%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT133^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n007.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 08:44:53 AM UTC 2026
% Result : Theorem 2.87s 1.95s
% Output : Refutation 2.87s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : DAT133^1 : TPTP v9.2.1. Released v7.0.0.
% 0.13/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n007.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 06:07:38 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.31/0.46 ---- Original DTF file ---
% 0.31/0.46 thf(spec,logic,$$dhol).
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 % File : DAT133^1 : TPTP v9.2.1. Released v7.0.0.
% 0.31/0.46 % Domain : Data Structures
% 0.31/0.46 % Problem : Coinductive list 4159
% 0.31/0.46 % Version : [Bla16] axioms : Especial.
% 0.31/0.46 % English :
% 0.31/0.46
% 0.31/0.46 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.31/0.46 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.31/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.31/0.46 % Source : [Bla16]
% 0.31/0.46 % Names : coinductive_list__4159.p [Bla16]
% 0.31/0.46
% 0.31/0.46 % Status : Theorem
% 0.31/0.46 % Rating : 0.67 v9.1.0, 0.33 v8.1.0, 0.25 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.31/0.46 % Syntax : Number of formulae : 335 ( 110 unt; 54 typ; 0 def)
% 0.31/0.46 % Number of atoms : 699 ( 214 equ; 0 cnn)
% 0.31/0.46 % Maximal formula atoms : 7 ( 2 avg)
% 0.31/0.46 % Number of connectives : 3529 ( 90 ~; 11 |; 54 &;3076 @)
% 0.31/0.46 % ( 0 <=>; 298 =>; 0 <=; 0 <~>)
% 0.31/0.46 % Maximal formula depth : 20 ( 8 avg)
% 0.31/0.46 % Number of types : 4 ( 3 usr)
% 0.31/0.46 % Number of type conns : 214 ( 214 >; 0 *; 0 +; 0 <<)
% 0.31/0.46 % Number of symbols : 53 ( 51 usr; 4 con; 0-4 aty)
% 0.31/0.46 % Number of variables : 987 ( 80 ^; 828 !; 36 ?; 987 :)
% 0.31/0.46 % ( 43 !>; 0 ?*; 0 @-; 0 @+)
% 0.31/0.46 % SPC : TH1_THM_EQU_NAR
% 0.31/0.46
% 0.31/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.31/0.46 % 2016-07-13 15:05:08.605
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 %----Could-be-implicit typings (6)
% 0.31/0.46 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
% 0.31/0.46 coindu1593790203_llist: $tType > $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Extended__Nat_Oenat,type,
% 0.31/0.46 extended_enat: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Set_Oset,type,
% 0.31/0.46 set: $tType > $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Nat_Onat,type,
% 0.31/0.46 nat: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_itself,type,
% 0.31/0.46 itself: $tType > $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_tf_a,type,
% 0.31/0.46 a: $tType ).
% 0.31/0.46
% 0.31/0.46 %----Explicit typings (48)
% 0.31/0.46 thf(sy_cl_HOL_Otype,type,
% 0.31/0.46 type:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Oone,type,
% 0.31/0.46 one:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Num_Oneg__numeral,type,
% 0.31/0.46 neg_numeral:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.31/0.46 group_add:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Lattices_Olattice,type,
% 0.31/0.46 lattice:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.31/0.46 finite_finite:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.31/0.46 semigroup_add:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.31/0.46 linordered_idom:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.31/0.46 ab_semigroup_add:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 0.31/0.46 semilattice_inf:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.31/0.46 linordered_semidom:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.31/0.46 cancel_semigroup_add:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.31/0.46 ordere779506340up_add:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.31/0.46 ordere236663937imp_le:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 0.31/0.46 ordere223160158up_add:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 0.31/0.46 strict2144017051up_add:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__llength,type,
% 0.31/0.46 coindu493225792length:
% 0.31/0.46 !>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldrop,type,
% 0.31/0.46 coindu191418589_ldrop:
% 0.31/0.46 !>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OldropWhile,type,
% 0.31/0.46 coindu438612276pWhile:
% 0.31/0.46 !>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldropn,type,
% 0.31/0.46 coindu531130065ldropn:
% 0.31/0.46 !>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfilter,type,
% 0.31/0.46 coindu1889960678filter:
% 0.31/0.46 !>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite,type,
% 0.31/0.46 coindu1213758845finite:
% 0.31/0.46 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollast,type,
% 0.31/0.46 coindu2000965700_llast:
% 0.31/0.46 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
% 0.31/0.46 coindu1018505716length:
% 0.31/0.46 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olhd,type,
% 0.31/0.46 coindu1046438764le_lhd:
% 0.31/0.46 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
% 0.31/0.46 coindu335574135_lnull:
% 0.31/0.46 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olset,type,
% 0.31/0.46 coindu1112613586e_lset:
% 0.31/0.46 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Oltl,type,
% 0.31/0.46 coindu1047225960le_ltl:
% 0.31/0.46 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olnth,type,
% 0.31/0.46 coindu749330388e_lnth:
% 0.31/0.46 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > nat > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OltakeWhile,type,
% 0.31/0.46 coindu721411036eWhile:
% 0.31/0.46 !>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Coinductive__Nat_Oenat__unfold,type,
% 0.31/0.46 coindu1491768222unfold:
% 0.31/0.46 !>[A: $tType] : ( ( A > $o ) > ( A > A ) > A > extended_enat ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Extended__Nat_Oenat,type,
% 0.31/0.46 extended_enat2: nat > extended_enat ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
% 0.31/0.46 extended_rec_enat:
% 0.31/0.46 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Extended__Nat_Othe__enat,type,
% 0.31/0.46 extended_the_enat: extended_enat > nat ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Finite__Set_Ofinite,type,
% 0.31/0.46 finite_finite2:
% 0.31/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.31/0.46 one_one:
% 0.31/0.46 !>[A: $tType] : A ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.31/0.46 plus_plus:
% 0.31/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.31/0.46 inf_inf:
% 0.31/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.31/0.46 ord_less:
% 0.31/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.31/0.46 ord_less_eq:
% 0.31/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Pure_Otype,type,
% 0.31/0.46 type2:
% 0.31/0.46 !>[A: $tType] : ( itself @ A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Set_OCollect,type,
% 0.31/0.46 collect:
% 0.31/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Set_Oimage,type,
% 0.31/0.46 image:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_member,type,
% 0.31/0.46 member:
% 0.31/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_v_P,type,
% 0.31/0.46 p: a > $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_v_n____,type,
% 0.31/0.46 n: nat ).
% 0.31/0.46
% 0.31/0.46 thf(sy_v_xs,type,
% 0.31/0.46 xs: coindu1593790203_llist @ a ).
% 0.31/0.46
% 0.31/0.46 thf(sy_v_xsa____,type,
% 0.31/0.46 xsa: coindu1593790203_llist @ a ).
% 0.31/0.46
% 0.31/0.46 %----Relevant facts (253)
% 0.31/0.46 thf(fact_0__092_060open_062finite_A_I_123n_O_Aenat_An_A_060_Allength_A_Iltl_A_IldropWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_J_125_A_092_060inter_062_A_123n_O_AP_A_Ilnth_A_Iltl_A_IldropWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_J_An_J_125_J_092_060close_062,axiom,
% 0.31/0.46 ( finite_finite2 @ nat
% 0.31/0.46 @ ( inf_inf @ ( set @ nat )
% 0.31/0.46 @ ( collect @ nat
% 0.31/0.46 @ ^ [N: nat] :
% 0.31/0.46 ( ord_less @ extended_enat @ ( extended_enat2 @ N )
% 0.31/0.46 @ ( coindu1018505716length @ a
% 0.31/0.46 @ ( coindu1047225960le_ltl @ a
% 0.31/0.46 @ ( coindu438612276pWhile @ a
% 0.31/0.46 @ ^ [X: a] :
% 0.31/0.46 ~ ( p @ X )
% 0.31/0.46 @ xsa ) ) ) ) )
% 0.31/0.46 @ ( collect @ nat
% 0.31/0.46 @ ^ [N: nat] :
% 0.31/0.46 ( p
% 0.31/0.46 @ ( coindu749330388e_lnth @ a
% 0.31/0.46 @ ( coindu1047225960le_ltl @ a
% 0.31/0.46 @ ( coindu438612276pWhile @ a
% 0.31/0.46 @ ^ [X: a] :
% 0.31/0.46 ~ ( p @ X )
% 0.31/0.46 @ xsa ) )
% 0.31/0.46 @ N ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % \<open>finite ({n. enat n < llength (ltl (ldropWhile (\<lambda>x. \<not> P x) xs))} \<inter> {n. P (lnth (ltl (ldropWhile (\<lambda>x. \<not> P x) xs)) n)})\<close>
% 0.31/0.46 thf(fact_1_LCons_Oprems,axiom,
% 0.31/0.46 ~ ( coindu1213758845finite @ a @ xsa ) ).
% 0.31/0.46
% 0.31/0.46 % LCons.prems
% 0.31/0.46 thf(fact_2_n,axiom,
% 0.31/0.46 ( ( coindu1018505716length @ a
% 0.31/0.46 @ ( coindu721411036eWhile @ a
% 0.31/0.46 @ ^ [X: a] :
% 0.31/0.46 ~ ( p @ X )
% 0.31/0.46 @ xsa ) )
% 0.31/0.46 = ( extended_enat2 @ n ) ) ).
% 0.31/0.46
% 0.31/0.46 % n
% 0.31/0.46 thf(fact_3_finite__Int,axiom,
% 0.31/0.46 ! [A: $tType,F: set @ A,G: set @ A] :
% 0.31/0.46 ( ( ( finite_finite2 @ A @ F )
% 0.31/0.46 | ( finite_finite2 @ A @ G ) )
% 0.31/0.46 => ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F @ G ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_Int
% 0.31/0.46 thf(fact_4_finite__imageI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,F: set @ A,H: A > B] :
% 0.31/0.46 ( ( finite_finite2 @ A @ F )
% 0.31/0.46 => ( finite_finite2 @ B @ ( image @ A @ B @ H @ F ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_imageI
% 0.31/0.46 thf(fact_5_add__less__cancel__left,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [C: A,A2: A,B2: A] :
% 0.31/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.31/0.46 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_less_cancel_left
% 0.31/0.46 thf(fact_6_add__less__cancel__right,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,C: A,B2: A] :
% 0.31/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.31/0.46 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_less_cancel_right
% 0.31/0.46 thf(fact_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_Allength_A_IltakeWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_A_061_Aenat_An_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.31/0.46 ~ ! [N2: nat] :
% 0.31/0.46 ( ( coindu1018505716length @ a
% 0.31/0.46 @ ( coindu721411036eWhile @ a
% 0.31/0.46 @ ^ [X: a] :
% 0.31/0.46 ~ ( p @ X )
% 0.31/0.46 @ xsa ) )
% 0.31/0.46 != ( extended_enat2 @ N2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % \<open>\<And>thesis. (\<And>n. llength (ltakeWhile (\<lambda>x. \<not> P x) xs) = enat n \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.31/0.46 thf(fact_8_finite__Collect__conjI,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.31/0.46 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
% 0.31/0.46 => ( finite_finite2 @ A
% 0.31/0.46 @ ( collect @ A
% 0.31/0.46 @ ^ [X: A] :
% 0.31/0.46 ( ( P @ X )
% 0.31/0.46 & ( Q @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_Collect_conjI
% 0.31/0.46 thf(fact_9_finite__Collect__disjI,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ( finite_finite2 @ A
% 0.31/0.46 @ ( collect @ A
% 0.31/0.46 @ ^ [X: A] :
% 0.31/0.46 ( ( P @ X )
% 0.31/0.46 | ( Q @ X ) ) ) )
% 0.31/0.46 = ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.31/0.46 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_Collect_disjI
% 0.31/0.46 thf(fact_10_image__ident,axiom,
% 0.31/0.46 ! [A: $tType,Y: set @ A] :
% 0.31/0.46 ( ( image @ A @ A
% 0.31/0.46 @ ^ [X: A] : X
% 0.31/0.46 @ Y )
% 0.31/0.46 = Y ) ).
