%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT152^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:44:57 AM UTC 2026
% Result : Theorem 2.61s 1.59s
% Output : Refutation 2.61s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.10 % Problem : DAT152^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.13/0.32 % Computer : n002.cluster.edu
% 0.13/0.32 % Model : x86_64 x86_64
% 0.13/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.32 % Memory : 8042.1875MB
% 0.13/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.32 % CPULimit : 300
% 0.13/0.32 % WCLimit : 300
% 0.13/0.32 % DateTime : Tue Feb 24 06:10:18 EST 2026
% 0.13/0.32 % CPUTime :
% 0.13/0.32 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.41 ---- Original DTF file ---
% 0.16/0.41 thf(spec,logic,$$dhol).
% 0.16/0.41 %------------------------------------------------------------------------------
% 0.16/0.41 % File : DAT152^1 : TPTP v9.2.1. Released v7.0.0.
% 0.16/0.41 % Domain : Data Structures
% 0.16/0.41 % Problem : Coinductive stream 429
% 0.16/0.41 % Version : [Bla16] axioms : Especial.
% 0.16/0.41 % English :
% 0.16/0.41
% 0.16/0.41 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.16/0.41 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.16/0.41 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.16/0.41 % Source : [Bla16]
% 0.16/0.41 % Names : coinductive_stream__429.p [Bla16]
% 0.16/0.41
% 0.16/0.41 % Status : Theorem
% 0.16/0.41 % Rating : 0.67 v9.1.0, 0.33 v8.1.0, 0.25 v7.5.0, 0.00 v7.1.0
% 0.16/0.41 % Syntax : Number of formulae : 323 ( 96 unt; 39 typ; 0 def)
% 0.16/0.41 % Number of atoms : 903 ( 179 equ; 0 cnn)
% 0.16/0.41 % Maximal formula atoms : 24 ( 3 avg)
% 0.16/0.41 % Number of connectives : 3478 ( 119 ~; 22 |; 45 &;2820 @)
% 0.16/0.41 % ( 0 <=>; 472 =>; 0 <=; 0 <~>)
% 0.16/0.41 % Maximal formula depth : 21 ( 8 avg)
% 0.16/0.41 % Number of types : 4 ( 3 usr)
% 0.16/0.41 % Number of type conns : 198 ( 198 >; 0 *; 0 +; 0 <<)
% 0.16/0.41 % Number of symbols : 39 ( 36 usr; 3 con; 0-4 aty)
% 0.16/0.41 % Number of variables : 991 ( 106 ^; 816 !; 38 ?; 991 :)
% 0.16/0.41 % ( 31 !>; 0 ?*; 0 @-; 0 @+)
% 0.16/0.41 % SPC : TH1_THM_EQU_NAR
% 0.16/0.41
% 0.16/0.41 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.16/0.41 % 2016-07-13 15:14:51.363
% 0.16/0.41 %------------------------------------------------------------------------------
% 0.16/0.41 %----Could-be-implicit typings (6)
% 0.16/0.41 thf(ty_t_Extended__Nat_Oenat,type,
% 0.16/0.41 extended_enat: $tType ).
% 0.16/0.41
% 0.16/0.41 thf(ty_t_Stream_Ostream,type,
% 0.16/0.41 stream: $tType > $tType ).
% 0.16/0.41
% 0.16/0.41 thf(ty_t_Set_Oset,type,
% 0.16/0.41 set: $tType > $tType ).
% 0.16/0.41
% 0.16/0.41 thf(ty_t_Nat_Onat,type,
% 0.16/0.41 nat: $tType ).
% 0.16/0.41
% 0.16/0.41 thf(ty_t_itself,type,
% 0.16/0.41 itself: $tType > $tType ).
% 0.16/0.41
% 0.16/0.41 thf(ty_tf_a,type,
% 0.16/0.41 a: $tType ).
% 0.16/0.41
% 0.16/0.41 %----Explicit typings (33)
% 0.16/0.41 thf(sy_cl_HOL_Otype,type,
% 0.16/0.41 type:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_cl_Orderings_Oord,type,
% 0.16/0.41 ord:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_cl_Orderings_Oorder,type,
% 0.16/0.41 order:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_cl_Orderings_Ono__bot,type,
% 0.16/0.41 no_bot:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_cl_Orderings_Ono__top,type,
% 0.16/0.41 no_top:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.16/0.41 finite_finite:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_cl_Orderings_Olinorder,type,
% 0.16/0.41 linorder:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_cl_Orderings_Opreorder,type,
% 0.16/0.41 preorder:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_cl_Orderings_Owellorder,type,
% 0.16/0.41 wellorder:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_cl_Orderings_Odense__order,type,
% 0.16/0.41 dense_order:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.16/0.41 linordered_idom:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.16/0.41 dense_linorder:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.16/0.41 condit1656338222tinuum:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.16/0.41 condit1037483654norder:
% 0.16/0.41 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Oscount,type,
% 0.16/0.41 coindu1365464361scount:
% 0.16/0.41 !>[S: $tType] : ( ( ( stream @ S ) > $o ) > ( stream @ S ) > extended_enat ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Extended__Nat_Oenat,type,
% 0.16/0.41 extended_enat2: nat > extended_enat ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
% 0.16/0.41 extended_case_enat:
% 0.16/0.41 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
% 0.16/0.41 extended_rec_enat:
% 0.16/0.41 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
% 0.16/0.41 extend1396239628finity:
% 0.16/0.41 !>[A: $tType] : A ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Finite__Set_Ocard,type,
% 0.16/0.41 finite_card:
% 0.16/0.41 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Finite__Set_Ofinite,type,
% 0.16/0.41 finite_finite2:
% 0.16/0.41 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Linear__Temporal__Logic__on__Streams_Oalw,type,
% 0.16/0.41 linear1386806755on_alw:
% 0.16/0.41 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Linear__Temporal__Logic__on__Streams_Oev,type,
% 0.16/0.41 linear505997466_on_ev:
% 0.16/0.41 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Linear__Temporal__Logic__on__Streams_Onxt,type,
% 0.16/0.41 linear1494993505on_nxt:
% 0.16/0.41 !>[A: $tType,B: $tType] : ( ( ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Linear__Temporal__Logic__on__Streams_Owait,type,
% 0.16/0.41 linear1335279038n_wait:
% 0.16/0.41 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > nat ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.16/0.41 ord_less:
% 0.16/0.41 !>[A: $tType] : ( A > A > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.16/0.41 ord_less_eq:
% 0.16/0.41 !>[A: $tType] : ( A > A > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Pure_Otype,type,
% 0.16/0.41 type2:
% 0.16/0.41 !>[A: $tType] : ( itself @ A ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Set_OCollect,type,
% 0.16/0.41 collect:
% 0.16/0.41 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_Stream_Osdrop,type,
% 0.16/0.41 sdrop:
% 0.16/0.41 !>[A: $tType] : ( nat > ( stream @ A ) > ( stream @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_c_member,type,
% 0.16/0.41 member:
% 0.16/0.41 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.16/0.41
% 0.16/0.41 thf(sy_v_P,type,
% 0.16/0.41 p: ( stream @ a ) > $o ).
% 0.16/0.41
% 0.16/0.41 thf(sy_v__092_060omega_062,type,
% 0.16/0.41 omega: stream @ a ).
% 0.16/0.41
% 0.16/0.41 %----Relevant facts (256)
% 0.16/0.41 thf(fact_0__092_060open_062ev_A_Ialw_A_I_092_060lambda_062xs_O_A_092_060not_062_AP_Axs_J_J_A_092_060omega_062_A_092_060Longrightarrow_062_Ascount_AP_A_092_060omega_062_A_061_Aenat_A_Icard_A_123i_O_AP_A_Isdrop_Ai_A_092_060omega_062_J_125_J_092_060close_062,axiom,
% 0.16/0.41 ( ( linear505997466_on_ev @ a
% 0.16/0.41 @ ( linear1386806755on_alw @ a
% 0.16/0.41 @ ^ [Xs: stream @ a] :
% 0.16/0.41 ~ ( p @ Xs ) )
% 0.16/0.41 @ omega )
% 0.16/0.41 => ( ( coindu1365464361scount @ a @ p @ omega )
% 0.16/0.41 = ( extended_enat2
% 0.16/0.41 @ ( finite_card @ nat
% 0.16/0.41 @ ( collect @ nat
% 0.16/0.41 @ ^ [I: nat] : ( p @ ( sdrop @ a @ I @ omega ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % \<open>ev (alw (\<lambda>xs. \<not> P xs)) \<omega> \<Longrightarrow> scount P \<omega> = enat (card {i. P (sdrop i \<omega>)})\<close>
% 0.16/0.41 thf(fact_1_not__ev__not,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( ^ [Xs: stream @ A] :
% 0.16/0.41 ~ ( linear505997466_on_ev @ A
% 0.16/0.41 @ ^ [Xt: stream @ A] :
% 0.16/0.41 ~ ( Phi @ Xt )
% 0.16/0.41 @ Xs ) )
% 0.16/0.41 = ( linear1386806755on_alw @ A @ Phi ) ) ).
% 0.16/0.41
% 0.16/0.41 % not_ev_not
% 0.16/0.41 thf(fact_2_not__alw__not,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( ^ [Xs: stream @ A] :
% 0.16/0.41 ~ ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xt: stream @ A] :
% 0.16/0.41 ~ ( Phi @ Xt )
% 0.16/0.41 @ Xs ) )
% 0.16/0.41 = ( linear505997466_on_ev @ A @ Phi ) ) ).
% 0.16/0.41
% 0.16/0.41 % not_alw_not
% 0.16/0.41 thf(fact_3_enat__ord__simps_I6_J,axiom,
% 0.16/0.41 ! [Q: extended_enat] :
% 0.16/0.41 ~ ( ord_less @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ Q ) ).
% 0.16/0.41
% 0.16/0.41 % enat_ord_simps(6)
% 0.16/0.41 thf(fact_4_enat__ord__simps_I4_J,axiom,
% 0.16/0.41 ! [Q: extended_enat] :
% 0.16/0.41 ( ( ord_less @ extended_enat @ Q @ ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 = ( Q
% 0.16/0.41 != ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat_ord_simps(4)
% 0.16/0.41 thf(fact_5_ev__ev,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A @ ( linear505997466_on_ev @ A @ Phi ) )
% 0.16/0.41 = ( linear505997466_on_ev @ A @ Phi ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev_ev
% 0.16/0.41 thf(fact_6_alw__alw,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ Phi ) )
% 0.16/0.41 = ( linear1386806755on_alw @ A @ Phi ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_alw
% 0.16/0.41 thf(fact_7_not__ev,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( ^ [Xs: stream @ A] :
% 0.16/0.41 ~ ( linear505997466_on_ev @ A @ Phi @ Xs ) )
% 0.16/0.41 = ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ~ ( Phi @ Xs ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % not_ev
% 0.16/0.41 thf(fact_8_not__alw,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( ^ [Xs: stream @ A] :
% 0.16/0.41 ~ ( linear1386806755on_alw @ A @ Phi @ Xs ) )
% 0.16/0.41 = ( linear505997466_on_ev @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ~ ( Phi @ Xs ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % not_alw
% 0.16/0.41 thf(fact_9_not__ev__iff,axiom,
% 0.16/0.41 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.16/0.41 ( ( ~ ( linear505997466_on_ev @ A @ P @ Omega ) )
% 0.16/0.41 = ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ~ ( P @ Xs )
% 0.16/0.41 @ Omega ) ) ).
% 0.16/0.41
% 0.16/0.41 % not_ev_iff
% 0.16/0.41 thf(fact_10_ev__alw__aand,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
% 0.16/0.41 => ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Psi ) @ Xs2 )
% 0.16/0.41 => ( linear505997466_on_ev @ A
% 0.16/0.41 @ ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs )
% 0.16/0.41 & ( Psi @ Xs ) ) )
% 0.16/0.41 @ Xs2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev_alw_aand
% 0.16/0.41 thf(fact_11_ev__alw__impl,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 0.16/0.41 => ( ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs )
% 0.16/0.41 => ( Psi @ Xs ) )
% 0.16/0.41 @ Xs2 )
% 0.16/0.41 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev_alw_impl
% 0.16/0.41 thf(fact_12_not__alw__iff,axiom,
% 0.16/0.41 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.16/0.41 ( ( ~ ( linear1386806755on_alw @ A @ P @ Omega ) )
% 0.16/0.41 = ( linear505997466_on_ev @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ~ ( P @ Xs )
% 0.16/0.41 @ Omega ) ) ).
% 0.16/0.41
% 0.16/0.41 % not_alw_iff
% 0.16/0.41 thf(fact_13_enat_Oinject,axiom,
% 0.16/0.41 ! [Nat: nat,Nat2: nat] :
% 0.16/0.41 ( ( ( extended_enat2 @ Nat )
% 0.16/0.41 = ( extended_enat2 @ Nat2 ) )
% 0.16/0.41 = ( Nat = Nat2 ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat.inject
% 0.16/0.41 thf(fact_14_not__infinity__eq,axiom,
% 0.16/0.41 ! [X: extended_enat] :
% 0.16/0.41 ( ( X
% 0.16/0.41 != ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 = ( ? [I: nat] :
% 0.16/0.41 ( X
% 0.16/0.41 = ( extended_enat2 @ I ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % not_infinity_eq
% 0.16/0.41 thf(fact_15_not__enat__eq,axiom,
% 0.16/0.41 ! [X: extended_enat] :
% 0.16/0.41 ( ( ! [Y: nat] :
% 0.16/0.41 ( X
% 0.16/0.41 != ( extended_enat2 @ Y ) ) )
% 0.16/0.41 = ( X
% 0.16/0.41 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % not_enat_eq
% 0.16/0.41 thf(fact_16_enat__ex__split,axiom,
% 0.16/0.41 ( ( ^ [P2: extended_enat > $o] :
% 0.16/0.41 ? [X2: extended_enat] : ( P2 @ X2 ) )
% 0.16/0.41 = ( ^ [P3: extended_enat > $o] :
% 0.16/0.41 ( ( P3 @ ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 | ? [X3: nat] : ( P3 @ ( extended_enat2 @ X3 ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat_ex_split
% 0.16/0.41 thf(fact_17_enat_Oinducts,axiom,
% 0.16/0.41 ! [P: extended_enat > $o,Enat: extended_enat] :
% 0.16/0.41 ( ! [Nat3: nat] : ( P @ ( extended_enat2 @ Nat3 ) )
% 0.16/0.41 => ( ( P @ ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 => ( P @ Enat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat.inducts
% 0.16/0.41 thf(fact_18_enat_Oexhaust,axiom,
% 0.16/0.41 ! [Y2: extended_enat] :
% 0.16/0.41 ( ! [Nat3: nat] :
% 0.16/0.41 ( Y2
% 0.16/0.41 != ( extended_enat2 @ Nat3 ) )
% 0.16/0.41 => ( Y2
% 0.16/0.41 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat.exhaust
% 0.16/0.41 thf(fact_19_enat3__cases,axiom,
% 0.16/0.41 ! [Y2: extended_enat,Ya: extended_enat,Yb: extended_enat] :
% 0.16/0.41 ( ( ? [Nat3: nat] :
% 0.16/0.41 ( Y2
% 0.16/0.41 = ( extended_enat2 @ Nat3 ) )
% 0.16/0.41 => ( ? [Nata: nat] :
% 0.16/0.41 ( Ya
% 0.16/0.41 = ( extended_enat2 @ Nata ) )
% 0.16/0.41 => ! [Natb: nat] :
% 0.16/0.41 ( Yb
% 0.16/0.41 != ( extended_enat2 @ Natb ) ) ) )
% 0.16/0.41 => ( ( ? [Nat3: nat] :
% 0.16/0.41 ( Y2
% 0.16/0.41 = ( extended_enat2 @ Nat3 ) )
% 0.16/0.41 => ( ? [Nata: nat] :
% 0.16/0.41 ( Ya
% 0.16/0.41 = ( extended_enat2 @ Nata ) )
% 0.16/0.41 => ( Yb
% 0.16/0.41 != ( extend1396239628finity @ extended_enat ) ) ) )
% 0.16/0.41 => ( ( ? [Nat3: nat] :
% 0.16/0.41 ( Y2
% 0.16/0.41 = ( extended_enat2 @ Nat3 ) )
% 0.16/0.41 => ( ( Ya
% 0.16/0.41 = ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 => ! [Nata: nat] :
% 0.16/0.41 ( Yb
% 0.16/0.41 != ( extended_enat2 @ Nata ) ) ) )
% 0.16/0.41 => ( ( ? [Nat3: nat] :
% 0.16/0.41 ( Y2
% 0.16/0.41 = ( extended_enat2 @ Nat3 ) )
% 0.16/0.41 => ( ( Ya
% 0.16/0.41 = ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 => ( Yb
% 0.16/0.41 != ( extend1396239628finity @ extended_enat ) ) ) )
% 0.16/0.41 => ( ( ( Y2
% 0.16/0.41 = ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 => ( ? [Nat3: nat] :
% 0.16/0.41 ( Ya
% 0.16/0.41 = ( extended_enat2 @ Nat3 ) )
% 0.16/0.41 => ! [Nata: nat] :
% 0.16/0.41 ( Yb
% 0.16/0.41 != ( extended_enat2 @ Nata ) ) ) )
% 0.16/0.41 => ( ( ( Y2
% 0.16/0.41 = ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 => ( ? [Nat3: nat] :
% 0.16/0.41 ( Ya
% 0.16/0.41 = ( extended_enat2 @ Nat3 ) )
% 0.16/0.41 => ( Yb
% 0.16/0.41 != ( extend1396239628finity @ extended_enat ) ) ) )
% 0.16/0.41 => ( ( ( Y2
% 0.16/0.41 = ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 => ( ( Ya
% 0.16/0.41 = ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 => ! [Nat3: nat] :
% 0.16/0.41 ( Yb
% 0.16/0.41 != ( extended_enat2 @ Nat3 ) ) ) )
% 0.16/0.41 => ~ ( ( Y2
% 0.16/0.41 = ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 => ( ( Ya
% 0.16/0.41 = ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 => ( Yb
% 0.16/0.41 != ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat3_cases
% 0.16/0.41 thf(fact_20_enat2__cases,axiom,
% 0.16/0.41 ! [Y2: extended_enat,Ya: extended_enat] :
% 0.16/0.41 ( ( ? [Nat3: nat] :
% 0.16/0.41 ( Y2
% 0.16/0.41 = ( extended_enat2 @ Nat3 ) )
% 0.16/0.41 => ! [Nata: nat] :
% 0.16/0.41 ( Ya
% 0.16/0.41 != ( extended_enat2 @ Nata ) ) )
% 0.16/0.41 => ( ( ? [Nat3: nat] :
% 0.16/0.41 ( Y2
% 0.16/0.41 = ( extended_enat2 @ Nat3 ) )
% 0.16/0.41 => ( Ya
% 0.16/0.41 != ( extend1396239628finity @ extended_enat ) ) )
% 0.16/0.41 => ( ( ( Y2
% 0.16/0.41 = ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 => ! [Nat3: nat] :
% 0.16/0.41 ( Ya
% 0.16/0.41 != ( extended_enat2 @ Nat3 ) ) )
% 0.16/0.41 => ~ ( ( Y2
% 0.16/0.41 = ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 => ( Ya
% 0.16/0.41 != ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat2_cases
% 0.16/0.41 thf(fact_21_enat_Odistinct_I1_J,axiom,
% 0.16/0.41 ! [Nat: nat] :
% 0.16/0.41 ( ( extended_enat2 @ Nat )
% 0.16/0.41 != ( extend1396239628finity @ extended_enat ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat.distinct(1)
% 0.16/0.41 thf(fact_22_enat_Odistinct_I2_J,axiom,
% 0.16/0.41 ! [Nat4: nat] :
% 0.16/0.41 ( ( extend1396239628finity @ extended_enat )
% 0.16/0.41 != ( extended_enat2 @ Nat4 ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat.distinct(2)
% 0.16/0.41 thf(fact_23_enat__iless,axiom,
% 0.16/0.41 ! [N: extended_enat,M: nat] :
% 0.16/0.41 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 0.16/0.41 => ? [K: nat] :
% 0.16/0.41 ( N
% 0.16/0.41 = ( extended_enat2 @ K ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat_iless
% 0.16/0.41 thf(fact_24_chain__incr,axiom,
% 0.16/0.41 ! [A: $tType,Y3: A > extended_enat,K2: nat] :
% 0.16/0.41 ( ! [I2: A] :
% 0.16/0.41 ? [J: A] : ( ord_less @ extended_enat @ ( Y3 @ I2 ) @ ( Y3 @ J ) )
% 0.16/0.41 => ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K2 ) @ ( Y3 @ J2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % chain_incr
% 0.16/0.41 thf(fact_25_alw__iff__sdrop,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A )
% 0.16/0.41 = ( ^ [P3: ( stream @ A ) > $o,Omega2: stream @ A] :
% 0.16/0.41 ! [M2: nat] : ( P3 @ ( sdrop @ A @ M2 @ Omega2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_iff_sdrop
% 0.16/0.41 thf(fact_26_ev__iff__sdrop,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A )
% 0.16/0.41 = ( ^ [P3: ( stream @ A ) > $o,Omega2: stream @ A] :
% 0.16/0.41 ? [M2: nat] : ( P3 @ ( sdrop @ A @ M2 @ Omega2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev_iff_sdrop
% 0.16/0.41 thf(fact_27_alw__sdrop,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,N: nat] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 0.16/0.41 => ( linear1386806755on_alw @ A @ Phi @ ( sdrop @ A @ N @ Xs2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_sdrop
% 0.16/0.41 thf(fact_28_all__imp__alw,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 0.16/0.41 ( ! [X1: stream @ A] : ( Phi @ X1 )
% 0.16/0.41 => ( linear1386806755on_alw @ A @ Phi @ Xs2 ) ) ).
% 0.16/0.41
% 0.16/0.41 % all_imp_alw
% 0.16/0.41 thf(fact_29_alw__mono,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 0.16/0.41 => ( ! [Xs3: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs3 )
% 0.16/0.41 => ( Psi @ Xs3 ) )
% 0.16/0.41 => ( linear1386806755on_alw @ A @ Psi @ Xs2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_mono
% 0.16/0.41 thf(fact_30_alw__cong,axiom,
% 0.16/0.41 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A,Q1: ( stream @ A ) > $o,Q2: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A @ P @ Omega )
% 0.16/0.41 => ( ! [Omega3: stream @ A] :
% 0.16/0.41 ( ( P @ Omega3 )
% 0.16/0.41 => ( ( Q1 @ Omega3 )
% 0.16/0.41 = ( Q2 @ Omega3 ) ) )
% 0.16/0.41 => ( ( linear1386806755on_alw @ A @ Q1 @ Omega )
% 0.16/0.41 = ( linear1386806755on_alw @ A @ Q2 @ Omega ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_cong
% 0.16/0.41 thf(fact_31_alw__alwD,axiom,
% 0.16/0.41 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A @ P @ Omega )
% 0.16/0.41 => ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ P ) @ Omega ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_alwD
% 0.16/0.41 thf(fact_32_ev__mono,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 0.16/0.41 => ( ! [Xs3: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs3 )
% 0.16/0.41 => ( Psi @ Xs3 ) )
% 0.16/0.41 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev_mono
% 0.16/0.41 thf(fact_33_ev_Obase,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs2 )
% 0.16/0.41 => ( linear505997466_on_ev @ A @ Phi @ Xs2 ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev.base
% 0.16/0.41 thf(fact_34_alwD,axiom,
% 0.16/0.41 ! [A: $tType,P: ( stream @ A ) > $o,X: stream @ A] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A @ P @ X )
% 0.16/0.41 => ( P @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % alwD
% 0.16/0.41 thf(fact_35_enat__less__induct,axiom,
% 0.16/0.41 ! [P: extended_enat > $o,N: extended_enat] :
% 0.16/0.41 ( ! [N2: extended_enat] :
% 0.16/0.41 ( ! [M3: extended_enat] :
% 0.16/0.41 ( ( ord_less @ extended_enat @ M3 @ N2 )
% 0.16/0.41 => ( P @ M3 ) )
% 0.16/0.41 => ( P @ N2 ) )
% 0.16/0.41 => ( P @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat_less_induct
% 0.16/0.41 thf(fact_36_infinity__ilessE,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ~ ( ord_less @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% 0.16/0.41
% 0.16/0.41 % infinity_ilessE
% 0.16/0.41 thf(fact_37_less__infinityE,axiom,
% 0.16/0.41 ! [N: extended_enat] :
% 0.16/0.41 ( ( ord_less @ extended_enat @ N @ ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 => ~ ! [K: nat] :
% 0.16/0.41 ( N
% 0.16/0.41 != ( extended_enat2 @ K ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % less_infinityE
% 0.16/0.41 thf(fact_38_enat__ord__code_I4_J,axiom,
% 0.16/0.41 ! [M: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat_ord_code(4)
% 0.16/0.41 thf(fact_39_alw__ev__sdrop,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,M: nat,Xs2: stream @ A] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ ( sdrop @ A @ M @ Xs2 ) )
% 0.16/0.41 => ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ Xs2 ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_ev_sdrop
% 0.16/0.41 thf(fact_40_alw__False,axiom,
% 0.16/0.41 ! [A: $tType,Omega: stream @ A] :
% 0.16/0.41 ~ ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [X3: stream @ A] : $false
% 0.16/0.41 @ Omega ) ).
% 0.16/0.41
% 0.16/0.41 % alw_False
% 0.16/0.41 thf(fact_41_ev__False,axiom,
% 0.16/0.41 ! [A: $tType,Omega: stream @ A] :
% 0.16/0.41 ~ ( linear505997466_on_ev @ A
% 0.16/0.41 @ ^ [X3: stream @ A] : $false
% 0.16/0.41 @ Omega ) ).
% 0.16/0.41
% 0.16/0.41 % ev_False
% 0.16/0.41 thf(fact_42_alw__aand,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs )
% 0.16/0.41 & ( Psi @ Xs ) ) )
% 0.16/0.41 = ( ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A @ Phi @ Xs )
% 0.16/0.41 & ( linear1386806755on_alw @ A @ Psi @ Xs ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_aand
% 0.16/0.41 thf(fact_43_alw__mp,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 0.16/0.41 => ( ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs )
% 0.16/0.41 => ( Psi @ Xs ) )
% 0.16/0.41 @ Xs2 )
% 0.16/0.41 => ( linear1386806755on_alw @ A @ Psi @ Xs2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_mp
% 0.16/0.41 thf(fact_44_ev__or,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs )
% 0.16/0.41 | ( Psi @ Xs ) ) )
% 0.16/0.41 = ( ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A @ Phi @ Xs )
% 0.16/0.41 | ( linear505997466_on_ev @ A @ Psi @ Xs ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev_or
% 0.16/0.41 thf(fact_45_mem__Collect__eq,axiom,
% 0.16/0.41 ! [A: $tType,A2: A,P: A > $o] :
% 0.16/0.41 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.16/0.41 = ( P @ A2 ) ) ).
% 0.16/0.41
% 0.16/0.41 % mem_Collect_eq
% 0.16/0.41 thf(fact_46_Collect__mem__eq,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A] :
% 0.16/0.41 ( ( collect @ A
% 0.16/0.41 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.16/0.41 = A3 ) ).
% 0.16/0.41
% 0.16/0.41 % Collect_mem_eq
% 0.16/0.41 thf(fact_47_Collect__cong,axiom,
% 0.16/0.41 ! [A: $tType,P: A > $o,Q3: A > $o] :
% 0.16/0.41 ( ! [X4: A] :
% 0.16/0.41 ( ( P @ X4 )
% 0.16/0.41 = ( Q3 @ X4 ) )
% 0.16/0.41 => ( ( collect @ A @ P )
% 0.16/0.41 = ( collect @ A @ Q3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Collect_cong
% 0.16/0.41 thf(fact_48_ext,axiom,
% 0.16/0.41 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.16/0.41 ( ! [X4: A] :
% 0.16/0.41 ( ( F @ X4 )
% 0.16/0.41 = ( G @ X4 ) )
% 0.16/0.41 => ( F = G ) ) ).
% 0.16/0.41
% 0.16/0.41 % ext
% 0.16/0.41 thf(fact_49_scount__eq__card,axiom,
% 0.16/0.41 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A
% 0.16/0.41 @ ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ~ ( P @ Xs ) )
% 0.16/0.41 @ Omega )
% 0.16/0.41 => ( ( coindu1365464361scount @ A @ P @ Omega )
% 0.16/0.41 = ( extended_enat2
% 0.16/0.41 @ ( finite_card @ nat
% 0.16/0.41 @ ( collect @ nat
% 0.16/0.41 @ ^ [I: nat] : ( P @ ( sdrop @ A @ I @ Omega ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % scount_eq_card
% 0.16/0.41 thf(fact_50_ev__alw__imp__alw__ev,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
% 0.16/0.41 => ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ Xs2 ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev_alw_imp_alw_ev
% 0.16/0.41 thf(fact_51_ev__cong,axiom,
% 0.16/0.41 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A,Q1: ( stream @ A ) > $o,Q2: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A @ P @ Omega )
% 0.16/0.41 => ( ! [Omega3: stream @ A] :
% 0.16/0.41 ( ( P @ Omega3 )
% 0.16/0.41 => ( ( Q1 @ Omega3 )
% 0.16/0.41 = ( Q2 @ Omega3 ) ) )
% 0.16/0.41 => ( ( linear505997466_on_ev @ A @ Q1 @ Omega )
% 0.16/0.41 = ( linear505997466_on_ev @ A @ Q2 @ Omega ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev_cong
% 0.16/0.41 thf(fact_52_alw__ev__imp__ev__alw,axiom,
% 0.16/0.41 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Omega )
% 0.16/0.41 => ( linear505997466_on_ev @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( P @ Xs )
% 0.16/0.41 & ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Xs ) )
% 0.16/0.41 @ Omega ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_ev_imp_ev_alw
% 0.16/0.41 thf(fact_53_ev__alw__alw__impl,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
% 0.16/0.41 => ( ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A @ Phi @ Xs )
% 0.16/0.41 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 0.16/0.41 @ Xs2 )
% 0.16/0.41 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev_alw_alw_impl
% 0.16/0.41 thf(fact_54_alw__impl__ev__alw,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs )
% 0.16/0.41 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 0.16/0.41 @ Xs2 )
% 0.16/0.41 => ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A @ Phi @ Xs )
% 0.16/0.41 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 0.16/0.41 @ Xs2 ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_impl_ev_alw
% 0.16/0.41 thf(fact_55_alw__alw__impl__ev,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A @ Phi @ Xs )
% 0.16/0.41 => ( linear505997466_on_ev @ A @ Psi @ Xs ) ) )
% 0.16/0.41 = ( ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs )
% 0.16/0.41 => ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Psi ) @ Xs ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_alw_impl_ev
% 0.16/0.41 thf(fact_56_ev__alw__impl__ev,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 0.16/0.41 => ( ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs )
% 0.16/0.41 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 0.16/0.41 @ Xs2 )
% 0.16/0.41 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev_alw_impl_ev
% 0.16/0.41 thf(fact_57_enat_Osimps_I7_J,axiom,
% 0.16/0.41 ! [T: $tType,F1: nat > T,F2: T] :
% 0.16/0.41 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 = F2 ) ).
% 0.16/0.41
% 0.16/0.41 % enat.simps(7)
% 0.16/0.41 thf(fact_58_enat_Osimps_I6_J,axiom,
% 0.16/0.41 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 0.16/0.41 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 0.16/0.41 = ( F1 @ Nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat.simps(6)
% 0.16/0.41 thf(fact_59_sdrop__wait,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 0.16/0.41 => ( Phi @ ( sdrop @ A @ ( linear1335279038n_wait @ A @ Phi @ Xs2 ) @ Xs2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % sdrop_wait
% 0.16/0.41 thf(fact_60_enat_Osimps_I5_J,axiom,
% 0.16/0.41 ! [T: $tType,F1: nat > T,F2: T] :
% 0.16/0.41 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
% 0.16/0.41 = F2 ) ).
% 0.16/0.41
% 0.16/0.41 % enat.simps(5)
% 0.16/0.41 thf(fact_61_enat_Osimps_I4_J,axiom,
% 0.16/0.41 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 0.16/0.41 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 0.16/0.41 = ( F1 @ Nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat.simps(4)
% 0.16/0.41 thf(fact_62_infinite__iff__alw__ev,axiom,
% 0.16/0.41 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.16/0.41 ( ( ~ ( finite_finite2 @ nat
% 0.16/0.41 @ ( collect @ nat
% 0.16/0.41 @ ^ [M2: nat] : ( P @ ( sdrop @ A @ M2 @ Omega ) ) ) ) )
% 0.16/0.41 = ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Omega ) ) ).
% 0.16/0.41
% 0.16/0.41 % infinite_iff_alw_ev
% 0.16/0.41 thf(fact_63_ev__alw__not__HLD__finite,axiom,
% 0.16/0.41 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A
% 0.16/0.41 @ ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ~ ( P @ Xs ) )
% 0.16/0.41 @ Omega )
% 0.16/0.41 => ( finite_finite2 @ nat
% 0.16/0.41 @ ( collect @ nat
% 0.16/0.41 @ ^ [I: nat] : ( P @ ( sdrop @ A @ I @ Omega ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev_alw_not_HLD_finite
% 0.16/0.41 thf(fact_64_ev__alw__imp__nxt,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 0.16/0.41 => ( ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs )
% 0.16/0.41 => ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs ) )
% 0.16/0.41 @ Xs2 )
% 0.16/0.41 => ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev_alw_imp_nxt
% 0.16/0.41 thf(fact_65_variance,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( Phi @ Xs2 )
% 0.16/0.41 => ( ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs )
% 0.16/0.41 => ( ( Psi @ Xs )
% 0.16/0.41 | ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs ) ) )
% 0.16/0.41 @ Xs2 )
% 0.16/0.41 => ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 0.16/0.41 | ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % variance
% 0.16/0.41 thf(fact_66_ex__gt__or__lt,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [A2: A] :
% 0.16/0.41 ? [B2: A] :
% 0.16/0.41 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.41 | ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ex_gt_or_lt
% 0.16/0.41 thf(fact_67_linorder__neqE__linordered__idom,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [X: A,Y2: A] :
% 0.16/0.41 ( ( X != Y2 )
% 0.16/0.41 => ( ~ ( ord_less @ A @ X @ Y2 )
% 0.16/0.41 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_neqE_linordered_idom
% 0.16/0.41 thf(fact_68_enat__ord__simps_I2_J,axiom,
% 0.16/0.41 ! [M: nat,N: nat] :
% 0.16/0.41 ( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.16/0.41 = ( ord_less @ nat @ M @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat_ord_simps(2)
% 0.16/0.41 thf(fact_69_nxt__mono,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.16/0.41 ( ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs2 )
% 0.16/0.41 => ( ! [Xs3: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs3 )
% 0.16/0.41 => ( Psi @ Xs3 ) )
% 0.16/0.41 => ( linear1494993505on_nxt @ A @ $o @ Psi @ Xs2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nxt_mono
% 0.16/0.41 thf(fact_70_case__enat__def,axiom,
% 0.16/0.41 ! [T: $tType] :
% 0.16/0.41 ( ( extended_case_enat @ T )
% 0.16/0.41 = ( extended_rec_enat @ T ) ) ).
