%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM172^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n023.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 08:41:50 AM UTC 2026
% Result : Theorem 3.00s 1.97s
% Output : Refutation 3.00s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : COM172^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n023.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
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 04:17:04 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.27/0.45 ---- Original DTF file ---
% 0.27/0.46 thf(spec,logic,$$dhol).
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 % File : COM172^1 : TPTP v9.2.1. Released v7.0.0.
% 0.27/0.46 % Domain : Computing Theory
% 0.27/0.46 % Problem : Koenig's lemma 47
% 0.27/0.46 % Version : [Bla16] axioms : Especial.
% 0.27/0.46 % English :
% 0.27/0.46
% 0.27/0.46 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.27/0.46 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.27/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.27/0.46 % Source : [Bla16]
% 0.27/0.46 % Names : koenigslemma__47.p [Bla16]
% 0.27/0.46
% 0.27/0.46 % Status : Theorem
% 0.27/0.46 % Rating : 0.33 v9.1.0, 0.00 v8.2.0, 0.33 v8.1.0, 0.25 v7.5.0, 0.67 v7.3.0, 1.00 v7.1.0
% 0.27/0.46 % Syntax : Number of formulae : 317 ( 135 unt; 47 typ; 0 def)
% 0.27/0.46 % Number of atoms : 747 ( 252 equ; 0 cnn)
% 0.27/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.27/0.46 % Number of connectives : 4075 ( 107 ~; 32 |; 78 &;3519 @)
% 0.27/0.46 % ( 0 <=>; 339 =>; 0 <=; 0 <~>)
% 0.27/0.46 % Maximal formula depth : 22 ( 8 avg)
% 0.27/0.46 % Number of types : 2 ( 1 usr)
% 0.27/0.46 % Number of type conns : 329 ( 329 >; 0 *; 0 +; 0 <<)
% 0.27/0.46 % Number of symbols : 49 ( 46 usr; 8 con; 0-8 aty)
% 0.27/0.46 % Number of variables : 1137 ( 92 ^; 952 !; 54 ?;1137 :)
% 0.27/0.46 % ( 39 !>; 0 ?*; 0 @-; 0 @+)
% 0.27/0.46 % SPC : TH1_THM_EQU_NAR
% 0.27/0.46
% 0.27/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.27/0.46 % 2016-07-13 14:45:38.000
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 %----Could-be-implicit typings (4)
% 0.27/0.46 thf(ty_t_Coinductive__List_Ollist,type,
% 0.27/0.46 coinductive_llist: $tType > $tType ).
% 0.27/0.46
% 0.27/0.46 thf(ty_t_Set_Oset,type,
% 0.27/0.46 set: $tType > $tType ).
% 0.27/0.46
% 0.27/0.46 thf(ty_t_itself,type,
% 0.27/0.46 itself: $tType > $tType ).
% 0.27/0.46
% 0.27/0.46 thf(ty_tf_a,type,
% 0.27/0.46 a: $tType ).
% 0.27/0.46
% 0.27/0.46 %----Explicit typings (43)
% 0.27/0.46 thf(sy_cl_HOL_Otype,type,
% 0.27/0.46 type:
% 0.27/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.27/0.46 ord:
% 0.27/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.27/0.46 order:
% 0.27/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.27/0.46 linorder:
% 0.27/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.27/0.46 preorder:
% 0.27/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
% 0.27/0.46 coindu328551480prefix:
% 0.27/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Ogen__lset,type,
% 0.27/0.46 coinductive_gen_lset:
% 0.27/0.46 !>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Olappend,type,
% 0.27/0.46 coinductive_lappend:
% 0.27/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_OldropWhile,type,
% 0.27/0.46 coindu218763757pWhile:
% 0.27/0.46 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Olfilter,type,
% 0.27/0.46 coinductive_lfilter:
% 0.27/0.46 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.27/0.46 coinductive_lfinite:
% 0.27/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Ollast,type,
% 0.27/0.46 coinductive_llast:
% 0.27/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 0.27/0.46 coinductive_LCons:
% 0.27/0.46 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 0.27/0.46 coinductive_LNil:
% 0.27/0.46 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Ollist_Ocase__llist,type,
% 0.27/0.46 coindu1381640503_llist:
% 0.27/0.46 !>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Ollist_Ocorec__llist,type,
% 0.27/0.46 coindu1259883913_llist:
% 0.27/0.46 !>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Ollist_Olhd,type,
% 0.27/0.46 coinductive_lhd:
% 0.27/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 0.27/0.46 coinductive_lnull:
% 0.27/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 0.27/0.46 coinductive_lset:
% 0.27/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Olmember,type,
% 0.27/0.46 coinductive_lmember:
% 0.27/0.46 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Olstrict__prefix,type,
% 0.27/0.46 coindu1478340336prefix:
% 0.27/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_OltakeWhile,type,
% 0.27/0.46 coindu501562517eWhile:
% 0.27/0.46 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Oord__class_Olsorted,type,
% 0.27/0.46 coindu63249387sorted:
% 0.27/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 0.27/0.46 coindu1441602521_llist:
% 0.27/0.46 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_HOL_Oundefined,type,
% 0.27/0.46 undefined:
% 0.27/0.46 !>[A: $tType] : A ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_If,type,
% 0.27/0.46 if:
% 0.27/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Opaths,type,
% 0.27/0.46 koenig916195507_paths:
% 0.27/0.46 !>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ ( coinductive_llist @ Node ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Opathsp,type,
% 0.27/0.46 koenig2031690877pathsp:
% 0.27/0.46 !>[Node: $tType] : ( ( Node > Node > $o ) > ( coinductive_llist @ Node ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__via,type,
% 0.27/0.46 koenig317145564le_via:
% 0.27/0.46 !>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ Node ) > Node > ( set @ Node ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__viap,type,
% 0.27/0.46 koenig1757754772e_viap:
% 0.27/0.46 !>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ Node ) > Node > Node > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.27/0.46 sup_sup:
% 0.27/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.27/0.46 ord_less_eq:
% 0.27/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Pure_Otype,type,
% 0.27/0.46 type2:
% 0.27/0.46 !>[A: $tType] : ( itself @ A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Relation_OPowp,type,
% 0.27/0.46 powp:
% 0.27/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Set_OCollect,type,
% 0.27/0.46 collect:
% 0.27/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_member,type,
% 0.27/0.46 member:
% 0.27/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_graph,type,
% 0.27/0.46 graph: a > a > $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_x,type,
% 0.27/0.46 x: coinductive_llist @ a ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_x21,type,
% 0.27/0.46 x21: a ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_x22,type,
% 0.27/0.46 x22: coinductive_llist @ a ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_xa,type,
% 0.27/0.46 xa: a ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_xs,type,
% 0.27/0.46 xs: coinductive_llist @ a ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_ys,type,
% 0.27/0.46 ys: coinductive_llist @ a ).
% 0.27/0.46
% 0.27/0.46 %----Relevant facts (254)
% 0.27/0.46 thf(fact_0_assms,axiom,
% 0.27/0.46 member @ ( coinductive_llist @ a ) @ ( coinductive_lappend @ a @ xs @ ys ) @ ( koenig916195507_paths @ a @ graph ) ).
% 0.27/0.46
% 0.27/0.46 % assms
% 0.27/0.46 thf(fact_1_paths_OEmpty,axiom,
% 0.27/0.46 ! [Node: $tType,Graph: Node > Node > $o] : ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LNil @ Node ) @ ( koenig916195507_paths @ Node @ Graph ) ) ).
% 0.27/0.46
% 0.27/0.46 % paths.Empty
% 0.27/0.46 thf(fact_2_paths_OLCons,axiom,
% 0.27/0.46 ! [Node: $tType,Graph: Node > Node > $o,X: Node,Y: Node,Xs: coinductive_llist @ Node] :
% 0.27/0.46 ( ( Graph @ X @ Y )
% 0.27/0.46 => ( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y @ Xs ) @ ( koenig916195507_paths @ Node @ Graph ) )
% 0.27/0.46 => ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ X @ ( coinductive_LCons @ Node @ Y @ Xs ) ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % paths.LCons
% 0.27/0.46 thf(fact_3_paths_Ocases,axiom,
% 0.27/0.46 ! [Node: $tType,A2: coinductive_llist @ Node,Graph: Node > Node > $o] :
% 0.27/0.46 ( ( member @ ( coinductive_llist @ Node ) @ A2 @ ( koenig916195507_paths @ Node @ Graph ) )
% 0.27/0.46 => ( ( A2
% 0.27/0.46 != ( coinductive_LNil @ Node ) )
% 0.27/0.46 => ( ! [X2: Node] :
% 0.27/0.46 ( A2
% 0.27/0.46 != ( coinductive_LCons @ Node @ X2 @ ( coinductive_LNil @ Node ) ) )
% 0.27/0.46 => ~ ! [X2: Node,Y2: Node,Xs2: coinductive_llist @ Node] :
% 0.27/0.46 ( ( A2
% 0.27/0.46 = ( coinductive_LCons @ Node @ X2 @ ( coinductive_LCons @ Node @ Y2 @ Xs2 ) ) )
% 0.27/0.46 => ( ( Graph @ X2 @ Y2 )
% 0.27/0.46 => ~ ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y2 @ Xs2 ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % paths.cases
% 0.27/0.46 thf(fact_4_paths_Osimps,axiom,
% 0.27/0.46 ! [Node: $tType,A2: coinductive_llist @ Node,Graph: Node > Node > $o] :
% 0.27/0.46 ( ( member @ ( coinductive_llist @ Node ) @ A2 @ ( koenig916195507_paths @ Node @ Graph ) )
% 0.27/0.46 = ( ( A2
% 0.27/0.46 = ( coinductive_LNil @ Node ) )
% 0.27/0.46 | ? [X3: Node] :
% 0.27/0.46 ( A2
% 0.27/0.46 = ( coinductive_LCons @ Node @ X3 @ ( coinductive_LNil @ Node ) ) )
% 0.27/0.46 | ? [X3: Node,Y3: Node,Xs3: coinductive_llist @ Node] :
% 0.27/0.46 ( ( A2
% 0.27/0.46 = ( coinductive_LCons @ Node @ X3 @ ( coinductive_LCons @ Node @ Y3 @ Xs3 ) ) )
% 0.27/0.46 & ( Graph @ X3 @ Y3 )
% 0.27/0.46 & ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y3 @ Xs3 ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % paths.simps
% 0.27/0.46 thf(fact_5_paths_OSingle,axiom,
% 0.27/0.46 ! [Node: $tType,X: Node,Graph: Node > Node > $o] : ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ X @ ( coinductive_LNil @ Node ) ) @ ( koenig916195507_paths @ Node @ Graph ) ) ).
% 0.27/0.46
% 0.27/0.46 % paths.Single
% 0.27/0.46 thf(fact_6_paths__LConsD,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Graph: A > A > $o] :
% 0.27/0.46 ( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X @ Xs ) @ ( koenig916195507_paths @ A @ Graph ) )
% 0.27/0.46 => ( member @ ( coinductive_llist @ A ) @ Xs @ ( koenig916195507_paths @ A @ Graph ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % paths_LConsD
% 0.27/0.46 thf(fact_7_paths_Ocoinduct,axiom,
% 0.27/0.46 ! [Node: $tType,X4: ( coinductive_llist @ Node ) > $o,X: coinductive_llist @ Node,Graph: Node > Node > $o] :
% 0.27/0.46 ( ( X4 @ X )
% 0.27/0.46 => ( ! [X2: coinductive_llist @ Node] :
% 0.27/0.46 ( ( X4 @ X2 )
% 0.27/0.46 => ( ( X2
% 0.27/0.46 = ( coinductive_LNil @ Node ) )
% 0.27/0.46 | ? [Xa: Node] :
% 0.27/0.46 ( X2
% 0.27/0.46 = ( coinductive_LCons @ Node @ Xa @ ( coinductive_LNil @ Node ) ) )
% 0.27/0.46 | ? [Xa: Node,Y4: Node,Xs4: coinductive_llist @ Node] :
% 0.27/0.46 ( ( X2
% 0.27/0.46 = ( coinductive_LCons @ Node @ Xa @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) ) )
% 0.27/0.46 & ( Graph @ Xa @ Y4 )
% 0.27/0.46 & ( ( X4 @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) )
% 0.27/0.46 | ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ) )
% 0.27/0.46 => ( member @ ( coinductive_llist @ Node ) @ X @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % paths.coinduct
% 0.27/0.46 thf(fact_8_lappend__code_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
% 0.27/0.46 = Ys ) ).
% 0.27/0.46
% 0.27/0.46 % lappend_code(1)
% 0.27/0.46 thf(fact_9_lappend__LNil2,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
% 0.27/0.46 = Xs ) ).
% 0.27/0.46
% 0.27/0.46 % lappend_LNil2
% 0.27/0.46 thf(fact_10_lappend__code_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,Xa2: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa2 @ X ) @ Ys )
% 0.27/0.46 = ( coinductive_LCons @ A @ Xa2 @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend_code(2)
% 0.27/0.46 thf(fact_11_lappend__snocL1__conv__LCons2,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
% 0.27/0.46 = ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend_snocL1_conv_LCons2
% 0.27/0.46 thf(fact_12_llist_Oinject,axiom,
% 0.27/0.46 ! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( coinductive_LCons @ A @ X21 @ X22 )
% 0.27/0.46 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 0.27/0.46 = ( ( X21 = Y21 )
% 0.27/0.46 & ( X22 = Y22 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.inject
% 0.27/0.46 thf(fact_13_lappend__LNil__LNil,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend_LNil_LNil
% 0.27/0.46 thf(fact_14_LNil__eq__lappend__iff,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( coinductive_LNil @ A )
% 0.27/0.46 = ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( ( Xs
% 0.27/0.46 = ( coinductive_LNil @ A ) )
% 0.27/0.46 & ( Ys
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % LNil_eq_lappend_iff
% 0.27/0.46 thf(fact_15_lappend__eq__LNil__iff,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.27/0.46 = ( coinductive_LNil @ A ) )
% 0.27/0.46 = ( ( Xs
% 0.27/0.46 = ( coinductive_LNil @ A ) )
% 0.27/0.46 & ( Ys
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend_eq_LNil_iff
% 0.27/0.46 thf(fact_16_llist_Odistinct_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_LNil @ A )
% 0.27/0.46 != ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.distinct(1)
% 0.27/0.46 thf(fact_17_llist_Oexhaust,axiom,
% 0.27/0.46 ! [A: $tType,Y: coinductive_llist @ A] :
% 0.27/0.46 ( ( Y
% 0.27/0.46 != ( coinductive_LNil @ A ) )
% 0.27/0.46 => ~ ! [X212: A,X222: coinductive_llist @ A] :
% 0.27/0.46 ( Y
% 0.27/0.46 != ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.exhaust
% 0.27/0.46 thf(fact_18_neq__LNil__conv,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( Xs
% 0.27/0.46 != ( coinductive_LNil @ A ) )
% 0.27/0.46 = ( ? [X3: A,Xs5: coinductive_llist @ A] :
% 0.27/0.46 ( Xs
% 0.27/0.46 = ( coinductive_LCons @ A @ X3 @ Xs5 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % neq_LNil_conv
% 0.27/0.46 thf(fact_19_lappend__assoc,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 0.27/0.46 = ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend_assoc
% 0.27/0.46 thf(fact_20_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.27/0.46 = ( ( Xs
% 0.27/0.46 = ( coinductive_LNil @ A ) )
% 0.27/0.46 | ? [Xs5: coinductive_llist @ A] :
% 0.27/0.46 ( ( Xs
% 0.27/0.46 = ( coinductive_LCons @ A @ Y @ Xs5 ) )
% 0.27/0.46 & ( coindu328551480prefix @ A @ Xs5 @ Ys ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Coinductive_List.finite_lprefix_nitpick_simps(3)
% 0.27/0.46 thf(fact_21_lstrict__prefix__code_I2_J,axiom,
% 0.27/0.46 ! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% 0.27/0.46
% 0.27/0.46 % lstrict_prefix_code(2)
% 0.27/0.46 thf(fact_22_lstrict__prefix__code_I3_J,axiom,
% 0.27/0.46 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 0.27/0.46 ~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% 0.27/0.46
% 0.27/0.46 % lstrict_prefix_code(3)
% 0.27/0.46 thf(fact_23_llast__singleton,axiom,
% 0.27/0.46 ! [A: $tType,X: A] :
% 0.27/0.46 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
% 0.27/0.46 = X ) ).
% 0.27/0.46
% 0.27/0.46 % llast_singleton
% 0.27/0.46 thf(fact_24_lmember__code_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,X: A] :
% 0.27/0.46 ~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lmember_code(1)
% 0.27/0.46 thf(fact_25_lmember__code_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.27/0.46 = ( ( X = Y )
% 0.27/0.46 | ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lmember_code(2)
% 0.27/0.46 thf(fact_26_pathsp_Ocoinduct,axiom,
% 0.27/0.46 ! [Node: $tType,X4: ( coinductive_llist @ Node ) > $o,X: coinductive_llist @ Node,Graph: Node > Node > $o] :
% 0.27/0.46 ( ( X4 @ X )
% 0.27/0.46 => ( ! [X2: coinductive_llist @ Node] :
% 0.27/0.46 ( ( X4 @ X2 )
% 0.27/0.46 => ( ( X2
% 0.27/0.46 = ( coinductive_LNil @ Node ) )
% 0.27/0.46 | ? [Xa: Node] :
% 0.27/0.46 ( X2
% 0.27/0.46 = ( coinductive_LCons @ Node @ Xa @ ( coinductive_LNil @ Node ) ) )
% 0.27/0.46 | ? [Xa: Node,Y4: Node,Xs4: coinductive_llist @ Node] :
% 0.27/0.46 ( ( X2
% 0.27/0.46 = ( coinductive_LCons @ Node @ Xa @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) ) )
% 0.27/0.46 & ( Graph @ Xa @ Y4 )
% 0.27/0.46 & ( ( X4 @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) )
% 0.27/0.46 | ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) ) ) ) ) )
% 0.27/0.46 => ( koenig2031690877pathsp @ Node @ Graph @ X ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % pathsp.coinduct
% 0.27/0.46 thf(fact_27_pathsp_OSingle,axiom,
% 0.27/0.46 ! [Node: $tType,Graph: Node > Node > $o,X: Node] : ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ X @ ( coinductive_LNil @ Node ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % pathsp.Single
% 0.27/0.46 thf(fact_28_pathsp_Osimps,axiom,
% 0.27/0.46 ! [Node: $tType] :
% 0.27/0.46 ( ( koenig2031690877pathsp @ Node )
% 0.27/0.46 = ( ^ [Graph2: Node > Node > $o,A3: coinductive_llist @ Node] :
% 0.27/0.46 ( ( A3
% 0.27/0.46 = ( coinductive_LNil @ Node ) )
% 0.27/0.46 | ? [X3: Node] :
% 0.27/0.46 ( A3
% 0.27/0.46 = ( coinductive_LCons @ Node @ X3 @ ( coinductive_LNil @ Node ) ) )
% 0.27/0.46 | ? [X3: Node,Y3: Node,Xs3: coinductive_llist @ Node] :
% 0.27/0.46 ( ( A3
% 0.27/0.46 = ( coinductive_LCons @ Node @ X3 @ ( coinductive_LCons @ Node @ Y3 @ Xs3 ) ) )
% 0.27/0.46 & ( Graph2 @ X3 @ Y3 )
% 0.27/0.46 & ( koenig2031690877pathsp @ Node @ Graph2 @ ( coinductive_LCons @ Node @ Y3 @ Xs3 ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % pathsp.simps
% 0.27/0.46 thf(fact_29_pathsp_Ocases,axiom,
% 0.27/0.46 ! [Node: $tType,Graph: Node > Node > $o,A2: coinductive_llist @ Node] :
% 0.27/0.46 ( ( koenig2031690877pathsp @ Node @ Graph @ A2 )
% 0.27/0.46 => ( ( A2
% 0.27/0.46 != ( coinductive_LNil @ Node ) )
% 0.27/0.46 => ( ! [X2: Node] :
% 0.27/0.46 ( A2
% 0.27/0.46 != ( coinductive_LCons @ Node @ X2 @ ( coinductive_LNil @ Node ) ) )
% 0.27/0.46 => ~ ! [X2: Node,Y2: Node,Xs2: coinductive_llist @ Node] :
% 0.27/0.46 ( ( A2
% 0.27/0.46 = ( coinductive_LCons @ Node @ X2 @ ( coinductive_LCons @ Node @ Y2 @ Xs2 ) ) )
% 0.27/0.46 => ( ( Graph @ X2 @ Y2 )
% 0.27/0.46 => ~ ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ Y2 @ Xs2 ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % pathsp.cases
% 0.27/0.46 thf(fact_30_llast__LCons2,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 0.27/0.46 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llast_LCons2
% 0.27/0.46 thf(fact_31_lstrict__prefix__code_I4_J,axiom,
% 0.27/0.46 ! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
% 0.27/0.46 ( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
% 0.27/0.46 = ( ( X = Y )
% 0.27/0.46 & ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lstrict_prefix_code(4)
% 0.27/0.46 thf(fact_32_lstrict__prefix__code_I1_J,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lstrict_prefix_code(1)
% 0.27/0.46 thf(fact_33_llist__less__induct,axiom,
% 0.27/0.46 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ! [Xs2: coinductive_llist @ A] :
% 0.27/0.46 ( ! [Ys2: coinductive_llist @ A] :
% 0.27/0.46 ( ( coindu1478340336prefix @ A @ Ys2 @ Xs2 )
% 0.27/0.46 => ( P @ Ys2 ) )
% 0.27/0.46 => ( P @ Xs2 ) )
% 0.27/0.46 => ( P @ Xs ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist_less_induct
% 0.27/0.46 thf(fact_34_pathsp_OLCons,axiom,
% 0.27/0.46 ! [Node: $tType,Graph: Node > Node > $o,X: Node,Y: Node,Xs: coinductive_llist @ Node] :
% 0.27/0.46 ( ( Graph @ X @ Y )
% 0.27/0.46 => ( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ Y @ Xs ) )
% 0.27/0.46 => ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ X @ ( coinductive_LCons @ Node @ Y @ Xs ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % pathsp.LCons
% 0.27/0.46 thf(fact_35_pathsp_OEmpty,axiom,
% 0.27/0.46 ! [Node: $tType,Graph: Node > Node > $o] : ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LNil @ Node ) ) ).
% 0.27/0.46
% 0.27/0.46 % pathsp.Empty
% 0.27/0.46 thf(fact_36_pathsp__paths__eq,axiom,
% 0.27/0.46 ! [Node: $tType] :
% 0.27/0.46 ( ( koenig2031690877pathsp @ Node )
% 0.27/0.46 = ( ^ [Graph2: Node > Node > $o,X3: coinductive_llist @ Node] : ( member @ ( coinductive_llist @ Node ) @ X3 @ ( koenig916195507_paths @ Node @ Graph2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % pathsp_paths_eq
% 0.27/0.46 thf(fact_37_paths__def,axiom,
% 0.27/0.46 ! [Node: $tType] :
% 0.27/0.46 ( ( koenig916195507_paths @ Node )
% 0.27/0.46 = ( ^ [Graph2: Node > Node > $o] : ( collect @ ( coinductive_llist @ Node ) @ ( koenig2031690877pathsp @ Node @ Graph2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % paths_def
% 0.27/0.46 thf(fact_38_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 0.27/0.46
% 0.27/0.46 % Coinductive_List.finite_lprefix_nitpick_simps(2)
% 0.27/0.46 thf(fact_39_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 0.27/0.46 = ( Xs
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Coinductive_List.finite_lprefix_nitpick_simps(1)
% 0.27/0.46 thf(fact_40_llast__lappend__LCons,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
% 0.27/0.46 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llast_lappend_LCons
% 0.27/0.46 thf(fact_41_llimit__induct,axiom,
% 0.27/0.46 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( P @ ( coinductive_LNil @ A ) )
% 0.27/0.46 => ( ! [X2: A,Xs2: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ Xs2 )
% 0.27/0.46 => ( ( P @ Xs2 )
% 0.27/0.46 => ( P @ ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) )
% 0.27/0.46 => ( ( ! [Ys2: coinductive_llist @ A] :
% 0.27/0.46 ( ( coindu1478340336prefix @ A @ Ys2 @ Xs )
% 0.27/0.46 => ( P @ Ys2 ) )
% 0.27/0.46 => ( P @ Xs ) )
% 0.27/0.46 => ( P @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llimit_induct
% 0.27/0.46 thf(fact_42_gen__lset__code_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A] :
% 0.27/0.46 ( ( coinductive_gen_lset @ A @ A4 @ ( coinductive_LNil @ A ) )
% 0.27/0.46 = A4 ) ).
% 0.27/0.46
% 0.27/0.46 % gen_lset_code(1)
% 0.27/0.46 thf(fact_43_lsorted__code_I2_J,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lsorted_code(2)
% 0.27/0.46 thf(fact_44_lfinite__rev__induct,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 => ( ( P @ ( coinductive_LNil @ A ) )
% 0.27/0.46 => ( ! [X2: A,Xs2: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ Xs2 )
% 0.27/0.46 => ( ( P @ Xs2 )
% 0.27/0.46 => ( P @ ( coinductive_lappend @ A @ Xs2 @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) ) ) )
% 0.27/0.46 => ( P @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfinite_rev_induct
% 0.27/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.27/0.46 ! [A: $tType,A2: A,P: A > $o] :
% 0.27/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.27/0.46 = ( P @ A2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % mem_Collect_eq
% 0.27/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A] :
% 0.27/0.46 ( ( collect @ A
% 0.27/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.27/0.46 = A4 ) ).
% 0.27/0.46
% 0.27/0.46 % Collect_mem_eq
% 0.27/0.46 thf(fact_47_Collect__cong,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.46 ( ! [X2: A] :
% 0.27/0.46 ( ( P @ X2 )
% 0.27/0.46 = ( Q @ X2 ) )
% 0.27/0.46 => ( ( collect @ A @ P )
% 0.27/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Collect_cong
% 0.27/0.46 thf(fact_48_ext,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.27/0.46 ( ! [X2: A] :
% 0.27/0.46 ( ( F @ X2 )
% 0.27/0.46 = ( G @ X2 ) )
% 0.27/0.46 => ( F = G ) ) ).
% 0.27/0.46
% 0.27/0.46 % ext
% 0.27/0.46 thf(fact_49_llist_Ocorec__code,axiom,
% 0.27/0.46 ! [A: $tType,C: $tType] :
% 0.27/0.46 ( ( coindu1259883913_llist @ C @ A )
% 0.27/0.46 = ( ^ [P2: C > $o,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C,A3: C] : ( if @ ( coinductive_llist @ A ) @ ( P2 @ A3 ) @ ( coinductive_LNil @ A ) @ ( coinductive_LCons @ A @ ( G21 @ A3 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1259883913_llist @ C @ A @ P2 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.corec_code
% 0.27/0.46 thf(fact_50_unfold__llist_Ocode,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType] :
% 0.27/0.46 ( ( coindu1441602521_llist @ A @ B )
% 0.27/0.46 = ( ^ [P2: A > $o,G21: A > B,G22: A > A,A3: A] : ( if @ ( coinductive_llist @ B ) @ ( P2 @ A3 ) @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ ( G21 @ A3 ) @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % unfold_llist.code
% 0.27/0.46 thf(fact_51_lappend_Ocode,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( coinductive_lappend @ A )
% 0.27/0.46 = ( ^ [Xs3: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
% 0.27/0.46 ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ Ys3
% 0.27/0.46 @ ^ [X3: A,Xs5: coinductive_llist @ A] : ( coinductive_LCons @ A @ X3 @ ( coinductive_lappend @ A @ Xs5 @ Ys3 ) )
% 0.27/0.46 @ Xs3 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend.code
% 0.27/0.46 thf(fact_52_ltakeWhile__LCons,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( P @ X )
% 0.27/0.46 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.27/0.46 = ( coinductive_LCons @ A @ X @ ( coindu501562517eWhile @ A @ P @ Xs ) ) ) )
% 0.27/0.46 & ( ~ ( P @ X )
% 0.27/0.46 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ltakeWhile_LCons
% 0.27/0.46 thf(fact_53_ltakeWhile__K__True,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coindu501562517eWhile @ A
% 0.27/0.46 @ ^ [Uu: A] : $true
% 0.27/0.46 @ Xs )
% 0.27/0.46 = Xs ) ).
% 0.27/0.46
% 0.27/0.46 % ltakeWhile_K_True
% 0.27/0.46 thf(fact_54_lfinite__LCons,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.27/0.46 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfinite_LCons
% 0.27/0.46 thf(fact_55_lfinite__code_I2_J,axiom,
% 0.27/0.46 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 0.27/0.46 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
% 0.27/0.46 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfinite_code(2)
% 0.27/0.46 thf(fact_56_lfinite__code_I1_J,axiom,
% 0.27/0.46 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfinite_code(1)
% 0.27/0.46 thf(fact_57_lfinite__lappend,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 & ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfinite_lappend
% 0.27/0.46 thf(fact_58_ltakeWhile__LNil,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o] :
% 0.27/0.46 ( ( coindu501562517eWhile @ A @ P @ ( coinductive_LNil @ A ) )
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % ltakeWhile_LNil
% 0.27/0.46 thf(fact_59_lsorted__code_I1_J,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.27/0.46 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lsorted_code(1)
% 0.27/0.46 thf(fact_60_unfold__llist__eq__LCons,axiom,
% 0.27/0.46 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
% 0.27/0.46 = ( coinductive_LCons @ A @ X @ Xs ) )
% 0.27/0.46 = ( ~ ( IS_LNIL @ B2 )
% 0.27/0.46 & ( X
% 0.27/0.46 = ( LHD @ B2 ) )
% 0.27/0.46 & ( Xs
% 0.27/0.46 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % unfold_llist_eq_LCons
% 0.27/0.46 thf(fact_61_ltakeWhile__K__False,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coindu501562517eWhile @ A
% 0.27/0.46 @ ^ [Uu: A] : $false
% 0.27/0.46 @ Xs )
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % ltakeWhile_K_False
% 0.27/0.46 thf(fact_62_corec__llist__never__stop,axiom,
% 0.27/0.46 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,MORE: B > ( coinductive_llist @ A ),LTL: B > B,X: B] :
% 0.27/0.46 ( ( coindu1259883913_llist @ B @ A @ IS_LNIL @ LHD
% 0.27/0.46 @ ^ [Uu: B] : $false
% 0.27/0.46 @ MORE
% 0.27/0.46 @ LTL
% 0.27/0.46 @ X )
% 0.27/0.46 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ X ) ) ).
% 0.27/0.46
% 0.27/0.46 % corec_llist_never_stop
% 0.27/0.46 thf(fact_63_llist_Ocase__distrib,axiom,
% 0.27/0.46 ! [B: $tType,C: $tType,A: $tType,H: B > C,F1: B,F2: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
% 0.27/0.46 ( ( H @ ( coindu1381640503_llist @ B @ A @ F1 @ F2 @ Llist ) )
% 0.27/0.46 = ( coindu1381640503_llist @ C @ A @ ( H @ F1 )
% 0.27/0.46 @ ^ [X1: A,X23: coinductive_llist @ A] : ( H @ ( F2 @ X1 @ X23 ) )
% 0.27/0.46 @ Llist ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.case_distrib
% 0.27/0.46 thf(fact_64_lfinite__LConsI,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfinite_LConsI
% 0.27/0.46 thf(fact_65_lfinite__LNil,axiom,
% 0.27/0.46 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfinite_LNil
% 0.27/0.46 thf(fact_66_lappend__inf,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.27/0.46 = Xs ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend_inf
% 0.27/0.46 thf(fact_67_LNil,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.27/0.46 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % LNil
% 0.27/0.46 thf(fact_68_lstrict__prefix__lfinite1,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coindu1478340336prefix @ A @ Xs @ Ys )
% 0.27/0.46 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.27/0.46
% 0.27/0.46 % lstrict_prefix_lfinite1
% 0.27/0.46 thf(fact_69_unfold__llist_Octr_I2_J,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType,P3: A > $o,A2: A,G212: A > B,G223: A > A] :
% 0.27/0.46 ( ~ ( P3 @ A2 )
% 0.27/0.46 => ( ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 )
% 0.27/0.46 = ( coinductive_LCons @ B @ ( G212 @ A2 ) @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ ( G223 @ A2 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % unfold_llist.ctr(2)
% 0.27/0.46 thf(fact_70_unfold__llist_Octr_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,B: $tType,P3: A > $o,A2: A,G212: A > B,G223: A > A] :
% 0.27/0.46 ( ( P3 @ A2 )
% 0.27/0.46 => ( ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 )
% 0.27/0.46 = ( coinductive_LNil @ B ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % unfold_llist.ctr(1)
% 0.27/0.46 thf(fact_71_llist_Ocorec_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,C: $tType,P3: C > $o,A2: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 0.27/0.46 ( ~ ( P3 @ A2 )
% 0.27/0.46 => ( ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 )
% 0.27/0.46 = ( coinductive_LCons @ A @ ( G212 @ A2 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q222 @ A2 ) @ ( G2212 @ A2 ) @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ ( G2222 @ A2 ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.corec(2)
% 0.27/0.46 thf(fact_72_llist_Ocorec_I1_J,axiom,
% 0.27/0.46 ! [C: $tType,A: $tType,P3: C > $o,A2: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 0.27/0.46 ( ( P3 @ A2 )
% 0.27/0.46 => ( ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 )
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.corec(1)
% 0.27/0.46 thf(fact_73_llist_Osimps_I5_J,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType,F1: B,F2: A > ( coinductive_llist @ A ) > B,X21: A,X22: coinductive_llist @ A] :
% 0.27/0.46 ( ( coindu1381640503_llist @ B @ A @ F1 @ F2 @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 0.27/0.46 = ( F2 @ X21 @ X22 ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.simps(5)
% 0.27/0.46 thf(fact_74_llist_Osimps_I4_J,axiom,
% 0.27/0.46 ! [A: $tType,B: $tType,F1: B,F2: A > ( coinductive_llist @ A ) > B] :
% 0.27/0.46 ( ( coindu1381640503_llist @ B @ A @ F1 @ F2 @ ( coinductive_LNil @ A ) )
% 0.27/0.46 = F1 ) ).
% 0.27/0.46
% 0.27/0.46 % llist.simps(4)
% 0.27/0.46 thf(fact_75_lfinite_Oinducts,axiom,
% 0.27/0.46 ! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ X )
% 0.27/0.46 => ( ( P @ ( coinductive_LNil @ A ) )
% 0.27/0.46 => ( ! [Xs2: coinductive_llist @ A,X2: A] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ Xs2 )
% 0.27/0.46 => ( ( P @ Xs2 )
% 0.27/0.46 => ( P @ ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) )
% 0.27/0.46 => ( P @ X ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfinite.inducts
% 0.27/0.46 thf(fact_76_lfinite_Osimps,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( coinductive_lfinite @ A )
% 0.27/0.46 = ( ^ [A3: coinductive_llist @ A] :
% 0.27/0.46 ( ( A3
% 0.27/0.46 = ( coinductive_LNil @ A ) )
% 0.27/0.46 | ? [Xs3: coinductive_llist @ A,X3: A] :
% 0.27/0.46 ( ( A3
% 0.27/0.46 = ( coinductive_LCons @ A @ X3 @ Xs3 ) )
% 0.27/0.46 & ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfinite.simps
% 0.27/0.46 thf(fact_77_lfinite_Ocases,axiom,
% 0.27/0.46 ! [A: $tType,A2: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ A2 )
% 0.27/0.46 => ( ( A2
% 0.27/0.46 != ( coinductive_LNil @ A ) )
% 0.27/0.46 => ~ ! [Xs2: coinductive_llist @ A] :
% 0.27/0.46 ( ? [X2: A] :
% 0.27/0.46 ( A2
% 0.27/0.46 = ( coinductive_LCons @ A @ X2 @ Xs2 ) )
% 0.27/0.46 => ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfinite.cases
% 0.27/0.46 thf(fact_78_Singleton,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Singleton
% 0.27/0.46 thf(fact_79_lfilter_Osimps,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( coinductive_lfilter @ A )
% 0.27/0.46 = ( ^ [P4: A > $o] :
% 0.27/0.46 ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_LNil @ A )
% 0.27/0.46 @ ^ [X3: A,Xs5: coinductive_llist @ A] : ( if @ ( coinductive_llist @ A ) @ ( P4 @ X3 ) @ ( coinductive_LCons @ A @ X3 @ ( coinductive_lfilter @ A @ P4 @ Xs5 ) ) @ ( coinductive_lfilter @ A @ P4 @ Xs5 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter.simps
% 0.27/0.46 thf(fact_80_lsorted_Ocoinduct,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
% 0.27/0.46 ( ( X4 @ X )
% 0.27/0.46 => ( ! [X2: coinductive_llist @ A] :
% 0.27/0.46 ( ( X4 @ X2 )
% 0.27/0.46 => ( ( X2
% 0.27/0.46 = ( coinductive_LNil @ A ) )
% 0.27/0.46 | ? [Xa: A] :
% 0.27/0.46 ( X2
% 0.27/0.46 = ( coinductive_LCons @ A @ Xa @ ( coinductive_LNil @ A ) ) )
% 0.27/0.46 | ? [Xa: A,Y4: A,Xs4: coinductive_llist @ A] :
% 0.27/0.46 ( ( X2
% 0.27/0.46 = ( coinductive_LCons @ A @ Xa @ ( coinductive_LCons @ A @ Y4 @ Xs4 ) ) )
% 0.27/0.46 & ( ord_less_eq @ A @ Xa @ Y4 )
% 0.27/0.46 & ( ( X4 @ ( coinductive_LCons @ A @ Y4 @ Xs4 ) )
% 0.27/0.46 | ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y4 @ Xs4 ) ) ) ) ) )
% 0.27/0.46 => ( coindu63249387sorted @ A @ X ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lsorted.coinduct
% 0.27/0.46 thf(fact_81_lsorted_Osimps,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.27/0.46 => ( ( coindu63249387sorted @ A )
% 0.27/0.46 = ( ^ [A3: coinductive_llist @ A] :
% 0.27/0.46 ( ( A3
% 0.27/0.46 = ( coinductive_LNil @ A ) )
% 0.27/0.46 | ? [X3: A] :
% 0.27/0.46 ( A3
% 0.27/0.46 = ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
% 0.27/0.46 | ? [X3: A,Y3: A,Xs3: coinductive_llist @ A] :
% 0.27/0.46 ( ( A3
% 0.27/0.46 = ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs3 ) ) )
% 0.27/0.46 & ( ord_less_eq @ A @ X3 @ Y3 )
% 0.27/0.46 & ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y3 @ Xs3 ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lsorted.simps
% 0.27/0.46 thf(fact_82_lsorted_Ocases,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [A2: coinductive_llist @ A] :
% 0.27/0.46 ( ( coindu63249387sorted @ A @ A2 )
% 0.27/0.46 => ( ( A2
% 0.27/0.46 != ( coinductive_LNil @ A ) )
% 0.27/0.46 => ( ! [X2: A] :
% 0.27/0.46 ( A2
% 0.27/0.46 != ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
% 0.27/0.46 => ~ ! [X2: A,Y2: A,Xs2: coinductive_llist @ A] :
% 0.27/0.46 ( ( A2
% 0.27/0.46 = ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y2 @ Xs2 ) ) )
% 0.27/0.46 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.27/0.46 => ~ ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lsorted.cases
% 0.27/0.46 thf(fact_83_lstrict__prefix__lappend__conv,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 & ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lstrict_prefix_lappend_conv
% 0.27/0.46 thf(fact_84_split__llist__first,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ? [Ys4: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 0.27/0.46 ( ( Xs
% 0.27/0.46 = ( coinductive_lappend @ A @ Ys4 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
% 0.27/0.46 & ( coinductive_lfinite @ A @ Ys4 )
% 0.27/0.46 & ~ ( member @ A @ X @ ( coinductive_lset @ A @ Ys4 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % split_llist_first
% 0.27/0.46 thf(fact_85_split__llist,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ? [Ys4: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 0.27/0.46 ( ( Xs
% 0.27/0.46 = ( coinductive_lappend @ A @ Ys4 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
% 0.27/0.46 & ( coinductive_lfinite @ A @ Ys4 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % split_llist
% 0.27/0.46 thf(fact_86_lappend__ltakeWhile__ldropWhile,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lappend @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 0.27/0.46 = Xs ) ).
