsed: 0.009 s
% 2.88/1.64 % (27105)Instructions burned: 20 (million)
% 2.88/1.64 % (27105)------------------------------
% 2.88/1.64 % (27105)------------------------------
% 2.88/1.64 % (27103)Instruction limit reached!
% 2.88/1.64 % (27103)------------------------------
% 2.88/1.64 % (27103)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.64 % (27103)Termination reason: Unknown
% 2.88/1.64 % (27103)Termination phase: shuffling
% 2.88/1.64
% 2.88/1.64 % (27103)Memory used [KB]: 1407
% 2.88/1.64 % (27103)Time elapsed: 0.004 s
% 2.88/1.64 % (27103)Instructions burned: 4 (million)
% 2.88/1.64 % (27103)------------------------------
% 2.88/1.64 % (27103)------------------------------
% 2.88/1.64 % (27106)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_26799 for (2999ds/710Mi)
% 2.88/1.64 % (27107)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_26799 for (2999ds/6Mi)
% 2.88/1.64 % (27107)Instruction limit reached!
% 2.88/1.64 % (27107)------------------------------
% 2.88/1.64 % (27107)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.64 % (27107)Termination reason: Unknown
% 2.88/1.64 % (27107)Termination phase: shuffling
% 2.88/1.64
% 2.88/1.64 % (27107)Memory used [KB]: 1407
% 2.88/1.64 % (27107)Time elapsed: 0.004 s
% 2.88/1.64 % (27107)Instructions burned: 6 (million)
% 2.88/1.64 % (27107)------------------------------
% 2.88/1.64 % (27107)------------------------------
% 2.88/1.65 % (27109)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_26799 for (2999ds/21Mi)
% 2.88/1.65 % (27108)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_26799 for (2999ds/902Mi)
% 2.88/1.65 % (27106)First to succeed.
% 2.88/1.65 % (27106)Refutation found. Thanks to Tanya!
% 2.88/1.65 % SZS status Theorem for DTF2THF_26799
% 2.88/1.65 % SZS output start Proof for DTF2THF_26799
% 2.88/1.65 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.88/1.65 thf(type_def_7, type, set: ($tType) > $tType).
% 2.88/1.65 thf(type_def_8, type, coinductive_llist: ($tType) > $tType).
% 2.88/1.65 thf(type_def_9, type, stream: ($tType) > $tType).
% 2.88/1.65 thf(type_def_10, type, a: $tType).
% 2.88/1.65 thf(func_def_0, type, coinductive_llist: ($tType) > $tType).
% 2.88/1.65 thf(func_def_1, type, stream: ($tType) > $tType).
% 2.88/1.65 thf(func_def_2, type, set: ($tType) > $tType).
% 2.88/1.65 thf(func_def_3, type, itself: ($tType) > $tType).
% 2.88/1.65 thf(func_def_4, type, a: $tType).
% 2.88/1.65 thf(func_def_5, type, bounded_lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.65 thf(func_def_6, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.65 thf(func_def_7, type, bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.65 thf(func_def_8, type, lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.65 thf(func_def_9, type, distrib_lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.65 thf(func_def_10, type, semilattice_inf: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.65 thf(func_def_11, type, semilattice_sup: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.65 thf(func_def_12, type, bounded_lattice_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.65 thf(func_def_13, type, bounde1808546759up_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.65 thf(func_def_14, type, coinductive_gen_lset: !>[X0: $tType]:(set @ X0 > coinductive_llist @ X0 > set @ X0)).
% 2.88/1.65 thf(func_def_15, type, coinductive_lappend: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.88/1.65 thf(func_def_16, type, coinductive_lconcat: !>[X0: $tType]:(coinductive_llist @ coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.88/1.65 thf(func_def_17, type, coindu351974385stinct: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.88/1.65 thf(func_def_18, type, coindu218763757pWhile: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.88/1.65 thf(func_def_19, type, coinductive_lfilter: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.88/1.65 thf(func_def_20, type, coinductive_lfinite: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.88/1.65 thf(func_def_21, type, coinductive_llast: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 2.88/1.65 thf(func_def_22, type, coinductive_LCons: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.88/1.65 thf(func_def_23, type, coinductive_LNil: !>[X0: $tType]:(coinductive_llist @ X0)).
% 2.88/1.65 thf(func_def_24, type, coindu1381640503_llist: !>[X0: $tType, X1: $tType]:(X0 > (X1 > coinductive_llist @ X1 > X0) > coinductive_llist @ X1 > X0)).