% 0.16/0.41
% 0.16/0.41 % case_enat_def
% 0.16/0.41 thf(fact_71_less__enat__def,axiom,
% 0.16/0.41 ( ( ord_less @ extended_enat )
% 0.16/0.41 = ( ^ [M2: extended_enat,N3: extended_enat] :
% 0.16/0.41 ( extended_case_enat @ $o
% 0.16/0.41 @ ^ [M1: nat] : ( extended_case_enat @ $o @ ( ord_less @ nat @ M1 ) @ $true @ N3 )
% 0.16/0.41 @ $false
% 0.16/0.41 @ M2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % less_enat_def
% 0.16/0.41 thf(fact_72_ev__nxt,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A )
% 0.16/0.41 = ( ^ [Phi2: ( stream @ A ) > $o,Xs: stream @ A] :
% 0.16/0.41 ( ( Phi2 @ Xs )
% 0.16/0.41 | ( linear1494993505on_nxt @ A @ $o @ ( linear505997466_on_ev @ A @ Phi2 ) @ Xs ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ev_nxt
% 0.16/0.41 thf(fact_73_alw__nxt,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( linear1386806755on_alw @ A )
% 0.16/0.41 = ( ^ [Phi2: ( stream @ A ) > $o,Xs: stream @ A] :
% 0.16/0.41 ( ( Phi2 @ Xs )
% 0.16/0.41 & ( linear1494993505on_nxt @ A @ $o @ ( linear1386806755on_alw @ A @ Phi2 ) @ Xs ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_nxt
% 0.16/0.41 thf(fact_74_alw__invar,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs2 )
% 0.16/0.41 => ( ( linear1386806755on_alw @ A
% 0.16/0.41 @ ^ [Xs: stream @ A] :
% 0.16/0.41 ( ( Phi @ Xs )
% 0.16/0.41 => ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs ) )
% 0.16/0.41 @ Xs2 )
% 0.16/0.41 => ( linear1386806755on_alw @ A @ Phi @ Xs2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % alw_invar
% 0.16/0.41 thf(fact_75_less__enatE,axiom,
% 0.16/0.41 ! [N: extended_enat,M: nat] :
% 0.16/0.41 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 0.16/0.41 => ~ ! [K: nat] :
% 0.16/0.41 ( ( N
% 0.16/0.41 = ( extended_enat2 @ K ) )
% 0.16/0.41 => ~ ( ord_less @ nat @ K @ M ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % less_enatE
% 0.16/0.41 thf(fact_76_card__Collect__less__nat,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( finite_card @ nat
% 0.16/0.41 @ ( collect @ nat
% 0.16/0.41 @ ^ [I: nat] : ( ord_less @ nat @ I @ N ) ) )
% 0.16/0.41 = N ) ).
% 0.16/0.41
% 0.16/0.41 % card_Collect_less_nat
% 0.16/0.41 thf(fact_77_finite__Collect__less__nat,axiom,
% 0.16/0.41 ! [K2: nat] :
% 0.16/0.41 ( finite_finite2 @ nat
% 0.16/0.41 @ ( collect @ nat
% 0.16/0.41 @ ^ [N3: nat] : ( ord_less @ nat @ N3 @ K2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_Collect_less_nat
% 0.16/0.41 thf(fact_78_finite__Collect__disjI,axiom,
% 0.16/0.41 ! [A: $tType,P: A > $o,Q3: A > $o] :
% 0.16/0.41 ( ( finite_finite2 @ A
% 0.16/0.41 @ ( collect @ A
% 0.16/0.41 @ ^ [X3: A] :
% 0.16/0.41 ( ( P @ X3 )
% 0.16/0.41 | ( Q3 @ X3 ) ) ) )
% 0.16/0.41 = ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.16/0.41 & ( finite_finite2 @ A @ ( collect @ A @ Q3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_Collect_disjI
% 0.16/0.41 thf(fact_79_finite__Collect__conjI,axiom,
% 0.16/0.41 ! [A: $tType,P: A > $o,Q3: A > $o] :
% 0.16/0.41 ( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.16/0.41 | ( finite_finite2 @ A @ ( collect @ A @ Q3 ) ) )
% 0.16/0.41 => ( finite_finite2 @ A
% 0.16/0.41 @ ( collect @ A
% 0.16/0.41 @ ^ [X3: A] :
% 0.16/0.41 ( ( P @ X3 )
% 0.16/0.41 & ( Q3 @ X3 ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_Collect_conjI
% 0.16/0.41 thf(fact_80_finite__code,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.16/0.41 => ( ( finite_finite2 @ A )
% 0.16/0.41 = ( ^ [A4: set @ A] : $true ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_code
% 0.16/0.41 thf(fact_81_psubset__card__mono,axiom,
% 0.16/0.41 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.16/0.41 ( ( finite_finite2 @ A @ B3 )
% 0.16/0.41 => ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % psubset_card_mono
% 0.16/0.41 thf(fact_82_finite__M__bounded__by__nat,axiom,
% 0.16/0.41 ! [P: nat > $o,I3: nat] :
% 0.16/0.41 ( finite_finite2 @ nat
% 0.16/0.41 @ ( collect @ nat
% 0.16/0.41 @ ^ [K3: nat] :
% 0.16/0.41 ( ( P @ K3 )
% 0.16/0.41 & ( ord_less @ nat @ K3 @ I3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_M_bounded_by_nat
% 0.16/0.41 thf(fact_83_finite__set__choice,axiom,
% 0.16/0.41 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 0.16/0.41 ( ( finite_finite2 @ A @ A3 )
% 0.16/0.41 => ( ! [X4: A] :
% 0.16/0.41 ( ( member @ A @ X4 @ A3 )
% 0.16/0.41 => ? [X12: B] : ( P @ X4 @ X12 ) )
% 0.16/0.41 => ? [F3: A > B] :
% 0.16/0.41 ! [X5: A] :
% 0.16/0.41 ( ( member @ A @ X5 @ A3 )
% 0.16/0.41 => ( P @ X5 @ ( F3 @ X5 ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_set_choice
% 0.16/0.41 thf(fact_84_finite,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite
% 0.16/0.41 thf(fact_85_finite__psubset__induct,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
% 0.16/0.41 ( ( finite_finite2 @ A @ A3 )
% 0.16/0.41 => ( ! [A5: set @ A] :
% 0.16/0.41 ( ( finite_finite2 @ A @ A5 )
% 0.16/0.41 => ( ! [B4: set @ A] :
% 0.16/0.41 ( ( ord_less @ ( set @ A ) @ B4 @ A5 )
% 0.16/0.41 => ( P @ B4 ) )
% 0.16/0.41 => ( P @ A5 ) ) )
% 0.16/0.41 => ( P @ A3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_psubset_induct
% 0.16/0.41 thf(fact_86_bounded__nat__set__is__finite,axiom,
% 0.16/0.41 ! [N4: set @ nat,N: nat] :
% 0.16/0.41 ( ! [X4: nat] :
% 0.16/0.41 ( ( member @ nat @ X4 @ N4 )
% 0.16/0.41 => ( ord_less @ nat @ X4 @ N ) )
% 0.16/0.41 => ( finite_finite2 @ nat @ N4 ) ) ).
% 0.16/0.41
% 0.16/0.41 % bounded_nat_set_is_finite
% 0.16/0.41 thf(fact_87_finite__nat__set__iff__bounded,axiom,
% 0.16/0.41 ( ( finite_finite2 @ nat )
% 0.16/0.41 = ( ^ [N5: set @ nat] :
% 0.16/0.41 ? [M2: nat] :
% 0.16/0.41 ! [X3: nat] :
% 0.16/0.41 ( ( member @ nat @ X3 @ N5 )
% 0.16/0.41 => ( ord_less @ nat @ X3 @ M2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_nat_set_iff_bounded
% 0.16/0.41 thf(fact_88_not__finite__existsD,axiom,
% 0.16/0.41 ! [A: $tType,P: A > $o] :
% 0.16/0.41 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.16/0.41 => ? [X1: A] : ( P @ X1 ) ) ).
% 0.16/0.41
% 0.16/0.41 % not_finite_existsD
% 0.16/0.41 thf(fact_89_pigeonhole__infinite__rel,axiom,
% 0.16/0.41 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R: A > B > $o] :
% 0.16/0.41 ( ~ ( finite_finite2 @ A @ A3 )
% 0.16/0.41 => ( ( finite_finite2 @ B @ B3 )
% 0.16/0.41 => ( ! [X4: A] :
% 0.16/0.41 ( ( member @ A @ X4 @ A3 )
% 0.16/0.41 => ? [Xa: B] :
% 0.16/0.41 ( ( member @ B @ Xa @ B3 )
% 0.16/0.41 & ( R @ X4 @ Xa ) ) )
% 0.16/0.41 => ? [X4: B] :
% 0.16/0.41 ( ( member @ B @ X4 @ B3 )
% 0.16/0.41 & ~ ( finite_finite2 @ A
% 0.16/0.41 @ ( collect @ A
% 0.16/0.41 @ ^ [A6: A] :
% 0.16/0.41 ( ( member @ A @ A6 @ A3 )
% 0.16/0.41 & ( R @ A6 @ X4 ) ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % pigeonhole_infinite_rel
% 0.16/0.41 thf(fact_90_unbounded__k__infinite,axiom,
% 0.16/0.41 ! [K2: nat,S2: set @ nat] :
% 0.16/0.41 ( ! [M4: nat] :
% 0.16/0.41 ( ( ord_less @ nat @ K2 @ M4 )
% 0.16/0.41 => ? [N6: nat] :
% 0.16/0.41 ( ( ord_less @ nat @ M4 @ N6 )
% 0.16/0.41 & ( member @ nat @ N6 @ S2 ) ) )
% 0.16/0.41 => ~ ( finite_finite2 @ nat @ S2 ) ) ).
% 0.16/0.41
% 0.16/0.41 % unbounded_k_infinite
% 0.16/0.41 thf(fact_91_infinite__nat__iff__unbounded,axiom,
% 0.16/0.41 ! [S2: set @ nat] :
% 0.16/0.41 ( ( ~ ( finite_finite2 @ nat @ S2 ) )
% 0.16/0.41 = ( ! [M2: nat] :
% 0.16/0.41 ? [N3: nat] :
% 0.16/0.41 ( ( ord_less @ nat @ M2 @ N3 )
% 0.16/0.41 & ( member @ nat @ N3 @ S2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % infinite_nat_iff_unbounded
% 0.16/0.41 thf(fact_92_less__set__def,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( ord_less @ ( set @ A ) )
% 0.16/0.41 = ( ^ [A4: set @ A,B5: set @ A] :
% 0.16/0.41 ( ord_less @ ( A > $o )
% 0.16/0.41 @ ^ [X3: A] : ( member @ A @ X3 @ A4 )
% 0.16/0.41 @ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % less_set_def
% 0.16/0.41 thf(fact_93_card__psubset,axiom,
% 0.16/0.41 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.16/0.41 ( ( finite_finite2 @ A @ B3 )
% 0.16/0.41 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) )
% 0.16/0.41 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % card_psubset
% 0.16/0.41 thf(fact_94_sdrop__wait__least,axiom,
% 0.16/0.41 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,N: nat] :
% 0.16/0.41 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 0.16/0.41 => ( ( Phi @ ( sdrop @ A @ N @ Xs2 ) )
% 0.16/0.41 => ( ord_less_eq @ nat @ ( linear1335279038n_wait @ A @ Phi @ Xs2 ) @ N ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % sdrop_wait_least
% 0.16/0.41 thf(fact_95_psubset__trans,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
% 0.16/0.41 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( ord_less @ ( set @ A ) @ B3 @ C )
% 0.16/0.41 => ( ord_less @ ( set @ A ) @ A3 @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % psubset_trans
% 0.16/0.41 thf(fact_96_subset__antisym,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 0.16/0.41 => ( A3 = B3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % subset_antisym
% 0.16/0.41 thf(fact_97_subsetI,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.16/0.41 ( ! [X4: A] :
% 0.16/0.41 ( ( member @ A @ X4 @ A3 )
% 0.16/0.41 => ( member @ A @ X4 @ B3 ) )
% 0.16/0.41 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.16/0.41
% 0.16/0.41 % subsetI
% 0.16/0.41 thf(fact_98_psubsetI,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( A3 != B3 )
% 0.16/0.41 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % psubsetI
% 0.16/0.41 thf(fact_99_finite__Collect__subsets,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A] :
% 0.16/0.41 ( ( finite_finite2 @ A @ A3 )
% 0.16/0.41 => ( finite_finite2 @ ( set @ A )
% 0.16/0.41 @ ( collect @ ( set @ A )
% 0.16/0.41 @ ^ [B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_Collect_subsets
% 0.16/0.41 thf(fact_100_finite__Collect__le__nat,axiom,
% 0.16/0.41 ! [K2: nat] :
% 0.16/0.41 ( finite_finite2 @ nat
% 0.16/0.41 @ ( collect @ nat
% 0.16/0.41 @ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ K2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_Collect_le_nat
% 0.16/0.41 thf(fact_101_infinite__nat__iff__unbounded__le,axiom,
% 0.16/0.41 ! [S2: set @ nat] :
% 0.16/0.41 ( ( ~ ( finite_finite2 @ nat @ S2 ) )
% 0.16/0.41 = ( ! [M2: nat] :
% 0.16/0.41 ? [N3: nat] :
% 0.16/0.41 ( ( ord_less_eq @ nat @ M2 @ N3 )
% 0.16/0.41 & ( member @ nat @ N3 @ S2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % infinite_nat_iff_unbounded_le
% 0.16/0.41 thf(fact_102_subset__iff__psubset__eq,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) )
% 0.16/0.41 = ( ^ [A4: set @ A,B5: set @ A] :
% 0.16/0.41 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 0.16/0.41 | ( A4 = B5 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % subset_iff_psubset_eq
% 0.16/0.41 thf(fact_103_subset__psubset__trans,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( ord_less @ ( set @ A ) @ B3 @ C )
% 0.16/0.41 => ( ord_less @ ( set @ A ) @ A3 @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % subset_psubset_trans
% 0.16/0.41 thf(fact_104_subset__not__subset__eq,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( ord_less @ ( set @ A ) )
% 0.16/0.41 = ( ^ [A4: set @ A,B5: set @ A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.16/0.41 & ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % subset_not_subset_eq
% 0.16/0.41 thf(fact_105_psubset__subset__trans,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
% 0.16/0.41 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C )
% 0.16/0.41 => ( ord_less @ ( set @ A ) @ A3 @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % psubset_subset_trans
% 0.16/0.41 thf(fact_106_psubset__imp__subset,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.16/0.41 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.16/0.41
% 0.16/0.41 % psubset_imp_subset
% 0.16/0.41 thf(fact_107_psubset__eq,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( ord_less @ ( set @ A ) )
% 0.16/0.41 = ( ^ [A4: set @ A,B5: set @ A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.16/0.41 & ( A4 != B5 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % psubset_eq
% 0.16/0.41 thf(fact_108_psubsetE,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.16/0.41 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % psubsetE
% 0.16/0.41 thf(fact_109_Collect__mono__iff,axiom,
% 0.16/0.41 ! [A: $tType,P: A > $o,Q3: A > $o] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q3 ) )
% 0.16/0.41 = ( ! [X3: A] :
% 0.16/0.41 ( ( P @ X3 )
% 0.16/0.41 => ( Q3 @ X3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Collect_mono_iff
% 0.16/0.41 thf(fact_110_contra__subsetD,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ~ ( member @ A @ C2 @ B3 )
% 0.16/0.41 => ~ ( member @ A @ C2 @ A3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % contra_subsetD
% 0.16/0.41 thf(fact_111_set__eq__subset,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( ^ [Y4: set @ A,Z: set @ A] : ( Y4 = Z ) )
% 0.16/0.41 = ( ^ [A4: set @ A,B5: set @ A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.16/0.41 & ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % set_eq_subset
% 0.16/0.41 thf(fact_112_subset__trans,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C )
% 0.16/0.41 => ( ord_less_eq @ ( set @ A ) @ A3 @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % subset_trans
% 0.16/0.41 thf(fact_113_Collect__mono,axiom,
% 0.16/0.41 ! [A: $tType,P: A > $o,Q3: A > $o] :
% 0.16/0.41 ( ! [X4: A] :
% 0.16/0.41 ( ( P @ X4 )
% 0.16/0.41 => ( Q3 @ X4 ) )
% 0.16/0.41 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Collect_mono
% 0.16/0.41 thf(fact_114_subset__refl,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.16/0.41
% 0.16/0.41 % subset_refl
% 0.16/0.41 thf(fact_115_rev__subsetD,axiom,
% 0.16/0.41 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.16/0.41 ( ( member @ A @ C2 @ A3 )
% 0.16/0.41 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % rev_subsetD
% 0.16/0.41 thf(fact_116_subset__iff,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) )
% 0.16/0.41 = ( ^ [A4: set @ A,B5: set @ A] :
% 0.16/0.41 ! [T2: A] :
% 0.16/0.41 ( ( member @ A @ T2 @ A4 )
% 0.16/0.41 => ( member @ A @ T2 @ B5 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % subset_iff
% 0.16/0.41 thf(fact_117_set__rev__mp,axiom,
% 0.16/0.41 ! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
% 0.16/0.41 ( ( member @ A @ X @ A3 )
% 0.16/0.41 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( member @ A @ X @ B3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % set_rev_mp
% 0.16/0.41 thf(fact_118_equalityD2,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.16/0.41 ( ( A3 = B3 )
% 0.16/0.41 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% 0.16/0.41
% 0.16/0.41 % equalityD2
% 0.16/0.41 thf(fact_119_equalityD1,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.16/0.41 ( ( A3 = B3 )
% 0.16/0.41 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.16/0.41
% 0.16/0.41 % equalityD1
% 0.16/0.41 thf(fact_120_subset__eq,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) )
% 0.16/0.41 = ( ^ [A4: set @ A,B5: set @ A] :
% 0.16/0.41 ! [X3: A] :
% 0.16/0.41 ( ( member @ A @ X3 @ A4 )
% 0.16/0.41 => ( member @ A @ X3 @ B5 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % subset_eq
% 0.16/0.41 thf(fact_121_equalityE,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.16/0.41 ( ( A3 = B3 )
% 0.16/0.41 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % equalityE
% 0.16/0.41 thf(fact_122_subsetCE,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( member @ A @ C2 @ A3 )
% 0.16/0.41 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % subsetCE
% 0.16/0.41 thf(fact_123_subsetD,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( member @ A @ C2 @ A3 )
% 0.16/0.41 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % subsetD
% 0.16/0.41 thf(fact_124_in__mono,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A,X: A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( member @ A @ X @ A3 )
% 0.16/0.41 => ( member @ A @ X @ B3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % in_mono
% 0.16/0.41 thf(fact_125_set__mp,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A,X: A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( member @ A @ X @ A3 )
% 0.16/0.41 => ( member @ A @ X @ B3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % set_mp
% 0.16/0.41 thf(fact_126_card__seteq,axiom,
% 0.16/0.41 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.16/0.41 ( ( finite_finite2 @ A @ B3 )
% 0.16/0.41 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B3 ) @ ( finite_card @ A @ A3 ) )
% 0.16/0.41 => ( A3 = B3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % card_seteq
% 0.16/0.41 thf(fact_127_card__mono,axiom,
% 0.16/0.41 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.16/0.41 ( ( finite_finite2 @ A @ B3 )
% 0.16/0.41 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % card_mono
% 0.16/0.41 thf(fact_128_complete__interval,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [A2: A,B6: A,P: A > $o] :
% 0.16/0.41 ( ( ord_less @ A @ A2 @ B6 )
% 0.16/0.41 => ( ( P @ A2 )
% 0.16/0.41 => ( ~ ( P @ B6 )
% 0.16/0.41 => ? [C3: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.16/0.41 & ( ord_less_eq @ A @ C3 @ B6 )
% 0.16/0.41 & ! [X5: A] :
% 0.16/0.41 ( ( ( ord_less_eq @ A @ A2 @ X5 )
% 0.16/0.41 & ( ord_less @ A @ X5 @ C3 ) )
% 0.16/0.41 => ( P @ X5 ) )
% 0.16/0.41 & ! [D: A] :
% 0.16/0.41 ( ! [X4: A] :
% 0.16/0.41 ( ( ( ord_less_eq @ A @ A2 @ X4 )
% 0.16/0.41 & ( ord_less @ A @ X4 @ D ) )
% 0.16/0.41 => ( P @ X4 ) )
% 0.16/0.41 => ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % complete_interval
% 0.16/0.41 thf(fact_129_finite__subset,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( finite_finite2 @ A @ B3 )
% 0.16/0.41 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_subset
% 0.16/0.41 thf(fact_130_infinite__super,axiom,
% 0.16/0.41 ! [A: $tType,S2: set @ A,T3: set @ A] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) @ S2 @ T3 )
% 0.16/0.41 => ( ~ ( finite_finite2 @ A @ S2 )
% 0.16/0.41 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % infinite_super
% 0.16/0.41 thf(fact_131_rev__finite__subset,axiom,
% 0.16/0.41 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.16/0.41 ( ( finite_finite2 @ A @ B3 )
% 0.16/0.41 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % rev_finite_subset
% 0.16/0.41 thf(fact_132_finite__nat__set__iff__bounded__le,axiom,
% 0.16/0.41 ( ( finite_finite2 @ nat )
% 0.16/0.41 = ( ^ [N5: set @ nat] :
% 0.16/0.41 ? [M2: nat] :
% 0.16/0.41 ! [X3: nat] :
% 0.16/0.41 ( ( member @ nat @ X3 @ N5 )
% 0.16/0.41 => ( ord_less_eq @ nat @ X3 @ M2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_nat_set_iff_bounded_le
% 0.16/0.41 thf(fact_133_finite__less__ub,axiom,
% 0.16/0.41 ! [F: nat > nat,U: nat] :
% 0.16/0.41 ( ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( F @ N2 ) )
% 0.16/0.41 => ( finite_finite2 @ nat
% 0.16/0.41 @ ( collect @ nat
% 0.16/0.41 @ ^ [N3: nat] : ( ord_less_eq @ nat @ ( F @ N3 ) @ U ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_less_ub
% 0.16/0.41 thf(fact_134_card__subset__eq,axiom,
% 0.16/0.41 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.16/0.41 ( ( finite_finite2 @ A @ B3 )
% 0.16/0.41 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( ( finite_card @ A @ A3 )
% 0.16/0.41 = ( finite_card @ A @ B3 ) )
% 0.16/0.41 => ( A3 = B3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % card_subset_eq
% 0.16/0.41 thf(fact_135_infinite__arbitrarily__large,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,N: nat] :
% 0.16/0.41 ( ~ ( finite_finite2 @ A @ A3 )
% 0.16/0.41 => ? [B7: set @ A] :
% 0.16/0.41 ( ( finite_finite2 @ A @ B7 )
% 0.16/0.41 & ( ( finite_card @ A @ B7 )
% 0.16/0.41 = N )
% 0.16/0.41 & ( ord_less_eq @ ( set @ A ) @ B7 @ A3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % infinite_arbitrarily_large
% 0.16/0.41 thf(fact_136_psubsetD,axiom,
% 0.16/0.41 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 0.16/0.41 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.16/0.41 => ( ( member @ A @ C2 @ A3 )
% 0.16/0.41 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % psubsetD
% 0.16/0.41 thf(fact_137_order__refl,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_refl
% 0.16/0.41 thf(fact_138_ex__has__greatest__nat,axiom,
% 0.16/0.41 ! [A: $tType,P: A > $o,K2: A,M: A > nat,B6: nat] :
% 0.16/0.41 ( ( P @ K2 )
% 0.16/0.41 => ( ! [Y5: A] :
% 0.16/0.41 ( ( P @ Y5 )
% 0.16/0.41 => ( ord_less @ nat @ ( M @ Y5 ) @ B6 ) )
% 0.16/0.41 => ? [X4: A] :
% 0.16/0.41 ( ( P @ X4 )
% 0.16/0.41 & ! [Y6: A] :
% 0.16/0.41 ( ( P @ Y6 )
% 0.16/0.41 => ( ord_less_eq @ nat @ ( M @ Y6 ) @ ( M @ X4 ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ex_has_greatest_nat
% 0.16/0.41 thf(fact_139_less__mono__imp__le__mono,axiom,
% 0.16/0.41 ! [F: nat > nat,I3: nat,J3: nat] :
% 0.16/0.41 ( ! [I2: nat,J2: nat] :
% 0.16/0.41 ( ( ord_less @ nat @ I2 @ J2 )
% 0.16/0.41 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.16/0.41 => ( ( ord_less_eq @ nat @ I3 @ J3 )
% 0.16/0.41 => ( ord_less_eq @ nat @ ( F @ I3 ) @ ( F @ J3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % less_mono_imp_le_mono
% 0.16/0.41 thf(fact_140_le__neq__implies__less,axiom,
% 0.16/0.41 ! [M: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq @ nat @ M @ N )
% 0.16/0.41 => ( ( M != N )
% 0.16/0.41 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_neq_implies_less
% 0.16/0.41 thf(fact_141_less__or__eq__imp__le,axiom,
% 0.16/0.41 ! [M: nat,N: nat] :
% 0.16/0.41 ( ( ( ord_less @ nat @ M @ N )
% 0.16/0.41 | ( M = N ) )
% 0.16/0.41 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % less_or_eq_imp_le
% 0.16/0.41 thf(fact_142_enat__ord__simps_I5_J,axiom,
% 0.16/0.41 ! [Q: extended_enat] :
% 0.16/0.41 ( ( ord_less_eq @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ Q )
% 0.16/0.41 = ( Q
% 0.16/0.41 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat_ord_simps(5)
% 0.16/0.41 thf(fact_143_enat__ord__code_I3_J,axiom,
% 0.16/0.41 ! [Q: extended_enat] : ( ord_less_eq @ extended_enat @ Q @ ( extend1396239628finity @ extended_enat ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat_ord_code(3)
% 0.16/0.41 thf(fact_144_enat__ord__simps_I1_J,axiom,
% 0.16/0.41 ! [M: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.16/0.41 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat_ord_simps(1)
% 0.16/0.41 thf(fact_145_less__fun__def,axiom,
% 0.16/0.41 ! [B: $tType,A: $tType] :
% 0.16/0.41 ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.41 => ( ( ord_less @ ( A > B ) )
% 0.16/0.41 = ( ^ [F4: A > B,G2: A > B] :
% 0.16/0.41 ( ( ord_less_eq @ ( A > B ) @ F4 @ G2 )
% 0.16/0.41 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F4 ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % less_fun_def
% 0.16/0.41 thf(fact_146_finite__enat__bounded,axiom,
% 0.16/0.41 ! [A3: set @ extended_enat,N: nat] :
% 0.16/0.41 ( ! [Y5: extended_enat] :
% 0.16/0.41 ( ( member @ extended_enat @ Y5 @ A3 )
% 0.16/0.41 => ( ord_less_eq @ extended_enat @ Y5 @ ( extended_enat2 @ N ) ) )
% 0.16/0.41 => ( finite_finite2 @ extended_enat @ A3 ) ) ).
% 0.16/0.41
% 0.16/0.41 % finite_enat_bounded
% 0.16/0.41 thf(fact_147_enat__ile,axiom,
% 0.16/0.41 ! [N: extended_enat,M: nat] :
% 0.16/0.41 ( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 0.16/0.41 => ? [K: nat] :
% 0.16/0.41 ( N
% 0.16/0.41 = ( extended_enat2 @ K ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat_ile
% 0.16/0.41 thf(fact_148_enat__ord__simps_I3_J,axiom,
% 0.16/0.41 ! [Q: extended_enat] : ( ord_less_eq @ extended_enat @ Q @ ( extend1396239628finity @ extended_enat ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat_ord_simps(3)
% 0.16/0.41 thf(fact_149_less__eq__set__def,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( ord_less_eq @ ( set @ A ) )
% 0.16/0.41 = ( ^ [A4: set @ A,B5: set @ A] :
% 0.16/0.41 ( ord_less_eq @ ( A > $o )
% 0.16/0.41 @ ^ [X3: A] : ( member @ A @ X3 @ A4 )
% 0.16/0.41 @ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % less_eq_set_def
% 0.16/0.41 thf(fact_150_infinity__ileE,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ~ ( ord_less_eq @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% 0.16/0.41
% 0.16/0.41 % infinity_ileE
% 0.16/0.41 thf(fact_151_enat__ord__code_I5_J,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ~ ( ord_less_eq @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % enat_ord_code(5)
% 0.16/0.41 thf(fact_152_le__funD,axiom,
% 0.16/0.41 ! [B: $tType,A: $tType] :
% 0.16/0.41 ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.41 => ! [F: A > B,G: A > B,X: A] :
% 0.16/0.41 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.16/0.41 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_funD
% 0.16/0.41 thf(fact_153_le__funE,axiom,
% 0.16/0.41 ! [B: $tType,A: $tType] :
% 0.16/0.41 ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.41 => ! [F: A > B,G: A > B,X: A] :
% 0.16/0.41 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.16/0.41 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_funE
% 0.16/0.41 thf(fact_154_le__funI,axiom,
% 0.16/0.41 ! [B: $tType,A: $tType] :
% 0.16/0.41 ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.41 => ! [F: A > B,G: A > B] :
% 0.16/0.41 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.16/0.41 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_funI
% 0.16/0.41 thf(fact_155_le__fun__def,axiom,
% 0.16/0.41 ! [B: $tType,A: $tType] :
% 0.16/0.41 ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.41 => ( ( ord_less_eq @ ( A > B ) )
% 0.16/0.41 = ( ^ [F4: A > B,G2: A > B] :
% 0.16/0.41 ! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_fun_def
% 0.16/0.41 thf(fact_156_order__subst1,axiom,
% 0.16/0.41 ! [A: $tType,B: $tType] :
% 0.16/0.41 ( ( ( order @ B @ ( type2 @ B ) )
% 0.16/0.41 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.41 => ! [A2: A,F: B > A,B6: B,C2: B] :
% 0.16/0.41 ( ( ord_less_eq @ A @ A2 @ ( F @ B6 ) )
% 0.16/0.41 => ( ( ord_less_eq @ B @ B6 @ C2 )
% 0.16/0.41 => ( ! [X4: B,Y5: B] :
% 0.16/0.41 ( ( ord_less_eq @ B @ X4 @ Y5 )
% 0.16/0.41 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.16/0.41 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_157_order__subst2,axiom,
% 0.16/0.41 ! [A: $tType,C4: $tType] :
% 0.16/0.41 ( ( ( order @ C4 @ ( type2 @ C4 ) )
% 0.16/0.41 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.41 => ! [A2: A,B6: A,F: A > C4,C2: C4] :
% 0.16/0.41 ( ( ord_less_eq @ A @ A2 @ B6 )
% 0.16/0.41 => ( ( ord_less_eq @ C4 @ ( F @ B6 ) @ C2 )
% 0.16/0.41 => ( ! [X4: A,Y5: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 0.16/0.41 => ( ord_less_eq @ C4 @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.16/0.41 => ( ord_less_eq @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_158_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: $tType,B: $tType] :
% 0.16/0.41 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.41 & ( ord @ A @ ( type2 @ A ) ) )
% 0.16/0.41 => ! [A2: A,F: B > A,B6: B,C2: B] :
% 0.16/0.41 ( ( A2
% 0.16/0.41 = ( F @ B6 ) )
% 0.16/0.41 => ( ( ord_less_eq @ B @ B6 @ C2 )
% 0.16/0.41 => ( ! [X4: B,Y5: B] :
% 0.16/0.41 ( ( ord_less_eq @ B @ X4 @ Y5 )
% 0.16/0.41 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.16/0.41 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_159_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: $tType,B: $tType] :
% 0.16/0.41 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.41 & ( ord @ A @ ( type2 @ A ) ) )
% 0.16/0.41 => ! [A2: A,B6: A,F: A > B,C2: B] :
% 0.16/0.41 ( ( ord_less_eq @ A @ A2 @ B6 )
% 0.16/0.41 => ( ( ( F @ B6 )
% 0.16/0.41 = C2 )
% 0.16/0.41 => ( ! [X4: A,Y5: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 0.16/0.41 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.16/0.41 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_160_eq__iff,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.41 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.16/0.41 = ( ^ [X3: A,Y: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.16/0.41 & ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_iff
% 0.16/0.41 thf(fact_161_antisym,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [X: A,Y2: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.41 => ( ( ord_less_eq @ A @ Y2 @ X )
% 0.16/0.41 => ( X = Y2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % antisym
% 0.16/0.41 thf(fact_162_linear,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [X: A,Y2: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.41 | ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % linear
% 0.16/0.41 thf(fact_163_eq__refl,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [X: A,Y2: A] :
% 0.16/0.41 ( ( X = Y2 )
% 0.16/0.41 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_refl
% 0.16/0.41 thf(fact_164_le__cases,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [X: A,Y2: A] :
% 0.16/0.41 ( ~ ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.41 => ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_cases
% 0.16/0.41 thf(fact_165_order_Otrans,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [A2: A,B6: A,C2: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ A2 @ B6 )
% 0.16/0.41 => ( ( ord_less_eq @ A @ B6 @ C2 )
% 0.16/0.41 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order.trans
% 0.16/0.41 thf(fact_166_le__cases3,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [X: A,Y2: A,Z2: A] :
% 0.16/0.41 ( ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.41 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 0.16/0.41 => ( ( ( ord_less_eq @ A @ Y2 @ X )
% 0.16/0.41 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.16/0.41 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.16/0.41 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 0.16/0.41 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 0.16/0.41 => ~ ( ord_less_eq @ A @ Y2 @ X ) )
% 0.16/0.41 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.16/0.41 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.16/0.41 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.16/0.41 => ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_cases3
% 0.16/0.41 thf(fact_167_antisym__conv,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [Y2: A,X: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ Y2 @ X )
% 0.16/0.41 => ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.41 = ( X = Y2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % antisym_conv
% 0.16/0.41 thf(fact_168_ord__eq__le__trans,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( ord @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [A2: A,B6: A,C2: A] :
% 0.16/0.41 ( ( A2 = B6 )
% 0.16/0.41 => ( ( ord_less_eq @ A @ B6 @ C2 )
% 0.16/0.41 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_trans
% 0.16/0.41 thf(fact_169_ord__le__eq__trans,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( ord @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [A2: A,B6: A,C2: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ A2 @ B6 )
% 0.16/0.41 => ( ( B6 = C2 )
% 0.16/0.41 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_trans
% 0.16/0.41 thf(fact_170_order__class_Oorder_Oantisym,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [A2: A,B6: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ A2 @ B6 )
% 0.16/0.41 => ( ( ord_less_eq @ A @ B6 @ A2 )
% 0.16/0.41 => ( A2 = B6 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_class.order.antisym
% 0.16/0.41 thf(fact_171_order__trans,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [X: A,Y2: A,Z2: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.41 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.16/0.41 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_trans
% 0.16/0.41 thf(fact_172_dual__order_Orefl,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.refl
% 0.16/0.41 thf(fact_173_linorder__wlog,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [P: A > A > $o,A2: A,B6: A] :
% 0.16/0.41 ( ! [A7: A,B2: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ A7 @ B2 )
% 0.16/0.41 => ( P @ A7 @ B2 ) )
% 0.16/0.41 => ( ! [A7: A,B2: A] :
% 0.16/0.41 ( ( P @ B2 @ A7 )
% 0.16/0.41 => ( P @ A7 @ B2 ) )
% 0.16/0.41 => ( P @ A2 @ B6 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_wlog
% 0.16/0.41 thf(fact_174_dual__order_Otrans,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [B6: A,A2: A,C2: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ B6 @ A2 )
% 0.16/0.41 => ( ( ord_less_eq @ A @ C2 @ B6 )
% 0.16/0.41 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.trans
% 0.16/0.41 thf(fact_175_dual__order_Oantisym,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [B6: A,A2: A] :
% 0.16/0.41 ( ( ord_less_eq @ A @ B6 @ A2 )
% 0.16/0.41 => ( ( ord_less_eq @ A @ A2 @ B6 )
% 0.16/0.41 => ( A2 = B6 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.antisym
% 0.16/0.41 thf(fact_176_dual__order_Ostrict__implies__not__eq,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [B6: A,A2: A] :
% 0.16/0.41 ( ( ord_less @ A @ B6 @ A2 )
% 0.16/0.41 => ( A2 != B6 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.strict_implies_not_eq
% 0.16/0.41 thf(fact_177_order_Ostrict__implies__not__eq,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [A2: A,B6: A] :
% 0.16/0.41 ( ( ord_less @ A @ A2 @ B6 )
% 0.16/0.41 => ( A2 != B6 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order.strict_implies_not_eq
% 0.16/0.41 thf(fact_178_not__less__iff__gr__or__eq,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [X: A,Y2: A] :
% 0.16/0.41 ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 0.16/0.41 = ( ( ord_less @ A @ Y2 @ X )
% 0.16/0.41 | ( X = Y2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % not_less_iff_gr_or_eq
% 0.16/0.41 thf(fact_179_dual__order_Ostrict__trans,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [B6: A,A2: A,C2: A] :
% 0.16/0.41 ( ( ord_less @ A @ B6 @ A2 )
% 0.16/0.41 => ( ( ord_less @ A @ C2 @ B6 )
% 0.16/0.41 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.strict_trans
% 0.16/0.41 thf(fact_180_less__imp__not__less,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [X: A,Y2: A] :
% 0.16/0.41 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.41 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % less_imp_not_less
% 0.16/0.41 thf(fact_181_order_Ostrict__trans,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [A2: A,B6: A,C2: A] :
% 0.16/0.41 ( ( ord_less @ A @ A2 @ B6 )
% 0.16/0.41 => ( ( ord_less @ A @ B6 @ C2 )
% 0.16/0.41 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order.strict_trans
% 0.16/0.41 thf(fact_182_dual__order_Oirrefl,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [A2: A] :
% 0.16/0.41 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.irrefl
% 0.16/0.41 thf(fact_183_linorder__cases,axiom,
% 0.16/0.41 ! [A: $tType] :
% 0.16/0.41 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.41 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ~ ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ( ( X != Y2 )
% 0.16/0.42 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_cases
% 0.16/0.42 thf(fact_184_less__imp__triv,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A,P: $o] :
% 0.16/0.42 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ( ( ord_less @ A @ Y2 @ X )
% 0.16/0.42 => P ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_imp_triv
% 0.16/0.42 thf(fact_185_less__imp__not__eq2,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ( Y2 != X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_imp_not_eq2
% 0.16/0.42 thf(fact_186_antisym__conv3,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [Y2: A,X: A] :
% 0.16/0.42 ( ~ ( ord_less @ A @ Y2 @ X )
% 0.16/0.42 => ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 0.16/0.42 = ( X = Y2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % antisym_conv3
% 0.16/0.42 thf(fact_187_less__induct,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [P: A > $o,A2: A] :
% 0.16/0.42 ( ! [X4: A] :
% 0.16/0.42 ( ! [Y6: A] :
% 0.16/0.42 ( ( ord_less @ A @ Y6 @ X4 )
% 0.16/0.42 => ( P @ Y6 ) )
% 0.16/0.42 => ( P @ X4 ) )
% 0.16/0.42 => ( P @ A2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_induct
% 0.16/0.42 thf(fact_188_less__not__sym,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_not_sym
% 0.16/0.42 thf(fact_189_less__imp__not__eq,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ( X != Y2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_imp_not_eq
% 0.16/0.42 thf(fact_190_dual__order_Oasym,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [B6: A,A2: A] :
% 0.16/0.42 ( ( ord_less @ A @ B6 @ A2 )
% 0.16/0.42 => ~ ( ord_less @ A @ A2 @ B6 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.asym
% 0.16/0.42 thf(fact_191_ord__less__eq__trans,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( ord @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [A2: A,B6: A,C2: A] :
% 0.16/0.42 ( ( ord_less @ A @ A2 @ B6 )
% 0.16/0.42 => ( ( B6 = C2 )
% 0.16/0.42 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_less_eq_trans
% 0.16/0.42 thf(fact_192_ord__eq__less__trans,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( ord @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [A2: A,B6: A,C2: A] :
% 0.16/0.42 ( ( A2 = B6 )
% 0.16/0.42 => ( ( ord_less @ A @ B6 @ C2 )
% 0.16/0.42 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_eq_less_trans
% 0.16/0.42 thf(fact_193_less__irrefl,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A] :
% 0.16/0.42 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_irrefl
% 0.16/0.42 thf(fact_194_less__linear,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 | ( X = Y2 )
% 0.16/0.42 | ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_linear
% 0.16/0.42 thf(fact_195_less__trans,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A,Z2: A] :
% 0.16/0.42 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.16/0.42 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_trans
% 0.16/0.42 thf(fact_196_less__asym_H,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [A2: A,B6: A] :
% 0.16/0.42 ( ( ord_less @ A @ A2 @ B6 )
% 0.16/0.42 => ~ ( ord_less @ A @ B6 @ A2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_asym'
% 0.16/0.42 thf(fact_197_less__asym,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_asym
% 0.16/0.42 thf(fact_198_less__imp__neq,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ( X != Y2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_imp_neq
% 0.16/0.42 thf(fact_199_dense,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ? [Z3: A] :
% 0.16/0.42 ( ( ord_less @ A @ X @ Z3 )
% 0.16/0.42 & ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dense
% 0.16/0.42 thf(fact_200_order_Oasym,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [A2: A,B6: A] :
% 0.16/0.42 ( ( ord_less @ A @ A2 @ B6 )
% 0.16/0.42 => ~ ( ord_less @ A @ B6 @ A2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.asym
% 0.16/0.42 thf(fact_201_neq__iff,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( X != Y2 )
% 0.16/0.42 = ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 | ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % neq_iff
% 0.16/0.42 thf(fact_202_neqE,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( X != Y2 )
% 0.16/0.42 => ( ~ ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % neqE
% 0.16/0.42 thf(fact_203_gt__ex,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( no_top @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A] :
% 0.16/0.42 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 0.16/0.42
% 0.16/0.42 % gt_ex
% 0.16/0.42 thf(fact_204_lt__ex,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A] :
% 0.16/0.42 ? [Y5: A] : ( ord_less @ A @ Y5 @ X ) ) ).