% 0.27/0.46
% 0.27/0.46 % lappend_ltakeWhile_ldropWhile
% 0.27/0.46 thf(fact_87_lfilter__idem,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 0.27/0.46 = ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter_idem
% 0.27/0.46 thf(fact_88_lfilter__K__True,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfilter @ A
% 0.27/0.46 @ ^ [Uu: A] : $true
% 0.27/0.46 @ Xs )
% 0.27/0.46 = Xs ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter_K_True
% 0.27/0.46 thf(fact_89_lappend_Odisc__iff_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 0.27/0.46 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend.disc_iff(2)
% 0.27/0.46 thf(fact_90_lnull__lappend,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 & ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lnull_lappend
% 0.27/0.46 thf(fact_91_lfilter__LCons,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( P @ X )
% 0.27/0.46 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.27/0.46 = ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
% 0.27/0.46 & ( ~ ( P @ X )
% 0.27/0.46 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.27/0.46 = ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter_LCons
% 0.27/0.46 thf(fact_92_lfilter__LNil,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o] :
% 0.27/0.46 ( ( coinductive_lfilter @ A @ P @ ( coinductive_LNil @ A ) )
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter_LNil
% 0.27/0.46 thf(fact_93_ldropWhile__LCons,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( P @ X )
% 0.27/0.46 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.27/0.46 = ( coindu218763757pWhile @ A @ P @ Xs ) ) )
% 0.27/0.46 & ( ~ ( P @ X )
% 0.27/0.46 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.27/0.46 = ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ldropWhile_LCons
% 0.27/0.46 thf(fact_94_ldropWhile__LNil,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o] :
% 0.27/0.46 ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LNil @ A ) )
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % ldropWhile_LNil
% 0.27/0.46 thf(fact_95_unfold__llist_Odisc__iff_I2_J,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType,P3: A > $o,G212: A > B,G223: A > A,A2: A] :
% 0.27/0.46 ( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 ) ) )
% 0.27/0.46 = ( ~ ( P3 @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % unfold_llist.disc_iff(2)
% 0.27/0.46 thf(fact_96_unfold__llist_Odisc__iff_I1_J,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType,P3: A > $o,G212: A > B,G223: A > A,A2: A] :
% 0.27/0.46 ( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 ) )
% 0.27/0.46 = ( P3 @ A2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % unfold_llist.disc_iff(1)
% 0.27/0.46 thf(fact_97_llist_Ocorec__disc__iff_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,C: $tType,P3: C > $o,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C,A2: C] :
% 0.27/0.46 ( ( ~ ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) ) )
% 0.27/0.46 = ( ~ ( P3 @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.corec_disc_iff(2)
% 0.27/0.46 thf(fact_98_llist_Ocorec__disc__iff_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,C: $tType,P3: C > $o,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C,A2: C] :
% 0.27/0.46 ( ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) )
% 0.27/0.46 = ( P3 @ A2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.corec_disc_iff(1)
% 0.27/0.46 thf(fact_99_lfitler__K__False,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfilter @ A
% 0.27/0.46 @ ^ [Uu: A] : $false
% 0.27/0.46 @ Xs )
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfitler_K_False
% 0.27/0.46 thf(fact_100_lset__lfilter,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lset @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 0.27/0.46 = ( collect @ A
% 0.27/0.46 @ ^ [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 & ( P @ X3 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lset_lfilter
% 0.27/0.46 thf(fact_101_ldropWhile__K__True,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coindu218763757pWhile @ A
% 0.27/0.46 @ ^ [Uu: A] : $true
% 0.27/0.46 @ Xs )
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % ldropWhile_K_True
% 0.27/0.46 thf(fact_102_lsorted__LCons__LCons,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [X: A,Y: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 0.27/0.46 = ( ( ord_less_eq @ A @ X @ Y )
% 0.27/0.46 & ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lsorted_LCons_LCons
% 0.27/0.46 thf(fact_103_diverge__lfilter__LNil,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.27/0.46 ( ! [X2: A] :
% 0.27/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ~ ( P @ X2 ) )
% 0.27/0.46 => ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % diverge_lfilter_LNil
% 0.27/0.46 thf(fact_104_lnull__lfilter,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lnull @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 0.27/0.46 = ( ! [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ~ ( P @ X3 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lnull_lfilter
% 0.27/0.46 thf(fact_105_lnull__ldropWhile,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lnull @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 0.27/0.46 = ( ! [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ( P @ X3 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lnull_ldropWhile
% 0.27/0.46 thf(fact_106_lfilter__lappend__lfinite,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter_lappend_lfinite
% 0.27/0.46 thf(fact_107_lfinite__ldropWhile,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 0.27/0.46 = ( ? [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 & ~ ( P @ X3 ) )
% 0.27/0.46 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfinite_ldropWhile
% 0.27/0.46 thf(fact_108_llist_Odisc__eq__case_I1_J,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( coinductive_lnull @ A )
% 0.27/0.46 = ( coindu1381640503_llist @ $o @ A @ $true
% 0.27/0.46 @ ^ [Uu: A,Uv: coinductive_llist @ A] : $false ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.disc_eq_case(1)
% 0.27/0.46 thf(fact_109_llist_Odisc__eq__case_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.27/0.46 ( ( ~ ( coinductive_lnull @ A @ Llist ) )
% 0.27/0.46 = ( coindu1381640503_llist @ $o @ A @ $false
% 0.27/0.46 @ ^ [Uu: A,Uv: coinductive_llist @ A] : $true
% 0.27/0.46 @ Llist ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.disc_eq_case(2)
% 0.27/0.46 thf(fact_110_lfilter__cong,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 0.27/0.46 ( ( Xs = Ys )
% 0.27/0.46 => ( ! [X2: A] :
% 0.27/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) )
% 0.27/0.46 => ( ( P @ X2 )
% 0.27/0.46 = ( Q @ X2 ) ) )
% 0.27/0.46 => ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.27/0.46 = ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter_cong
% 0.27/0.46 thf(fact_111_lzip_Oexhaust,axiom,
% 0.27/0.46 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.27/0.46 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 | ( coinductive_lnull @ B @ Ys ) )
% 0.27/0.46 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lzip.exhaust
% 0.27/0.46 thf(fact_112_lappend_Oexhaust,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 => ~ ( coinductive_lnull @ A @ Ys ) )
% 0.27/0.46 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend.exhaust
% 0.27/0.46 thf(fact_113_ldropWhile__cong,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 0.27/0.46 ( ( Xs = Ys )
% 0.27/0.46 => ( ! [X2: A] :
% 0.27/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) )
% 0.27/0.46 => ( ( P @ X2 )
% 0.27/0.46 = ( Q @ X2 ) ) )
% 0.27/0.46 => ( ( coindu218763757pWhile @ A @ P @ Xs )
% 0.27/0.46 = ( coindu218763757pWhile @ A @ Q @ Ys ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ldropWhile_cong
% 0.27/0.46 thf(fact_114_lfilter__id__conv,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.27/0.46 = Xs )
% 0.27/0.46 = ( ! [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ( P @ X3 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter_id_conv
% 0.27/0.46 thf(fact_115_wlog__linorder__le,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [P: A > A > $o,B2: A,A2: A] :
% 0.27/0.46 ( ! [A5: A,B3: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.27/0.46 => ( P @ A5 @ B3 ) )
% 0.27/0.46 => ( ( ( P @ B2 @ A2 )
% 0.27/0.46 => ( P @ A2 @ B2 ) )
% 0.27/0.46 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % wlog_linorder_le
% 0.27/0.46 thf(fact_116_lfilter__empty__conv,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.27/0.46 = ( coinductive_LNil @ A ) )
% 0.27/0.46 = ( ! [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ~ ( P @ X3 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter_empty_conv
% 0.27/0.46 thf(fact_117_in__lset__ldropWhileD,axiom,
% 0.27/0.46 ! [A: $tType,X: A,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) )
% 0.27/0.46 => ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % in_lset_ldropWhileD
% 0.27/0.46 thf(fact_118_lset__ldropWhile__subset,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) @ ( coinductive_lset @ A @ Xs ) ) ).
% 0.27/0.46
% 0.27/0.46 % lset_ldropWhile_subset
% 0.27/0.46 thf(fact_119_lset__lappend1,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lset_lappend1
% 0.27/0.46 thf(fact_120_lfilter__lfilter,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Q: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ Q @ Xs ) )
% 0.27/0.46 = ( coinductive_lfilter @ A
% 0.27/0.46 @ ^ [X3: A] :
% 0.27/0.46 ( ( P @ X3 )
% 0.27/0.46 & ( Q @ X3 ) )
% 0.27/0.46 @ Xs ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter_lfilter
% 0.27/0.46 thf(fact_121_ldropWhile__eq__LNil__iff,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( coindu218763757pWhile @ A @ P @ Xs )
% 0.27/0.46 = ( coinductive_LNil @ A ) )
% 0.27/0.46 = ( ! [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ( P @ X3 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ldropWhile_eq_LNil_iff
% 0.27/0.46 thf(fact_122_ltakeWhile__all__conv,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( coindu501562517eWhile @ A @ P @ Xs )
% 0.27/0.46 = Xs )
% 0.27/0.46 = ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( collect @ A @ P ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ltakeWhile_all_conv
% 0.27/0.46 thf(fact_123_lsorted__LCons,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [X: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.27/0.46 = ( ( coindu63249387sorted @ A @ Xs )
% 0.27/0.46 & ! [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lsorted_LCons
% 0.27/0.46 thf(fact_124_lfilter__LCons__seek,axiom,
% 0.27/0.46 ! [A: $tType,P3: A > $o,X: A,L: coinductive_llist @ A] :
% 0.27/0.46 ( ~ ( P3 @ X )
% 0.27/0.46 => ( ( coinductive_lfilter @ A @ P3 @ ( coinductive_LCons @ A @ X @ L ) )
% 0.27/0.46 = ( coinductive_lfilter @ A @ P3 @ L ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter_LCons_seek
% 0.27/0.46 thf(fact_125_lfilter__LCons__found,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( P @ X )
% 0.27/0.46 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.27/0.46 = ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter_LCons_found
% 0.27/0.46 thf(fact_126_lset__intros_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,X5: A] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X5 @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lset_intros(2)
% 0.27/0.46 thf(fact_127_lset__intros_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lset_intros(1)
% 0.27/0.46 thf(fact_128_llist_Oset__intros_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
% 0.27/0.46 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.set_intros(2)
% 0.27/0.46 thf(fact_129_llist_Oset__intros_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.set_intros(1)
% 0.27/0.46 thf(fact_130_lset__cases,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ( ! [Xs6: coinductive_llist @ A] :
% 0.27/0.46 ( Xs
% 0.27/0.46 != ( coinductive_LCons @ A @ X @ Xs6 ) )
% 0.27/0.46 => ~ ! [X6: A,Xs6: coinductive_llist @ A] :
% 0.27/0.46 ( ( Xs
% 0.27/0.46 = ( coinductive_LCons @ A @ X6 @ Xs6 ) )
% 0.27/0.46 => ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs6 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lset_cases
% 0.27/0.46 thf(fact_131_lset__induct,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
% 0.27/0.46 => ( ! [X6: A,Xs2: coinductive_llist @ A] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
% 0.27/0.46 => ( ( X != X6 )
% 0.27/0.46 => ( ( P @ Xs2 )
% 0.27/0.46 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) ) )
% 0.27/0.46 => ( P @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lset_induct
% 0.27/0.46 thf(fact_132_lset__induct_H,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
% 0.27/0.46 => ( ! [X6: A,Xs2: coinductive_llist @ A] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
% 0.27/0.46 => ( ( P @ Xs2 )
% 0.27/0.46 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) )
% 0.27/0.46 => ( P @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lset_induct'
% 0.27/0.46 thf(fact_133_llist_Oset__cases,axiom,
% 0.27/0.46 ! [A: $tType,E: A,A2: coinductive_llist @ A] :
% 0.27/0.46 ( ( member @ A @ E @ ( coinductive_lset @ A @ A2 ) )
% 0.27/0.46 => ( ! [Z2: coinductive_llist @ A] :
% 0.27/0.46 ( A2
% 0.27/0.46 != ( coinductive_LCons @ A @ E @ Z2 ) )
% 0.27/0.46 => ~ ! [Z1: A,Z2: coinductive_llist @ A] :
% 0.27/0.46 ( ( A2
% 0.27/0.46 = ( coinductive_LCons @ A @ Z1 @ Z2 ) )
% 0.27/0.46 => ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z2 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.set_cases
% 0.27/0.46 thf(fact_134_llist_Oset__induct,axiom,
% 0.27/0.46 ! [A: $tType,X: A,A2: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) )
% 0.27/0.46 => ( ! [Z1: A,Z2: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) )
% 0.27/0.46 => ( ! [Z1: A,Z2: coinductive_llist @ A,Xa3: A] :
% 0.27/0.46 ( ( member @ A @ Xa3 @ ( coinductive_lset @ A @ Z2 ) )
% 0.27/0.46 => ( ( P @ Xa3 @ Z2 )
% 0.27/0.46 => ( P @ Xa3 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) ) ) )
% 0.27/0.46 => ( P @ X @ A2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.set_induct
% 0.27/0.46 thf(fact_135_llist_Odisc_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.27/0.46 ~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.disc(2)
% 0.27/0.46 thf(fact_136_llist_OdiscI_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
% 0.27/0.46 ( ( Llist
% 0.27/0.46 = ( coinductive_LCons @ A @ X21 @ X22 ) )
% 0.27/0.46 => ~ ( coinductive_lnull @ A @ Llist ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.discI(2)
% 0.27/0.46 thf(fact_137_not__lnull__conv,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ~ ( coinductive_lnull @ A @ Xs ) )
% 0.27/0.46 = ( ? [X3: A,Xs5: coinductive_llist @ A] :
% 0.27/0.46 ( Xs
% 0.27/0.46 = ( coinductive_LCons @ A @ X3 @ Xs5 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % not_lnull_conv
% 0.27/0.46 thf(fact_138_llist_Odisc_I1_J,axiom,
% 0.27/0.46 ! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.disc(1)
% 0.27/0.46 thf(fact_139_llist_OdiscI_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.27/0.46 ( ( Llist
% 0.27/0.46 = ( coinductive_LNil @ A ) )
% 0.27/0.46 => ( coinductive_lnull @ A @ Llist ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.discI(1)
% 0.27/0.46 thf(fact_140_llist_Ocollapse_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lnull @ A @ Llist )
% 0.27/0.46 => ( Llist
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.collapse(1)
% 0.27/0.46 thf(fact_141_lnull__def,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( coinductive_lnull @ A )
% 0.27/0.46 = ( ^ [Llist2: coinductive_llist @ A] :
% 0.27/0.46 ( Llist2
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lnull_def
% 0.27/0.46 thf(fact_142_lfinite__lfilterI,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 => ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfinite_lfilterI
% 0.27/0.46 thf(fact_143_lappend_Odisc_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 | ~ ( coinductive_lnull @ A @ Ys ) )
% 0.27/0.46 => ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend.disc(2)
% 0.27/0.46 thf(fact_144_lappend_Odisc_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_lnull @ A @ Ys )
% 0.27/0.46 => ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend.disc(1)
% 0.27/0.46 thf(fact_145_lappend__lnull1,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.27/0.46 = Ys ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend_lnull1
% 0.27/0.46 thf(fact_146_lappend__lnull2,axiom,
% 0.27/0.46 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lnull @ A @ Ys )
% 0.27/0.46 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.27/0.46 = Xs ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend_lnull2
% 0.27/0.46 thf(fact_147_lnull__imp__lfinite,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.27/0.46
% 0.27/0.46 % lnull_imp_lfinite
% 0.27/0.46 thf(fact_148_ltakeWhile__all,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.27/0.46 ( ! [X2: A] :
% 0.27/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ( P @ X2 ) )
% 0.27/0.46 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 0.27/0.46 = Xs ) ) ).
% 0.27/0.46
% 0.27/0.46 % ltakeWhile_all
% 0.27/0.46 thf(fact_149_ltakeWhile__cong,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 0.27/0.46 ( ( Xs = Ys )
% 0.27/0.46 => ( ! [X2: A] :
% 0.27/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) )
% 0.27/0.46 => ( ( P @ X2 )
% 0.27/0.46 = ( Q @ X2 ) ) )
% 0.27/0.46 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 0.27/0.46 = ( coindu501562517eWhile @ A @ Q @ Ys ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ltakeWhile_cong
% 0.27/0.46 thf(fact_150_lset__ltakeWhileD,axiom,
% 0.27/0.46 ! [A: $tType,X: A,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
% 0.27/0.46 => ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 & ( P @ X ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lset_ltakeWhileD
% 0.27/0.46 thf(fact_151_lsorted__lfilterI,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [Xs: coinductive_llist @ A,P: A > $o] :
% 0.27/0.46 ( ( coindu63249387sorted @ A @ Xs )
% 0.27/0.46 => ( coindu63249387sorted @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lsorted_lfilterI
% 0.27/0.46 thf(fact_152_ltakeWhile__lappend2,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( collect @ A @ P ) )
% 0.27/0.46 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( coinductive_lappend @ A @ Xs @ ( coindu501562517eWhile @ A @ P @ Ys ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ltakeWhile_lappend2
% 0.27/0.46 thf(fact_153_unfold__llist_Odisc_I2_J,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType,P3: A > $o,A2: A,G212: A > B,G223: A > A] :
% 0.27/0.46 ( ~ ( P3 @ A2 )
% 0.27/0.46 => ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % unfold_llist.disc(2)
% 0.27/0.46 thf(fact_154_unfold__llist_Odisc_I1_J,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType,P3: A > $o,A2: A,G212: A > B,G223: A > A] :
% 0.27/0.46 ( ( P3 @ A2 )
% 0.27/0.46 => ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % unfold_llist.disc(1)
% 0.27/0.46 thf(fact_155_llist_Ocorec__disc_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,C: $tType,P3: C > $o,A2: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 0.27/0.46 ( ~ ( P3 @ A2 )
% 0.27/0.46 => ~ ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.corec_disc(2)
% 0.27/0.46 thf(fact_156_llist_Ocorec__disc_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,C: $tType,P3: C > $o,A2: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 0.27/0.46 ( ( P3 @ A2 )
% 0.27/0.46 => ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llist.corec_disc(1)
% 0.27/0.46 thf(fact_157_lfilter__eq__LConsD,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( coinductive_lfilter @ A @ P @ Ys )
% 0.27/0.46 = ( coinductive_LCons @ A @ X @ Xs ) )
% 0.27/0.46 => ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 0.27/0.46 ( ( Ys
% 0.27/0.46 = ( coinductive_lappend @ A @ Us @ ( coinductive_LCons @ A @ X @ Vs ) ) )
% 0.27/0.46 & ( coinductive_lfinite @ A @ Us )
% 0.27/0.46 & ! [X7: A] :
% 0.27/0.46 ( ( member @ A @ X7 @ ( coinductive_lset @ A @ Us ) )
% 0.27/0.46 => ~ ( P @ X7 ) )
% 0.27/0.46 & ( P @ X )
% 0.27/0.46 & ( Xs
% 0.27/0.46 = ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter_eq_LConsD
% 0.27/0.46 thf(fact_158_lsorted__lfilter__same,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [C2: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( coindu63249387sorted @ A
% 0.27/0.46 @ ( coinductive_lfilter @ A
% 0.27/0.46 @ ^ [X3: A] : ( X3 = C2 )
% 0.27/0.46 @ Xs ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lsorted_lfilter_same
% 0.27/0.46 thf(fact_159_lset__lmember,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 = ( coinductive_lmember @ A @ X @ Xs ) ) ).
% 0.27/0.46
% 0.27/0.46 % lset_lmember
% 0.27/0.46 thf(fact_160_lfilter__eq__lappend__lfiniteD,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.27/0.46 = ( coinductive_lappend @ A @ Ys @ Zs ) )
% 0.27/0.46 => ( ( coinductive_lfinite @ A @ Ys )
% 0.27/0.46 => ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 0.27/0.46 ( ( Xs
% 0.27/0.46 = ( coinductive_lappend @ A @ Us @ Vs ) )
% 0.27/0.46 & ( coinductive_lfinite @ A @ Us )
% 0.27/0.46 & ( Ys
% 0.27/0.46 = ( coinductive_lfilter @ A @ P @ Us ) )
% 0.27/0.46 & ( Zs
% 0.27/0.46 = ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfilter_eq_lappend_lfiniteD
% 0.27/0.46 thf(fact_161_in__lset__lappend__iff,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 0.27/0.46 = ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 | ( ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 & ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % in_lset_lappend_iff
% 0.27/0.46 thf(fact_162_lappend_Octr_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_lnull @ A @ Ys )
% 0.27/0.46 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend.ctr(1)
% 0.27/0.46 thf(fact_163_ltakeWhile__lappend1,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 => ( ~ ( P @ X )
% 0.27/0.46 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ltakeWhile_lappend1
% 0.27/0.46 thf(fact_164_lfinite__ltakeWhile,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 0.27/0.46 = ( ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 | ? [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 & ~ ( P @ X3 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lfinite_ltakeWhile
% 0.27/0.46 thf(fact_165_llast__LCons,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 0.27/0.46 ( ( ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.27/0.46 = X ) )
% 0.27/0.46 & ( ~ ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.27/0.46 = ( coinductive_llast @ A @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llast_LCons
% 0.27/0.46 thf(fact_166_LCons__LCons,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [X: A,Y: A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.27/0.46 => ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) )
% 0.27/0.46 => ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % LCons_LCons
% 0.27/0.46 thf(fact_167_ldropWhile_Osimps,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( coindu218763757pWhile @ A )
% 0.27/0.46 = ( ^ [P4: A > $o,Xs3: coinductive_llist @ A] :
% 0.27/0.46 ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_LNil @ A )
% 0.27/0.46 @ ^ [X3: A,Xs5: coinductive_llist @ A] : ( if @ ( coinductive_llist @ A ) @ ( P4 @ X3 ) @ ( coindu218763757pWhile @ A @ P4 @ Xs5 ) @ Xs3 )
% 0.27/0.46 @ Xs3 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ldropWhile.simps
% 0.27/0.46 thf(fact_168_order__refl,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.27/0.46
% 0.27/0.46 % order_refl
% 0.27/0.46 thf(fact_169_reachable__viap_Ocases,axiom,
% 0.27/0.46 ! [Node: $tType,Graph: Node > Node > $o,Ns: set @ Node,N: Node,A2: Node] :
% 0.27/0.46 ( ( koenig1757754772e_viap @ Node @ Graph @ Ns @ N @ A2 )
% 0.27/0.46 => ~ ! [Xs2: coinductive_llist @ Node] :
% 0.27/0.46 ( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ N @ Xs2 ) )
% 0.27/0.46 => ( ( member @ Node @ A2 @ ( coinductive_lset @ Node @ Xs2 ) )
% 0.27/0.46 => ~ ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % reachable_viap.cases
% 0.27/0.46 thf(fact_170_reachable__viap_Osimps,axiom,
% 0.27/0.46 ! [Node: $tType] :
% 0.27/0.46 ( ( koenig1757754772e_viap @ Node )
% 0.27/0.46 = ( ^ [Graph2: Node > Node > $o,Ns2: set @ Node,N2: Node,A3: Node] :
% 0.27/0.46 ? [Xs3: coinductive_llist @ Node,N3: Node] :
% 0.27/0.46 ( ( A3 = N3 )
% 0.27/0.46 & ( koenig2031690877pathsp @ Node @ Graph2 @ ( coinductive_LCons @ Node @ N2 @ Xs3 ) )
% 0.27/0.46 & ( member @ Node @ N3 @ ( coinductive_lset @ Node @ Xs3 ) )
% 0.27/0.46 & ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs3 ) @ Ns2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % reachable_viap.simps
% 0.27/0.46 thf(fact_171_reachable__viap_Ointros,axiom,
% 0.27/0.46 ! [Node: $tType,Graph: Node > Node > $o,N: Node,Xs: coinductive_llist @ Node,N4: Node,Ns: set @ Node] :
% 0.27/0.46 ( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ N @ Xs ) )
% 0.27/0.46 => ( ( member @ Node @ N4 @ ( coinductive_lset @ Node @ Xs ) )
% 0.27/0.46 => ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs ) @ Ns )
% 0.27/0.46 => ( koenig1757754772e_viap @ Node @ Graph @ Ns @ N @ N4 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % reachable_viap.intros
% 0.27/0.46 thf(fact_172_reachable__viap_Oinducts,axiom,
% 0.27/0.46 ! [Node: $tType,Graph: Node > Node > $o,Ns: set @ Node,N: Node,X: Node,P: Node > $o] :
% 0.27/0.46 ( ( koenig1757754772e_viap @ Node @ Graph @ Ns @ N @ X )
% 0.27/0.46 => ( ! [Xs2: coinductive_llist @ Node,N5: Node] :
% 0.27/0.46 ( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ N @ Xs2 ) )
% 0.27/0.46 => ( ( member @ Node @ N5 @ ( coinductive_lset @ Node @ Xs2 ) )
% 0.27/0.46 => ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns )
% 0.27/0.46 => ( P @ N5 ) ) ) )
% 0.27/0.46 => ( P @ X ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % reachable_viap.inducts
% 0.27/0.46 thf(fact_173_reachable__via_Ocases,axiom,
% 0.27/0.46 ! [Node: $tType,A2: Node,Graph: Node > Node > $o,Ns: set @ Node,N: Node] :
% 0.27/0.46 ( ( member @ Node @ A2 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N ) )
% 0.27/0.46 => ~ ! [Xs2: coinductive_llist @ Node] :
% 0.27/0.46 ( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N @ Xs2 ) @ ( koenig916195507_paths @ Node @ Graph ) )
% 0.27/0.46 => ( ( member @ Node @ A2 @ ( coinductive_lset @ Node @ Xs2 ) )
% 0.27/0.46 => ~ ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % reachable_via.cases
% 0.27/0.46 thf(fact_174_reachable__viap__reachable__via__eq,axiom,
% 0.27/0.46 ! [Node: $tType] :
% 0.27/0.46 ( ( koenig1757754772e_viap @ Node )
% 0.27/0.46 = ( ^ [Graph2: Node > Node > $o,Ns2: set @ Node,N2: Node,X3: Node] : ( member @ Node @ X3 @ ( koenig317145564le_via @ Node @ Graph2 @ Ns2 @ N2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % reachable_viap_reachable_via_eq
% 0.27/0.46 thf(fact_175_reachable__via__def,axiom,
% 0.27/0.46 ! [Node: $tType] :
% 0.27/0.46 ( ( koenig317145564le_via @ Node )
% 0.27/0.46 = ( ^ [Graph2: Node > Node > $o,Ns2: set @ Node,N2: Node] : ( collect @ Node @ ( koenig1757754772e_viap @ Node @ Graph2 @ Ns2 @ N2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % reachable_via_def
% 0.27/0.46 thf(fact_176_le__funD,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType] :
% 0.27/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.27/0.46 => ! [F: A > B,G: A > B,X: A] :
% 0.27/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.27/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % le_funD
% 0.27/0.46 thf(fact_177_le__funE,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType] :
% 0.27/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.27/0.46 => ! [F: A > B,G: A > B,X: A] :
% 0.27/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.27/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % le_funE
% 0.27/0.46 thf(fact_178_le__funI,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType] :
% 0.27/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.27/0.46 => ! [F: A > B,G: A > B] :
% 0.27/0.46 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.27/0.46 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % le_funI
% 0.27/0.46 thf(fact_179_le__fun__def,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType] :
% 0.27/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.27/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.27/0.46 = ( ^ [F3: A > B,G2: A > B] :
% 0.27/0.46 ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % le_fun_def
% 0.27/0.46 thf(fact_180_order__subst1,axiom,
% 0.27/0.46 ! [A: $tType,B: $tType] :
% 0.27/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.27/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.27/0.46 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.27/0.46 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.27/0.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.27/0.46 => ( ! [X2: B,Y2: B] :
% 0.27/0.46 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.27/0.46 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.27/0.46 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % order_subst1
% 0.27/0.46 thf(fact_181_order__subst2,axiom,
% 0.27/0.46 ! [A: $tType,C: $tType] :
% 0.27/0.46 ( ( ( order @ C @ ( type2 @ C ) )
% 0.27/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.27/0.46 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.27/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.27/0.46 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.27/0.46 => ( ! [X2: A,Y2: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.27/0.46 => ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.27/0.46 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % order_subst2
% 0.27/0.46 thf(fact_182_ord__eq__le__subst,axiom,
% 0.27/0.46 ! [A: $tType,B: $tType] :
% 0.27/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.27/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.27/0.46 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.27/0.46 ( ( A2
% 0.27/0.46 = ( F @ B2 ) )
% 0.27/0.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.27/0.46 => ( ! [X2: B,Y2: B] :
% 0.27/0.46 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.27/0.46 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.27/0.46 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ord_eq_le_subst
% 0.27/0.46 thf(fact_183_ord__le__eq__subst,axiom,
% 0.27/0.46 ! [A: $tType,B: $tType] :
% 0.27/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.27/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.27/0.46 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 0.27/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.27/0.46 => ( ( ( F @ B2 )
% 0.27/0.46 = C2 )
% 0.27/0.46 => ( ! [X2: A,Y2: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.27/0.46 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.27/0.46 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ord_le_eq_subst
% 0.27/0.46 thf(fact_184_eq__iff,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.27/0.46 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.27/0.46 = ( ^ [X3: A,Y3: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.27/0.46 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % eq_iff
% 0.27/0.46 thf(fact_185_antisym,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [X: A,Y: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.27/0.46 => ( ( ord_less_eq @ A @ Y @ X )
% 0.27/0.46 => ( X = Y ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % antisym
% 0.27/0.46 thf(fact_186_linear,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [X: A,Y: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.27/0.46 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % linear
% 0.27/0.46 thf(fact_187_eq__refl,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [X: A,Y: A] :
% 0.27/0.46 ( ( X = Y )
% 0.27/0.46 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % eq_refl
% 0.27/0.46 thf(fact_188_le__cases,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [X: A,Y: A] :
% 0.27/0.46 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.27/0.46 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % le_cases
% 0.27/0.46 thf(fact_189_order_Otrans,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.27/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.27/0.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % order.trans
% 0.27/0.46 thf(fact_190_le__cases3,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [X: A,Y: A,Z3: A] :
% 0.27/0.46 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.27/0.46 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 0.27/0.46 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.27/0.46 => ~ ( ord_less_eq @ A @ X @ Z3 ) )
% 0.27/0.46 => ( ( ( ord_less_eq @ A @ X @ Z3 )
% 0.27/0.46 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 0.27/0.46 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 0.27/0.46 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.27/0.46 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.27/0.46 => ~ ( ord_less_eq @ A @ Z3 @ X ) )
% 0.27/0.46 => ~ ( ( ord_less_eq @ A @ Z3 @ X )
% 0.27/0.46 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % le_cases3
% 0.27/0.46 thf(fact_191_antisym__conv,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [Y: A,X: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.27/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.27/0.46 = ( X = Y ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % antisym_conv
% 0.27/0.46 thf(fact_192_ord__eq__le__trans,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.27/0.46 ( ( A2 = B2 )
% 0.27/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.27/0.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ord_eq_le_trans
% 0.27/0.46 thf(fact_193_ord__le__eq__trans,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.27/0.46 => ( ( B2 = C2 )
% 0.27/0.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ord_le_eq_trans
% 0.27/0.46 thf(fact_194_order__class_Oorder_Oantisym,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [A2: A,B2: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.27/0.46 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.27/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % order_class.order.antisym
% 0.27/0.46 thf(fact_195_order__trans,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [X: A,Y: A,Z3: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.27/0.46 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.27/0.46 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % order_trans
% 0.27/0.46 thf(fact_196_dual__order_Orefl,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % dual_order.refl
% 0.27/0.46 thf(fact_197_linorder__wlog,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.27/0.46 ( ! [A5: A,B3: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.27/0.46 => ( P @ A5 @ B3 ) )
% 0.27/0.46 => ( ! [A5: A,B3: A] :
% 0.27/0.46 ( ( P @ B3 @ A5 )
% 0.27/0.46 => ( P @ A5 @ B3 ) )
% 0.27/0.46 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % linorder_wlog
% 0.27/0.46 thf(fact_198_dual__order_Otrans,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [B2: A,A2: A,C2: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.27/0.46 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.27/0.46 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % dual_order.trans
% 0.27/0.46 thf(fact_199_dual__order_Oantisym,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.27/0.46 => ! [B2: A,A2: A] :
% 0.27/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.27/0.46 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.27/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % dual_order.antisym
% 0.27/0.46 thf(fact_200_reachable__via_Oinducts,axiom,
% 0.27/0.46 ! [Node: $tType,X: Node,Graph: Node > Node > $o,Ns: set @ Node,N: Node,P: Node > $o] :
% 0.27/0.46 ( ( member @ Node @ X @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N ) )
% 0.27/0.46 => ( ! [Xs2: coinductive_llist @ Node,N5: Node] :
% 0.27/0.46 ( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N @ Xs2 ) @ ( koenig916195507_paths @ Node @ Graph ) )
% 0.27/0.46 => ( ( member @ Node @ N5 @ ( coinductive_lset @ Node @ Xs2 ) )
% 0.27/0.46 => ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns )
% 0.27/0.46 => ( P @ N5 ) ) ) )
% 0.27/0.46 => ( P @ X ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % reachable_via.inducts
% 0.27/0.46 thf(fact_201_reachable__via_Ointros,axiom,
% 0.27/0.46 ! [Node: $tType,N: Node,Xs: coinductive_llist @ Node,Graph: Node > Node > $o,N4: Node,Ns: set @ Node] :
% 0.27/0.46 ( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N @ Xs ) @ ( koenig916195507_paths @ Node @ Graph ) )
% 0.27/0.46 => ( ( member @ Node @ N4 @ ( coinductive_lset @ Node @ Xs ) )
% 0.27/0.46 => ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs ) @ Ns )
% 0.27/0.46 => ( member @ Node @ N4 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % reachable_via.intros
% 0.27/0.46 thf(fact_202_reachable__via_Osimps,axiom,
% 0.27/0.46 ! [Node: $tType,A2: Node,Graph: Node > Node > $o,Ns: set @ Node,N: Node] :
% 0.27/0.46 ( ( member @ Node @ A2 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N ) )
% 0.27/0.46 = ( ? [Xs3: coinductive_llist @ Node,N3: Node] :
% 0.27/0.46 ( ( A2 = N3 )
% 0.27/0.46 & ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N @ Xs3 ) @ ( koenig916195507_paths @ Node @ Graph ) )
% 0.27/0.46 & ( member @ Node @ N3 @ ( coinductive_lset @ Node @ Xs3 ) )
% 0.27/0.46 & ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs3 ) @ Ns ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % reachable_via.simps
% 0.27/0.46 thf(fact_203_subset__antisym,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.27/0.46 => ( A4 = B4 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % subset_antisym
% 0.27/0.46 thf(fact_204_subsetI,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.27/0.46 ( ! [X2: A] :
% 0.27/0.46 ( ( member @ A @ X2 @ A4 )
% 0.27/0.46 => ( member @ A @ X2 @ B4 ) )
% 0.27/0.46 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.27/0.46
% 0.27/0.46 % subsetI
% 0.27/0.46 thf(fact_205_ldropWhile__lappend,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( ? [X7: A] :
% 0.27/0.46 ( ( member @ A @ X7 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 & ~ ( P @ X7 ) )
% 0.27/0.46 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( coinductive_lappend @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) @ Ys ) ) )
% 0.27/0.46 & ( ~ ? [X2: A] :
% 0.27/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 & ~ ( P @ X2 ) )
% 0.27/0.46 => ( ( ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( coindu218763757pWhile @ A @ P @ Ys ) ) )
% 0.27/0.46 & ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ldropWhile_lappend
% 0.27/0.46 thf(fact_206_set__mp,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.46 => ( ( member @ A @ X @ A4 )
% 0.27/0.46 => ( member @ A @ X @ B4 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % set_mp
% 0.27/0.46 thf(fact_207_in__mono,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.46 => ( ( member @ A @ X @ A4 )
% 0.27/0.46 => ( member @ A @ X @ B4 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % in_mono
% 0.27/0.46 thf(fact_208_subsetD,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.46 => ( ( member @ A @ C2 @ A4 )
% 0.27/0.46 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % subsetD
% 0.27/0.46 thf(fact_209_subsetCE,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.46 => ( ( member @ A @ C2 @ A4 )
% 0.27/0.46 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % subsetCE
% 0.27/0.46 thf(fact_210_equalityE,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.27/0.46 ( ( A4 = B4 )
% 0.27/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % equalityE
% 0.27/0.46 thf(fact_211_subset__eq,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.27/0.46 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.27/0.46 ! [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ A6 )
% 0.27/0.46 => ( member @ A @ X3 @ B5 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % subset_eq
% 0.27/0.46 thf(fact_212_equalityD1,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.27/0.46 ( ( A4 = B4 )
% 0.27/0.46 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.27/0.46
% 0.27/0.46 % equalityD1
% 0.27/0.46 thf(fact_213_equalityD2,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.27/0.46 ( ( A4 = B4 )
% 0.27/0.46 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 0.27/0.46
% 0.27/0.46 % equalityD2
% 0.27/0.46 thf(fact_214_set__rev__mp,axiom,
% 0.27/0.46 ! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
% 0.27/0.46 ( ( member @ A @ X @ A4 )
% 0.27/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.46 => ( member @ A @ X @ B4 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % set_rev_mp
% 0.27/0.46 thf(fact_215_subset__iff,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.27/0.46 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.27/0.46 ! [T: A] :
% 0.27/0.46 ( ( member @ A @ T @ A6 )
% 0.27/0.46 => ( member @ A @ T @ B5 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % subset_iff
% 0.27/0.46 thf(fact_216_rev__subsetD,axiom,
% 0.27/0.46 ! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
% 0.27/0.46 ( ( member @ A @ C2 @ A4 )
% 0.27/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.46 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % rev_subsetD
% 0.27/0.46 thf(fact_217_subset__refl,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 0.27/0.46
% 0.27/0.46 % subset_refl
% 0.27/0.46 thf(fact_218_Collect__mono,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.46 ( ! [X2: A] :
% 0.27/0.46 ( ( P @ X2 )
% 0.27/0.46 => ( Q @ X2 ) )
% 0.27/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Collect_mono
% 0.27/0.46 thf(fact_219_subset__trans,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 0.27/0.46 => ( ord_less_eq @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % subset_trans
% 0.27/0.46 thf(fact_220_set__eq__subset,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ^ [Y5: set @ A,Z: set @ A] : ( Y5 = Z ) )
% 0.27/0.46 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
% 0.27/0.46 & ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % set_eq_subset
% 0.27/0.46 thf(fact_221_contra__subsetD,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.46 => ( ~ ( member @ A @ C2 @ B4 )
% 0.27/0.46 => ~ ( member @ A @ C2 @ A4 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % contra_subsetD
% 0.27/0.46 thf(fact_222_Collect__mono__iff,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.27/0.46 = ( ! [X3: A] :
% 0.27/0.46 ( ( P @ X3 )
% 0.27/0.46 => ( Q @ X3 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Collect_mono_iff
% 0.27/0.46 thf(fact_223_ltakeWhile__lappend,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( ? [X7: A] :
% 0.27/0.46 ( ( member @ A @ X7 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 & ~ ( P @ X7 ) )
% 0.27/0.46 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( coindu501562517eWhile @ A @ P @ Xs ) ) )
% 0.27/0.46 & ( ~ ? [X2: A] :
% 0.27/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.27/0.46 & ~ ( P @ X2 ) )
% 0.27/0.46 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( coinductive_lappend @ A @ Xs @ ( coindu501562517eWhile @ A @ P @ Ys ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ltakeWhile_lappend
% 0.27/0.46 thf(fact_224_less__eq__set__def,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.27/0.46 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.27/0.46 ( ord_less_eq @ ( A > $o )
% 0.27/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A6 )
% 0.27/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % less_eq_set_def
% 0.27/0.46 thf(fact_225_pred__subset__eq,axiom,
% 0.27/0.46 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.27/0.46 ( ( ord_less_eq @ ( A > $o )
% 0.27/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 0.27/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ S ) )
% 0.27/0.46 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 0.27/0.46
% 0.27/0.46 % pred_subset_eq
% 0.27/0.46 thf(fact_226_conj__subset__def,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,P: A > $o,Q: A > $o] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4
% 0.27/0.46 @ ( collect @ A
% 0.27/0.46 @ ^ [X3: A] :
% 0.27/0.46 ( ( P @ X3 )
% 0.27/0.46 & ( Q @ X3 ) ) ) )
% 0.27/0.46 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P ) )
% 0.27/0.46 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % conj_subset_def
% 0.27/0.46 thf(fact_227_predicate1I,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.46 ( ! [X2: A] :
% 0.27/0.46 ( ( P @ X2 )
% 0.27/0.46 => ( Q @ X2 ) )
% 0.27/0.46 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.27/0.46
% 0.27/0.46 % predicate1I
% 0.27/0.46 thf(fact_228_rev__predicate1D,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,X: A,Q: A > $o] :
% 0.27/0.46 ( ( P @ X )
% 0.27/0.46 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.27/0.46 => ( Q @ X ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % rev_predicate1D
% 0.27/0.46 thf(fact_229_predicate1D,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Q: A > $o,X: A] :
% 0.27/0.46 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.27/0.46 => ( ( P @ X )
% 0.27/0.46 => ( Q @ X ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % predicate1D
% 0.27/0.46 thf(fact_230_subset__Collect__iff,axiom,
% 0.27/0.46 ! [A: $tType,B4: set @ A,A4: set @ A,P: A > $o] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.27/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B4
% 0.27/0.46 @ ( collect @ A
% 0.27/0.46 @ ^ [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ A4 )
% 0.27/0.46 & ( P @ X3 ) ) ) )
% 0.27/0.46 = ( ! [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ B4 )
% 0.27/0.46 => ( P @ X3 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % subset_Collect_iff
% 0.27/0.46 thf(fact_231_subset__CollectI,axiom,
% 0.27/0.46 ! [A: $tType,B4: set @ A,A4: set @ A,Q: A > $o,P: A > $o] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.27/0.46 => ( ! [X2: A] :
% 0.27/0.46 ( ( member @ A @ X2 @ B4 )
% 0.27/0.46 => ( ( Q @ X2 )
% 0.27/0.46 => ( P @ X2 ) ) )
% 0.27/0.46 => ( ord_less_eq @ ( set @ A )
% 0.27/0.46 @ ( collect @ A
% 0.27/0.46 @ ^ [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ B4 )
% 0.27/0.46 & ( Q @ X3 ) ) )
% 0.27/0.46 @ ( collect @ A
% 0.27/0.46 @ ^ [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ A4 )
% 0.27/0.46 & ( P @ X3 ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % subset_CollectI
% 0.27/0.46 thf(fact_232_prop__restrict,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Z4: set @ A,X4: set @ A,P: A > $o] :
% 0.27/0.46 ( ( member @ A @ X @ Z4 )
% 0.27/0.46 => ( ( ord_less_eq @ ( set @ A ) @ Z4
% 0.27/0.46 @ ( collect @ A
% 0.27/0.46 @ ^ [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ X4 )
% 0.27/0.46 & ( P @ X3 ) ) ) )
% 0.27/0.46 => ( P @ X ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % prop_restrict
% 0.27/0.46 thf(fact_233_Collect__restrict,axiom,
% 0.27/0.46 ! [A: $tType,X4: set @ A,P: A > $o] :
% 0.27/0.46 ( ord_less_eq @ ( set @ A )
% 0.27/0.46 @ ( collect @ A
% 0.27/0.46 @ ^ [X3: A] :
% 0.27/0.46 ( ( member @ A @ X3 @ X4 )
% 0.27/0.46 & ( P @ X3 ) ) )
% 0.27/0.46 @ X4 ) ).