% 2.88/1.65 thf(func_def_25, type, coinductive_lhd: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 2.88/1.65 thf(func_def_26, type, coinductive_lnull: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.88/1.65 thf(func_def_27, type, coinductive_lset: !>[X0: $tType]:(coinductive_llist @ X0 > set @ X0)).
% 2.88/1.65 thf(func_def_28, type, coinductive_ltl: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.88/1.65 thf(func_def_29, type, coinductive_lmember: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > $o)).
% 2.88/1.65 thf(func_def_30, type, coinductive_lsetp: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > $o)).
% 2.88/1.65 thf(func_def_31, type, coindu1478340336prefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 2.88/1.65 thf(func_def_32, type, coindu501562517eWhile: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.88/1.65 thf(func_def_33, type, coindu1441602521_llist: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > X1) > (X0 > X0) > X0 > coinductive_llist @ X1)).
% 2.88/1.65 thf(func_def_34, type, coindu1724414836stream: !>[X0: $tType]:(stream @ X0 > coinductive_llist @ X0)).
% 2.88/1.65 thf(func_def_35, type, coindu1183105481stream: !>[X0: $tType]:(coinductive_llist @ X0 > stream @ X0 > $o)).
% 2.88/1.65 thf(func_def_36, type, coindu2010755910_llist: !>[X0: $tType]:(coinductive_llist @ X0 > stream @ X0)).
% 2.88/1.65 thf(func_def_37, type, coindu139217191stream: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > X0) > X0 > stream @ X1)).
% 2.88/1.65 thf(func_def_38, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.88/1.65 thf(func_def_39, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.88/1.65 thf(func_def_40, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.88/1.65 thf(func_def_41, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.88/1.65 thf(func_def_42, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.88/1.65 thf(func_def_43, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.88/1.65 thf(func_def_44, type, disjnt: !>[X0: $tType]:(set @ X0 > set @ X0 > $o)).
% 2.88/1.65 thf(func_def_45, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.88/1.65 thf(func_def_46, type, sinterleave: !>[X0: $tType]:(stream @ X0 > stream @ X0 > stream @ X0)).
% 2.88/1.65 thf(func_def_47, type, smember: !>[X0: $tType]:(X0 > stream @ X0 > $o)).
% 2.88/1.65 thf(func_def_48, type, shd: !>[X0: $tType]:(stream @ X0 > X0)).
% 2.88/1.65 thf(func_def_49, type, sset: !>[X0: $tType]:(stream @ X0 > set @ X0)).
% 2.88/1.65 thf(func_def_50, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.88/1.65 thf(func_def_51, type, x: a).
% 2.88/1.65 thf(func_def_52, type, xs: stream @ a).
% 2.88/1.65 thf(func_def_53, type, xsa: stream @ a).
% 2.88/1.65 thf(func_def_69, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.88/1.65 thf(f795,plain,(
% 2.88/1.65 $false),
% 2.88/1.65 inference(avatar_sat_refutation,[status(thm)],[f789,f791,f794])).
% 2.88/1.65 thf(f794,plain,(
% 2.88/1.65 ~spl1_1),
% 2.88/1.65 inference(avatar_contradiction_clause,[status(thm)],[f793])).
% 2.88/1.65 thf(f793,plain,(
% 2.88/1.65 $false | ~spl1_1),
% 2.88/1.65 inference(subsumption_resolution,[status(thm)],[f792,f773])).
% 2.88/1.65 thf(f773,plain,(
% 2.88/1.65 ((member @ a @ (shd @ a @ xsa) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa))) != $true)),
% 2.88/1.65 inference(cnf_transformation,[status(thm)],[f749])).
% 2.88/1.65 thf(f749,plain,(
% 2.88/1.65 ((member @ a @ (shd @ a @ xsa) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa))) != $true)),
% 2.88/1.65 inference(flattening,[status(thm)],[f610])).
% 2.88/1.65 thf(f610,plain,(
% 2.88/1.65 ~((member @ a @ (shd @ a @ xsa) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa))) = $true)),
% 2.88/1.65 inference(fool_elimination,[status(thm)],[f609])).
% 2.88/1.65 thf(f609,plain,(
% 2.88/1.65 ~(member @ a @ (shd @ a @ xsa) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa)))),
% 2.88/1.65 inference(rectify,[status(thm)],[f283])).