% 0.16/0.42
% 0.16/0.42 % lt_ex
% 0.16/0.42 thf(fact_205_order__less__subst2,axiom,
% 0.16/0.42 ! [A: $tType,C4: $tType] :
% 0.16/0.42 ( ( ( order @ C4 @ ( type2 @ C4 ) )
% 0.16/0.42 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.42 => ! [A2: A,B6: A,F: A > C4,C2: C4] :
% 0.16/0.42 ( ( ord_less @ A @ A2 @ B6 )
% 0.16/0.42 => ( ( ord_less @ C4 @ ( F @ B6 ) @ C2 )
% 0.16/0.42 => ( ! [X4: A,Y5: A] :
% 0.16/0.42 ( ( ord_less @ A @ X4 @ Y5 )
% 0.16/0.42 => ( ord_less @ C4 @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.16/0.42 => ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order_less_subst2
% 0.16/0.42 thf(fact_206_order__less__subst1,axiom,
% 0.16/0.42 ! [A: $tType,B: $tType] :
% 0.16/0.42 ( ( ( order @ B @ ( type2 @ B ) )
% 0.16/0.42 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.42 => ! [A2: A,F: B > A,B6: B,C2: B] :
% 0.16/0.42 ( ( ord_less @ A @ A2 @ ( F @ B6 ) )
% 0.16/0.42 => ( ( ord_less @ B @ B6 @ C2 )
% 0.16/0.42 => ( ! [X4: B,Y5: B] :
% 0.16/0.42 ( ( ord_less @ B @ X4 @ Y5 )
% 0.16/0.42 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.16/0.42 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order_less_subst1
% 0.16/0.42 thf(fact_207_ord__less__eq__subst,axiom,
% 0.16/0.42 ! [A: $tType,B: $tType] :
% 0.16/0.42 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.42 & ( ord @ A @ ( type2 @ A ) ) )
% 0.16/0.42 => ! [A2: A,B6: A,F: A > B,C2: B] :
% 0.16/0.42 ( ( ord_less @ A @ A2 @ B6 )
% 0.16/0.42 => ( ( ( F @ B6 )
% 0.16/0.42 = C2 )
% 0.16/0.42 => ( ! [X4: A,Y5: A] :
% 0.16/0.42 ( ( ord_less @ A @ X4 @ Y5 )
% 0.16/0.42 => ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.16/0.42 => ( ord_less @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_less_eq_subst
% 0.16/0.42 thf(fact_208_ord__eq__less__subst,axiom,
% 0.16/0.42 ! [A: $tType,B: $tType] :
% 0.16/0.42 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.16/0.42 & ( ord @ A @ ( type2 @ A ) ) )
% 0.16/0.42 => ! [A2: A,F: B > A,B6: B,C2: B] :
% 0.16/0.42 ( ( A2
% 0.16/0.42 = ( F @ B6 ) )
% 0.16/0.42 => ( ( ord_less @ B @ B6 @ C2 )
% 0.16/0.42 => ( ! [X4: B,Y5: B] :
% 0.16/0.42 ( ( ord_less @ B @ X4 @ Y5 )
% 0.16/0.42 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.16/0.42 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_eq_less_subst
% 0.16/0.42 thf(fact_209_less__eq__enat__def,axiom,
% 0.16/0.42 ( ( ord_less_eq @ extended_enat )
% 0.16/0.42 = ( ^ [M2: extended_enat] :
% 0.16/0.42 ( extended_case_enat @ $o
% 0.16/0.42 @ ^ [N1: nat] :
% 0.16/0.42 ( extended_case_enat @ $o
% 0.16/0.42 @ ^ [M1: nat] : ( ord_less_eq @ nat @ M1 @ N1 )
% 0.16/0.42 @ $false
% 0.16/0.42 @ M2 )
% 0.16/0.42 @ $true ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_eq_enat_def
% 0.16/0.42 thf(fact_210_infinite__descent__measure,axiom,
% 0.16/0.42 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 0.16/0.42 ( ! [X4: A] :
% 0.16/0.42 ( ~ ( P @ X4 )
% 0.16/0.42 => ? [Y6: A] :
% 0.16/0.42 ( ( ord_less @ nat @ ( V @ Y6 ) @ ( V @ X4 ) )
% 0.16/0.42 & ~ ( P @ Y6 ) ) )
% 0.16/0.42 => ( P @ X ) ) ).
% 0.16/0.42
% 0.16/0.42 % infinite_descent_measure
% 0.16/0.42 thf(fact_211_measure__induct__rule,axiom,
% 0.16/0.42 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.16/0.42 ( ! [X4: A] :
% 0.16/0.42 ( ! [Y6: A] :
% 0.16/0.42 ( ( ord_less @ nat @ ( F @ Y6 ) @ ( F @ X4 ) )
% 0.16/0.42 => ( P @ Y6 ) )
% 0.16/0.42 => ( P @ X4 ) )
% 0.16/0.42 => ( P @ A2 ) ) ).
% 0.16/0.42
% 0.16/0.42 % measure_induct_rule
% 0.16/0.42 thf(fact_212_linorder__neqE__nat,axiom,
% 0.16/0.42 ! [X: nat,Y2: nat] :
% 0.16/0.42 ( ( X != Y2 )
% 0.16/0.42 => ( ~ ( ord_less @ nat @ X @ Y2 )
% 0.16/0.42 => ( ord_less @ nat @ Y2 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_neqE_nat
% 0.16/0.42 thf(fact_213_infinite__descent,axiom,
% 0.16/0.42 ! [P: nat > $o,N: nat] :
% 0.16/0.42 ( ! [N2: nat] :
% 0.16/0.42 ( ~ ( P @ N2 )
% 0.16/0.42 => ? [M3: nat] :
% 0.16/0.42 ( ( ord_less @ nat @ M3 @ N2 )
% 0.16/0.42 & ~ ( P @ M3 ) ) )
% 0.16/0.42 => ( P @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % infinite_descent
% 0.16/0.42 thf(fact_214_nat__less__induct,axiom,
% 0.16/0.42 ! [P: nat > $o,N: nat] :
% 0.16/0.42 ( ! [N2: nat] :
% 0.16/0.42 ( ! [M3: nat] :
% 0.16/0.42 ( ( ord_less @ nat @ M3 @ N2 )
% 0.16/0.42 => ( P @ M3 ) )
% 0.16/0.42 => ( P @ N2 ) )
% 0.16/0.42 => ( P @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % nat_less_induct
% 0.16/0.42 thf(fact_215_less__irrefl__nat,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ~ ( ord_less @ nat @ N @ N ) ).
% 0.16/0.42
% 0.16/0.42 % less_irrefl_nat
% 0.16/0.42 thf(fact_216_measure__induct,axiom,
% 0.16/0.42 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.16/0.42 ( ! [X4: A] :
% 0.16/0.42 ( ! [Y6: A] :
% 0.16/0.42 ( ( ord_less @ nat @ ( F @ Y6 ) @ ( F @ X4 ) )
% 0.16/0.42 => ( P @ Y6 ) )
% 0.16/0.42 => ( P @ X4 ) )
% 0.16/0.42 => ( P @ A2 ) ) ).
% 0.16/0.42
% 0.16/0.42 % measure_induct
% 0.16/0.42 thf(fact_217_less__not__refl3,axiom,
% 0.16/0.42 ! [S3: nat,T4: nat] :
% 0.16/0.42 ( ( ord_less @ nat @ S3 @ T4 )
% 0.16/0.42 => ( S3 != T4 ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_not_refl3
% 0.16/0.42 thf(fact_218_less__not__refl2,axiom,
% 0.16/0.42 ! [N: nat,M: nat] :
% 0.16/0.42 ( ( ord_less @ nat @ N @ M )
% 0.16/0.42 => ( M != N ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_not_refl2
% 0.16/0.42 thf(fact_219_less__not__refl,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ~ ( ord_less @ nat @ N @ N ) ).
% 0.16/0.42
% 0.16/0.42 % less_not_refl
% 0.16/0.42 thf(fact_220_nat__neq__iff,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( M != N )
% 0.16/0.42 = ( ( ord_less @ nat @ M @ N )
% 0.16/0.42 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nat_neq_iff
% 0.16/0.42 thf(fact_221_le__refl,axiom,
% 0.16/0.42 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.16/0.42
% 0.16/0.42 % le_refl
% 0.16/0.42 thf(fact_222_le__trans,axiom,
% 0.16/0.42 ! [I3: nat,J3: nat,K2: nat] :
% 0.16/0.42 ( ( ord_less_eq @ nat @ I3 @ J3 )
% 0.16/0.42 => ( ( ord_less_eq @ nat @ J3 @ K2 )
% 0.16/0.42 => ( ord_less_eq @ nat @ I3 @ K2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_trans
% 0.16/0.42 thf(fact_223_eq__imp__le,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( M = N )
% 0.16/0.42 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % eq_imp_le
% 0.16/0.42 thf(fact_224_le__antisym,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_eq @ nat @ M @ N )
% 0.16/0.42 => ( ( ord_less_eq @ nat @ N @ M )
% 0.16/0.42 => ( M = N ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_antisym
% 0.16/0.42 thf(fact_225_nat__le__linear,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_eq @ nat @ M @ N )
% 0.16/0.42 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.16/0.42
% 0.16/0.42 % nat_le_linear
% 0.16/0.42 thf(fact_226_ex__has__least__nat,axiom,
% 0.16/0.42 ! [A: $tType,P: A > $o,K2: A,M: A > nat] :
% 0.16/0.42 ( ( P @ K2 )
% 0.16/0.42 => ? [X4: A] :
% 0.16/0.42 ( ( P @ X4 )
% 0.16/0.42 & ! [Y6: A] :
% 0.16/0.42 ( ( P @ Y6 )
% 0.16/0.42 => ( ord_less_eq @ nat @ ( M @ X4 ) @ ( M @ Y6 ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ex_has_least_nat
% 0.16/0.42 thf(fact_227_order_Onot__eq__order__implies__strict,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [A2: A,B6: A] :
% 0.16/0.42 ( ( A2 != B6 )
% 0.16/0.42 => ( ( ord_less_eq @ A @ A2 @ B6 )
% 0.16/0.42 => ( ord_less @ A @ A2 @ B6 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.not_eq_order_implies_strict
% 0.16/0.42 thf(fact_228_dual__order_Ostrict__implies__order,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [B6: A,A2: A] :
% 0.16/0.42 ( ( ord_less @ A @ B6 @ A2 )
% 0.16/0.42 => ( ord_less_eq @ A @ B6 @ A2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_implies_order
% 0.16/0.42 thf(fact_229_dual__order_Ostrict__iff__order,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ( ( ord_less @ A )
% 0.16/0.42 = ( ^ [B8: A,A6: A] :
% 0.16/0.42 ( ( ord_less_eq @ A @ B8 @ A6 )
% 0.16/0.42 & ( A6 != B8 ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_iff_order
% 0.16/0.42 thf(fact_230_dual__order_Oorder__iff__strict,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ( ( ord_less_eq @ A )
% 0.16/0.42 = ( ^ [B8: A,A6: A] :
% 0.16/0.42 ( ( ord_less @ A @ B8 @ A6 )
% 0.16/0.42 | ( A6 = B8 ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.order_iff_strict
% 0.16/0.42 thf(fact_231_order_Ostrict__implies__order,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [A2: A,B6: A] :
% 0.16/0.42 ( ( ord_less @ A @ A2 @ B6 )
% 0.16/0.42 => ( ord_less_eq @ A @ A2 @ B6 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_implies_order
% 0.16/0.42 thf(fact_232_dense__le__bounded,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A,Z2: A] :
% 0.16/0.42 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ( ! [W: A] :
% 0.16/0.42 ( ( ord_less @ A @ X @ W )
% 0.16/0.42 => ( ( ord_less @ A @ W @ Y2 )
% 0.16/0.42 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.16/0.42 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dense_le_bounded
% 0.16/0.42 thf(fact_233_dense__ge__bounded,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [Z2: A,X: A,Y2: A] :
% 0.16/0.42 ( ( ord_less @ A @ Z2 @ X )
% 0.16/0.42 => ( ! [W: A] :
% 0.16/0.42 ( ( ord_less @ A @ Z2 @ W )
% 0.16/0.42 => ( ( ord_less @ A @ W @ X )
% 0.16/0.42 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 0.16/0.42 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dense_ge_bounded
% 0.16/0.42 thf(fact_234_dual__order_Ostrict__trans2,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [B6: A,A2: A,C2: A] :
% 0.16/0.42 ( ( ord_less @ A @ B6 @ A2 )
% 0.16/0.42 => ( ( ord_less_eq @ A @ C2 @ B6 )
% 0.16/0.42 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_trans2
% 0.16/0.42 thf(fact_235_dual__order_Ostrict__trans1,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [B6: A,A2: A,C2: A] :
% 0.16/0.42 ( ( ord_less_eq @ A @ B6 @ A2 )
% 0.16/0.42 => ( ( ord_less @ A @ C2 @ B6 )
% 0.16/0.42 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_trans1
% 0.16/0.42 thf(fact_236_order_Ostrict__iff__order,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ( ( ord_less @ A )
% 0.16/0.42 = ( ^ [A6: A,B8: A] :
% 0.16/0.42 ( ( ord_less_eq @ A @ A6 @ B8 )
% 0.16/0.42 & ( A6 != B8 ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_iff_order
% 0.16/0.42 thf(fact_237_order_Oorder__iff__strict,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ( ( ord_less_eq @ A )
% 0.16/0.42 = ( ^ [A6: A,B8: A] :
% 0.16/0.42 ( ( ord_less @ A @ A6 @ B8 )
% 0.16/0.42 | ( A6 = B8 ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.order_iff_strict
% 0.16/0.42 thf(fact_238_order_Ostrict__trans2,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [A2: A,B6: A,C2: A] :
% 0.16/0.42 ( ( ord_less @ A @ A2 @ B6 )
% 0.16/0.42 => ( ( ord_less_eq @ A @ B6 @ C2 )
% 0.16/0.42 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_trans2
% 0.16/0.42 thf(fact_239_order_Ostrict__trans1,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [A2: A,B6: A,C2: A] :
% 0.16/0.42 ( ( ord_less_eq @ A @ A2 @ B6 )
% 0.16/0.42 => ( ( ord_less @ A @ B6 @ C2 )
% 0.16/0.42 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_trans1
% 0.16/0.42 thf(fact_240_not__le__imp__less,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [Y2: A,X: A] :
% 0.16/0.42 ( ~ ( ord_less_eq @ A @ Y2 @ X )
% 0.16/0.42 => ( ord_less @ A @ X @ Y2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % not_le_imp_less
% 0.16/0.42 thf(fact_241_less__le__not__le,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ( ( ord_less @ A )
% 0.16/0.42 = ( ^ [X3: A,Y: A] :
% 0.16/0.42 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.16/0.42 & ~ ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_le_not_le
% 0.16/0.42 thf(fact_242_le__imp__less__or__eq,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.42 => ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 | ( X = Y2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_imp_less_or_eq
% 0.16/0.42 thf(fact_243_le__less__linear,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.42 | ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_less_linear
% 0.16/0.42 thf(fact_244_dense__le,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [Y2: A,Z2: A] :
% 0.16/0.42 ( ! [X4: A] :
% 0.16/0.42 ( ( ord_less @ A @ X4 @ Y2 )
% 0.16/0.42 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 0.16/0.42 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dense_le
% 0.16/0.42 thf(fact_245_dense__ge,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [Z2: A,Y2: A] :
% 0.16/0.42 ( ! [X4: A] :
% 0.16/0.42 ( ( ord_less @ A @ Z2 @ X4 )
% 0.16/0.42 => ( ord_less_eq @ A @ Y2 @ X4 ) )
% 0.16/0.42 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dense_ge
% 0.16/0.42 thf(fact_246_less__le__trans,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A,Z2: A] :
% 0.16/0.42 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.16/0.42 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_le_trans
% 0.16/0.42 thf(fact_247_le__less__trans,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A,Z2: A] :
% 0.16/0.42 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.42 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.16/0.42 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_less_trans
% 0.16/0.42 thf(fact_248_antisym__conv2,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.42 => ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 0.16/0.42 = ( X = Y2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % antisym_conv2
% 0.16/0.42 thf(fact_249_antisym__conv1,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ~ ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.42 = ( X = Y2 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % antisym_conv1
% 0.16/0.42 thf(fact_250_less__imp__le,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.42 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_imp_le
% 0.16/0.42 thf(fact_251_le__neq__trans,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [A2: A,B6: A] :
% 0.16/0.42 ( ( ord_less_eq @ A @ A2 @ B6 )
% 0.16/0.42 => ( ( A2 != B6 )
% 0.16/0.42 => ( ord_less @ A @ A2 @ B6 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_neq_trans
% 0.16/0.42 thf(fact_252_not__less,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 0.16/0.42 = ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % not_less
% 0.16/0.42 thf(fact_253_not__le,axiom,
% 0.16/0.42 ! [A: $tType] :
% 0.16/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.42 => ! [X: A,Y2: A] :
% 0.16/0.42 ( ( ~ ( ord_less_eq @ A @ X @ Y2 ) )
% 0.16/0.42 = ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % not_le
% 0.16/0.42 thf(fact_254_order__less__le__subst2,axiom,
% 0.16/0.42 ! [A: $tType,C4: $tType] :
% 0.16/0.42 ( ( ( order @ C4 @ ( type2 @ C4 ) )
% 0.16/0.42 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.42 => ! [A2: A,B6: A,F: A > C4,C2: C4] :
% 0.16/0.42 ( ( ord_less @ A @ A2 @ B6 )
% 0.16/0.42 => ( ( ord_less_eq @ C4 @ ( F @ B6 ) @ C2 )
% 0.16/0.42 => ( ! [X4: A,Y5: A] :
% 0.16/0.42 ( ( ord_less @ A @ X4 @ Y5 )
% 0.16/0.42 => ( ord_less @ C4 @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.16/0.42 => ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order_less_le_subst2
% 0.16/0.42 thf(fact_255_order__less__le__subst1,axiom,
% 0.16/0.42 ! [A: $tType,B: $tType] :
% 0.16/0.42 ( ( ( order @ B @ ( type2 @ B ) )
% 0.16/0.42 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.42 => ! [A2: A,F: B > A,B6: B,C2: B] :
% 0.16/0.42 ( ( ord_less @ A @ A2 @ ( F @ B6 ) )
% 0.16/0.42 => ( ( ord_less_eq @ B @ B6 @ C2 )
% 0.16/0.42 => ( ! [X4: B,Y5: B] :
% 0.16/0.42 ( ( ord_less_eq @ B @ X4 @ Y5 )
% 0.16/0.42 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.16/0.42 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order_less_le_subst1
% 0.16/0.42
% 0.16/0.42 %----Type constructors (26)
% 0.16/0.42 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.16/0.42 ! [A8: $tType,A9: $tType] :
% 0.16/0.42 ( ( preorder @ A9 @ ( type2 @ A9 ) )
% 0.16/0.42 => ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.16/0.42 ! [A8: $tType,A9: $tType] :
% 0.16/0.42 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.16/0.42 & ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
% 0.16/0.42 => ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.16/0.42 ! [A8: $tType,A9: $tType] :
% 0.16/0.42 ( ( order @ A9 @ ( type2 @ A9 ) )
% 0.16/0.42 => ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_fun___Orderings_Oord,axiom,
% 0.16/0.42 ! [A8: $tType,A9: $tType] :
% 0.16/0.42 ( ( ord @ A9 @ ( type2 @ A9 ) )
% 0.16/0.42 => ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.16/0.42 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.16/0.42 wellorder @ nat @ ( type2 @ nat ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.16/0.42 preorder @ nat @ ( type2 @ nat ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.16/0.42 linorder @ nat @ ( type2 @ nat ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.16/0.42 no_top @ nat @ ( type2 @ nat ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.16/0.42 order @ nat @ ( type2 @ nat ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.16/0.42 ord @ nat @ ( type2 @ nat ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.16/0.42 ! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 0.16/0.42 ! [A8: $tType] :
% 0.16/0.42 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.16/0.42 => ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.16/0.42 ! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 0.16/0.42 ! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
% 0.16/0.42 preorder @ $o @ ( type2 @ $o ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_HOL_Obool___Orderings_Olinorder_9,axiom,
% 0.16/0.42 linorder @ $o @ ( type2 @ $o ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_HOL_Obool___Finite__Set_Ofinite_10,axiom,
% 0.16/0.42 finite_finite @ $o @ ( type2 @ $o ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 0.16/0.42 order @ $o @ ( type2 @ $o ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 0.16/0.42 ord @ $o @ ( type2 @ $o ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_13,axiom,
% 0.16/0.42 condit1037483654norder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_14,axiom,
% 0.16/0.42 wellorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_15,axiom,
% 0.16/0.42 preorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_16,axiom,
% 0.16/0.42 linorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_17,axiom,
% 0.16/0.42 order @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.16/0.42
% 0.16/0.42 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_18,axiom,
% 0.16/0.42 ord @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.16/0.42
% 0.16/0.42 %----Conjectures (2)
% 0.16/0.42 thf(conj_0,hypothesis,
% 0.16/0.42 ( linear505997466_on_ev @ a
% 0.16/0.42 @ ( linear1386806755on_alw @ a
% 0.16/0.42 @ ^ [Xs: stream @ a] :
% 0.16/0.42 ~ ( p @ Xs ) )
% 0.16/0.42 @ omega ) ).
% 0.16/0.42
% 0.16/0.42 thf(conj_1,conjecture,
% 0.16/0.42 ord_less @ extended_enat @ ( coindu1365464361scount @ a @ p @ omega ) @ ( extend1396239628finity @ extended_enat ) ).
% 0.16/0.42
% 0.16/0.42 %------------------------------------------------------------------------------
% 0.16/0.42 ------------------------
% 2.33/1.37 ---- Embedded in THF ---
% 2.33/1.37 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.33/1.37 %%% Generated on Tue Feb 24 06:10:19 EST 2026
% 2.33/1.37 %%% using '$$dhol' embedding, version 1.3.0.
% 2.33/1.37 %%% Logic specification used:
% 2.33/1.37 %%% thf(spec, logic, $$dhol).
% 2.33/1.37
% 2.33/1.37 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.HepLMUEemU/DTF2DTF20348.p
% 2.33/1.37 thf(ty_t_Extended__Nat_Oenat, type, extended_enat: $tType).
% 2.33/1.37 thf(ty_t_Stream_Ostream, type, stream: ($tType > $tType)).
% 2.33/1.37 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.33/1.37 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.33/1.37 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.33/1.37 thf(ty_tf_a, type, a: $tType).
% 2.33/1.37 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.33/1.37 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Oscount, type, coindu1365464361scount: (!> [S:$tType]: ((((stream @ S) > $o) > ((stream @ S) > extended_enat))))).
% 2.33/1.37 thf(sy_c_Extended__Nat_Oenat, type, extended_enat2: (nat > extended_enat)).
% 2.33/1.37 thf(sy_c_Extended__Nat_Oenat_Ocase__enat, type, extended_case_enat: (!> [T:$tType]: (((nat > T) > (T > (extended_enat > T)))))).
% 2.33/1.37 thf(sy_c_Extended__Nat_Oenat_Orec__enat, type, extended_rec_enat: (!> [T:$tType]: (((nat > T) > (T > (extended_enat > T)))))).
% 2.33/1.37 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity, type, extend1396239628finity: (!> [A:$tType]: (A))).
% 2.33/1.37 thf(sy_c_Finite__Set_Ocard, type, finite_card: (!> [B:$tType]: (((set @ B) > nat)))).
% 2.33/1.37 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.33/1.37 thf(sy_c_Linear__Temporal__Logic__on__Streams_Oalw, type, linear1386806755on_alw: (!> [A:$tType]: ((((stream @ A) > $o) > ((stream @ A) > $o))))).
% 2.33/1.37 thf(sy_c_Linear__Temporal__Logic__on__Streams_Oev, type, linear505997466_on_ev: (!> [A:$tType]: ((((stream @ A) > $o) > ((stream @ A) > $o))))).
% 2.33/1.37 thf(sy_c_Linear__Temporal__Logic__on__Streams_Onxt, type, linear1494993505on_nxt: (!> [A:$tType,B:$tType]: ((((stream @ A) > B) > ((stream @ A) > B))))).
% 2.33/1.37 thf(sy_c_Linear__Temporal__Logic__on__Streams_Owait, type, linear1335279038n_wait: (!> [A:$tType]: ((((stream @ A) > $o) > ((stream @ A) > nat))))).
% 2.33/1.37 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.33/1.37 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.33/1.37 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.33/1.37 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.33/1.37 thf(sy_c_Stream_Osdrop, type, sdrop: (!> [A:$tType]: ((nat > ((stream @ A) > (stream @ A)))))).
% 2.33/1.37 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.33/1.37 thf(sy_v_P, type, p: ((stream @ a) > $o)).
% 2.33/1.37 thf(sy_v__092_060omega_062, type, omega: (stream @ a)).
% 2.33/1.37 thf(fact_0__092_060open_062ev_A_Ialw_A_I_092_060lambda_062xs_O_A_092_060not_062_AP_Axs_J_J_A_092_060omega_062_A_092_060Longrightarrow_062_Ascount_AP_A_092_060omega_062_A_061_Aenat_A_Icard_A_123i_O_AP_A_Isdrop_Ai_A_092_060omega_062_J_125_J_092_060close_062, axiom, ((((linear505997466_on_ev @ a) @ ((linear1386806755on_alw @ a) @ (^ [Xs:(stream @ a)]: ((~ ((p @ Xs))))))) @ omega) => ((((coindu1365464361scount @ a) @ p) @ omega) = (extended_enat2 @ ((finite_card @ nat) @ ((collect @ nat) @ (^ [I:nat]: ((p @ (((sdrop @ a) @ I) @ omega)))))))))).
% 2.33/1.37 thf(fact_1_not__ev__not, axiom, (! [A:$tType,Phi:((stream @ A) > $o)]: (((^ [Xs:(stream @ A)]: ((~ ((((linear505997466_on_ev @ A) @ (^ [Xt:(stream @ A)]: ((~ ((Phi @ Xt)))))) @ Xs))))) = ((linear1386806755on_alw @ A) @ Phi))))).
% 2.33/1.37 thf(fact_2_not__alw__not, axiom, (! [A:$tType,Phi:((stream @ A) > $o)]: (((^ [Xs:(stream @ A)]: ((~ ((((linear1386806755on_alw @ A) @ (^ [Xt:(stream @ A)]: ((~ ((Phi @ Xt)))))) @ Xs))))) = ((linear505997466_on_ev @ A) @ Phi))))).
% 2.33/1.37 thf(fact_3_enat__ord__simps_I6_J, axiom, (! [Q:extended_enat]: ((~ ((((ord_less @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ Q)))))).
% 2.33/1.37 thf(fact_4_enat__ord__simps_I4_J, axiom, (! [Q:extended_enat]: (((((ord_less @ extended_enat) @ Q) @ (extend1396239628finity @ extended_enat)) = (Q != (extend1396239628finity @ extended_enat)))))).
% 2.33/1.37 thf(fact_5_ev__ev, axiom, (! [A:$tType,Phi:((stream @ A) > $o)]: ((((linear505997466_on_ev @ A) @ ((linear505997466_on_ev @ A) @ Phi)) = ((linear505997466_on_ev @ A) @ Phi))))).
% 2.33/1.37 thf(fact_6_alw__alw, axiom, (! [A:$tType,Phi:((stream @ A) > $o)]: ((((linear1386806755on_alw @ A) @ ((linear1386806755on_alw @ A) @ Phi)) = ((linear1386806755on_alw @ A) @ Phi))))).
% 2.33/1.37 thf(fact_7_not__ev, axiom, (! [A:$tType,Phi:((stream @ A) > $o)]: (((^ [Xs:(stream @ A)]: ((~ ((((linear505997466_on_ev @ A) @ Phi) @ Xs))))) = ((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: ((~ ((Phi @ Xs)))))))))).
% 2.33/1.37 thf(fact_8_not__alw, axiom, (! [A:$tType,Phi:((stream @ A) > $o)]: (((^ [Xs:(stream @ A)]: ((~ ((((linear1386806755on_alw @ A) @ Phi) @ Xs))))) = ((linear505997466_on_ev @ A) @ (^ [Xs:(stream @ A)]: ((~ ((Phi @ Xs)))))))))).
% 2.33/1.37 thf(fact_9_not__ev__iff, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((~ ((((linear505997466_on_ev @ A) @ P) @ Omega))) = (((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: ((~ ((P @ Xs)))))) @ Omega))))).
% 2.33/1.37 thf(fact_10_ev__alw__aand, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ Phi)) @ Xs2) => ((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ Psi)) @ Xs2) => (((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) & (Psi @ Xs)))))) @ Xs2)))))).
% 2.33/1.37 thf(fact_11_ev__alw__impl, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs2) => ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) => (Psi @ Xs))))) @ Xs2) => (((linear505997466_on_ev @ A) @ Psi) @ Xs2)))))).
% 2.33/1.37 thf(fact_12_not__alw__iff, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((~ ((((linear1386806755on_alw @ A) @ P) @ Omega))) = (((linear505997466_on_ev @ A) @ (^ [Xs:(stream @ A)]: ((~ ((P @ Xs)))))) @ Omega))))).
% 2.33/1.37 thf(fact_13_enat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((extended_enat2 @ Nat) = (extended_enat2 @ Nat2)) = (Nat = Nat2))))).
% 2.33/1.37 thf(fact_14_not__infinity__eq, axiom, (! [X:extended_enat]: (((X != (extend1396239628finity @ extended_enat)) = (? [I:nat]: ((X = (extended_enat2 @ I)))))))).
% 2.33/1.37 thf(fact_15_not__enat__eq, axiom, (! [X:extended_enat]: (((! [Y:nat]: ((X != (extended_enat2 @ Y)))) = (X = (extend1396239628finity @ extended_enat)))))).
% 2.33/1.37 thf(fact_16_enat__ex__split, axiom, ((^ [P2:(extended_enat > $o)]: ((? [X2:extended_enat]: ((P2 @ X2))))) = (^ [P3:(extended_enat > $o)]: (((P3 @ (extend1396239628finity @ extended_enat)) | (? [X3:nat]: ((P3 @ (extended_enat2 @ X3))))))))).
% 2.33/1.37 thf(fact_17_enat_Oinducts, axiom, (! [P:(extended_enat > $o),Enat:extended_enat]: (((! [Nat3:nat]: ((P @ (extended_enat2 @ Nat3)))) => ((P @ (extend1396239628finity @ extended_enat)) => (P @ Enat)))))).
% 2.33/1.37 thf(fact_18_enat_Oexhaust, axiom, (! [Y2:extended_enat]: (((! [Nat3:nat]: ((Y2 != (extended_enat2 @ Nat3)))) => (Y2 = (extend1396239628finity @ extended_enat)))))).
% 2.33/1.37 thf(fact_19_enat3__cases, axiom, (! [Y2:extended_enat,Ya:extended_enat,Yb:extended_enat]: ((((? [Nat3:nat]: ((Y2 = (extended_enat2 @ Nat3)))) => ((? [Nata:nat]: ((Ya = (extended_enat2 @ Nata)))) => (! [Natb:nat]: ((Yb != (extended_enat2 @ Natb)))))) => (((? [Nat3:nat]: ((Y2 = (extended_enat2 @ Nat3)))) => ((? [Nata:nat]: ((Ya = (extended_enat2 @ Nata)))) => (Yb != (extend1396239628finity @ extended_enat)))) => (((? [Nat3:nat]: ((Y2 = (extended_enat2 @ Nat3)))) => ((Ya = (extend1396239628finity @ extended_enat)) => (! [Nata:nat]: ((Yb != (extended_enat2 @ Nata)))))) => (((? [Nat3:nat]: ((Y2 = (extended_enat2 @ Nat3)))) => ((Ya = (extend1396239628finity @ extended_enat)) => (Yb != (extend1396239628finity @ extended_enat)))) => (((Y2 = (extend1396239628finity @ extended_enat)) => ((? [Nat3:nat]: ((Ya = (extended_enat2 @ Nat3)))) => (! [Nata:nat]: ((Yb != (extended_enat2 @ Nata)))))) => (((Y2 = (extend1396239628finity @ extended_enat)) => ((? [Nat3:nat]: ((Ya = (extended_enat2 @ Nat3)))) => (Yb != (extend1396239628finity @ extended_enat)))) => (((Y2 = (extend1396239628finity @ extended_enat)) => ((Ya = (extend1396239628finity @ extended_enat)) => (! [Nat3:nat]: ((Yb != (extended_enat2 @ Nat3)))))) => (~ (((Y2 = (extend1396239628finity @ extended_enat)) => ((Ya = (extend1396239628finity @ extended_enat)) => (Yb != (extend1396239628finity @ extended_enat)))))))))))))))).