% 0.31/0.46
% 0.31/0.46 % image_ident
% 0.31/0.46 thf(fact_11_less__add__one,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_add_one
% 0.31/0.46 thf(fact_12_exP,axiom,
% 0.31/0.46 ? [X2: a] :
% 0.31/0.46 ( ( member @ a @ X2 @ ( coindu1112613586e_lset @ a @ xsa ) )
% 0.31/0.46 & ( p @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % exP
% 0.31/0.46 thf(fact_13_enat_Oinject,axiom,
% 0.31/0.46 ! [Nat: nat,Nat2: nat] :
% 0.31/0.46 ( ( ( extended_enat2 @ Nat )
% 0.31/0.46 = ( extended_enat2 @ Nat2 ) )
% 0.31/0.46 = ( Nat = Nat2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % enat.inject
% 0.31/0.46 thf(fact_14_LCons_Ohyps_I3_J,axiom,
% 0.31/0.46 ! [Xs: coindu1593790203_llist @ a] :
% 0.31/0.46 ( ( ( coindu1047225960le_ltl @ a @ ( coindu1889960678filter @ a @ p @ xsa ) )
% 0.31/0.46 = ( coindu1889960678filter @ a @ p @ Xs ) )
% 0.31/0.46 => ( ~ ( coindu1213758845finite @ a @ Xs )
% 0.31/0.46 => ( finite_finite2 @ nat
% 0.31/0.46 @ ( collect @ nat
% 0.31/0.46 @ ^ [N: nat] :
% 0.31/0.46 ( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ a @ Xs ) )
% 0.31/0.46 & ( p @ ( coindu749330388e_lnth @ a @ Xs @ N ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % LCons.hyps(3)
% 0.31/0.46 thf(fact_15__092_060open_062_092_060not_062_Alfinite_Axs_092_060close_062,axiom,
% 0.31/0.46 ~ ( coindu1213758845finite @ a @ xs ) ).
% 0.31/0.46
% 0.31/0.46 % \<open>\<not> lfinite xs\<close>
% 0.31/0.46 thf(fact_16_LCons_Ohyps_I1_J,axiom,
% 0.31/0.46 coindu1213758845finite @ a @ ( coindu1889960678filter @ a @ p @ xsa ) ).
% 0.31/0.46
% 0.31/0.46 % LCons.hyps(1)
% 0.31/0.46 thf(fact_17__092_060open_062lfinite_A_Ilfilter_AP_Axs_J_092_060close_062,axiom,
% 0.31/0.46 coindu1213758845finite @ a @ ( coindu1889960678filter @ a @ p @ xs ) ).
% 0.31/0.46
% 0.31/0.46 % \<open>lfinite (lfilter P xs)\<close>
% 0.31/0.46 thf(fact_18_add__right__cancel,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [B2: A,A2: A,C: A] :
% 0.31/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.31/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.31/0.46 = ( B2 = C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_right_cancel
% 0.31/0.46 thf(fact_19_add__left__cancel,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A,C: A] :
% 0.31/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.31/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.31/0.46 = ( B2 = C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_left_cancel
% 0.31/0.46 thf(fact_20_image__eqI,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,B2: A,F2: B > A,X3: B,A3: set @ B] :
% 0.31/0.46 ( ( B2
% 0.31/0.46 = ( F2 @ X3 ) )
% 0.31/0.46 => ( ( member @ B @ X3 @ A3 )
% 0.31/0.46 => ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % image_eqI
% 0.31/0.46 thf(fact_21_finite__code,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.31/0.46 => ( ( finite_finite2 @ A )
% 0.31/0.46 = ( ^ [A4: set @ A] : $true ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_code
% 0.31/0.46 thf(fact_22_Int__iff,axiom,
% 0.31/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.31/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 0.31/0.46 = ( ( member @ A @ C @ A3 )
% 0.31/0.46 & ( member @ A @ C @ B3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Int_iff
% 0.31/0.46 thf(fact_23_IntI,axiom,
% 0.31/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.31/0.46 ( ( member @ A @ C @ A3 )
% 0.31/0.46 => ( ( member @ A @ C @ B3 )
% 0.31/0.46 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % IntI
% 0.31/0.46 thf(fact_24_lfilter__idem,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu1889960678filter @ A @ P @ ( coindu1889960678filter @ A @ P @ Xs ) )
% 0.31/0.46 = ( coindu1889960678filter @ A @ P @ Xs ) ) ).
% 0.31/0.46
% 0.31/0.46 % lfilter_idem
% 0.31/0.46 thf(fact_25_ltakeWhile__K__True,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu721411036eWhile @ A
% 0.31/0.46 @ ^ [Uu: A] : $true
% 0.31/0.46 @ Xs )
% 0.31/0.46 = Xs ) ).
% 0.31/0.46
% 0.31/0.46 % ltakeWhile_K_True
% 0.31/0.46 thf(fact_26_finite__Collect__less__nat,axiom,
% 0.31/0.46 ! [K: nat] :
% 0.31/0.46 ( finite_finite2 @ nat
% 0.31/0.46 @ ( collect @ nat
% 0.31/0.46 @ ^ [N: nat] : ( ord_less @ nat @ N @ K ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_Collect_less_nat
% 0.31/0.46 thf(fact_27_lfilter__K__True,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu1889960678filter @ A
% 0.31/0.46 @ ^ [Uu: A] : $true
% 0.31/0.46 @ Xs )
% 0.31/0.46 = Xs ) ).
% 0.31/0.46
% 0.31/0.46 % lfilter_K_True
% 0.31/0.46 thf(fact_28_plus__enat__simps_I1_J,axiom,
% 0.31/0.46 ! [M: nat,N3: nat] :
% 0.31/0.46 ( ( plus_plus @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N3 ) )
% 0.31/0.46 = ( extended_enat2 @ ( plus_plus @ nat @ M @ N3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % plus_enat_simps(1)
% 0.31/0.46 thf(fact_29_lfinite__ltl,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu1213758845finite @ A @ ( coindu1047225960le_ltl @ A @ Xs ) )
% 0.31/0.46 = ( coindu1213758845finite @ A @ Xs ) ) ).
% 0.31/0.46
% 0.31/0.46 % lfinite_ltl
% 0.31/0.46 thf(fact_30_enat__ord__simps_I2_J,axiom,
% 0.31/0.46 ! [M: nat,N3: nat] :
% 0.31/0.46 ( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N3 ) )
% 0.31/0.46 = ( ord_less @ nat @ M @ N3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % enat_ord_simps(2)
% 0.31/0.46 thf(fact_31_lset__lfilter,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu1112613586e_lset @ A @ ( coindu1889960678filter @ A @ P @ Xs ) )
% 0.31/0.46 = ( collect @ A
% 0.31/0.46 @ ^ [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
% 0.31/0.46 & ( P @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lset_lfilter
% 0.31/0.46 thf(fact_32_LCons_Ohyps_I2_J,axiom,
% 0.31/0.46 ~ ( coindu335574135_lnull @ a @ ( coindu1889960678filter @ a @ p @ xsa ) ) ).
% 0.31/0.46
% 0.31/0.46 % LCons.hyps(2)
% 0.31/0.46 thf(fact_33_lfilter__cong,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ( Xs = Ys )
% 0.31/0.46 => ( ! [X2: A] :
% 0.31/0.46 ( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Ys ) )
% 0.31/0.46 => ( ( P @ X2 )
% 0.31/0.46 = ( Q @ X2 ) ) )
% 0.31/0.46 => ( ( coindu1889960678filter @ A @ P @ Xs )
% 0.31/0.46 = ( coindu1889960678filter @ A @ Q @ Ys ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lfilter_cong
% 0.31/0.46 thf(fact_34_ltakeWhile__all,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.31/0.46 ( ! [X2: A] :
% 0.31/0.46 ( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 0.31/0.46 => ( P @ X2 ) )
% 0.31/0.46 => ( ( coindu721411036eWhile @ A @ P @ Xs )
% 0.31/0.46 = Xs ) ) ).
% 0.31/0.46
% 0.31/0.46 % ltakeWhile_all
% 0.31/0.46 thf(fact_35_ltakeWhile__cong,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ( Xs = Ys )
% 0.31/0.46 => ( ! [X2: A] :
% 0.31/0.46 ( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Ys ) )
% 0.31/0.46 => ( ( P @ X2 )
% 0.31/0.46 = ( Q @ X2 ) ) )
% 0.31/0.46 => ( ( coindu721411036eWhile @ A @ P @ Xs )
% 0.31/0.46 = ( coindu721411036eWhile @ A @ Q @ Ys ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ltakeWhile_cong
% 0.31/0.46 thf(fact_36_lfinite__lfilterI,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.31/0.46 ( ( coindu1213758845finite @ A @ Xs )
% 0.31/0.46 => ( coindu1213758845finite @ A @ ( coindu1889960678filter @ A @ P @ Xs ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lfinite_lfilterI
% 0.31/0.46 thf(fact_37_lset__ltakeWhileD,axiom,
% 0.31/0.46 ! [A: $tType,X3: A,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
% 0.31/0.46 => ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 0.31/0.46 & ( P @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lset_ltakeWhileD
% 0.31/0.46 thf(fact_38_lfinite__ltakeWhile,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu1213758845finite @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 0.31/0.46 = ( ( coindu1213758845finite @ A @ Xs )
% 0.31/0.46 | ? [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
% 0.31/0.46 & ~ ( P @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lfinite_ltakeWhile
% 0.31/0.46 thf(fact_39_lfinite__imp__finite__lset,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu1213758845finite @ A @ Xs )
% 0.31/0.46 => ( finite_finite2 @ A @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lfinite_imp_finite_lset
% 0.31/0.46 thf(fact_40_lfinite__ldropWhile,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu1213758845finite @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) )
% 0.31/0.46 = ( ? [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
% 0.31/0.46 & ~ ( P @ X ) )
% 0.31/0.46 => ( coindu1213758845finite @ A @ Xs ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lfinite_ldropWhile
% 0.31/0.46 thf(fact_41_finite__psubset__induct,axiom,
% 0.31/0.46 ! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
% 0.31/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.31/0.46 => ( ! [A5: set @ A] :
% 0.31/0.46 ( ( finite_finite2 @ A @ A5 )
% 0.31/0.46 => ( ! [B4: set @ A] :
% 0.31/0.46 ( ( ord_less @ ( set @ A ) @ B4 @ A5 )
% 0.31/0.46 => ( P @ B4 ) )
% 0.31/0.46 => ( P @ A5 ) ) )
% 0.31/0.46 => ( P @ A3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_psubset_induct
% 0.31/0.46 thf(fact_42_lfilter__lfilter,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Q: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu1889960678filter @ A @ P @ ( coindu1889960678filter @ A @ Q @ Xs ) )
% 0.31/0.46 = ( coindu1889960678filter @ A
% 0.31/0.46 @ ^ [X: A] :
% 0.31/0.46 ( ( P @ X )
% 0.31/0.46 & ( Q @ X ) )
% 0.31/0.46 @ Xs ) ) ).