% 0.27/0.46
% 0.27/0.46 % Collect_restrict
% 0.27/0.46 thf(fact_234_llast__lappend,axiom,
% 0.27/0.46 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( coinductive_lnull @ A @ Ys )
% 0.27/0.46 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( coinductive_llast @ A @ Xs ) ) )
% 0.27/0.46 & ( ~ ( coinductive_lnull @ A @ Ys )
% 0.27/0.46 => ( ( ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( coinductive_llast @ A @ Ys ) ) )
% 0.27/0.46 & ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( undefined @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llast_lappend
% 0.27/0.46 thf(fact_235_Powp__mono,axiom,
% 0.27/0.46 ! [A: $tType,A4: A > $o,B4: A > $o] :
% 0.27/0.46 ( ( ord_less_eq @ ( A > $o ) @ A4 @ B4 )
% 0.27/0.46 => ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A4 ) @ ( powp @ A @ B4 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Powp_mono
% 0.27/0.46 thf(fact_236_llast__linfinite,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_llast @ A @ Xs )
% 0.27/0.46 = ( undefined @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % llast_linfinite
% 0.27/0.46 thf(fact_237_llast__LNil,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
% 0.27/0.46 = ( undefined @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % llast_LNil
% 0.27/0.46 thf(fact_238_lappend_Octr_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 | ~ ( coinductive_lnull @ A @ Ys ) )
% 0.27/0.46 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.27/0.46 = ( coinductive_LCons @ A
% 0.27/0.46 @ ( coindu1381640503_llist @ A @ A @ ( coinductive_lhd @ A @ Ys )
% 0.27/0.46 @ ^ [X3: A,Xs5: coinductive_llist @ A] : X3
% 0.27/0.46 @ Xs )
% 0.27/0.46 @ ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A
% 0.27/0.46 @ ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( undefined @ ( coinductive_llist @ A ) )
% 0.27/0.46 @ ^ [Uu2: A,Uv2: coinductive_llist @ A] : Uv2
% 0.27/0.46 @ Ys )
% 0.27/0.46 @ ^ [X3: A,Xs5: coinductive_llist @ A] : ( coinductive_lappend @ A @ Xs5 @ Ys )
% 0.27/0.46 @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lappend.ctr(2)
% 0.27/0.46 thf(fact_239_lset__lappend__lfinite,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lset_lappend_lfinite
% 0.27/0.46 thf(fact_240_Un__subset__iff,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 )
% 0.27/0.46 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 0.27/0.46 & ( ord_less_eq @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Un_subset_iff
% 0.27/0.46 thf(fact_241_lhd__lappend,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.27/0.46 ( ( ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( coinductive_lhd @ A @ Ys ) ) )
% 0.27/0.46 & ( ~ ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 => ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.27/0.46 = ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lhd_lappend
% 0.27/0.46 thf(fact_242_lnull__ltakeWhile,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 0.27/0.46 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 => ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % lnull_ltakeWhile
% 0.27/0.46 thf(fact_243_ltakeWhile_Odisc__iff_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 0.27/0.46 = ( ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 | ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ltakeWhile.disc_iff(1)
% 0.27/0.46 thf(fact_244_ltakeWhile_Odisc__iff_I2_J,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.27/0.46 ( ( ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
% 0.27/0.46 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 & ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ltakeWhile.disc_iff(2)
% 0.27/0.46 thf(fact_245_subset__Un__eq,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.27/0.46 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.27/0.46 ( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
% 0.27/0.46 = B5 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % subset_Un_eq
% 0.27/0.46 thf(fact_246_Un__absorb2,axiom,
% 0.27/0.46 ! [A: $tType,B4: set @ A,A4: set @ A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.27/0.46 => ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.46 = A4 ) ) ).
% 0.27/0.46
% 0.27/0.46 % Un_absorb2
% 0.27/0.46 thf(fact_247_Un__absorb1,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.46 => ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.46 = B4 ) ) ).
% 0.27/0.46
% 0.27/0.46 % Un_absorb1
% 0.27/0.46 thf(fact_248_Un__upper2,axiom,
% 0.27/0.46 ! [A: $tType,B4: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.27/0.46
% 0.27/0.46 % Un_upper2
% 0.27/0.46 thf(fact_249_Un__upper1,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.27/0.46
% 0.27/0.46 % Un_upper1
% 0.27/0.46 thf(fact_250_Un__least,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,C3: set @ A,B4: set @ A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 0.27/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 0.27/0.46 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Un_least
% 0.27/0.46 thf(fact_251_Un__mono,axiom,
% 0.27/0.46 ! [A: $tType,A4: set @ A,C3: set @ A,B4: set @ A,D: set @ A] :
% 0.27/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 0.27/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ D )
% 0.27/0.46 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Un_mono
% 0.27/0.46 thf(fact_252_lhd__def,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( coinductive_lhd @ A )
% 0.27/0.46 = ( coindu1381640503_llist @ A @ A @ ( undefined @ A )
% 0.27/0.46 @ ^ [X213: A,X223: coinductive_llist @ A] : X213 ) ) ).
% 0.27/0.46
% 0.27/0.46 % lhd_def
% 0.27/0.46 thf(fact_253_ltakeWhile_Octr_I1_J,axiom,
% 0.27/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.27/0.46 ( ( ( coinductive_lnull @ A @ Xs )
% 0.27/0.46 | ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
% 0.27/0.46 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 0.27/0.46 = ( coinductive_LNil @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ltakeWhile.ctr(1)
% 0.27/0.46
% 0.27/0.46 %----Type constructors (10)
% 0.27/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.27/0.46 ! [A7: $tType,A8: $tType] :
% 0.27/0.46 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 0.27/0.46 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.27/0.46 ! [A7: $tType,A8: $tType] :
% 0.27/0.46 ( ( order @ A8 @ ( type2 @ A8 ) )
% 0.27/0.46 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.27/0.46 ! [A7: $tType,A8: $tType] :
% 0.27/0.46 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 0.27/0.46 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
% 0.27/0.46 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 0.27/0.46 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
% 0.27/0.46 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
% 0.27/0.46 preorder @ $o @ ( type2 @ $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.27/0.46 linorder @ $o @ ( type2 @ $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
% 0.27/0.46 order @ $o @ ( type2 @ $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
% 0.27/0.46 ord @ $o @ ( type2 @ $o ) ).
% 0.27/0.46
% 0.27/0.46 %----Helper facts (3)
% 0.27/0.46 thf(help_If_3_1_T,axiom,
% 0.27/0.46 ! [P: $o] :
% 0.27/0.46 ( ( P = $true )
% 0.27/0.46 | ( P = $false ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(help_If_2_1_T,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Y: A] :
% 0.27/0.46 ( ( if @ A @ $false @ X @ Y )
% 0.27/0.46 = Y ) ).
% 0.27/0.46
% 0.27/0.46 thf(help_If_1_1_T,axiom,
% 0.27/0.46 ! [A: $tType,X: A,Y: A] :
% 0.27/0.46 ( ( if @ A @ $true @ X @ Y )
% 0.27/0.46 = X ) ).
% 0.27/0.46
% 0.27/0.46 %----Conjectures (3)
% 0.27/0.46 thf(conj_0,hypothesis,
% 0.27/0.46 ( ( coinductive_lappend @ a @ x @ ys )
% 0.27/0.46 = ( coinductive_LCons @ a @ xa @ ( coinductive_LNil @ a ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(conj_1,hypothesis,
% 0.27/0.46 ( x
% 0.27/0.46 = ( coinductive_LCons @ a @ x21 @ x22 ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(conj_2,conjecture,
% 0.27/0.46 ( ( x
% 0.27/0.46 = ( coinductive_LNil @ a ) )
% 0.27/0.46 | ? [X7: a] :
% 0.27/0.46 ( x
% 0.27/0.46 = ( coinductive_LCons @ a @ X7 @ ( coinductive_LNil @ a ) ) )
% 0.27/0.46 | ? [X7: a,Y4: a,Xs4: coinductive_llist @ a] :
% 0.27/0.46 ( ( x
% 0.27/0.46 = ( coinductive_LCons @ a @ X7 @ ( coinductive_LCons @ a @ Y4 @ Xs4 ) ) )
% 0.27/0.46 & ( graph @ X7 @ Y4 )
% 0.27/0.46 & ( ( member @ ( coinductive_llist @ a ) @ ( coinductive_lappend @ a @ ( coinductive_LCons @ a @ Y4 @ Xs4 ) @ ys ) @ ( koenig916195507_paths @ a @ graph ) )
% 0.27/0.46 | ( member @ ( coinductive_llist @ a ) @ ( coinductive_LCons @ a @ Y4 @ Xs4 ) @ ( koenig916195507_paths @ a @ graph ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 ------------------------
% 2.93/1.52 ---- Embedded in THF ---
% 2.93/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.93/1.52 %%% Generated on Tue Feb 24 04:17:05 EST 2026
% 2.93/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.93/1.52 %%% Logic specification used:
% 2.93/1.52 %%% thf(spec, logic, $$dhol).
% 2.93/1.52
% 2.93/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.5ITWujfIVW/DTF2DTF22687.p
% 2.93/1.52 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 2.93/1.52 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.93/1.52 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.93/1.52 thf(ty_tf_a, type, a: $tType).
% 2.93/1.52 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.93/1.52 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.93/1.52 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.93/1.52 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.93/1.52 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Ofinite__lprefix, type, coindu328551480prefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Ogen__lset, type, coinductive_gen_lset: (!> [A:$tType]: (((set @ A) > ((coinductive_llist @ A) > (set @ A)))))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Olappend, type, coinductive_lappend: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.93/1.52 thf(sy_c_Coinductive__List_OldropWhile, type, coindu218763757pWhile: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Olfilter, type, coinductive_lfilter: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Olfinite, type, coinductive_lfinite: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Ollast, type, coinductive_llast: (!> [A:$tType]: (((coinductive_llist @ A) > A)))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Ollist_OLCons, type, coinductive_LCons: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Ollist_OLNil, type, coinductive_LNil: (!> [A:$tType]: ((coinductive_llist @ A)))).
% 2.93/1.52 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)))))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Ollist_Ocorec__llist, type, coindu1259883913_llist: (!> [C:$tType,A:$tType]: (((C > $o) > ((C > A) > ((C > $o) > ((C > (coinductive_llist @ A)) > ((C > C) > (C > (coinductive_llist @ A)))))))))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Ollist_Olhd, type, coinductive_lhd: (!> [A:$tType]: (((coinductive_llist @ A) > A)))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Ollist_Olnull, type, coinductive_lnull: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Ollist_Olset, type, coinductive_lset: (!> [A:$tType]: (((coinductive_llist @ A) > (set @ A))))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Olmember, type, coinductive_lmember: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > $o))))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Olstrict__prefix, type, coindu1478340336prefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 2.93/1.52 thf(sy_c_Coinductive__List_OltakeWhile, type, coindu501562517eWhile: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Oord__class_Olsorted, type, coindu63249387sorted: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.93/1.52 thf(sy_c_Coinductive__List_Ounfold__llist, type, coindu1441602521_llist: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > B) > ((A > A) > (A > (coinductive_llist @ B)))))))).
% 2.93/1.52 thf(sy_c_HOL_Oundefined, type, undefined: (!> [A:$tType]: (A))).
% 2.93/1.52 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.93/1.52 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Opaths, type, koenig916195507_paths: (!> [Node:$tType]: (((Node > (Node > $o)) > (set @ (coinductive_llist @ Node)))))).
% 2.93/1.52 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Opathsp, type, koenig2031690877pathsp: (!> [Node:$tType]: (((Node > (Node > $o)) > ((coinductive_llist @ Node) > $o))))).
% 2.93/1.52 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__via, type, koenig317145564le_via: (!> [Node:$tType]: (((Node > (Node > $o)) > ((set @ Node) > (Node > (set @ Node))))))).
% 2.93/1.52 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__viap, type, koenig1757754772e_viap: (!> [Node:$tType]: (((Node > (Node > $o)) > ((set @ Node) > (Node > (Node > $o))))))).
% 2.93/1.52 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.93/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.93/1.52 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.93/1.52 thf(sy_c_Relation_OPowp, type, powp: (!> [A:$tType]: (((A > $o) > ((set @ A) > $o))))).
% 2.93/1.52 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.93/1.52 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.93/1.52 thf(sy_v_graph, type, graph: (a > (a > $o))).
% 2.93/1.52 thf(sy_v_x, type, x: (coinductive_llist @ a)).
% 2.93/1.52 thf(sy_v_x21, type, x21: a).
% 2.93/1.52 thf(sy_v_x22, type, x22: (coinductive_llist @ a)).
% 2.93/1.52 thf(sy_v_xa, type, xa: a).
% 2.93/1.52 thf(sy_v_xs, type, xs: (coinductive_llist @ a)).
% 2.93/1.52 thf(sy_v_ys, type, ys: (coinductive_llist @ a)).
% 2.93/1.52 thf(fact_0_assms, axiom, (((member @ (coinductive_llist @ a)) @ (((coinductive_lappend @ a) @ xs) @ ys)) @ ((koenig916195507_paths @ a) @ graph))).
% 2.93/1.52 thf(fact_1_paths_OEmpty, axiom, (! [Node:$tType,Graph:(Node > (Node > $o))]: ((((member @ (coinductive_llist @ Node)) @ (coinductive_LNil @ Node)) @ ((koenig916195507_paths @ Node) @ Graph))))).
% 2.93/1.52 thf(fact_2_paths_OLCons, axiom, (! [Node:$tType,Graph:(Node > (Node > $o)),X:Node,Y:Node,Xs:(coinductive_llist @ Node)]: ((((Graph @ X) @ Y) => ((((member @ (coinductive_llist @ Node)) @ (((coinductive_LCons @ Node) @ Y) @ Xs)) @ ((koenig916195507_paths @ Node) @ Graph)) => (((member @ (coinductive_llist @ Node)) @ (((coinductive_LCons @ Node) @ X) @ (((coinductive_LCons @ Node) @ Y) @ Xs))) @ ((koenig916195507_paths @ Node) @ Graph))))))).
% 2.93/1.52 thf(fact_3_paths_Ocases, axiom, (! [Node:$tType,A2:(coinductive_llist @ Node),Graph:(Node > (Node > $o))]: (((((member @ (coinductive_llist @ Node)) @ A2) @ ((koenig916195507_paths @ Node) @ Graph)) => ((A2 != (coinductive_LNil @ Node)) => ((! [X2:Node]: ((A2 != (((coinductive_LCons @ Node) @ X2) @ (coinductive_LNil @ Node))))) => (~ ((! [X2:Node,Y2:Node,Xs2:(coinductive_llist @ Node)]: (((A2 = (((coinductive_LCons @ Node) @ X2) @ (((coinductive_LCons @ Node) @ Y2) @ Xs2))) => (((Graph @ X2) @ Y2) => (~ ((((member @ (coinductive_llist @ Node)) @ (((coinductive_LCons @ Node) @ Y2) @ Xs2)) @ ((koenig916195507_paths @ Node) @ Graph)))))))))))))))).
% 2.93/1.52 thf(fact_4_paths_Osimps, axiom, (! [Node:$tType,A2:(coinductive_llist @ Node),Graph:(Node > (Node > $o))]: (((((member @ (coinductive_llist @ Node)) @ A2) @ ((koenig916195507_paths @ Node) @ Graph)) = ((A2 = (coinductive_LNil @ Node)) | ((? [X3:Node]: ((A2 = (((coinductive_LCons @ Node) @ X3) @ (coinductive_LNil @ Node))))) | (? [X3:Node,Y3:Node,Xs3:(coinductive_llist @ Node)]: (((A2 = (((coinductive_LCons @ Node) @ X3) @ (((coinductive_LCons @ Node) @ Y3) @ Xs3))) & (((Graph @ X3) @ Y3) & (((member @ (coinductive_llist @ Node)) @ (((coinductive_LCons @ Node) @ Y3) @ Xs3)) @ ((koenig916195507_paths @ Node) @ Graph)))))))))))).
% 2.93/1.52 thf(fact_5_paths_OSingle, axiom, (! [Node:$tType,X:Node,Graph:(Node > (Node > $o))]: ((((member @ (coinductive_llist @ Node)) @ (((coinductive_LCons @ Node) @ X) @ (coinductive_LNil @ Node))) @ ((koenig916195507_paths @ Node) @ Graph))))).
% 2.93/1.52 thf(fact_6_paths__LConsD, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),Graph:(A > (A > $o))]: (((((member @ (coinductive_llist @ A)) @ (((coinductive_LCons @ A) @ X) @ Xs)) @ ((koenig916195507_paths @ A) @ Graph)) => (((member @ (coinductive_llist @ A)) @ Xs) @ ((koenig916195507_paths @ A) @ Graph)))))).
% 2.93/1.52 thf(fact_7_paths_Ocoinduct, axiom, (! [Node:$tType,X4:((coinductive_llist @ Node) > $o),X:(coinductive_llist @ Node),Graph:(Node > (Node > $o))]: (((X4 @ X) => ((! [X2:(coinductive_llist @ Node)]: (((X4 @ X2) => ((X2 = (coinductive_LNil @ Node)) | ((? [Xa:Node]: ((X2 = (((coinductive_LCons @ Node) @ Xa) @ (coinductive_LNil @ Node))))) | (? [Xa:Node,Y4:Node,Xs4:(coinductive_llist @ Node)]: (((X2 = (((coinductive_LCons @ Node) @ Xa) @ (((coinductive_LCons @ Node) @ Y4) @ Xs4))) & (((Graph @ Xa) @ Y4) & ((X4 @ (((coinductive_LCons @ Node) @ Y4) @ Xs4)) | (((member @ (coinductive_llist @ Node)) @ (((coinductive_LCons @ Node) @ Y4) @ Xs4)) @ ((koenig916195507_paths @ Node) @ Graph)))))))))))) => (((member @ (coinductive_llist @ Node)) @ X) @ ((koenig916195507_paths @ Node) @ Graph))))))).
% 2.93/1.52 thf(fact_8_lappend__code_I1_J, axiom, (! [A:$tType,Ys:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (coinductive_LNil @ A)) @ Ys) = Ys)))).
% 2.93/1.52 thf(fact_9_lappend__LNil2, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ Xs) @ (coinductive_LNil @ A)) = Xs)))).
% 2.93/1.52 thf(fact_10_lappend__code_I2_J, axiom, (! [A:$tType,Xa2:A,X:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coinductive_LCons @ A) @ Xa2) @ X)) @ Ys) = (((coinductive_LCons @ A) @ Xa2) @ (((coinductive_lappend @ A) @ X) @ Ys)))))).
% 2.93/1.52 thf(fact_11_lappend__snocL1__conv__LCons2, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Y:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coinductive_lappend @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y) @ (coinductive_LNil @ A)))) @ Ys) = (((coinductive_lappend @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y) @ Ys)))))).
% 2.93/1.52 thf(fact_12_llist_Oinject, axiom, (! [A:$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)))))).
% 2.93/1.52 thf(fact_13_lappend__LNil__LNil, axiom, (! [A:$tType]: (((((coinductive_lappend @ A) @ (coinductive_LNil @ A)) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.93/1.52 thf(fact_14_LNil__eq__lappend__iff, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_LNil @ A) = (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((Xs = (coinductive_LNil @ A)) & (Ys = (coinductive_LNil @ A))))))).
% 2.93/1.52 thf(fact_15_lappend__eq__LNil__iff, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((((coinductive_lappend @ A) @ Xs) @ Ys) = (coinductive_LNil @ A)) = ((Xs = (coinductive_LNil @ A)) & (Ys = (coinductive_LNil @ A))))))).
% 2.93/1.52 thf(fact_16_llist_Odistinct_I1_J, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: (((coinductive_LNil @ A) != (((coinductive_LCons @ A) @ X21) @ X22))))).
% 2.93/1.52 thf(fact_17_llist_Oexhaust, axiom, (! [A:$tType,Y:(coinductive_llist @ A)]: (((Y != (coinductive_LNil @ A)) => (~ ((! [X212:A,X222:(coinductive_llist @ A)]: ((Y != (((coinductive_LCons @ A) @ X212) @ X222)))))))))).
% 2.93/1.52 thf(fact_18_neq__LNil__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((Xs != (coinductive_LNil @ A)) = (? [X3:A,Xs5:(coinductive_llist @ A)]: ((Xs = (((coinductive_LCons @ A) @ X3) @ Xs5)))))))).
% 2.93/1.52 thf(fact_19_lappend__assoc, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) @ Zs) = (((coinductive_lappend @ A) @ Xs) @ (((coinductive_lappend @ A) @ Ys) @ Zs)))))).
% 2.93/1.52 thf(fact_20_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Y:A,Ys:(coinductive_llist @ A)]: (((((coindu328551480prefix @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y) @ Ys)) = ((Xs = (coinductive_LNil @ A)) | (? [Xs5:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ Y) @ Xs5)) & (((coindu328551480prefix @ A) @ Xs5) @ Ys))))))))).
% 2.93/1.52 thf(fact_21_lstrict__prefix__code_I2_J, axiom, (! [B:$tType,Y:B,Ys:(coinductive_llist @ B)]: ((((coindu1478340336prefix @ B) @ (coinductive_LNil @ B)) @ (((coinductive_LCons @ B) @ Y) @ Ys))))).
% 2.93/1.52 thf(fact_22_lstrict__prefix__code_I3_J, axiom, (! [B:$tType,X:B,Xs:(coinductive_llist @ B)]: ((~ ((((coindu1478340336prefix @ B) @ (((coinductive_LCons @ B) @ X) @ Xs)) @ (coinductive_LNil @ B))))))).
% 2.93/1.52 thf(fact_23_llast__singleton, axiom, (! [A:$tType,X:A]: ((((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X) @ (coinductive_LNil @ A))) = X)))).
% 2.93/1.52 thf(fact_24_lmember__code_I1_J, axiom, (! [A:$tType,X:A]: ((~ ((((coinductive_lmember @ A) @ X) @ (coinductive_LNil @ A))))))).
% 2.93/1.52 thf(fact_25_lmember__code_I2_J, axiom, (! [A:$tType,X:A,Y:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lmember @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Ys)) = ((X = Y) | (((coinductive_lmember @ A) @ X) @ Ys)))))).
% 2.93/1.52 thf(fact_26_pathsp_Ocoinduct, axiom, (! [Node:$tType,X4:((coinductive_llist @ Node) > $o),X:(coinductive_llist @ Node),Graph:(Node > (Node > $o))]: (((X4 @ X) => ((! [X2:(coinductive_llist @ Node)]: (((X4 @ X2) => ((X2 = (coinductive_LNil @ Node)) | ((? [Xa:Node]: ((X2 = (((coinductive_LCons @ Node) @ Xa) @ (coinductive_LNil @ Node))))) | (? [Xa:Node,Y4:Node,Xs4:(coinductive_llist @ Node)]: (((X2 = (((coinductive_LCons @ Node) @ Xa) @ (((coinductive_LCons @ Node) @ Y4) @ Xs4))) & (((Graph @ Xa) @ Y4) & ((X4 @ (((coinductive_LCons @ Node) @ Y4) @ Xs4)) | (((koenig2031690877pathsp @ Node) @ Graph) @ (((coinductive_LCons @ Node) @ Y4) @ Xs4)))))))))))) => (((koenig2031690877pathsp @ Node) @ Graph) @ X)))))).
% 2.93/1.52 thf(fact_27_pathsp_OSingle, axiom, (! [Node:$tType,Graph:(Node > (Node > $o)),X:Node]: ((((koenig2031690877pathsp @ Node) @ Graph) @ (((coinductive_LCons @ Node) @ X) @ (coinductive_LNil @ Node)))))).
% 2.93/1.52 thf(fact_28_pathsp_Osimps, axiom, (! [Node:$tType]: (((koenig2031690877pathsp @ Node) = (^ [Graph2:(Node > (Node > $o)),A3:(coinductive_llist @ Node)]: (((A3 = (coinductive_LNil @ Node)) | ((? [X3:Node]: ((A3 = (((coinductive_LCons @ Node) @ X3) @ (coinductive_LNil @ Node))))) | (? [X3:Node,Y3:Node,Xs3:(coinductive_llist @ Node)]: (((A3 = (((coinductive_LCons @ Node) @ X3) @ (((coinductive_LCons @ Node) @ Y3) @ Xs3))) & (((Graph2 @ X3) @ Y3) & (((koenig2031690877pathsp @ Node) @ Graph2) @ (((coinductive_LCons @ Node) @ Y3) @ Xs3)))))))))))))).
% 2.93/1.52 thf(fact_29_pathsp_Ocases, axiom, (! [Node:$tType,Graph:(Node > (Node > $o)),A2:(coinductive_llist @ Node)]: (((((koenig2031690877pathsp @ Node) @ Graph) @ A2) => ((A2 != (coinductive_LNil @ Node)) => ((! [X2:Node]: ((A2 != (((coinductive_LCons @ Node) @ X2) @ (coinductive_LNil @ Node))))) => (~ ((! [X2:Node,Y2:Node,Xs2:(coinductive_llist @ Node)]: (((A2 = (((coinductive_LCons @ Node) @ X2) @ (((coinductive_LCons @ Node) @ Y2) @ Xs2))) => (((Graph @ X2) @ Y2) => (~ ((((koenig2031690877pathsp @ Node) @ Graph) @ (((coinductive_LCons @ Node) @ Y2) @ Xs2)))))))))))))))).
% 2.93/1.52 thf(fact_30_llast__LCons2, axiom, (! [A:$tType,X:A,Y:A,Xs:(coinductive_llist @ A)]: ((((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Xs))) = ((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ Y) @ Xs)))))).
% 2.93/1.52 thf(fact_31_lstrict__prefix__code_I4_J, axiom, (! [B:$tType,X:B,Xs:(coinductive_llist @ B),Y:B,Ys:(coinductive_llist @ B)]: (((((coindu1478340336prefix @ B) @ (((coinductive_LCons @ B) @ X) @ Xs)) @ (((coinductive_LCons @ B) @ Y) @ Ys)) = ((X = Y) & (((coindu1478340336prefix @ B) @ Xs) @ Ys)))))).
% 2.93/1.52 thf(fact_32_lstrict__prefix__code_I1_J, axiom, (! [A:$tType]: ((~ ((((coindu1478340336prefix @ A) @ (coinductive_LNil @ A)) @ (coinductive_LNil @ A))))))).
% 2.93/1.52 thf(fact_33_llist__less__induct, axiom, (! [A:$tType,P:((coinductive_llist @ A) > $o),Xs:(coinductive_llist @ A)]: (((! [Xs2:(coinductive_llist @ A)]: (((! [Ys2:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Ys2) @ Xs2) => (P @ Ys2)))) => (P @ Xs2)))) => (P @ Xs))))).
% 2.93/1.52 thf(fact_34_pathsp_OLCons, axiom, (! [Node:$tType,Graph:(Node > (Node > $o)),X:Node,Y:Node,Xs:(coinductive_llist @ Node)]: ((((Graph @ X) @ Y) => ((((koenig2031690877pathsp @ Node) @ Graph) @ (((coinductive_LCons @ Node) @ Y) @ Xs)) => (((koenig2031690877pathsp @ Node) @ Graph) @ (((coinductive_LCons @ Node) @ X) @ (((coinductive_LCons @ Node) @ Y) @ Xs)))))))).
% 2.93/1.52 thf(fact_35_pathsp_OEmpty, axiom, (! [Node:$tType,Graph:(Node > (Node > $o))]: ((((koenig2031690877pathsp @ Node) @ Graph) @ (coinductive_LNil @ Node))))).
% 2.93/1.52 thf(fact_36_pathsp__paths__eq, axiom, (! [Node:$tType]: (((koenig2031690877pathsp @ Node) = (^ [Graph2:(Node > (Node > $o)),X3:(coinductive_llist @ Node)]: ((((member @ (coinductive_llist @ Node)) @ X3) @ ((koenig916195507_paths @ Node) @ Graph2)))))))).
% 2.93/1.52 thf(fact_37_paths__def, axiom, (! [Node:$tType]: (((koenig916195507_paths @ Node) = (^ [Graph2:(Node > (Node > $o))]: (((collect @ (coinductive_llist @ Node)) @ ((koenig2031690877pathsp @ Node) @ Graph2)))))))).
% 2.93/1.52 thf(fact_38_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coindu328551480prefix @ A) @ (coinductive_LNil @ A)) @ Xs)))).
% 2.93/1.52 thf(fact_39_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coindu328551480prefix @ A) @ Xs) @ (coinductive_LNil @ A)) = (Xs = (coinductive_LNil @ A)))))).
% 2.93/1.52 thf(fact_40_llast__lappend__LCons, axiom, (! [A:$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))))))).
% 2.93/1.52 thf(fact_41_llimit__induct, axiom, (! [A:$tType,P:((coinductive_llist @ A) > $o),Xs:(coinductive_llist @ A)]: (((P @ (coinductive_LNil @ A)) => ((! [X2:A,Xs2:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs2) => ((P @ Xs2) => (P @ (((coinductive_LCons @ A) @ X2) @ Xs2)))))) => (((! [Ys2:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Ys2) @ Xs) => (P @ Ys2)))) => (P @ Xs)) => (P @ Xs))))))).
% 2.93/1.52 thf(fact_42_gen__lset__code_I1_J, axiom, (! [A:$tType,A4:(set @ A)]: (((((coinductive_gen_lset @ A) @ A4) @ (coinductive_LNil @ A)) = A4)))).
% 2.93/1.52 thf(fact_43_lsorted__code_I2_J, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A]: (((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ (coinductive_LNil @ A))))))))).
% 2.93/1.52 thf(fact_44_lfinite__rev__induct, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: ((((coinductive_lfinite @ A) @ Xs) => ((P @ (coinductive_LNil @ A)) => ((! [X2:A,Xs2:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs2) => ((P @ Xs2) => (P @ (((coinductive_lappend @ A) @ Xs2) @ (((coinductive_LCons @ A) @ X2) @ (coinductive_LNil @ A)))))))) => (P @ Xs))))))).
% 2.93/1.52 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.93/1.52 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A4)))) = A4)))).
% 2.93/1.52 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.93/1.52 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 2.93/1.52 thf(fact_49_llist_Ocorec__code, axiom, (! [A:$tType,C:$tType]: ((((coindu1259883913_llist @ C) @ A) = (^ [P2:(C > $o),G21:(C > A),Q22:(C > $o),G221:(C > (coinductive_llist @ A)),G222:(C > C),A3:C]: (((((if @ (coinductive_llist @ A)) @ (P2 @ A3)) @ (coinductive_LNil @ A)) @ (((coinductive_LCons @ A) @ (G21 @ A3)) @ ((((if @ (coinductive_llist @ A)) @ (Q22 @ A3)) @ (G221 @ A3)) @ ((((((((coindu1259883913_llist @ C) @ A) @ P2) @ G21) @ Q22) @ G221) @ G222) @ (G222 @ A3))))))))))).
% 2.93/1.52 thf(fact_50_unfold__llist_Ocode, axiom, (! [B:$tType,A:$tType]: ((((coindu1441602521_llist @ A) @ B) = (^ [P2:(A > $o),G21:(A > B),G22:(A > A),A3:A]: (((((if @ (coinductive_llist @ B)) @ (P2 @ A3)) @ (coinductive_LNil @ B)) @ (((coinductive_LCons @ B) @ (G21 @ A3)) @ ((((((coindu1441602521_llist @ A) @ B) @ P2) @ G21) @ G22) @ (G22 @ A3)))))))))).