% 2.33/1.37 thf(fact_20_enat2__cases, axiom, (! [Y2:extended_enat,Ya:extended_enat]: ((((? [Nat3:nat]: ((Y2 = (extended_enat2 @ Nat3)))) => (! [Nata:nat]: ((Ya != (extended_enat2 @ Nata))))) => (((? [Nat3:nat]: ((Y2 = (extended_enat2 @ Nat3)))) => (Ya != (extend1396239628finity @ extended_enat))) => (((Y2 = (extend1396239628finity @ extended_enat)) => (! [Nat3:nat]: ((Ya != (extended_enat2 @ Nat3))))) => (~ (((Y2 = (extend1396239628finity @ extended_enat)) => (Ya != (extend1396239628finity @ extended_enat))))))))))).
% 2.33/1.37 thf(fact_21_enat_Odistinct_I1_J, axiom, (! [Nat:nat]: (((extended_enat2 @ Nat) != (extend1396239628finity @ extended_enat))))).
% 2.33/1.37 thf(fact_22_enat_Odistinct_I2_J, axiom, (! [Nat4:nat]: (((extend1396239628finity @ extended_enat) != (extended_enat2 @ Nat4))))).
% 2.33/1.37 thf(fact_23_enat__iless, axiom, (! [N:extended_enat,M:nat]: (((((ord_less @ extended_enat) @ N) @ (extended_enat2 @ M)) => (? [K:nat]: ((N = (extended_enat2 @ K)))))))).
% 2.33/1.37 thf(fact_24_chain__incr, axiom, (! [A:$tType,Y3:(A > extended_enat),K2:nat]: (((! [I2:A]: ((? [J:A]: ((((ord_less @ extended_enat) @ (Y3 @ I2)) @ (Y3 @ J)))))) => (? [J2:A]: ((((ord_less @ extended_enat) @ (extended_enat2 @ K2)) @ (Y3 @ J2)))))))).
% 2.33/1.37 thf(fact_25_alw__iff__sdrop, axiom, (! [A:$tType]: (((linear1386806755on_alw @ A) = (^ [P3:((stream @ A) > $o),Omega2:(stream @ A)]: ((! [M2:nat]: ((P3 @ (((sdrop @ A) @ M2) @ Omega2)))))))))).
% 2.33/1.37 thf(fact_26_ev__iff__sdrop, axiom, (! [A:$tType]: (((linear505997466_on_ev @ A) = (^ [P3:((stream @ A) > $o),Omega2:(stream @ A)]: ((? [M2:nat]: ((P3 @ (((sdrop @ A) @ M2) @ Omega2)))))))))).
% 2.33/1.37 thf(fact_27_alw__sdrop, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),N:nat]: (((((linear1386806755on_alw @ A) @ Phi) @ Xs2) => (((linear1386806755on_alw @ A) @ Phi) @ (((sdrop @ A) @ N) @ Xs2)))))).
% 2.33/1.37 thf(fact_28_all__imp__alw, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A)]: (((! [X1:(stream @ A)]: ((Phi @ X1))) => (((linear1386806755on_alw @ A) @ Phi) @ Xs2))))).
% 2.33/1.37 thf(fact_29_alw__mono, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear1386806755on_alw @ A) @ Phi) @ Xs2) => ((! [Xs3:(stream @ A)]: (((Phi @ Xs3) => (Psi @ Xs3)))) => (((linear1386806755on_alw @ A) @ Psi) @ Xs2)))))).
% 2.33/1.37 thf(fact_30_alw__cong, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A),Q1:((stream @ A) > $o),Q2:((stream @ A) > $o)]: (((((linear1386806755on_alw @ A) @ P) @ Omega) => ((! [Omega3:(stream @ A)]: (((P @ Omega3) => ((Q1 @ Omega3) = (Q2 @ Omega3))))) => ((((linear1386806755on_alw @ A) @ Q1) @ Omega) = (((linear1386806755on_alw @ A) @ Q2) @ Omega))))))).
% 2.33/1.37 thf(fact_31_alw__alwD, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((((linear1386806755on_alw @ A) @ P) @ Omega) => (((linear1386806755on_alw @ A) @ ((linear1386806755on_alw @ A) @ P)) @ Omega))))).
% 2.33/1.37 thf(fact_32_ev__mono, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs2) => ((! [Xs3:(stream @ A)]: (((Phi @ Xs3) => (Psi @ Xs3)))) => (((linear505997466_on_ev @ A) @ Psi) @ Xs2)))))).
% 2.33/1.37 thf(fact_33_ev_Obase, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A)]: (((Phi @ Xs2) => (((linear505997466_on_ev @ A) @ Phi) @ Xs2))))).
% 2.33/1.37 thf(fact_34_alwD, axiom, (! [A:$tType,P:((stream @ A) > $o),X:(stream @ A)]: (((((linear1386806755on_alw @ A) @ P) @ X) => (P @ X))))).
% 2.33/1.37 thf(fact_35_enat__less__induct, axiom, (! [P:(extended_enat > $o),N:extended_enat]: (((! [N2:extended_enat]: (((! [M3:extended_enat]: (((((ord_less @ extended_enat) @ M3) @ N2) => (P @ M3)))) => (P @ N2)))) => (P @ N))))).
% 2.33/1.37 thf(fact_36_infinity__ilessE, axiom, (! [M:nat]: ((~ ((((ord_less @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ (extended_enat2 @ M))))))).
% 2.33/1.37 thf(fact_37_less__infinityE, axiom, (! [N:extended_enat]: (((((ord_less @ extended_enat) @ N) @ (extend1396239628finity @ extended_enat)) => (~ ((! [K:nat]: ((N != (extended_enat2 @ K)))))))))).
% 2.33/1.37 thf(fact_38_enat__ord__code_I4_J, axiom, (! [M:nat]: ((((ord_less @ extended_enat) @ (extended_enat2 @ M)) @ (extend1396239628finity @ extended_enat))))).
% 2.33/1.37 thf(fact_39_alw__ev__sdrop, axiom, (! [A:$tType,Phi:((stream @ A) > $o),M:nat,Xs2:(stream @ A)]: (((((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ Phi)) @ (((sdrop @ A) @ M) @ Xs2)) => (((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ Phi)) @ Xs2))))).
% 2.33/1.37 thf(fact_40_alw__False, axiom, (! [A:$tType,Omega:(stream @ A)]: ((~ ((((linear1386806755on_alw @ A) @ (^ [X3:(stream @ A)]: ($false))) @ Omega)))))).
% 2.33/1.37 thf(fact_41_ev__False, axiom, (! [A:$tType,Omega:(stream @ A)]: ((~ ((((linear505997466_on_ev @ A) @ (^ [X3:(stream @ A)]: ($false))) @ Omega)))))).
% 2.33/1.37 thf(fact_42_alw__aand, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Psi:((stream @ A) > $o)]: ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) & (Psi @ Xs))))) = (^ [Xs:(stream @ A)]: (((((linear1386806755on_alw @ A) @ Phi) @ Xs) & (((linear1386806755on_alw @ A) @ Psi) @ Xs)))))))).
% 2.33/1.37 thf(fact_43_alw__mp, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear1386806755on_alw @ A) @ Phi) @ Xs2) => ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) => (Psi @ Xs))))) @ Xs2) => (((linear1386806755on_alw @ A) @ Psi) @ Xs2)))))).
% 2.33/1.37 thf(fact_44_ev__or, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Psi:((stream @ A) > $o)]: ((((linear505997466_on_ev @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) | (Psi @ Xs))))) = (^ [Xs:(stream @ A)]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs) | (((linear505997466_on_ev @ A) @ Psi) @ Xs)))))))).
% 2.33/1.37 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.33/1.37 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A3)))) = A3)))).
% 2.33/1.37 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q3:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q3 @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q3)))))).
% 2.33/1.37 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 2.33/1.37 thf(fact_49_scount__eq__card, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: ((~ ((P @ Xs))))))) @ Omega) => ((((coindu1365464361scount @ A) @ P) @ Omega) = (extended_enat2 @ ((finite_card @ nat) @ ((collect @ nat) @ (^ [I:nat]: ((P @ (((sdrop @ A) @ I) @ Omega)))))))))))).
% 2.33/1.37 thf(fact_50_ev__alw__imp__alw__ev, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A)]: (((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ Phi)) @ Xs2) => (((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ Phi)) @ Xs2))))).
% 2.33/1.37 thf(fact_51_ev__cong, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A),Q1:((stream @ A) > $o),Q2:((stream @ A) > $o)]: (((((linear1386806755on_alw @ A) @ P) @ Omega) => ((! [Omega3:(stream @ A)]: (((P @ Omega3) => ((Q1 @ Omega3) = (Q2 @ Omega3))))) => ((((linear505997466_on_ev @ A) @ Q1) @ Omega) = (((linear505997466_on_ev @ A) @ Q2) @ Omega))))))).
% 2.33/1.37 thf(fact_52_alw__ev__imp__ev__alw, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ P)) @ Omega) => (((linear505997466_on_ev @ A) @ (^ [Xs:(stream @ A)]: (((P @ Xs) & (((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ P)) @ Xs))))) @ Omega))))).
% 2.33/1.37 thf(fact_53_ev__alw__alw__impl, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ Phi)) @ Xs2) => ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((((linear1386806755on_alw @ A) @ Phi) @ Xs) => (((linear505997466_on_ev @ A) @ Psi) @ Xs))))) @ Xs2) => (((linear505997466_on_ev @ A) @ Psi) @ Xs2)))))).
% 2.33/1.37 thf(fact_54_alw__impl__ev__alw, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Psi:((stream @ A) > $o),Xs2:(stream @ A)]: (((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) => (((linear505997466_on_ev @ A) @ Psi) @ Xs))))) @ Xs2) => (((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs) => (((linear505997466_on_ev @ A) @ Psi) @ Xs))))) @ Xs2))))).
% 2.33/1.37 thf(fact_55_alw__alw__impl__ev, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Psi:((stream @ A) > $o)]: ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((((linear1386806755on_alw @ A) @ Phi) @ Xs) => (((linear505997466_on_ev @ A) @ Psi) @ Xs))))) = (^ [Xs:(stream @ A)]: (((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ Phi)) @ Xs) => (((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ Psi)) @ Xs)))))))).
% 2.33/1.37 thf(fact_56_ev__alw__impl__ev, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs2) => ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) => (((linear505997466_on_ev @ A) @ Psi) @ Xs))))) @ Xs2) => (((linear505997466_on_ev @ A) @ Psi) @ Xs2)))))).
% 2.33/1.37 thf(fact_57_enat_Osimps_I7_J, axiom, (! [T:$tType,F1:(nat > T),F2:T]: ((((((extended_rec_enat @ T) @ F1) @ F2) @ (extend1396239628finity @ extended_enat)) = F2)))).
% 2.33/1.37 thf(fact_58_enat_Osimps_I6_J, axiom, (! [T:$tType,F1:(nat > T),F2:T,Nat:nat]: ((((((extended_rec_enat @ T) @ F1) @ F2) @ (extended_enat2 @ Nat)) = (F1 @ Nat))))).
% 2.33/1.37 thf(fact_59_sdrop__wait, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A)]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs2) => (Phi @ (((sdrop @ A) @ (((linear1335279038n_wait @ A) @ Phi) @ Xs2)) @ Xs2)))))).
% 2.33/1.37 thf(fact_60_enat_Osimps_I5_J, axiom, (! [T:$tType,F1:(nat > T),F2:T]: ((((((extended_case_enat @ T) @ F1) @ F2) @ (extend1396239628finity @ extended_enat)) = F2)))).
% 2.33/1.37 thf(fact_61_enat_Osimps_I4_J, axiom, (! [T:$tType,F1:(nat > T),F2:T,Nat:nat]: ((((((extended_case_enat @ T) @ F1) @ F2) @ (extended_enat2 @ Nat)) = (F1 @ Nat))))).
% 2.33/1.37 thf(fact_62_infinite__iff__alw__ev, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((~ (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [M2:nat]: ((P @ (((sdrop @ A) @ M2) @ Omega)))))))) = (((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ P)) @ Omega))))).
% 2.33/1.37 thf(fact_63_ev__alw__not__HLD__finite, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: ((~ ((P @ Xs))))))) @ Omega) => ((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [I:nat]: ((P @ (((sdrop @ A) @ I) @ Omega)))))))))).
% 2.33/1.37 thf(fact_64_ev__alw__imp__nxt, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A)]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs2) => ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) => ((((linear1494993505on_nxt @ A) @ $o) @ Phi) @ Xs))))) @ Xs2) => (((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ Phi)) @ Xs2)))))).
% 2.33/1.37 thf(fact_65_variance, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((Phi @ Xs2) => ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) => ((Psi @ Xs) | ((((linear1494993505on_nxt @ A) @ $o) @ Phi) @ Xs)))))) @ Xs2) => ((((linear1386806755on_alw @ A) @ Phi) @ Xs2) | (((linear505997466_on_ev @ A) @ Psi) @ Xs2))))))).
% 2.33/1.37 thf(fact_66_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A2:A]: ((? [B2:A]: (((((ord_less @ A) @ A2) @ B2) | (((ord_less @ A) @ B2) @ A2)))))))))).
% 2.33/1.37 thf(fact_67_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((X != Y2) => ((~ ((((ord_less @ A) @ X) @ Y2))) => (((ord_less @ A) @ Y2) @ X))))))))).
% 2.33/1.37 thf(fact_68_enat__ord__simps_I2_J, axiom, (! [M:nat,N:nat]: (((((ord_less @ extended_enat) @ (extended_enat2 @ M)) @ (extended_enat2 @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 2.33/1.37 thf(fact_69_nxt__mono, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: ((((((linear1494993505on_nxt @ A) @ $o) @ Phi) @ Xs2) => ((! [Xs3:(stream @ A)]: (((Phi @ Xs3) => (Psi @ Xs3)))) => ((((linear1494993505on_nxt @ A) @ $o) @ Psi) @ Xs2)))))).
% 2.33/1.37 thf(fact_70_case__enat__def, axiom, (! [T:$tType]: (((extended_case_enat @ T) = (extended_rec_enat @ T))))).
% 2.33/1.37 thf(fact_71_less__enat__def, axiom, ((ord_less @ extended_enat) = (^ [M2:extended_enat,N3:extended_enat]: (((((extended_case_enat @ $o) @ (^ [M1:nat]: (((((extended_case_enat @ $o) @ ((ord_less @ nat) @ M1)) @ $true) @ N3)))) @ $false) @ M2))))).
% 2.33/1.37 thf(fact_72_ev__nxt, axiom, (! [A:$tType]: (((linear505997466_on_ev @ A) = (^ [Phi2:((stream @ A) > $o),Xs:(stream @ A)]: (((Phi2 @ Xs) | ((((linear1494993505on_nxt @ A) @ $o) @ ((linear505997466_on_ev @ A) @ Phi2)) @ Xs)))))))).
% 2.33/1.37 thf(fact_73_alw__nxt, axiom, (! [A:$tType]: (((linear1386806755on_alw @ A) = (^ [Phi2:((stream @ A) > $o),Xs:(stream @ A)]: (((Phi2 @ Xs) & ((((linear1494993505on_nxt @ A) @ $o) @ ((linear1386806755on_alw @ A) @ Phi2)) @ Xs)))))))).
% 2.33/1.37 thf(fact_74_alw__invar, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A)]: (((Phi @ Xs2) => ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) => ((((linear1494993505on_nxt @ A) @ $o) @ Phi) @ Xs))))) @ Xs2) => (((linear1386806755on_alw @ A) @ Phi) @ Xs2)))))).
% 2.33/1.37 thf(fact_75_less__enatE, axiom, (! [N:extended_enat,M:nat]: (((((ord_less @ extended_enat) @ N) @ (extended_enat2 @ M)) => (~ ((! [K:nat]: (((N = (extended_enat2 @ K)) => (~ ((((ord_less @ nat) @ K) @ M)))))))))))).
% 2.33/1.37 thf(fact_76_card__Collect__less__nat, axiom, (! [N:nat]: ((((finite_card @ nat) @ ((collect @ nat) @ (^ [I:nat]: ((((ord_less @ nat) @ I) @ N))))) = N)))).
% 2.33/1.37 thf(fact_77_finite__Collect__less__nat, axiom, (! [K2:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N3:nat]: ((((ord_less @ nat) @ N3) @ K2)))))))).
% 2.33/1.37 thf(fact_78_finite__Collect__disjI, axiom, (! [A:$tType,P:(A > $o),Q3:(A > $o)]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [X3:A]: (((P @ X3) | (Q3 @ X3)))))) = (((finite_finite2 @ A) @ ((collect @ A) @ P)) & ((finite_finite2 @ A) @ ((collect @ A) @ Q3))))))).
% 2.33/1.37 thf(fact_79_finite__Collect__conjI, axiom, (! [A:$tType,P:(A > $o),Q3:(A > $o)]: (((((finite_finite2 @ A) @ ((collect @ A) @ P)) | ((finite_finite2 @ A) @ ((collect @ A) @ Q3))) => ((finite_finite2 @ A) @ ((collect @ A) @ (^ [X3:A]: (((P @ X3) & (Q3 @ X3)))))))))).
% 2.33/1.37 thf(fact_80_finite__code, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => ((finite_finite2 @ A) = (^ [A4:(set @ A)]: ($true))))))).
% 2.33/1.37 thf(fact_81_psubset__card__mono, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less @ (set @ A)) @ A3) @ B3) => (((ord_less @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3))))))).
% 2.33/1.37 thf(fact_82_finite__M__bounded__by__nat, axiom, (! [P:(nat > $o),I3:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((P @ K3) & (((ord_less @ nat) @ K3) @ I3))))))))).
% 2.33/1.37 thf(fact_83_finite__set__choice, axiom, (! [B:$tType,A:$tType,A3:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X4:A]: (((((member @ A) @ X4) @ A3) => (? [X12:B]: (((P @ X4) @ X12)))))) => (? [F3:(A > B)]: ((! [X5:A]: (((((member @ A) @ X5) @ A3) => ((P @ X5) @ (F3 @ X5)))))))))))).
% 2.33/1.37 thf(fact_84_finite, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => (! [A3:(set @ A)]: (((finite_finite2 @ A) @ A3))))))).
% 2.33/1.37 thf(fact_85_finite__psubset__induct, axiom, (! [A:$tType,A3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A3) => ((! [A5:(set @ A)]: ((((finite_finite2 @ A) @ A5) => ((! [B4:(set @ A)]: (((((ord_less @ (set @ A)) @ B4) @ A5) => (P @ B4)))) => (P @ A5))))) => (P @ A3)))))).
% 2.33/1.37 thf(fact_86_bounded__nat__set__is__finite, axiom, (! [N4:(set @ nat),N:nat]: (((! [X4:nat]: (((((member @ nat) @ X4) @ N4) => (((ord_less @ nat) @ X4) @ N)))) => ((finite_finite2 @ nat) @ N4))))).
% 2.33/1.37 thf(fact_87_finite__nat__set__iff__bounded, axiom, ((finite_finite2 @ nat) = (^ [N5:(set @ nat)]: ((? [M2:nat]: ((! [X3:nat]: (((((member @ nat) @ X3) @ N5) => (((ord_less @ nat) @ X3) @ M2)))))))))).
% 2.33/1.37 thf(fact_88_not__finite__existsD, axiom, (! [A:$tType,P:(A > $o)]: (((~ (((finite_finite2 @ A) @ ((collect @ A) @ P)))) => (? [X1:A]: ((P @ X1))))))).
% 2.33/1.37 thf(fact_89_pigeonhole__infinite__rel, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B),R:(A > (B > $o))]: (((~ (((finite_finite2 @ A) @ A3))) => (((finite_finite2 @ B) @ B3) => ((! [X4:A]: (((((member @ A) @ X4) @ A3) => (? [Xa:B]: (((((member @ B) @ Xa) @ B3) & ((R @ X4) @ Xa))))))) => (? [X4:B]: (((((member @ B) @ X4) @ B3) & (~ (((finite_finite2 @ A) @ ((collect @ A) @ (^ [A6:A]: (((((member @ A) @ A6) @ A3) & ((R @ A6) @ X4))))))))))))))))).
% 2.33/1.37 thf(fact_90_unbounded__k__infinite, axiom, (! [K2:nat,S2:(set @ nat)]: (((! [M4:nat]: (((((ord_less @ nat) @ K2) @ M4) => (? [N6:nat]: (((((ord_less @ nat) @ M4) @ N6) & (((member @ nat) @ N6) @ S2))))))) => (~ (((finite_finite2 @ nat) @ S2))))))).
% 2.33/1.37 thf(fact_91_infinite__nat__iff__unbounded, axiom, (! [S2:(set @ nat)]: (((~ (((finite_finite2 @ nat) @ S2))) = (! [M2:nat]: ((? [N3:nat]: (((((ord_less @ nat) @ M2) @ N3) & (((member @ nat) @ N3) @ S2)))))))))).
% 2.33/1.37 thf(fact_92_less__set__def, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A4:(set @ A),B5:(set @ A)]: ((((ord_less @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A4)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B5)))))))))).
% 2.33/1.37 thf(fact_93_card__psubset, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3)) => (((ord_less @ (set @ A)) @ A3) @ B3))))))).
% 2.33/1.37 thf(fact_94_sdrop__wait__least, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),N:nat]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs2) => ((Phi @ (((sdrop @ A) @ N) @ Xs2)) => (((ord_less_eq @ nat) @ (((linear1335279038n_wait @ A) @ Phi) @ Xs2)) @ N)))))).
% 2.33/1.37 thf(fact_95_psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) => ((((ord_less @ (set @ A)) @ B3) @ C) => (((ord_less @ (set @ A)) @ A3) @ C)))))).
% 2.33/1.37 thf(fact_96_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => (A3 = B3)))))).
% 2.33/1.37 thf(fact_97_subsetI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A3) => (((member @ A) @ X4) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 2.33/1.37 thf(fact_98_psubsetI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((A3 != B3) => (((ord_less @ (set @ A)) @ A3) @ B3)))))).
% 2.33/1.37 thf(fact_99_finite__Collect__subsets, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ (set @ A)) @ ((collect @ (set @ A)) @ (^ [B5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B5) @ A3))))))))).
% 2.33/1.37 thf(fact_100_finite__Collect__le__nat, axiom, (! [K2:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N3:nat]: ((((ord_less_eq @ nat) @ N3) @ K2)))))))).
% 2.33/1.37 thf(fact_101_infinite__nat__iff__unbounded__le, axiom, (! [S2:(set @ nat)]: (((~ (((finite_finite2 @ nat) @ S2))) = (! [M2:nat]: ((? [N3:nat]: (((((ord_less_eq @ nat) @ M2) @ N3) & (((member @ nat) @ N3) @ S2)))))))))).
% 2.33/1.37 thf(fact_102_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B5:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B5) | (A4 = B5)))))))).
% 2.33/1.37 thf(fact_103_subset__psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less @ (set @ A)) @ B3) @ C) => (((ord_less @ (set @ A)) @ A3) @ C)))))).
% 2.33/1.37 thf(fact_104_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A4:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) & (~ ((((ord_less_eq @ (set @ A)) @ B5) @ A4)))))))))).
% 2.33/1.37 thf(fact_105_psubset__subset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C) => (((ord_less @ (set @ A)) @ A3) @ C)))))).
% 2.33/1.37 thf(fact_106_psubset__imp__subset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 2.33/1.37 thf(fact_107_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A4:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) & (A4 != B5)))))))).
% 2.33/1.37 thf(fact_108_psubsetE, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ (set @ A)) @ B3) @ A3)))))))).
% 2.33/1.37 thf(fact_109_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q3:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q3)) = (! [X3:A]: (((P @ X3) => (Q3 @ X3)))))))).
% 2.33/1.37 thf(fact_110_contra__subsetD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((~ ((((member @ A) @ C2) @ B3))) => (~ ((((member @ A) @ C2) @ A3)))))))).
% 2.33/1.37 thf(fact_111_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z:(set @ A)]: ((Y4 = Z))) = (^ [A4:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) & (((ord_less_eq @ (set @ A)) @ B5) @ A4)))))))).
% 2.33/1.37 thf(fact_112_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C) => (((ord_less_eq @ (set @ A)) @ A3) @ C)))))).
% 2.33/1.37 thf(fact_113_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q3:(A > $o)]: (((! [X4:A]: (((P @ X4) => (Q3 @ X4)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q3)))))).
% 2.33/1.37 thf(fact_114_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 2.33/1.37 thf(fact_115_rev__subsetD, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ A3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((member @ A) @ C2) @ B3)))))).
% 2.33/1.37 thf(fact_116_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B5:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A4) => (((member @ A) @ T2) @ B5)))))))))).
% 2.33/1.37 thf(fact_117_set__rev__mp, axiom, (! [A:$tType,X:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ X) @ A3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((member @ A) @ X) @ B3)))))).
% 2.33/1.37 thf(fact_118_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (((ord_less_eq @ (set @ A)) @ B3) @ A3))))).
% 2.33/1.37 thf(fact_119_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 2.33/1.37 thf(fact_120_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B5:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A4) => (((member @ A) @ X3) @ B5)))))))))).
% 2.33/1.37 thf(fact_121_equalityE, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (~ ((((ord_less_eq @ (set @ A)) @ B3) @ A3)))))))))).
% 2.33/1.37 thf(fact_122_subsetCE, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))).
% 2.33/1.37 thf(fact_123_subsetD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))).
% 2.33/1.37 thf(fact_124_in__mono, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ X) @ A3) => (((member @ A) @ X) @ B3)))))).
% 2.33/1.37 thf(fact_125_set__mp, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ X) @ A3) => (((member @ A) @ X) @ B3)))))).
% 2.33/1.37 thf(fact_126_card__seteq, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ B3)) @ ((finite_card @ A) @ A3)) => (A3 = B3))))))).
% 2.33/1.37 thf(fact_127_card__mono, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3))))))).
% 2.33/1.37 thf(fact_128_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A2:A,B6:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B6) => ((P @ A2) => ((~ ((P @ B6))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B6) & ((! [X5:A]: ((((((ord_less_eq @ A) @ A2) @ X5) & (((ord_less @ A) @ X5) @ C3)) => (P @ X5)))) & (! [D:A]: (((! [X4:A]: ((((((ord_less_eq @ A) @ A2) @ X4) & (((ord_less @ A) @ X4) @ D)) => (P @ X4)))) => (((ord_less_eq @ A) @ D) @ C3)))))))))))))))))).
% 2.33/1.37 thf(fact_129_finite__subset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((finite_finite2 @ A) @ B3) => ((finite_finite2 @ A) @ A3)))))).
% 2.33/1.37 thf(fact_130_infinite__super, axiom, (! [A:$tType,S2:(set @ A),T3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S2) @ T3) => ((~ (((finite_finite2 @ A) @ S2))) => (~ (((finite_finite2 @ A) @ T3)))))))).
% 2.33/1.37 thf(fact_131_rev__finite__subset, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((finite_finite2 @ A) @ A3)))))).
% 2.33/1.37 thf(fact_132_finite__nat__set__iff__bounded__le, axiom, ((finite_finite2 @ nat) = (^ [N5:(set @ nat)]: ((? [M2:nat]: ((! [X3:nat]: (((((member @ nat) @ X3) @ N5) => (((ord_less_eq @ nat) @ X3) @ M2)))))))))).
% 2.33/1.37 thf(fact_133_finite__less__ub, axiom, (! [F:(nat > nat),U:nat]: (((! [N2:nat]: ((((ord_less_eq @ nat) @ N2) @ (F @ N2)))) => ((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N3:nat]: ((((ord_less_eq @ nat) @ (F @ N3)) @ U))))))))).
% 2.33/1.37 thf(fact_134_card__subset__eq, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((finite_card @ A) @ A3) = ((finite_card @ A) @ B3)) => (A3 = B3))))))).
% 2.33/1.37 thf(fact_135_infinite__arbitrarily__large, axiom, (! [A:$tType,A3:(set @ A),N:nat]: (((~ (((finite_finite2 @ A) @ A3))) => (? [B7:(set @ A)]: ((((finite_finite2 @ A) @ B7) & ((((finite_card @ A) @ B7) = N) & (((ord_less_eq @ (set @ A)) @ B7) @ A3))))))))).
% 2.33/1.37 thf(fact_136_psubsetD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:A]: (((((ord_less @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))).
% 2.33/1.37 thf(fact_137_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.33/1.37 thf(fact_138_ex__has__greatest__nat, axiom, (! [A:$tType,P:(A > $o),K2:A,M:(A > nat),B6:nat]: (((P @ K2) => ((! [Y5:A]: (((P @ Y5) => (((ord_less @ nat) @ (M @ Y5)) @ B6)))) => (? [X4:A]: (((P @ X4) & (! [Y6:A]: (((P @ Y6) => (((ord_less_eq @ nat) @ (M @ Y6)) @ (M @ X4))))))))))))).
% 2.33/1.37 thf(fact_139_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I3:nat,J3:nat]: (((! [I2:nat,J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (((ord_less @ nat) @ (F @ I2)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I3) @ J3) => (((ord_less_eq @ nat) @ (F @ I3)) @ (F @ J3))))))).
% 2.33/1.37 thf(fact_140_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 2.33/1.37 thf(fact_141_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.33/1.37 thf(fact_142_enat__ord__simps_I5_J, axiom, (! [Q:extended_enat]: (((((ord_less_eq @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ Q) = (Q = (extend1396239628finity @ extended_enat)))))).
% 2.33/1.37 thf(fact_143_enat__ord__code_I3_J, axiom, (! [Q:extended_enat]: ((((ord_less_eq @ extended_enat) @ Q) @ (extend1396239628finity @ extended_enat))))).
% 2.33/1.37 thf(fact_144_enat__ord__simps_I1_J, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ extended_enat) @ (extended_enat2 @ M)) @ (extended_enat2 @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.33/1.37 thf(fact_145_less__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less @ (A > B)) = (^ [F4:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F4) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F4))))))))))).
% 2.33/1.37 thf(fact_146_finite__enat__bounded, axiom, (! [A3:(set @ extended_enat),N:nat]: (((! [Y5:extended_enat]: (((((member @ extended_enat) @ Y5) @ A3) => (((ord_less_eq @ extended_enat) @ Y5) @ (extended_enat2 @ N))))) => ((finite_finite2 @ extended_enat) @ A3))))).
% 2.33/1.37 thf(fact_147_enat__ile, axiom, (! [N:extended_enat,M:nat]: (((((ord_less_eq @ extended_enat) @ N) @ (extended_enat2 @ M)) => (? [K:nat]: ((N = (extended_enat2 @ K)))))))).
% 2.33/1.37 thf(fact_148_enat__ord__simps_I3_J, axiom, (! [Q:extended_enat]: ((((ord_less_eq @ extended_enat) @ Q) @ (extend1396239628finity @ extended_enat))))).
% 2.33/1.37 thf(fact_149_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B5:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A4)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B5)))))))))).
% 2.33/1.37 thf(fact_150_infinity__ileE, axiom, (! [M:nat]: ((~ ((((ord_less_eq @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ (extended_enat2 @ M))))))).
% 2.33/1.37 thf(fact_151_enat__ord__code_I5_J, axiom, (! [N:nat]: ((~ ((((ord_less_eq @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ (extended_enat2 @ N))))))).
% 2.33/1.37 thf(fact_152_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.33/1.37 thf(fact_153_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.33/1.37 thf(fact_154_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X4:A]: ((((ord_less_eq @ B) @ (F @ X4)) @ (G @ X4)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.33/1.37 thf(fact_155_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F4:(A > B),G2:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F4 @ X3)) @ (G2 @ X3))))))))))).
% 2.33/1.38 thf(fact_156_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B6:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B6)) => ((((ord_less_eq @ B) @ B6) @ C2) => ((! [X4:B,Y5:B]: (((((ord_less_eq @ B) @ X4) @ Y5) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y5))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.33/1.38 thf(fact_157_order__subst2, axiom, (! [A:$tType,C4:$tType]: (((((order @ C4) @ (type2 @ C4)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B6:A,F:(A > C4),C2:C4]: (((((ord_less_eq @ A) @ A2) @ B6) => ((((ord_less_eq @ C4) @ (F @ B6)) @ C2) => ((! [X4:A,Y5:A]: (((((ord_less_eq @ A) @ X4) @ Y5) => (((ord_less_eq @ C4) @ (F @ X4)) @ (F @ Y5))))) => (((ord_less_eq @ C4) @ (F @ A2)) @ C2)))))))))).
% 2.33/1.38 thf(fact_158_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B6:B,C2:B]: (((A2 = (F @ B6)) => ((((ord_less_eq @ B) @ B6) @ C2) => ((! [X4:B,Y5:B]: (((((ord_less_eq @ B) @ X4) @ Y5) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y5))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.33/1.38 thf(fact_159_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B6:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B6) => (((F @ B6) = C2) => ((! [X4:A,Y5:A]: (((((ord_less_eq @ A) @ X4) @ Y5) => (((ord_less_eq @ B) @ (F @ X4)) @ (F @ Y5))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C2)))))))))).
% 2.33/1.38 thf(fact_160_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) & (((ord_less_eq @ A) @ Y) @ X3))))))))).
% 2.33/1.38 thf(fact_161_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X) => (X = Y2))))))))).
% 2.33/1.38 thf(fact_162_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X)))))))).
% 2.33/1.38 thf(fact_163_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((X = Y2) => (((ord_less_eq @ A) @ X) @ Y2)))))))).
% 2.33/1.38 thf(fact_164_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X)))))))).
% 2.33/1.38 thf(fact_165_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B6:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B6) => ((((ord_less_eq @ A) @ B6) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.33/1.38 thf(fact_166_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z2:A]: ((((((ord_less_eq @ A) @ X) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z2)))) => (((((ord_less_eq @ A) @ Y2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z2)))) => (((((ord_less_eq @ A) @ X) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y2)))) => (((((ord_less_eq @ A) @ Z2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X)))) => (((((ord_less_eq @ A) @ Y2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y2))))))))))))))))).
% 2.33/1.38 thf(fact_167_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y2:A,X:A]: (((((ord_less_eq @ A) @ Y2) @ X) => ((((ord_less_eq @ A) @ X) @ Y2) = (X = Y2))))))))).
% 2.33/1.38 thf(fact_168_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B6:A,C2:A]: (((A2 = B6) => ((((ord_less_eq @ A) @ B6) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.33/1.38 thf(fact_169_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B6:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B6) => ((B6 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.33/1.38 thf(fact_170_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B6:A]: (((((ord_less_eq @ A) @ A2) @ B6) => ((((ord_less_eq @ A) @ B6) @ A2) => (A2 = B6))))))))).
% 2.33/1.38 thf(fact_171_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 2.33/1.38 thf(fact_172_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.33/1.38 thf(fact_173_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B6:A]: (((! [A7:A,B2:A]: (((((ord_less_eq @ A) @ A7) @ B2) => ((P @ A7) @ B2)))) => ((! [A7:A,B2:A]: ((((P @ B2) @ A7) => ((P @ A7) @ B2)))) => ((P @ A2) @ B6))))))))).
% 2.33/1.38 thf(fact_174_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B6:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B6) @ A2) => ((((ord_less_eq @ A) @ C2) @ B6) => (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 2.33/1.38 thf(fact_175_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B6:A,A2:A]: (((((ord_less_eq @ A) @ B6) @ A2) => ((((ord_less_eq @ A) @ A2) @ B6) => (A2 = B6))))))))).
% 2.33/1.38 thf(fact_176_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B6:A,A2:A]: (((((ord_less @ A) @ B6) @ A2) => (A2 != B6)))))))).