% 0.31/0.46
% 0.31/0.46 % lfilter_lfilter
% 0.31/0.46 thf(fact_43_llength__ltakeWhile__lt__iff,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( ord_less @ extended_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) @ ( coindu1018505716length @ A @ Xs ) )
% 0.31/0.46 = ( ? [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
% 0.31/0.46 & ~ ( P @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % llength_ltakeWhile_lt_iff
% 0.31/0.46 thf(fact_44_mem__Collect__eq,axiom,
% 0.31/0.46 ! [A: $tType,A2: A,P: A > $o] :
% 0.31/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.31/0.46 = ( P @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % mem_Collect_eq
% 0.31/0.46 thf(fact_45_Collect__mem__eq,axiom,
% 0.31/0.46 ! [A: $tType,A3: set @ A] :
% 0.31/0.46 ( ( collect @ A
% 0.31/0.46 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 0.31/0.46 = A3 ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mem_eq
% 0.31/0.46 thf(fact_46_Collect__cong,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ! [X2: A] :
% 0.31/0.46 ( ( P @ X2 )
% 0.31/0.46 = ( Q @ X2 ) )
% 0.31/0.46 => ( ( collect @ A @ P )
% 0.31/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_cong
% 0.31/0.46 thf(fact_47_ext,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 0.31/0.46 ( ! [X2: A] :
% 0.31/0.46 ( ( F2 @ X2 )
% 0.31/0.46 = ( G2 @ X2 ) )
% 0.31/0.46 => ( F2 = G2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % ext
% 0.31/0.46 thf(fact_48_in__lset__ltlD,axiom,
% 0.31/0.46 ! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) )
% 0.31/0.46 => ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % in_lset_ltlD
% 0.31/0.46 thf(fact_49_llength__ltakeWhile__all,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 0.31/0.46 = ( coindu1018505716length @ A @ Xs ) )
% 0.31/0.46 = ( ( coindu721411036eWhile @ A @ P @ Xs )
% 0.31/0.46 = Xs ) ) ).
% 0.31/0.46
% 0.31/0.46 % llength_ltakeWhile_all
% 0.31/0.46 thf(fact_50_in__lset__ldropWhileD,axiom,
% 0.31/0.46 ! [A: $tType,X3: A,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) ) )
% 0.31/0.46 => ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % in_lset_ldropWhileD
% 0.31/0.46 thf(fact_51_ldropWhile__cong,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ( Xs = Ys )
% 0.31/0.46 => ( ! [X2: A] :
% 0.31/0.46 ( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Ys ) )
% 0.31/0.46 => ( ( P @ X2 )
% 0.31/0.46 = ( Q @ X2 ) ) )
% 0.31/0.46 => ( ( coindu438612276pWhile @ A @ P @ Xs )
% 0.31/0.46 = ( coindu438612276pWhile @ A @ Q @ Ys ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ldropWhile_cong
% 0.31/0.46 thf(fact_52_one__enat__def,axiom,
% 0.31/0.46 ( ( one_one @ extended_enat )
% 0.31/0.46 = ( extended_enat2 @ ( one_one @ nat ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % one_enat_def
% 0.31/0.46 thf(fact_53_enat__1__iff_I1_J,axiom,
% 0.31/0.46 ! [X3: nat] :
% 0.31/0.46 ( ( ( extended_enat2 @ X3 )
% 0.31/0.46 = ( one_one @ extended_enat ) )
% 0.31/0.46 = ( X3
% 0.31/0.46 = ( one_one @ nat ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % enat_1_iff(1)
% 0.31/0.46 thf(fact_54_enat__1__iff_I2_J,axiom,
% 0.31/0.46 ! [X3: nat] :
% 0.31/0.46 ( ( ( one_one @ extended_enat )
% 0.31/0.46 = ( extended_enat2 @ X3 ) )
% 0.31/0.46 = ( X3
% 0.31/0.46 = ( one_one @ nat ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % enat_1_iff(2)
% 0.31/0.46 thf(fact_55_less__enatE,axiom,
% 0.31/0.46 ! [N3: extended_enat,M: nat] :
% 0.31/0.46 ( ( ord_less @ extended_enat @ N3 @ ( extended_enat2 @ M ) )
% 0.31/0.46 => ~ ! [K2: nat] :
% 0.31/0.46 ( ( N3
% 0.31/0.46 = ( extended_enat2 @ K2 ) )
% 0.31/0.46 => ~ ( ord_less @ nat @ K2 @ M ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_enatE
% 0.31/0.46 thf(fact_56_finite__conv__nat__seg__image,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( finite_finite2 @ A )
% 0.31/0.46 = ( ^ [A4: set @ A] :
% 0.31/0.46 ? [N: nat,F3: nat > A] :
% 0.31/0.46 ( A4
% 0.31/0.46 = ( image @ nat @ A @ F3
% 0.31/0.46 @ ( collect @ nat
% 0.31/0.46 @ ^ [I: nat] : ( ord_less @ nat @ I @ N ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_conv_nat_seg_image
% 0.31/0.46 thf(fact_57_nat__seg__image__imp__finite,axiom,
% 0.31/0.46 ! [A: $tType,A3: set @ A,F2: nat > A,N3: nat] :
% 0.31/0.46 ( ( A3
% 0.31/0.46 = ( image @ nat @ A @ F2
% 0.31/0.46 @ ( collect @ nat
% 0.31/0.46 @ ^ [I: nat] : ( ord_less @ nat @ I @ N3 ) ) ) )
% 0.31/0.46 => ( finite_finite2 @ A @ A3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat_seg_image_imp_finite
% 0.31/0.46 thf(fact_58_lfinite__conv__llength__enat,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( coindu1213758845finite @ A )
% 0.31/0.46 = ( ^ [Xs2: coindu1593790203_llist @ A] :
% 0.31/0.46 ? [N: nat] :
% 0.31/0.46 ( ( coindu1018505716length @ A @ Xs2 )
% 0.31/0.46 = ( extended_enat2 @ N ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lfinite_conv_llength_enat
% 0.31/0.46 thf(fact_59_llength__eq__enat__lfiniteD,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A,N3: nat] :
% 0.31/0.46 ( ( ( coindu1018505716length @ A @ Xs )
% 0.31/0.46 = ( extended_enat2 @ N3 ) )
% 0.31/0.46 => ( coindu1213758845finite @ A @ Xs ) ) ).
% 0.31/0.46
% 0.31/0.46 % llength_eq_enat_lfiniteD
% 0.31/0.46 thf(fact_60_lfinite__llength__enat,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu1213758845finite @ A @ Xs )
% 0.31/0.46 => ? [N2: nat] :
% 0.31/0.46 ( ( coindu1018505716length @ A @ Xs )
% 0.31/0.46 = ( extended_enat2 @ N2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lfinite_llength_enat
% 0.31/0.46 thf(fact_61_linorder__neqE__linordered__idom,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X3: A,Y2: A] :
% 0.31/0.46 ( ( X3 != Y2 )
% 0.31/0.46 => ( ~ ( ord_less @ A @ X3 @ Y2 )
% 0.31/0.46 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_neqE_linordered_idom
% 0.31/0.46 thf(fact_62_add__right__imp__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [B2: A,A2: A,C: A] :
% 0.31/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.31/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.31/0.46 => ( B2 = C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_right_imp_eq
% 0.31/0.46 thf(fact_63_add__left__imp__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A,C: A] :
% 0.31/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.31/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.31/0.46 => ( B2 = C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_left_imp_eq
% 0.31/0.46 thf(fact_64_add_Oleft__commute,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [B2: A,A2: A,C: A] :
% 0.31/0.46 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
% 0.31/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add.left_commute
% 0.31/0.46 thf(fact_65_add_Ocommute,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ( ( plus_plus @ A )
% 0.31/0.46 = ( ^ [A6: A,B5: A] : ( plus_plus @ A @ B5 @ A6 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add.commute
% 0.31/0.46 thf(fact_66_add_Oright__cancel,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( group_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [B2: A,A2: A,C: A] :
% 0.31/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.31/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.31/0.46 = ( B2 = C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add.right_cancel
% 0.31/0.46 thf(fact_67_add_Oleft__cancel,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( group_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A,C: A] :
% 0.31/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.31/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.31/0.46 = ( B2 = C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add.left_cancel
% 0.31/0.46 thf(fact_68_add_Oassoc,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semigroup_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A,C: A] :
% 0.31/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.31/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add.assoc
% 0.31/0.46 thf(fact_69_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [I2: A,J: A,K: A,L: A] :
% 0.31/0.46 ( ( ( I2 = J )
% 0.31/0.46 & ( K = L ) )
% 0.31/0.46 => ( ( plus_plus @ A @ I2 @ K )
% 0.31/0.46 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_mono_thms_linordered_semiring(4)
% 0.31/0.46 thf(fact_70_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A,C: A] :
% 0.31/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.31/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.31/0.46 thf(fact_71_one__reorient,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( one @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X3: A] :
% 0.31/0.46 ( ( ( one_one @ A )
% 0.31/0.46 = X3 )
% 0.31/0.46 = ( X3
% 0.31/0.46 = ( one_one @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % one_reorient
% 0.31/0.46 thf(fact_72_rev__image__eqI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,X3: A,A3: set @ A,B2: B,F2: A > B] :
% 0.31/0.46 ( ( member @ A @ X3 @ A3 )
% 0.31/0.46 => ( ( B2
% 0.31/0.46 = ( F2 @ X3 ) )
% 0.31/0.46 => ( member @ B @ B2 @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rev_image_eqI
% 0.31/0.46 thf(fact_73_ball__imageD,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
% 0.31/0.46 ( ! [X2: A] :
% 0.31/0.46 ( ( member @ A @ X2 @ ( image @ B @ A @ F2 @ A3 ) )
% 0.31/0.46 => ( P @ X2 ) )
% 0.31/0.46 => ! [X4: B] :
% 0.31/0.46 ( ( member @ B @ X4 @ A3 )
% 0.31/0.46 => ( P @ ( F2 @ X4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ball_imageD
% 0.31/0.46 thf(fact_74_image__cong,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,M2: set @ A,N4: set @ A,F2: A > B,G2: A > B] :
% 0.31/0.46 ( ( M2 = N4 )
% 0.31/0.46 => ( ! [X2: A] :
% 0.31/0.46 ( ( member @ A @ X2 @ N4 )
% 0.31/0.46 => ( ( F2 @ X2 )
% 0.31/0.46 = ( G2 @ X2 ) ) )
% 0.31/0.46 => ( ( image @ A @ B @ F2 @ M2 )
% 0.31/0.46 = ( image @ A @ B @ G2 @ N4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % image_cong
% 0.31/0.46 thf(fact_75_bex__imageD,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
% 0.31/0.46 ( ? [X4: A] :
% 0.31/0.46 ( ( member @ A @ X4 @ ( image @ B @ A @ F2 @ A3 ) )
% 0.31/0.46 & ( P @ X4 ) )
% 0.31/0.46 => ? [X2: B] :
% 0.31/0.46 ( ( member @ B @ X2 @ A3 )
% 0.31/0.46 & ( P @ ( F2 @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bex_imageD
% 0.31/0.46 thf(fact_76_image__iff,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,Z: A,F2: B > A,A3: set @ B] :
% 0.31/0.46 ( ( member @ A @ Z @ ( image @ B @ A @ F2 @ A3 ) )
% 0.31/0.46 = ( ? [X: B] :
% 0.31/0.46 ( ( member @ B @ X @ A3 )
% 0.31/0.46 & ( Z
% 0.31/0.46 = ( F2 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % image_iff
% 0.31/0.46 thf(fact_77_imageI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,X3: A,A3: set @ A,F2: A > B] :
% 0.31/0.46 ( ( member @ A @ X3 @ A3 )
% 0.31/0.46 => ( member @ B @ ( F2 @ X3 ) @ ( image @ A @ B @ F2 @ A3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % imageI
% 0.31/0.46 thf(fact_78_finite,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite
% 0.31/0.46 thf(fact_79_finite__set__choice,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 0.31/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.31/0.46 => ( ! [X2: A] :
% 0.31/0.46 ( ( member @ A @ X2 @ A3 )
% 0.31/0.46 => ? [X1: B] : ( P @ X2 @ X1 ) )
% 0.31/0.46 => ? [F4: A > B] :
% 0.31/0.46 ! [X4: A] :
% 0.31/0.46 ( ( member @ A @ X4 @ A3 )
% 0.31/0.46 => ( P @ X4 @ ( F4 @ X4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_set_choice
% 0.31/0.46 thf(fact_80_Int__left__commute,axiom,
% 0.31/0.46 ! [A: $tType,A3: set @ A,B3: set @ A,C2: set @ A] :
% 0.31/0.46 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) )
% 0.31/0.46 = ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Int_left_commute
% 0.31/0.46 thf(fact_81_Int__left__absorb,axiom,
% 0.31/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.31/0.46 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 0.31/0.46 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % Int_left_absorb
% 0.31/0.46 thf(fact_82_Int__commute,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( inf_inf @ ( set @ A ) )
% 0.31/0.46 = ( ^ [A4: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Int_commute
% 0.31/0.46 thf(fact_83_Int__absorb,axiom,
% 0.31/0.46 ! [A: $tType,A3: set @ A] :
% 0.31/0.46 ( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
% 0.31/0.46 = A3 ) ).