% 2.93/1.52 thf(fact_51_lappend_Ocode, axiom, (! [A:$tType]: (((coinductive_lappend @ A) = (^ [Xs3:(coinductive_llist @ A),Ys3:(coinductive_llist @ A)]: ((((((coindu1381640503_llist @ (coinductive_llist @ A)) @ A) @ Ys3) @ (^ [X3:A,Xs5:(coinductive_llist @ A)]: ((((coinductive_LCons @ A) @ X3) @ (((coinductive_lappend @ A) @ Xs5) @ Ys3))))) @ Xs3))))))).
% 2.93/1.52 thf(fact_52_ltakeWhile__LCons, axiom, (! [A:$tType,P:(A > $o),X:A,Xs:(coinductive_llist @ A)]: ((((P @ X) => ((((coindu501562517eWhile @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_LCons @ A) @ X) @ (((coindu501562517eWhile @ A) @ P) @ Xs)))) & ((~ ((P @ X))) => ((((coindu501562517eWhile @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (coinductive_LNil @ A))))))).
% 2.93/1.52 thf(fact_53_ltakeWhile__K__True, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coindu501562517eWhile @ A) @ (^ [Uu:A]: ($true))) @ Xs) = Xs)))).
% 2.93/1.52 thf(fact_54_lfinite__LCons, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = ((coinductive_lfinite @ A) @ Xs))))).
% 2.93/1.52 thf(fact_55_lfinite__code_I2_J, axiom, (! [B:$tType,X:B,Xs:(coinductive_llist @ B)]: ((((coinductive_lfinite @ B) @ (((coinductive_LCons @ B) @ X) @ Xs)) = ((coinductive_lfinite @ B) @ Xs))))).
% 2.93/1.52 thf(fact_56_lfinite__code_I1_J, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) @ (coinductive_LNil @ A))))).
% 2.93/1.52 thf(fact_57_lfinite__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lfinite @ A) @ Xs) & ((coinductive_lfinite @ A) @ Ys)))))).
% 2.93/1.52 thf(fact_58_ltakeWhile__LNil, axiom, (! [A:$tType,P:(A > $o)]: (((((coindu501562517eWhile @ A) @ P) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.93/1.52 thf(fact_59_lsorted__code_I1_J, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => ((coindu63249387sorted @ A) @ (coinductive_LNil @ A)))))).
% 2.93/1.52 thf(fact_60_unfold__llist__eq__LCons, axiom, (! [A:$tType,B:$tType,IS_LNIL:(B > $o),LHD:(B > A),LTL:(B > B),B2:B,X:A,Xs:(coinductive_llist @ A)]: (((((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ B2) = (((coinductive_LCons @ A) @ X) @ Xs)) = ((~ ((IS_LNIL @ B2))) & ((X = (LHD @ B2)) & (Xs = ((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ (LTL @ B2))))))))).
% 2.93/1.52 thf(fact_61_ltakeWhile__K__False, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coindu501562517eWhile @ A) @ (^ [Uu:A]: ($false))) @ Xs) = (coinductive_LNil @ A))))).
% 2.93/1.52 thf(fact_62_corec__llist__never__stop, axiom, (! [A:$tType,B:$tType,IS_LNIL:(B > $o),LHD:(B > A),MORE:(B > (coinductive_llist @ A)),LTL:(B > B),X:B]: ((((((((((coindu1259883913_llist @ B) @ A) @ IS_LNIL) @ LHD) @ (^ [Uu:B]: ($false))) @ MORE) @ LTL) @ X) = ((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ X))))).
% 2.93/1.52 thf(fact_63_llist_Ocase__distrib, axiom, (! [B:$tType,C:$tType,A:$tType,H:(B > C),F1:B,F2:(A > ((coinductive_llist @ A) > B)),Llist:(coinductive_llist @ A)]: (((H @ (((((coindu1381640503_llist @ B) @ A) @ F1) @ F2) @ Llist)) = (((((coindu1381640503_llist @ C) @ A) @ (H @ F1)) @ (^ [X1:A,X23:(coinductive_llist @ A)]: ((H @ ((F2 @ X1) @ X23))))) @ Llist))))).
% 2.93/1.52 thf(fact_64_lfinite__LConsI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),X:A]: ((((coinductive_lfinite @ A) @ Xs) => ((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)))))).
% 2.93/1.52 thf(fact_65_lfinite__LNil, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) @ (coinductive_LNil @ A))))).
% 2.93/1.52 thf(fact_66_lappend__inf, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = Xs))))).
% 2.93/1.52 thf(fact_67_LNil, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => ((coindu63249387sorted @ A) @ (coinductive_LNil @ A)))))).
% 2.93/1.52 thf(fact_68_lstrict__prefix__lfinite1, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Xs) @ Ys) => ((coinductive_lfinite @ A) @ Xs))))).
% 2.93/1.52 thf(fact_69_unfold__llist_Octr_I2_J, axiom, (! [B:$tType,A:$tType,P3:(A > $o),A2:A,G212:(A > B),G223:(A > A)]: (((~ ((P3 @ A2))) => (((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ A2) = (((coinductive_LCons @ B) @ (G212 @ A2)) @ ((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ (G223 @ A2)))))))).
% 2.93/1.52 thf(fact_70_unfold__llist_Octr_I1_J, axiom, (! [A:$tType,B:$tType,P3:(A > $o),A2:A,G212:(A > B),G223:(A > A)]: (((P3 @ A2) => (((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ A2) = (coinductive_LNil @ B)))))).
% 2.93/1.52 thf(fact_71_llist_Ocorec_I2_J, axiom, (! [A:$tType,C:$tType,P3:(C > $o),A2:C,G212:(C > A),Q222:(C > $o),G2212:(C > (coinductive_llist @ A)),G2222:(C > C)]: (((~ ((P3 @ A2))) => (((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ A2) = (((coinductive_LCons @ A) @ (G212 @ A2)) @ ((((if @ (coinductive_llist @ A)) @ (Q222 @ A2)) @ (G2212 @ A2)) @ ((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ (G2222 @ A2))))))))).
% 2.93/1.52 thf(fact_72_llist_Ocorec_I1_J, axiom, (! [C:$tType,A:$tType,P3:(C > $o),A2:C,G212:(C > A),Q222:(C > $o),G2212:(C > (coinductive_llist @ A)),G2222:(C > C)]: (((P3 @ A2) => (((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ A2) = (coinductive_LNil @ A)))))).
% 2.93/1.52 thf(fact_73_llist_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,F1:B,F2:(A > ((coinductive_llist @ A) > B)),X21:A,X22:(coinductive_llist @ A)]: (((((((coindu1381640503_llist @ B) @ A) @ F1) @ F2) @ (((coinductive_LCons @ A) @ X21) @ X22)) = ((F2 @ X21) @ X22))))).
% 2.93/1.52 thf(fact_74_llist_Osimps_I4_J, axiom, (! [A:$tType,B:$tType,F1:B,F2:(A > ((coinductive_llist @ A) > B))]: (((((((coindu1381640503_llist @ B) @ A) @ F1) @ F2) @ (coinductive_LNil @ A)) = F1)))).
% 2.93/1.52 thf(fact_75_lfinite_Oinducts, axiom, (! [A:$tType,X:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: ((((coinductive_lfinite @ A) @ X) => ((P @ (coinductive_LNil @ A)) => ((! [Xs2:(coinductive_llist @ A),X2:A]: ((((coinductive_lfinite @ A) @ Xs2) => ((P @ Xs2) => (P @ (((coinductive_LCons @ A) @ X2) @ Xs2)))))) => (P @ X))))))).
% 2.93/1.52 thf(fact_76_lfinite_Osimps, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) = (^ [A3:(coinductive_llist @ A)]: (((A3 = (coinductive_LNil @ A)) | (? [Xs3:(coinductive_llist @ A),X3:A]: (((A3 = (((coinductive_LCons @ A) @ X3) @ Xs3)) & ((coinductive_lfinite @ A) @ Xs3))))))))))).
% 2.93/1.52 thf(fact_77_lfinite_Ocases, axiom, (! [A:$tType,A2:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ A2) => ((A2 != (coinductive_LNil @ A)) => (~ ((! [Xs2:(coinductive_llist @ A)]: (((? [X2:A]: ((A2 = (((coinductive_LCons @ A) @ X2) @ Xs2)))) => (~ (((coinductive_lfinite @ A) @ Xs2))))))))))))).
% 2.93/1.52 thf(fact_78_Singleton, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A]: (((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ (coinductive_LNil @ A))))))))).
% 2.93/1.52 thf(fact_79_lfilter_Osimps, axiom, (! [A:$tType]: (((coinductive_lfilter @ A) = (^ [P4:(A > $o)]: (((((coindu1381640503_llist @ (coinductive_llist @ A)) @ A) @ (coinductive_LNil @ A)) @ (^ [X3:A,Xs5:(coinductive_llist @ A)]: (((((if @ (coinductive_llist @ A)) @ (P4 @ X3)) @ (((coinductive_LCons @ A) @ X3) @ (((coinductive_lfilter @ A) @ P4) @ Xs5))) @ (((coinductive_lfilter @ A) @ P4) @ Xs5))))))))))).
% 2.93/1.52 thf(fact_80_lsorted_Ocoinduct, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X4:((coinductive_llist @ A) > $o),X:(coinductive_llist @ A)]: (((X4 @ X) => ((! [X2:(coinductive_llist @ A)]: (((X4 @ X2) => ((X2 = (coinductive_LNil @ A)) | ((? [Xa:A]: ((X2 = (((coinductive_LCons @ A) @ Xa) @ (coinductive_LNil @ A))))) | (? [Xa:A,Y4:A,Xs4:(coinductive_llist @ A)]: (((X2 = (((coinductive_LCons @ A) @ Xa) @ (((coinductive_LCons @ A) @ Y4) @ Xs4))) & ((((ord_less_eq @ A) @ Xa) @ Y4) & ((X4 @ (((coinductive_LCons @ A) @ Y4) @ Xs4)) | ((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y4) @ Xs4)))))))))))) => ((coindu63249387sorted @ A) @ X))))))))).
% 2.93/1.52 thf(fact_81_lsorted_Osimps, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => ((coindu63249387sorted @ A) = (^ [A3:(coinductive_llist @ A)]: (((A3 = (coinductive_LNil @ A)) | ((? [X3:A]: ((A3 = (((coinductive_LCons @ A) @ X3) @ (coinductive_LNil @ A))))) | (? [X3:A,Y3:A,Xs3:(coinductive_llist @ A)]: (((A3 = (((coinductive_LCons @ A) @ X3) @ (((coinductive_LCons @ A) @ Y3) @ Xs3))) & ((((ord_less_eq @ A) @ X3) @ Y3) & ((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y3) @ Xs3))))))))))))))).
% 2.93/1.52 thf(fact_82_lsorted_Ocases, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:(coinductive_llist @ A)]: ((((coindu63249387sorted @ A) @ A2) => ((A2 != (coinductive_LNil @ A)) => ((! [X2:A]: ((A2 != (((coinductive_LCons @ A) @ X2) @ (coinductive_LNil @ A))))) => (~ ((! [X2:A,Y2:A,Xs2:(coinductive_llist @ A)]: (((A2 = (((coinductive_LCons @ A) @ X2) @ (((coinductive_LCons @ A) @ Y2) @ Xs2))) => ((((ord_less_eq @ A) @ X2) @ Y2) => (~ (((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y2) @ Xs2))))))))))))))))))).
% 2.93/1.52 thf(fact_83_lstrict__prefix__lappend__conv, axiom, (! [A:$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)))))))).
% 2.93/1.52 thf(fact_84_split__llist__first, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (? [Ys4:(coinductive_llist @ A),Zs2:(coinductive_llist @ A)]: (((Xs = (((coinductive_lappend @ A) @ Ys4) @ (((coinductive_LCons @ A) @ X) @ Zs2))) & (((coinductive_lfinite @ A) @ Ys4) & (~ ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys4)))))))))))).
% 2.93/1.52 thf(fact_85_split__llist, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (? [Ys4:(coinductive_llist @ A),Zs2:(coinductive_llist @ A)]: (((Xs = (((coinductive_lappend @ A) @ Ys4) @ (((coinductive_LCons @ A) @ X) @ Zs2))) & ((coinductive_lfinite @ A) @ Ys4)))))))).
% 2.93/1.52 thf(fact_86_lappend__ltakeWhile__ldropWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) @ (((coindu218763757pWhile @ A) @ P) @ Xs)) = Xs)))).
% 2.93/1.52 thf(fact_87_lfilter__idem, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((coinductive_lfilter @ A) @ P) @ (((coinductive_lfilter @ A) @ P) @ Xs)) = (((coinductive_lfilter @ A) @ P) @ Xs))))).
% 2.93/1.52 thf(fact_88_lfilter__K__True, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_lfilter @ A) @ (^ [Uu:A]: ($true))) @ Xs) = Xs)))).
% 2.93/1.52 thf(fact_89_lappend_Odisc__iff_I2_J, axiom, (! [A:$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)))))))).
% 2.93/1.52 thf(fact_90_lnull__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lnull @ A) @ Xs) & ((coinductive_lnull @ A) @ Ys)))))).
% 2.93/1.52 thf(fact_91_lfilter__LCons, axiom, (! [A:$tType,P:(A > $o),X:A,Xs:(coinductive_llist @ A)]: ((((P @ X) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_LCons @ A) @ X) @ (((coinductive_lfilter @ A) @ P) @ Xs)))) & ((~ ((P @ X))) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_lfilter @ A) @ P) @ Xs))))))).
% 2.93/1.52 thf(fact_92_lfilter__LNil, axiom, (! [A:$tType,P:(A > $o)]: (((((coinductive_lfilter @ A) @ P) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.93/1.52 thf(fact_93_ldropWhile__LCons, axiom, (! [A:$tType,P:(A > $o),X:A,Xs:(coinductive_llist @ A)]: ((((P @ X) => ((((coindu218763757pWhile @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coindu218763757pWhile @ A) @ P) @ Xs))) & ((~ ((P @ X))) => ((((coindu218763757pWhile @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_LCons @ A) @ X) @ Xs))))))).
% 2.93/1.52 thf(fact_94_ldropWhile__LNil, axiom, (! [A:$tType,P:(A > $o)]: (((((coindu218763757pWhile @ A) @ P) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.93/1.52 thf(fact_95_unfold__llist_Odisc__iff_I2_J, axiom, (! [B:$tType,A:$tType,P3:(A > $o),G212:(A > B),G223:(A > A),A2:A]: (((~ (((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ A2)))) = (~ ((P3 @ A2))))))).
% 2.93/1.52 thf(fact_96_unfold__llist_Odisc__iff_I1_J, axiom, (! [B:$tType,A:$tType,P3:(A > $o),G212:(A > B),G223:(A > A),A2:A]: ((((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ A2)) = (P3 @ A2))))).
% 2.93/1.52 thf(fact_97_llist_Ocorec__disc__iff_I2_J, axiom, (! [A:$tType,C:$tType,P3:(C > $o),G212:(C > A),Q222:(C > $o),G2212:(C > (coinductive_llist @ A)),G2222:(C > C),A2:C]: (((~ (((coinductive_lnull @ A) @ ((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ A2)))) = (~ ((P3 @ A2))))))).
% 2.93/1.52 thf(fact_98_llist_Ocorec__disc__iff_I1_J, axiom, (! [A:$tType,C:$tType,P3:(C > $o),G212:(C > A),Q222:(C > $o),G2212:(C > (coinductive_llist @ A)),G2222:(C > C),A2:C]: ((((coinductive_lnull @ A) @ ((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ A2)) = (P3 @ A2))))).
% 2.93/1.52 thf(fact_99_lfitler__K__False, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_lfilter @ A) @ (^ [Uu:A]: ($false))) @ Xs) = (coinductive_LNil @ A))))).
% 2.93/1.52 thf(fact_100_lset__lfilter, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lset @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)) = ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) & (P @ X3))))))))).
% 2.93/1.52 thf(fact_101_ldropWhile__K__True, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coindu218763757pWhile @ A) @ (^ [Uu:A]: ($true))) @ Xs) = (coinductive_LNil @ A))))).
% 2.93/1.52 thf(fact_102_lsorted__LCons__LCons, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A,Y:A,Xs:(coinductive_llist @ A)]: ((((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Xs))) = ((((ord_less_eq @ A) @ X) @ Y) & ((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y) @ Xs)))))))))).
% 2.93/1.52 thf(fact_103_diverge__lfilter__LNil, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => (~ ((P @ X2)))))) => ((((coinductive_lfilter @ A) @ P) @ Xs) = (coinductive_LNil @ A)))))).
% 2.93/1.52 thf(fact_104_lnull__lfilter, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)) = (! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => (~ ((P @ X3)))))))))).
% 2.93/1.52 thf(fact_105_lnull__ldropWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coindu218763757pWhile @ A) @ P) @ Xs)) = (! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => (P @ X3)))))))).
% 2.93/1.52 thf(fact_106_lfilter__lappend__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lappend @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)) @ (((coinductive_lfilter @ A) @ P) @ Ys))))))).
% 2.93/1.52 thf(fact_107_lfinite__ldropWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coindu218763757pWhile @ A) @ P) @ Xs)) = ((? [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P @ X3)))))) => ((coinductive_lfinite @ A) @ Xs)))))).
% 2.93/1.52 thf(fact_108_llist_Odisc__eq__case_I1_J, axiom, (! [A:$tType]: (((coinductive_lnull @ A) = ((((coindu1381640503_llist @ $o) @ A) @ $true) @ (^ [Uu:A,Uv:(coinductive_llist @ A)]: ($false))))))).
% 2.93/1.52 thf(fact_109_llist_Odisc__eq__case_I2_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ Llist))) = (((((coindu1381640503_llist @ $o) @ A) @ $false) @ (^ [Uu:A,Uv:(coinductive_llist @ A)]: ($true))) @ Llist))))).
% 2.93/1.52 thf(fact_110_lfilter__cong, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Ys)) => ((P @ X2) = (Q @ X2))))) => ((((coinductive_lfilter @ A) @ P) @ Xs) = (((coinductive_lfilter @ A) @ Q) @ Ys))))))).
% 2.93/1.52 thf(fact_111_lzip_Oexhaust, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((~ ((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ B) @ Ys)))) => (~ (((~ (((coinductive_lnull @ A) @ Xs))) => ((coinductive_lnull @ B) @ Ys)))))))).
% 2.93/1.52 thf(fact_112_lappend_Oexhaust, axiom, (! [A:$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)))))))).
% 2.93/1.52 thf(fact_113_ldropWhile__cong, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Ys)) => ((P @ X2) = (Q @ X2))))) => ((((coindu218763757pWhile @ A) @ P) @ Xs) = (((coindu218763757pWhile @ A) @ Q) @ Ys))))))).
% 2.93/1.52 thf(fact_114_lfilter__id__conv, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Xs) = Xs) = (! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => (P @ X3)))))))).
% 2.93/1.52 thf(fact_115_wlog__linorder__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),B2:A,A2:A]: (((! [A5:A,B3:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((P @ A5) @ B3)))) => ((((P @ B2) @ A2) => ((P @ A2) @ B2)) => ((P @ A2) @ B2))))))))).
% 2.93/1.52 thf(fact_116_lfilter__empty__conv, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Xs) = (coinductive_LNil @ A)) = (! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => (~ ((P @ X3)))))))))).
% 2.93/1.52 thf(fact_117_in__lset__ldropWhileD, axiom, (! [A:$tType,X:A,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coindu218763757pWhile @ A) @ P) @ Xs))) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)))))).
% 2.93/1.52 thf(fact_118_lset__ldropWhile__subset, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((ord_less_eq @ (set @ A)) @ ((coinductive_lset @ A) @ (((coindu218763757pWhile @ A) @ P) @ Xs))) @ ((coinductive_lset @ A) @ Xs))))).
% 2.93/1.52 thf(fact_119_lset__lappend1, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((ord_less_eq @ (set @ A)) @ ((coinductive_lset @ A) @ Xs)) @ ((coinductive_lset @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)))))).
% 2.93/1.52 thf(fact_120_lfilter__lfilter, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o),Xs:(coinductive_llist @ A)]: (((((coinductive_lfilter @ A) @ P) @ (((coinductive_lfilter @ A) @ Q) @ Xs)) = (((coinductive_lfilter @ A) @ (^ [X3:A]: (((P @ X3) & (Q @ X3))))) @ Xs))))).
% 2.93/1.52 thf(fact_121_ldropWhile__eq__LNil__iff, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((((coindu218763757pWhile @ A) @ P) @ Xs) = (coinductive_LNil @ A)) = (! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => (P @ X3)))))))).
% 2.93/1.52 thf(fact_122_ltakeWhile__all__conv, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((((coindu501562517eWhile @ A) @ P) @ Xs) = Xs) = (((ord_less_eq @ (set @ A)) @ ((coinductive_lset @ A) @ Xs)) @ ((collect @ A) @ P)))))).
% 2.93/1.52 thf(fact_123_lsorted__LCons, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Xs:(coinductive_llist @ A)]: ((((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coindu63249387sorted @ A) @ Xs) & (! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => (((ord_less_eq @ A) @ X) @ X3)))))))))))).
% 2.93/1.52 thf(fact_124_lfilter__LCons__seek, axiom, (! [A:$tType,P3:(A > $o),X:A,L:(coinductive_llist @ A)]: (((~ ((P3 @ X))) => ((((coinductive_lfilter @ A) @ P3) @ (((coinductive_LCons @ A) @ X) @ L)) = (((coinductive_lfilter @ A) @ P3) @ L)))))).
% 2.93/1.52 thf(fact_125_lfilter__LCons__found, axiom, (! [A:$tType,P:(A > $o),X:A,Xs:(coinductive_llist @ A)]: (((P @ X) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_LCons @ A) @ X) @ (((coinductive_lfilter @ A) @ P) @ Xs))))))).
% 2.93/1.52 thf(fact_126_lset__intros_I2_J, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),X5:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X5) @ Xs))))))).
% 2.93/1.52 thf(fact_127_lset__intros_I1_J, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)))))).
% 2.93/1.52 thf(fact_128_llist_Oset__intros_I2_J, axiom, (! [A:$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))))))).
% 2.93/1.52 thf(fact_129_llist_Oset__intros_I1_J, axiom, (! [A:$tType,A1:A,A22:(coinductive_llist @ A)]: ((((member @ A) @ A1) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ A1) @ A22)))))).
% 2.93/1.52 thf(fact_130_lset__cases, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs6:(coinductive_llist @ A)]: ((Xs != (((coinductive_LCons @ A) @ X) @ Xs6)))) => (~ ((! [X6:A,Xs6:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ X6) @ Xs6)) => (~ ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs6)))))))))))))).
% 2.93/1.52 thf(fact_131_lset__induct, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs2:(coinductive_llist @ A)]: ((P @ (((coinductive_LCons @ A) @ X) @ Xs2)))) => ((! [X6:A,Xs2:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs2)) => ((X != X6) => ((P @ Xs2) => (P @ (((coinductive_LCons @ A) @ X6) @ Xs2))))))) => (P @ Xs))))))).
% 2.93/1.52 thf(fact_132_lset__induct_H, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs2:(coinductive_llist @ A)]: ((P @ (((coinductive_LCons @ A) @ X) @ Xs2)))) => ((! [X6:A,Xs2:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs2)) => ((P @ Xs2) => (P @ (((coinductive_LCons @ A) @ X6) @ Xs2)))))) => (P @ Xs))))))).
% 2.93/1.52 thf(fact_133_llist_Oset__cases, axiom, (! [A:$tType,E:A,A2:(coinductive_llist @ A)]: (((((member @ A) @ E) @ ((coinductive_lset @ A) @ A2)) => ((! [Z2:(coinductive_llist @ A)]: ((A2 != (((coinductive_LCons @ A) @ E) @ Z2)))) => (~ ((! [Z1:A,Z2:(coinductive_llist @ A)]: (((A2 = (((coinductive_LCons @ A) @ Z1) @ Z2)) => (~ ((((member @ A) @ E) @ ((coinductive_lset @ A) @ Z2)))))))))))))).
% 2.93/1.52 thf(fact_134_llist_Oset__induct, axiom, (! [A:$tType,X:A,A2:(coinductive_llist @ A),P:(A > ((coinductive_llist @ A) > $o))]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ A2)) => ((! [Z1:A,Z2:(coinductive_llist @ A)]: (((P @ Z1) @ (((coinductive_LCons @ A) @ Z1) @ Z2)))) => ((! [Z1:A,Z2:(coinductive_llist @ A),Xa3:A]: (((((member @ A) @ Xa3) @ ((coinductive_lset @ A) @ Z2)) => (((P @ Xa3) @ Z2) => ((P @ Xa3) @ (((coinductive_LCons @ A) @ Z1) @ Z2)))))) => ((P @ X) @ A2))))))).
% 2.93/1.52 thf(fact_135_llist_Odisc_I2_J, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: ((~ (((coinductive_lnull @ A) @ (((coinductive_LCons @ A) @ X21) @ X22))))))).
% 2.93/1.52 thf(fact_136_llist_OdiscI_I2_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A),X21:A,X22:(coinductive_llist @ A)]: (((Llist = (((coinductive_LCons @ A) @ X21) @ X22)) => (~ (((coinductive_lnull @ A) @ Llist))))))).
% 2.93/1.52 thf(fact_137_not__lnull__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ Xs))) = (? [X3:A,Xs5:(coinductive_llist @ A)]: ((Xs = (((coinductive_LCons @ A) @ X3) @ Xs5)))))))).
% 2.93/1.52 thf(fact_138_llist_Odisc_I1_J, axiom, (! [A:$tType]: (((coinductive_lnull @ A) @ (coinductive_LNil @ A))))).
% 2.93/1.52 thf(fact_139_llist_OdiscI_I1_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: (((Llist = (coinductive_LNil @ A)) => ((coinductive_lnull @ A) @ Llist))))).
% 2.93/1.52 thf(fact_140_llist_Ocollapse_I1_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Llist) => (Llist = (coinductive_LNil @ A)))))).
% 2.93/1.52 thf(fact_141_lnull__def, axiom, (! [A:$tType]: (((coinductive_lnull @ A) = (^ [Llist2:(coinductive_llist @ A)]: ((Llist2 = (coinductive_LNil @ A)))))))).
% 2.93/1.52 thf(fact_142_lfinite__lfilterI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: ((((coinductive_lfinite @ A) @ Xs) => ((coinductive_lfinite @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)))))).
% 2.93/1.52 thf(fact_143_lappend_Odisc_I2_J, axiom, (! [A:$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)))))))).
% 2.93/1.52 thf(fact_144_lappend_Odisc_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => (((coinductive_lnull @ A) @ Ys) => ((coinductive_lnull @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys))))))).
% 2.93/1.52 thf(fact_145_lappend__lnull1, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = Ys))))).
% 2.93/1.52 thf(fact_146_lappend__lnull2, axiom, (! [A:$tType,Ys:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Ys) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = Xs))))).
% 2.93/1.52 thf(fact_147_lnull__imp__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((coinductive_lfinite @ A) @ Xs))))).
% 2.93/1.52 thf(fact_148_ltakeWhile__all, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => (P @ X2)))) => ((((coindu501562517eWhile @ A) @ P) @ Xs) = Xs))))).
% 2.93/1.52 thf(fact_149_ltakeWhile__cong, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Ys)) => ((P @ X2) = (Q @ X2))))) => ((((coindu501562517eWhile @ A) @ P) @ Xs) = (((coindu501562517eWhile @ A) @ Q) @ Ys))))))).
% 2.93/1.52 thf(fact_150_lset__ltakeWhileD, axiom, (! [A:$tType,X:A,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs))) => ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) & (P @ X)))))).
% 2.93/1.52 thf(fact_151_lsorted__lfilterI, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),P:(A > $o)]: ((((coindu63249387sorted @ A) @ Xs) => ((coindu63249387sorted @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs))))))))).
% 2.93/1.52 thf(fact_152_ltakeWhile__lappend2, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o),Ys:(coinductive_llist @ A)]: (((((ord_less_eq @ (set @ A)) @ ((coinductive_lset @ A) @ Xs)) @ ((collect @ A) @ P)) => ((((coindu501562517eWhile @ A) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lappend @ A) @ Xs) @ (((coindu501562517eWhile @ A) @ P) @ Ys))))))).
% 2.93/1.52 thf(fact_153_unfold__llist_Odisc_I2_J, axiom, (! [B:$tType,A:$tType,P3:(A > $o),A2:A,G212:(A > B),G223:(A > A)]: (((~ ((P3 @ A2))) => (~ (((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ A2)))))))).
% 2.93/1.52 thf(fact_154_unfold__llist_Odisc_I1_J, axiom, (! [B:$tType,A:$tType,P3:(A > $o),A2:A,G212:(A > B),G223:(A > A)]: (((P3 @ A2) => ((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ A2)))))).
% 2.93/1.52 thf(fact_155_llist_Ocorec__disc_I2_J, axiom, (! [A:$tType,C:$tType,P3:(C > $o),A2:C,G212:(C > A),Q222:(C > $o),G2212:(C > (coinductive_llist @ A)),G2222:(C > C)]: (((~ ((P3 @ A2))) => (~ (((coinductive_lnull @ A) @ ((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ A2)))))))).
% 2.93/1.52 thf(fact_156_llist_Ocorec__disc_I1_J, axiom, (! [A:$tType,C:$tType,P3:(C > $o),A2:C,G212:(C > A),Q222:(C > $o),G2212:(C > (coinductive_llist @ A)),G2222:(C > C)]: (((P3 @ A2) => ((coinductive_lnull @ A) @ ((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ A2)))))).
% 2.93/1.52 thf(fact_157_lfilter__eq__LConsD, axiom, (! [A:$tType,P:(A > $o),Ys:(coinductive_llist @ A),X:A,Xs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Ys) = (((coinductive_LCons @ A) @ X) @ Xs)) => (? [Us:(coinductive_llist @ A),Vs:(coinductive_llist @ A)]: (((Ys = (((coinductive_lappend @ A) @ Us) @ (((coinductive_LCons @ A) @ X) @ Vs))) & (((coinductive_lfinite @ A) @ Us) & ((! [X7:A]: (((((member @ A) @ X7) @ ((coinductive_lset @ A) @ Us)) => (~ ((P @ X7)))))) & ((P @ X) & (Xs = (((coinductive_lfilter @ A) @ P) @ Vs)))))))))))).
% 2.93/1.52 thf(fact_158_lsorted__lfilter__same, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [C2:A,Xs:(coinductive_llist @ A)]: (((coindu63249387sorted @ A) @ (((coinductive_lfilter @ A) @ (^ [X3:A]: ((X3 = C2)))) @ Xs)))))))).
% 2.93/1.52 thf(fact_159_lset__lmember, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) = (((coinductive_lmember @ A) @ X) @ Xs))))).
% 2.93/1.52 thf(fact_160_lfilter__eq__lappend__lfiniteD, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Xs) = (((coinductive_lappend @ A) @ Ys) @ Zs)) => (((coinductive_lfinite @ A) @ Ys) => (? [Us:(coinductive_llist @ A),Vs:(coinductive_llist @ A)]: (((Xs = (((coinductive_lappend @ A) @ Us) @ Vs)) & (((coinductive_lfinite @ A) @ Us) & ((Ys = (((coinductive_lfilter @ A) @ P) @ Us)) & (Zs = (((coinductive_lfilter @ A) @ P) @ Vs)))))))))))).
% 2.93/1.52 thf(fact_161_in__lset__lappend__iff, axiom, (! [A:$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)))))))).
% 2.93/1.52 thf(fact_162_lappend_Octr_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => (((coinductive_lnull @ A) @ Ys) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = (coinductive_LNil @ A))))))).
% 2.93/1.52 thf(fact_163_ltakeWhile__lappend1, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),P:(A > $o),Ys:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((~ ((P @ X))) => ((((coindu501562517eWhile @ A) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coindu501562517eWhile @ A) @ P) @ Xs))))))).
% 2.93/1.52 thf(fact_164_lfinite__ltakeWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) = (((coinductive_lfinite @ A) @ Xs) | (? [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P @ X3))))))))))).
% 2.93/1.52 thf(fact_165_llast__LCons, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),X:A]: (((((coinductive_lnull @ A) @ Xs) => (((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = X)) & ((~ (((coinductive_lnull @ A) @ Xs))) => (((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = ((coinductive_llast @ A) @ Xs))))))).
% 2.93/1.52 thf(fact_166_LCons__LCons, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A,Y:A,Xs:(coinductive_llist @ A)]: (((((ord_less_eq @ A) @ X) @ Y) => (((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y) @ Xs)) => ((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Xs))))))))))).
% 2.93/1.52 thf(fact_167_ldropWhile_Osimps, axiom, (! [A:$tType]: (((coindu218763757pWhile @ A) = (^ [P4:(A > $o),Xs3:(coinductive_llist @ A)]: ((((((coindu1381640503_llist @ (coinductive_llist @ A)) @ A) @ (coinductive_LNil @ A)) @ (^ [X3:A,Xs5:(coinductive_llist @ A)]: (((((if @ (coinductive_llist @ A)) @ (P4 @ X3)) @ (((coindu218763757pWhile @ A) @ P4) @ Xs5)) @ Xs3)))) @ Xs3))))))).
% 2.93/1.52 thf(fact_168_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.93/1.52 thf(fact_169_reachable__viap_Ocases, axiom, (! [Node:$tType,Graph:(Node > (Node > $o)),Ns:(set @ Node),N:Node,A2:Node]: (((((((koenig1757754772e_viap @ Node) @ Graph) @ Ns) @ N) @ A2) => (~ ((! [Xs2:(coinductive_llist @ Node)]: (((((koenig2031690877pathsp @ Node) @ Graph) @ (((coinductive_LCons @ Node) @ N) @ Xs2)) => ((((member @ Node) @ A2) @ ((coinductive_lset @ Node) @ Xs2)) => (~ ((((ord_less_eq @ (set @ Node)) @ ((coinductive_lset @ Node) @ Xs2)) @ Ns))))))))))))).
% 2.93/1.52 thf(fact_170_reachable__viap_Osimps, axiom, (! [Node:$tType]: (((koenig1757754772e_viap @ Node) = (^ [Graph2:(Node > (Node > $o)),Ns2:(set @ Node),N2:Node,A3:Node]: ((? [Xs3:(coinductive_llist @ Node),N3:Node]: (((A3 = N3) & ((((koenig2031690877pathsp @ Node) @ Graph2) @ (((coinductive_LCons @ Node) @ N2) @ Xs3)) & ((((member @ Node) @ N3) @ ((coinductive_lset @ Node) @ Xs3)) & (((ord_less_eq @ (set @ Node)) @ ((coinductive_lset @ Node) @ Xs3)) @ Ns2)))))))))))).
% 2.93/1.52 thf(fact_171_reachable__viap_Ointros, axiom, (! [Node:$tType,Graph:(Node > (Node > $o)),N:Node,Xs:(coinductive_llist @ Node),N4:Node,Ns:(set @ Node)]: (((((koenig2031690877pathsp @ Node) @ Graph) @ (((coinductive_LCons @ Node) @ N) @ Xs)) => ((((member @ Node) @ N4) @ ((coinductive_lset @ Node) @ Xs)) => ((((ord_less_eq @ (set @ Node)) @ ((coinductive_lset @ Node) @ Xs)) @ Ns) => (((((koenig1757754772e_viap @ Node) @ Graph) @ Ns) @ N) @ N4))))))).
% 2.93/1.52 thf(fact_172_reachable__viap_Oinducts, axiom, (! [Node:$tType,Graph:(Node > (Node > $o)),Ns:(set @ Node),N:Node,X:Node,P:(Node > $o)]: (((((((koenig1757754772e_viap @ Node) @ Graph) @ Ns) @ N) @ X) => ((! [Xs2:(coinductive_llist @ Node),N5:Node]: (((((koenig2031690877pathsp @ Node) @ Graph) @ (((coinductive_LCons @ Node) @ N) @ Xs2)) => ((((member @ Node) @ N5) @ ((coinductive_lset @ Node) @ Xs2)) => ((((ord_less_eq @ (set @ Node)) @ ((coinductive_lset @ Node) @ Xs2)) @ Ns) => (P @ N5)))))) => (P @ X)))))).
% 2.93/1.52 thf(fact_173_reachable__via_Ocases, axiom, (! [Node:$tType,A2:Node,Graph:(Node > (Node > $o)),Ns:(set @ Node),N:Node]: (((((member @ Node) @ A2) @ ((((koenig317145564le_via @ Node) @ Graph) @ Ns) @ N)) => (~ ((! [Xs2:(coinductive_llist @ Node)]: (((((member @ (coinductive_llist @ Node)) @ (((coinductive_LCons @ Node) @ N) @ Xs2)) @ ((koenig916195507_paths @ Node) @ Graph)) => ((((member @ Node) @ A2) @ ((coinductive_lset @ Node) @ Xs2)) => (~ ((((ord_less_eq @ (set @ Node)) @ ((coinductive_lset @ Node) @ Xs2)) @ Ns))))))))))))).
% 2.93/1.52 thf(fact_174_reachable__viap__reachable__via__eq, axiom, (! [Node:$tType]: (((koenig1757754772e_viap @ Node) = (^ [Graph2:(Node > (Node > $o)),Ns2:(set @ Node),N2:Node,X3:Node]: ((((member @ Node) @ X3) @ ((((koenig317145564le_via @ Node) @ Graph2) @ Ns2) @ N2)))))))).
% 2.93/1.52 thf(fact_175_reachable__via__def, axiom, (! [Node:$tType]: (((koenig317145564le_via @ Node) = (^ [Graph2:(Node > (Node > $o)),Ns2:(set @ Node),N2:Node]: (((collect @ Node) @ ((((koenig1757754772e_viap @ Node) @ Graph2) @ Ns2) @ N2)))))))).
% 2.93/1.52 thf(fact_176_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.93/1.52 thf(fact_177_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.93/1.52 thf(fact_178_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.93/1.52 thf(fact_179_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F3 @ X3)) @ (G2 @ X3))))))))))).
% 2.93/1.52 thf(fact_180_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.93/1.52 thf(fact_181_order__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ C) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ C) @ (F @ A2)) @ C2)))))))))).