% 2.33/1.38 thf(fact_177_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B6:A]: (((((ord_less @ A) @ A2) @ B6) => (A2 != B6)))))))).
% 2.33/1.38 thf(fact_178_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less @ A) @ X) @ Y2))) = ((((ord_less @ A) @ Y2) @ X) | (X = Y2))))))))).
% 2.33/1.38 thf(fact_179_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B6:A,A2:A,C2:A]: (((((ord_less @ A) @ B6) @ A2) => ((((ord_less @ A) @ C2) @ B6) => (((ord_less @ A) @ C2) @ A2))))))))).
% 2.33/1.38 thf(fact_180_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X)))))))))).
% 2.33/1.38 thf(fact_181_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B6:A,C2:A]: (((((ord_less @ A) @ A2) @ B6) => ((((ord_less @ A) @ B6) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.33/1.38 thf(fact_182_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.33/1.38 thf(fact_183_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less @ A) @ X) @ Y2))) => ((X != Y2) => (((ord_less @ A) @ Y2) @ X))))))))).
% 2.33/1.38 thf(fact_184_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,P:$o]: (((((ord_less @ A) @ X) @ Y2) => ((((ord_less @ A) @ Y2) @ X) => P)))))))).
% 2.33/1.38 thf(fact_185_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (Y2 != X)))))))).
% 2.33/1.38 thf(fact_186_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y2:A,X:A]: (((~ ((((ord_less @ A) @ Y2) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y2))) = (X = Y2))))))))).
% 2.33/1.38 thf(fact_187_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A]: (((! [X4:A]: (((! [Y6:A]: (((((ord_less @ A) @ Y6) @ X4) => (P @ Y6)))) => (P @ X4)))) => (P @ A2)))))))).
% 2.33/1.38 thf(fact_188_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X)))))))))).
% 2.33/1.38 thf(fact_189_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (X != Y2)))))))).
% 2.33/1.38 thf(fact_190_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B6:A,A2:A]: (((((ord_less @ A) @ B6) @ A2) => (~ ((((ord_less @ A) @ A2) @ B6)))))))))).
% 2.33/1.38 thf(fact_191_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B6:A,C2:A]: (((((ord_less @ A) @ A2) @ B6) => ((B6 = C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.33/1.38 thf(fact_192_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B6:A,C2:A]: (((A2 = B6) => ((((ord_less @ A) @ B6) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.33/1.38 thf(fact_193_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 2.33/1.38 thf(fact_194_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) | ((X = Y2) | (((ord_less @ A) @ Y2) @ X))))))))).
% 2.33/1.38 thf(fact_195_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.33/1.38 thf(fact_196_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:A,B6:A]: (((((ord_less @ A) @ A2) @ B6) => (~ ((((ord_less @ A) @ B6) @ A2)))))))))).
% 2.33/1.38 thf(fact_197_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X)))))))))).
% 2.33/1.38 thf(fact_198_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (X != Y2)))))))).
% 2.33/1.38 thf(fact_199_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (? [Z3:A]: (((((ord_less @ A) @ X) @ Z3) & (((ord_less @ A) @ Z3) @ Y2))))))))))).
% 2.33/1.38 thf(fact_200_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B6:A]: (((((ord_less @ A) @ A2) @ B6) => (~ ((((ord_less @ A) @ B6) @ A2)))))))))).
% 2.33/1.38 thf(fact_201_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((X != Y2) = ((((ord_less @ A) @ X) @ Y2) | (((ord_less @ A) @ Y2) @ X))))))))).
% 2.33/1.38 thf(fact_202_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((X != Y2) => ((~ ((((ord_less @ A) @ X) @ Y2))) => (((ord_less @ A) @ Y2) @ X))))))))).
% 2.33/1.38 thf(fact_203_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X:A]: ((? [X1:A]: ((((ord_less @ A) @ X) @ X1))))))))).
% 2.33/1.38 thf(fact_204_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X:A]: ((? [Y5:A]: ((((ord_less @ A) @ Y5) @ X))))))))).
% 2.33/1.38 thf(fact_205_order__less__subst2, axiom, (! [A:$tType,C4:$tType]: (((((order @ C4) @ (type2 @ C4)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B6:A,F:(A > C4),C2:C4]: (((((ord_less @ A) @ A2) @ B6) => ((((ord_less @ C4) @ (F @ B6)) @ C2) => ((! [X4:A,Y5:A]: (((((ord_less @ A) @ X4) @ Y5) => (((ord_less @ C4) @ (F @ X4)) @ (F @ Y5))))) => (((ord_less @ C4) @ (F @ A2)) @ C2)))))))))).
% 2.33/1.38 thf(fact_206_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B6:B,C2:B]: (((((ord_less @ A) @ A2) @ (F @ B6)) => ((((ord_less @ B) @ B6) @ C2) => ((! [X4:B,Y5:B]: (((((ord_less @ B) @ X4) @ Y5) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y5))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.33/1.38 thf(fact_207_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B6:A,F:(A > B),C2:B]: (((((ord_less @ A) @ A2) @ B6) => (((F @ B6) = C2) => ((! [X4:A,Y5:A]: (((((ord_less @ A) @ X4) @ Y5) => (((ord_less @ B) @ (F @ X4)) @ (F @ Y5))))) => (((ord_less @ B) @ (F @ A2)) @ C2)))))))))).
% 2.33/1.38 thf(fact_208_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B6:B,C2:B]: (((A2 = (F @ B6)) => ((((ord_less @ B) @ B6) @ C2) => ((! [X4:B,Y5:B]: (((((ord_less @ B) @ X4) @ Y5) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y5))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.33/1.38 thf(fact_209_less__eq__enat__def, axiom, ((ord_less_eq @ extended_enat) = (^ [M2:extended_enat]: ((((extended_case_enat @ $o) @ (^ [N1:nat]: (((((extended_case_enat @ $o) @ (^ [M1:nat]: ((((ord_less_eq @ nat) @ M1) @ N1)))) @ $false) @ M2)))) @ $true))))).
% 2.33/1.38 thf(fact_210_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X:A]: (((! [X4:A]: (((~ ((P @ X4))) => (? [Y6:A]: (((((ord_less @ nat) @ (V @ Y6)) @ (V @ X4)) & (~ ((P @ Y6))))))))) => (P @ X))))).
% 2.33/1.38 thf(fact_211_measure__induct__rule, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X4:A]: (((! [Y6:A]: (((((ord_less @ nat) @ (F @ Y6)) @ (F @ X4)) => (P @ Y6)))) => (P @ X4)))) => (P @ A2))))).
% 2.33/1.38 thf(fact_212_linorder__neqE__nat, axiom, (! [X:nat,Y2:nat]: (((X != Y2) => ((~ ((((ord_less @ nat) @ X) @ Y2))) => (((ord_less @ nat) @ Y2) @ X)))))).
% 2.33/1.38 thf(fact_213_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((~ ((P @ N2))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N2) & (~ ((P @ M3))))))))) => (P @ N))))).
% 2.33/1.38 thf(fact_214_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N2) => (P @ M3)))) => (P @ N2)))) => (P @ N))))).
% 2.33/1.38 thf(fact_215_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.33/1.38 thf(fact_216_measure__induct, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X4:A]: (((! [Y6:A]: (((((ord_less @ nat) @ (F @ Y6)) @ (F @ X4)) => (P @ Y6)))) => (P @ X4)))) => (P @ A2))))).
% 2.33/1.38 thf(fact_217_less__not__refl3, axiom, (! [S3:nat,T4:nat]: (((((ord_less @ nat) @ S3) @ T4) => (S3 != T4))))).
% 2.33/1.38 thf(fact_218_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 2.33/1.38 thf(fact_219_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.33/1.38 thf(fact_220_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 2.33/1.38 thf(fact_221_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.33/1.38 thf(fact_222_le__trans, axiom, (! [I3:nat,J3:nat,K2:nat]: (((((ord_less_eq @ nat) @ I3) @ J3) => ((((ord_less_eq @ nat) @ J3) @ K2) => (((ord_less_eq @ nat) @ I3) @ K2)))))).
% 2.33/1.38 thf(fact_223_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.33/1.38 thf(fact_224_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.33/1.38 thf(fact_225_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.33/1.38 thf(fact_226_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K2:A,M:(A > nat)]: (((P @ K2) => (? [X4:A]: (((P @ X4) & (! [Y6:A]: (((P @ Y6) => (((ord_less_eq @ nat) @ (M @ X4)) @ (M @ Y6)))))))))))).
% 2.33/1.38 thf(fact_227_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B6:A]: (((A2 != B6) => ((((ord_less_eq @ A) @ A2) @ B6) => (((ord_less @ A) @ A2) @ B6))))))))).
% 2.33/1.38 thf(fact_228_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B6:A,A2:A]: (((((ord_less @ A) @ B6) @ A2) => (((ord_less_eq @ A) @ B6) @ A2)))))))).
% 2.33/1.38 thf(fact_229_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B8:A,A6:A]: (((((ord_less_eq @ A) @ B8) @ A6) & (A6 != B8))))))))).
% 2.33/1.38 thf(fact_230_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B8:A,A6:A]: (((((ord_less @ A) @ B8) @ A6) | (A6 = B8))))))))).
% 2.33/1.38 thf(fact_231_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B6:A]: (((((ord_less @ A) @ A2) @ B6) => (((ord_less_eq @ A) @ A2) @ B6)))))))).
% 2.33/1.38 thf(fact_232_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X) @ Y2) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y2) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.33/1.38 thf(fact_233_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,X:A,Y2:A]: (((((ord_less @ A) @ Z2) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y2) @ W))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.33/1.38 thf(fact_234_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B6:A,A2:A,C2:A]: (((((ord_less @ A) @ B6) @ A2) => ((((ord_less_eq @ A) @ C2) @ B6) => (((ord_less @ A) @ C2) @ A2))))))))).
% 2.33/1.38 thf(fact_235_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B6:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B6) @ A2) => ((((ord_less @ A) @ C2) @ B6) => (((ord_less @ A) @ C2) @ A2))))))))).
% 2.33/1.38 thf(fact_236_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A6:A,B8:A]: (((((ord_less_eq @ A) @ A6) @ B8) & (A6 != B8))))))))).
% 2.33/1.38 thf(fact_237_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A6:A,B8:A]: (((((ord_less @ A) @ A6) @ B8) | (A6 = B8))))))))).
% 2.33/1.38 thf(fact_238_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B6:A,C2:A]: (((((ord_less @ A) @ A2) @ B6) => ((((ord_less_eq @ A) @ B6) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.33/1.38 thf(fact_239_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B6:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B6) => ((((ord_less @ A) @ B6) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.33/1.38 thf(fact_240_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y2:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y2) @ X))) => (((ord_less @ A) @ X) @ Y2)))))))).
% 2.33/1.38 thf(fact_241_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) & (~ ((((ord_less_eq @ A) @ Y) @ X3))))))))))).
% 2.33/1.38 thf(fact_242_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less @ A) @ X) @ Y2) | (X = Y2))))))))).
% 2.33/1.38 thf(fact_243_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) | (((ord_less @ A) @ Y2) @ X)))))))).
% 2.33/1.38 thf(fact_244_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y2:A,Z2:A]: (((! [X4:A]: (((((ord_less @ A) @ X4) @ Y2) => (((ord_less_eq @ A) @ X4) @ Z2)))) => (((ord_less_eq @ A) @ Y2) @ Z2)))))))).
% 2.33/1.38 thf(fact_245_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,Y2:A]: (((! [X4:A]: (((((ord_less @ A) @ Z2) @ X4) => (((ord_less_eq @ A) @ Y2) @ X4)))) => (((ord_less_eq @ A) @ Y2) @ Z2)))))))).
% 2.33/1.38 thf(fact_246_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.33/1.38 thf(fact_247_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.33/1.38 thf(fact_248_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((~ ((((ord_less @ A) @ X) @ Y2))) = (X = Y2))))))))).
% 2.33/1.38 thf(fact_249_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less @ A) @ X) @ Y2))) => ((((ord_less_eq @ A) @ X) @ Y2) = (X = Y2))))))))).
% 2.33/1.38 thf(fact_250_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (((ord_less_eq @ A) @ X) @ Y2)))))))).
% 2.33/1.38 thf(fact_251_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B6:A]: (((((ord_less_eq @ A) @ A2) @ B6) => ((A2 != B6) => (((ord_less @ A) @ A2) @ B6))))))))).
% 2.33/1.38 thf(fact_252_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less @ A) @ X) @ Y2))) = (((ord_less_eq @ A) @ Y2) @ X)))))))).
% 2.33/1.38 thf(fact_253_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y2))) = (((ord_less @ A) @ Y2) @ X)))))))).
% 2.33/1.38 thf(fact_254_order__less__le__subst2, axiom, (! [A:$tType,C4:$tType]: (((((order @ C4) @ (type2 @ C4)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B6:A,F:(A > C4),C2:C4]: (((((ord_less @ A) @ A2) @ B6) => ((((ord_less_eq @ C4) @ (F @ B6)) @ C2) => ((! [X4:A,Y5:A]: (((((ord_less @ A) @ X4) @ Y5) => (((ord_less @ C4) @ (F @ X4)) @ (F @ Y5))))) => (((ord_less @ C4) @ (F @ A2)) @ C2)))))))))).
% 2.33/1.38 thf(fact_255_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B6:B,C2:B]: (((((ord_less @ A) @ A2) @ (F @ B6)) => ((((ord_less_eq @ B) @ B6) @ C2) => ((! [X4:B,Y5:B]: (((((ord_less_eq @ B) @ X4) @ Y5) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y5))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.33/1.38 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: ((((preorder @ A9) @ (type2 @ A9)) => ((preorder @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.33/1.38 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A8:$tType,A9:$tType]: (((((finite_finite @ A8) @ (type2 @ A8)) & ((finite_finite @ A9) @ (type2 @ A9))) => ((finite_finite @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.33/1.38 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: ((((order @ A9) @ (type2 @ A9)) => ((order @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.33/1.38 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: ((((ord @ A9) @ (type2 @ A9)) => ((ord @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.33/1.38 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, ((condit1037483654norder @ nat) @ (type2 @ nat))).
% 2.33/1.38 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, ((wellorder @ nat) @ (type2 @ nat))).
% 2.33/1.38 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 2.33/1.38 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 2.33/1.38 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, ((no_top @ nat) @ (type2 @ nat))).
% 2.33/1.38 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, ((order @ nat) @ (type2 @ nat))).
% 2.33/1.38 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, ((ord @ nat) @ (type2 @ nat))).
% 2.33/1.38 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A8:$tType]: (((preorder @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.33/1.38 thf(tcon_Set_Oset___Finite__Set_Ofinite_5, axiom, (! [A8:$tType]: ((((finite_finite @ A8) @ (type2 @ A8)) => ((finite_finite @ (set @ A8)) @ (type2 @ (set @ A8))))))).
% 2.33/1.38 thf(tcon_Set_Oset___Orderings_Oorder_6, axiom, (! [A8:$tType]: (((order @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.33/1.38 thf(tcon_Set_Oset___Orderings_Oord_7, axiom, (! [A8:$tType]: (((ord @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.33/1.38 thf(tcon_HOL_Obool___Orderings_Opreorder_8, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.33/1.38 thf(tcon_HOL_Obool___Orderings_Olinorder_9, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.33/1.38 thf(tcon_HOL_Obool___Finite__Set_Ofinite_10, axiom, ((finite_finite @ $o) @ (type2 @ $o))).
% 2.33/1.38 thf(tcon_HOL_Obool___Orderings_Oorder_11, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.33/1.38 thf(tcon_HOL_Obool___Orderings_Oord_12, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.33/1.38 thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_13, axiom, ((condit1037483654norder @ extended_enat) @ (type2 @ extended_enat))).
% 2.33/1.38 thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_14, axiom, ((wellorder @ extended_enat) @ (type2 @ extended_enat))).
% 2.33/1.38 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_15, axiom, ((preorder @ extended_enat) @ (type2 @ extended_enat))).
% 2.33/1.38 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_16, axiom, ((linorder @ extended_enat) @ (type2 @ extended_enat))).
% 2.33/1.38 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_17, axiom, ((order @ extended_enat) @ (type2 @ extended_enat))).
% 2.33/1.38 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_18, axiom, ((ord @ extended_enat) @ (type2 @ extended_enat))).
% 2.33/1.38 thf(conj_0, hypothesis, (((linear505997466_on_ev @ a) @ ((linear1386806755on_alw @ a) @ (^ [Xs:(stream @ a)]: ((~ ((p @ Xs))))))) @ omega)).
% 2.33/1.38 thf(conj_1, conjecture, (((ord_less @ extended_enat) @ (((coindu1365464361scount @ a) @ p) @ omega)) @ (extend1396239628finity @ extended_enat))).
% 2.33/1.38 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.HepLMUEemU/DTF2DTF20348.p
% 2.33/1.38 ------------------------
% 2.59/1.40 ---- Cleaned THF ---
% 2.59/1.40 thf(ty_t_Extended__Nat_Oenat,type,
% 2.59/1.40 extended_enat: $tType ).
% 2.59/1.40
% 2.59/1.40 thf(ty_t_Stream_Ostream,type,
% 2.59/1.40 stream: $tType > $tType ).
% 2.59/1.40
% 2.59/1.40 thf(ty_t_Set_Oset,type,
% 2.59/1.40 set: $tType > $tType ).
% 2.59/1.40
% 2.59/1.40 thf(ty_t_Nat_Onat,type,
% 2.59/1.40 nat: $tType ).
% 2.59/1.40
% 2.59/1.40 thf(ty_t_itself,type,
% 2.59/1.40 itself: $tType > $tType ).
% 2.59/1.40
% 2.59/1.40 thf(ty_tf_a,type,
% 2.59/1.40 a: $tType ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_HOL_Otype,type,
% 2.59/1.40 type:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_Orderings_Oord,type,
% 2.59/1.40 ord:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_Orderings_Oorder,type,
% 2.59/1.40 order:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_Orderings_Ono__bot,type,
% 2.59/1.40 no_bot:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_Orderings_Ono__top,type,
% 2.59/1.40 no_top:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.59/1.40 finite_finite:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_Orderings_Olinorder,type,
% 2.59/1.40 linorder:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_Orderings_Opreorder,type,
% 2.59/1.40 preorder:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_Orderings_Owellorder,type,
% 2.59/1.40 wellorder:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_Orderings_Odense__order,type,
% 2.59/1.40 dense_order:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.59/1.40 linordered_idom:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.59/1.40 dense_linorder:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.59/1.40 condit1656338222tinuum:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.59/1.40 condit1037483654norder:
% 2.59/1.40 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Oscount,type,
% 2.59/1.40 coindu1365464361scount:
% 2.59/1.40 !>[S: $tType] : ( ( ( stream @ S ) > $o ) > ( stream @ S ) > extended_enat ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Extended__Nat_Oenat,type,
% 2.59/1.40 extended_enat2: nat > extended_enat ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
% 2.59/1.40 extended_case_enat:
% 2.59/1.40 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
% 2.59/1.40 extended_rec_enat:
% 2.59/1.40 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
% 2.59/1.40 extend1396239628finity:
% 2.59/1.40 !>[A: $tType] : A ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Finite__Set_Ocard,type,
% 2.59/1.40 finite_card:
% 2.59/1.40 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Finite__Set_Ofinite,type,
% 2.59/1.40 finite_finite2:
% 2.59/1.40 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Linear__Temporal__Logic__on__Streams_Oalw,type,
% 2.59/1.40 linear1386806755on_alw:
% 2.59/1.40 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Linear__Temporal__Logic__on__Streams_Oev,type,
% 2.59/1.40 linear505997466_on_ev:
% 2.59/1.40 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Linear__Temporal__Logic__on__Streams_Onxt,type,
% 2.59/1.40 linear1494993505on_nxt:
% 2.59/1.40 !>[A: $tType,B: $tType] : ( ( ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Linear__Temporal__Logic__on__Streams_Owait,type,
% 2.59/1.40 linear1335279038n_wait:
% 2.59/1.40 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > nat ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.59/1.40 ord_less:
% 2.59/1.40 !>[A: $tType] : ( A > A > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.59/1.40 ord_less_eq:
% 2.59/1.40 !>[A: $tType] : ( A > A > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Pure_Otype,type,
% 2.59/1.40 type2:
% 2.59/1.40 !>[A: $tType] : ( itself @ A ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Set_OCollect,type,
% 2.59/1.40 collect:
% 2.59/1.40 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_Stream_Osdrop,type,
% 2.59/1.40 sdrop:
% 2.59/1.40 !>[A: $tType] : ( nat > ( stream @ A ) > ( stream @ A ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_c_member,type,
% 2.59/1.40 member:
% 2.59/1.40 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(sy_v_P,type,
% 2.59/1.40 p: ( stream @ a ) > $o ).
% 2.59/1.40
% 2.59/1.40 thf(sy_v__092_060omega_062,type,
% 2.59/1.40 omega: stream @ a ).
% 2.59/1.40
% 2.59/1.40 thf(fact_0__092_060open_062ev_A_Ialw_A_I_092_060lambda_062xs_O_A_092_060not_062_AP_Axs_J_J_A_092_060omega_062_A_092_060Longrightarrow_062_Ascount_AP_A_092_060omega_062_A_061_Aenat_A_Icard_A_123i_O_AP_A_Isdrop_Ai_A_092_060omega_062_J_125_J_092_060close_062,axiom,
% 2.59/1.40 ( ( linear505997466_on_ev @ a
% 2.59/1.40 @ ( linear1386806755on_alw @ a
% 2.59/1.40 @ ^ [Xs: stream @ a] :
% 2.59/1.40 ~ ( p @ Xs ) )
% 2.59/1.40 @ omega )
% 2.59/1.40 => ( ( coindu1365464361scount @ a @ p @ omega )
% 2.59/1.40 = ( extended_enat2
% 2.59/1.40 @ ( finite_card @ nat
% 2.59/1.40 @ ( collect @ nat
% 2.59/1.40 @ ^ [I: nat] : ( p @ ( sdrop @ a @ I @ omega ) ) ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_1_not__ev__not,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( ^ [Xs: stream @ A] :
% 2.59/1.40 ~ ( linear505997466_on_ev @ A
% 2.59/1.40 @ ^ [Xt: stream @ A] :
% 2.59/1.40 ~ ( Phi @ Xt )
% 2.59/1.40 @ Xs ) )
% 2.59/1.40 = ( linear1386806755on_alw @ A @ Phi ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_2_not__alw__not,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( ^ [Xs: stream @ A] :
% 2.59/1.40 ~ ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xt: stream @ A] :
% 2.59/1.40 ~ ( Phi @ Xt )
% 2.59/1.40 @ Xs ) )
% 2.59/1.40 = ( linear505997466_on_ev @ A @ Phi ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_3_enat__ord__simps_I6_J,axiom,
% 2.59/1.40 ! [Q: extended_enat] :
% 2.59/1.40 ~ ( ord_less @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ Q ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_4_enat__ord__simps_I4_J,axiom,
% 2.59/1.40 ! [Q: extended_enat] :
% 2.59/1.40 ( ( ord_less @ extended_enat @ Q @ ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 = ( Q
% 2.59/1.40 != ( extend1396239628finity @ extended_enat ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_5_ev__ev,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A @ ( linear505997466_on_ev @ A @ Phi ) )
% 2.59/1.40 = ( linear505997466_on_ev @ A @ Phi ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_6_alw__alw,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ Phi ) )
% 2.59/1.40 = ( linear1386806755on_alw @ A @ Phi ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_7_not__ev,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( ^ [Xs: stream @ A] :
% 2.59/1.40 ~ ( linear505997466_on_ev @ A @ Phi @ Xs ) )
% 2.59/1.40 = ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ~ ( Phi @ Xs ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_8_not__alw,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( ^ [Xs: stream @ A] :
% 2.59/1.40 ~ ( linear1386806755on_alw @ A @ Phi @ Xs ) )
% 2.59/1.40 = ( linear505997466_on_ev @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ~ ( Phi @ Xs ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_9_not__ev__iff,axiom,
% 2.59/1.40 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.59/1.40 ( ( ~ ( linear505997466_on_ev @ A @ P @ Omega ) )
% 2.59/1.40 = ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ~ ( P @ Xs )
% 2.59/1.40 @ Omega ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_10_ev__alw__aand,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
% 2.59/1.40 => ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Psi ) @ Xs2 )
% 2.59/1.40 => ( linear505997466_on_ev @ A
% 2.59/1.40 @ ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs )
% 2.59/1.40 & ( Psi @ Xs ) ) )
% 2.59/1.40 @ Xs2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_11_ev__alw__impl,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 2.59/1.40 => ( ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs )
% 2.59/1.40 => ( Psi @ Xs ) )
% 2.59/1.40 @ Xs2 )
% 2.59/1.40 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_12_not__alw__iff,axiom,
% 2.59/1.40 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.59/1.40 ( ( ~ ( linear1386806755on_alw @ A @ P @ Omega ) )
% 2.59/1.40 = ( linear505997466_on_ev @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ~ ( P @ Xs )
% 2.59/1.40 @ Omega ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_13_enat_Oinject,axiom,
% 2.59/1.40 ! [Nat: nat,Nat2: nat] :
% 2.59/1.40 ( ( ( extended_enat2 @ Nat )
% 2.59/1.40 = ( extended_enat2 @ Nat2 ) )
% 2.59/1.40 = ( Nat = Nat2 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_14_not__infinity__eq,axiom,
% 2.59/1.40 ! [X: extended_enat] :
% 2.59/1.40 ( ( X
% 2.59/1.40 != ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 = ( ? [I: nat] :
% 2.59/1.40 ( X
% 2.59/1.40 = ( extended_enat2 @ I ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_15_not__enat__eq,axiom,
% 2.59/1.40 ! [X: extended_enat] :
% 2.59/1.40 ( ( ! [Y: nat] :
% 2.59/1.40 ( X
% 2.59/1.40 != ( extended_enat2 @ Y ) ) )
% 2.59/1.40 = ( X
% 2.59/1.40 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_16_enat__ex__split,axiom,
% 2.59/1.40 ( ( ^ [P2: extended_enat > $o] :
% 2.59/1.40 ? [X2: extended_enat] : ( P2 @ X2 ) )
% 2.59/1.40 = ( ^ [P3: extended_enat > $o] :
% 2.59/1.40 ( ( P3 @ ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 | ? [X3: nat] : ( P3 @ ( extended_enat2 @ X3 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_17_enat_Oinducts,axiom,
% 2.59/1.40 ! [P: extended_enat > $o,Enat: extended_enat] :
% 2.59/1.40 ( ! [Nat3: nat] : ( P @ ( extended_enat2 @ Nat3 ) )
% 2.59/1.40 => ( ( P @ ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 => ( P @ Enat ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_18_enat_Oexhaust,axiom,
% 2.59/1.40 ! [Y2: extended_enat] :
% 2.59/1.40 ( ! [Nat3: nat] :
% 2.59/1.40 ( Y2
% 2.59/1.40 != ( extended_enat2 @ Nat3 ) )
% 2.59/1.40 => ( Y2
% 2.59/1.40 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_19_enat3__cases,axiom,
% 2.59/1.40 ! [Y2: extended_enat,Ya: extended_enat,Yb: extended_enat] :
% 2.59/1.40 ( ( ? [Nat3: nat] :
% 2.59/1.40 ( Y2
% 2.59/1.40 = ( extended_enat2 @ Nat3 ) )
% 2.59/1.40 => ( ? [Nata: nat] :
% 2.59/1.40 ( Ya
% 2.59/1.40 = ( extended_enat2 @ Nata ) )
% 2.59/1.40 => ! [Natb: nat] :
% 2.59/1.40 ( Yb
% 2.59/1.40 != ( extended_enat2 @ Natb ) ) ) )
% 2.59/1.40 => ( ( ? [Nat3: nat] :
% 2.59/1.40 ( Y2
% 2.59/1.40 = ( extended_enat2 @ Nat3 ) )
% 2.59/1.40 => ( ? [Nata: nat] :
% 2.59/1.40 ( Ya
% 2.59/1.40 = ( extended_enat2 @ Nata ) )
% 2.59/1.40 => ( Yb
% 2.59/1.40 != ( extend1396239628finity @ extended_enat ) ) ) )
% 2.59/1.40 => ( ( ? [Nat3: nat] :
% 2.59/1.40 ( Y2
% 2.59/1.40 = ( extended_enat2 @ Nat3 ) )
% 2.59/1.40 => ( ( Ya
% 2.59/1.40 = ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 => ! [Nata: nat] :
% 2.59/1.40 ( Yb
% 2.59/1.40 != ( extended_enat2 @ Nata ) ) ) )
% 2.59/1.40 => ( ( ? [Nat3: nat] :
% 2.59/1.40 ( Y2
% 2.59/1.40 = ( extended_enat2 @ Nat3 ) )
% 2.59/1.40 => ( ( Ya
% 2.59/1.40 = ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 => ( Yb
% 2.59/1.40 != ( extend1396239628finity @ extended_enat ) ) ) )
% 2.59/1.40 => ( ( ( Y2
% 2.59/1.40 = ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 => ( ? [Nat3: nat] :
% 2.59/1.40 ( Ya
% 2.59/1.40 = ( extended_enat2 @ Nat3 ) )
% 2.59/1.40 => ! [Nata: nat] :
% 2.59/1.40 ( Yb
% 2.59/1.40 != ( extended_enat2 @ Nata ) ) ) )
% 2.59/1.40 => ( ( ( Y2
% 2.59/1.40 = ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 => ( ? [Nat3: nat] :
% 2.59/1.40 ( Ya
% 2.59/1.40 = ( extended_enat2 @ Nat3 ) )
% 2.59/1.40 => ( Yb
% 2.59/1.40 != ( extend1396239628finity @ extended_enat ) ) ) )
% 2.59/1.40 => ( ( ( Y2
% 2.59/1.40 = ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 => ( ( Ya
% 2.59/1.40 = ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 => ! [Nat3: nat] :
% 2.59/1.40 ( Yb
% 2.59/1.40 != ( extended_enat2 @ Nat3 ) ) ) )
% 2.59/1.40 => ~ ( ( Y2
% 2.59/1.40 = ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 => ( ( Ya
% 2.59/1.40 = ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 => ( Yb
% 2.59/1.40 != ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_20_enat2__cases,axiom,
% 2.59/1.40 ! [Y2: extended_enat,Ya: extended_enat] :
% 2.59/1.40 ( ( ? [Nat3: nat] :
% 2.59/1.40 ( Y2
% 2.59/1.40 = ( extended_enat2 @ Nat3 ) )
% 2.59/1.40 => ! [Nata: nat] :
% 2.59/1.40 ( Ya
% 2.59/1.40 != ( extended_enat2 @ Nata ) ) )
% 2.59/1.40 => ( ( ? [Nat3: nat] :
% 2.59/1.40 ( Y2
% 2.59/1.40 = ( extended_enat2 @ Nat3 ) )
% 2.59/1.40 => ( Ya
% 2.59/1.40 != ( extend1396239628finity @ extended_enat ) ) )
% 2.59/1.40 => ( ( ( Y2
% 2.59/1.40 = ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 => ! [Nat3: nat] :
% 2.59/1.40 ( Ya
% 2.59/1.40 != ( extended_enat2 @ Nat3 ) ) )
% 2.59/1.40 => ~ ( ( Y2
% 2.59/1.40 = ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 => ( Ya
% 2.59/1.40 != ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_21_enat_Odistinct_I1_J,axiom,
% 2.59/1.40 ! [Nat: nat] :
% 2.59/1.40 ( ( extended_enat2 @ Nat )
% 2.59/1.40 != ( extend1396239628finity @ extended_enat ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_22_enat_Odistinct_I2_J,axiom,
% 2.59/1.40 ! [Nat4: nat] :
% 2.59/1.40 ( ( extend1396239628finity @ extended_enat )
% 2.59/1.40 != ( extended_enat2 @ Nat4 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_23_enat__iless,axiom,
% 2.59/1.40 ! [N: extended_enat,M: nat] :
% 2.59/1.40 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 2.59/1.40 => ? [K: nat] :
% 2.59/1.40 ( N
% 2.59/1.40 = ( extended_enat2 @ K ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_24_chain__incr,axiom,
% 2.59/1.40 ! [A: $tType,Y3: A > extended_enat,K2: nat] :
% 2.59/1.40 ( ! [I2: A] :
% 2.59/1.40 ? [J: A] : ( ord_less @ extended_enat @ ( Y3 @ I2 ) @ ( Y3 @ J ) )
% 2.59/1.40 => ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K2 ) @ ( Y3 @ J2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_25_alw__iff__sdrop,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A )
% 2.59/1.40 = ( ^ [P3: ( stream @ A ) > $o,Omega2: stream @ A] :
% 2.59/1.40 ! [M2: nat] : ( P3 @ ( sdrop @ A @ M2 @ Omega2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_26_ev__iff__sdrop,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A )
% 2.59/1.40 = ( ^ [P3: ( stream @ A ) > $o,Omega2: stream @ A] :
% 2.59/1.40 ? [M2: nat] : ( P3 @ ( sdrop @ A @ M2 @ Omega2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_27_alw__sdrop,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,N: nat] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 2.59/1.40 => ( linear1386806755on_alw @ A @ Phi @ ( sdrop @ A @ N @ Xs2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_28_all__imp__alw,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 2.59/1.40 ( ! [X1: stream @ A] : ( Phi @ X1 )
% 2.59/1.40 => ( linear1386806755on_alw @ A @ Phi @ Xs2 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_29_alw__mono,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 2.59/1.40 => ( ! [Xs3: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs3 )
% 2.59/1.40 => ( Psi @ Xs3 ) )
% 2.59/1.40 => ( linear1386806755on_alw @ A @ Psi @ Xs2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_30_alw__cong,axiom,
% 2.59/1.40 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A,Q1: ( stream @ A ) > $o,Q2: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A @ P @ Omega )
% 2.59/1.40 => ( ! [Omega3: stream @ A] :
% 2.59/1.40 ( ( P @ Omega3 )
% 2.59/1.40 => ( ( Q1 @ Omega3 )
% 2.59/1.40 = ( Q2 @ Omega3 ) ) )
% 2.59/1.40 => ( ( linear1386806755on_alw @ A @ Q1 @ Omega )
% 2.59/1.40 = ( linear1386806755on_alw @ A @ Q2 @ Omega ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_31_alw__alwD,axiom,
% 2.59/1.40 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A @ P @ Omega )
% 2.59/1.40 => ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ P ) @ Omega ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_32_ev__mono,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 2.59/1.40 => ( ! [Xs3: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs3 )
% 2.59/1.40 => ( Psi @ Xs3 ) )
% 2.59/1.40 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_33_ev_Obase,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs2 )
% 2.59/1.40 => ( linear505997466_on_ev @ A @ Phi @ Xs2 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_34_alwD,axiom,
% 2.59/1.40 ! [A: $tType,P: ( stream @ A ) > $o,X: stream @ A] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A @ P @ X )
% 2.59/1.40 => ( P @ X ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_35_enat__less__induct,axiom,
% 2.59/1.40 ! [P: extended_enat > $o,N: extended_enat] :
% 2.59/1.40 ( ! [N2: extended_enat] :
% 2.59/1.40 ( ! [M3: extended_enat] :
% 2.59/1.40 ( ( ord_less @ extended_enat @ M3 @ N2 )
% 2.59/1.40 => ( P @ M3 ) )
% 2.59/1.40 => ( P @ N2 ) )
% 2.59/1.40 => ( P @ N ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_36_infinity__ilessE,axiom,
% 2.59/1.40 ! [M: nat] :
% 2.59/1.40 ~ ( ord_less @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_37_less__infinityE,axiom,
% 2.59/1.40 ! [N: extended_enat] :
% 2.59/1.40 ( ( ord_less @ extended_enat @ N @ ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 => ~ ! [K: nat] :
% 2.59/1.40 ( N
% 2.59/1.40 != ( extended_enat2 @ K ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_38_enat__ord__code_I4_J,axiom,
% 2.59/1.40 ! [M: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_39_alw__ev__sdrop,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,M: nat,Xs2: stream @ A] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ ( sdrop @ A @ M @ Xs2 ) )
% 2.59/1.40 => ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ Xs2 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_40_alw__False,axiom,
% 2.59/1.40 ! [A: $tType,Omega: stream @ A] :
% 2.59/1.40 ~ ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [X3: stream @ A] : $false
% 2.59/1.40 @ Omega ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_41_ev__False,axiom,
% 2.59/1.40 ! [A: $tType,Omega: stream @ A] :
% 2.59/1.40 ~ ( linear505997466_on_ev @ A
% 2.59/1.40 @ ^ [X3: stream @ A] : $false
% 2.59/1.40 @ Omega ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_42_alw__aand,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs )
% 2.59/1.40 & ( Psi @ Xs ) ) )
% 2.59/1.40 = ( ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A @ Phi @ Xs )
% 2.59/1.40 & ( linear1386806755on_alw @ A @ Psi @ Xs ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_43_alw__mp,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 2.59/1.40 => ( ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs )
% 2.59/1.40 => ( Psi @ Xs ) )
% 2.59/1.40 @ Xs2 )
% 2.59/1.40 => ( linear1386806755on_alw @ A @ Psi @ Xs2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_44_ev__or,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs )
% 2.59/1.40 | ( Psi @ Xs ) ) )
% 2.59/1.40 = ( ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A @ Phi @ Xs )
% 2.59/1.40 | ( linear505997466_on_ev @ A @ Psi @ Xs ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_45_mem__Collect__eq,axiom,
% 2.59/1.40 ! [A: $tType,A2: A,P: A > $o] :
% 2.59/1.40 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.59/1.40 = ( P @ A2 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_46_Collect__mem__eq,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A] :
% 2.59/1.40 ( ( collect @ A
% 2.59/1.40 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 2.59/1.40 = A3 ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_47_Collect__cong,axiom,