% 0.31/0.46
% 0.31/0.46 % Int_absorb
% 0.31/0.46 thf(fact_84_Int__assoc,axiom,
% 0.31/0.46 ! [A: $tType,A3: set @ A,B3: set @ A,C2: set @ A] :
% 0.31/0.46 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C2 )
% 0.31/0.46 = ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Int_assoc
% 0.31/0.46 thf(fact_85_IntD2,axiom,
% 0.31/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.31/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 0.31/0.46 => ( member @ A @ C @ B3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % IntD2
% 0.31/0.46 thf(fact_86_IntD1,axiom,
% 0.31/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.31/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 0.31/0.46 => ( member @ A @ C @ A3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % IntD1
% 0.31/0.46 thf(fact_87_IntE,axiom,
% 0.31/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.31/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 0.31/0.46 => ~ ( ( member @ A @ C @ A3 )
% 0.31/0.46 => ~ ( member @ A @ C @ B3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % IntE
% 0.31/0.46 thf(fact_88_enat__less__induct,axiom,
% 0.31/0.46 ! [P: extended_enat > $o,N3: extended_enat] :
% 0.31/0.46 ( ! [N2: extended_enat] :
% 0.31/0.46 ( ! [M3: extended_enat] :
% 0.31/0.46 ( ( ord_less @ extended_enat @ M3 @ N2 )
% 0.31/0.46 => ( P @ M3 ) )
% 0.31/0.46 => ( P @ N2 ) )
% 0.31/0.46 => ( P @ N3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % enat_less_induct
% 0.31/0.46 thf(fact_89_in__lset__conv__lnth,axiom,
% 0.31/0.46 ! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 0.31/0.46 = ( ? [N: nat] :
% 0.31/0.46 ( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs ) )
% 0.31/0.46 & ( ( coindu749330388e_lnth @ A @ Xs @ N )
% 0.31/0.46 = X3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % in_lset_conv_lnth
% 0.31/0.46 thf(fact_90_ltakeWhile__nth,axiom,
% 0.31/0.46 ! [A: $tType,I2: nat,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( ord_less @ extended_enat @ ( extended_enat2 @ I2 ) @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
% 0.31/0.46 => ( ( coindu749330388e_lnth @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) @ I2 )
% 0.31/0.46 = ( coindu749330388e_lnth @ A @ Xs @ I2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ltakeWhile_nth
% 0.31/0.46 thf(fact_91_lfinite__finite__index,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu1213758845finite @ A @ Xs )
% 0.31/0.46 => ( finite_finite2 @ nat
% 0.31/0.46 @ ( collect @ nat
% 0.31/0.46 @ ^ [N: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lfinite_finite_index
% 0.31/0.46 thf(fact_92_Compr__image__eq,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
% 0.31/0.46 ( ( collect @ A
% 0.31/0.46 @ ^ [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ ( image @ B @ A @ F2 @ A3 ) )
% 0.31/0.46 & ( P @ X ) ) )
% 0.31/0.46 = ( image @ B @ A @ F2
% 0.31/0.46 @ ( collect @ B
% 0.31/0.46 @ ^ [X: B] :
% 0.31/0.46 ( ( member @ B @ X @ A3 )
% 0.31/0.46 & ( P @ ( F2 @ X ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Compr_image_eq
% 0.31/0.46 thf(fact_93_image__image,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,C3: $tType,F2: B > A,G2: C3 > B,A3: set @ C3] :
% 0.31/0.46 ( ( image @ B @ A @ F2 @ ( image @ C3 @ B @ G2 @ A3 ) )
% 0.31/0.46 = ( image @ C3 @ A
% 0.31/0.46 @ ^ [X: C3] : ( F2 @ ( G2 @ X ) )
% 0.31/0.46 @ A3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % image_image
% 0.31/0.46 thf(fact_94_imageE,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,B2: A,F2: B > A,A3: set @ B] :
% 0.31/0.46 ( ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) )
% 0.31/0.46 => ~ ! [X2: B] :
% 0.31/0.46 ( ( B2
% 0.31/0.46 = ( F2 @ X2 ) )
% 0.31/0.46 => ~ ( member @ B @ X2 @ A3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % imageE
% 0.31/0.46 thf(fact_95_pigeonhole__infinite__rel,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R: A > B > $o] :
% 0.31/0.46 ( ~ ( finite_finite2 @ A @ A3 )
% 0.31/0.46 => ( ( finite_finite2 @ B @ B3 )
% 0.31/0.46 => ( ! [X2: A] :
% 0.31/0.46 ( ( member @ A @ X2 @ A3 )
% 0.31/0.46 => ? [Xa: B] :
% 0.31/0.46 ( ( member @ B @ Xa @ B3 )
% 0.31/0.46 & ( R @ X2 @ Xa ) ) )
% 0.31/0.46 => ? [X2: B] :
% 0.31/0.46 ( ( member @ B @ X2 @ B3 )
% 0.31/0.46 & ~ ( finite_finite2 @ A
% 0.31/0.46 @ ( collect @ A
% 0.31/0.46 @ ^ [A6: A] :
% 0.31/0.46 ( ( member @ A @ A6 @ A3 )
% 0.31/0.46 & ( R @ A6 @ X2 ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pigeonhole_infinite_rel
% 0.31/0.46 thf(fact_96_not__finite__existsD,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o] :
% 0.31/0.46 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.31/0.46 => ? [X12: A] : ( P @ X12 ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_finite_existsD
% 0.31/0.46 thf(fact_97_Collect__conj__eq,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ( collect @ A
% 0.31/0.46 @ ^ [X: A] :
% 0.31/0.46 ( ( P @ X )
% 0.31/0.46 & ( Q @ X ) ) )
% 0.31/0.46 = ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_conj_eq
% 0.31/0.46 thf(fact_98_Int__Collect,axiom,
% 0.31/0.46 ! [A: $tType,X3: A,A3: set @ A,P: A > $o] :
% 0.31/0.46 ( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) )
% 0.31/0.46 = ( ( member @ A @ X3 @ A3 )
% 0.31/0.46 & ( P @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Int_Collect
% 0.31/0.46 thf(fact_99_Int__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( inf_inf @ ( set @ A ) )
% 0.31/0.46 = ( ^ [A4: set @ A,B6: set @ A] :
% 0.31/0.46 ( collect @ A
% 0.31/0.46 @ ^ [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ A4 )
% 0.31/0.46 & ( member @ A @ X @ B6 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Int_def
% 0.31/0.46 thf(fact_100_add__less__imp__less__right,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,C: A,B2: A] :
% 0.31/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.31/0.46 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_less_imp_less_right
% 0.31/0.46 thf(fact_101_add__less__imp__less__left,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [C: A,A2: A,B2: A] :
% 0.31/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.31/0.46 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_less_imp_less_left
% 0.31/0.46 thf(fact_102_add__strict__right__mono,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A,C: A] :
% 0.31/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_strict_right_mono
% 0.31/0.46 thf(fact_103_add__strict__left__mono,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A,C: A] :
% 0.31/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.46 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_strict_left_mono
% 0.31/0.46 thf(fact_104_add__strict__mono,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( strict2144017051up_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A,C: A,D: A] :
% 0.31/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less @ A @ C @ D )
% 0.31/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_strict_mono
% 0.31/0.46 thf(fact_105_add__mono__thms__linordered__field_I1_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [I2: A,J: A,K: A,L: A] :
% 0.31/0.46 ( ( ( ord_less @ A @ I2 @ J )
% 0.31/0.46 & ( K = L ) )
% 0.31/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_mono_thms_linordered_field(1)
% 0.31/0.46 thf(fact_106_add__mono__thms__linordered__field_I2_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [I2: A,J: A,K: A,L: A] :
% 0.31/0.46 ( ( ( I2 = J )
% 0.31/0.46 & ( ord_less @ A @ K @ L ) )
% 0.31/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_mono_thms_linordered_field(2)
% 0.31/0.46 thf(fact_107_add__mono__thms__linordered__field_I5_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [I2: A,J: A,K: A,L: A] :
% 0.31/0.46 ( ( ( ord_less @ A @ I2 @ J )
% 0.31/0.46 & ( ord_less @ A @ K @ L ) )
% 0.31/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_mono_thms_linordered_field(5)
% 0.31/0.46 thf(fact_108_enat__iless,axiom,
% 0.31/0.46 ! [N3: extended_enat,M: nat] :
% 0.31/0.46 ( ( ord_less @ extended_enat @ N3 @ ( extended_enat2 @ M ) )
% 0.31/0.46 => ? [K2: nat] :
% 0.31/0.46 ( N3
% 0.31/0.46 = ( extended_enat2 @ K2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % enat_iless
% 0.31/0.46 thf(fact_109_chain__incr,axiom,
% 0.31/0.46 ! [A: $tType,Y: A > extended_enat,K: nat] :
% 0.31/0.46 ( ! [I3: A] :
% 0.31/0.46 ? [J2: A] : ( ord_less @ extended_enat @ ( Y @ I3 ) @ ( Y @ J2 ) )
% 0.31/0.46 => ? [J3: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K ) @ ( Y @ J3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % chain_incr
% 0.31/0.46 thf(fact_110_pigeonhole__infinite,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
% 0.31/0.46 ( ~ ( finite_finite2 @ A @ A3 )
% 0.31/0.46 => ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A3 ) )
% 0.31/0.46 => ? [X2: A] :
% 0.31/0.46 ( ( member @ A @ X2 @ A3 )
% 0.31/0.46 & ~ ( finite_finite2 @ A
% 0.31/0.46 @ ( collect @ A
% 0.31/0.46 @ ^ [A6: A] :
% 0.31/0.46 ( ( member @ A @ A6 @ A3 )
% 0.31/0.46 & ( ( F2 @ A6 )
% 0.31/0.46 = ( F2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pigeonhole_infinite
% 0.31/0.46 thf(fact_111_enat__add__mono,axiom,
% 0.31/0.46 ! [X3: nat,Y2: extended_enat,Z: extended_enat] :
% 0.31/0.46 ( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Y2 ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Z ) )
% 0.31/0.46 = ( ord_less @ extended_enat @ Y2 @ Z ) ) ).