% 2.93/1.52 thf(fact_182_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.93/1.52 thf(fact_183_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C2) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C2)))))))))).
% 2.93/1.52 thf(fact_184_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X3))))))))).
% 2.93/1.52 thf(fact_185_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.93/1.52 thf(fact_186_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.93/1.52 thf(fact_187_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.93/1.52 thf(fact_188_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.93/1.52 thf(fact_189_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.93/1.52 thf(fact_190_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z3)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z3)))) => (((((ord_less_eq @ A) @ X) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ Y)))) => (((((ord_less_eq @ A) @ Z3) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ X)))) => (~ (((((ord_less_eq @ A) @ Z3) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.93/1.52 thf(fact_191_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.93/1.52 thf(fact_192_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.93/1.52 thf(fact_193_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.93/1.52 thf(fact_194_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.93/1.52 thf(fact_195_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z3) => (((ord_less_eq @ A) @ X) @ Z3))))))))).
% 2.93/1.52 thf(fact_196_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.93/1.52 thf(fact_197_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A5:A,B3:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((P @ A5) @ B3)))) => ((! [A5:A,B3:A]: ((((P @ B3) @ A5) => ((P @ A5) @ B3)))) => ((P @ A2) @ B2))))))))).
% 2.93/1.52 thf(fact_198_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 2.93/1.52 thf(fact_199_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.93/1.52 thf(fact_200_reachable__via_Oinducts, axiom, (! [Node:$tType,X:Node,Graph:(Node > (Node > $o)),Ns:(set @ Node),N:Node,P:(Node > $o)]: (((((member @ Node) @ X) @ ((((koenig317145564le_via @ Node) @ Graph) @ Ns) @ N)) => ((! [Xs2:(coinductive_llist @ Node),N5:Node]: (((((member @ (coinductive_llist @ Node)) @ (((coinductive_LCons @ Node) @ N) @ Xs2)) @ ((koenig916195507_paths @ Node) @ Graph)) => ((((member @ Node) @ N5) @ ((coinductive_lset @ Node) @ Xs2)) => ((((ord_less_eq @ (set @ Node)) @ ((coinductive_lset @ Node) @ Xs2)) @ Ns) => (P @ N5)))))) => (P @ X)))))).
% 2.93/1.52 thf(fact_201_reachable__via_Ointros, axiom, (! [Node:$tType,N:Node,Xs:(coinductive_llist @ Node),Graph:(Node > (Node > $o)),N4:Node,Ns:(set @ Node)]: (((((member @ (coinductive_llist @ Node)) @ (((coinductive_LCons @ Node) @ N) @ Xs)) @ ((koenig916195507_paths @ Node) @ Graph)) => ((((member @ Node) @ N4) @ ((coinductive_lset @ Node) @ Xs)) => ((((ord_less_eq @ (set @ Node)) @ ((coinductive_lset @ Node) @ Xs)) @ Ns) => (((member @ Node) @ N4) @ ((((koenig317145564le_via @ Node) @ Graph) @ Ns) @ N)))))))).
% 2.93/1.52 thf(fact_202_reachable__via_Osimps, axiom, (! [Node:$tType,A2:Node,Graph:(Node > (Node > $o)),Ns:(set @ Node),N:Node]: (((((member @ Node) @ A2) @ ((((koenig317145564le_via @ Node) @ Graph) @ Ns) @ N)) = (? [Xs3:(coinductive_llist @ Node),N3:Node]: (((A2 = N3) & ((((member @ (coinductive_llist @ Node)) @ (((coinductive_LCons @ Node) @ N) @ Xs3)) @ ((koenig916195507_paths @ Node) @ Graph)) & ((((member @ Node) @ N3) @ ((coinductive_lset @ Node) @ Xs3)) & (((ord_less_eq @ (set @ Node)) @ ((coinductive_lset @ Node) @ Xs3)) @ Ns)))))))))).
% 2.93/1.52 thf(fact_203_subset__antisym, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ A4) => (A4 = B4)))))).
% 2.93/1.52 thf(fact_204_subsetI, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A4) => (((member @ A) @ X2) @ B4)))) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 2.93/1.52 thf(fact_205_ldropWhile__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o),Ys:(coinductive_llist @ A)]: ((((? [X7:A]: (((((member @ A) @ X7) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P @ X7)))))) => ((((coindu218763757pWhile @ A) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lappend @ A) @ (((coindu218763757pWhile @ A) @ P) @ Xs)) @ Ys))) & ((~ ((? [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P @ X2)))))))) => ((((coinductive_lfinite @ A) @ Xs) => ((((coindu218763757pWhile @ A) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coindu218763757pWhile @ A) @ P) @ Ys))) & ((~ (((coinductive_lfinite @ A) @ Xs))) => ((((coindu218763757pWhile @ A) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (coinductive_LNil @ A))))))))).
% 2.93/1.52 thf(fact_206_set__mp, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ X) @ A4) => (((member @ A) @ X) @ B4)))))).
% 2.93/1.52 thf(fact_207_in__mono, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ X) @ A4) => (((member @ A) @ X) @ B4)))))).
% 2.93/1.52 thf(fact_208_subsetD, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ C2) @ A4) => (((member @ A) @ C2) @ B4)))))).
% 2.93/1.52 thf(fact_209_subsetCE, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ C2) @ A4) => (((member @ A) @ C2) @ B4)))))).
% 2.93/1.52 thf(fact_210_equalityE, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (~ ((((ord_less_eq @ (set @ A)) @ B4) @ A4)))))))))).
% 2.93/1.52 thf(fact_211_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((member @ A) @ X3) @ B5)))))))))).
% 2.93/1.52 thf(fact_212_equalityD1, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 2.93/1.52 thf(fact_213_equalityD2, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (((ord_less_eq @ (set @ A)) @ B4) @ A4))))).
% 2.93/1.52 thf(fact_214_set__rev__mp, axiom, (! [A:$tType,X:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ X) @ A4) => ((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((member @ A) @ X) @ B4)))))).
% 2.93/1.52 thf(fact_215_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A6) => (((member @ A) @ T) @ B5)))))))))).
% 2.93/1.52 thf(fact_216_rev__subsetD, axiom, (! [A:$tType,C2:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ C2) @ A4) => ((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((member @ A) @ C2) @ B4)))))).
% 2.93/1.52 thf(fact_217_subset__refl, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A4) @ A4)))).
% 2.93/1.52 thf(fact_218_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.93/1.52 thf(fact_219_subset__trans, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ C3) => (((ord_less_eq @ (set @ A)) @ A4) @ C3)))))).
% 2.93/1.52 thf(fact_220_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z:(set @ A)]: ((Y5 = Z))) = (^ [A6:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B5) & (((ord_less_eq @ (set @ A)) @ B5) @ A6)))))))).
% 2.93/1.52 thf(fact_221_contra__subsetD, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((~ ((((member @ A) @ C2) @ B4))) => (~ ((((member @ A) @ C2) @ A4)))))))).
% 2.93/1.52 thf(fact_222_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 2.93/1.52 thf(fact_223_ltakeWhile__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o),Ys:(coinductive_llist @ A)]: ((((? [X7:A]: (((((member @ A) @ X7) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P @ X7)))))) => ((((coindu501562517eWhile @ A) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coindu501562517eWhile @ A) @ P) @ Xs))) & ((~ ((? [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P @ X2)))))))) => ((((coindu501562517eWhile @ A) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lappend @ A) @ Xs) @ (((coindu501562517eWhile @ A) @ P) @ Ys)))))))).
% 2.93/1.52 thf(fact_224_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A6)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B5)))))))))).
% 2.93/1.52 thf(fact_225_pred__subset__eq, axiom, (! [A:$tType,R:(set @ A),S:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ R)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ S)))) = (((ord_less_eq @ (set @ A)) @ R) @ S))))).
% 2.93/1.52 thf(fact_226_conj__subset__def, axiom, (! [A:$tType,A4:(set @ A),P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A4) @ ((collect @ A) @ (^ [X3:A]: (((P @ X3) & (Q @ X3)))))) = ((((ord_less_eq @ (set @ A)) @ A4) @ ((collect @ A) @ P)) & (((ord_less_eq @ (set @ A)) @ A4) @ ((collect @ A) @ Q))))))).
% 2.93/1.52 thf(fact_227_predicate1I, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (((ord_less_eq @ (A > $o)) @ P) @ Q))))).
% 2.93/1.52 thf(fact_228_rev__predicate1D, axiom, (! [A:$tType,P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((((ord_less_eq @ (A > $o)) @ P) @ Q) => (Q @ X)))))).
% 2.93/1.52 thf(fact_229_predicate1D, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o),X:A]: (((((ord_less_eq @ (A > $o)) @ P) @ Q) => ((P @ X) => (Q @ X)))))).
% 2.93/1.52 thf(fact_230_subset__Collect__iff, axiom, (! [A:$tType,B4:(set @ A),A4:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B4) @ A4) => ((((ord_less_eq @ (set @ A)) @ B4) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A4) & (P @ X3)))))) = (! [X3:A]: (((((member @ A) @ X3) @ B4) => (P @ X3))))))))).
% 2.93/1.52 thf(fact_231_subset__CollectI, axiom, (! [A:$tType,B4:(set @ A),A4:(set @ A),Q:(A > $o),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B4) @ A4) => ((! [X2:A]: (((((member @ A) @ X2) @ B4) => ((Q @ X2) => (P @ X2))))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ B4) & (Q @ X3)))))) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A4) & (P @ X3))))))))))).
% 2.93/1.52 thf(fact_232_prop__restrict, axiom, (! [A:$tType,X:A,Z4:(set @ A),X4:(set @ A),P:(A > $o)]: (((((member @ A) @ X) @ Z4) => ((((ord_less_eq @ (set @ A)) @ Z4) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ X4) & (P @ X3)))))) => (P @ X)))))).
% 2.93/1.52 thf(fact_233_Collect__restrict, axiom, (! [A:$tType,X4:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ X4) & (P @ X3)))))) @ X4)))).
% 2.93/1.52 thf(fact_234_llast__lappend, axiom, (! [A:$tType,Ys:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Ys) => (((coinductive_llast @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((coinductive_llast @ A) @ Xs))) & ((~ (((coinductive_lnull @ A) @ Ys))) => ((((coinductive_lfinite @ A) @ Xs) => (((coinductive_llast @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((coinductive_llast @ A) @ Ys))) & ((~ (((coinductive_lfinite @ A) @ Xs))) => (((coinductive_llast @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (undefined @ A))))))))).
% 2.93/1.52 thf(fact_235_Powp__mono, axiom, (! [A:$tType,A4:(A > $o),B4:(A > $o)]: (((((ord_less_eq @ (A > $o)) @ A4) @ B4) => (((ord_less_eq @ ((set @ A) > $o)) @ ((powp @ A) @ A4)) @ ((powp @ A) @ B4)))))).
% 2.93/1.52 thf(fact_236_llast__linfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => (((coinductive_llast @ A) @ Xs) = (undefined @ A)))))).
% 2.93/1.52 thf(fact_237_llast__LNil, axiom, (! [A:$tType]: ((((coinductive_llast @ A) @ (coinductive_LNil @ A)) = (undefined @ A))))).
% 2.93/1.52 thf(fact_238_lappend_Octr_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Ys)))) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = (((coinductive_LCons @ A) @ (((((coindu1381640503_llist @ A) @ A) @ ((coinductive_lhd @ A) @ Ys)) @ (^ [X3:A,Xs5:(coinductive_llist @ A)]: (X3))) @ Xs)) @ (((((coindu1381640503_llist @ (coinductive_llist @ A)) @ A) @ (((((coindu1381640503_llist @ (coinductive_llist @ A)) @ A) @ (undefined @ (coinductive_llist @ A))) @ (^ [Uu2:A,Uv2:(coinductive_llist @ A)]: (Uv2))) @ Ys)) @ (^ [X3:A,Xs5:(coinductive_llist @ A)]: ((((coinductive_lappend @ A) @ Xs5) @ Ys)))) @ Xs))))))).
% 2.93/1.52 thf(fact_239_lset__lappend__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs) => (((coinductive_lset @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((sup_sup @ (set @ A)) @ ((coinductive_lset @ A) @ Xs)) @ ((coinductive_lset @ A) @ Ys))))))).
% 2.93/1.52 thf(fact_240_Un__subset__iff, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B4)) @ C3) = ((((ord_less_eq @ (set @ A)) @ A4) @ C3) & (((ord_less_eq @ (set @ A)) @ B4) @ C3)))))).
% 2.93/1.52 thf(fact_241_lhd__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Xs) => (((coinductive_lhd @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((coinductive_lhd @ A) @ Ys))) & ((~ (((coinductive_lnull @ A) @ Xs))) => (((coinductive_lhd @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((coinductive_lhd @ A) @ Xs))))))).
% 2.93/1.52 thf(fact_242_lnull__ltakeWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) = ((~ (((coinductive_lnull @ A) @ Xs))) => (~ ((P @ ((coinductive_lhd @ A) @ Xs))))))))).
% 2.93/1.52 thf(fact_243_ltakeWhile_Odisc__iff_I1_J, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) = (((coinductive_lnull @ A) @ Xs) | (~ ((P @ ((coinductive_lhd @ A) @ Xs))))))))).
% 2.93/1.52 thf(fact_244_ltakeWhile_Odisc__iff_I2_J, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)))) = ((~ (((coinductive_lnull @ A) @ Xs))) & (P @ ((coinductive_lhd @ A) @ Xs))))))).
% 2.93/1.52 thf(fact_245_subset__Un__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: (((((sup_sup @ (set @ A)) @ A6) @ B5) = B5))))))).
% 2.93/1.52 thf(fact_246_Un__absorb2, axiom, (! [A:$tType,B4:(set @ A),A4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B4) @ A4) => ((((sup_sup @ (set @ A)) @ A4) @ B4) = A4))))).
% 2.93/1.52 thf(fact_247_Un__absorb1, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((sup_sup @ (set @ A)) @ A4) @ B4) = B4))))).
% 2.93/1.52 thf(fact_248_Un__upper2, axiom, (! [A:$tType,B4:(set @ A),A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B4) @ (((sup_sup @ (set @ A)) @ A4) @ B4))))).
% 2.93/1.52 thf(fact_249_Un__upper1, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ A4) @ B4))))).
% 2.93/1.52 thf(fact_250_Un__least, axiom, (! [A:$tType,A4:(set @ A),C3:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ C3) => ((((ord_less_eq @ (set @ A)) @ B4) @ C3) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B4)) @ C3)))))).
% 2.93/1.52 thf(fact_251_Un__mono, axiom, (! [A:$tType,A4:(set @ A),C3:(set @ A),B4:(set @ A),D:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ C3) => ((((ord_less_eq @ (set @ A)) @ B4) @ D) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B4)) @ (((sup_sup @ (set @ A)) @ C3) @ D))))))).
% 2.93/1.52 thf(fact_252_lhd__def, axiom, (! [A:$tType]: (((coinductive_lhd @ A) = ((((coindu1381640503_llist @ A) @ A) @ (undefined @ A)) @ (^ [X213:A,X223:(coinductive_llist @ A)]: (X213))))))).
% 2.93/1.52 thf(fact_253_ltakeWhile_Octr_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((((coinductive_lnull @ A) @ Xs) | (~ ((P @ ((coinductive_lhd @ A) @ Xs))))) => ((((coindu501562517eWhile @ A) @ P) @ Xs) = (coinductive_LNil @ A)))))).
% 2.93/1.52 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: ((((preorder @ A8) @ (type2 @ A8)) => ((preorder @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.93/1.52 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: ((((order @ A8) @ (type2 @ A8)) => ((order @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.93/1.52 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: ((((ord @ A8) @ (type2 @ A8)) => ((ord @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.93/1.52 thf(tcon_Set_Oset___Orderings_Opreorder_1, axiom, (! [A7:$tType]: (((preorder @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.93/1.52 thf(tcon_Set_Oset___Orderings_Oorder_2, axiom, (! [A7:$tType]: (((order @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.93/1.52 thf(tcon_Set_Oset___Orderings_Oord_3, axiom, (! [A7:$tType]: (((ord @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.93/1.52 thf(tcon_HOL_Obool___Orderings_Opreorder_4, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.93/1.52 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.93/1.52 thf(tcon_HOL_Obool___Orderings_Oorder_5, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.93/1.52 thf(tcon_HOL_Obool___Orderings_Oord_6, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.93/1.52 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.93/1.52 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.93/1.52 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.93/1.52 thf(conj_0, hypothesis, ((((coinductive_lappend @ a) @ x) @ ys) = (((coinductive_LCons @ a) @ xa) @ (coinductive_LNil @ a)))).
% 2.93/1.52 thf(conj_1, hypothesis, (x = (((coinductive_LCons @ a) @ x21) @ x22))).
% 2.93/1.52 thf(conj_2, conjecture, ((x = (coinductive_LNil @ a)) | ((? [X7:a]: ((x = (((coinductive_LCons @ a) @ X7) @ (coinductive_LNil @ a))))) | (? [X7:a,Y4:a,Xs4:(coinductive_llist @ a)]: (((x = (((coinductive_LCons @ a) @ X7) @ (((coinductive_LCons @ a) @ Y4) @ Xs4))) & (((graph @ X7) @ Y4) & ((((member @ (coinductive_llist @ a)) @ (((coinductive_lappend @ a) @ (((coinductive_LCons @ a) @ Y4) @ Xs4)) @ ys)) @ ((koenig916195507_paths @ a) @ graph)) | (((member @ (coinductive_llist @ a)) @ (((coinductive_LCons @ a) @ Y4) @ Xs4)) @ ((koenig916195507_paths @ a) @ graph)))))))))).
% 2.93/1.52 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.5ITWujfIVW/DTF2DTF22687.p
% 2.93/1.52 ------------------------
% 2.93/1.55 ---- Cleaned THF ---
% 2.93/1.55 thf(ty_t_Coinductive__List_Ollist,type,
% 2.93/1.55 coinductive_llist: $tType > $tType ).
% 2.93/1.55
% 2.93/1.55 thf(ty_t_Set_Oset,type,
% 2.93/1.55 set: $tType > $tType ).
% 2.93/1.55
% 2.93/1.55 thf(ty_t_itself,type,
% 2.93/1.55 itself: $tType > $tType ).
% 2.93/1.55
% 2.93/1.55 thf(ty_tf_a,type,
% 2.93/1.55 a: $tType ).
% 2.93/1.55
% 2.93/1.55 thf(sy_cl_HOL_Otype,type,
% 2.93/1.55 type:
% 2.93/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_cl_Orderings_Oord,type,
% 2.93/1.55 ord:
% 2.93/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_cl_Orderings_Oorder,type,
% 2.93/1.55 order:
% 2.93/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_cl_Orderings_Olinorder,type,
% 2.93/1.55 linorder:
% 2.93/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_cl_Orderings_Opreorder,type,
% 2.93/1.55 preorder:
% 2.93/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
% 2.93/1.55 coindu328551480prefix:
% 2.93/1.55 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Ogen__lset,type,
% 2.93/1.55 coinductive_gen_lset:
% 2.93/1.55 !>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Olappend,type,
% 2.93/1.55 coinductive_lappend:
% 2.93/1.55 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_OldropWhile,type,
% 2.93/1.55 coindu218763757pWhile:
% 2.93/1.55 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Olfilter,type,
% 2.93/1.55 coinductive_lfilter:
% 2.93/1.55 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Olfinite,type,
% 2.93/1.55 coinductive_lfinite:
% 2.93/1.55 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Ollast,type,
% 2.93/1.55 coinductive_llast:
% 2.93/1.55 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 2.93/1.55 coinductive_LCons:
% 2.93/1.55 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 2.93/1.55 coinductive_LNil:
% 2.93/1.55 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Ollist_Ocase__llist,type,
% 2.93/1.55 coindu1381640503_llist:
% 2.93/1.55 !>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Ollist_Ocorec__llist,type,
% 2.93/1.55 coindu1259883913_llist:
% 2.93/1.55 !>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Ollist_Olhd,type,
% 2.93/1.55 coinductive_lhd:
% 2.93/1.55 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 2.93/1.55 coinductive_lnull:
% 2.93/1.55 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 2.93/1.55 coinductive_lset:
% 2.93/1.55 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Olmember,type,
% 2.93/1.55 coinductive_lmember:
% 2.93/1.55 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Olstrict__prefix,type,
% 2.93/1.55 coindu1478340336prefix:
% 2.93/1.55 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_OltakeWhile,type,
% 2.93/1.55 coindu501562517eWhile:
% 2.93/1.55 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Oord__class_Olsorted,type,
% 2.93/1.55 coindu63249387sorted:
% 2.93/1.55 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 2.93/1.55 coindu1441602521_llist:
% 2.93/1.55 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_HOL_Oundefined,type,
% 2.93/1.55 undefined:
% 2.93/1.55 !>[A: $tType] : A ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_If,type,
% 2.93/1.55 if:
% 2.93/1.55 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Opaths,type,
% 2.93/1.55 koenig916195507_paths:
% 2.93/1.55 !>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ ( coinductive_llist @ Node ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Opathsp,type,
% 2.93/1.55 koenig2031690877pathsp:
% 2.93/1.55 !>[Node: $tType] : ( ( Node > Node > $o ) > ( coinductive_llist @ Node ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__via,type,
% 2.93/1.55 koenig317145564le_via:
% 2.93/1.55 !>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ Node ) > Node > ( set @ Node ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__viap,type,
% 2.93/1.55 koenig1757754772e_viap:
% 2.93/1.55 !>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ Node ) > Node > Node > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.93/1.55 sup_sup:
% 2.93/1.55 !>[A: $tType] : ( A > A > A ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.93/1.55 ord_less_eq:
% 2.93/1.55 !>[A: $tType] : ( A > A > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Pure_Otype,type,
% 2.93/1.55 type2:
% 2.93/1.55 !>[A: $tType] : ( itself @ A ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Relation_OPowp,type,
% 2.93/1.55 powp:
% 2.93/1.55 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_Set_OCollect,type,
% 2.93/1.55 collect:
% 2.93/1.55 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_c_member,type,
% 2.93/1.55 member:
% 2.93/1.55 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(sy_v_graph,type,
% 2.93/1.55 graph: a > a > $o ).
% 2.93/1.55
% 2.93/1.55 thf(sy_v_x,type,
% 2.93/1.55 x: coinductive_llist @ a ).
% 2.93/1.55
% 2.93/1.55 thf(sy_v_x21,type,
% 2.93/1.55 x21: a ).
% 2.93/1.55
% 2.93/1.55 thf(sy_v_x22,type,
% 2.93/1.55 x22: coinductive_llist @ a ).
% 2.93/1.55
% 2.93/1.55 thf(sy_v_xa,type,
% 2.93/1.55 xa: a ).
% 2.93/1.55
% 2.93/1.55 thf(sy_v_xs,type,
% 2.93/1.55 xs: coinductive_llist @ a ).
% 2.93/1.55
% 2.93/1.55 thf(sy_v_ys,type,
% 2.93/1.55 ys: coinductive_llist @ a ).
% 2.93/1.55
% 2.93/1.55 thf(fact_0_assms,axiom,
% 2.93/1.55 member @ ( coinductive_llist @ a ) @ ( coinductive_lappend @ a @ xs @ ys ) @ ( koenig916195507_paths @ a @ graph ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_1_paths_OEmpty,axiom,
% 2.93/1.55 ! [Node: $tType,Graph: Node > Node > $o] : ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LNil @ Node ) @ ( koenig916195507_paths @ Node @ Graph ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_2_paths_OLCons,axiom,
% 2.93/1.55 ! [Node: $tType,Graph: Node > Node > $o,X: Node,Y: Node,Xs: coinductive_llist @ Node] :
% 2.93/1.55 ( ( Graph @ X @ Y )
% 2.93/1.55 => ( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y @ Xs ) @ ( koenig916195507_paths @ Node @ Graph ) )
% 2.93/1.55 => ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ X @ ( coinductive_LCons @ Node @ Y @ Xs ) ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_3_paths_Ocases,axiom,
% 2.93/1.55 ! [Node: $tType,A2: coinductive_llist @ Node,Graph: Node > Node > $o] :
% 2.93/1.55 ( ( member @ ( coinductive_llist @ Node ) @ A2 @ ( koenig916195507_paths @ Node @ Graph ) )
% 2.93/1.55 => ( ( A2
% 2.93/1.55 != ( coinductive_LNil @ Node ) )
% 2.93/1.55 => ( ! [X2: Node] :
% 2.93/1.55 ( A2
% 2.93/1.55 != ( coinductive_LCons @ Node @ X2 @ ( coinductive_LNil @ Node ) ) )
% 2.93/1.55 => ~ ! [X2: Node,Y2: Node,Xs2: coinductive_llist @ Node] :
% 2.93/1.55 ( ( A2
% 2.93/1.55 = ( coinductive_LCons @ Node @ X2 @ ( coinductive_LCons @ Node @ Y2 @ Xs2 ) ) )
% 2.93/1.55 => ( ( Graph @ X2 @ Y2 )
% 2.93/1.55 => ~ ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y2 @ Xs2 ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_4_paths_Osimps,axiom,
% 2.93/1.55 ! [Node: $tType,A2: coinductive_llist @ Node,Graph: Node > Node > $o] :
% 2.93/1.55 ( ( member @ ( coinductive_llist @ Node ) @ A2 @ ( koenig916195507_paths @ Node @ Graph ) )
% 2.93/1.55 = ( ( A2
% 2.93/1.55 = ( coinductive_LNil @ Node ) )
% 2.93/1.55 | ? [X3: Node] :
% 2.93/1.55 ( A2
% 2.93/1.55 = ( coinductive_LCons @ Node @ X3 @ ( coinductive_LNil @ Node ) ) )
% 2.93/1.55 | ? [X3: Node,Y3: Node,Xs3: coinductive_llist @ Node] :
% 2.93/1.55 ( ( A2
% 2.93/1.55 = ( coinductive_LCons @ Node @ X3 @ ( coinductive_LCons @ Node @ Y3 @ Xs3 ) ) )
% 2.93/1.55 & ( Graph @ X3 @ Y3 )
% 2.93/1.55 & ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y3 @ Xs3 ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_5_paths_OSingle,axiom,
% 2.93/1.55 ! [Node: $tType,X: Node,Graph: Node > Node > $o] : ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ X @ ( coinductive_LNil @ Node ) ) @ ( koenig916195507_paths @ Node @ Graph ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_6_paths__LConsD,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Graph: A > A > $o] :
% 2.93/1.55 ( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X @ Xs ) @ ( koenig916195507_paths @ A @ Graph ) )
% 2.93/1.55 => ( member @ ( coinductive_llist @ A ) @ Xs @ ( koenig916195507_paths @ A @ Graph ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_7_paths_Ocoinduct,axiom,
% 2.93/1.55 ! [Node: $tType,X4: ( coinductive_llist @ Node ) > $o,X: coinductive_llist @ Node,Graph: Node > Node > $o] :
% 2.93/1.55 ( ( X4 @ X )
% 2.93/1.55 => ( ! [X2: coinductive_llist @ Node] :
% 2.93/1.55 ( ( X4 @ X2 )
% 2.93/1.55 => ( ( X2
% 2.93/1.55 = ( coinductive_LNil @ Node ) )
% 2.93/1.55 | ? [Xa: Node] :
% 2.93/1.55 ( X2
% 2.93/1.55 = ( coinductive_LCons @ Node @ Xa @ ( coinductive_LNil @ Node ) ) )
% 2.93/1.55 | ? [Xa: Node,Y4: Node,Xs4: coinductive_llist @ Node] :
% 2.93/1.55 ( ( X2
% 2.93/1.55 = ( coinductive_LCons @ Node @ Xa @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) ) )
% 2.93/1.55 & ( Graph @ Xa @ Y4 )
% 2.93/1.55 & ( ( X4 @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) )
% 2.93/1.55 | ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ) )
% 2.93/1.55 => ( member @ ( coinductive_llist @ Node ) @ X @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_8_lappend__code_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
% 2.93/1.55 = Ys ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_9_lappend__LNil2,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
% 2.93/1.55 = Xs ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_10_lappend__code_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,Xa2: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa2 @ X ) @ Ys )
% 2.93/1.55 = ( coinductive_LCons @ A @ Xa2 @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_11_lappend__snocL1__conv__LCons2,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
% 2.93/1.55 = ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_12_llist_Oinject,axiom,
% 2.93/1.55 ! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( coinductive_LCons @ A @ X21 @ X22 )
% 2.93/1.55 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 2.93/1.55 = ( ( X21 = Y21 )
% 2.93/1.55 & ( X22 = Y22 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_13_lappend__LNil__LNil,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_14_LNil__eq__lappend__iff,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( coinductive_LNil @ A )
% 2.93/1.55 = ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( ( Xs
% 2.93/1.55 = ( coinductive_LNil @ A ) )
% 2.93/1.55 & ( Ys
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_15_lappend__eq__LNil__iff,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.93/1.55 = ( coinductive_LNil @ A ) )
% 2.93/1.55 = ( ( Xs
% 2.93/1.55 = ( coinductive_LNil @ A ) )
% 2.93/1.55 & ( Ys
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_16_llist_Odistinct_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_LNil @ A )
% 2.93/1.55 != ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_17_llist_Oexhaust,axiom,
% 2.93/1.55 ! [A: $tType,Y: coinductive_llist @ A] :
% 2.93/1.55 ( ( Y
% 2.93/1.55 != ( coinductive_LNil @ A ) )
% 2.93/1.55 => ~ ! [X212: A,X222: coinductive_llist @ A] :
% 2.93/1.55 ( Y
% 2.93/1.55 != ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_18_neq__LNil__conv,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( Xs
% 2.93/1.55 != ( coinductive_LNil @ A ) )
% 2.93/1.55 = ( ? [X3: A,Xs5: coinductive_llist @ A] :
% 2.93/1.55 ( Xs
% 2.93/1.55 = ( coinductive_LCons @ A @ X3 @ Xs5 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_19_lappend__assoc,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 2.93/1.55 = ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_20_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 2.93/1.55 = ( ( Xs
% 2.93/1.55 = ( coinductive_LNil @ A ) )
% 2.93/1.55 | ? [Xs5: coinductive_llist @ A] :
% 2.93/1.55 ( ( Xs
% 2.93/1.55 = ( coinductive_LCons @ A @ Y @ Xs5 ) )
% 2.93/1.55 & ( coindu328551480prefix @ A @ Xs5 @ Ys ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_21_lstrict__prefix__code_I2_J,axiom,
% 2.93/1.55 ! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_22_lstrict__prefix__code_I3_J,axiom,
% 2.93/1.55 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 2.93/1.55 ~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_23_llast__singleton,axiom,
% 2.93/1.55 ! [A: $tType,X: A] :
% 2.93/1.55 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
% 2.93/1.55 = X ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_24_lmember__code_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,X: A] :
% 2.93/1.55 ~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_25_lmember__code_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 2.93/1.55 = ( ( X = Y )
% 2.93/1.55 | ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_26_pathsp_Ocoinduct,axiom,
% 2.93/1.55 ! [Node: $tType,X4: ( coinductive_llist @ Node ) > $o,X: coinductive_llist @ Node,Graph: Node > Node > $o] :
% 2.93/1.55 ( ( X4 @ X )
% 2.93/1.55 => ( ! [X2: coinductive_llist @ Node] :
% 2.93/1.55 ( ( X4 @ X2 )
% 2.93/1.55 => ( ( X2
% 2.93/1.55 = ( coinductive_LNil @ Node ) )
% 2.93/1.55 | ? [Xa: Node] :
% 2.93/1.55 ( X2
% 2.93/1.55 = ( coinductive_LCons @ Node @ Xa @ ( coinductive_LNil @ Node ) ) )
% 2.93/1.55 | ? [Xa: Node,Y4: Node,Xs4: coinductive_llist @ Node] :
% 2.93/1.55 ( ( X2
% 2.93/1.55 = ( coinductive_LCons @ Node @ Xa @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) ) )
% 2.93/1.55 & ( Graph @ Xa @ Y4 )
% 2.93/1.55 & ( ( X4 @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) )
% 2.93/1.55 | ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) ) ) ) ) )
% 2.93/1.55 => ( koenig2031690877pathsp @ Node @ Graph @ X ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_27_pathsp_OSingle,axiom,
% 2.93/1.55 ! [Node: $tType,Graph: Node > Node > $o,X: Node] : ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ X @ ( coinductive_LNil @ Node ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_28_pathsp_Osimps,axiom,
% 2.93/1.55 ! [Node: $tType] :
% 2.93/1.55 ( ( koenig2031690877pathsp @ Node )
% 2.93/1.55 = ( ^ [Graph2: Node > Node > $o,A3: coinductive_llist @ Node] :
% 2.93/1.55 ( ( A3
% 2.93/1.55 = ( coinductive_LNil @ Node ) )
% 2.93/1.55 | ? [X3: Node] :
% 2.93/1.55 ( A3
% 2.93/1.55 = ( coinductive_LCons @ Node @ X3 @ ( coinductive_LNil @ Node ) ) )
% 2.93/1.55 | ? [X3: Node,Y3: Node,Xs3: coinductive_llist @ Node] :
% 2.93/1.55 ( ( A3
% 2.93/1.55 = ( coinductive_LCons @ Node @ X3 @ ( coinductive_LCons @ Node @ Y3 @ Xs3 ) ) )
% 2.93/1.55 & ( Graph2 @ X3 @ Y3 )
% 2.93/1.55 & ( koenig2031690877pathsp @ Node @ Graph2 @ ( coinductive_LCons @ Node @ Y3 @ Xs3 ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_29_pathsp_Ocases,axiom,
% 2.93/1.55 ! [Node: $tType,Graph: Node > Node > $o,A2: coinductive_llist @ Node] :
% 2.93/1.55 ( ( koenig2031690877pathsp @ Node @ Graph @ A2 )
% 2.93/1.55 => ( ( A2
% 2.93/1.55 != ( coinductive_LNil @ Node ) )
% 2.93/1.55 => ( ! [X2: Node] :
% 2.93/1.55 ( A2
% 2.93/1.55 != ( coinductive_LCons @ Node @ X2 @ ( coinductive_LNil @ Node ) ) )
% 2.93/1.55 => ~ ! [X2: Node,Y2: Node,Xs2: coinductive_llist @ Node] :
% 2.93/1.55 ( ( A2
% 2.93/1.55 = ( coinductive_LCons @ Node @ X2 @ ( coinductive_LCons @ Node @ Y2 @ Xs2 ) ) )
% 2.93/1.55 => ( ( Graph @ X2 @ Y2 )
% 2.93/1.55 => ~ ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ Y2 @ Xs2 ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_30_llast__LCons2,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 2.93/1.55 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_31_lstrict__prefix__code_I4_J,axiom,
% 2.93/1.55 ! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
% 2.93/1.55 ( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
% 2.93/1.55 = ( ( X = Y )
% 2.93/1.55 & ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_32_lstrict__prefix__code_I1_J,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_33_llist__less__induct,axiom,
% 2.93/1.55 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ! [Xs2: coinductive_llist @ A] :
% 2.93/1.55 ( ! [Ys2: coinductive_llist @ A] :
% 2.93/1.55 ( ( coindu1478340336prefix @ A @ Ys2 @ Xs2 )
% 2.93/1.55 => ( P @ Ys2 ) )
% 2.93/1.55 => ( P @ Xs2 ) )
% 2.93/1.55 => ( P @ Xs ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_34_pathsp_OLCons,axiom,
% 2.93/1.55 ! [Node: $tType,Graph: Node > Node > $o,X: Node,Y: Node,Xs: coinductive_llist @ Node] :
% 2.93/1.55 ( ( Graph @ X @ Y )
% 2.93/1.55 => ( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ Y @ Xs ) )
% 2.93/1.55 => ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ X @ ( coinductive_LCons @ Node @ Y @ Xs ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_35_pathsp_OEmpty,axiom,
% 2.93/1.55 ! [Node: $tType,Graph: Node > Node > $o] : ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LNil @ Node ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_36_pathsp__paths__eq,axiom,
% 2.93/1.55 ! [Node: $tType] :
% 2.93/1.55 ( ( koenig2031690877pathsp @ Node )
% 2.93/1.55 = ( ^ [Graph2: Node > Node > $o,X3: coinductive_llist @ Node] : ( member @ ( coinductive_llist @ Node ) @ X3 @ ( koenig916195507_paths @ Node @ Graph2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_37_paths__def,axiom,
% 2.93/1.55 ! [Node: $tType] :
% 2.93/1.55 ( ( koenig916195507_paths @ Node )
% 2.93/1.55 = ( ^ [Graph2: Node > Node > $o] : ( collect @ ( coinductive_llist @ Node ) @ ( koenig2031690877pathsp @ Node @ Graph2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_38_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_39_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 2.93/1.55 = ( Xs
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_40_llast__lappend__LCons,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
% 2.93/1.55 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_41_llimit__induct,axiom,
% 2.93/1.55 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( P @ ( coinductive_LNil @ A ) )
% 2.93/1.55 => ( ! [X2: A,Xs2: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ Xs2 )
% 2.93/1.55 => ( ( P @ Xs2 )
% 2.93/1.55 => ( P @ ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) )
% 2.93/1.55 => ( ( ! [Ys2: coinductive_llist @ A] :
% 2.93/1.55 ( ( coindu1478340336prefix @ A @ Ys2 @ Xs )
% 2.93/1.55 => ( P @ Ys2 ) )
% 2.93/1.55 => ( P @ Xs ) )
% 2.93/1.55 => ( P @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_42_gen__lset__code_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A] :
% 2.93/1.55 ( ( coinductive_gen_lset @ A @ A4 @ ( coinductive_LNil @ A ) )
% 2.93/1.55 = A4 ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_43_lsorted__code_I2_J,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_44_lfinite__rev__induct,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 => ( ( P @ ( coinductive_LNil @ A ) )
% 2.93/1.55 => ( ! [X2: A,Xs2: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ Xs2 )
% 2.93/1.55 => ( ( P @ Xs2 )
% 2.93/1.55 => ( P @ ( coinductive_lappend @ A @ Xs2 @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) ) ) )
% 2.93/1.55 => ( P @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_45_mem__Collect__eq,axiom,
% 2.93/1.55 ! [A: $tType,A2: A,P: A > $o] :
% 2.93/1.55 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.93/1.55 = ( P @ A2 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_46_Collect__mem__eq,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A] :
% 2.93/1.55 ( ( collect @ A
% 2.93/1.55 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 2.93/1.55 = A4 ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_47_Collect__cong,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.93/1.55 ( ! [X2: A] :
% 2.93/1.55 ( ( P @ X2 )
% 2.93/1.55 = ( Q @ X2 ) )
% 2.93/1.55 => ( ( collect @ A @ P )
% 2.93/1.55 = ( collect @ A @ Q ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_48_ext,axiom,
% 2.93/1.55 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.93/1.55 ( ! [X2: A] :
% 2.93/1.55 ( ( F @ X2 )
% 2.93/1.55 = ( G @ X2 ) )
% 2.93/1.55 => ( F = G ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_49_llist_Ocorec__code,axiom,
% 2.93/1.55 ! [A: $tType,C: $tType] :
% 2.93/1.55 ( ( coindu1259883913_llist @ C @ A )
% 2.93/1.55 = ( ^ [P2: C > $o,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C,A3: C] : ( if @ ( coinductive_llist @ A ) @ ( P2 @ A3 ) @ ( coinductive_LNil @ A ) @ ( coinductive_LCons @ A @ ( G21 @ A3 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1259883913_llist @ C @ A @ P2 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_50_unfold__llist_Ocode,axiom,
% 2.93/1.55 ! [B: $tType,A: $tType] :
% 2.93/1.55 ( ( coindu1441602521_llist @ A @ B )
% 2.93/1.55 = ( ^ [P2: A > $o,G21: A > B,G22: A > A,A3: A] : ( if @ ( coinductive_llist @ B ) @ ( P2 @ A3 ) @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ ( G21 @ A3 ) @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_51_lappend_Ocode,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( coinductive_lappend @ A )
% 2.93/1.55 = ( ^ [Xs3: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
% 2.93/1.55 ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ Ys3
% 2.93/1.55 @ ^ [X3: A,Xs5: coinductive_llist @ A] : ( coinductive_LCons @ A @ X3 @ ( coinductive_lappend @ A @ Xs5 @ Ys3 ) )
% 2.93/1.55 @ Xs3 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_52_ltakeWhile__LCons,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( P @ X )
% 2.93/1.55 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.93/1.55 = ( coinductive_LCons @ A @ X @ ( coindu501562517eWhile @ A @ P @ Xs ) ) ) )
% 2.93/1.55 & ( ~ ( P @ X )
% 2.93/1.55 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_53_ltakeWhile__K__True,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coindu501562517eWhile @ A
% 2.93/1.55 @ ^ [Uu: A] : $true
% 2.93/1.55 @ Xs )
% 2.93/1.55 = Xs ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_54_lfinite__LCons,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.93/1.55 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_55_lfinite__code_I2_J,axiom,
% 2.93/1.55 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 2.93/1.55 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
% 2.93/1.55 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_56_lfinite__code_I1_J,axiom,
% 2.93/1.55 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_57_lfinite__lappend,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 & ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_58_ltakeWhile__LNil,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o] :
% 2.93/1.55 ( ( coindu501562517eWhile @ A @ P @ ( coinductive_LNil @ A ) )