% 2.88/1.65 thf(f283,negated_conjecture,(
% 2.88/1.65 ~(member @ a @ (shd @ a @ xsa) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa)))),
% 2.88/1.65 inference(negated_conjecture,[status(cth)],[f282])).
% 2.88/1.65 thf(f282,conjecture,(
% 2.88/1.65 (member @ a @ (shd @ a @ xsa) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa)))),
% 2.88/1.65 file('/export/starexec/sandbox/tmp/tmp.61YHUtgMPz/DTF2THF_26799.p',conj_0)).
% 2.88/1.65 thf(f792,plain,(
% 2.88/1.65 ((member @ a @ (shd @ a @ xsa) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa))) = $true) | ~spl1_1),
% 2.88/1.65 inference(forward_demodulation,[status(thm)],[f784,f768])).
% 2.88/1.65 thf(f768,plain,(
% 2.88/1.65 ( ! [X0 : $tType,X1 : stream @ X0] : (((shd @ X0 @ X1) = (coinductive_lhd @ X0 @ (coindu1724414836stream @ X0 @ X1)))) )),
% 2.88/1.65 inference(cnf_transformation,[status(thm)],[f753])).
% 2.88/1.65 thf(f753,plain,(
% 2.88/1.65 ! [X0 : $tType,X1 : stream @ X0] : ((shd @ X0 @ X1) = (coinductive_lhd @ X0 @ (coindu1724414836stream @ X0 @ X1)))),
% 2.88/1.65 inference(rectify,[status(thm)],[f3])).
% 2.88/1.65 thf(f3,axiom,(
% 2.88/1.65 ! [X0 : $tType,X4 : stream @ X0] : ((coinductive_lhd @ X0 @ (coindu1724414836stream @ X0 @ X4)) = (shd @ X0 @ X4))),
% 2.88/1.65 file('/export/starexec/sandbox/tmp/tmp.61YHUtgMPz/DTF2THF_26799.p',fact_2_lhd__llist__of__stream)).
% 2.88/1.65 thf(f784,plain,(
% 2.88/1.65 ((member @ a @ (coinductive_lhd @ a @ (coindu1724414836stream @ a @ xsa)) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa))) = $true) | ~spl1_1),
% 2.88/1.65 inference(avatar_component_clause,[status(thm)],[f782])).
% 2.88/1.65 thf(f782,definition,(
% 2.88/1.65 spl1_1 <=> ((member @ a @ (coinductive_lhd @ a @ (coindu1724414836stream @ a @ xsa)) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa))) = $true)),
% 2.88/1.65 introduced(definition,[new_symbols(naming,[spl1_1])],[avatar_definition])).
% 2.88/1.65 thf(f791,plain,(
% 2.88/1.65 ~spl1_2),
% 2.88/1.65 inference(avatar_contradiction_clause,[status(thm)],[f790])).
% 2.88/1.65 thf(f790,plain,(
% 2.88/1.65 $false | ~spl1_2),
% 2.88/1.65 inference(subsumption_resolution,[status(thm)],[f788,f772])).
% 2.88/1.65 thf(f772,plain,(
% 2.88/1.65 ( ! [X0 : $tType,X1 : stream @ X0] : (($true != (coinductive_lnull @ X0 @ (coindu1724414836stream @ X0 @ X1)))) )),
% 2.88/1.65 inference(cnf_transformation,[status(thm)],[f751])).
% 2.88/1.65 thf(f751,plain,(
% 2.88/1.65 ! [X0 : $tType,X1 : stream @ X0] : ($true != (coinductive_lnull @ X0 @ (coindu1724414836stream @ X0 @ X1)))),
% 2.88/1.65 inference(flattening,[status(thm)],[f458])).
% 2.88/1.65 thf(f458,plain,(
% 2.88/1.65 ! [X0 : $tType,X1 : stream @ X0] : ~($true = (coinductive_lnull @ X0 @ (coindu1724414836stream @ X0 @ X1)))),
% 2.88/1.65 inference(fool_elimination,[status(thm)],[f457])).
% 2.88/1.65 thf(f457,plain,(
% 2.88/1.65 ! [X0 : $tType,X1 : stream @ X0] : ~(coinductive_lnull @ X0 @ (coindu1724414836stream @ X0 @ X1))),
% 2.88/1.65 inference(rectify,[status(thm)],[f4])).