% 0.31/0.46
% 0.31/0.46 % enat_add_mono
% 0.31/0.46 thf(fact_112_enat__less__enat__plusI2,axiom,
% 0.31/0.46 ! [Y2: nat,Z: extended_enat,X3: nat] :
% 0.31/0.46 ( ( ord_less @ extended_enat @ ( extended_enat2 @ Y2 ) @ Z )
% 0.31/0.46 => ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ X3 @ Y2 ) ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Z ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % enat_less_enat_plusI2
% 0.31/0.46 thf(fact_113_enat__add2__eq,axiom,
% 0.31/0.46 ! [Y2: extended_enat,X3: nat,Z: extended_enat] :
% 0.31/0.46 ( ( ( plus_plus @ extended_enat @ Y2 @ ( extended_enat2 @ X3 ) )
% 0.31/0.46 = ( plus_plus @ extended_enat @ Z @ ( extended_enat2 @ X3 ) ) )
% 0.31/0.46 = ( Y2 = Z ) ) ).
% 0.31/0.46
% 0.31/0.46 % enat_add2_eq
% 0.31/0.46 thf(fact_114_enat__add1__eq,axiom,
% 0.31/0.46 ! [X3: nat,Y2: extended_enat,Z: extended_enat] :
% 0.31/0.46 ( ( ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Y2 )
% 0.31/0.46 = ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Z ) )
% 0.31/0.46 = ( Y2 = Z ) ) ).
% 0.31/0.46
% 0.31/0.46 % enat_add1_eq
% 0.31/0.46 thf(fact_115_nat__add__left__cancel__less,axiom,
% 0.31/0.46 ! [K: nat,M: nat,N3: nat] :
% 0.31/0.46 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N3 ) )
% 0.31/0.46 = ( ord_less @ nat @ M @ N3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat_add_left_cancel_less
% 0.31/0.46 thf(fact_116_inf__right__idem,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X3: A,Y2: A] :
% 0.31/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Y2 )
% 0.31/0.46 = ( inf_inf @ A @ X3 @ Y2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf_right_idem
% 0.31/0.46 thf(fact_117_inf_Oright__idem,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A] :
% 0.31/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
% 0.31/0.46 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf.right_idem
% 0.31/0.46 thf(fact_118_inf__left__idem,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X3: A,Y2: A] :
% 0.31/0.46 ( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ X3 @ Y2 ) )
% 0.31/0.46 = ( inf_inf @ A @ X3 @ Y2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf_left_idem
% 0.31/0.46 thf(fact_119_inf_Oleft__idem,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A] :
% 0.31/0.46 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
% 0.31/0.46 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf.left_idem
% 0.31/0.46 thf(fact_120_inf__idem,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X3: A] :
% 0.31/0.46 ( ( inf_inf @ A @ X3 @ X3 )
% 0.31/0.46 = X3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf_idem
% 0.31/0.46 thf(fact_121_inf__apply,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 0.31/0.46 => ( ( inf_inf @ ( A > B ) )
% 0.31/0.46 = ( ^ [F3: A > B,G3: A > B,X: A] : ( inf_inf @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf_apply
% 0.31/0.46 thf(fact_122_inf_Oidem,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A] :
% 0.31/0.46 ( ( inf_inf @ A @ A2 @ A2 )
% 0.31/0.46 = A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf.idem
% 0.31/0.46 thf(fact_123_lnull__lfilter,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu335574135_lnull @ A @ ( coindu1889960678filter @ A @ P @ Xs ) )
% 0.31/0.46 = ( ! [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
% 0.31/0.46 => ~ ( P @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lnull_lfilter
% 0.31/0.46 thf(fact_124_lnull__ldropWhile,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu335574135_lnull @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) )
% 0.31/0.46 = ( ! [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
% 0.31/0.46 => ( P @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lnull_ldropWhile
% 0.31/0.46 thf(fact_125_psubsetD,axiom,
% 0.31/0.46 ! [A: $tType,A3: set @ A,B3: set @ A,C: A] :
% 0.31/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.46 => ( ( member @ A @ C @ A3 )
% 0.31/0.46 => ( member @ A @ C @ B3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % psubsetD
% 0.31/0.46 thf(fact_126_nat__neq__iff,axiom,
% 0.31/0.46 ! [M: nat,N3: nat] :
% 0.31/0.46 ( ( M != N3 )
% 0.31/0.46 = ( ( ord_less @ nat @ M @ N3 )
% 0.31/0.46 | ( ord_less @ nat @ N3 @ M ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat_neq_iff
% 0.31/0.46 thf(fact_127_less__set__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord_less @ ( set @ A ) )
% 0.31/0.46 = ( ^ [A4: set @ A,B6: set @ A] :
% 0.31/0.46 ( ord_less @ ( A > $o )
% 0.31/0.46 @ ^ [X: A] : ( member @ A @ X @ A4 )
% 0.31/0.46 @ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_set_def
% 0.31/0.46 thf(fact_128_less__not__refl,axiom,
% 0.31/0.46 ! [N3: nat] :
% 0.31/0.46 ~ ( ord_less @ nat @ N3 @ N3 ) ).
% 0.31/0.46
% 0.31/0.46 % less_not_refl
% 0.31/0.46 thf(fact_129_psubset__trans,axiom,
% 0.31/0.46 ! [A: $tType,A3: set @ A,B3: set @ A,C2: set @ A] :
% 0.31/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.46 => ( ( ord_less @ ( set @ A ) @ B3 @ C2 )
% 0.31/0.46 => ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % psubset_trans
% 0.31/0.46 thf(fact_130_less__not__refl2,axiom,
% 0.31/0.46 ! [N3: nat,M: nat] :
% 0.31/0.46 ( ( ord_less @ nat @ N3 @ M )
% 0.31/0.46 => ( M != N3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_not_refl2
% 0.31/0.46 thf(fact_131_less__not__refl3,axiom,
% 0.31/0.46 ! [S: nat,T2: nat] :
% 0.31/0.46 ( ( ord_less @ nat @ S @ T2 )
% 0.31/0.46 => ( S != T2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_not_refl3
% 0.31/0.46 thf(fact_132_measure__induct,axiom,
% 0.31/0.46 ! [A: $tType,F2: A > nat,P: A > $o,A2: A] :
% 0.31/0.46 ( ! [X2: A] :
% 0.31/0.46 ( ! [Y3: A] :
% 0.31/0.46 ( ( ord_less @ nat @ ( F2 @ Y3 ) @ ( F2 @ X2 ) )
% 0.31/0.46 => ( P @ Y3 ) )
% 0.31/0.46 => ( P @ X2 ) )
% 0.31/0.46 => ( P @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % measure_induct
% 0.31/0.46 thf(fact_133_less__irrefl__nat,axiom,
% 0.31/0.46 ! [N3: nat] :
% 0.31/0.46 ~ ( ord_less @ nat @ N3 @ N3 ) ).
% 0.31/0.46
% 0.31/0.46 % less_irrefl_nat
% 0.31/0.46 thf(fact_134_nat__less__induct,axiom,
% 0.31/0.46 ! [P: nat > $o,N3: nat] :
% 0.31/0.46 ( ! [N2: nat] :
% 0.31/0.46 ( ! [M3: nat] :
% 0.31/0.46 ( ( ord_less @ nat @ M3 @ N2 )
% 0.31/0.46 => ( P @ M3 ) )
% 0.31/0.46 => ( P @ N2 ) )
% 0.31/0.46 => ( P @ N3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat_less_induct
% 0.31/0.46 thf(fact_135_infinite__descent,axiom,
% 0.31/0.46 ! [P: nat > $o,N3: nat] :
% 0.31/0.46 ( ! [N2: nat] :
% 0.31/0.46 ( ~ ( P @ N2 )
% 0.31/0.46 => ? [M3: nat] :
% 0.31/0.46 ( ( ord_less @ nat @ M3 @ N2 )
% 0.31/0.46 & ~ ( P @ M3 ) ) )
% 0.31/0.46 => ( P @ N3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % infinite_descent
% 0.31/0.46 thf(fact_136_linorder__neqE__nat,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ( X3 != Y2 )
% 0.31/0.46 => ( ~ ( ord_less @ nat @ X3 @ Y2 )
% 0.31/0.46 => ( ord_less @ nat @ Y2 @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_neqE_nat
% 0.31/0.46 thf(fact_137_measure__induct__rule,axiom,
% 0.31/0.46 ! [A: $tType,F2: A > nat,P: A > $o,A2: A] :
% 0.31/0.46 ( ! [X2: A] :
% 0.31/0.46 ( ! [Y3: A] :
% 0.31/0.46 ( ( ord_less @ nat @ ( F2 @ Y3 ) @ ( F2 @ X2 ) )
% 0.31/0.46 => ( P @ Y3 ) )
% 0.31/0.46 => ( P @ X2 ) )
% 0.31/0.46 => ( P @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % measure_induct_rule
% 0.31/0.46 thf(fact_138_infinite__descent__measure,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,V: A > nat,X3: A] :
% 0.31/0.46 ( ! [X2: A] :
% 0.31/0.46 ( ~ ( P @ X2 )
% 0.31/0.46 => ? [Y3: A] :
% 0.31/0.46 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
% 0.31/0.46 & ~ ( P @ Y3 ) ) )
% 0.31/0.46 => ( P @ X3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % infinite_descent_measure
% 0.31/0.46 thf(fact_139_lappend_Oexhaust,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.46 => ~ ( coindu335574135_lnull @ A @ Ys ) )
% 0.31/0.46 => ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.46 | ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lappend.exhaust
% 0.31/0.46 thf(fact_140_lzip_Oexhaust,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
% 0.31/0.46 ( ~ ( ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.46 | ( coindu335574135_lnull @ B @ Ys ) )
% 0.31/0.46 => ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.46 => ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lzip.exhaust
% 0.31/0.46 thf(fact_141_lnull__ltlI,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.46 => ( coindu335574135_lnull @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lnull_ltlI
% 0.31/0.46 thf(fact_142_lnull__imp__lfinite,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.46 => ( coindu1213758845finite @ A @ Xs ) ) ).