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_59_lsorted__code_I1_J,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.55 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_60_unfold__llist__eq__LCons,axiom,
% 2.93/1.55 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
% 2.93/1.55 = ( coinductive_LCons @ A @ X @ Xs ) )
% 2.93/1.55 = ( ~ ( IS_LNIL @ B2 )
% 2.93/1.55 & ( X
% 2.93/1.55 = ( LHD @ B2 ) )
% 2.93/1.55 & ( Xs
% 2.93/1.55 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_61_ltakeWhile__K__False,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coindu501562517eWhile @ A
% 2.93/1.55 @ ^ [Uu: A] : $false
% 2.93/1.55 @ Xs )
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_62_corec__llist__never__stop,axiom,
% 2.93/1.55 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,MORE: B > ( coinductive_llist @ A ),LTL: B > B,X: B] :
% 2.93/1.55 ( ( coindu1259883913_llist @ B @ A @ IS_LNIL @ LHD
% 2.93/1.55 @ ^ [Uu: B] : $false
% 2.93/1.55 @ MORE
% 2.93/1.55 @ LTL
% 2.93/1.55 @ X )
% 2.93/1.55 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ X ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_63_llist_Ocase__distrib,axiom,
% 2.93/1.55 ! [B: $tType,C: $tType,A: $tType,H: B > C,F1: B,F2: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
% 2.93/1.55 ( ( H @ ( coindu1381640503_llist @ B @ A @ F1 @ F2 @ Llist ) )
% 2.93/1.55 = ( coindu1381640503_llist @ C @ A @ ( H @ F1 )
% 2.93/1.55 @ ^ [X1: A,X23: coinductive_llist @ A] : ( H @ ( F2 @ X1 @ X23 ) )
% 2.93/1.55 @ Llist ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_64_lfinite__LConsI,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_65_lfinite__LNil,axiom,
% 2.93/1.55 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_66_lappend__inf,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.93/1.55 = Xs ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_67_LNil,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.55 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_68_lstrict__prefix__lfinite1,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coindu1478340336prefix @ A @ Xs @ Ys )
% 2.93/1.55 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_69_unfold__llist_Octr_I2_J,axiom,
% 2.93/1.55 ! [B: $tType,A: $tType,P3: A > $o,A2: A,G212: A > B,G223: A > A] :
% 2.93/1.55 ( ~ ( P3 @ A2 )
% 2.93/1.55 => ( ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 )
% 2.93/1.55 = ( coinductive_LCons @ B @ ( G212 @ A2 ) @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ ( G223 @ A2 ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_70_unfold__llist_Octr_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,B: $tType,P3: A > $o,A2: A,G212: A > B,G223: A > A] :
% 2.93/1.55 ( ( P3 @ A2 )
% 2.93/1.55 => ( ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 )
% 2.93/1.55 = ( coinductive_LNil @ B ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_71_llist_Ocorec_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,C: $tType,P3: C > $o,A2: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 2.93/1.55 ( ~ ( P3 @ A2 )
% 2.93/1.55 => ( ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 )
% 2.93/1.55 = ( coinductive_LCons @ A @ ( G212 @ A2 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q222 @ A2 ) @ ( G2212 @ A2 ) @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ ( G2222 @ A2 ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_72_llist_Ocorec_I1_J,axiom,
% 2.93/1.55 ! [C: $tType,A: $tType,P3: C > $o,A2: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 2.93/1.55 ( ( P3 @ A2 )
% 2.93/1.55 => ( ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 )
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_73_llist_Osimps_I5_J,axiom,
% 2.93/1.55 ! [B: $tType,A: $tType,F1: B,F2: A > ( coinductive_llist @ A ) > B,X21: A,X22: coinductive_llist @ A] :
% 2.93/1.55 ( ( coindu1381640503_llist @ B @ A @ F1 @ F2 @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 2.93/1.55 = ( F2 @ X21 @ X22 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_74_llist_Osimps_I4_J,axiom,
% 2.93/1.55 ! [A: $tType,B: $tType,F1: B,F2: A > ( coinductive_llist @ A ) > B] :
% 2.93/1.55 ( ( coindu1381640503_llist @ B @ A @ F1 @ F2 @ ( coinductive_LNil @ A ) )
% 2.93/1.55 = F1 ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_75_lfinite_Oinducts,axiom,
% 2.93/1.55 ! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ X )
% 2.93/1.55 => ( ( P @ ( coinductive_LNil @ A ) )
% 2.93/1.55 => ( ! [Xs2: coinductive_llist @ A,X2: A] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ Xs2 )
% 2.93/1.55 => ( ( P @ Xs2 )
% 2.93/1.55 => ( P @ ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) )
% 2.93/1.55 => ( P @ X ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_76_lfinite_Osimps,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( coinductive_lfinite @ A )
% 2.93/1.55 = ( ^ [A3: coinductive_llist @ A] :
% 2.93/1.55 ( ( A3
% 2.93/1.55 = ( coinductive_LNil @ A ) )
% 2.93/1.55 | ? [Xs3: coinductive_llist @ A,X3: A] :
% 2.93/1.55 ( ( A3
% 2.93/1.55 = ( coinductive_LCons @ A @ X3 @ Xs3 ) )
% 2.93/1.55 & ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_77_lfinite_Ocases,axiom,
% 2.93/1.55 ! [A: $tType,A2: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ A2 )
% 2.93/1.55 => ( ( A2
% 2.93/1.55 != ( coinductive_LNil @ A ) )
% 2.93/1.55 => ~ ! [Xs2: coinductive_llist @ A] :
% 2.93/1.55 ( ? [X2: A] :
% 2.93/1.55 ( A2
% 2.93/1.55 = ( coinductive_LCons @ A @ X2 @ Xs2 ) )
% 2.93/1.55 => ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_78_Singleton,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_79_lfilter_Osimps,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( coinductive_lfilter @ A )
% 2.93/1.55 = ( ^ [P4: A > $o] :
% 2.93/1.55 ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_LNil @ A )
% 2.93/1.55 @ ^ [X3: A,Xs5: coinductive_llist @ A] : ( if @ ( coinductive_llist @ A ) @ ( P4 @ X3 ) @ ( coinductive_LCons @ A @ X3 @ ( coinductive_lfilter @ A @ P4 @ Xs5 ) ) @ ( coinductive_lfilter @ A @ P4 @ Xs5 ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_80_lsorted_Ocoinduct,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
% 2.93/1.55 ( ( X4 @ X )
% 2.93/1.55 => ( ! [X2: coinductive_llist @ A] :
% 2.93/1.55 ( ( X4 @ X2 )
% 2.93/1.55 => ( ( X2
% 2.93/1.55 = ( coinductive_LNil @ A ) )
% 2.93/1.55 | ? [Xa: A] :
% 2.93/1.55 ( X2
% 2.93/1.55 = ( coinductive_LCons @ A @ Xa @ ( coinductive_LNil @ A ) ) )
% 2.93/1.55 | ? [Xa: A,Y4: A,Xs4: coinductive_llist @ A] :
% 2.93/1.55 ( ( X2
% 2.93/1.55 = ( coinductive_LCons @ A @ Xa @ ( coinductive_LCons @ A @ Y4 @ Xs4 ) ) )
% 2.93/1.55 & ( ord_less_eq @ A @ Xa @ Y4 )
% 2.93/1.55 & ( ( X4 @ ( coinductive_LCons @ A @ Y4 @ Xs4 ) )
% 2.93/1.55 | ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y4 @ Xs4 ) ) ) ) ) )
% 2.93/1.55 => ( coindu63249387sorted @ A @ X ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_81_lsorted_Osimps,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.55 => ( ( coindu63249387sorted @ A )
% 2.93/1.55 = ( ^ [A3: coinductive_llist @ A] :
% 2.93/1.55 ( ( A3
% 2.93/1.55 = ( coinductive_LNil @ A ) )
% 2.93/1.55 | ? [X3: A] :
% 2.93/1.55 ( A3
% 2.93/1.55 = ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
% 2.93/1.55 | ? [X3: A,Y3: A,Xs3: coinductive_llist @ A] :
% 2.93/1.55 ( ( A3
% 2.93/1.55 = ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs3 ) ) )
% 2.93/1.55 & ( ord_less_eq @ A @ X3 @ Y3 )
% 2.93/1.55 & ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y3 @ Xs3 ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_82_lsorted_Ocases,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [A2: coinductive_llist @ A] :
% 2.93/1.55 ( ( coindu63249387sorted @ A @ A2 )
% 2.93/1.55 => ( ( A2
% 2.93/1.55 != ( coinductive_LNil @ A ) )
% 2.93/1.55 => ( ! [X2: A] :
% 2.93/1.55 ( A2
% 2.93/1.55 != ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
% 2.93/1.55 => ~ ! [X2: A,Y2: A,Xs2: coinductive_llist @ A] :
% 2.93/1.55 ( ( A2
% 2.93/1.55 = ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y2 @ Xs2 ) ) )
% 2.93/1.55 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.93/1.55 => ~ ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_83_lstrict__prefix__lappend__conv,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 & ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_84_split__llist__first,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ? [Ys4: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 2.93/1.55 ( ( Xs
% 2.93/1.55 = ( coinductive_lappend @ A @ Ys4 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
% 2.93/1.55 & ( coinductive_lfinite @ A @ Ys4 )
% 2.93/1.55 & ~ ( member @ A @ X @ ( coinductive_lset @ A @ Ys4 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_85_split__llist,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ? [Ys4: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 2.93/1.55 ( ( Xs
% 2.93/1.55 = ( coinductive_lappend @ A @ Ys4 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
% 2.93/1.55 & ( coinductive_lfinite @ A @ Ys4 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_86_lappend__ltakeWhile__ldropWhile,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lappend @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 2.93/1.55 = Xs ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_87_lfilter__idem,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 2.93/1.55 = ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_88_lfilter__K__True,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfilter @ A
% 2.93/1.55 @ ^ [Uu: A] : $true
% 2.93/1.55 @ Xs )
% 2.93/1.55 = Xs ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_89_lappend_Odisc__iff_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 2.93/1.55 = ( ~ ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_90_lnull__lappend,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 & ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_91_lfilter__LCons,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( P @ X )
% 2.93/1.55 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.93/1.55 = ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
% 2.93/1.55 & ( ~ ( P @ X )
% 2.93/1.55 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.93/1.55 = ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_92_lfilter__LNil,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o] :
% 2.93/1.55 ( ( coinductive_lfilter @ A @ P @ ( coinductive_LNil @ A ) )
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_93_ldropWhile__LCons,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( P @ X )
% 2.93/1.55 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.93/1.55 = ( coindu218763757pWhile @ A @ P @ Xs ) ) )
% 2.93/1.55 & ( ~ ( P @ X )
% 2.93/1.55 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.93/1.55 = ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_94_ldropWhile__LNil,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o] :
% 2.93/1.55 ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LNil @ A ) )
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_95_unfold__llist_Odisc__iff_I2_J,axiom,
% 2.93/1.55 ! [B: $tType,A: $tType,P3: A > $o,G212: A > B,G223: A > A,A2: A] :
% 2.93/1.55 ( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 ) ) )
% 2.93/1.55 = ( ~ ( P3 @ A2 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_96_unfold__llist_Odisc__iff_I1_J,axiom,
% 2.93/1.55 ! [B: $tType,A: $tType,P3: A > $o,G212: A > B,G223: A > A,A2: A] :
% 2.93/1.55 ( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 ) )
% 2.93/1.55 = ( P3 @ A2 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_97_llist_Ocorec__disc__iff_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,C: $tType,P3: C > $o,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C,A2: C] :
% 2.93/1.55 ( ( ~ ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) ) )
% 2.93/1.55 = ( ~ ( P3 @ A2 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_98_llist_Ocorec__disc__iff_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,C: $tType,P3: C > $o,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C,A2: C] :
% 2.93/1.55 ( ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) )
% 2.93/1.55 = ( P3 @ A2 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_99_lfitler__K__False,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfilter @ A
% 2.93/1.55 @ ^ [Uu: A] : $false
% 2.93/1.55 @ Xs )
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_100_lset__lfilter,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lset @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 2.93/1.55 = ( collect @ A
% 2.93/1.55 @ ^ [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 & ( P @ X3 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_101_ldropWhile__K__True,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coindu218763757pWhile @ A
% 2.93/1.55 @ ^ [Uu: A] : $true
% 2.93/1.55 @ Xs )
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_102_lsorted__LCons__LCons,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [X: A,Y: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 2.93/1.55 = ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.55 & ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_103_diverge__lfilter__LNil,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.93/1.55 ( ! [X2: A] :
% 2.93/1.55 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ~ ( P @ X2 ) )
% 2.93/1.55 => ( ( coinductive_lfilter @ A @ P @ Xs )
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_104_lnull__lfilter,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lnull @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 2.93/1.55 = ( ! [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ~ ( P @ X3 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_105_lnull__ldropWhile,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lnull @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 2.93/1.55 = ( ! [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ( P @ X3 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_106_lfilter__lappend__lfinite,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_107_lfinite__ldropWhile,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 2.93/1.55 = ( ? [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 & ~ ( P @ X3 ) )
% 2.93/1.55 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_108_llist_Odisc__eq__case_I1_J,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( coinductive_lnull @ A )
% 2.93/1.55 = ( coindu1381640503_llist @ $o @ A @ $true
% 2.93/1.55 @ ^ [Uu: A,Uv: coinductive_llist @ A] : $false ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_109_llist_Odisc__eq__case_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,Llist: coinductive_llist @ A] :
% 2.93/1.55 ( ( ~ ( coinductive_lnull @ A @ Llist ) )
% 2.93/1.55 = ( coindu1381640503_llist @ $o @ A @ $false
% 2.93/1.55 @ ^ [Uu: A,Uv: coinductive_llist @ A] : $true
% 2.93/1.55 @ Llist ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_110_lfilter__cong,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 2.93/1.55 ( ( Xs = Ys )
% 2.93/1.55 => ( ! [X2: A] :
% 2.93/1.55 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) )
% 2.93/1.55 => ( ( P @ X2 )
% 2.93/1.55 = ( Q @ X2 ) ) )
% 2.93/1.55 => ( ( coinductive_lfilter @ A @ P @ Xs )
% 2.93/1.55 = ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_111_lzip_Oexhaust,axiom,
% 2.93/1.55 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.93/1.55 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 | ( coinductive_lnull @ B @ Ys ) )
% 2.93/1.55 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_112_lappend_Oexhaust,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 => ~ ( coinductive_lnull @ A @ Ys ) )
% 2.93/1.55 => ( ~ ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_113_ldropWhile__cong,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 2.93/1.55 ( ( Xs = Ys )
% 2.93/1.55 => ( ! [X2: A] :
% 2.93/1.55 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) )
% 2.93/1.55 => ( ( P @ X2 )
% 2.93/1.55 = ( Q @ X2 ) ) )
% 2.93/1.55 => ( ( coindu218763757pWhile @ A @ P @ Xs )
% 2.93/1.55 = ( coindu218763757pWhile @ A @ Q @ Ys ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_114_lfilter__id__conv,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 2.93/1.55 = Xs )
% 2.93/1.55 = ( ! [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ( P @ X3 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_115_wlog__linorder__le,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [P: A > A > $o,B2: A,A2: A] :
% 2.93/1.55 ( ! [A5: A,B3: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ A5 @ B3 )
% 2.93/1.55 => ( P @ A5 @ B3 ) )
% 2.93/1.55 => ( ( ( P @ B2 @ A2 )
% 2.93/1.55 => ( P @ A2 @ B2 ) )
% 2.93/1.55 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_116_lfilter__empty__conv,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 2.93/1.55 = ( coinductive_LNil @ A ) )
% 2.93/1.55 = ( ! [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ~ ( P @ X3 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_117_in__lset__ldropWhileD,axiom,
% 2.93/1.55 ! [A: $tType,X: A,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) )
% 2.93/1.55 => ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_118_lset__ldropWhile__subset,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) @ ( coinductive_lset @ A @ Xs ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_119_lset__lappend1,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_120_lfilter__lfilter,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Q: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ Q @ Xs ) )
% 2.93/1.55 = ( coinductive_lfilter @ A
% 2.93/1.55 @ ^ [X3: A] :
% 2.93/1.55 ( ( P @ X3 )
% 2.93/1.55 & ( Q @ X3 ) )
% 2.93/1.55 @ Xs ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_121_ldropWhile__eq__LNil__iff,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( coindu218763757pWhile @ A @ P @ Xs )
% 2.93/1.55 = ( coinductive_LNil @ A ) )
% 2.93/1.55 = ( ! [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ( P @ X3 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_122_ltakeWhile__all__conv,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( coindu501562517eWhile @ A @ P @ Xs )
% 2.93/1.55 = Xs )
% 2.93/1.55 = ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( collect @ A @ P ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_123_lsorted__LCons,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [X: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.93/1.55 = ( ( coindu63249387sorted @ A @ Xs )
% 2.93/1.55 & ! [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_124_lfilter__LCons__seek,axiom,
% 2.93/1.55 ! [A: $tType,P3: A > $o,X: A,L: coinductive_llist @ A] :
% 2.93/1.55 ( ~ ( P3 @ X )
% 2.93/1.55 => ( ( coinductive_lfilter @ A @ P3 @ ( coinductive_LCons @ A @ X @ L ) )
% 2.93/1.55 = ( coinductive_lfilter @ A @ P3 @ L ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_125_lfilter__LCons__found,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( P @ X )
% 2.93/1.55 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.93/1.55 = ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_126_lset__intros_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Xs: coinductive_llist @ A,X5: A] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X5 @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_127_lset__intros_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_128_llist_Oset__intros_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
% 2.93/1.55 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_129_llist_Oset__intros_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_130_lset__cases,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ( ! [Xs6: coinductive_llist @ A] :
% 2.93/1.55 ( Xs
% 2.93/1.55 != ( coinductive_LCons @ A @ X @ Xs6 ) )
% 2.93/1.55 => ~ ! [X6: A,Xs6: coinductive_llist @ A] :
% 2.93/1.55 ( ( Xs
% 2.93/1.55 = ( coinductive_LCons @ A @ X6 @ Xs6 ) )
% 2.93/1.55 => ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs6 ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_131_lset__induct,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
% 2.93/1.55 => ( ! [X6: A,Xs2: coinductive_llist @ A] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
% 2.93/1.55 => ( ( X != X6 )
% 2.93/1.55 => ( ( P @ Xs2 )
% 2.93/1.55 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) ) )
% 2.93/1.55 => ( P @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_132_lset__induct_H,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
% 2.93/1.55 => ( ! [X6: A,Xs2: coinductive_llist @ A] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
% 2.93/1.55 => ( ( P @ Xs2 )
% 2.93/1.55 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) )
% 2.93/1.55 => ( P @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_133_llist_Oset__cases,axiom,
% 2.93/1.55 ! [A: $tType,E: A,A2: coinductive_llist @ A] :
% 2.93/1.55 ( ( member @ A @ E @ ( coinductive_lset @ A @ A2 ) )
% 2.93/1.55 => ( ! [Z2: coinductive_llist @ A] :
% 2.93/1.55 ( A2
% 2.93/1.55 != ( coinductive_LCons @ A @ E @ Z2 ) )
% 2.93/1.55 => ~ ! [Z1: A,Z2: coinductive_llist @ A] :
% 2.93/1.55 ( ( A2
% 2.93/1.55 = ( coinductive_LCons @ A @ Z1 @ Z2 ) )
% 2.93/1.55 => ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z2 ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_134_llist_Oset__induct,axiom,
% 2.93/1.55 ! [A: $tType,X: A,A2: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) )
% 2.93/1.55 => ( ! [Z1: A,Z2: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) )
% 2.93/1.55 => ( ! [Z1: A,Z2: coinductive_llist @ A,Xa3: A] :
% 2.93/1.55 ( ( member @ A @ Xa3 @ ( coinductive_lset @ A @ Z2 ) )
% 2.93/1.55 => ( ( P @ Xa3 @ Z2 )
% 2.93/1.55 => ( P @ Xa3 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) ) ) )
% 2.93/1.55 => ( P @ X @ A2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_135_llist_Odisc_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 2.93/1.55 ~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_136_llist_OdiscI_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
% 2.93/1.55 ( ( Llist
% 2.93/1.55 = ( coinductive_LCons @ A @ X21 @ X22 ) )
% 2.93/1.55 => ~ ( coinductive_lnull @ A @ Llist ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_137_not__lnull__conv,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ~ ( coinductive_lnull @ A @ Xs ) )
% 2.93/1.55 = ( ? [X3: A,Xs5: coinductive_llist @ A] :
% 2.93/1.55 ( Xs
% 2.93/1.55 = ( coinductive_LCons @ A @ X3 @ Xs5 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_138_llist_Odisc_I1_J,axiom,
% 2.93/1.55 ! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_139_llist_OdiscI_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,Llist: coinductive_llist @ A] :
% 2.93/1.55 ( ( Llist
% 2.93/1.55 = ( coinductive_LNil @ A ) )
% 2.93/1.55 => ( coinductive_lnull @ A @ Llist ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_140_llist_Ocollapse_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,Llist: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lnull @ A @ Llist )
% 2.93/1.55 => ( Llist
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_141_lnull__def,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( coinductive_lnull @ A )
% 2.93/1.55 = ( ^ [Llist2: coinductive_llist @ A] :
% 2.93/1.55 ( Llist2
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_142_lfinite__lfilterI,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 => ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_143_lappend_Odisc_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 | ~ ( coinductive_lnull @ A @ Ys ) )
% 2.93/1.55 => ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_144_lappend_Odisc_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_lnull @ A @ Ys )
% 2.93/1.55 => ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_145_lappend__lnull1,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.93/1.55 = Ys ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_146_lappend__lnull2,axiom,
% 2.93/1.55 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lnull @ A @ Ys )
% 2.93/1.55 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.93/1.55 = Xs ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_147_lnull__imp__lfinite,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_148_ltakeWhile__all,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.93/1.55 ( ! [X2: A] :
% 2.93/1.55 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ( P @ X2 ) )
% 2.93/1.55 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 2.93/1.55 = Xs ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_149_ltakeWhile__cong,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 2.93/1.55 ( ( Xs = Ys )
% 2.93/1.55 => ( ! [X2: A] :
% 2.93/1.55 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) )
% 2.93/1.55 => ( ( P @ X2 )
% 2.93/1.55 = ( Q @ X2 ) ) )
% 2.93/1.55 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 2.93/1.55 = ( coindu501562517eWhile @ A @ Q @ Ys ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_150_lset__ltakeWhileD,axiom,
% 2.93/1.55 ! [A: $tType,X: A,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
% 2.93/1.55 => ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 & ( P @ X ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_151_lsorted__lfilterI,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [Xs: coinductive_llist @ A,P: A > $o] :
% 2.93/1.55 ( ( coindu63249387sorted @ A @ Xs )
% 2.93/1.55 => ( coindu63249387sorted @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_152_ltakeWhile__lappend2,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( collect @ A @ P ) )
% 2.93/1.55 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( coinductive_lappend @ A @ Xs @ ( coindu501562517eWhile @ A @ P @ Ys ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_153_unfold__llist_Odisc_I2_J,axiom,
% 2.93/1.55 ! [B: $tType,A: $tType,P3: A > $o,A2: A,G212: A > B,G223: A > A] :
% 2.93/1.55 ( ~ ( P3 @ A2 )
% 2.93/1.55 => ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_154_unfold__llist_Odisc_I1_J,axiom,
% 2.93/1.55 ! [B: $tType,A: $tType,P3: A > $o,A2: A,G212: A > B,G223: A > A] :
% 2.93/1.55 ( ( P3 @ A2 )
% 2.93/1.55 => ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_155_llist_Ocorec__disc_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,C: $tType,P3: C > $o,A2: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 2.93/1.55 ( ~ ( P3 @ A2 )
% 2.93/1.55 => ~ ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_156_llist_Ocorec__disc_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,C: $tType,P3: C > $o,A2: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 2.93/1.55 ( ( P3 @ A2 )
% 2.93/1.55 => ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_157_lfilter__eq__LConsD,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( coinductive_lfilter @ A @ P @ Ys )
% 2.93/1.55 = ( coinductive_LCons @ A @ X @ Xs ) )
% 2.93/1.55 => ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 2.93/1.55 ( ( Ys
% 2.93/1.55 = ( coinductive_lappend @ A @ Us @ ( coinductive_LCons @ A @ X @ Vs ) ) )
% 2.93/1.55 & ( coinductive_lfinite @ A @ Us )
% 2.93/1.55 & ! [X7: A] :
% 2.93/1.55 ( ( member @ A @ X7 @ ( coinductive_lset @ A @ Us ) )
% 2.93/1.55 => ~ ( P @ X7 ) )
% 2.93/1.55 & ( P @ X )
% 2.93/1.55 & ( Xs
% 2.93/1.55 = ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_158_lsorted__lfilter__same,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [C2: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( coindu63249387sorted @ A
% 2.93/1.55 @ ( coinductive_lfilter @ A
% 2.93/1.55 @ ^ [X3: A] : ( X3 = C2 )
% 2.93/1.55 @ Xs ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_159_lset__lmember,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 = ( coinductive_lmember @ A @ X @ Xs ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_160_lfilter__eq__lappend__lfiniteD,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 2.93/1.55 = ( coinductive_lappend @ A @ Ys @ Zs ) )
% 2.93/1.55 => ( ( coinductive_lfinite @ A @ Ys )
% 2.93/1.55 => ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 2.93/1.55 ( ( Xs
% 2.93/1.55 = ( coinductive_lappend @ A @ Us @ Vs ) )
% 2.93/1.55 & ( coinductive_lfinite @ A @ Us )
% 2.93/1.55 & ( Ys
% 2.93/1.55 = ( coinductive_lfilter @ A @ P @ Us ) )
% 2.93/1.55 & ( Zs
% 2.93/1.55 = ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_161_in__lset__lappend__iff,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 2.93/1.55 = ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 | ( ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 & ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_162_lappend_Octr_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_lnull @ A @ Ys )
% 2.93/1.55 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_163_ltakeWhile__lappend1,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 => ( ~ ( P @ X )
% 2.93/1.55 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_164_lfinite__ltakeWhile,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 2.93/1.55 = ( ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 | ? [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 & ~ ( P @ X3 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_165_llast__LCons,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 2.93/1.55 ( ( ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.93/1.55 = X ) )
% 2.93/1.55 & ( ~ ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.93/1.55 = ( coinductive_llast @ A @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_166_LCons__LCons,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [X: A,Y: A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.55 => ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) )
% 2.93/1.55 => ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_167_ldropWhile_Osimps,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( coindu218763757pWhile @ A )
% 2.93/1.55 = ( ^ [P4: A > $o,Xs3: coinductive_llist @ A] :
% 2.93/1.55 ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_LNil @ A )
% 2.93/1.55 @ ^ [X3: A,Xs5: coinductive_llist @ A] : ( if @ ( coinductive_llist @ A ) @ ( P4 @ X3 ) @ ( coindu218763757pWhile @ A @ P4 @ Xs5 ) @ Xs3 )
% 2.93/1.55 @ Xs3 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_168_order__refl,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_169_reachable__viap_Ocases,axiom,
% 2.93/1.55 ! [Node: $tType,Graph: Node > Node > $o,Ns: set @ Node,N: Node,A2: Node] :
% 2.93/1.55 ( ( koenig1757754772e_viap @ Node @ Graph @ Ns @ N @ A2 )
% 2.93/1.55 => ~ ! [Xs2: coinductive_llist @ Node] :
% 2.93/1.55 ( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ N @ Xs2 ) )
% 2.93/1.55 => ( ( member @ Node @ A2 @ ( coinductive_lset @ Node @ Xs2 ) )
% 2.93/1.55 => ~ ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_170_reachable__viap_Osimps,axiom,
% 2.93/1.55 ! [Node: $tType] :
% 2.93/1.55 ( ( koenig1757754772e_viap @ Node )
% 2.93/1.55 = ( ^ [Graph2: Node > Node > $o,Ns2: set @ Node,N2: Node,A3: Node] :
% 2.93/1.55 ? [Xs3: coinductive_llist @ Node,N3: Node] :
% 2.93/1.55 ( ( A3 = N3 )
% 2.93/1.55 & ( koenig2031690877pathsp @ Node @ Graph2 @ ( coinductive_LCons @ Node @ N2 @ Xs3 ) )
% 2.93/1.55 & ( member @ Node @ N3 @ ( coinductive_lset @ Node @ Xs3 ) )
% 2.93/1.55 & ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs3 ) @ Ns2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_171_reachable__viap_Ointros,axiom,
% 2.93/1.55 ! [Node: $tType,Graph: Node > Node > $o,N: Node,Xs: coinductive_llist @ Node,N4: Node,Ns: set @ Node] :
% 2.93/1.55 ( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ N @ Xs ) )
% 2.93/1.55 => ( ( member @ Node @ N4 @ ( coinductive_lset @ Node @ Xs ) )
% 2.93/1.55 => ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs ) @ Ns )
% 2.93/1.55 => ( koenig1757754772e_viap @ Node @ Graph @ Ns @ N @ N4 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_172_reachable__viap_Oinducts,axiom,
% 2.93/1.55 ! [Node: $tType,Graph: Node > Node > $o,Ns: set @ Node,N: Node,X: Node,P: Node > $o] :
% 2.93/1.55 ( ( koenig1757754772e_viap @ Node @ Graph @ Ns @ N @ X )
% 2.93/1.55 => ( ! [Xs2: coinductive_llist @ Node,N5: Node] :
% 2.93/1.55 ( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ N @ Xs2 ) )
% 2.93/1.55 => ( ( member @ Node @ N5 @ ( coinductive_lset @ Node @ Xs2 ) )
% 2.93/1.55 => ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns )
% 2.93/1.55 => ( P @ N5 ) ) ) )
% 2.93/1.55 => ( P @ X ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_173_reachable__via_Ocases,axiom,
% 2.93/1.55 ! [Node: $tType,A2: Node,Graph: Node > Node > $o,Ns: set @ Node,N: Node] :
% 2.93/1.55 ( ( member @ Node @ A2 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N ) )
% 2.93/1.55 => ~ ! [Xs2: coinductive_llist @ Node] :
% 2.93/1.55 ( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N @ Xs2 ) @ ( koenig916195507_paths @ Node @ Graph ) )
% 2.93/1.55 => ( ( member @ Node @ A2 @ ( coinductive_lset @ Node @ Xs2 ) )
% 2.93/1.55 => ~ ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_174_reachable__viap__reachable__via__eq,axiom,
% 2.93/1.55 ! [Node: $tType] :
% 2.93/1.55 ( ( koenig1757754772e_viap @ Node )
% 2.93/1.55 = ( ^ [Graph2: Node > Node > $o,Ns2: set @ Node,N2: Node,X3: Node] : ( member @ Node @ X3 @ ( koenig317145564le_via @ Node @ Graph2 @ Ns2 @ N2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_175_reachable__via__def,axiom,
% 2.93/1.55 ! [Node: $tType] :
% 2.93/1.55 ( ( koenig317145564le_via @ Node )
% 2.93/1.55 = ( ^ [Graph2: Node > Node > $o,Ns2: set @ Node,N2: Node] : ( collect @ Node @ ( koenig1757754772e_viap @ Node @ Graph2 @ Ns2 @ N2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_176_le__funD,axiom,
% 2.93/1.55 ! [B: $tType,A: $tType] :
% 2.93/1.55 ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.55 => ! [F: A > B,G: A > B,X: A] :
% 2.93/1.55 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.93/1.55 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_177_le__funE,axiom,
% 2.93/1.55 ! [B: $tType,A: $tType] :
% 2.93/1.55 ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.55 => ! [F: A > B,G: A > B,X: A] :
% 2.93/1.55 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.93/1.55 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_178_le__funI,axiom,
% 2.93/1.55 ! [B: $tType,A: $tType] :
% 2.93/1.55 ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.55 => ! [F: A > B,G: A > B] :
% 2.93/1.55 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 2.93/1.55 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_179_le__fun__def,axiom,
% 2.93/1.55 ! [B: $tType,A: $tType] :
% 2.93/1.55 ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.55 => ( ( ord_less_eq @ ( A > B ) )
% 2.93/1.55 = ( ^ [F3: A > B,G2: A > B] :
% 2.93/1.55 ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_180_order__subst1,axiom,
% 2.93/1.55 ! [A: $tType,B: $tType] :
% 2.93/1.55 ( ( ( order @ B @ ( type2 @ B ) )
% 2.93/1.55 & ( order @ A @ ( type2 @ A ) ) )
% 2.93/1.55 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 2.93/1.55 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.93/1.55 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.93/1.55 => ( ! [X2: B,Y2: B] :
% 2.93/1.55 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 2.93/1.55 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.93/1.55 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_181_order__subst2,axiom,
% 2.93/1.55 ! [A: $tType,C: $tType] :
% 2.93/1.55 ( ( ( order @ C @ ( type2 @ C ) )
% 2.93/1.55 & ( order @ A @ ( type2 @ A ) ) )
% 2.93/1.55 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 2.93/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.55 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 2.93/1.55 => ( ! [X2: A,Y2: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.93/1.55 => ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.93/1.55 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_182_ord__eq__le__subst,axiom,
% 2.93/1.55 ! [A: $tType,B: $tType] :
% 2.93/1.55 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.55 & ( ord @ A @ ( type2 @ A ) ) )
% 2.93/1.55 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 2.93/1.55 ( ( A2
% 2.93/1.55 = ( F @ B2 ) )
% 2.93/1.55 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.93/1.55 => ( ! [X2: B,Y2: B] :
% 2.93/1.55 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 2.93/1.55 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.93/1.55 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_183_ord__le__eq__subst,axiom,
% 2.93/1.55 ! [A: $tType,B: $tType] :
% 2.93/1.55 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.93/1.55 & ( ord @ A @ ( type2 @ A ) ) )
% 2.93/1.55 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 2.93/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.55 => ( ( ( F @ B2 )
% 2.93/1.55 = C2 )
% 2.93/1.55 => ( ! [X2: A,Y2: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.93/1.55 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.93/1.55 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_184_eq__iff,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.55 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 2.93/1.55 = ( ^ [X3: A,Y3: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.93/1.55 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_185_antisym,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [X: A,Y: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.55 => ( ( ord_less_eq @ A @ Y @ X )
% 2.93/1.55 => ( X = Y ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_186_linear,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [X: A,Y: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.55 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_187_eq__refl,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [X: A,Y: A] :
% 2.93/1.55 ( ( X = Y )
% 2.93/1.55 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_188_le__cases,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [X: A,Y: A] :
% 2.93/1.55 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.93/1.55 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_189_order_Otrans,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [A2: A,B2: A,C2: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.55 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.93/1.55 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_190_le__cases3,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [X: A,Y: A,Z3: A] :
% 2.93/1.55 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.55 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 2.93/1.55 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.93/1.55 => ~ ( ord_less_eq @ A @ X @ Z3 ) )
% 2.93/1.55 => ( ( ( ord_less_eq @ A @ X @ Z3 )
% 2.93/1.55 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 2.93/1.55 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 2.93/1.55 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.93/1.55 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.93/1.55 => ~ ( ord_less_eq @ A @ Z3 @ X ) )
% 2.93/1.55 => ~ ( ( ord_less_eq @ A @ Z3 @ X )
% 2.93/1.55 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_191_antisym__conv,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [Y: A,X: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ Y @ X )
% 2.93/1.55 => ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.55 = ( X = Y ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_192_ord__eq__le__trans,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [A2: A,B2: A,C2: A] :
% 2.93/1.55 ( ( A2 = B2 )
% 2.93/1.55 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.93/1.55 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_193_ord__le__eq__trans,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [A2: A,B2: A,C2: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.55 => ( ( B2 = C2 )