% 2.59/1.40 ! [A: $tType,P: A > $o,Q3: A > $o] :
% 2.59/1.40 ( ! [X4: A] :
% 2.59/1.40 ( ( P @ X4 )
% 2.59/1.40 = ( Q3 @ X4 ) )
% 2.59/1.40 => ( ( collect @ A @ P )
% 2.59/1.40 = ( collect @ A @ Q3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_48_ext,axiom,
% 2.59/1.40 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.59/1.40 ( ! [X4: A] :
% 2.59/1.40 ( ( F @ X4 )
% 2.59/1.40 = ( G @ X4 ) )
% 2.59/1.40 => ( F = G ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_49_scount__eq__card,axiom,
% 2.59/1.40 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A
% 2.59/1.40 @ ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ~ ( P @ Xs ) )
% 2.59/1.40 @ Omega )
% 2.59/1.40 => ( ( coindu1365464361scount @ A @ P @ Omega )
% 2.59/1.40 = ( extended_enat2
% 2.59/1.40 @ ( finite_card @ nat
% 2.59/1.40 @ ( collect @ nat
% 2.59/1.40 @ ^ [I: nat] : ( P @ ( sdrop @ A @ I @ Omega ) ) ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_50_ev__alw__imp__alw__ev,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
% 2.59/1.40 => ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ Xs2 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_51_ev__cong,axiom,
% 2.59/1.40 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A,Q1: ( stream @ A ) > $o,Q2: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A @ P @ Omega )
% 2.59/1.40 => ( ! [Omega3: stream @ A] :
% 2.59/1.40 ( ( P @ Omega3 )
% 2.59/1.40 => ( ( Q1 @ Omega3 )
% 2.59/1.40 = ( Q2 @ Omega3 ) ) )
% 2.59/1.40 => ( ( linear505997466_on_ev @ A @ Q1 @ Omega )
% 2.59/1.40 = ( linear505997466_on_ev @ A @ Q2 @ Omega ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_52_alw__ev__imp__ev__alw,axiom,
% 2.59/1.40 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Omega )
% 2.59/1.40 => ( linear505997466_on_ev @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( P @ Xs )
% 2.59/1.40 & ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Xs ) )
% 2.59/1.40 @ Omega ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_53_ev__alw__alw__impl,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
% 2.59/1.40 => ( ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A @ Phi @ Xs )
% 2.59/1.40 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 2.59/1.40 @ Xs2 )
% 2.59/1.40 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_54_alw__impl__ev__alw,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs )
% 2.59/1.40 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 2.59/1.40 @ Xs2 )
% 2.59/1.40 => ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A @ Phi @ Xs )
% 2.59/1.40 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 2.59/1.40 @ Xs2 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_55_alw__alw__impl__ev,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A @ Phi @ Xs )
% 2.59/1.40 => ( linear505997466_on_ev @ A @ Psi @ Xs ) ) )
% 2.59/1.40 = ( ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs )
% 2.59/1.40 => ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Psi ) @ Xs ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_56_ev__alw__impl__ev,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 2.59/1.40 => ( ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs )
% 2.59/1.40 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 2.59/1.40 @ Xs2 )
% 2.59/1.40 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_57_enat_Osimps_I7_J,axiom,
% 2.59/1.40 ! [T: $tType,F1: nat > T,F2: T] :
% 2.59/1.40 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 = F2 ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_58_enat_Osimps_I6_J,axiom,
% 2.59/1.40 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 2.59/1.40 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 2.59/1.40 = ( F1 @ Nat ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_59_sdrop__wait,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 2.59/1.40 => ( Phi @ ( sdrop @ A @ ( linear1335279038n_wait @ A @ Phi @ Xs2 ) @ Xs2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_60_enat_Osimps_I5_J,axiom,
% 2.59/1.40 ! [T: $tType,F1: nat > T,F2: T] :
% 2.59/1.40 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
% 2.59/1.40 = F2 ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_61_enat_Osimps_I4_J,axiom,
% 2.59/1.40 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 2.59/1.40 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 2.59/1.40 = ( F1 @ Nat ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_62_infinite__iff__alw__ev,axiom,
% 2.59/1.40 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.59/1.40 ( ( ~ ( finite_finite2 @ nat
% 2.59/1.40 @ ( collect @ nat
% 2.59/1.40 @ ^ [M2: nat] : ( P @ ( sdrop @ A @ M2 @ Omega ) ) ) ) )
% 2.59/1.40 = ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Omega ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_63_ev__alw__not__HLD__finite,axiom,
% 2.59/1.40 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A
% 2.59/1.40 @ ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ~ ( P @ Xs ) )
% 2.59/1.40 @ Omega )
% 2.59/1.40 => ( finite_finite2 @ nat
% 2.59/1.40 @ ( collect @ nat
% 2.59/1.40 @ ^ [I: nat] : ( P @ ( sdrop @ A @ I @ Omega ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_64_ev__alw__imp__nxt,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 2.59/1.40 => ( ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs )
% 2.59/1.40 => ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs ) )
% 2.59/1.40 @ Xs2 )
% 2.59/1.40 => ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_65_variance,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( Phi @ Xs2 )
% 2.59/1.40 => ( ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs )
% 2.59/1.40 => ( ( Psi @ Xs )
% 2.59/1.40 | ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs ) ) )
% 2.59/1.40 @ Xs2 )
% 2.59/1.40 => ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 2.59/1.40 | ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_66_ex__gt__or__lt,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A] :
% 2.59/1.40 ? [B2: A] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ B2 )
% 2.59/1.40 | ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_67_linorder__neqE__linordered__idom,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( X != Y2 )
% 2.59/1.40 => ( ~ ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_68_enat__ord__simps_I2_J,axiom,
% 2.59/1.40 ! [M: nat,N: nat] :
% 2.59/1.40 ( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 2.59/1.40 = ( ord_less @ nat @ M @ N ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_69_nxt__mono,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.59/1.40 ( ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs2 )
% 2.59/1.40 => ( ! [Xs3: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs3 )
% 2.59/1.40 => ( Psi @ Xs3 ) )
% 2.59/1.40 => ( linear1494993505on_nxt @ A @ $o @ Psi @ Xs2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_70_case__enat__def,axiom,
% 2.59/1.40 ! [T: $tType] :
% 2.59/1.40 ( ( extended_case_enat @ T )
% 2.59/1.40 = ( extended_rec_enat @ T ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_71_less__enat__def,axiom,
% 2.59/1.40 ( ( ord_less @ extended_enat )
% 2.59/1.40 = ( ^ [M2: extended_enat,N3: extended_enat] :
% 2.59/1.40 ( extended_case_enat @ $o
% 2.59/1.40 @ ^ [M1: nat] : ( extended_case_enat @ $o @ ( ord_less @ nat @ M1 ) @ $true @ N3 )
% 2.59/1.40 @ $false
% 2.59/1.40 @ M2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_72_ev__nxt,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A )
% 2.59/1.40 = ( ^ [Phi2: ( stream @ A ) > $o,Xs: stream @ A] :
% 2.59/1.40 ( ( Phi2 @ Xs )
% 2.59/1.40 | ( linear1494993505on_nxt @ A @ $o @ ( linear505997466_on_ev @ A @ Phi2 ) @ Xs ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_73_alw__nxt,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linear1386806755on_alw @ A )
% 2.59/1.40 = ( ^ [Phi2: ( stream @ A ) > $o,Xs: stream @ A] :
% 2.59/1.40 ( ( Phi2 @ Xs )
% 2.59/1.40 & ( linear1494993505on_nxt @ A @ $o @ ( linear1386806755on_alw @ A @ Phi2 ) @ Xs ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_74_alw__invar,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs2 )
% 2.59/1.40 => ( ( linear1386806755on_alw @ A
% 2.59/1.40 @ ^ [Xs: stream @ A] :
% 2.59/1.40 ( ( Phi @ Xs )
% 2.59/1.40 => ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs ) )
% 2.59/1.40 @ Xs2 )
% 2.59/1.40 => ( linear1386806755on_alw @ A @ Phi @ Xs2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_75_less__enatE,axiom,
% 2.59/1.40 ! [N: extended_enat,M: nat] :
% 2.59/1.40 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 2.59/1.40 => ~ ! [K: nat] :
% 2.59/1.40 ( ( N
% 2.59/1.40 = ( extended_enat2 @ K ) )
% 2.59/1.40 => ~ ( ord_less @ nat @ K @ M ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_76_card__Collect__less__nat,axiom,
% 2.59/1.40 ! [N: nat] :
% 2.59/1.40 ( ( finite_card @ nat
% 2.59/1.40 @ ( collect @ nat
% 2.59/1.40 @ ^ [I: nat] : ( ord_less @ nat @ I @ N ) ) )
% 2.59/1.40 = N ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_77_finite__Collect__less__nat,axiom,
% 2.59/1.40 ! [K2: nat] :
% 2.59/1.40 ( finite_finite2 @ nat
% 2.59/1.40 @ ( collect @ nat
% 2.59/1.40 @ ^ [N3: nat] : ( ord_less @ nat @ N3 @ K2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_78_finite__Collect__disjI,axiom,
% 2.59/1.40 ! [A: $tType,P: A > $o,Q3: A > $o] :
% 2.59/1.40 ( ( finite_finite2 @ A
% 2.59/1.40 @ ( collect @ A
% 2.59/1.40 @ ^ [X3: A] :
% 2.59/1.40 ( ( P @ X3 )
% 2.59/1.40 | ( Q3 @ X3 ) ) ) )
% 2.59/1.40 = ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.59/1.40 & ( finite_finite2 @ A @ ( collect @ A @ Q3 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_79_finite__Collect__conjI,axiom,
% 2.59/1.40 ! [A: $tType,P: A > $o,Q3: A > $o] :
% 2.59/1.40 ( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.59/1.40 | ( finite_finite2 @ A @ ( collect @ A @ Q3 ) ) )
% 2.59/1.40 => ( finite_finite2 @ A
% 2.59/1.40 @ ( collect @ A
% 2.59/1.40 @ ^ [X3: A] :
% 2.59/1.40 ( ( P @ X3 )
% 2.59/1.40 & ( Q3 @ X3 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_80_finite__code,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( finite_finite @ A @ ( type2 @ A ) )
% 2.59/1.40 => ( ( finite_finite2 @ A )
% 2.59/1.40 = ( ^ [A4: set @ A] : $true ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_81_psubset__card__mono,axiom,
% 2.59/1.40 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.59/1.40 ( ( finite_finite2 @ A @ B3 )
% 2.59/1.40 => ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_82_finite__M__bounded__by__nat,axiom,
% 2.59/1.40 ! [P: nat > $o,I3: nat] :
% 2.59/1.40 ( finite_finite2 @ nat
% 2.59/1.40 @ ( collect @ nat
% 2.59/1.40 @ ^ [K3: nat] :
% 2.59/1.40 ( ( P @ K3 )
% 2.59/1.40 & ( ord_less @ nat @ K3 @ I3 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_83_finite__set__choice,axiom,
% 2.59/1.40 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 2.59/1.40 ( ( finite_finite2 @ A @ A3 )
% 2.59/1.40 => ( ! [X4: A] :
% 2.59/1.40 ( ( member @ A @ X4 @ A3 )
% 2.59/1.40 => ? [X12: B] : ( P @ X4 @ X12 ) )
% 2.59/1.40 => ? [F3: A > B] :
% 2.59/1.40 ! [X5: A] :
% 2.59/1.40 ( ( member @ A @ X5 @ A3 )
% 2.59/1.40 => ( P @ X5 @ ( F3 @ X5 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_84_finite,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( finite_finite @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_85_finite__psubset__induct,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
% 2.59/1.40 ( ( finite_finite2 @ A @ A3 )
% 2.59/1.40 => ( ! [A5: set @ A] :
% 2.59/1.40 ( ( finite_finite2 @ A @ A5 )
% 2.59/1.40 => ( ! [B4: set @ A] :
% 2.59/1.40 ( ( ord_less @ ( set @ A ) @ B4 @ A5 )
% 2.59/1.40 => ( P @ B4 ) )
% 2.59/1.40 => ( P @ A5 ) ) )
% 2.59/1.40 => ( P @ A3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_86_bounded__nat__set__is__finite,axiom,
% 2.59/1.40 ! [N4: set @ nat,N: nat] :
% 2.59/1.40 ( ! [X4: nat] :
% 2.59/1.40 ( ( member @ nat @ X4 @ N4 )
% 2.59/1.40 => ( ord_less @ nat @ X4 @ N ) )
% 2.59/1.40 => ( finite_finite2 @ nat @ N4 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_87_finite__nat__set__iff__bounded,axiom,
% 2.59/1.40 ( ( finite_finite2 @ nat )
% 2.59/1.40 = ( ^ [N5: set @ nat] :
% 2.59/1.40 ? [M2: nat] :
% 2.59/1.40 ! [X3: nat] :
% 2.59/1.40 ( ( member @ nat @ X3 @ N5 )
% 2.59/1.40 => ( ord_less @ nat @ X3 @ M2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_88_not__finite__existsD,axiom,
% 2.59/1.40 ! [A: $tType,P: A > $o] :
% 2.59/1.40 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.59/1.40 => ? [X1: A] : ( P @ X1 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_89_pigeonhole__infinite__rel,axiom,
% 2.59/1.40 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R: A > B > $o] :
% 2.59/1.40 ( ~ ( finite_finite2 @ A @ A3 )
% 2.59/1.40 => ( ( finite_finite2 @ B @ B3 )
% 2.59/1.40 => ( ! [X4: A] :
% 2.59/1.40 ( ( member @ A @ X4 @ A3 )
% 2.59/1.40 => ? [Xa: B] :
% 2.59/1.40 ( ( member @ B @ Xa @ B3 )
% 2.59/1.40 & ( R @ X4 @ Xa ) ) )
% 2.59/1.40 => ? [X4: B] :
% 2.59/1.40 ( ( member @ B @ X4 @ B3 )
% 2.59/1.40 & ~ ( finite_finite2 @ A
% 2.59/1.40 @ ( collect @ A
% 2.59/1.40 @ ^ [A6: A] :
% 2.59/1.40 ( ( member @ A @ A6 @ A3 )
% 2.59/1.40 & ( R @ A6 @ X4 ) ) ) ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_90_unbounded__k__infinite,axiom,
% 2.59/1.40 ! [K2: nat,S2: set @ nat] :
% 2.59/1.40 ( ! [M4: nat] :
% 2.59/1.40 ( ( ord_less @ nat @ K2 @ M4 )
% 2.59/1.40 => ? [N6: nat] :
% 2.59/1.40 ( ( ord_less @ nat @ M4 @ N6 )
% 2.59/1.40 & ( member @ nat @ N6 @ S2 ) ) )
% 2.59/1.40 => ~ ( finite_finite2 @ nat @ S2 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_91_infinite__nat__iff__unbounded,axiom,
% 2.59/1.40 ! [S2: set @ nat] :
% 2.59/1.40 ( ( ~ ( finite_finite2 @ nat @ S2 ) )
% 2.59/1.40 = ( ! [M2: nat] :
% 2.59/1.40 ? [N3: nat] :
% 2.59/1.40 ( ( ord_less @ nat @ M2 @ N3 )
% 2.59/1.40 & ( member @ nat @ N3 @ S2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_92_less__set__def,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( ord_less @ ( set @ A ) )
% 2.59/1.40 = ( ^ [A4: set @ A,B5: set @ A] :
% 2.59/1.40 ( ord_less @ ( A > $o )
% 2.59/1.40 @ ^ [X3: A] : ( member @ A @ X3 @ A4 )
% 2.59/1.40 @ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_93_card__psubset,axiom,
% 2.59/1.40 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.59/1.40 ( ( finite_finite2 @ A @ B3 )
% 2.59/1.40 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) )
% 2.59/1.40 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_94_sdrop__wait__least,axiom,
% 2.59/1.40 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,N: nat] :
% 2.59/1.40 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 2.59/1.40 => ( ( Phi @ ( sdrop @ A @ N @ Xs2 ) )
% 2.59/1.40 => ( ord_less_eq @ nat @ ( linear1335279038n_wait @ A @ Phi @ Xs2 ) @ N ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_95_psubset__trans,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
% 2.59/1.40 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( ord_less @ ( set @ A ) @ B3 @ C )
% 2.59/1.40 => ( ord_less @ ( set @ A ) @ A3 @ C ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_96_subset__antisym,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 2.59/1.40 => ( A3 = B3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_97_subsetI,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.59/1.40 ( ! [X4: A] :
% 2.59/1.40 ( ( member @ A @ X4 @ A3 )
% 2.59/1.40 => ( member @ A @ X4 @ B3 ) )
% 2.59/1.40 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_98_psubsetI,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( A3 != B3 )
% 2.59/1.40 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_99_finite__Collect__subsets,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A] :
% 2.59/1.40 ( ( finite_finite2 @ A @ A3 )
% 2.59/1.40 => ( finite_finite2 @ ( set @ A )
% 2.59/1.40 @ ( collect @ ( set @ A )
% 2.59/1.40 @ ^ [B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_100_finite__Collect__le__nat,axiom,
% 2.59/1.40 ! [K2: nat] :
% 2.59/1.40 ( finite_finite2 @ nat
% 2.59/1.40 @ ( collect @ nat
% 2.59/1.40 @ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ K2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_101_infinite__nat__iff__unbounded__le,axiom,
% 2.59/1.40 ! [S2: set @ nat] :
% 2.59/1.40 ( ( ~ ( finite_finite2 @ nat @ S2 ) )
% 2.59/1.40 = ( ! [M2: nat] :
% 2.59/1.40 ? [N3: nat] :
% 2.59/1.40 ( ( ord_less_eq @ nat @ M2 @ N3 )
% 2.59/1.40 & ( member @ nat @ N3 @ S2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_102_subset__iff__psubset__eq,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) )
% 2.59/1.40 = ( ^ [A4: set @ A,B5: set @ A] :
% 2.59/1.40 ( ( ord_less @ ( set @ A ) @ A4 @ B5 )
% 2.59/1.40 | ( A4 = B5 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_103_subset__psubset__trans,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( ord_less @ ( set @ A ) @ B3 @ C )
% 2.59/1.40 => ( ord_less @ ( set @ A ) @ A3 @ C ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_104_subset__not__subset__eq,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( ord_less @ ( set @ A ) )
% 2.59/1.40 = ( ^ [A4: set @ A,B5: set @ A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 2.59/1.40 & ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_105_psubset__subset__trans,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
% 2.59/1.40 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C )
% 2.59/1.40 => ( ord_less @ ( set @ A ) @ A3 @ C ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_106_psubset__imp__subset,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.59/1.40 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_107_psubset__eq,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( ord_less @ ( set @ A ) )
% 2.59/1.40 = ( ^ [A4: set @ A,B5: set @ A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 2.59/1.40 & ( A4 != B5 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_108_psubsetE,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.59/1.40 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_109_Collect__mono__iff,axiom,
% 2.59/1.40 ! [A: $tType,P: A > $o,Q3: A > $o] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q3 ) )
% 2.59/1.40 = ( ! [X3: A] :
% 2.59/1.40 ( ( P @ X3 )
% 2.59/1.40 => ( Q3 @ X3 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_110_contra__subsetD,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ~ ( member @ A @ C2 @ B3 )
% 2.59/1.40 => ~ ( member @ A @ C2 @ A3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_111_set__eq__subset,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( ^ [Y4: set @ A,Z: set @ A] : ( Y4 = Z ) )
% 2.59/1.40 = ( ^ [A4: set @ A,B5: set @ A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 2.59/1.40 & ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_112_subset__trans,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C )
% 2.59/1.40 => ( ord_less_eq @ ( set @ A ) @ A3 @ C ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_113_Collect__mono,axiom,
% 2.59/1.40 ! [A: $tType,P: A > $o,Q3: A > $o] :
% 2.59/1.40 ( ! [X4: A] :
% 2.59/1.40 ( ( P @ X4 )
% 2.59/1.40 => ( Q3 @ X4 ) )
% 2.59/1.40 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_114_subset__refl,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_115_rev__subsetD,axiom,
% 2.59/1.40 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 2.59/1.40 ( ( member @ A @ C2 @ A3 )
% 2.59/1.40 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_116_subset__iff,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) )
% 2.59/1.40 = ( ^ [A4: set @ A,B5: set @ A] :
% 2.59/1.40 ! [T2: A] :
% 2.59/1.40 ( ( member @ A @ T2 @ A4 )
% 2.59/1.40 => ( member @ A @ T2 @ B5 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_117_set__rev__mp,axiom,
% 2.59/1.40 ! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
% 2.59/1.40 ( ( member @ A @ X @ A3 )
% 2.59/1.40 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( member @ A @ X @ B3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_118_equalityD2,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.59/1.40 ( ( A3 = B3 )
% 2.59/1.40 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_119_equalityD1,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.59/1.40 ( ( A3 = B3 )
% 2.59/1.40 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_120_subset__eq,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) )
% 2.59/1.40 = ( ^ [A4: set @ A,B5: set @ A] :
% 2.59/1.40 ! [X3: A] :
% 2.59/1.40 ( ( member @ A @ X3 @ A4 )
% 2.59/1.40 => ( member @ A @ X3 @ B5 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_121_equalityE,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.59/1.40 ( ( A3 = B3 )
% 2.59/1.40 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_122_subsetCE,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( member @ A @ C2 @ A3 )
% 2.59/1.40 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_123_subsetD,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( member @ A @ C2 @ A3 )
% 2.59/1.40 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_124_in__mono,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A,X: A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( member @ A @ X @ A3 )
% 2.59/1.40 => ( member @ A @ X @ B3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_125_set__mp,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A,X: A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( member @ A @ X @ A3 )
% 2.59/1.40 => ( member @ A @ X @ B3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_126_card__seteq,axiom,
% 2.59/1.40 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.59/1.40 ( ( finite_finite2 @ A @ B3 )
% 2.59/1.40 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B3 ) @ ( finite_card @ A @ A3 ) )
% 2.59/1.40 => ( A3 = B3 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_127_card__mono,axiom,
% 2.59/1.40 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.59/1.40 ( ( finite_finite2 @ A @ B3 )
% 2.59/1.40 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_128_complete__interval,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A,P: A > $o] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( P @ A2 )
% 2.59/1.40 => ( ~ ( P @ B6 )
% 2.59/1.40 => ? [C3: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ A2 @ C3 )
% 2.59/1.40 & ( ord_less_eq @ A @ C3 @ B6 )
% 2.59/1.40 & ! [X5: A] :
% 2.59/1.40 ( ( ( ord_less_eq @ A @ A2 @ X5 )
% 2.59/1.40 & ( ord_less @ A @ X5 @ C3 ) )
% 2.59/1.40 => ( P @ X5 ) )
% 2.59/1.40 & ! [D: A] :
% 2.59/1.40 ( ! [X4: A] :
% 2.59/1.40 ( ( ( ord_less_eq @ A @ A2 @ X4 )
% 2.59/1.40 & ( ord_less @ A @ X4 @ D ) )
% 2.59/1.40 => ( P @ X4 ) )
% 2.59/1.40 => ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_129_finite__subset,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( finite_finite2 @ A @ B3 )
% 2.59/1.40 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_130_infinite__super,axiom,
% 2.59/1.40 ! [A: $tType,S2: set @ A,T3: set @ A] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) @ S2 @ T3 )
% 2.59/1.40 => ( ~ ( finite_finite2 @ A @ S2 )
% 2.59/1.40 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_131_rev__finite__subset,axiom,
% 2.59/1.40 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.59/1.40 ( ( finite_finite2 @ A @ B3 )
% 2.59/1.40 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_132_finite__nat__set__iff__bounded__le,axiom,
% 2.59/1.40 ( ( finite_finite2 @ nat )
% 2.59/1.40 = ( ^ [N5: set @ nat] :
% 2.59/1.40 ? [M2: nat] :
% 2.59/1.40 ! [X3: nat] :
% 2.59/1.40 ( ( member @ nat @ X3 @ N5 )
% 2.59/1.40 => ( ord_less_eq @ nat @ X3 @ M2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_133_finite__less__ub,axiom,
% 2.59/1.40 ! [F: nat > nat,U: nat] :
% 2.59/1.40 ( ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( F @ N2 ) )
% 2.59/1.40 => ( finite_finite2 @ nat
% 2.59/1.40 @ ( collect @ nat
% 2.59/1.40 @ ^ [N3: nat] : ( ord_less_eq @ nat @ ( F @ N3 ) @ U ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_134_card__subset__eq,axiom,
% 2.59/1.40 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.59/1.40 ( ( finite_finite2 @ A @ B3 )
% 2.59/1.40 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( ( finite_card @ A @ A3 )
% 2.59/1.40 = ( finite_card @ A @ B3 ) )
% 2.59/1.40 => ( A3 = B3 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_135_infinite__arbitrarily__large,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,N: nat] :
% 2.59/1.40 ( ~ ( finite_finite2 @ A @ A3 )
% 2.59/1.40 => ? [B7: set @ A] :
% 2.59/1.40 ( ( finite_finite2 @ A @ B7 )
% 2.59/1.40 & ( ( finite_card @ A @ B7 )
% 2.59/1.40 = N )
% 2.59/1.40 & ( ord_less_eq @ ( set @ A ) @ B7 @ A3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_136_psubsetD,axiom,
% 2.59/1.40 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 2.59/1.40 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.59/1.40 => ( ( member @ A @ C2 @ A3 )
% 2.59/1.40 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_137_order__refl,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_138_ex__has__greatest__nat,axiom,
% 2.59/1.40 ! [A: $tType,P: A > $o,K2: A,M: A > nat,B6: nat] :
% 2.59/1.40 ( ( P @ K2 )
% 2.59/1.40 => ( ! [Y5: A] :
% 2.59/1.40 ( ( P @ Y5 )
% 2.59/1.40 => ( ord_less @ nat @ ( M @ Y5 ) @ B6 ) )
% 2.59/1.40 => ? [X4: A] :
% 2.59/1.40 ( ( P @ X4 )
% 2.59/1.40 & ! [Y6: A] :
% 2.59/1.40 ( ( P @ Y6 )
% 2.59/1.40 => ( ord_less_eq @ nat @ ( M @ Y6 ) @ ( M @ X4 ) ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_139_less__mono__imp__le__mono,axiom,
% 2.59/1.40 ! [F: nat > nat,I3: nat,J3: nat] :
% 2.59/1.40 ( ! [I2: nat,J2: nat] :
% 2.59/1.40 ( ( ord_less @ nat @ I2 @ J2 )
% 2.59/1.40 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.59/1.40 => ( ( ord_less_eq @ nat @ I3 @ J3 )
% 2.59/1.40 => ( ord_less_eq @ nat @ ( F @ I3 ) @ ( F @ J3 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_140_le__neq__implies__less,axiom,
% 2.59/1.40 ! [M: nat,N: nat] :
% 2.59/1.40 ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.40 => ( ( M != N )
% 2.59/1.40 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_141_less__or__eq__imp__le,axiom,
% 2.59/1.40 ! [M: nat,N: nat] :
% 2.59/1.40 ( ( ( ord_less @ nat @ M @ N )
% 2.59/1.40 | ( M = N ) )
% 2.59/1.40 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_142_enat__ord__simps_I5_J,axiom,
% 2.59/1.40 ! [Q: extended_enat] :
% 2.59/1.40 ( ( ord_less_eq @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ Q )
% 2.59/1.40 = ( Q
% 2.59/1.40 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_143_enat__ord__code_I3_J,axiom,
% 2.59/1.40 ! [Q: extended_enat] : ( ord_less_eq @ extended_enat @ Q @ ( extend1396239628finity @ extended_enat ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_144_enat__ord__simps_I1_J,axiom,
% 2.59/1.40 ! [M: nat,N: nat] :
% 2.59/1.40 ( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 2.59/1.40 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_145_less__fun__def,axiom,
% 2.59/1.40 ! [B: $tType,A: $tType] :
% 2.59/1.40 ( ( ord @ B @ ( type2 @ B ) )
% 2.59/1.40 => ( ( ord_less @ ( A > B ) )
% 2.59/1.40 = ( ^ [F4: A > B,G2: A > B] :
% 2.59/1.40 ( ( ord_less_eq @ ( A > B ) @ F4 @ G2 )
% 2.59/1.40 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F4 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_146_finite__enat__bounded,axiom,
% 2.59/1.40 ! [A3: set @ extended_enat,N: nat] :
% 2.59/1.40 ( ! [Y5: extended_enat] :
% 2.59/1.40 ( ( member @ extended_enat @ Y5 @ A3 )
% 2.59/1.40 => ( ord_less_eq @ extended_enat @ Y5 @ ( extended_enat2 @ N ) ) )
% 2.59/1.40 => ( finite_finite2 @ extended_enat @ A3 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_147_enat__ile,axiom,
% 2.59/1.40 ! [N: extended_enat,M: nat] :
% 2.59/1.40 ( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 2.59/1.40 => ? [K: nat] :
% 2.59/1.40 ( N
% 2.59/1.40 = ( extended_enat2 @ K ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_148_enat__ord__simps_I3_J,axiom,
% 2.59/1.40 ! [Q: extended_enat] : ( ord_less_eq @ extended_enat @ Q @ ( extend1396239628finity @ extended_enat ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_149_less__eq__set__def,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( ord_less_eq @ ( set @ A ) )
% 2.59/1.40 = ( ^ [A4: set @ A,B5: set @ A] :
% 2.59/1.40 ( ord_less_eq @ ( A > $o )
% 2.59/1.40 @ ^ [X3: A] : ( member @ A @ X3 @ A4 )
% 2.59/1.40 @ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_150_infinity__ileE,axiom,
% 2.59/1.40 ! [M: nat] :
% 2.59/1.40 ~ ( ord_less_eq @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_151_enat__ord__code_I5_J,axiom,
% 2.59/1.40 ! [N: nat] :
% 2.59/1.40 ~ ( ord_less_eq @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ N ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_152_le__funD,axiom,
% 2.59/1.40 ! [B: $tType,A: $tType] :
% 2.59/1.40 ( ( ord @ B @ ( type2 @ B ) )
% 2.59/1.40 => ! [F: A > B,G: A > B,X: A] :
% 2.59/1.40 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.59/1.40 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_153_le__funE,axiom,
% 2.59/1.40 ! [B: $tType,A: $tType] :
% 2.59/1.40 ( ( ord @ B @ ( type2 @ B ) )
% 2.59/1.40 => ! [F: A > B,G: A > B,X: A] :
% 2.59/1.40 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.59/1.40 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_154_le__funI,axiom,
% 2.59/1.40 ! [B: $tType,A: $tType] :
% 2.59/1.40 ( ( ord @ B @ ( type2 @ B ) )
% 2.59/1.40 => ! [F: A > B,G: A > B] :
% 2.59/1.40 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
% 2.59/1.40 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_155_le__fun__def,axiom,
% 2.59/1.40 ! [B: $tType,A: $tType] :
% 2.59/1.40 ( ( ord @ B @ ( type2 @ B ) )
% 2.59/1.40 => ( ( ord_less_eq @ ( A > B ) )
% 2.59/1.40 = ( ^ [F4: A > B,G2: A > B] :
% 2.59/1.40 ! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_156_order__subst1,axiom,
% 2.59/1.40 ! [A: $tType,B: $tType] :
% 2.59/1.40 ( ( ( order @ B @ ( type2 @ B ) )
% 2.59/1.40 & ( order @ A @ ( type2 @ A ) ) )
% 2.59/1.40 => ! [A2: A,F: B > A,B6: B,C2: B] :
% 2.59/1.40 ( ( ord_less_eq @ A @ A2 @ ( F @ B6 ) )
% 2.59/1.40 => ( ( ord_less_eq @ B @ B6 @ C2 )
% 2.59/1.40 => ( ! [X4: B,Y5: B] :
% 2.59/1.40 ( ( ord_less_eq @ B @ X4 @ Y5 )
% 2.59/1.40 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 2.59/1.40 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_157_order__subst2,axiom,
% 2.59/1.40 ! [A: $tType,C4: $tType] :
% 2.59/1.40 ( ( ( order @ C4 @ ( type2 @ C4 ) )
% 2.59/1.40 & ( order @ A @ ( type2 @ A ) ) )
% 2.59/1.40 => ! [A2: A,B6: A,F: A > C4,C2: C4] :
% 2.59/1.40 ( ( ord_less_eq @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( ord_less_eq @ C4 @ ( F @ B6 ) @ C2 )
% 2.59/1.40 => ( ! [X4: A,Y5: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 2.59/1.40 => ( ord_less_eq @ C4 @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 2.59/1.40 => ( ord_less_eq @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_158_ord__eq__le__subst,axiom,
% 2.59/1.40 ! [A: $tType,B: $tType] :
% 2.59/1.40 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.59/1.40 & ( ord @ A @ ( type2 @ A ) ) )
% 2.59/1.40 => ! [A2: A,F: B > A,B6: B,C2: B] :
% 2.59/1.40 ( ( A2
% 2.59/1.40 = ( F @ B6 ) )
% 2.59/1.40 => ( ( ord_less_eq @ B @ B6 @ C2 )
% 2.59/1.40 => ( ! [X4: B,Y5: B] :
% 2.59/1.40 ( ( ord_less_eq @ B @ X4 @ Y5 )
% 2.59/1.40 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 2.59/1.40 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_159_ord__le__eq__subst,axiom,
% 2.59/1.40 ! [A: $tType,B: $tType] :
% 2.59/1.40 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.59/1.40 & ( ord @ A @ ( type2 @ A ) ) )
% 2.59/1.40 => ! [A2: A,B6: A,F: A > B,C2: B] :
% 2.59/1.40 ( ( ord_less_eq @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( ( F @ B6 )
% 2.59/1.40 = C2 )
% 2.59/1.40 => ( ! [X4: A,Y5: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 2.59/1.40 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 2.59/1.40 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_160_eq__iff,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 2.59/1.40 = ( ^ [X3: A,Y: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.59/1.40 & ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_161_antisym,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.59/1.40 => ( ( ord_less_eq @ A @ Y2 @ X )
% 2.59/1.40 => ( X = Y2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_162_linear,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.59/1.40 | ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_163_eq__refl,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( X = Y2 )
% 2.59/1.40 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_164_le__cases,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ~ ( ord_less_eq @ A @ X @ Y2 )
% 2.59/1.40 => ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_165_order_Otrans,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A,C2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( ord_less_eq @ A @ B6 @ C2 )
% 2.59/1.40 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_166_le__cases3,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A,Z2: A] :
% 2.59/1.40 ( ( ( ord_less_eq @ A @ X @ Y2 )
% 2.59/1.40 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 2.59/1.40 => ( ( ( ord_less_eq @ A @ Y2 @ X )
% 2.59/1.40 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 2.59/1.40 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 2.59/1.40 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 2.59/1.40 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 2.59/1.40 => ~ ( ord_less_eq @ A @ Y2 @ X ) )
% 2.59/1.40 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.59/1.40 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 2.59/1.40 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 2.59/1.40 => ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_167_antisym__conv,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [Y2: A,X: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ Y2 @ X )
% 2.59/1.40 => ( ( ord_less_eq @ A @ X @ Y2 )
% 2.59/1.40 = ( X = Y2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_168_ord__eq__le__trans,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( ord @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A,C2: A] :
% 2.59/1.40 ( ( A2 = B6 )
% 2.59/1.40 => ( ( ord_less_eq @ A @ B6 @ C2 )
% 2.59/1.40 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_169_ord__le__eq__trans,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( ord @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A,C2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( B6 = C2 )
% 2.59/1.40 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_170_order__class_Oorder_Oantisym,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( ord_less_eq @ A @ B6 @ A2 )
% 2.59/1.40 => ( A2 = B6 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_171_order__trans,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A,Z2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.59/1.40 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.59/1.40 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_172_dual__order_Orefl,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_173_linorder__wlog,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [P: A > A > $o,A2: A,B6: A] :
% 2.59/1.40 ( ! [A7: A,B2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ A7 @ B2 )
% 2.59/1.40 => ( P @ A7 @ B2 ) )
% 2.59/1.40 => ( ! [A7: A,B2: A] :
% 2.59/1.40 ( ( P @ B2 @ A7 )
% 2.59/1.40 => ( P @ A7 @ B2 ) )
% 2.59/1.40 => ( P @ A2 @ B6 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_174_dual__order_Otrans,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [B6: A,A2: A,C2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ B6 @ A2 )
% 2.59/1.40 => ( ( ord_less_eq @ A @ C2 @ B6 )
% 2.59/1.40 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_175_dual__order_Oantisym,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [B6: A,A2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ B6 @ A2 )