% 0.31/0.46
% 0.31/0.46 % lnull_imp_lfinite
% 0.31/0.46 thf(fact_143_inf__set__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( inf_inf @ ( set @ A ) )
% 0.31/0.46 = ( ^ [A4: set @ A,B6: set @ A] :
% 0.31/0.46 ( collect @ A
% 0.31/0.46 @ ( inf_inf @ ( A > $o )
% 0.31/0.46 @ ^ [X: A] : ( member @ A @ X @ A4 )
% 0.31/0.46 @ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf_set_def
% 0.31/0.46 thf(fact_144_llist_Oset__sel_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,A2: coindu1593790203_llist @ A,X3: A] :
% 0.31/0.46 ( ~ ( coindu335574135_lnull @ A @ A2 )
% 0.31/0.46 => ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ A2 ) ) )
% 0.31/0.46 => ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ A2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % llist.set_sel(2)
% 0.31/0.46 thf(fact_145_lfinite__induct,axiom,
% 0.31/0.46 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
% 0.31/0.46 ( ( coindu1213758845finite @ A @ Xs )
% 0.31/0.46 => ( ! [Xs3: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu335574135_lnull @ A @ Xs3 )
% 0.31/0.46 => ( P @ Xs3 ) )
% 0.31/0.46 => ( ! [Xs3: coindu1593790203_llist @ A] :
% 0.31/0.46 ( ( coindu1213758845finite @ A @ Xs3 )
% 0.31/0.46 => ( ~ ( coindu335574135_lnull @ A @ Xs3 )
% 0.31/0.46 => ( ( P @ ( coindu1047225960le_ltl @ A @ Xs3 ) )
% 0.31/0.46 => ( P @ Xs3 ) ) ) )
% 0.31/0.46 => ( P @ Xs ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % lfinite_induct
% 0.31/0.46 thf(fact_146_inf__sup__aci_I4_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X3: A,Y2: A] :
% 0.31/0.46 ( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ X3 @ Y2 ) )
% 0.31/0.46 = ( inf_inf @ A @ X3 @ Y2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf_sup_aci(4)
% 0.31/0.46 thf(fact_147_inf__sup__aci_I3_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X3: A,Y2: A,Z: A] :
% 0.31/0.46 ( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) )
% 0.31/0.46 = ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X3 @ Z ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf_sup_aci(3)
% 0.31/0.46 thf(fact_148_inf__sup__aci_I2_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X3: A,Y2: A,Z: A] :
% 0.31/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Z )
% 0.31/0.46 = ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf_sup_aci(2)
% 0.31/0.46 thf(fact_149_inf__sup__aci_I1_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.46 => ( ( inf_inf @ A )
% 0.31/0.46 = ( ^ [X: A,Y4: A] : ( inf_inf @ A @ Y4 @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf_sup_aci(1)
% 0.31/0.46 thf(fact_150_inf__fun__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 0.31/0.46 => ( ( inf_inf @ ( A > B ) )
% 0.31/0.46 = ( ^ [F3: A > B,G3: A > B,X: A] : ( inf_inf @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf_fun_def
% 0.31/0.46 thf(fact_151_inf_Oassoc,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A,C: A] :
% 0.31/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C )
% 0.31/0.46 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf.assoc
% 0.31/0.46 thf(fact_152_inf__assoc,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X3: A,Y2: A,Z: A] :
% 0.31/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Z )
% 0.31/0.46 = ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf_assoc
% 0.31/0.46 thf(fact_153_inf_Ocommute,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ( ( inf_inf @ A )
% 0.31/0.46 = ( ^ [A6: A,B5: A] : ( inf_inf @ A @ B5 @ A6 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf.commute
% 0.31/0.46 thf(fact_154_inf__commute,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ( ( inf_inf @ A )
% 0.31/0.46 = ( ^ [X: A,Y4: A] : ( inf_inf @ A @ Y4 @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf_commute
% 0.31/0.46 thf(fact_155_inf_Oleft__commute,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [B2: A,A2: A,C: A] :
% 0.31/0.46 ( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C ) )
% 0.31/0.46 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf.left_commute
% 0.31/0.46 thf(fact_156_inf__left__commute,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X3: A,Y2: A,Z: A] :
% 0.31/0.46 ( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) )
% 0.31/0.46 = ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X3 @ Z ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf_left_commute
% 0.31/0.46 thf(fact_157_nat__add__left__cancel,axiom,
% 0.31/0.46 ! [K: nat,M: nat,N3: nat] :
% 0.31/0.46 ( ( ( plus_plus @ nat @ K @ M )
% 0.31/0.46 = ( plus_plus @ nat @ K @ N3 ) )
% 0.31/0.46 = ( M = N3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat_add_left_cancel
% 0.31/0.46 thf(fact_158_nat__add__right__cancel,axiom,
% 0.31/0.46 ! [M: nat,K: nat,N3: nat] :
% 0.31/0.46 ( ( ( plus_plus @ nat @ M @ K )
% 0.31/0.46 = ( plus_plus @ nat @ N3 @ K ) )
% 0.31/0.46 = ( M = N3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nat_add_right_cancel
% 0.31/0.46 thf(fact_159_add__lessD1,axiom,
% 0.31/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.31/0.46 ( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
% 0.31/0.46 => ( ord_less @ nat @ I2 @ K ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_lessD1
% 0.31/0.46 thf(fact_160_add__less__mono,axiom,
% 0.31/0.46 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.31/0.46 ( ( ord_less @ nat @ I2 @ J )
% 0.31/0.46 => ( ( ord_less @ nat @ K @ L )
% 0.31/0.46 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_less_mono
% 0.31/0.46 thf(fact_161_not__add__less1,axiom,
% 0.31/0.46 ! [I2: nat,J: nat] :
% 0.31/0.46 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ I2 ) ).
% 0.31/0.46
% 0.31/0.46 % not_add_less1
% 0.31/0.46 thf(fact_162_not__add__less2,axiom,
% 0.31/0.46 ! [J: nat,I2: nat] :
% 0.31/0.46 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I2 ) @ I2 ) ).
% 0.31/0.46
% 0.31/0.46 % not_add_less2
% 0.31/0.46 thf(fact_163_add__less__mono1,axiom,
% 0.31/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.31/0.46 ( ( ord_less @ nat @ I2 @ J )
% 0.31/0.46 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % add_less_mono1
% 0.31/0.46 thf(fact_164_trans__less__add1,axiom,
% 0.31/0.46 ! [I2: nat,J: nat,M: nat] :
% 0.31/0.46 ( ( ord_less @ nat @ I2 @ J )
% 0.31/0.46 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % trans_less_add1
% 0.31/0.46 thf(fact_165_trans__less__add2,axiom,
% 0.31/0.46 ! [I2: nat,J: nat,M: nat] :
% 0.31/0.46 ( ( ord_less @ nat @ I2 @ J )
% 0.31/0.46 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % trans_less_add2
% 0.31/0.46 thf(fact_166_less__add__eq__less,axiom,
% 0.31/0.46 ! [K: nat,L: nat,M: nat,N3: nat] :
% 0.31/0.46 ( ( ord_less @ nat @ K @ L )
% 0.31/0.46 => ( ( ( plus_plus @ nat @ M @ L )
% 0.31/0.46 = ( plus_plus @ nat @ K @ N3 ) )
% 0.31/0.46 => ( ord_less @ nat @ M @ N3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_add_eq_less
% 0.31/0.46 thf(fact_167_less__infI1,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,X3: A,B2: A] :
% 0.31/0.46 ( ( ord_less @ A @ A2 @ X3 )
% 0.31/0.46 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_infI1
% 0.31/0.46 thf(fact_168_less__infI2,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [B2: A,X3: A,A2: A] :
% 0.31/0.46 ( ( ord_less @ A @ B2 @ X3 )
% 0.31/0.46 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_infI2
% 0.31/0.46 thf(fact_169_inf_Ostrict__boundedE,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,B2: A,C: A] :
% 0.31/0.46 ( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
% 0.31/0.46 => ~ ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.46 => ~ ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf.strict_boundedE
% 0.31/0.46 thf(fact_170_inf_Ostrict__order__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ( ( ord_less @ A )
% 0.31/0.46 = ( ^ [A6: A,B5: A] :
% 0.31/0.46 ( ( A6
% 0.31/0.46 = ( inf_inf @ A @ A6 @ B5 ) )
% 0.31/0.46 & ( A6 != B5 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inf.strict_order_iff
% 0.31/0.46 thf(fact_171_inf_Ostrict__coboundedI1,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A2: A,C: A,B2: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ C )
% 0.31/0.47 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf.strict_coboundedI1
% 0.31/0.47 thf(fact_172_inf_Ostrict__coboundedI2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,C: A,A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ B2 @ C )
% 0.31/0.47 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf.strict_coboundedI2
% 0.31/0.47 thf(fact_173_enat__less__enat__plusI,axiom,
% 0.31/0.47 ! [X3: nat,Y2: nat,Z: extended_enat] :
% 0.31/0.47 ( ( ord_less @ nat @ X3 @ Y2 )
% 0.31/0.47 => ( ord_less @ extended_enat @ ( extended_enat2 @ X3 ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ Y2 ) @ Z ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % enat_less_enat_plusI
% 0.31/0.47 thf(fact_174_fold__atLeastAtMost__nat_Oinduct,axiom,
% 0.31/0.47 ! [A: $tType,P: ( nat > A > A ) > nat > nat > A > $o,A0: nat > A > A,A1: nat,A22: nat,A32: A] :
% 0.31/0.47 ( ! [F4: nat > A > A,A7: nat,B7: nat,Acc: A] :
% 0.31/0.47 ( ( ~ ( ord_less @ nat @ B7 @ A7 )
% 0.31/0.47 => ( P @ F4 @ ( plus_plus @ nat @ A7 @ ( one_one @ nat ) ) @ B7 @ ( F4 @ A7 @ Acc ) ) )
% 0.31/0.47 => ( P @ F4 @ A7 @ B7 @ Acc ) )
% 0.31/0.47 => ( P @ A0 @ A1 @ A22 @ A32 ) ) ).