% 2.93/1.55 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_194_order__class_Oorder_Oantisym,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [A2: A,B2: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.55 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.93/1.55 => ( A2 = B2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_195_order__trans,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( preorder @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [X: A,Y: A,Z3: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 2.93/1.55 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.93/1.55 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_196_dual__order_Orefl,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_197_linorder__wlog,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( linorder @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.93/1.55 ( ! [A5: A,B3: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ A5 @ B3 )
% 2.93/1.55 => ( P @ A5 @ B3 ) )
% 2.93/1.55 => ( ! [A5: A,B3: A] :
% 2.93/1.55 ( ( P @ B3 @ A5 )
% 2.93/1.55 => ( P @ A5 @ B3 ) )
% 2.93/1.55 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_198_dual__order_Otrans,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [B2: A,A2: A,C2: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.93/1.55 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 2.93/1.55 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_199_dual__order_Oantisym,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.93/1.55 => ! [B2: A,A2: A] :
% 2.93/1.55 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.93/1.55 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.93/1.55 => ( A2 = B2 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_200_reachable__via_Oinducts,axiom,
% 2.93/1.55 ! [Node: $tType,X: Node,Graph: Node > Node > $o,Ns: set @ Node,N: Node,P: Node > $o] :
% 2.93/1.55 ( ( member @ Node @ X @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N ) )
% 2.93/1.55 => ( ! [Xs2: coinductive_llist @ Node,N5: Node] :
% 2.93/1.55 ( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N @ Xs2 ) @ ( koenig916195507_paths @ Node @ Graph ) )
% 2.93/1.55 => ( ( member @ Node @ N5 @ ( coinductive_lset @ Node @ Xs2 ) )
% 2.93/1.55 => ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns )
% 2.93/1.55 => ( P @ N5 ) ) ) )
% 2.93/1.55 => ( P @ X ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_201_reachable__via_Ointros,axiom,
% 2.93/1.55 ! [Node: $tType,N: Node,Xs: coinductive_llist @ Node,Graph: Node > Node > $o,N4: Node,Ns: set @ Node] :
% 2.93/1.55 ( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N @ Xs ) @ ( koenig916195507_paths @ Node @ Graph ) )
% 2.93/1.55 => ( ( member @ Node @ N4 @ ( coinductive_lset @ Node @ Xs ) )
% 2.93/1.55 => ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs ) @ Ns )
% 2.93/1.55 => ( member @ Node @ N4 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_202_reachable__via_Osimps,axiom,
% 2.93/1.55 ! [Node: $tType,A2: Node,Graph: Node > Node > $o,Ns: set @ Node,N: Node] :
% 2.93/1.55 ( ( member @ Node @ A2 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N ) )
% 2.93/1.55 = ( ? [Xs3: coinductive_llist @ Node,N3: Node] :
% 2.93/1.55 ( ( A2 = N3 )
% 2.93/1.55 & ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N @ Xs3 ) @ ( koenig916195507_paths @ Node @ Graph ) )
% 2.93/1.55 & ( member @ Node @ N3 @ ( coinductive_lset @ Node @ Xs3 ) )
% 2.93/1.55 & ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs3 ) @ Ns ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_203_subset__antisym,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.93/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 2.93/1.55 => ( A4 = B4 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_204_subsetI,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.93/1.55 ( ! [X2: A] :
% 2.93/1.55 ( ( member @ A @ X2 @ A4 )
% 2.93/1.55 => ( member @ A @ X2 @ B4 ) )
% 2.93/1.55 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_205_ldropWhile__lappend,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( ? [X7: A] :
% 2.93/1.55 ( ( member @ A @ X7 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 & ~ ( P @ X7 ) )
% 2.93/1.55 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( coinductive_lappend @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) @ Ys ) ) )
% 2.93/1.55 & ( ~ ? [X2: A] :
% 2.93/1.55 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 & ~ ( P @ X2 ) )
% 2.93/1.55 => ( ( ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( coindu218763757pWhile @ A @ P @ Ys ) ) )
% 2.93/1.55 & ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_206_set__mp,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.93/1.55 => ( ( member @ A @ X @ A4 )
% 2.93/1.55 => ( member @ A @ X @ B4 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_207_in__mono,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.93/1.55 => ( ( member @ A @ X @ A4 )
% 2.93/1.55 => ( member @ A @ X @ B4 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_208_subsetD,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.93/1.55 => ( ( member @ A @ C2 @ A4 )
% 2.93/1.55 => ( member @ A @ C2 @ B4 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_209_subsetCE,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.93/1.55 => ( ( member @ A @ C2 @ A4 )
% 2.93/1.55 => ( member @ A @ C2 @ B4 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_210_equalityE,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.93/1.55 ( ( A4 = B4 )
% 2.93/1.55 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.93/1.55 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_211_subset__eq,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.55 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.93/1.55 ! [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ A6 )
% 2.93/1.55 => ( member @ A @ X3 @ B5 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_212_equalityD1,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.93/1.55 ( ( A4 = B4 )
% 2.93/1.55 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_213_equalityD2,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.93/1.55 ( ( A4 = B4 )
% 2.93/1.55 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_214_set__rev__mp,axiom,
% 2.93/1.55 ! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
% 2.93/1.55 ( ( member @ A @ X @ A4 )
% 2.93/1.55 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.93/1.55 => ( member @ A @ X @ B4 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_215_subset__iff,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.55 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.93/1.55 ! [T: A] :
% 2.93/1.55 ( ( member @ A @ T @ A6 )
% 2.93/1.55 => ( member @ A @ T @ B5 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_216_rev__subsetD,axiom,
% 2.93/1.55 ! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
% 2.93/1.55 ( ( member @ A @ C2 @ A4 )
% 2.93/1.55 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.93/1.55 => ( member @ A @ C2 @ B4 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_217_subset__refl,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_218_Collect__mono,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.93/1.55 ( ! [X2: A] :
% 2.93/1.55 ( ( P @ X2 )
% 2.93/1.55 => ( Q @ X2 ) )
% 2.93/1.55 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_219_subset__trans,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.93/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 2.93/1.55 => ( ord_less_eq @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_220_set__eq__subset,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ^ [Y5: set @ A,Z: set @ A] : ( Y5 = Z ) )
% 2.93/1.55 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
% 2.93/1.55 & ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_221_contra__subsetD,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.93/1.55 => ( ~ ( member @ A @ C2 @ B4 )
% 2.93/1.55 => ~ ( member @ A @ C2 @ A4 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_222_Collect__mono__iff,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.93/1.55 = ( ! [X3: A] :
% 2.93/1.55 ( ( P @ X3 )
% 2.93/1.55 => ( Q @ X3 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_223_ltakeWhile__lappend,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( ? [X7: A] :
% 2.93/1.55 ( ( member @ A @ X7 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 & ~ ( P @ X7 ) )
% 2.93/1.55 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( coindu501562517eWhile @ A @ P @ Xs ) ) )
% 2.93/1.55 & ( ~ ? [X2: A] :
% 2.93/1.55 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 2.93/1.55 & ~ ( P @ X2 ) )
% 2.93/1.55 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( coinductive_lappend @ A @ Xs @ ( coindu501562517eWhile @ A @ P @ Ys ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_224_less__eq__set__def,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.55 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.93/1.55 ( ord_less_eq @ ( A > $o )
% 2.93/1.55 @ ^ [X3: A] : ( member @ A @ X3 @ A6 )
% 2.93/1.55 @ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_225_pred__subset__eq,axiom,
% 2.93/1.55 ! [A: $tType,R: set @ A,S: set @ A] :
% 2.93/1.55 ( ( ord_less_eq @ ( A > $o )
% 2.93/1.55 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 2.93/1.55 @ ^ [X3: A] : ( member @ A @ X3 @ S ) )
% 2.93/1.55 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_226_conj__subset__def,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,P: A > $o,Q: A > $o] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4
% 2.93/1.55 @ ( collect @ A
% 2.93/1.55 @ ^ [X3: A] :
% 2.93/1.55 ( ( P @ X3 )
% 2.93/1.55 & ( Q @ X3 ) ) ) )
% 2.93/1.55 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P ) )
% 2.93/1.55 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_227_predicate1I,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.93/1.55 ( ! [X2: A] :
% 2.93/1.55 ( ( P @ X2 )
% 2.93/1.55 => ( Q @ X2 ) )
% 2.93/1.55 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_228_rev__predicate1D,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,X: A,Q: A > $o] :
% 2.93/1.55 ( ( P @ X )
% 2.93/1.55 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 2.93/1.55 => ( Q @ X ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_229_predicate1D,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Q: A > $o,X: A] :
% 2.93/1.55 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 2.93/1.55 => ( ( P @ X )
% 2.93/1.55 => ( Q @ X ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_230_subset__Collect__iff,axiom,
% 2.93/1.55 ! [A: $tType,B4: set @ A,A4: set @ A,P: A > $o] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 2.93/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B4
% 2.93/1.55 @ ( collect @ A
% 2.93/1.55 @ ^ [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ A4 )
% 2.93/1.55 & ( P @ X3 ) ) ) )
% 2.93/1.55 = ( ! [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ B4 )
% 2.93/1.55 => ( P @ X3 ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_231_subset__CollectI,axiom,
% 2.93/1.55 ! [A: $tType,B4: set @ A,A4: set @ A,Q: A > $o,P: A > $o] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 2.93/1.55 => ( ! [X2: A] :
% 2.93/1.55 ( ( member @ A @ X2 @ B4 )
% 2.93/1.55 => ( ( Q @ X2 )
% 2.93/1.55 => ( P @ X2 ) ) )
% 2.93/1.55 => ( ord_less_eq @ ( set @ A )
% 2.93/1.55 @ ( collect @ A
% 2.93/1.55 @ ^ [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ B4 )
% 2.93/1.55 & ( Q @ X3 ) ) )
% 2.93/1.55 @ ( collect @ A
% 2.93/1.55 @ ^ [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ A4 )
% 2.93/1.55 & ( P @ X3 ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_232_prop__restrict,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Z4: set @ A,X4: set @ A,P: A > $o] :
% 2.93/1.55 ( ( member @ A @ X @ Z4 )
% 2.93/1.55 => ( ( ord_less_eq @ ( set @ A ) @ Z4
% 2.93/1.55 @ ( collect @ A
% 2.93/1.55 @ ^ [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ X4 )
% 2.93/1.55 & ( P @ X3 ) ) ) )
% 2.93/1.55 => ( P @ X ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_233_Collect__restrict,axiom,
% 2.93/1.55 ! [A: $tType,X4: set @ A,P: A > $o] :
% 2.93/1.55 ( ord_less_eq @ ( set @ A )
% 2.93/1.55 @ ( collect @ A
% 2.93/1.55 @ ^ [X3: A] :
% 2.93/1.55 ( ( member @ A @ X3 @ X4 )
% 2.93/1.55 & ( P @ X3 ) ) )
% 2.93/1.55 @ X4 ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_234_llast__lappend,axiom,
% 2.93/1.55 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( coinductive_lnull @ A @ Ys )
% 2.93/1.55 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( coinductive_llast @ A @ Xs ) ) )
% 2.93/1.55 & ( ~ ( coinductive_lnull @ A @ Ys )
% 2.93/1.55 => ( ( ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( coinductive_llast @ A @ Ys ) ) )
% 2.93/1.55 & ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( undefined @ A ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_235_Powp__mono,axiom,
% 2.93/1.55 ! [A: $tType,A4: A > $o,B4: A > $o] :
% 2.93/1.55 ( ( ord_less_eq @ ( A > $o ) @ A4 @ B4 )
% 2.93/1.55 => ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A4 ) @ ( powp @ A @ B4 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_236_llast__linfinite,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_llast @ A @ Xs )
% 2.93/1.55 = ( undefined @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_237_llast__LNil,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
% 2.93/1.55 = ( undefined @ A ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_238_lappend_Octr_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 | ~ ( coinductive_lnull @ A @ Ys ) )
% 2.93/1.55 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.93/1.55 = ( coinductive_LCons @ A
% 2.93/1.55 @ ( coindu1381640503_llist @ A @ A @ ( coinductive_lhd @ A @ Ys )
% 2.93/1.55 @ ^ [X3: A,Xs5: coinductive_llist @ A] : X3
% 2.93/1.55 @ Xs )
% 2.93/1.55 @ ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A
% 2.93/1.55 @ ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( undefined @ ( coinductive_llist @ A ) )
% 2.93/1.55 @ ^ [Uu2: A,Uv2: coinductive_llist @ A] : Uv2
% 2.93/1.55 @ Ys )
% 2.93/1.55 @ ^ [X3: A,Xs5: coinductive_llist @ A] : ( coinductive_lappend @ A @ Xs5 @ Ys )
% 2.93/1.55 @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_239_lset__lappend__lfinite,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lfinite @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_240_Un__subset__iff,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 )
% 2.93/1.55 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 2.93/1.55 & ( ord_less_eq @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_241_lhd__lappend,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.93/1.55 ( ( ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( coinductive_lhd @ A @ Ys ) ) )
% 2.93/1.55 & ( ~ ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 => ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.93/1.55 = ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_242_lnull__ltakeWhile,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 2.93/1.55 = ( ~ ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 => ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_243_ltakeWhile_Odisc__iff_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 2.93/1.55 = ( ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 | ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_244_ltakeWhile_Odisc__iff_I2_J,axiom,
% 2.93/1.55 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.93/1.55 ( ( ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
% 2.93/1.55 = ( ~ ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 & ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_245_subset__Un__eq,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.55 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.93/1.55 ( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
% 2.93/1.55 = B5 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_246_Un__absorb2,axiom,
% 2.93/1.55 ! [A: $tType,B4: set @ A,A4: set @ A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 2.93/1.55 => ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
% 2.93/1.55 = A4 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_247_Un__absorb1,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.93/1.55 => ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
% 2.93/1.55 = B4 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_248_Un__upper2,axiom,
% 2.93/1.55 ! [A: $tType,B4: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_249_Un__upper1,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_250_Un__least,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,C3: set @ A,B4: set @ A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 2.93/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 2.93/1.55 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_251_Un__mono,axiom,
% 2.93/1.55 ! [A: $tType,A4: set @ A,C3: set @ A,B4: set @ A,D: set @ A] :
% 2.93/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
% 2.93/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ D )
% 2.93/1.55 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_252_lhd__def,axiom,
% 2.93/1.55 ! [A: $tType] :
% 2.93/1.55 ( ( coinductive_lhd @ A )
% 2.93/1.55 = ( coindu1381640503_llist @ A @ A @ ( undefined @ A )
% 2.93/1.55 @ ^ [X213: A,X223: coinductive_llist @ A] : X213 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(fact_253_ltakeWhile_Octr_I1_J,axiom,
% 2.93/1.55 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.93/1.55 ( ( ( coinductive_lnull @ A @ Xs )
% 2.93/1.55 | ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
% 2.93/1.55 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 2.93/1.55 = ( coinductive_LNil @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.93/1.55 ! [A7: $tType,A8: $tType] :
% 2.93/1.55 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 2.93/1.55 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.93/1.55 ! [A7: $tType,A8: $tType] :
% 2.93/1.55 ( ( order @ A8 @ ( type2 @ A8 ) )
% 2.93/1.55 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(tcon_fun___Orderings_Oord,axiom,
% 2.93/1.55 ! [A7: $tType,A8: $tType] :
% 2.93/1.55 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 2.93/1.55 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
% 2.93/1.55 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 2.93/1.55 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
% 2.93/1.55 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
% 2.93/1.55 preorder @ $o @ ( type2 @ $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.93/1.55 linorder @ $o @ ( type2 @ $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
% 2.93/1.55 order @ $o @ ( type2 @ $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
% 2.93/1.55 ord @ $o @ ( type2 @ $o ) ).
% 2.93/1.55
% 2.93/1.55 thf(help_If_3_1_T,axiom,
% 2.93/1.55 ! [P: $o] :
% 2.93/1.55 ( ( P = $true )
% 2.93/1.55 | ( P = $false ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(help_If_2_1_T,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Y: A] :
% 2.93/1.55 ( ( if @ A @ $false @ X @ Y )
% 2.93/1.55 = Y ) ).
% 2.93/1.55
% 2.93/1.55 thf(help_If_1_1_T,axiom,
% 2.93/1.55 ! [A: $tType,X: A,Y: A] :
% 2.93/1.55 ( ( if @ A @ $true @ X @ Y )
% 2.93/1.55 = X ) ).
% 2.93/1.55
% 2.93/1.55 thf(conj_0,hypothesis,
% 2.93/1.55 ( ( coinductive_lappend @ a @ x @ ys )
% 2.93/1.55 = ( coinductive_LCons @ a @ xa @ ( coinductive_LNil @ a ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(conj_1,hypothesis,
% 2.93/1.55 ( x
% 2.93/1.55 = ( coinductive_LCons @ a @ x21 @ x22 ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(conj_2,conjecture,
% 2.93/1.55 ( ( x
% 2.93/1.55 = ( coinductive_LNil @ a ) )
% 2.93/1.55 | ? [X7: a] :
% 2.93/1.55 ( x
% 2.93/1.55 = ( coinductive_LCons @ a @ X7 @ ( coinductive_LNil @ a ) ) )
% 2.93/1.55 | ? [X7: a,Y4: a,Xs4: coinductive_llist @ a] :
% 2.93/1.55 ( ( x
% 2.93/1.55 = ( coinductive_LCons @ a @ X7 @ ( coinductive_LCons @ a @ Y4 @ Xs4 ) ) )
% 2.93/1.55 & ( graph @ X7 @ Y4 )
% 2.93/1.55 & ( ( member @ ( coinductive_llist @ a ) @ ( coinductive_lappend @ a @ ( coinductive_LCons @ a @ Y4 @ Xs4 ) @ ys ) @ ( koenig916195507_paths @ a @ graph ) )
% 2.93/1.55 | ( member @ ( coinductive_llist @ a ) @ ( coinductive_LCons @ a @ Y4 @ Xs4 ) @ ( koenig916195507_paths @ a @ graph ) ) ) ) ) ).
% 2.93/1.55 ------------------------
% 2.93/1.58 % (22846)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22687 for (2999ds/183Mi)
% 2.93/1.58 % (22852)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22687 for (2999ds/18Mi)
% 2.93/1.58 % (22847)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_22687 for (2999ds/4Mi)
% 2.93/1.58 % (22849)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22687 for (2999ds/2Mi)
% 2.93/1.58 % (22848)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_22687 for (2999ds/27Mi)
% 2.93/1.58 % (22849)Instruction limit reached!
% 2.93/1.58 % (22849)------------------------------
% 2.93/1.58 % (22849)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.58 % (22849)Termination reason: Unknown
% 2.93/1.58 % (22849)Termination phase: shuffling
% 2.93/1.58
% 2.93/1.58 % (22849)Memory used [KB]: 1407
% 2.93/1.58 % (22849)Time elapsed: 0.003 s
% 2.93/1.58 % (22849)Instructions burned: 3 (million)
% 2.93/1.58 % (22849)------------------------------
% 2.93/1.58 % (22849)------------------------------
% 2.93/1.58 % (22847)Instruction limit reached!
% 2.93/1.58 % (22847)------------------------------
% 2.93/1.58 % (22847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.58 % (22847)Termination reason: Unknown
% 2.93/1.58 % (22847)Termination phase: shuffling
% 2.93/1.58
% 2.93/1.58 % (22847)Memory used [KB]: 1407
% 2.93/1.58 % (22847)Time elapsed: 0.004 s
% 2.93/1.58 % (22847)Instructions burned: 5 (million)
% 2.93/1.58 % (22847)------------------------------
% 2.93/1.58 % (22847)------------------------------
% 2.93/1.58 % (22852)Instruction limit reached!
% 2.93/1.58 % (22852)------------------------------
% 2.93/1.58 % (22852)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.58 % (22852)Termination reason: Unknown
% 2.93/1.58 % (22852)Termination phase: shuffling
% 2.93/1.58
% 2.93/1.58 % (22852)Memory used [KB]: 1663
% 2.93/1.58 % (22852)Time elapsed: 0.008 s
% 2.93/1.58 % (22852)Instructions burned: 18 (million)
% 2.93/1.58 % (22852)------------------------------
% 2.93/1.58 % (22852)------------------------------
% 2.93/1.59 % (22851)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_22687 for (2999ds/275Mi)
% 2.93/1.59 % (22848)Instruction limit reached!
% 2.93/1.59 % (22848)------------------------------
% 2.93/1.59 % (22848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (22848)Termination reason: Unknown
% 2.93/1.59 % (22848)Termination phase: shuffling
% 2.93/1.59
% 2.93/1.59 % (22848)Memory used [KB]: 1918
% 2.93/1.59 % (22848)Time elapsed: 0.013 s
% 2.93/1.59 % (22848)Instructions burned: 27 (million)
% 2.93/1.59 % (22848)------------------------------
% 2.93/1.59 % (22848)------------------------------
% 2.93/1.59 % (22850)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_22687 for (2999ds/2Mi)
% 2.93/1.59 % (22855)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22687 for (2999ds/3Mi)
% 2.93/1.59 % (22850)Instruction limit reached!
% 2.93/1.59 % (22850)------------------------------
% 2.93/1.59 % (22850)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (22850)Termination reason: Unknown
% 2.93/1.59 % (22850)Termination phase: shuffling
% 2.93/1.59
% 2.93/1.59 % (22850)Memory used [KB]: 1407
% 2.93/1.59 % (22850)Time elapsed: 0.004 s
% 2.93/1.59 % (22850)Instructions burned: 2 (million)
% 2.93/1.59 % (22850)------------------------------
% 2.93/1.59 % (22850)------------------------------
% 2.93/1.60 % (22855)Instruction limit reached!
% 2.93/1.60 % (22855)------------------------------
% 2.93/1.60 % (22855)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.60 % (22855)Termination reason: Unknown
% 2.93/1.60 % (22855)Termination phase: shuffling
% 2.93/1.60
% 2.93/1.60 % (22855)Memory used [KB]: 1407
% 2.93/1.60 % (22855)Time elapsed: 0.003 s
% 2.93/1.60 % (22855)Instructions burned: 3 (million)
% 2.93/1.60 % (22855)------------------------------
% 2.93/1.60 % (22855)------------------------------
% 3.00/1.60 % (22857)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_22687 for (2999ds/37Mi)
% 3.00/1.61 % (22860)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_22687 for (2999ds/3Mi)
% 3.00/1.61 % (22859)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_22687 for (2999ds/15Mi)
% 3.00/1.61 % (22860)Instruction limit reached!
% 3.00/1.61 % (22860)------------------------------
% 3.00/1.61 % (22860)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.61 % (22860)Termination reason: Unknown
% 3.00/1.61 % (22860)Termination phase: shuffling
% 3.00/1.61
% 3.00/1.61 % (22860)Memory used [KB]: 1407
% 3.00/1.61 % (22860)Time elapsed: 0.003 s
% 3.00/1.61 % (22860)Instructions burned: 3 (million)
% 3.00/1.61 % (22860)------------------------------
% 3.00/1.61 % (22860)------------------------------
% 3.00/1.61 % (22862)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22687 for (2999ds/7Mi)
% 3.00/1.61 % (22859)Instruction limit reached!
% 3.00/1.61 % (22859)------------------------------
% 3.00/1.61 % (22859)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.61 % (22859)Termination reason: Unknown
% 3.00/1.61 % (22859)Termination phase: shuffling
% 3.00/1.61
% 3.00/1.61 % (22859)Memory used [KB]: 1663
% 3.00/1.61 % (22859)Time elapsed: 0.009 s
% 3.00/1.61 % (22859)Instructions burned: 15 (million)
% 3.00/1.61 % (22859)------------------------------
% 3.00/1.61 % (22859)------------------------------
% 3.00/1.61 % (22862)Instruction limit reached!
% 3.00/1.61 % (22862)------------------------------
% 3.00/1.61 % (22862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.61 % (22862)Termination reason: Unknown
% 3.00/1.61 % (22862)Termination phase: shuffling
% 3.00/1.61
% 3.00/1.61 % (22862)Memory used [KB]: 1535
% 3.00/1.61 % (22862)Time elapsed: 0.004 s
% 3.00/1.61 % (22862)Instructions burned: 8 (million)
% 3.00/1.61 % (22862)------------------------------
% 3.00/1.61 % (22862)------------------------------
% 3.00/1.62 % (22861)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_22687 for (2999ds/1041Mi)
% 3.00/1.62 % (22857)Instruction limit reached!
% 3.00/1.62 % (22857)------------------------------
% 3.00/1.62 % (22857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.62 % (22857)Termination reason: Unknown
% 3.00/1.62 % (22857)Termination phase: shuffling
% 3.00/1.62
% 3.00/1.62 % (22857)Memory used [KB]: 2174
% 3.00/1.62 % (22857)Time elapsed: 0.020 s
% 3.00/1.62 % (22857)Instructions burned: 38 (million)
% 3.00/1.62 % (22857)------------------------------
% 3.00/1.62 % (22857)------------------------------
% 3.00/1.62 % (22865)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22687 for (2999ds/16Mi)
% 3.00/1.62 % (22869)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_22687 for (2999ds/3Mi)
% 3.00/1.63 % (22869)Instruction limit reached!
% 3.00/1.63 % (22869)------------------------------
% 3.00/1.63 % (22869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.63 % (22869)Termination reason: Unknown
% 3.00/1.63 % (22869)Termination phase: shuffling
% 3.00/1.63
% 3.00/1.63 % (22869)Memory used [KB]: 1407
% 3.00/1.63 % (22869)Time elapsed: 0.004 s
% 3.00/1.63 % (22869)Instructions burned: 4 (million)
% 3.00/1.63 % (22869)------------------------------
% 3.00/1.63 % (22869)------------------------------
% 3.00/1.63 % (22870)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_22687 for (2999ds/3Mi)
% 3.00/1.63 % (22870)Instruction limit reached!
% 3.00/1.63 % (22870)------------------------------
% 3.00/1.63 % (22870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.63 % (22870)Termination reason: Unknown
% 3.00/1.63 % (22870)Termination phase: shuffling
% 3.00/1.63
% 3.00/1.63 % (22870)Memory used [KB]: 1407
% 3.00/1.63 % (22870)Time elapsed: 0.003 s
% 3.00/1.63 % (22865)Instruction limit reached!
% 3.00/1.63 % (22865)------------------------------
% 3.00/1.63 % (22865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.63 % (22870)Instructions burned: 3 (million)
% 3.00/1.63 % (22865)Termination reason: Unknown
% 3.00/1.63 % (22870)------------------------------
% 3.00/1.63 % (22870)------------------------------
% 3.00/1.63 % (22865)Termination phase: shuffling
% 3.00/1.63
% 3.00/1.63 % (22865)Memory used [KB]: 1663
% 3.00/1.63 % (22865)Time elapsed: 0.009 s
% 3.00/1.63 % (22865)Instructions burned: 16 (million)
% 3.00/1.63 % (22865)------------------------------
% 3.00/1.63 % (22865)------------------------------
% 3.00/1.63 % (22872)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_22687 for (2999ds/7Mi)
% 3.00/1.64 % (22872)Instruction limit reached!
% 3.00/1.64 % (22872)------------------------------
% 3.00/1.64 % (22872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.64 % (22872)Termination reason: Unknown
% 3.00/1.64 % (22872)Termination phase: shuffling
% 3.00/1.64
% 3.00/1.64 % (22872)Memory used [KB]: 1535
% 3.00/1.64 % (22872)Time elapsed: 0.005 s
% 3.00/1.64 % (22872)Instructions burned: 7 (million)
% 3.00/1.64 % (22872)------------------------------
% 3.00/1.64 % (22872)------------------------------
% 3.00/1.64 % (22877)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_22687 for (2999ds/4Mi)
% 3.00/1.64 % (22877)Instruction limit reached!
% 3.00/1.64 % (22877)------------------------------
% 3.00/1.64 % (22877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.64 % (22877)Termination reason: Unknown
% 3.00/1.64 % (22877)Termination phase: shuffling
% 3.00/1.64
% 3.00/1.64 % (22877)Memory used [KB]: 1535
% 3.00/1.64 % (22877)Time elapsed: 0.004 s
% 3.00/1.64 % (22877)Instructions burned: 6 (million)
% 3.00/1.64 % (22877)------------------------------
% 3.00/1.64 % (22877)------------------------------
% 3.00/1.65 % (22878)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_22687 for (2999ds/18Mi)
% 3.00/1.65 % (22880)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_22687 for (2999ds/710Mi)
% 3.00/1.65 % (22881)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22687 for (2999ds/6Mi)
% 3.00/1.66 % (22878)Instruction limit reached!
% 3.00/1.66 % (22878)------------------------------
% 3.00/1.66 % (22878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.66 % (22878)Termination reason: Unknown
% 3.00/1.66 % (22878)Termination phase: shuffling
% 3.00/1.66
% 3.00/1.66 % (22878)Memory used [KB]: 1791
% 3.00/1.66 % (22878)Time elapsed: 0.011 s
% 3.00/1.66 % (22878)Instructions burned: 19 (million)
% 3.00/1.66 % (22878)------------------------------
% 3.00/1.66 % (22878)------------------------------
% 3.00/1.66 % (22881)Instruction limit reached!
% 3.00/1.66 % (22881)------------------------------
% 3.00/1.66 % (22881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.66 % (22881)Termination reason: Unknown
% 3.00/1.66 % (22881)Termination phase: shuffling
% 3.00/1.66
% 3.00/1.66 % (22881)Memory used [KB]: 1407
% 3.00/1.66 % (22881)Time elapsed: 0.004 s
% 3.00/1.66 % (22881)Instructions burned: 6 (million)
% 3.00/1.66 % (22881)------------------------------
% 3.00/1.66 % (22881)------------------------------
% 3.00/1.66 % (22876)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_22687 for (2999ds/3Mi)
% 3.00/1.66 % (22876)Instruction limit reached!
% 3.00/1.66 % (22876)------------------------------
% 3.00/1.66 % (22876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.66 % (22876)Termination reason: Unknown
% 3.00/1.66 % (22876)Termination phase: shuffling
% 3.00/1.66
% 3.00/1.66 % (22876)Memory used [KB]: 1407
% 3.00/1.66 % (22876)Time elapsed: 0.004 s
% 3.00/1.66 % (22876)Instructions burned: 4 (million)
% 3.00/1.66 % (22876)------------------------------
% 3.00/1.66 % (22876)------------------------------
% 3.00/1.66 % (22846)Instruction limit reached!
% 3.00/1.66 % (22846)------------------------------
% 3.00/1.66 % (22846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.66 % (22846)Termination reason: Unknown
% 3.00/1.66 % (22846)Termination phase: Saturation
% 3.00/1.66
% 3.00/1.66 % (22846)Memory used [KB]: 7291
% 3.00/1.66 % (22846)Time elapsed: 0.089 s
% 3.00/1.66 % (22846)Instructions burned: 184 (million)
% 3.00/1.66 % (22846)------------------------------
% 3.00/1.66 % (22846)------------------------------
% 3.00/1.67 % (22886)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_22687 for (2999ds/21Mi)
% 3.00/1.68 % (22886)Instruction limit reached!
% 3.00/1.68 % (22886)------------------------------
% 3.00/1.68 % (22886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.68 % (22886)Termination reason: Unknown
% 3.00/1.68 % (22886)Termination phase: shuffling
% 3.00/1.68
% 3.00/1.68 % (22886)Memory used [KB]: 1791
% 3.00/1.68 % (22886)Time elapsed: 0.010 s
% 3.00/1.68 % (22886)Instructions burned: 22 (million)
% 3.00/1.68 % (22886)------------------------------
% 3.00/1.68 % (22886)------------------------------
% 3.00/1.68 % (22885)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_22687 for (2999ds/902Mi)
% 3.00/1.68 % (22889)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_22687 for (2998ds/6Mi)
% 3.00/1.68 % (22888)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_22687 for (2999ds/5Mi)
% 3.00/1.68 % (22888)Instruction limit reached!
% 3.00/1.68 % (22888)------------------------------
% 3.00/1.68 % (22888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.68 % (22888)Termination reason: Unknown
% 3.00/1.68 % (22888)Termination phase: shuffling
% 3.00/1.68
% 3.00/1.68 % (22888)Memory used [KB]: 1407
% 3.00/1.68 % (22888)Time elapsed: 0.004 s
% 3.00/1.68 % (22888)Instructions burned: 6 (million)
% 3.00/1.68 % (22888)------------------------------
% 3.00/1.68 % (22888)------------------------------
% 3.00/1.68 % (22889)Instruction limit reached!
% 3.00/1.68 % (22889)------------------------------
% 3.00/1.68 % (22889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.68 % (22889)Termination reason: Unknown
% 3.00/1.68 % (22889)Termination phase: shuffling
% 3.00/1.68
% 3.00/1.68 % (22889)Memory used [KB]: 1535
% 3.00/1.68 % (22889)Time elapsed: 0.005 s
% 3.00/1.68 % (22889)Instructions burned: 7 (million)
% 3.00/1.68 % (22889)------------------------------
% 3.00/1.68 % (22889)------------------------------
% 3.00/1.69 % (22891)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_22687 for (2998ds/377Mi)
% 3.00/1.70 % (22892)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_22687 for (2998ds/779Mi)
% 3.00/1.70 % (22893)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_22687 for (2998ds/19Mi)
% 3.00/1.71 % (22893)Instruction limit reached!
% 3.00/1.71 % (22893)------------------------------
% 3.00/1.71 % (22893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.71 % (22893)Termination reason: Unknown
% 3.00/1.71 % (22893)Termination phase: shuffling
% 3.00/1.71
% 3.00/1.71 % (22893)Memory used [KB]: 1663
% 3.00/1.71 % (22893)Time elapsed: 0.010 s
% 3.00/1.71 % (22893)Instructions burned: 19 (million)
% 3.00/1.71 % (22893)------------------------------
% 3.00/1.71 % (22893)------------------------------
% 3.00/1.72 % (22851)Instruction limit reached!
% 3.00/1.72 % (22851)------------------------------
% 3.00/1.72 % (22851)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.72 % (22851)Termination reason: Unknown
% 3.00/1.72 % (22851)Termination phase: Saturation
% 3.00/1.72
% 3.00/1.72 % (22851)Memory used [KB]: 7164
% 3.00/1.72 % (22851)Time elapsed: 0.129 s
% 3.00/1.72 % (22851)Instructions burned: 276 (million)
% 3.00/1.72 % (22851)------------------------------
% 3.00/1.72 % (22851)------------------------------
% 3.00/1.72 % (22895)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_22687 for (2998ds/879Mi)
% 3.00/1.73 % (22896)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_22687 for (2998ds/17Mi)
% 3.00/1.74 % (22896)Instruction limit reached!
% 3.00/1.74 % (22896)------------------------------
% 3.00/1.74 % (22896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.74 % (22896)Termination reason: Unknown
% 3.00/1.74 % (22896)Termination phase: shuffling
% 3.00/1.74
% 3.00/1.74 % (22896)Memory used [KB]: 1791
% 3.00/1.74 % (22896)Time elapsed: 0.011 s
% 3.00/1.74 % (22896)Instructions burned: 18 (million)
% 3.00/1.74 % (22896)------------------------------
% 3.00/1.74 % (22896)------------------------------
% 3.00/1.76 % (22898)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_22687 for (2998ds/3Mi)
% 3.00/1.76 % (22898)Instruction limit reached!
% 3.00/1.76 % (22898)------------------------------
% 3.00/1.76 % (22898)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.76 % (22898)Termination reason: Unknown
% 3.00/1.76 % (22898)Termination phase: shuffling
% 3.00/1.76
% 3.00/1.76 % (22898)Memory used [KB]: 1407
% 3.00/1.76 % (22898)Time elapsed: 0.004 s
% 3.00/1.76 % (22898)Instructions burned: 4 (million)
% 3.00/1.76 % (22898)------------------------------
% 3.00/1.76 % (22898)------------------------------
% 3.00/1.78 % (22900)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_22687 for (2997ds/30Mi)
% 3.00/1.80 % (22900)Instruction limit reached!
% 3.00/1.80 % (22900)------------------------------
% 3.00/1.80 % (22900)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.80 % (22900)Termination reason: Unknown
% 3.00/1.80 % (22900)Termination phase: Property scanning
% 3.00/1.80
% 3.00/1.80 % (22900)Memory used [KB]: 1918
% 3.00/1.80 % (22900)Time elapsed: 0.018 s
% 3.00/1.80 % (22900)Instructions burned: 32 (million)
% 3.00/1.80 % (22900)------------------------------
% 3.00/1.80 % (22900)------------------------------
% 3.00/1.81 % (22903)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_22687 for (2997ds/127Mi)
% 3.00/1.85 % (22891)Instruction limit reached!
% 3.00/1.85 % (22891)------------------------------
% 3.00/1.85 % (22891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.85 % (22891)Termination reason: Unknown
% 3.00/1.85 % (22891)Termination phase: Saturation
% 3.00/1.85
% 3.00/1.85 % (22891)Memory used [KB]: 8059
% 3.00/1.85 % (22891)Time elapsed: 0.162 s
% 3.00/1.85 % (22891)Instructions burned: 378 (million)
% 3.00/1.85 % (22891)------------------------------
% 3.00/1.85 % (22891)------------------------------
% 3.00/1.86 % (22906)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_22687 for (2997ds/100Mi)
% 3.00/1.87 % (22903)Instruction limit reached!
% 3.00/1.87 % (22903)------------------------------
% 3.00/1.87 % (22903)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.87 % (22903)Termination reason: Unknown
% 3.00/1.87 % (22903)Termination phase: Saturation
% 3.00/1.87
% 3.00/1.87 % (22903)Memory used [KB]: 6908
% 3.00/1.87 % (22903)Time elapsed: 0.062 s
% 3.00/1.87 % (22903)Instructions burned: 127 (million)
% 3.00/1.87 % (22903)------------------------------
% 3.00/1.87 % (22903)------------------------------
% 3.00/1.89 % (22906)Instruction limit reached!
% 3.00/1.89 % (22906)------------------------------
% 3.00/1.89 % (22906)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.89 % (22906)Termination reason: Unknown
% 3.00/1.89 % (22906)Termination phase: Property scanning
% 3.00/1.89
% 3.00/1.89 % (22906)Memory used [KB]: 2430
% 3.00/1.89 % (22906)Time elapsed: 0.030 s
% 3.00/1.89 % (22906)Instructions burned: 102 (million)
% 3.00/1.89 % (22906)------------------------------
% 3.00/1.89 % (22906)------------------------------
% 3.00/1.90 % (22909)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_22687 for (2996ds/3Mi)
% 3.00/1.90 % (22909)Instruction limit reached!