% 2.59/1.40 => ( ( ord_less_eq @ A @ A2 @ B6 )
% 2.59/1.40 => ( A2 = B6 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_176_dual__order_Ostrict__implies__not__eq,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [B6: A,A2: A] :
% 2.59/1.40 ( ( ord_less @ A @ B6 @ A2 )
% 2.59/1.40 => ( A2 != B6 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_177_order_Ostrict__implies__not__eq,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ B6 )
% 2.59/1.40 => ( A2 != B6 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_178_not__less__iff__gr__or__eq,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 2.59/1.40 = ( ( ord_less @ A @ Y2 @ X )
% 2.59/1.40 | ( X = Y2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_179_dual__order_Ostrict__trans,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [B6: A,A2: A,C2: A] :
% 2.59/1.40 ( ( ord_less @ A @ B6 @ A2 )
% 2.59/1.40 => ( ( ord_less @ A @ C2 @ B6 )
% 2.59/1.40 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_180_less__imp__not__less,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_181_order_Ostrict__trans,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A,C2: A] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( ord_less @ A @ B6 @ C2 )
% 2.59/1.40 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_182_dual__order_Oirrefl,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A] :
% 2.59/1.40 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_183_linorder__cases,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ~ ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ( ( X != Y2 )
% 2.59/1.40 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_184_less__imp__triv,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A,P: $o] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ( ( ord_less @ A @ Y2 @ X )
% 2.59/1.40 => P ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_185_less__imp__not__eq2,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ( Y2 != X ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_186_antisym__conv3,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [Y2: A,X: A] :
% 2.59/1.40 ( ~ ( ord_less @ A @ Y2 @ X )
% 2.59/1.40 => ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 2.59/1.40 = ( X = Y2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_187_less__induct,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [P: A > $o,A2: A] :
% 2.59/1.40 ( ! [X4: A] :
% 2.59/1.40 ( ! [Y6: A] :
% 2.59/1.40 ( ( ord_less @ A @ Y6 @ X4 )
% 2.59/1.40 => ( P @ Y6 ) )
% 2.59/1.40 => ( P @ X4 ) )
% 2.59/1.40 => ( P @ A2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_188_less__not__sym,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_189_less__imp__not__eq,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ( X != Y2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_190_dual__order_Oasym,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [B6: A,A2: A] :
% 2.59/1.40 ( ( ord_less @ A @ B6 @ A2 )
% 2.59/1.40 => ~ ( ord_less @ A @ A2 @ B6 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_191_ord__less__eq__trans,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( ord @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A,C2: A] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( B6 = C2 )
% 2.59/1.40 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_192_ord__eq__less__trans,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( ord @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A,C2: A] :
% 2.59/1.40 ( ( A2 = B6 )
% 2.59/1.40 => ( ( ord_less @ A @ B6 @ C2 )
% 2.59/1.40 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_193_less__irrefl,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A] :
% 2.59/1.40 ~ ( ord_less @ A @ X @ X ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_194_less__linear,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 | ( X = Y2 )
% 2.59/1.40 | ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_195_less__trans,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A,Z2: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.59/1.40 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_196_less__asym_H,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ B6 )
% 2.59/1.40 => ~ ( ord_less @ A @ B6 @ A2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_197_less__asym,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_198_less__imp__neq,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ( X != Y2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_199_dense,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( dense_order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ? [Z3: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Z3 )
% 2.59/1.40 & ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_200_order_Oasym,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ B6 )
% 2.59/1.40 => ~ ( ord_less @ A @ B6 @ A2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_201_neq__iff,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( X != Y2 )
% 2.59/1.40 = ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 | ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_202_neqE,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( X != Y2 )
% 2.59/1.40 => ( ~ ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_203_gt__ex,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( no_top @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A] :
% 2.59/1.40 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_204_lt__ex,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A] :
% 2.59/1.40 ? [Y5: A] : ( ord_less @ A @ Y5 @ X ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_205_order__less__subst2,axiom,
% 2.59/1.40 ! [A: $tType,C4: $tType] :
% 2.59/1.40 ( ( ( order @ C4 @ ( type2 @ C4 ) )
% 2.59/1.40 & ( order @ A @ ( type2 @ A ) ) )
% 2.59/1.40 => ! [A2: A,B6: A,F: A > C4,C2: C4] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( ord_less @ C4 @ ( F @ B6 ) @ C2 )
% 2.59/1.40 => ( ! [X4: A,Y5: A] :
% 2.59/1.40 ( ( ord_less @ A @ X4 @ Y5 )
% 2.59/1.40 => ( ord_less @ C4 @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 2.59/1.40 => ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_206_order__less__subst1,axiom,
% 2.59/1.40 ! [A: $tType,B: $tType] :
% 2.59/1.40 ( ( ( order @ B @ ( type2 @ B ) )
% 2.59/1.40 & ( order @ A @ ( type2 @ A ) ) )
% 2.59/1.40 => ! [A2: A,F: B > A,B6: B,C2: B] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ ( F @ B6 ) )
% 2.59/1.40 => ( ( ord_less @ B @ B6 @ C2 )
% 2.59/1.40 => ( ! [X4: B,Y5: B] :
% 2.59/1.40 ( ( ord_less @ B @ X4 @ Y5 )
% 2.59/1.40 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 2.59/1.40 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_207_ord__less__eq__subst,axiom,
% 2.59/1.40 ! [A: $tType,B: $tType] :
% 2.59/1.40 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.59/1.40 & ( ord @ A @ ( type2 @ A ) ) )
% 2.59/1.40 => ! [A2: A,B6: A,F: A > B,C2: B] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( ( F @ B6 )
% 2.59/1.40 = C2 )
% 2.59/1.40 => ( ! [X4: A,Y5: A] :
% 2.59/1.40 ( ( ord_less @ A @ X4 @ Y5 )
% 2.59/1.40 => ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 2.59/1.40 => ( ord_less @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_208_ord__eq__less__subst,axiom,
% 2.59/1.40 ! [A: $tType,B: $tType] :
% 2.59/1.40 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.59/1.40 & ( ord @ A @ ( type2 @ A ) ) )
% 2.59/1.40 => ! [A2: A,F: B > A,B6: B,C2: B] :
% 2.59/1.40 ( ( A2
% 2.59/1.40 = ( F @ B6 ) )
% 2.59/1.40 => ( ( ord_less @ B @ B6 @ C2 )
% 2.59/1.40 => ( ! [X4: B,Y5: B] :
% 2.59/1.40 ( ( ord_less @ B @ X4 @ Y5 )
% 2.59/1.40 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 2.59/1.40 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_209_less__eq__enat__def,axiom,
% 2.59/1.40 ( ( ord_less_eq @ extended_enat )
% 2.59/1.40 = ( ^ [M2: extended_enat] :
% 2.59/1.40 ( extended_case_enat @ $o
% 2.59/1.40 @ ^ [N1: nat] :
% 2.59/1.40 ( extended_case_enat @ $o
% 2.59/1.40 @ ^ [M1: nat] : ( ord_less_eq @ nat @ M1 @ N1 )
% 2.59/1.40 @ $false
% 2.59/1.40 @ M2 )
% 2.59/1.40 @ $true ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_210_infinite__descent__measure,axiom,
% 2.59/1.40 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 2.59/1.40 ( ! [X4: A] :
% 2.59/1.40 ( ~ ( P @ X4 )
% 2.59/1.40 => ? [Y6: A] :
% 2.59/1.40 ( ( ord_less @ nat @ ( V @ Y6 ) @ ( V @ X4 ) )
% 2.59/1.40 & ~ ( P @ Y6 ) ) )
% 2.59/1.40 => ( P @ X ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_211_measure__induct__rule,axiom,
% 2.59/1.40 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 2.59/1.40 ( ! [X4: A] :
% 2.59/1.40 ( ! [Y6: A] :
% 2.59/1.40 ( ( ord_less @ nat @ ( F @ Y6 ) @ ( F @ X4 ) )
% 2.59/1.40 => ( P @ Y6 ) )
% 2.59/1.40 => ( P @ X4 ) )
% 2.59/1.40 => ( P @ A2 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_212_linorder__neqE__nat,axiom,
% 2.59/1.40 ! [X: nat,Y2: nat] :
% 2.59/1.40 ( ( X != Y2 )
% 2.59/1.40 => ( ~ ( ord_less @ nat @ X @ Y2 )
% 2.59/1.40 => ( ord_less @ nat @ Y2 @ X ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_213_infinite__descent,axiom,
% 2.59/1.40 ! [P: nat > $o,N: nat] :
% 2.59/1.40 ( ! [N2: nat] :
% 2.59/1.40 ( ~ ( P @ N2 )
% 2.59/1.40 => ? [M3: nat] :
% 2.59/1.40 ( ( ord_less @ nat @ M3 @ N2 )
% 2.59/1.40 & ~ ( P @ M3 ) ) )
% 2.59/1.40 => ( P @ N ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_214_nat__less__induct,axiom,
% 2.59/1.40 ! [P: nat > $o,N: nat] :
% 2.59/1.40 ( ! [N2: nat] :
% 2.59/1.40 ( ! [M3: nat] :
% 2.59/1.40 ( ( ord_less @ nat @ M3 @ N2 )
% 2.59/1.40 => ( P @ M3 ) )
% 2.59/1.40 => ( P @ N2 ) )
% 2.59/1.40 => ( P @ N ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_215_less__irrefl__nat,axiom,
% 2.59/1.40 ! [N: nat] :
% 2.59/1.40 ~ ( ord_less @ nat @ N @ N ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_216_measure__induct,axiom,
% 2.59/1.40 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 2.59/1.40 ( ! [X4: A] :
% 2.59/1.40 ( ! [Y6: A] :
% 2.59/1.40 ( ( ord_less @ nat @ ( F @ Y6 ) @ ( F @ X4 ) )
% 2.59/1.40 => ( P @ Y6 ) )
% 2.59/1.40 => ( P @ X4 ) )
% 2.59/1.40 => ( P @ A2 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_217_less__not__refl3,axiom,
% 2.59/1.40 ! [S3: nat,T4: nat] :
% 2.59/1.40 ( ( ord_less @ nat @ S3 @ T4 )
% 2.59/1.40 => ( S3 != T4 ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_218_less__not__refl2,axiom,
% 2.59/1.40 ! [N: nat,M: nat] :
% 2.59/1.40 ( ( ord_less @ nat @ N @ M )
% 2.59/1.40 => ( M != N ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_219_less__not__refl,axiom,
% 2.59/1.40 ! [N: nat] :
% 2.59/1.40 ~ ( ord_less @ nat @ N @ N ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_220_nat__neq__iff,axiom,
% 2.59/1.40 ! [M: nat,N: nat] :
% 2.59/1.40 ( ( M != N )
% 2.59/1.40 = ( ( ord_less @ nat @ M @ N )
% 2.59/1.40 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_221_le__refl,axiom,
% 2.59/1.40 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_222_le__trans,axiom,
% 2.59/1.40 ! [I3: nat,J3: nat,K2: nat] :
% 2.59/1.40 ( ( ord_less_eq @ nat @ I3 @ J3 )
% 2.59/1.40 => ( ( ord_less_eq @ nat @ J3 @ K2 )
% 2.59/1.40 => ( ord_less_eq @ nat @ I3 @ K2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_223_eq__imp__le,axiom,
% 2.59/1.40 ! [M: nat,N: nat] :
% 2.59/1.40 ( ( M = N )
% 2.59/1.40 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_224_le__antisym,axiom,
% 2.59/1.40 ! [M: nat,N: nat] :
% 2.59/1.40 ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.40 => ( ( ord_less_eq @ nat @ N @ M )
% 2.59/1.40 => ( M = N ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_225_nat__le__linear,axiom,
% 2.59/1.40 ! [M: nat,N: nat] :
% 2.59/1.40 ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.40 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_226_ex__has__least__nat,axiom,
% 2.59/1.40 ! [A: $tType,P: A > $o,K2: A,M: A > nat] :
% 2.59/1.40 ( ( P @ K2 )
% 2.59/1.40 => ? [X4: A] :
% 2.59/1.40 ( ( P @ X4 )
% 2.59/1.40 & ! [Y6: A] :
% 2.59/1.40 ( ( P @ Y6 )
% 2.59/1.40 => ( ord_less_eq @ nat @ ( M @ X4 ) @ ( M @ Y6 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_227_order_Onot__eq__order__implies__strict,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A] :
% 2.59/1.40 ( ( A2 != B6 )
% 2.59/1.40 => ( ( ord_less_eq @ A @ A2 @ B6 )
% 2.59/1.40 => ( ord_less @ A @ A2 @ B6 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_228_dual__order_Ostrict__implies__order,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [B6: A,A2: A] :
% 2.59/1.40 ( ( ord_less @ A @ B6 @ A2 )
% 2.59/1.40 => ( ord_less_eq @ A @ B6 @ A2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_229_dual__order_Ostrict__iff__order,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ( ( ord_less @ A )
% 2.59/1.40 = ( ^ [B8: A,A6: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ B8 @ A6 )
% 2.59/1.40 & ( A6 != B8 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_230_dual__order_Oorder__iff__strict,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ( ( ord_less_eq @ A )
% 2.59/1.40 = ( ^ [B8: A,A6: A] :
% 2.59/1.40 ( ( ord_less @ A @ B8 @ A6 )
% 2.59/1.40 | ( A6 = B8 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_231_order_Ostrict__implies__order,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ B6 )
% 2.59/1.40 => ( ord_less_eq @ A @ A2 @ B6 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_232_dense__le__bounded,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A,Z2: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ( ! [W: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ W )
% 2.59/1.40 => ( ( ord_less @ A @ W @ Y2 )
% 2.59/1.40 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 2.59/1.40 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_233_dense__ge__bounded,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [Z2: A,X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less @ A @ Z2 @ X )
% 2.59/1.40 => ( ! [W: A] :
% 2.59/1.40 ( ( ord_less @ A @ Z2 @ W )
% 2.59/1.40 => ( ( ord_less @ A @ W @ X )
% 2.59/1.40 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 2.59/1.40 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_234_dual__order_Ostrict__trans2,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [B6: A,A2: A,C2: A] :
% 2.59/1.40 ( ( ord_less @ A @ B6 @ A2 )
% 2.59/1.40 => ( ( ord_less_eq @ A @ C2 @ B6 )
% 2.59/1.40 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_235_dual__order_Ostrict__trans1,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [B6: A,A2: A,C2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ B6 @ A2 )
% 2.59/1.40 => ( ( ord_less @ A @ C2 @ B6 )
% 2.59/1.40 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_236_order_Ostrict__iff__order,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ( ( ord_less @ A )
% 2.59/1.40 = ( ^ [A6: A,B8: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ A6 @ B8 )
% 2.59/1.40 & ( A6 != B8 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_237_order_Oorder__iff__strict,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ( ( ord_less_eq @ A )
% 2.59/1.40 = ( ^ [A6: A,B8: A] :
% 2.59/1.40 ( ( ord_less @ A @ A6 @ B8 )
% 2.59/1.40 | ( A6 = B8 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_238_order_Ostrict__trans2,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A,C2: A] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( ord_less_eq @ A @ B6 @ C2 )
% 2.59/1.40 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_239_order_Ostrict__trans1,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A,C2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( ord_less @ A @ B6 @ C2 )
% 2.59/1.40 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_240_not__le__imp__less,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [Y2: A,X: A] :
% 2.59/1.40 ( ~ ( ord_less_eq @ A @ Y2 @ X )
% 2.59/1.40 => ( ord_less @ A @ X @ Y2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_241_less__le__not__le,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ( ( ord_less @ A )
% 2.59/1.40 = ( ^ [X3: A,Y: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.59/1.40 & ~ ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_242_le__imp__less__or__eq,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.59/1.40 => ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 | ( X = Y2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_243_le__less__linear,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.59/1.40 | ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_244_dense__le,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [Y2: A,Z2: A] :
% 2.59/1.40 ( ! [X4: A] :
% 2.59/1.40 ( ( ord_less @ A @ X4 @ Y2 )
% 2.59/1.40 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 2.59/1.40 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_245_dense__ge,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [Z2: A,Y2: A] :
% 2.59/1.40 ( ! [X4: A] :
% 2.59/1.40 ( ( ord_less @ A @ Z2 @ X4 )
% 2.59/1.40 => ( ord_less_eq @ A @ Y2 @ X4 ) )
% 2.59/1.40 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_246_less__le__trans,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A,Z2: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.59/1.40 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_247_le__less__trans,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A,Z2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.59/1.40 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.59/1.40 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_248_antisym__conv2,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.59/1.40 => ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 2.59/1.40 = ( X = Y2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_249_antisym__conv1,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ~ ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ( ( ord_less_eq @ A @ X @ Y2 )
% 2.59/1.40 = ( X = Y2 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_250_less__imp__le,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( preorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ord_less @ A @ X @ Y2 )
% 2.59/1.40 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_251_le__neq__trans,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( order @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [A2: A,B6: A] :
% 2.59/1.40 ( ( ord_less_eq @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( A2 != B6 )
% 2.59/1.40 => ( ord_less @ A @ A2 @ B6 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_252_not__less,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 2.59/1.40 = ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_253_not__le,axiom,
% 2.59/1.40 ! [A: $tType] :
% 2.59/1.40 ( ( linorder @ A @ ( type2 @ A ) )
% 2.59/1.40 => ! [X: A,Y2: A] :
% 2.59/1.40 ( ( ~ ( ord_less_eq @ A @ X @ Y2 ) )
% 2.59/1.40 = ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_254_order__less__le__subst2,axiom,
% 2.59/1.40 ! [A: $tType,C4: $tType] :
% 2.59/1.40 ( ( ( order @ C4 @ ( type2 @ C4 ) )
% 2.59/1.40 & ( order @ A @ ( type2 @ A ) ) )
% 2.59/1.40 => ! [A2: A,B6: A,F: A > C4,C2: C4] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ B6 )
% 2.59/1.40 => ( ( ord_less_eq @ C4 @ ( F @ B6 ) @ C2 )
% 2.59/1.40 => ( ! [X4: A,Y5: A] :
% 2.59/1.40 ( ( ord_less @ A @ X4 @ Y5 )
% 2.59/1.40 => ( ord_less @ C4 @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 2.59/1.40 => ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(fact_255_order__less__le__subst1,axiom,
% 2.59/1.40 ! [A: $tType,B: $tType] :
% 2.59/1.40 ( ( ( order @ B @ ( type2 @ B ) )
% 2.59/1.40 & ( order @ A @ ( type2 @ A ) ) )
% 2.59/1.40 => ! [A2: A,F: B > A,B6: B,C2: B] :
% 2.59/1.40 ( ( ord_less @ A @ A2 @ ( F @ B6 ) )
% 2.59/1.40 => ( ( ord_less_eq @ B @ B6 @ C2 )
% 2.59/1.40 => ( ! [X4: B,Y5: B] :
% 2.59/1.40 ( ( ord_less_eq @ B @ X4 @ Y5 )
% 2.59/1.40 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 2.59/1.40 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.59/1.40 ! [A8: $tType,A9: $tType] :
% 2.59/1.40 ( ( preorder @ A9 @ ( type2 @ A9 ) )
% 2.59/1.40 => ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.59/1.40 ! [A8: $tType,A9: $tType] :
% 2.59/1.40 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 2.59/1.40 & ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
% 2.59/1.40 => ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.59/1.40 ! [A8: $tType,A9: $tType] :
% 2.59/1.40 ( ( order @ A9 @ ( type2 @ A9 ) )
% 2.59/1.40 => ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_fun___Orderings_Oord,axiom,
% 2.59/1.40 ! [A8: $tType,A9: $tType] :
% 2.59/1.40 ( ( ord @ A9 @ ( type2 @ A9 ) )
% 2.59/1.40 => ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 2.59/1.40 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.59/1.40 wellorder @ nat @ ( type2 @ nat ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.59/1.40 preorder @ nat @ ( type2 @ nat ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.59/1.40 linorder @ nat @ ( type2 @ nat ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 2.59/1.40 no_top @ nat @ ( type2 @ nat ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.59/1.40 order @ nat @ ( type2 @ nat ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.59/1.40 ord @ nat @ ( type2 @ nat ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 2.59/1.40 ! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 2.59/1.40 ! [A8: $tType] :
% 2.59/1.40 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 2.59/1.40 => ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 2.59/1.40 ! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 2.59/1.40 ! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
% 2.59/1.40 preorder @ $o @ ( type2 @ $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_HOL_Obool___Orderings_Olinorder_9,axiom,
% 2.59/1.40 linorder @ $o @ ( type2 @ $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_HOL_Obool___Finite__Set_Ofinite_10,axiom,
% 2.59/1.40 finite_finite @ $o @ ( type2 @ $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 2.59/1.40 order @ $o @ ( type2 @ $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 2.59/1.40 ord @ $o @ ( type2 @ $o ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_13,axiom,
% 2.59/1.40 condit1037483654norder @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_14,axiom,
% 2.59/1.40 wellorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_15,axiom,
% 2.59/1.40 preorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_16,axiom,
% 2.59/1.40 linorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_17,axiom,
% 2.59/1.40 order @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.59/1.40
% 2.59/1.40 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_18,axiom,
% 2.59/1.40 ord @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.59/1.40
% 2.59/1.40 thf(conj_0,hypothesis,
% 2.59/1.40 ( linear505997466_on_ev @ a
% 2.59/1.40 @ ( linear1386806755on_alw @ a
% 2.59/1.40 @ ^ [Xs: stream @ a] :
% 2.59/1.40 ~ ( p @ Xs ) )
% 2.59/1.40 @ omega ) ).
% 2.59/1.40
% 2.59/1.40 thf(conj_1,conjecture,
% 2.59/1.40 ord_less @ extended_enat @ ( coindu1365464361scount @ a @ p @ omega ) @ ( extend1396239628finity @ extended_enat ) ).
% 2.59/1.40 ------------------------
% 2.59/1.42 % (20507)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_20348 for (2999ds/4Mi)
% 2.59/1.42 % (20511)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_20348 for (2999ds/275Mi)
% 2.59/1.42 % (20512)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/18Mi)
% 2.59/1.42 % (20510)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_20348 for (2999ds/2Mi)
% 2.59/1.42 % (20506)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/183Mi)
% 2.59/1.42 % (20508)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_20348 for (2999ds/27Mi)
% 2.59/1.42 % (20509)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/2Mi)
% 2.59/1.42 % (20510)Instruction limit reached!
% 2.59/1.42 % (20510)------------------------------
% 2.59/1.42 % (20510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.42 % (20510)Termination reason: Unknown
% 2.59/1.42 % (20510)Termination phase: shuffling
% 2.59/1.42
% 2.59/1.42 % (20510)Memory used [KB]: 1279
% 2.59/1.42 % (20510)Time elapsed: 0.002 s
% 2.59/1.42 % (20510)Instructions burned: 2 (million)
% 2.59/1.42 % (20510)------------------------------
% 2.59/1.42 % (20510)------------------------------
% 2.59/1.42 % (20507)Instruction limit reached!
% 2.59/1.42 % (20507)------------------------------
% 2.59/1.42 % (20507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.42 % (20507)Termination reason: Unknown
% 2.59/1.42 % (20507)Termination phase: shuffling
% 2.59/1.42
% 2.59/1.42 % (20507)Memory used [KB]: 1407
% 2.59/1.42 % (20507)Time elapsed: 0.003 s
% 2.59/1.42 % (20507)Instructions burned: 5 (million)
% 2.59/1.42 % (20507)------------------------------
% 2.59/1.42 % (20507)------------------------------
% 2.59/1.42 % (20509)Instruction limit reached!
% 2.59/1.42 % (20509)------------------------------
% 2.59/1.42 % (20509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.42 % (20509)Termination reason: Unknown
% 2.59/1.42 % (20509)Termination phase: shuffling
% 2.59/1.42
% 2.59/1.42 % (20509)Memory used [KB]: 1407
% 2.59/1.42 % (20509)Time elapsed: 0.003 s
% 2.59/1.42 % (20509)Instructions burned: 4 (million)
% 2.59/1.42 % (20509)------------------------------
% 2.59/1.42 % (20509)------------------------------
% 2.59/1.42 % (20512)Instruction limit reached!
% 2.59/1.42 % (20512)------------------------------
% 2.59/1.42 % (20512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.42 % (20512)Termination reason: Unknown
% 2.59/1.42 % (20512)Termination phase: shuffling
% 2.59/1.42
% 2.59/1.42 % (20512)Memory used [KB]: 1791
% 2.59/1.42 % (20512)Time elapsed: 0.008 s
% 2.59/1.42 % (20512)Instructions burned: 19 (million)
% 2.59/1.42 % (20512)------------------------------
% 2.59/1.42 % (20512)------------------------------
% 2.59/1.43 % (20508)Instruction limit reached!
% 2.59/1.43 % (20508)------------------------------
% 2.59/1.43 % (20508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.43 % (20508)Termination reason: Unknown
% 2.59/1.43 % (20508)Termination phase: Property scanning
% 2.59/1.43
% 2.59/1.43 % (20508)Memory used [KB]: 1791
% 2.59/1.43 % (20508)Time elapsed: 0.011 s
% 2.59/1.43 % (20508)Instructions burned: 30 (million)
% 2.59/1.43 % (20508)------------------------------
% 2.59/1.43 % (20508)------------------------------
% 2.59/1.43 % (20517)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/3Mi)
% 2.59/1.43 % (20518)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_20348 for (2999ds/37Mi)
% 2.59/1.43 % (20519)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_20348 for (2999ds/15Mi)
% 2.59/1.43 % (20517)Instruction limit reached!
% 2.59/1.43 % (20517)------------------------------
% 2.59/1.43 % (20517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.43 % (20517)Termination reason: Unknown
% 2.59/1.43 % (20517)Termination phase: shuffling
% 2.59/1.43
% 2.59/1.43 % (20517)Memory used [KB]: 1407
% 2.59/1.43 % (20517)Time elapsed: 0.003 s
% 2.59/1.43 % (20517)Instructions burned: 5 (million)
% 2.59/1.43 % (20517)------------------------------
% 2.59/1.43 % (20517)------------------------------
% 2.59/1.43 % (20521)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_20348 for (2999ds/3Mi)
% 2.59/1.43 % (20519)Instruction limit reached!
% 2.59/1.43 % (20519)------------------------------
% 2.59/1.43 % (20519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.43 % (20519)Termination reason: Unknown
% 2.59/1.43 % (20519)Termination phase: shuffling
% 2.59/1.43
% 2.59/1.43 % (20519)Memory used [KB]: 1791
% 2.59/1.43 % (20519)Time elapsed: 0.007 s
% 2.59/1.43 % (20519)Instructions burned: 17 (million)
% 2.59/1.43 % (20519)------------------------------
% 2.59/1.43 % (20519)------------------------------
% 2.59/1.43 % (20521)Instruction limit reached!
% 2.59/1.43 % (20521)------------------------------
% 2.59/1.43 % (20521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.43 % (20521)Termination reason: Unknown
% 2.59/1.43 % (20521)Termination phase: shuffling
% 2.59/1.43
% 2.59/1.43 % (20521)Memory used [KB]: 1407
% 2.59/1.43 % (20521)Time elapsed: 0.002 s
% 2.59/1.43 % (20521)Instructions burned: 3 (million)
% 2.59/1.43 % (20521)------------------------------
% 2.59/1.43 % (20521)------------------------------
% 2.59/1.44 % (20522)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_20348 for (2999ds/1041Mi)
% 2.59/1.44 % (20524)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/7Mi)
% 2.59/1.44 % (20518)Instruction limit reached!
% 2.59/1.44 % (20518)------------------------------
% 2.59/1.44 % (20518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.44 % (20518)Termination reason: Unknown
% 2.59/1.44 % (20518)Termination phase: Property scanning
% 2.59/1.44
% 2.59/1.44 % (20518)Memory used [KB]: 2174
% 2.59/1.44 % (20518)Time elapsed: 0.012 s
% 2.59/1.44 % (20518)Instructions burned: 39 (million)
% 2.59/1.44 % (20518)------------------------------
% 2.59/1.44 % (20518)------------------------------
% 2.59/1.44 % (20524)Instruction limit reached!
% 2.59/1.44 % (20524)------------------------------
% 2.59/1.44 % (20524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.44 % (20524)Termination reason: Unknown
% 2.59/1.44 % (20524)Termination phase: shuffling
% 2.59/1.44
% 2.59/1.44 % (20524)Memory used [KB]: 1535
% 2.59/1.44 % (20524)Time elapsed: 0.004 s
% 2.59/1.44 % (20524)Instructions burned: 9 (million)
% 2.59/1.44 % (20524)------------------------------
% 2.59/1.44 % (20524)------------------------------
% 2.59/1.44 % (20527)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/16Mi)
% 2.59/1.44 % (20528)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_20348 for (2999ds/3Mi)
% 2.59/1.44 % (20528)Instruction limit reached!
% 2.59/1.44 % (20528)------------------------------
% 2.59/1.44 % (20528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.44 % (20528)Termination reason: Unknown
% 2.59/1.44 % (20528)Termination phase: shuffling
% 2.59/1.44
% 2.59/1.44 % (20528)Memory used [KB]: 1407
% 2.59/1.44 % (20528)Time elapsed: 0.002 s
% 2.59/1.44 % (20528)Instructions burned: 3 (million)
% 2.59/1.44 % (20528)------------------------------
% 2.59/1.44 % (20528)------------------------------
% 2.59/1.45 % (20533)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_20348 for (2999ds/3Mi)
% 2.59/1.45 % (20533)Instruction limit reached!
% 2.59/1.45 % (20533)------------------------------
% 2.59/1.45 % (20533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.45 % (20533)Termination reason: Unknown
% 2.59/1.45 % (20533)Termination phase: shuffling
% 2.59/1.45
% 2.59/1.45 % (20533)Memory used [KB]: 1407
% 2.59/1.45 % (20533)Time elapsed: 0.002 s
% 2.59/1.45 % (20533)Instructions burned: 3 (million)
% 2.59/1.45 % (20533)------------------------------
% 2.59/1.45 % (20533)------------------------------
% 2.59/1.45 % (20527)Instruction limit reached!
% 2.59/1.45 % (20527)------------------------------
% 2.59/1.45 % (20527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.45 % (20527)Termination reason: Unknown
% 2.59/1.45 % (20527)Termination phase: shuffling
% 2.59/1.45
% 2.59/1.45 % (20527)Memory used [KB]: 1791
% 2.59/1.45 % (20527)Time elapsed: 0.007 s
% 2.59/1.45 % (20527)Instructions burned: 18 (million)
% 2.59/1.45 % (20527)------------------------------
% 2.59/1.45 % (20527)------------------------------
% 2.59/1.45 % (20534)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/7Mi)
% 2.59/1.45 % (20537)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/3Mi)
% 2.59/1.45 % (20534)Instruction limit reached!
% 2.59/1.45 % (20534)------------------------------
% 2.59/1.45 % (20534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.45 % (20534)Termination reason: Unknown
% 2.59/1.45 % (20534)Termination phase: shuffling
% 2.59/1.45
% 2.59/1.45 % (20534)Memory used [KB]: 1535
% 2.59/1.45 % (20534)Time elapsed: 0.004 s
% 2.59/1.45 % (20534)Instructions burned: 9 (million)
% 2.59/1.45 % (20534)------------------------------
% 2.59/1.45 % (20534)------------------------------
% 2.59/1.46 % (20537)Instruction limit reached!
% 2.59/1.46 % (20537)------------------------------
% 2.59/1.46 % (20537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.46 % (20537)Termination reason: Unknown
% 2.59/1.46 % (20537)Termination phase: shuffling
% 2.59/1.46
% 2.59/1.46 % (20537)Memory used [KB]: 1407
% 2.59/1.46 % (20537)Time elapsed: 0.002 s
% 2.59/1.46 % (20537)Instructions burned: 4 (million)
% 2.59/1.46 % (20537)------------------------------
% 2.59/1.46 % (20537)------------------------------
% 2.59/1.46 % (20538)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/4Mi)
% 2.59/1.46 % (20539)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/18Mi)
% 2.59/1.46 % (20538)Instruction limit reached!
% 2.59/1.46 % (20538)------------------------------
% 2.59/1.46 % (20538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.46 % (20538)Termination reason: Unknown
% 2.59/1.46 % (20538)Termination phase: shuffling
% 2.59/1.46
% 2.59/1.46 % (20538)Memory used [KB]: 1535
% 2.59/1.46 % (20538)Time elapsed: 0.003 s
% 2.59/1.46 % (20538)Instructions burned: 6 (million)
% 2.59/1.46 % (20538)------------------------------
% 2.59/1.46 % (20538)------------------------------
% 2.59/1.46 % (20540)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/710Mi)
% 2.59/1.46 % (20541)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/6Mi)
% 2.59/1.46 % (20539)Instruction limit reached!
% 2.59/1.46 % (20539)------------------------------
% 2.59/1.46 % (20539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.46 % (20539)Termination reason: Unknown
% 2.59/1.46 % (20539)Termination phase: shuffling
% 2.59/1.46
% 2.59/1.46 % (20539)Memory used [KB]: 1791
% 2.59/1.46 % (20539)Time elapsed: 0.007 s
% 2.59/1.46 % (20539)Instructions burned: 18 (million)
% 2.59/1.46 % (20539)------------------------------
% 2.59/1.46 % (20539)------------------------------
% 2.59/1.47 % (20541)Instruction limit reached!
% 2.59/1.47 % (20541)------------------------------
% 2.59/1.47 % (20541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.47 % (20541)Termination reason: Unknown
% 2.59/1.47 % (20541)Termination phase: shuffling
% 2.59/1.47
% 2.59/1.47 % (20541)Memory used [KB]: 1535
% 2.59/1.47 % (20541)Time elapsed: 0.004 s
% 2.59/1.47 % (20541)Instructions burned: 8 (million)
% 2.59/1.47 % (20541)------------------------------
% 2.59/1.47 % (20541)------------------------------
% 2.59/1.47 % (20543)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_20348 for (2999ds/902Mi)
% 2.59/1.47 % (20506)Instruction limit reached!
% 2.59/1.47 % (20506)------------------------------
% 2.59/1.47 % (20506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.47 % (20506)Termination reason: Unknown
% 2.59/1.47 % (20506)Termination phase: Saturation
% 2.59/1.47
% 2.59/1.47 % (20506)Memory used [KB]: 7291
% 2.59/1.47 % (20506)Time elapsed: 0.058 s
% 2.59/1.47 % (20506)Instructions burned: 185 (million)
% 2.59/1.47 % (20506)------------------------------
% 2.59/1.47 % (20506)------------------------------
% 2.59/1.47 % (20548)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_20348 for (2999ds/21Mi)
% 2.59/1.48 % (20552)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/5Mi)
% 2.59/1.48 % (20552)Instruction limit reached!
% 2.59/1.48 % (20552)------------------------------
% 2.59/1.48 % (20552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.48 % (20552)Termination reason: Unknown
% 2.59/1.48 % (20552)Termination phase: shuffling
% 2.59/1.48
% 2.59/1.48 % (20552)Memory used [KB]: 1407
% 2.59/1.48 % (20552)Time elapsed: 0.003 s
% 2.59/1.48 % (20552)Instructions burned: 6 (million)
% 2.59/1.48 % (20552)------------------------------
% 2.59/1.48 % (20552)------------------------------
% 2.59/1.48 % (20548)Instruction limit reached!