% 0.31/0.47
% 0.31/0.47 % fold_atLeastAtMost_nat.induct
% 0.31/0.47 thf(fact_175_finite__M__bounded__by__nat,axiom,
% 0.31/0.47 ! [P: nat > $o,I2: nat] :
% 0.31/0.47 ( finite_finite2 @ nat
% 0.31/0.47 @ ( collect @ nat
% 0.31/0.47 @ ^ [K3: nat] :
% 0.31/0.47 ( ( P @ K3 )
% 0.31/0.47 & ( ord_less @ nat @ K3 @ I2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_M_bounded_by_nat
% 0.31/0.47 thf(fact_176_gen__llength__def,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( coindu493225792length @ A )
% 0.31/0.47 = ( ^ [N: nat,Xs2: coindu1593790203_llist @ A] : ( plus_plus @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % gen_llength_def
% 0.31/0.47 thf(fact_177_unbounded__k__infinite,axiom,
% 0.31/0.47 ! [K: nat,S2: set @ nat] :
% 0.31/0.47 ( ! [M4: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ K @ M4 )
% 0.31/0.47 => ? [N5: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ M4 @ N5 )
% 0.31/0.47 & ( member @ nat @ N5 @ S2 ) ) )
% 0.31/0.47 => ~ ( finite_finite2 @ nat @ S2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % unbounded_k_infinite
% 0.31/0.47 thf(fact_178_inf1I,axiom,
% 0.31/0.47 ! [A: $tType,A3: A > $o,X3: A,B3: A > $o] :
% 0.31/0.47 ( ( A3 @ X3 )
% 0.31/0.47 => ( ( B3 @ X3 )
% 0.31/0.47 => ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf1I
% 0.31/0.47 thf(fact_179_inf1E,axiom,
% 0.31/0.47 ! [A: $tType,A3: A > $o,B3: A > $o,X3: A] :
% 0.31/0.47 ( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X3 )
% 0.31/0.47 => ~ ( ( A3 @ X3 )
% 0.31/0.47 => ~ ( B3 @ X3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf1E
% 0.31/0.47 thf(fact_180_inf1D1,axiom,
% 0.31/0.47 ! [A: $tType,A3: A > $o,B3: A > $o,X3: A] :
% 0.31/0.47 ( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X3 )
% 0.31/0.47 => ( A3 @ X3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf1D1
% 0.31/0.47 thf(fact_181_inf1D2,axiom,
% 0.31/0.47 ! [A: $tType,A3: A > $o,B3: A > $o,X3: A] :
% 0.31/0.47 ( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X3 )
% 0.31/0.47 => ( B3 @ X3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf1D2
% 0.31/0.47 thf(fact_182_finite__nat__set__iff__bounded,axiom,
% 0.31/0.47 ( ( finite_finite2 @ nat )
% 0.31/0.47 = ( ^ [N6: set @ nat] :
% 0.31/0.47 ? [M5: nat] :
% 0.31/0.47 ! [X: nat] :
% 0.31/0.47 ( ( member @ nat @ X @ N6 )
% 0.31/0.47 => ( ord_less @ nat @ X @ M5 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_nat_set_iff_bounded
% 0.31/0.47 thf(fact_183_infinite__nat__iff__unbounded,axiom,
% 0.31/0.47 ! [S2: set @ nat] :
% 0.31/0.47 ( ( ~ ( finite_finite2 @ nat @ S2 ) )
% 0.31/0.47 = ( ! [M5: nat] :
% 0.31/0.47 ? [N: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ M5 @ N )
% 0.31/0.47 & ( member @ nat @ N @ S2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % infinite_nat_iff_unbounded
% 0.31/0.47 thf(fact_184_bounded__nat__set__is__finite,axiom,
% 0.31/0.47 ! [N4: set @ nat,N3: nat] :
% 0.31/0.47 ( ! [X2: nat] :
% 0.31/0.47 ( ( member @ nat @ X2 @ N4 )
% 0.31/0.47 => ( ord_less @ nat @ X2 @ N3 ) )
% 0.31/0.47 => ( finite_finite2 @ nat @ N4 ) ) ).
% 0.31/0.47
% 0.31/0.47 % bounded_nat_set_is_finite
% 0.31/0.47 thf(fact_185_inf__Int__eq,axiom,
% 0.31/0.47 ! [A: $tType,R: set @ A,S2: set @ A] :
% 0.31/0.47 ( ( inf_inf @ ( A > $o )
% 0.31/0.47 @ ^ [X: A] : ( member @ A @ X @ R )
% 0.31/0.47 @ ^ [X: A] : ( member @ A @ X @ S2 ) )
% 0.31/0.47 = ( ^ [X: A] : ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ R @ S2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_Int_eq
% 0.31/0.47 thf(fact_186_lnth__llength__ltakeWhile,axiom,
% 0.31/0.47 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( ord_less @ extended_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) @ ( coindu1018505716length @ A @ Xs ) )
% 0.31/0.47 => ~ ( P @ ( coindu749330388e_lnth @ A @ Xs @ ( extended_the_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % lnth_llength_ltakeWhile
% 0.31/0.47 thf(fact_187_lnth__ldropn,axiom,
% 0.31/0.47 ! [A: $tType,N3: nat,M: nat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ N3 @ M ) ) @ ( coindu1018505716length @ A @ Xs ) )
% 0.31/0.47 => ( ( coindu749330388e_lnth @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) @ M )
% 0.31/0.47 = ( coindu749330388e_lnth @ A @ Xs @ ( plus_plus @ nat @ M @ N3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % lnth_ldropn
% 0.31/0.47 thf(fact_188_ldropn__ldropn,axiom,
% 0.31/0.47 ! [A: $tType,N3: nat,M: nat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( coindu531130065ldropn @ A @ N3 @ ( coindu531130065ldropn @ A @ M @ Xs ) )
% 0.31/0.47 = ( coindu531130065ldropn @ A @ ( plus_plus @ nat @ N3 @ M ) @ Xs ) ) ).
% 0.31/0.47
% 0.31/0.47 % ldropn_ldropn
% 0.31/0.47 thf(fact_189_lfinite__ldropn,axiom,
% 0.31/0.47 ! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( coindu1213758845finite @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
% 0.31/0.47 = ( coindu1213758845finite @ A @ Xs ) ) ).
% 0.31/0.47
% 0.31/0.47 % lfinite_ldropn
% 0.31/0.47 thf(fact_190_the__enat_Osimps,axiom,
% 0.31/0.47 ! [N3: nat] :
% 0.31/0.47 ( ( extended_the_enat @ ( extended_enat2 @ N3 ) )
% 0.31/0.47 = N3 ) ).
% 0.31/0.47
% 0.31/0.47 % the_enat.simps
% 0.31/0.47 thf(fact_191_ltl__ldropn,axiom,
% 0.31/0.47 ! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( coindu1047225960le_ltl @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
% 0.31/0.47 = ( coindu531130065ldropn @ A @ N3 @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ltl_ldropn
% 0.31/0.47 thf(fact_192_in__lset__ldropnD,axiom,
% 0.31/0.47 ! [A: $tType,X3: A,N3: nat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) ) )
% 0.31/0.47 => ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % in_lset_ldropnD
% 0.31/0.47 thf(fact_193_lnth__ldrop,axiom,
% 0.31/0.47 ! [A: $tType,N3: extended_enat,M: nat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ N3 @ ( extended_enat2 @ M ) ) @ ( coindu1018505716length @ A @ Xs ) )
% 0.31/0.47 => ( ( coindu749330388e_lnth @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) @ M )
% 0.31/0.47 = ( coindu749330388e_lnth @ A @ Xs @ ( plus_plus @ nat @ M @ ( extended_the_enat @ N3 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % lnth_ldrop
% 0.31/0.47 thf(fact_194_less__numeral__extra_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_numeral_extra(4)
% 0.31/0.47 thf(fact_195_ldrop__ldrop,axiom,
% 0.31/0.47 ! [A: $tType,N3: extended_enat,M: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( coindu191418589_ldrop @ A @ N3 @ ( coindu191418589_ldrop @ A @ M @ Xs ) )
% 0.31/0.47 = ( coindu191418589_ldrop @ A @ ( plus_plus @ extended_enat @ N3 @ M ) @ Xs ) ) ).
% 0.31/0.47
% 0.31/0.47 % ldrop_ldrop
% 0.31/0.47 thf(fact_196_in__lset__ldropD,axiom,
% 0.31/0.47 ! [A: $tType,X3: A,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) ) )
% 0.31/0.47 => ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % in_lset_ldropD
% 0.31/0.47 thf(fact_197_ltl__ldrop,axiom,
% 0.31/0.47 ! [A: $tType,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( coindu1047225960le_ltl @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) )
% 0.31/0.47 = ( coindu191418589_ldrop @ A @ N3 @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ltl_ldrop
% 0.31/0.47 thf(fact_198_ldrop__enat,axiom,
% 0.31/0.47 ! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( coindu191418589_ldrop @ A @ ( extended_enat2 @ N3 ) @ Xs )
% 0.31/0.47 = ( coindu531130065ldropn @ A @ N3 @ Xs ) ) ).