% 3.00/1.90 % (22909)------------------------------
% 3.00/1.90 % (22909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.90 % (22909)Termination reason: Unknown
% 3.00/1.90 % (22909)Termination phase: shuffling
% 3.00/1.90
% 3.00/1.90 % (22909)Memory used [KB]: 1407
% 3.00/1.90 % (22909)Time elapsed: 0.003 s
% 3.00/1.90 % (22909)Instructions burned: 3 (million)
% 3.00/1.90 % (22909)------------------------------
% 3.00/1.90 % (22909)------------------------------
% 3.00/1.90 % (22910)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_22687 for (2996ds/20Mi)
% 3.00/1.91 % (22910)Instruction limit reached!
% 3.00/1.91 % (22910)------------------------------
% 3.00/1.91 % (22910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.91 % (22910)Termination reason: Unknown
% 3.00/1.91 % (22910)Termination phase: shuffling
% 3.00/1.91
% 3.00/1.91 % (22910)Memory used [KB]: 1791
% 3.00/1.91 % (22910)Time elapsed: 0.008 s
% 3.00/1.91 % (22910)Instructions burned: 20 (million)
% 3.00/1.91 % (22910)------------------------------
% 3.00/1.91 % (22910)------------------------------
% 3.00/1.92 % (22912)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_22687 for (2996ds/107Mi)
% 3.00/1.92 % (22911)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_22687 for (2996ds/86Mi)
% 3.00/1.96 % (22911)Instruction limit reached!
% 3.00/1.96 % (22911)------------------------------
% 3.00/1.96 % (22911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.96 % (22911)Termination reason: Unknown
% 3.00/1.96 % (22911)Termination phase: Property scanning
% 3.00/1.96
% 3.00/1.96 % (22911)Memory used [KB]: 2302
% 3.00/1.96 % (22911)Time elapsed: 0.047 s
% 3.00/1.96 % (22911)Instructions burned: 87 (million)
% 3.00/1.96 % (22911)------------------------------
% 3.00/1.96 % (22911)------------------------------
% 3.00/1.97 % (22880)First to succeed.
% 3.00/1.97 % (22912)Instruction limit reached!
% 3.00/1.97 % (22912)------------------------------
% 3.00/1.97 % (22912)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.97 % (22912)Termination reason: Unknown
% 3.00/1.97 % (22912)Termination phase: Saturation
% 3.00/1.97
% 3.00/1.97 % (22912)Memory used [KB]: 6780
% 3.00/1.97 % (22912)Time elapsed: 0.053 s
% 3.00/1.97 % (22912)Instructions burned: 108 (million)
% 3.00/1.97 % (22912)------------------------------
% 3.00/1.97 % (22912)------------------------------
% 3.00/1.97 % (22880)Refutation found. Thanks to Tanya!
% 3.00/1.97 % SZS status Theorem for DTF2THF_22687
% 3.00/1.97 % SZS output start Proof for DTF2THF_22687
% 3.00/1.97 thf(type_def_5, type, itself: ($tType) > $tType).
% 3.00/1.97 thf(type_def_7, type, coinductive_llist: ($tType) > $tType).
% 3.00/1.97 thf(type_def_8, type, set: ($tType) > $tType).
% 3.00/1.97 thf(type_def_9, type, a: $tType).
% 3.00/1.97 thf(func_def_0, type, coinductive_llist: ($tType) > $tType).
% 3.00/1.97 thf(func_def_1, type, set: ($tType) > $tType).
% 3.00/1.97 thf(func_def_2, type, itself: ($tType) > $tType).
% 3.00/1.97 thf(func_def_3, type, a: $tType).
% 3.00/1.97 thf(func_def_4, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.00/1.97 thf(func_def_5, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.00/1.97 thf(func_def_6, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.00/1.97 thf(func_def_7, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.00/1.97 thf(func_def_8, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.00/1.97 thf(func_def_9, type, coindu328551480prefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 3.00/1.97 thf(func_def_10, type, coinductive_gen_lset: !>[X0: $tType]:(set @ X0 > coinductive_llist @ X0 > set @ X0)).
% 3.00/1.97 thf(func_def_11, type, coinductive_lappend: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.00/1.97 thf(func_def_12, type, coindu218763757pWhile: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.00/1.97 thf(func_def_13, type, coinductive_lfilter: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.00/1.97 thf(func_def_14, type, coinductive_lfinite: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 3.00/1.97 thf(func_def_15, type, coinductive_llast: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 3.00/1.97 thf(func_def_16, type, coinductive_LCons: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.00/1.97 thf(func_def_17, type, coinductive_LNil: !>[X0: $tType]:(coinductive_llist @ X0)).
% 3.00/1.97 thf(func_def_18, type, coindu1381640503_llist: !>[X0: $tType, X1: $tType]:(X0 > (X1 > coinductive_llist @ X1 > X0) > coinductive_llist @ X1 > X0)).
% 3.00/1.97 thf(func_def_19, type, coindu1259883913_llist: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > X1) > (X0 > $o) > (X0 > coinductive_llist @ X1) > (X0 > X0) > X0 > coinductive_llist @ X1)).
% 3.00/1.97 thf(func_def_20, type, coinductive_lhd: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 3.00/1.97 thf(func_def_21, type, coinductive_lnull: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 3.00/1.97 thf(func_def_22, type, coinductive_lset: !>[X0: $tType]:(coinductive_llist @ X0 > set @ X0)).
% 3.00/1.97 thf(func_def_23, type, coinductive_lmember: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > $o)).
% 3.00/1.97 thf(func_def_24, type, coindu1478340336prefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 3.00/1.97 thf(func_def_25, type, coindu501562517eWhile: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.00/1.97 thf(func_def_26, type, coindu63249387sorted: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 3.00/1.97 thf(func_def_27, type, coindu1441602521_llist: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > X1) > (X0 > X0) > X0 > coinductive_llist @ X1)).
% 3.00/1.97 thf(func_def_28, type, undefined: !>[X0: $tType]:(X0)).
% 3.00/1.97 thf(func_def_29, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.00/1.97 thf(func_def_30, type, koenig916195507_paths: !>[X0: $tType]:((X0 > X0 > $o) > set @ coinductive_llist @ X0)).
% 3.00/1.97 thf(func_def_31, type, koenig2031690877pathsp: !>[X0: $tType]:((X0 > X0 > $o) > coinductive_llist @ X0 > $o)).
% 3.00/1.97 thf(func_def_32, type, koenig317145564le_via: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > X0 > set @ X0)).
% 3.00/1.97 thf(func_def_33, type, koenig1757754772e_viap: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > X0 > X0 > $o)).
% 3.00/1.97 thf(func_def_34, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.00/1.97 thf(func_def_35, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.00/1.97 thf(func_def_36, type, type2: !>[X0: $tType]:(itself @ X0)).
% 3.00/1.97 thf(func_def_37, type, powp: !>[X0: $tType]:((X0 > $o) > set @ X0 > $o)).
% 3.00/1.97 thf(func_def_38, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.00/1.97 thf(func_def_39, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.00/1.97 thf(func_def_40, type, graph: a > a > $o).
% 3.00/1.97 thf(func_def_41, type, x: coinductive_llist @ a).
% 3.00/1.97 thf(func_def_42, type, x21: a).
% 3.00/1.97 thf(func_def_43, type, x22: coinductive_llist @ a).
% 3.00/1.97 thf(func_def_44, type, xa: a).
% 3.00/1.97 thf(func_def_45, type, xs: coinductive_llist @ a).
% 3.00/1.97 thf(func_def_46, type, ys: coinductive_llist @ a).
% 3.00/1.97 thf(func_def_63, type, sK0: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 3.00/1.97 thf(func_def_64, type, sK1: !>[X0: $tType]:((X0 > X0 > $o) > coinductive_llist @ X0 > X0)).
% 3.00/1.97 thf(func_def_65, type, sK2: !>[X0: $tType]:((X0 > X0 > $o) > coinductive_llist @ X0 > X0)).
% 3.00/1.97 thf(func_def_66, type, sK3: !>[X0: $tType]:((X0 > X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.00/1.97 thf(func_def_67, type, sK4: !>[X0: $tType]:((X0 > X0 > $o) > (coinductive_llist @ X0 > $o) > coinductive_llist @ X0)).
% 3.00/1.97 thf(func_def_68, type, sK5: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 3.00/1.97 thf(func_def_69, type, sK6: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.00/1.97 thf(func_def_70, type, sK7: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 3.00/1.97 thf(func_def_71, type, sK8: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.00/1.97 thf(func_def_72, type, sK9: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 3.00/1.97 thf(func_def_73, type, sK10: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.00/1.97 thf(func_def_74, type, sK11: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 3.00/1.97 thf(func_def_75, type, sK12: !>[X0: $tType]:((X0 > X0 > $o) > coinductive_llist @ X0 > X0)).
% 3.00/1.97 thf(func_def_76, type, sK13: !>[X0: $tType]:((X0 > X0 > $o) > coinductive_llist @ X0 > X0)).
% 3.00/1.97 thf(func_def_77, type, sK14: !>[X0: $tType]:((X0 > X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.00/1.97 thf(func_def_80, type, ph16: !>[X0: $tType]:(X0)).
% 3.00/1.97 thf(f1921,plain,(
% 3.00/1.97 $false),
% 3.00/1.97 inference(avatar_sat_refutation,[status(thm)],[f1787,f1920])).
% 3.00/1.97 thf(f1920,plain,(
% 3.00/1.97 spl15_12),
% 3.00/1.97 inference(avatar_contradiction_clause,[status(thm)],[f1919])).
% 3.00/1.97 thf(f1919,plain,(
% 3.00/1.97 $false | spl15_12),
% 3.00/1.97 inference(subsumption_resolution,[status(thm)],[f1918,f1903])).
% 3.00/1.97 thf(f1903,plain,(
% 3.00/1.97 (ys != x22) | spl15_12),
% 3.00/1.97 inference(backward_demodulation,[status(thm)],[f1747,f1898])).
% 3.00/1.97 thf(f1898,plain,(
% 3.00/1.97 (coinductive_LNil @ a = x22)),
% 3.00/1.97 inference(trivial_inequality_removal,[status(thm)],[f1895])).
% 3.00/1.97 thf(f1895,plain,(
% 3.00/1.97 (coinductive_LNil @ a != coinductive_LNil @ a) | (coinductive_LNil @ a = x22)),
% 3.00/1.97 inference(superposition,[status(thm)],[f862,f1885])).
% 3.00/1.97 thf(f1885,plain,(
% 3.00/1.97 (coinductive_LNil @ a = (coinductive_lappend @ a @ x22 @ ys))),
% 3.00/1.97 inference(forward_demodulation,[status(thm)],[f1883,f1205])).
% 3.00/1.97 thf(f1205,plain,(
% 3.00/1.97 ( ! [X0 : $tType,X2 : coinductive_llist @ X0,X1 : X0] : (((sK8 @ X0 @ (coinductive_LCons @ X0 @ X1 @ X2)) = X2)) )),
% 3.00/1.97 inference(subsumption_resolution,[status(thm)],[f1204,f885])).
% 3.00/1.97 thf(f885,plain,(
% 3.00/1.97 ( ! [X0 : $tType,X4 : X0,X5 : coinductive_llist @ X0] : ((coinductive_LNil @ X0 != (coinductive_LCons @ X0 @ X4 @ X5))) )),
% 3.00/1.97 inference(equality_resolution,[status(thm)],[f852])).
% 3.00/1.97 thf(f852,plain,(
% 3.00/1.97 ( ! [X0 : $tType,X1 : coinductive_llist @ X0,X4 : X0,X5 : coinductive_llist @ X0] : ((coinductive_LNil @ X0 != X1) | ((coinductive_LCons @ X0 @ X4 @ X5) != X1)) )),
% 3.00/1.97 inference(cnf_transformation,[status(thm)],[f806])).
% 3.00/1.97 thf(f806,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0] : ((((coinductive_LCons @ X0 @ (sK7 @ X0 @ X1) @ (sK8 @ X0 @ X1)) = X1) | (coinductive_LNil @ X0 = X1)) & ((coinductive_LNil @ X0 != X1) | ! [X4 : X0,X5 : coinductive_llist @ X0] : ((coinductive_LCons @ X0 @ X4 @ X5) != X1)))),
% 3.00/1.97 inference(skolemisation,[status(esa),new_symbols(skolem,[sK7,sK8])],[f804,f805])).
% 3.00/1.97 thf(f805,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0] : (? [X2 : X0,X3 : coinductive_llist @ X0] : ((coinductive_LCons @ X0 @ X2 @ X3) = X1) => ((coinductive_LCons @ X0 @ (sK7 @ X0 @ X1) @ (sK8 @ X0 @ X1)) = X1))),
% 3.00/1.97 introduced(definition,[],[choice_axiom])).
% 3.00/1.97 thf(f804,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0] : ((? [X2 : X0,X3 : coinductive_llist @ X0] : ((coinductive_LCons @ X0 @ X2 @ X3) = X1) | (coinductive_LNil @ X0 = X1)) & ((coinductive_LNil @ X0 != X1) | ! [X4 : X0,X5 : coinductive_llist @ X0] : ((coinductive_LCons @ X0 @ X4 @ X5) != X1)))),
% 3.00/1.97 inference(rectify,[status(thm)],[f803])).
% 3.00/1.97 thf(f803,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0] : ((? [X2 : X0,X3 : coinductive_llist @ X0] : ((coinductive_LCons @ X0 @ X2 @ X3) = X1) | (coinductive_LNil @ X0 = X1)) & ((coinductive_LNil @ X0 != X1) | ! [X2 : X0,X3 : coinductive_llist @ X0] : ((coinductive_LCons @ X0 @ X2 @ X3) != X1)))),
% 3.00/1.97 inference(nnf_transformation,[status(thm)],[f414])).
% 3.00/1.97 thf(f414,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0] : (? [X2 : X0,X3 : coinductive_llist @ X0] : ((coinductive_LCons @ X0 @ X2 @ X3) = X1) <=> (coinductive_LNil @ X0 != X1))),
% 3.00/1.97 inference(fool_elimination,[status(thm)],[f413])).
% 3.00/1.97 thf(f413,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0] : (? [X2 : X0,X3 : coinductive_llist @ X0] : ((coinductive_LCons @ X0 @ X2 @ X3) = X1) = (coinductive_LNil @ X0 != X1))),
% 3.00/1.97 inference(rectify,[status(thm)],[f19])).
% 3.00/1.97 thf(f19,axiom,(
% 3.00/1.97 ! [X0 : $tType,X7 : coinductive_llist @ X0] : (? [X12 : X0,X27 : coinductive_llist @ X0] : ((coinductive_LCons @ X0 @ X12 @ X27) = X7) = (coinductive_LNil @ X0 != X7))),
% 3.00/1.97 file('/export/starexec/sandbox2/tmp/tmp.5ITWujfIVW/DTF2THF_22687.p',fact_18_neq__LNil__conv)).
% 3.00/1.97 thf(f1204,plain,(
% 3.00/1.97 ( ! [X0 : $tType,X2 : coinductive_llist @ X0,X1 : X0] : ((coinductive_LNil @ X0 = (coinductive_LCons @ X0 @ X1 @ X2)) | ((sK8 @ X0 @ (coinductive_LCons @ X0 @ X1 @ X2)) = X2)) )),
% 3.00/1.97 inference(equality_resolution,[status(thm)],[f914])).
% 3.00/1.97 thf(f914,plain,(
% 3.00/1.97 ( ! [X0 : $tType,X2 : X0,X3 : coinductive_llist @ X0,X1 : coinductive_llist @ X0] : (((coinductive_LCons @ X0 @ X2 @ X3) != X1) | ((sK8 @ X0 @ X1) = X3) | (coinductive_LNil @ X0 = X1)) )),
% 3.00/1.97 inference(superposition,[status(thm)],[f849,f853])).
% 3.00/1.97 thf(f853,plain,(
% 3.00/1.97 ( ! [X0 : $tType,X1 : coinductive_llist @ X0] : (((coinductive_LCons @ X0 @ (sK7 @ X0 @ X1) @ (sK8 @ X0 @ X1)) = X1) | (coinductive_LNil @ X0 = X1)) )),
% 3.00/1.97 inference(cnf_transformation,[status(thm)],[f806])).
% 3.00/1.97 thf(f849,plain,(
% 3.00/1.97 ( ! [X0 : $tType,X2 : coinductive_llist @ X0,X3 : X0,X1 : X0,X4 : coinductive_llist @ X0] : (((coinductive_LCons @ X0 @ X3 @ X4) != (coinductive_LCons @ X0 @ X1 @ X2)) | (X2 = X4)) )),
% 3.00/1.97 inference(cnf_transformation,[status(thm)],[f802])).
% 3.00/1.97 thf(f802,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : X0,X2 : coinductive_llist @ X0,X3 : X0,X4 : coinductive_llist @ X0] : ((((coinductive_LCons @ X0 @ X3 @ X4) = (coinductive_LCons @ X0 @ X1 @ X2)) | (X1 != X3) | (X2 != X4)) & (((X1 = X3) & (X2 = X4)) | ((coinductive_LCons @ X0 @ X3 @ X4) != (coinductive_LCons @ X0 @ X1 @ X2))))),
% 3.00/1.97 inference(flattening,[status(thm)],[f801])).
% 3.00/1.97 thf(f801,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : X0,X2 : coinductive_llist @ X0,X3 : X0,X4 : coinductive_llist @ X0] : ((((coinductive_LCons @ X0 @ X3 @ X4) = (coinductive_LCons @ X0 @ X1 @ X2)) | ((X1 != X3) | (X2 != X4))) & (((X1 = X3) & (X2 = X4)) | ((coinductive_LCons @ X0 @ X3 @ X4) != (coinductive_LCons @ X0 @ X1 @ X2))))),
% 3.00/1.97 inference(nnf_transformation,[status(thm)],[f356])).
% 3.00/1.97 thf(f356,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : X0,X2 : coinductive_llist @ X0,X3 : X0,X4 : coinductive_llist @ X0] : (((coinductive_LCons @ X0 @ X3 @ X4) = (coinductive_LCons @ X0 @ X1 @ X2)) <=> ((X1 = X3) & (X2 = X4)))),
% 3.00/1.97 inference(fool_elimination,[status(thm)],[f355])).
% 3.00/1.97 thf(f355,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : X0,X2 : coinductive_llist @ X0,X3 : X0,X4 : coinductive_llist @ X0] : (((coinductive_LCons @ X0 @ X3 @ X4) = (coinductive_LCons @ X0 @ X1 @ X2)) = (X1 = X3) & (X2 = X4))),
% 3.00/1.97 inference(rectify,[status(thm)],[f13])).
% 3.00/1.97 thf(f13,axiom,(
% 3.00/1.97 ! [X0 : $tType,X21 : X0,X22 : coinductive_llist @ X0,X23 : X0,X24 : coinductive_llist @ X0] : (((coinductive_LCons @ X0 @ X21 @ X22) = (coinductive_LCons @ X0 @ X23 @ X24)) = (X21 = X23) & (X22 = X24))),
% 3.00/1.97 file('/export/starexec/sandbox2/tmp/tmp.5ITWujfIVW/DTF2THF_22687.p',fact_12_llist_Oinject)).
% 3.00/1.97 thf(f1883,plain,(
% 3.00/1.97 ((sK8 @ a @ (coinductive_LCons @ a @ xa @ coinductive_LNil @ a)) = (coinductive_lappend @ a @ x22 @ ys))),
% 3.00/1.97 inference(superposition,[status(thm)],[f1205,f1065])).
% 3.00/1.97 thf(f1065,plain,(
% 3.00/1.97 ((coinductive_LCons @ a @ xa @ coinductive_LNil @ a) = (coinductive_LCons @ a @ x21 @ (coinductive_lappend @ a @ x22 @ ys)))),
% 3.00/1.97 inference(superposition,[status(thm)],[f879,f843])).
% 3.00/1.97 thf(f843,plain,(
% 3.00/1.97 ( ! [X0 : $tType,X2 : coinductive_llist @ X0,X3 : coinductive_llist @ X0,X1 : X0] : (((coinductive_lappend @ X0 @ (coinductive_LCons @ X0 @ X1 @ X2) @ X3) = (coinductive_LCons @ X0 @ X1 @ (coinductive_lappend @ X0 @ X2 @ X3)))) )),
% 3.00/1.97 inference(cnf_transformation,[status(thm)],[f755])).
% 3.00/1.97 thf(f755,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : X0,X2 : coinductive_llist @ X0,X3 : coinductive_llist @ X0] : ((coinductive_lappend @ X0 @ (coinductive_LCons @ X0 @ X1 @ X2) @ X3) = (coinductive_LCons @ X0 @ X1 @ (coinductive_lappend @ X0 @ X2 @ X3)))),
% 3.00/1.97 inference(rectify,[status(thm)],[f11])).
% 3.00/1.97 thf(f11,axiom,(
% 3.00/1.97 ! [X0 : $tType,X20 : X0,X5 : coinductive_llist @ X0,X19 : coinductive_llist @ X0] : ((coinductive_lappend @ X0 @ (coinductive_LCons @ X0 @ X20 @ X5) @ X19) = (coinductive_LCons @ X0 @ X20 @ (coinductive_lappend @ X0 @ X5 @ X19)))),
% 3.00/1.97 file('/export/starexec/sandbox2/tmp/tmp.5ITWujfIVW/DTF2THF_22687.p',fact_10_lappend__code_I2_J)).
% 3.00/1.97 thf(f879,plain,(
% 3.00/1.97 ((coinductive_LCons @ a @ xa @ coinductive_LNil @ a) = (coinductive_lappend @ a @ (coinductive_LCons @ a @ x21 @ x22) @ ys))),
% 3.00/1.97 inference(definition_unfolding,[status(thm)],[f844,f854])).
% 3.00/1.97 thf(f854,plain,(
% 3.00/1.97 (x = (coinductive_LCons @ a @ x21 @ x22))),
% 3.00/1.97 inference(cnf_transformation,[status(thm)],[f269])).
% 3.00/1.97 thf(f269,axiom,(
% 3.00/1.97 (x = (coinductive_LCons @ a @ x21 @ x22))),
% 3.00/1.97 file('/export/starexec/sandbox2/tmp/tmp.5ITWujfIVW/DTF2THF_22687.p',conj_1)).
% 3.00/1.97 thf(f844,plain,(
% 3.00/1.97 ((coinductive_lappend @ a @ x @ ys) = (coinductive_LCons @ a @ xa @ coinductive_LNil @ a))),
% 3.00/1.97 inference(cnf_transformation,[status(thm)],[f268])).
% 3.00/1.97 thf(f268,axiom,(
% 3.00/1.97 ((coinductive_lappend @ a @ x @ ys) = (coinductive_LCons @ a @ xa @ coinductive_LNil @ a))),
% 3.00/1.97 file('/export/starexec/sandbox2/tmp/tmp.5ITWujfIVW/DTF2THF_22687.p',conj_0)).
% 3.00/1.97 thf(f862,plain,(
% 3.00/1.97 ( ! [X0 : $tType,X2 : coinductive_llist @ X0,X1 : coinductive_llist @ X0] : ((coinductive_LNil @ X0 != (coinductive_lappend @ X0 @ X1 @ X2)) | (coinductive_LNil @ X0 = X1)) )),
% 3.00/1.97 inference(cnf_transformation,[status(thm)],[f810])).
% 3.00/1.97 thf(f810,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0,X2 : coinductive_llist @ X0] : (((coinductive_LNil @ X0 = (coinductive_lappend @ X0 @ X1 @ X2)) | (coinductive_LNil @ X0 != X2) | (coinductive_LNil @ X0 != X1)) & (((coinductive_LNil @ X0 = X2) & (coinductive_LNil @ X0 = X1)) | (coinductive_LNil @ X0 != (coinductive_lappend @ X0 @ X1 @ X2))))),
% 3.00/1.97 inference(flattening,[status(thm)],[f809])).
% 3.00/1.97 thf(f809,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0,X2 : coinductive_llist @ X0] : (((coinductive_LNil @ X0 = (coinductive_lappend @ X0 @ X1 @ X2)) | ((coinductive_LNil @ X0 != X2) | (coinductive_LNil @ X0 != X1))) & (((coinductive_LNil @ X0 = X2) & (coinductive_LNil @ X0 = X1)) | (coinductive_LNil @ X0 != (coinductive_lappend @ X0 @ X1 @ X2))))),
% 3.00/1.97 inference(nnf_transformation,[status(thm)],[f430])).
% 3.00/1.97 thf(f430,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0,X2 : coinductive_llist @ X0] : ((coinductive_LNil @ X0 = (coinductive_lappend @ X0 @ X1 @ X2)) <=> ((coinductive_LNil @ X0 = X2) & (coinductive_LNil @ X0 = X1)))),
% 3.00/1.97 inference(fool_elimination,[status(thm)],[f429])).
% 3.00/1.97 thf(f429,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0,X2 : coinductive_llist @ X0] : ((coinductive_LNil @ X0 = X1) & (coinductive_LNil @ X0 = X2) = (coinductive_LNil @ X0 = (coinductive_lappend @ X0 @ X1 @ X2)))),
% 3.00/1.97 inference(rectify,[status(thm)],[f15])).
% 3.00/1.97 thf(f15,axiom,(
% 3.00/1.97 ! [X0 : $tType,X7 : coinductive_llist @ X0,X19 : coinductive_llist @ X0] : ((coinductive_LNil @ X0 = X7) & (coinductive_LNil @ X0 = X19) = (coinductive_LNil @ X0 = (coinductive_lappend @ X0 @ X7 @ X19)))),
% 3.00/1.97 file('/export/starexec/sandbox2/tmp/tmp.5ITWujfIVW/DTF2THF_22687.p',fact_14_LNil__eq__lappend__iff)).
% 3.00/1.97 thf(f1747,plain,(
% 3.00/1.97 (ys != coinductive_LNil @ a) | spl15_12),
% 3.00/1.97 inference(avatar_component_clause,[status(thm)],[f1746])).
% 3.00/1.97 thf(f1746,definition,(
% 3.00/1.97 spl15_12 <=> (ys = coinductive_LNil @ a)),
% 3.00/1.97 introduced(definition,[new_symbols(naming,[spl15_12])],[avatar_definition])).
% 3.00/1.97 thf(f1918,plain,(
% 3.00/1.97 (ys = x22)),
% 3.00/1.97 inference(forward_demodulation,[status(thm)],[f1897,f1898])).
% 3.00/1.97 thf(f1897,plain,(
% 3.00/1.97 (ys = coinductive_LNil @ a)),
% 3.00/1.97 inference(trivial_inequality_removal,[status(thm)],[f1896])).
% 3.00/1.97 thf(f1896,plain,(
% 3.00/1.97 (ys = coinductive_LNil @ a) | (coinductive_LNil @ a != coinductive_LNil @ a)),
% 3.00/1.97 inference(superposition,[status(thm)],[f827,f1885])).
% 3.00/1.97 thf(f827,plain,(
% 3.00/1.97 ( ! [X0 : $tType,X2 : coinductive_llist @ X0,X1 : coinductive_llist @ X0] : ((coinductive_LNil @ X0 != (coinductive_lappend @ X0 @ X1 @ X2)) | (coinductive_LNil @ X0 = X2)) )),
% 3.00/1.97 inference(cnf_transformation,[status(thm)],[f796])).
% 3.00/1.97 thf(f796,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0,X2 : coinductive_llist @ X0] : (((coinductive_LNil @ X0 = (coinductive_lappend @ X0 @ X1 @ X2)) | (coinductive_LNil @ X0 != X2) | (coinductive_LNil @ X0 != X1)) & (((coinductive_LNil @ X0 = X2) & (coinductive_LNil @ X0 = X1)) | (coinductive_LNil @ X0 != (coinductive_lappend @ X0 @ X1 @ X2))))),
% 3.00/1.97 inference(flattening,[status(thm)],[f795])).
% 3.00/1.97 thf(f795,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0,X2 : coinductive_llist @ X0] : (((coinductive_LNil @ X0 = (coinductive_lappend @ X0 @ X1 @ X2)) | ((coinductive_LNil @ X0 != X2) | (coinductive_LNil @ X0 != X1))) & (((coinductive_LNil @ X0 = X2) & (coinductive_LNil @ X0 = X1)) | (coinductive_LNil @ X0 != (coinductive_lappend @ X0 @ X1 @ X2))))),
% 3.00/1.97 inference(nnf_transformation,[status(thm)],[f634])).
% 3.00/1.97 thf(f634,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0,X2 : coinductive_llist @ X0] : ((coinductive_LNil @ X0 = (coinductive_lappend @ X0 @ X1 @ X2)) <=> ((coinductive_LNil @ X0 = X2) & (coinductive_LNil @ X0 = X1)))),
% 3.00/1.97 inference(fool_elimination,[status(thm)],[f633])).
% 3.00/1.97 thf(f633,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0,X2 : coinductive_llist @ X0] : ((coinductive_LNil @ X0 = X2) & (coinductive_LNil @ X0 = X1) = (coinductive_LNil @ X0 = (coinductive_lappend @ X0 @ X1 @ X2)))),
% 3.00/1.97 inference(rectify,[status(thm)],[f16])).
% 3.00/1.97 thf(f16,axiom,(
% 3.00/1.97 ! [X0 : $tType,X7 : coinductive_llist @ X0,X19 : coinductive_llist @ X0] : ((coinductive_LNil @ X0 = X19) & (coinductive_LNil @ X0 = X7) = (coinductive_LNil @ X0 = (coinductive_lappend @ X0 @ X7 @ X19)))),
% 3.00/1.97 file('/export/starexec/sandbox2/tmp/tmp.5ITWujfIVW/DTF2THF_22687.p',fact_15_lappend__eq__LNil__iff)).
% 3.00/1.97 thf(f1787,plain,(
% 3.00/1.97 ~spl15_12),
% 3.00/1.97 inference(avatar_contradiction_clause,[status(thm)],[f1786])).
% 3.00/1.97 thf(f1786,plain,(
% 3.00/1.97 $false | ~spl15_12),
% 3.00/1.97 inference(subsumption_resolution,[status(thm)],[f1783,f1753])).
% 3.00/1.97 thf(f1753,plain,(
% 3.00/1.97 ( ! [X3 : a] : (((coinductive_LCons @ a @ x21 @ x22) != (coinductive_LCons @ a @ X3 @ ys))) ) | ~spl15_12),
% 3.00/1.97 inference(backward_demodulation,[status(thm)],[f878,f1748])).
% 3.00/1.97 thf(f1748,plain,(
% 3.00/1.97 (ys = coinductive_LNil @ a) | ~spl15_12),
% 3.00/1.97 inference(avatar_component_clause,[status(thm)],[f1746])).
% 3.00/1.97 thf(f878,plain,(
% 3.00/1.97 ( ! [X3 : a] : (((coinductive_LCons @ a @ x21 @ x22) != (coinductive_LCons @ a @ X3 @ coinductive_LNil @ a))) )),
% 3.00/1.97 inference(definition_unfolding,[status(thm)],[f832,f854])).
% 3.00/1.97 thf(f832,plain,(
% 3.00/1.97 ( ! [X3 : a] : ((x != (coinductive_LCons @ a @ X3 @ coinductive_LNil @ a))) )),
% 3.00/1.97 inference(cnf_transformation,[status(thm)],[f780])).
% 3.00/1.97 thf(f780,plain,(
% 3.00/1.97 ! [X0 : a,X1 : a,X2 : coinductive_llist @ a] : ((x != (coinductive_LCons @ a @ X0 @ (coinductive_LCons @ a @ X1 @ X2))) | (((member @ coinductive_llist @ a @ (coinductive_lappend @ a @ (coinductive_LCons @ a @ X1 @ X2) @ ys) @ (koenig916195507_paths @ a @ graph)) != $true) & ($true != (member @ coinductive_llist @ a @ (coinductive_LCons @ a @ X1 @ X2) @ (koenig916195507_paths @ a @ graph)))) | ((graph @ X0 @ X1) != $true)) & (x != coinductive_LNil @ a) & ! [X3 : a] : (x != (coinductive_LCons @ a @ X3 @ coinductive_LNil @ a))),
% 3.00/1.97 inference(ennf_transformation,[status(thm)],[f438])).
% 3.00/1.97 thf(f438,plain,(
% 3.00/1.97 ~(? [X3 : a] : (x = (coinductive_LCons @ a @ X3 @ coinductive_LNil @ a)) | ? [X0 : a,X1 : a,X2 : coinductive_llist @ a] : ((x = (coinductive_LCons @ a @ X0 @ (coinductive_LCons @ a @ X1 @ X2))) & (($true = (member @ coinductive_llist @ a @ (coinductive_LCons @ a @ X1 @ X2) @ (koenig916195507_paths @ a @ graph))) | ((member @ coinductive_llist @ a @ (coinductive_lappend @ a @ (coinductive_LCons @ a @ X1 @ X2) @ ys) @ (koenig916195507_paths @ a @ graph)) = $true)) & ((graph @ X0 @ X1) = $true)) | (x = coinductive_LNil @ a))),
% 3.00/1.97 inference(fool_elimination,[status(thm)],[f437])).
% 3.00/1.97 thf(f437,plain,(
% 3.00/1.97 ~(? [X0 : a,X1 : a,X2 : coinductive_llist @ a] : ((graph @ X0 @ X1) & ((member @ coinductive_llist @ a @ (coinductive_LCons @ a @ X1 @ X2) @ (koenig916195507_paths @ a @ graph)) | (member @ coinductive_llist @ a @ (coinductive_lappend @ a @ (coinductive_LCons @ a @ X1 @ X2) @ ys) @ (koenig916195507_paths @ a @ graph))) & (x = (coinductive_LCons @ a @ X0 @ (coinductive_LCons @ a @ X1 @ X2)))) | ? [X3 : a] : (x = (coinductive_LCons @ a @ X3 @ coinductive_LNil @ a)) | (x = coinductive_LNil @ a))),
% 3.00/1.97 inference(rectify,[status(thm)],[f271])).
% 3.00/1.97 thf(f271,negated_conjecture,(
% 3.00/1.97 ~(? [X81 : a,X17 : a,X18 : coinductive_llist @ a] : ((graph @ X81 @ X17) & ((member @ coinductive_llist @ a @ (coinductive_LCons @ a @ X17 @ X18) @ (koenig916195507_paths @ a @ graph)) | (member @ coinductive_llist @ a @ (coinductive_lappend @ a @ (coinductive_LCons @ a @ X17 @ X18) @ ys) @ (koenig916195507_paths @ a @ graph))) & (x = (coinductive_LCons @ a @ X81 @ (coinductive_LCons @ a @ X17 @ X18)))) | ? [X81 : a] : (x = (coinductive_LCons @ a @ X81 @ coinductive_LNil @ a)) | (x = coinductive_LNil @ a))),
% 3.00/1.97 inference(negated_conjecture,[status(cth)],[f270])).
% 3.00/1.97 thf(f270,conjecture,(
% 3.00/1.97 ? [X81 : a,X17 : a,X18 : coinductive_llist @ a] : ((graph @ X81 @ X17) & ((member @ coinductive_llist @ a @ (coinductive_LCons @ a @ X17 @ X18) @ (koenig916195507_paths @ a @ graph)) | (member @ coinductive_llist @ a @ (coinductive_lappend @ a @ (coinductive_LCons @ a @ X17 @ X18) @ ys) @ (koenig916195507_paths @ a @ graph))) & (x = (coinductive_LCons @ a @ X81 @ (coinductive_LCons @ a @ X17 @ X18)))) | ? [X81 : a] : (x = (coinductive_LCons @ a @ X81 @ coinductive_LNil @ a)) | (x = coinductive_LNil @ a)),
% 3.00/1.97 file('/export/starexec/sandbox2/tmp/tmp.5ITWujfIVW/DTF2THF_22687.p',conj_2)).
% 3.00/1.97 thf(f1783,plain,(
% 3.00/1.97 ((coinductive_LCons @ a @ x21 @ x22) = (coinductive_LCons @ a @ xa @ ys)) | ~spl15_12),
% 3.00/1.97 inference(backward_demodulation,[status(thm)],[f1754,f1767])).
% 3.00/1.97 thf(f1767,plain,(
% 3.00/1.97 ( ! [X0 : coinductive_llist @ a] : (((coinductive_lappend @ a @ X0 @ ys) = X0)) ) | ~spl15_12),
% 3.00/1.97 inference(superposition,[status(thm)],[f847,f1748])).
% 3.00/1.97 thf(f847,plain,(
% 3.00/1.97 ( ! [X0 : $tType,X1 : coinductive_llist @ X0] : (((coinductive_lappend @ X0 @ X1 @ coinductive_LNil @ X0) = X1)) )),
% 3.00/1.97 inference(cnf_transformation,[status(thm)],[f757])).
% 3.00/1.97 thf(f757,plain,(
% 3.00/1.97 ! [X0 : $tType,X1 : coinductive_llist @ X0] : ((coinductive_lappend @ X0 @ X1 @ coinductive_LNil @ X0) = X1)),
% 3.00/1.97 inference(rectify,[status(thm)],[f10])).
% 3.00/1.97 thf(f10,axiom,(
% 3.00/1.97 ! [X0 : $tType,X7 : coinductive_llist @ X0] : ((coinductive_lappend @ X0 @ X7 @ coinductive_LNil @ X0) = X7)),
% 3.00/1.97 file('/export/starexec/sandbox2/tmp/tmp.5ITWujfIVW/DTF2THF_22687.p',fact_9_lappend__LNil2)).
% 3.00/1.97 thf(f1754,plain,(
% 3.00/1.97 ((coinductive_lappend @ a @ (coinductive_LCons @ a @ x21 @ x22) @ ys) = (coinductive_LCons @ a @ xa @ ys)) | ~spl15_12),
% 3.00/1.97 inference(backward_demodulation,[status(thm)],[f879,f1748])).
% 3.00/1.97 % SZS output end Proof for DTF2THF_22687
% 3.00/1.97 % (22880)------------------------------
% 3.00/1.97 % (22880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.97 % (22880)Termination reason: Refutation
% 3.00/1.97
% 3.00/1.97 % (22880)Memory used [KB]: 7547
% 3.00/1.97 % (22880)Time elapsed: 0.317 s
% 3.00/1.97 % (22880)Instructions burned: 671 (million)
% 3.00/1.97 % (22880)------------------------------
% 3.00/1.97 % (22880)------------------------------
% 3.00/1.97 % (22841)Success in time 0.416 s
%------------------------------------------------------------------------------