% 2.88/1.65 thf(f4,axiom,(
% 2.88/1.65 ! [X0 : $tType,X4 : stream @ X0] : ~(coinductive_lnull @ X0 @ (coindu1724414836stream @ X0 @ X4))),
% 2.88/1.65 file('/export/starexec/sandbox/tmp/tmp.61YHUtgMPz/DTF2THF_26799.p',fact_3_lnull__llist__of__stream)).
% 2.88/1.65 thf(f788,plain,(
% 2.88/1.65 ((coinductive_lnull @ a @ (coindu1724414836stream @ a @ xsa)) = $true) | ~spl1_2),
% 2.88/1.65 inference(avatar_component_clause,[status(thm)],[f786])).
% 2.88/1.65 thf(f786,definition,(
% 2.88/1.65 spl1_2 <=> ((coinductive_lnull @ a @ (coindu1724414836stream @ a @ xsa)) = $true)),
% 2.88/1.65 introduced(definition,[new_symbols(naming,[spl1_2])],[avatar_definition])).
% 2.88/1.65 thf(f789,plain,(
% 2.88/1.65 spl1_1 | spl1_2),
% 2.88/1.65 inference(avatar_split_clause,[status(thm)],[f774,f786,f782])).
% 2.88/1.65 thf(f774,plain,(
% 2.88/1.65 ((coinductive_lnull @ a @ (coindu1724414836stream @ a @ xsa)) = $true) | ((member @ a @ (coinductive_lhd @ a @ (coindu1724414836stream @ a @ xsa)) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa))) = $true)),
% 2.88/1.65 inference(cnf_transformation,[status(thm)],[f761])).
% 2.88/1.65 thf(f761,plain,(
% 2.88/1.65 ((coinductive_lnull @ a @ (coindu1724414836stream @ a @ xsa)) = $true) | ((member @ a @ (coinductive_lhd @ a @ (coindu1724414836stream @ a @ xsa)) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa))) = $true)),
% 2.88/1.65 inference(ennf_transformation,[status(thm)],[f750])).
% 2.88/1.65 thf(f750,plain,(
% 2.88/1.65 ((coinductive_lnull @ a @ (coindu1724414836stream @ a @ xsa)) != $true) => ((member @ a @ (coinductive_lhd @ a @ (coindu1724414836stream @ a @ xsa)) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa))) = $true)),
% 2.88/1.65 inference(flattening,[status(thm)],[f682])).
% 2.88/1.65 thf(f682,plain,(
% 2.88/1.65 ~((coinductive_lnull @ a @ (coindu1724414836stream @ a @ xsa)) = $true) => ((member @ a @ (coinductive_lhd @ a @ (coindu1724414836stream @ a @ xsa)) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa))) = $true)),
% 2.88/1.65 inference(fool_elimination,[status(thm)],[f681])).
% 2.88/1.65 thf(f681,plain,(
% 2.88/1.65 ~(coinductive_lnull @ a @ (coindu1724414836stream @ a @ xsa)) => (member @ a @ (coinductive_lhd @ a @ (coindu1724414836stream @ a @ xsa)) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa)))),
% 2.88/1.65 inference(rectify,[status(thm)],[f1])).
% 2.88/1.65 thf(f1,axiom,(
% 2.88/1.65 ~(coinductive_lnull @ a @ (coindu1724414836stream @ a @ xsa)) => (member @ a @ (coinductive_lhd @ a @ (coindu1724414836stream @ a @ xsa)) @ (coinductive_lset @ a @ (coindu1724414836stream @ a @ xsa)))),
% 2.88/1.65 file('/export/starexec/sandbox/tmp/tmp.61YHUtgMPz/DTF2THF_26799.p',fact_0__092_060open_062_092_060not_062_Alnull_A_Illist__of__stream_Axs_J_A_092_060Longrightarrow_062_Alhd_A_Illist__of__stream_Axs_J_A_092_060in_062_Alset_A_Illist__of__stream_Axs_J_092_060close_062)).
% 2.88/1.65 % SZS output end Proof for DTF2THF_26799
% 2.88/1.65 % (27106)------------------------------
% 2.88/1.65 % (27106)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.65 % (27106)Termination reason: Refutation
% 2.88/1.65
% 2.88/1.65 % (27106)Memory used [KB]: 6268
% 2.88/1.65 % (27106)Time elapsed: 0.013 s
% 2.88/1.65 % (27106)Instructions burned: 37 (million)
% 2.88/1.65 % (27106)------------------------------
% 2.88/1.65 % (27106)------------------------------
% 2.88/1.65 % (27085)Success in time 0.077 s
%------------------------------------------------------------------------------