% 2.59/1.48 % (20548)------------------------------
% 2.59/1.48 % (20548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.48 % (20548)Termination reason: Unknown
% 2.59/1.48 % (20548)Termination phase: Property scanning
% 2.59/1.48
% 2.59/1.48 % (20548)Memory used [KB]: 1791
% 2.59/1.48 % (20548)Time elapsed: 0.009 s
% 2.59/1.48 % (20548)Instructions burned: 24 (million)
% 2.59/1.48 % (20548)------------------------------
% 2.59/1.48 % (20548)------------------------------
% 2.59/1.48 % (20557)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_20348 for (2999ds/6Mi)
% 2.59/1.49 % (20557)Instruction limit reached!
% 2.59/1.49 % (20557)------------------------------
% 2.59/1.49 % (20557)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.49 % (20557)Termination reason: Unknown
% 2.59/1.49 % (20557)Termination phase: shuffling
% 2.59/1.49
% 2.59/1.49 % (20557)Memory used [KB]: 1407
% 2.59/1.49 % (20557)Time elapsed: 0.003 s
% 2.59/1.49 % (20557)Instructions burned: 6 (million)
% 2.59/1.49 % (20557)------------------------------
% 2.59/1.49 % (20557)------------------------------
% 2.59/1.49 % (20560)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/377Mi)
% 2.59/1.49 % (20565)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_20348 for (2999ds/779Mi)
% 2.59/1.49 % (20567)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_20348 for (2999ds/19Mi)
% 2.61/1.50 % (20567)Instruction limit reached!
% 2.61/1.50 % (20567)------------------------------
% 2.61/1.50 % (20567)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.50 % (20567)Termination reason: Unknown
% 2.61/1.50 % (20567)Termination phase: shuffling
% 2.61/1.50
% 2.61/1.50 % (20567)Memory used [KB]: 1791
% 2.61/1.50 % (20567)Time elapsed: 0.007 s
% 2.61/1.50 % (20567)Instructions burned: 21 (million)
% 2.61/1.50 % (20567)------------------------------
% 2.61/1.50 % (20567)------------------------------
% 2.61/1.50 % (20511)Instruction limit reached!
% 2.61/1.50 % (20511)------------------------------
% 2.61/1.50 % (20511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.50 % (20511)Termination reason: Unknown
% 2.61/1.50 % (20511)Termination phase: Saturation
% 2.61/1.50
% 2.61/1.50 % (20511)Memory used [KB]: 6908
% 2.61/1.50 % (20511)Time elapsed: 0.087 s
% 2.61/1.50 % (20511)Instructions burned: 276 (million)
% 2.61/1.50 % (20511)------------------------------
% 2.61/1.50 % (20511)------------------------------
% 2.61/1.51 % (20569)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/879Mi)
% 2.61/1.51 % (20572)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_20348 for (2999ds/17Mi)
% 2.61/1.52 % (20572)Instruction limit reached!
% 2.61/1.52 % (20572)------------------------------
% 2.61/1.52 % (20572)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.52 % (20572)Termination reason: Unknown
% 2.61/1.52 % (20572)Termination phase: shuffling
% 2.61/1.52
% 2.61/1.52 % (20572)Memory used [KB]: 1791
% 2.61/1.52 % (20572)Time elapsed: 0.007 s
% 2.61/1.52 % (20572)Instructions burned: 18 (million)
% 2.61/1.52 % (20572)------------------------------
% 2.61/1.52 % (20572)------------------------------
% 2.61/1.53 % (20587)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_20348 for (2998ds/3Mi)
% 2.61/1.53 % (20587)Instruction limit reached!
% 2.61/1.53 % (20587)------------------------------
% 2.61/1.53 % (20587)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.53 % (20587)Termination reason: Unknown
% 2.61/1.53 % (20587)Termination phase: shuffling
% 2.61/1.53
% 2.61/1.53 % (20587)Memory used [KB]: 1407
% 2.61/1.53 % (20587)Time elapsed: 0.002 s
% 2.61/1.53 % (20587)Instructions burned: 3 (million)
% 2.61/1.53 % (20587)------------------------------
% 2.61/1.53 % (20587)------------------------------
% 2.61/1.54 % (20594)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_20348 for (2998ds/30Mi)
% 2.61/1.55 % (20594)Instruction limit reached!
% 2.61/1.55 % (20594)------------------------------
% 2.61/1.55 % (20594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.55 % (20594)Termination reason: Unknown
% 2.61/1.55 % (20594)Termination phase: Property scanning
% 2.61/1.55
% 2.61/1.55 % (20594)Memory used [KB]: 1791
% 2.61/1.55 % (20594)Time elapsed: 0.011 s
% 2.61/1.55 % (20594)Instructions burned: 30 (million)
% 2.61/1.55 % (20594)------------------------------
% 2.61/1.55 % (20594)------------------------------
% 2.61/1.56 % (20598)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_20348 for (2998ds/127Mi)
% 2.61/1.59 % (20569)First to succeed.
% 2.61/1.59 % (20522)Also succeeded, but the first one will report.
% 2.61/1.59 % (20569)Refutation found. Thanks to Tanya!
% 2.61/1.59 % SZS status Theorem for DTF2THF_20348
% 2.61/1.59 % SZS output start Proof for DTF2THF_20348
% 2.61/1.59 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.61/1.59 thf(type_def_7, type, stream: ($tType) > $tType).
% 2.61/1.59 thf(type_def_8, type, extended_enat: $tType).
% 2.61/1.59 thf(type_def_9, type, nat: $tType).
% 2.61/1.59 thf(type_def_10, type, set: ($tType) > $tType).
% 2.61/1.59 thf(type_def_11, type, a: $tType).
% 2.61/1.59 thf(func_def_0, type, extended_enat: $tType).
% 2.61/1.59 thf(func_def_1, type, stream: ($tType) > $tType).
% 2.61/1.59 thf(func_def_2, type, set: ($tType) > $tType).
% 2.61/1.59 thf(func_def_3, type, nat: $tType).
% 2.61/1.59 thf(func_def_4, type, itself: ($tType) > $tType).
% 2.61/1.59 thf(func_def_5, type, a: $tType).
% 2.61/1.59 thf(func_def_6, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_7, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_8, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_9, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_10, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_11, type, finite_finite: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_12, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_13, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_14, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_15, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_16, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_17, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_18, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_19, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.61/1.59 thf(func_def_20, type, coindu1365464361scount: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > extended_enat)).
% 2.61/1.59 thf(func_def_21, type, extended_enat2: nat > extended_enat).
% 2.61/1.59 thf(func_def_22, type, extended_case_enat: !>[X0: $tType]:((nat > X0) > X0 > extended_enat > X0)).
% 2.61/1.59 thf(func_def_23, type, extended_rec_enat: !>[X0: $tType]:((nat > X0) > X0 > extended_enat > X0)).
% 2.61/1.59 thf(func_def_24, type, extend1396239628finity: !>[X0: $tType]:(X0)).
% 2.61/1.59 thf(func_def_25, type, finite_card: !>[X0: $tType]:(set @ X0 > nat)).
% 2.61/1.59 thf(func_def_26, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 2.61/1.59 thf(func_def_27, type, linear1386806755on_alw: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > $o)).
% 2.61/1.59 thf(func_def_28, type, linear505997466_on_ev: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > $o)).
% 2.61/1.59 thf(func_def_29, type, linear1494993505on_nxt: !>[X0: $tType, X1: $tType]:((stream @ X0 > X1) > stream @ X0 > X1)).
% 2.61/1.59 thf(func_def_30, type, linear1335279038n_wait: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_31, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.61/1.59 thf(func_def_32, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.61/1.59 thf(func_def_33, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.61/1.59 thf(func_def_34, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.61/1.59 thf(func_def_35, type, sdrop: !>[X0: $tType]:(nat > stream @ X0 > stream @ X0)).
% 2.61/1.59 thf(func_def_36, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.61/1.59 thf(func_def_37, type, p: stream @ a > $o).
% 2.61/1.59 thf(func_def_38, type, omega: stream @ a).
% 2.61/1.59 thf(func_def_53, type, sP0: extended_enat > extended_enat > $o).
% 2.61/1.59 thf(func_def_54, type, sP1: extended_enat > extended_enat > extended_enat > $o).
% 2.61/1.59 thf(func_def_55, type, sP2: extended_enat > extended_enat > extended_enat > $o).
% 2.61/1.59 thf(func_def_56, type, sP3: extended_enat > extended_enat > extended_enat > $o).
% 2.61/1.59 thf(func_def_57, type, sP4: extended_enat > extended_enat > extended_enat > $o).
% 2.61/1.59 thf(func_def_58, type, sP5: extended_enat > extended_enat > extended_enat > $o).
% 2.61/1.59 thf(func_def_59, type, sP6: extended_enat > extended_enat > extended_enat > $o).
% 2.61/1.59 thf(func_def_60, type, sP7: extended_enat > extended_enat > extended_enat > $o).
% 2.61/1.59 thf(func_def_61, type, sK8: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0)).
% 2.61/1.59 thf(func_def_62, type, sK9: nat > extended_enat > nat).
% 2.61/1.59 thf(func_def_63, type, sK10: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0)).
% 2.61/1.59 thf(func_def_64, type, sK11: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0)).
% 2.61/1.59 thf(func_def_65, type, sK12: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0)).
% 2.61/1.59 thf(func_def_66, type, sK13: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.61/1.59 thf(func_def_67, type, sK14: extended_enat > nat).
% 2.61/1.59 thf(func_def_68, type, sK15: extended_enat > nat).
% 2.61/1.59 thf(func_def_69, type, sK16: extended_enat > nat).
% 2.61/1.59 thf(func_def_70, type, sK17: extended_enat > nat).
% 2.61/1.59 thf(func_def_71, type, sK18: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0)).
% 2.61/1.59 thf(func_def_72, type, sK19: (extended_enat > $o) > extended_enat).
% 2.61/1.59 thf(func_def_73, type, sK20: extended_enat > nat).
% 2.61/1.59 thf(func_def_74, type, sK21: (extended_enat > $o) > nat).
% 2.61/1.59 thf(func_def_75, type, sK22: extended_enat > nat).
% 2.61/1.59 thf(func_def_76, type, sK23: extended_enat > nat).
% 2.61/1.59 thf(func_def_77, type, sK24: extended_enat > nat).
% 2.61/1.59 thf(func_def_78, type, sK25: extended_enat > nat).
% 2.61/1.59 thf(func_def_79, type, sK26: extended_enat > nat).
% 2.61/1.59 thf(func_def_80, type, sK27: extended_enat > nat).
% 2.61/1.59 thf(func_def_81, type, sK28: extended_enat > nat).
% 2.61/1.59 thf(func_def_82, type, sK29: extended_enat > nat).
% 2.61/1.59 thf(func_def_83, type, sK30: extended_enat > nat).
% 2.61/1.59 thf(func_def_84, type, sK31: extended_enat > nat).
% 2.61/1.59 thf(func_def_85, type, sK32: extended_enat > nat).
% 2.61/1.59 thf(func_def_86, type, sK33: extended_enat > nat).
% 2.61/1.59 thf(func_def_87, type, sK34: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0)).
% 2.61/1.59 thf(func_def_88, type, sK35: nat > set @ extended_enat > extended_enat).
% 2.61/1.59 thf(func_def_89, type, sK36: !>[X0: $tType]:(nat > (X0 > extended_enat) > X0)).
% 2.61/1.59 thf(func_def_90, type, sK37: !>[X0: $tType]:((X0 > extended_enat) > X0)).
% 2.61/1.59 thf(func_def_91, type, sK38: extended_enat > nat).
% 2.61/1.59 thf(func_def_92, type, sK39: extended_enat > nat).
% 2.61/1.59 thf(func_def_93, type, sK40: extended_enat > nat).
% 2.61/1.59 thf(func_def_96, type, sK42: nat > nat).
% 2.61/1.59 thf(func_def_97, type, sK43: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_98, type, sK44: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_99, type, sK45: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_100, type, sK46: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_101, type, sK47: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_102, type, sK48: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_103, type, sK49: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_104, type, sK50: !>[X0: $tType]:((stream @ X0 > $o) > nat > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_105, type, sK51: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_106, type, sK52: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_107, type, sK53: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_108, type, sK54: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_109, type, sK55: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_110, type, sK56: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat > nat)).
% 2.61/1.59 thf(func_def_111, type, sK57: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat > nat)).
% 2.61/1.59 thf(func_def_112, type, sK58: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_113, type, sK59: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat > nat)).
% 2.61/1.59 thf(func_def_114, type, sK60: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_115, type, sK61: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_116, type, sK62: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_117, type, sK63: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_118, type, sK64: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_119, type, sK65: !>[X0: $tType]:((stream @ X0 > $o) > nat > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_120, type, sK66: !>[X0: $tType]:((stream @ X0 > $o) > nat > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_121, type, sK67: set @ nat > nat).
% 2.61/1.59 thf(func_def_122, type, sK68: nat > set @ nat > nat).
% 2.61/1.59 thf(func_def_123, type, sK69: extended_enat > nat).
% 2.61/1.59 thf(func_def_124, type, sK70: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_125, type, sK71: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_126, type, sK72: nat).
% 2.61/1.59 thf(func_def_127, type, sK73: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_128, type, sK74: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_129, type, sK75: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_130, type, sK76: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_131, type, sK77: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_132, type, sK78: extended_enat > nat).
% 2.61/1.59 thf(func_def_133, type, sK79: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_134, type, sK80: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_135, type, sK81: !>[X0: $tType]:((stream @ X0 > $o) > nat > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_136, type, sK82: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_137, type, sK83: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_138, type, sK84: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_139, type, sK85: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_140, type, sK86: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_141, type, sK87: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_142, type, sK88: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_143, type, sK89: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_144, type, sK90: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_145, type, sK91: set @ nat > nat).
% 2.61/1.59 thf(func_def_146, type, sK92: nat > set @ nat > nat).
% 2.61/1.59 thf(func_def_147, type, sK93: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_148, type, sK94: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_149, type, sK95: nat).
% 2.61/1.59 thf(func_def_150, type, sK96: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_151, type, sK97: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_152, type, sK98: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_153, type, sK99: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat)).
% 2.61/1.59 thf(func_def_154, type, sK100: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_155, type, sK101: !>[X0: $tType]:((stream @ X0 > $o) > nat > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_156, type, sK102: !>[X0: $tType]:((stream @ X0 > $o) > nat > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_157, type, sK103: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_158, type, sK104: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_159, type, sK105: !>[X0: $tType]:((stream @ X0 > $o) > nat > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_160, type, sK106: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_161, type, sK107: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_162, type, sK108: !>[X0: $tType]:((stream @ X0 > $o) > nat > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_163, type, sK109: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_164, type, sK110: !>[X0: $tType]:((stream @ X0 > $o) > nat > stream @ X0 > nat)).
% 2.61/1.59 thf(func_def_165, type, sK111: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat > nat)).
% 2.61/1.59 thf(func_def_166, type, sK112: !>[X0: $tType]:(stream @ X0 > (stream @ X0 > $o) > nat > nat)).
% 2.61/1.59 thf(func_def_167, type, ph113: !>[X0: $tType]:(X0)).
% 2.61/1.59 thf(f1958,plain,(
% 2.61/1.59 $false),
% 2.61/1.59 inference(avatar_sat_refutation,[status(thm)],[f1677,f1933,f1955])).
% 2.61/1.59 thf(f1955,plain,(
% 2.61/1.59 ~spl41_1),
% 2.61/1.59 inference(avatar_contradiction_clause,[status(thm)],[f1954])).
% 2.61/1.59 thf(f1954,plain,(
% 2.61/1.59 $false | ~spl41_1),
% 2.61/1.59 inference(trivial_inequality_removal,[status(thm)],[f1948])).
% 2.61/1.59 thf(f1948,plain,(
% 2.61/1.59 ($true = $false) | ~spl41_1),
% 2.61/1.59 inference(superposition,[status(thm)],[f1672,f1560])).
% 2.61/1.59 thf(f1560,plain,(
% 2.61/1.59 ( ! [X1 : nat] : (($false = (p @ (sdrop @ a @ X1 @ (sdrop @ a @ sK95 @ omega))))) )),
% 2.61/1.59 inference(not_proxy_clausification,[status(thm)],[f1559])).
% 2.61/1.59 thf(f1559,plain,(
% 2.61/1.59 ( ! [X1 : nat] : (($true = (~ (p @ (sdrop @ a @ X1 @ (sdrop @ a @ sK95 @ omega)))))) )),
% 2.61/1.59 inference(beta_eta_normalization,[status(thm)],[f1558])).
% 2.61/1.59 thf(f1558,plain,(
% 2.61/1.59 ( ! [X1 : nat] : (($true = ((^[Y0 : nat]: (~ (p @ (sdrop @ a @ Y0 @ (sdrop @ a @ sK95 @ omega))))) @ X1))) )),
% 2.61/1.59 inference(pi_clausification,[status(thm)],[f1557])).
% 2.61/1.59 thf(f1557,plain,(
% 2.61/1.59 ($true = (!! @ nat @ (^[Y0 : nat]: (~ (p @ (sdrop @ a @ Y0 @ (sdrop @ a @ sK95 @ omega)))))))),
% 2.61/1.59 inference(beta_eta_normalization,[status(thm)],[f1556])).
% 2.61/1.59 thf(f1556,plain,(
% 2.61/1.59 ($true = ((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (~ (p @ (sdrop @ a @ Y1 @ (sdrop @ a @ Y0 @ omega))))))) @ sK95))),
% 2.61/1.59 inference(sigma_clausification,[status(thm)],[f1555])).
% 2.61/1.59 thf(f1555,plain,(
% 2.61/1.59 ($true = (?? @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (~ (p @ (sdrop @ a @ Y1 @ (sdrop @ a @ Y0 @ omega)))))))))),
% 2.61/1.59 inference(beta_eta_normalization,[status(thm)],[f1154])).
% 2.61/1.59 thf(f1154,plain,(
% 2.61/1.59 ($true = ((^[Y0 : stream @ a > $o]: ((^[Y1 : stream @ a]: (?? @ nat @ (^[Y2 : nat]: (Y0 @ (sdrop @ a @ Y2 @ Y1))))))) @ ((^[Y0 : stream @ a > $o]: ((^[Y1 : stream @ a]: (!! @ nat @ (^[Y2 : nat]: (Y0 @ (sdrop @ a @ Y2 @ Y1))))))) @ (^[Y0 : stream @ a]: (~ (p @ Y0)))) @ omega))),
% 2.61/1.59 inference(definition_unfolding,[status(thm)],[f1030,f1123,f1016])).
% 2.61/1.59 thf(f1016,plain,(
% 2.61/1.59 ( ! [X0 : $tType] : ((linear1386806755on_alw @ X0 = (^[Y0 : stream @ X0 > $o]: ((^[Y1 : stream @ X0]: (!! @ nat @ (^[Y2 : nat]: (Y0 @ (sdrop @ X0 @ Y2 @ Y1))))))))) )),
% 2.61/1.59 inference(cnf_transformation,[status(thm)],[f414])).
% 2.61/1.59 thf(f414,plain,(
% 2.61/1.59 ! [X0 : $tType] : (linear1386806755on_alw @ X0 = (^[Y0 : stream @ X0 > $o]: ((^[Y1 : stream @ X0]: (!! @ nat @ (^[Y2 : nat]: (Y0 @ (sdrop @ X0 @ Y2 @ Y1))))))))),
% 2.61/1.59 inference(fool_elimination,[status(thm)],[f413])).
% 2.61/1.59 thf(f413,plain,(
% 2.61/1.59 ! [X0 : $tType] : ((^[X1 : stream @ X0 > $o, X2 : stream @ X0] : (! [X3 : nat] : (X1 @ (sdrop @ X0 @ X3 @ X2)))) = linear1386806755on_alw @ X0)),
% 2.61/1.59 inference(rectify,[status(thm)],[f26])).
% 2.61/1.59 thf(f26,axiom,(
% 2.61/1.59 ! [X0 : $tType] : ((^[X19 : stream @ X0 > $o, X37 : stream @ X0] : (! [X38 : nat] : (X19 @ (sdrop @ X0 @ X38 @ X37)))) = linear1386806755on_alw @ X0)),
% 2.61/1.59 file('/export/starexec/sandbox/tmp/tmp.HepLMUEemU/DTF2THF_20348.p',fact_25_alw__iff__sdrop)).
% 2.61/1.59 thf(f1123,plain,(
% 2.61/1.59 ( ! [X0 : $tType] : ((linear505997466_on_ev @ X0 = (^[Y0 : stream @ X0 > $o]: ((^[Y1 : stream @ X0]: (?? @ nat @ (^[Y2 : nat]: (Y0 @ (sdrop @ X0 @ Y2 @ Y1))))))))) )),
% 2.61/1.59 inference(cnf_transformation,[status(thm)],[f580])).
% 2.61/1.59 thf(f580,plain,(
% 2.61/1.59 ! [X0 : $tType] : (linear505997466_on_ev @ X0 = (^[Y0 : stream @ X0 > $o]: ((^[Y1 : stream @ X0]: (?? @ nat @ (^[Y2 : nat]: (Y0 @ (sdrop @ X0 @ Y2 @ Y1))))))))),
% 2.61/1.59 inference(fool_elimination,[status(thm)],[f579])).
% 2.61/1.59 thf(f579,plain,(
% 2.61/1.59 ! [X0 : $tType] : (linear505997466_on_ev @ X0 = (^[X1 : stream @ X0 > $o, X2 : stream @ X0] : (? [X3 : nat] : (X1 @ (sdrop @ X0 @ X3 @ X2)))))),
% 2.61/1.59 inference(rectify,[status(thm)],[f27])).
% 2.61/1.59 thf(f27,axiom,(
% 2.61/1.59 ! [X0 : $tType] : (linear505997466_on_ev @ X0 = (^[X19 : stream @ X0 > $o, X37 : stream @ X0] : (? [X38 : nat] : (X19 @ (sdrop @ X0 @ X38 @ X37)))))),
% 2.61/1.59 file('/export/starexec/sandbox/tmp/tmp.HepLMUEemU/DTF2THF_20348.p',fact_26_ev__iff__sdrop)).
% 2.61/1.59 thf(f1030,plain,(
% 2.61/1.59 ($true = (linear505997466_on_ev @ a @ (linear1386806755on_alw @ a @ (^[Y0 : stream @ a]: (~ (p @ Y0)))) @ omega))),
% 2.61/1.59 inference(cnf_transformation,[status(thm)],[f340])).
% 2.61/1.59 thf(f340,plain,(
% 2.61/1.59 ($true = (linear505997466_on_ev @ a @ (linear1386806755on_alw @ a @ (^[Y0 : stream @ a]: (~ (p @ Y0)))) @ omega))),
% 2.61/1.59 inference(fool_elimination,[status(thm)],[f339])).
% 2.61/1.59 thf(f339,plain,(
% 2.61/1.59 (linear505997466_on_ev @ a @ (linear1386806755on_alw @ a @ (^[X0 : stream @ a] : (~(p @ X0)))) @ omega)),
% 2.61/1.59 inference(rectify,[status(thm)],[f283])).
% 2.61/1.59 thf(f283,axiom,(
% 2.61/1.59 (linear505997466_on_ev @ a @ (linear1386806755on_alw @ a @ (^[X4 : stream @ a] : (~(p @ X4)))) @ omega)),
% 2.61/1.59 file('/export/starexec/sandbox/tmp/tmp.HepLMUEemU/DTF2THF_20348.p',conj_0)).
% 2.61/1.59 thf(f1672,plain,(
% 2.61/1.59 ( ! [X1 : nat] : (($true = (p @ (sdrop @ a @ (sK42 @ X1) @ (sdrop @ a @ X1 @ omega))))) ) | ~spl41_1),
% 2.61/1.59 inference(avatar_component_clause,[status(thm)],[f1671])).
% 2.61/1.59 thf(f1671,definition,(
% 2.61/1.59 spl41_1 <=> ! [X1 : nat] : ($true = (p @ (sdrop @ a @ (sK42 @ X1) @ (sdrop @ a @ X1 @ omega))))),
% 2.61/1.59 introduced(definition,[new_symbols(naming,[spl41_1])],[avatar_definition])).
% 2.61/1.59 thf(f1933,plain,(
% 2.61/1.59 ~spl41_2),
% 2.61/1.59 inference(avatar_contradiction_clause,[status(thm)],[f1932])).
% 2.61/1.59 thf(f1932,plain,(
% 2.61/1.59 $false | ~spl41_2),
% 2.61/1.59 inference(subsumption_resolution,[status(thm)],[f1916,f1066])).
% 2.61/1.59 thf(f1066,plain,(
% 2.61/1.59 ((ord_less @ extended_enat @ (coindu1365464361scount @ a @ p @ omega) @ extend1396239628finity @ extended_enat) != $true)),
% 2.61/1.59 inference(cnf_transformation,[status(thm)],[f821])).
% 2.61/1.59 thf(f821,plain,(
% 2.61/1.59 ((ord_less @ extended_enat @ (coindu1365464361scount @ a @ p @ omega) @ extend1396239628finity @ extended_enat) != $true)),
% 2.61/1.59 inference(flattening,[status(thm)],[f492])).
% 2.61/1.59 thf(f492,plain,(
% 2.61/1.59 ~((ord_less @ extended_enat @ (coindu1365464361scount @ a @ p @ omega) @ extend1396239628finity @ extended_enat) = $true)),
% 2.61/1.59 inference(fool_elimination,[status(thm)],[f491])).
% 2.61/1.59 thf(f491,plain,(
% 2.61/1.59 ~(ord_less @ extended_enat @ (coindu1365464361scount @ a @ p @ omega) @ extend1396239628finity @ extended_enat)),
% 2.61/1.59 inference(rectify,[status(thm)],[f285])).
% 2.61/1.59 thf(f285,negated_conjecture,(
% 2.61/1.59 ~(ord_less @ extended_enat @ (coindu1365464361scount @ a @ p @ omega) @ extend1396239628finity @ extended_enat)),
% 2.61/1.59 inference(negated_conjecture,[status(cth)],[f284])).
% 2.61/1.59 thf(f284,conjecture,(
% 2.61/1.59 (ord_less @ extended_enat @ (coindu1365464361scount @ a @ p @ omega) @ extend1396239628finity @ extended_enat)),
% 2.61/1.59 file('/export/starexec/sandbox/tmp/tmp.HepLMUEemU/DTF2THF_20348.p',conj_1)).
% 2.61/1.59 thf(f1916,plain,(
% 2.61/1.59 ((ord_less @ extended_enat @ (coindu1365464361scount @ a @ p @ omega) @ extend1396239628finity @ extended_enat) = $true) | ~spl41_2),
% 2.61/1.59 inference(superposition,[status(thm)],[f1059,f1676])).
% 2.61/1.59 thf(f1676,plain,(
% 2.61/1.59 ((coindu1365464361scount @ a @ p @ omega) = (extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[Y0 : nat]: (p @ (sdrop @ a @ Y0 @ omega))))))) | ~spl41_2),
% 2.61/1.59 inference(avatar_component_clause,[status(thm)],[f1674])).
% 2.61/1.59 thf(f1674,definition,(
% 2.61/1.59 spl41_2 <=> ((coindu1365464361scount @ a @ p @ omega) = (extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[Y0 : nat]: (p @ (sdrop @ a @ Y0 @ omega)))))))),
% 2.61/1.59 introduced(definition,[new_symbols(naming,[spl41_2])],[avatar_definition])).
% 2.61/1.59 thf(f1059,plain,(
% 2.61/1.59 ( ! [X0 : nat] : (($true = (ord_less @ extended_enat @ (extended_enat2 @ X0) @ extend1396239628finity @ extended_enat))) )),
% 2.61/1.59 inference(cnf_transformation,[status(thm)],[f638])).
% 2.61/1.59 thf(f638,plain,(
% 2.61/1.59 ! [X0 : nat] : ($true = (ord_less @ extended_enat @ (extended_enat2 @ X0) @ extend1396239628finity @ extended_enat))),
% 2.61/1.59 inference(fool_elimination,[status(thm)],[f637])).
% 2.61/1.59 thf(f637,plain,(
% 2.61/1.59 ! [X0 : nat] : (ord_less @ extended_enat @ (extended_enat2 @ X0) @ extend1396239628finity @ extended_enat)),
% 2.61/1.59 inference(rectify,[status(thm)],[f39])).
% 2.61/1.59 thf(f39,axiom,(
% 2.61/1.59 ! [X30 : nat] : (ord_less @ extended_enat @ (extended_enat2 @ X30) @ extend1396239628finity @ extended_enat)),
% 2.61/1.59 file('/export/starexec/sandbox/tmp/tmp.HepLMUEemU/DTF2THF_20348.p',fact_38_enat__ord__code_I4_J)).
% 2.61/1.59 thf(f1677,plain,(
% 2.61/1.59 spl41_1 | spl41_2),
% 2.61/1.59 inference(avatar_split_clause,[status(thm)],[f1193,f1674,f1671])).
% 2.61/1.59 thf(f1193,plain,(
% 2.61/1.59 ( ! [X1 : nat] : (((coindu1365464361scount @ a @ p @ omega) = (extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[Y0 : nat]: (p @ (sdrop @ a @ Y0 @ omega))))))) | ($true = (p @ (sdrop @ a @ (sK42 @ X1) @ (sdrop @ a @ X1 @ omega))))) )),
% 2.61/1.59 inference(not_proxy_clausification,[status(thm)],[f1192])).
% 2.61/1.59 thf(f1192,plain,(
% 2.61/1.59 ( ! [X1 : nat] : (((~ (p @ (sdrop @ a @ (sK42 @ X1) @ (sdrop @ a @ X1 @ omega)))) = $false) | ((coindu1365464361scount @ a @ p @ omega) = (extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[Y0 : nat]: (p @ (sdrop @ a @ Y0 @ omega)))))))) )),
% 2.61/1.59 inference(beta_eta_normalization,[status(thm)],[f1191])).
% 2.61/1.59 thf(f1191,plain,(
% 2.61/1.59 ( ! [X1 : nat] : ((((^[Y0 : nat]: (~ (p @ (sdrop @ a @ Y0 @ (sdrop @ a @ X1 @ omega))))) @ (sK42 @ X1)) = $false) | ((coindu1365464361scount @ a @ p @ omega) = (extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[Y0 : nat]: (p @ (sdrop @ a @ Y0 @ omega)))))))) )),
% 2.61/1.59 inference(sigma_clausification,[status(thm)],[f1190])).
% 2.61/1.59 thf(f1190,plain,(
% 2.61/1.59 ( ! [X1 : nat] : (((coindu1365464361scount @ a @ p @ omega) = (extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[Y0 : nat]: (p @ (sdrop @ a @ Y0 @ omega))))))) | ((!! @ nat @ (^[Y0 : nat]: (~ (p @ (sdrop @ a @ Y0 @ (sdrop @ a @ X1 @ omega)))))) = $false)) )),
% 2.61/1.59 inference(beta_eta_normalization,[status(thm)],[f1189])).
% 2.61/1.59 thf(f1189,plain,(
% 2.61/1.59 ( ! [X1 : nat] : (((coindu1365464361scount @ a @ p @ omega) = (extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[Y0 : nat]: (p @ (sdrop @ a @ Y0 @ omega))))))) | ($false = ((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (~ (p @ (sdrop @ a @ Y1 @ (sdrop @ a @ Y0 @ omega))))))) @ X1))) )),
% 2.61/1.59 inference(pi_clausification,[status(thm)],[f1188])).
% 2.61/1.59 thf(f1188,plain,(
% 2.61/1.59 ($true != (?? @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (~ (p @ (sdrop @ a @ Y1 @ (sdrop @ a @ Y0 @ omega))))))))) | ((coindu1365464361scount @ a @ p @ omega) = (extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[Y0 : nat]: (p @ (sdrop @ a @ Y0 @ omega)))))))),
% 2.61/1.59 inference(beta_eta_normalization,[status(thm)],[f1143])).
% 2.61/1.59 thf(f1143,plain,(
% 2.61/1.59 ((coindu1365464361scount @ a @ p @ omega) = (extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[Y0 : nat]: (p @ (sdrop @ a @ Y0 @ omega))))))) | ($true != ((^[Y0 : stream @ a > $o]: ((^[Y1 : stream @ a]: (?? @ nat @ (^[Y2 : nat]: (Y0 @ (sdrop @ a @ Y2 @ Y1))))))) @ ((^[Y0 : stream @ a > $o]: ((^[Y1 : stream @ a]: (!! @ nat @ (^[Y2 : nat]: (Y0 @ (sdrop @ a @ Y2 @ Y1))))))) @ (^[Y0 : stream @ a]: (~ (p @ Y0)))) @ omega))),
% 2.61/1.59 inference(definition_unfolding,[status(thm)],[f1004,f1123,f1016])).
% 2.61/1.59 thf(f1004,plain,(
% 2.61/1.59 ($true != (linear505997466_on_ev @ a @ (linear1386806755on_alw @ a @ (^[Y0 : stream @ a]: (~ (p @ Y0)))) @ omega)) | ((coindu1365464361scount @ a @ p @ omega) = (extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[Y0 : nat]: (p @ (sdrop @ a @ Y0 @ omega)))))))),
% 2.61/1.59 inference(cnf_transformation,[status(thm)],[f884])).
% 2.61/1.59 thf(f884,plain,(
% 2.61/1.59 ($true != (linear505997466_on_ev @ a @ (linear1386806755on_alw @ a @ (^[Y0 : stream @ a]: (~ (p @ Y0)))) @ omega)) | ((coindu1365464361scount @ a @ p @ omega) = (extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[Y0 : nat]: (p @ (sdrop @ a @ Y0 @ omega)))))))),
% 2.61/1.59 inference(ennf_transformation,[status(thm)],[f616])).
% 2.61/1.59 thf(f616,plain,(
% 2.61/1.59 ($true = (linear505997466_on_ev @ a @ (linear1386806755on_alw @ a @ (^[Y0 : stream @ a]: (~ (p @ Y0)))) @ omega)) => ((coindu1365464361scount @ a @ p @ omega) = (extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[Y0 : nat]: (p @ (sdrop @ a @ Y0 @ omega)))))))),
% 2.61/1.59 inference(fool_elimination,[status(thm)],[f615])).
% 2.61/1.59 thf(f615,plain,(
% 2.61/1.59 (linear505997466_on_ev @ a @ (linear1386806755on_alw @ a @ (^[X0 : stream @ a] : (~(p @ X0)))) @ omega) => ((extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[X1 : nat] : (p @ (sdrop @ a @ X1 @ omega)))))) = (coindu1365464361scount @ a @ p @ omega))),
% 2.61/1.59 inference(rectify,[status(thm)],[f1])).
% 2.61/1.59 thf(f1,axiom,(
% 2.61/1.59 (linear505997466_on_ev @ a @ (linear1386806755on_alw @ a @ (^[X4 : stream @ a] : (~(p @ X4)))) @ omega) => ((extended_enat2 @ (finite_card @ nat @ (collect @ nat @ (^[X5 : nat] : (p @ (sdrop @ a @ X5 @ omega)))))) = (coindu1365464361scount @ a @ p @ omega))),
% 2.61/1.59 file('/export/starexec/sandbox/tmp/tmp.HepLMUEemU/DTF2THF_20348.p',fact_0__092_060open_062ev_A_Ialw_A_I_092_060lambda_062xs_O_A_092_060not_062_AP_Axs_J_J_A_092_060omega_062_A_092_060Longrightarrow_062_Ascount_AP_A_092_060omega_062_A_061_Aenat_A_Icard_A_123i_O_AP_A_Isdrop_Ai_A_092_060omega_062_J_125_J_092_060close_062)).
% 2.61/1.59 % SZS output end Proof for DTF2THF_20348
% 2.61/1.59 % (20569)------------------------------
% 2.61/1.59 % (20569)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.59 % (20569)Termination reason: Refutation
% 2.61/1.59
% 2.61/1.59 % (20569)Memory used [KB]: 7419
% 2.61/1.59 % (20569)Time elapsed: 0.084 s
% 2.61/1.59 % (20569)Instructions burned: 258 (million)
% 2.61/1.59 % (20569)------------------------------
% 2.61/1.59 % (20569)------------------------------
% 2.61/1.59 % (20501)Success in time 0.179 s
%------------------------------------------------------------------------------