% 0.31/0.47
% 0.31/0.47 % ldrop_enat
% 0.31/0.47 thf(fact_199_is__num__normalize_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( neg_numeral @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.31/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % is_num_normalize(1)
% 0.31/0.47 thf(fact_200_ldropWhile__eq__ldrop,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( coindu438612276pWhile @ A )
% 0.31/0.47 = ( ^ [P2: A > $o,Xs2: coindu1593790203_llist @ A] : ( coindu191418589_ldrop @ A @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P2 @ Xs2 ) ) @ Xs2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ldropWhile_eq_ldrop
% 0.31/0.47 thf(fact_201_lhd__ldropWhile__conv__lnth,axiom,
% 0.31/0.47 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.31/0.47 ( ? [X4: A] :
% 0.31/0.47 ( ( member @ A @ X4 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 0.31/0.47 & ~ ( P @ X4 ) )
% 0.31/0.47 => ( ( coindu1046438764le_lhd @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) )
% 0.31/0.47 = ( coindu749330388e_lnth @ A @ Xs @ ( extended_the_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % lhd_ldropWhile_conv_lnth
% 0.31/0.47 thf(fact_202_llast__ldropn,axiom,
% 0.31/0.47 ! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( ord_less @ extended_enat @ ( extended_enat2 @ N3 ) @ ( coindu1018505716length @ A @ Xs ) )
% 0.31/0.47 => ( ( coindu2000965700_llast @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
% 0.31/0.47 = ( coindu2000965700_llast @ A @ Xs ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % llast_ldropn
% 0.31/0.47 thf(fact_203_lnull__ltakeWhile,axiom,
% 0.31/0.47 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 0.31/0.47 = ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.47 => ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % lnull_ltakeWhile
% 0.31/0.47 thf(fact_204_ltakeWhile_Odisc__iff_I1_J,axiom,
% 0.31/0.47 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 0.31/0.47 = ( ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.47 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ltakeWhile.disc_iff(1)
% 0.31/0.47 thf(fact_205_ltakeWhile_Odisc__iff_I2_J,axiom,
% 0.31/0.47 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
% 0.31/0.47 = ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.47 & ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ltakeWhile.disc_iff(2)
% 0.31/0.47 thf(fact_206_ltakeWhile_Oexhaust,axiom,
% 0.31/0.47 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.31/0.47 ( ~ ( ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.47 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
% 0.31/0.47 => ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.47 => ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ltakeWhile.exhaust
% 0.31/0.47 thf(fact_207_llist_Oset__sel_I1_J,axiom,
% 0.31/0.47 ! [A: $tType,A2: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ~ ( coindu335574135_lnull @ A @ A2 )
% 0.31/0.47 => ( member @ A @ ( coindu1046438764le_lhd @ A @ A2 ) @ ( coindu1112613586e_lset @ A @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % llist.set_sel(1)
% 0.31/0.47 thf(fact_208_llist_Ocoinduct__strong,axiom,
% 0.31/0.47 ! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist2: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( R @ Llist @ Llist2 )
% 0.31/0.47 => ( ! [Llist3: coindu1593790203_llist @ A,Llist4: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( R @ Llist3 @ Llist4 )
% 0.31/0.47 => ( ( ( coindu335574135_lnull @ A @ Llist3 )
% 0.31/0.47 = ( coindu335574135_lnull @ A @ Llist4 ) )
% 0.31/0.47 & ( ~ ( coindu335574135_lnull @ A @ Llist3 )
% 0.31/0.47 => ( ~ ( coindu335574135_lnull @ A @ Llist4 )
% 0.31/0.47 => ( ( ( coindu1046438764le_lhd @ A @ Llist3 )
% 0.31/0.47 = ( coindu1046438764le_lhd @ A @ Llist4 ) )
% 0.31/0.47 & ( ( R @ ( coindu1047225960le_ltl @ A @ Llist3 ) @ ( coindu1047225960le_ltl @ A @ Llist4 ) )
% 0.31/0.47 | ( ( coindu1047225960le_ltl @ A @ Llist3 )
% 0.31/0.47 = ( coindu1047225960le_ltl @ A @ Llist4 ) ) ) ) ) ) ) )
% 0.31/0.47 => ( Llist = Llist2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % llist.coinduct_strong
% 0.31/0.47 thf(fact_209_llist_Ocoinduct,axiom,
% 0.31/0.47 ! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist2: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( R @ Llist @ Llist2 )
% 0.31/0.47 => ( ! [Llist3: coindu1593790203_llist @ A,Llist4: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( R @ Llist3 @ Llist4 )
% 0.31/0.47 => ( ( ( coindu335574135_lnull @ A @ Llist3 )
% 0.31/0.47 = ( coindu335574135_lnull @ A @ Llist4 ) )
% 0.31/0.47 & ( ~ ( coindu335574135_lnull @ A @ Llist3 )
% 0.31/0.47 => ( ~ ( coindu335574135_lnull @ A @ Llist4 )
% 0.31/0.47 => ( ( ( coindu1046438764le_lhd @ A @ Llist3 )
% 0.31/0.47 = ( coindu1046438764le_lhd @ A @ Llist4 ) )
% 0.31/0.47 & ( R @ ( coindu1047225960le_ltl @ A @ Llist3 ) @ ( coindu1047225960le_ltl @ A @ Llist4 ) ) ) ) ) ) )
% 0.31/0.47 => ( Llist = Llist2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % llist.coinduct
% 0.31/0.47 thf(fact_210_llist_Oexpand,axiom,
% 0.31/0.47 ! [A: $tType,Llist: coindu1593790203_llist @ A,Llist2: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( ( coindu335574135_lnull @ A @ Llist )
% 0.31/0.47 = ( coindu335574135_lnull @ A @ Llist2 ) )
% 0.31/0.47 => ( ( ~ ( coindu335574135_lnull @ A @ Llist )
% 0.31/0.47 => ( ~ ( coindu335574135_lnull @ A @ Llist2 )
% 0.31/0.47 => ( ( ( coindu1046438764le_lhd @ A @ Llist )
% 0.31/0.47 = ( coindu1046438764le_lhd @ A @ Llist2 ) )
% 0.31/0.47 & ( ( coindu1047225960le_ltl @ A @ Llist )
% 0.31/0.47 = ( coindu1047225960le_ltl @ A @ Llist2 ) ) ) ) )
% 0.31/0.47 => ( Llist = Llist2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % llist.expand
% 0.31/0.47 thf(fact_211_ltakeWhile_Odisc_I2_J,axiom,
% 0.31/0.47 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.31/0.47 ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.47 => ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
% 0.31/0.47 => ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ltakeWhile.disc(2)
% 0.31/0.47 thf(fact_212_ltakeWhile_Odisc_I1_J,axiom,
% 0.31/0.47 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.31/0.47 ( ( ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.47 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
% 0.31/0.47 => ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ltakeWhile.disc(1)
% 0.31/0.47 thf(fact_213_lhd__ltakeWhile,axiom,
% 0.31/0.47 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.31/0.47 ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.47 => ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
% 0.31/0.47 => ( ( coindu1046438764le_lhd @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 0.31/0.47 = ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % lhd_ltakeWhile
% 0.31/0.47 thf(fact_214_lhd__ldropWhile__in__lset,axiom,
% 0.31/0.47 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.31/0.47 ( ? [X4: A] :
% 0.31/0.47 ( ( member @ A @ X4 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 0.31/0.47 & ~ ( P @ X4 ) )
% 0.31/0.47 => ( member @ A @ ( coindu1046438764le_lhd @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) ) @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % lhd_ldropWhile_in_lset
% 0.31/0.47 thf(fact_215_lhd__ldropWhile,axiom,
% 0.31/0.47 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.31/0.47 ( ? [X4: A] :
% 0.31/0.47 ( ( member @ A @ X4 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 0.31/0.47 & ~ ( P @ X4 ) )
% 0.31/0.47 => ~ ( P @ ( coindu1046438764le_lhd @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % lhd_ldropWhile
% 0.31/0.47 thf(fact_216_llist__set__induct,axiom,
% 0.31/0.47 ! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A,P: A > ( coindu1593790203_llist @ A ) > $o] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 0.31/0.47 => ( ! [Xs3: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ~ ( coindu335574135_lnull @ A @ Xs3 )
% 0.31/0.47 => ( P @ ( coindu1046438764le_lhd @ A @ Xs3 ) @ Xs3 ) )
% 0.31/0.47 => ( ! [Xs3: coindu1593790203_llist @ A,Y5: A] :
% 0.31/0.47 ( ~ ( coindu335574135_lnull @ A @ Xs3 )
% 0.31/0.47 => ( ( member @ A @ Y5 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs3 ) ) )
% 0.31/0.47 => ( ( P @ Y5 @ ( coindu1047225960le_ltl @ A @ Xs3 ) )
% 0.31/0.47 => ( P @ Y5 @ Xs3 ) ) ) )
% 0.31/0.47 => ( P @ X3 @ Xs ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % llist_set_induct
% 0.31/0.47 thf(fact_217_ltakeWhile_Osimps_I4_J,axiom,
% 0.31/0.47 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.31/0.47 ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.31/0.47 => ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
% 0.31/0.47 => ( ( coindu1047225960le_ltl @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 0.31/0.47 = ( coindu721411036eWhile @ A @ P @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ltakeWhile.simps(4)
% 0.31/0.47 thf(fact_218_llast__ldrop,axiom,
% 0.31/0.47 ! [A: $tType,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( ord_less @ extended_enat @ N3 @ ( coindu1018505716length @ A @ Xs ) )
% 0.31/0.47 => ( ( coindu2000965700_llast @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) )
% 0.31/0.47 = ( coindu2000965700_llast @ A @ Xs ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % llast_ldrop
% 0.31/0.47 thf(fact_219_lhd__ldropn,axiom,
% 0.31/0.47 ! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( ord_less @ extended_enat @ ( extended_enat2 @ N3 ) @ ( coindu1018505716length @ A @ Xs ) )
% 0.31/0.47 => ( ( coindu1046438764le_lhd @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
% 0.31/0.47 = ( coindu749330388e_lnth @ A @ Xs @ N3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % lhd_ldropn
% 0.31/0.47 thf(fact_220_lhd__ldrop,axiom,
% 0.31/0.47 ! [A: $tType,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( ord_less @ extended_enat @ N3 @ ( coindu1018505716length @ A @ Xs ) )
% 0.31/0.47 => ( ( coindu1046438764le_lhd @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) )
% 0.31/0.47 = ( coindu749330388e_lnth @ A @ Xs @ ( extended_the_enat @ N3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % lhd_ldrop
% 0.31/0.47 thf(fact_221_enat_Osimps_I6_J,axiom,
% 0.31/0.47 ! [T: $tType,F1: nat > T,F22: T,Nat: nat] :
% 0.31/0.47 ( ( extended_rec_enat @ T @ F1 @ F22 @ ( extended_enat2 @ Nat ) )
% 0.31/0.47 = ( F1 @ Nat ) ) ).
% 0.31/0.47
% 0.31/0.47 % enat.simps(6)
% 0.31/0.47 thf(fact_222_lset__conv__lnth,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( coindu1112613586e_lset @ A )
% 0.31/0.47 = ( ^ [Xs2: coindu1593790203_llist @ A] :
% 0.31/0.47 ( collect @ A
% 0.31/0.47 @ ^ [Uu: A] :
% 0.31/0.47 ? [N: nat] :
% 0.31/0.47 ( ( Uu
% 0.31/0.47 = ( coindu749330388e_lnth @ A @ Xs2 @ N ) )
% 0.31/0.47 & ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % lset_conv_lnth
% 0.31/0.47 thf(fact_223_finite__Collect__bounded__ex,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,P: A > $o,Q: B > A > $o] :
% 0.31/0.47 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.31/0.47 => ( ( finite_finite2 @ B
% 0.31/0.47 @ ( collect @ B
% 0.31/0.47 @ ^ [X: B] :
% 0.31/0.47 ? [Y4: A] :
% 0.31/0.47 ( ( P @ Y4 )
% 0.31/0.47 & ( Q @ X @ Y4 ) ) ) )
% 0.31/0.47 = ( ! [Y4: A] :
% 0.31/0.47 ( ( P @ Y4 )
% 0.31/0.47 => ( finite_finite2 @ B
% 0.31/0.47 @ ( collect @ B
% 0.31/0.47 @ ^ [X: B] : ( Q @ X @ Y4 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_Collect_bounded_ex
% 0.31/0.47 thf(fact_224_finite__image__set,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,P: A > $o,F2: A > B] :
% 0.31/0.47 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.31/0.47 => ( finite_finite2 @ B
% 0.31/0.47 @ ( collect @ B
% 0.31/0.47 @ ^ [Uu: B] :
% 0.31/0.47 ? [X: A] :
% 0.31/0.47 ( ( Uu
% 0.31/0.47 = ( F2 @ X ) )
% 0.31/0.47 & ( P @ X ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_image_set
% 0.31/0.47 thf(fact_225_finite__image__set2,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,C3: $tType,P: A > $o,Q: B > $o,F2: A > B > C3] :
% 0.31/0.47 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.31/0.47 => ( ( finite_finite2 @ B @ ( collect @ B @ Q ) )
% 0.31/0.47 => ( finite_finite2 @ C3
% 0.31/0.47 @ ( collect @ C3
% 0.31/0.47 @ ^ [Uu: C3] :
% 0.31/0.47 ? [X: A,Y4: B] :
% 0.31/0.47 ( ( Uu
% 0.31/0.47 = ( F2 @ X @ Y4 ) )
% 0.31/0.47 & ( P @ X )
% 0.31/0.47 & ( Q @ Y4 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_image_set2
% 0.31/0.47 thf(fact_226_Setcompr__eq__image,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
% 0.31/0.47 ( ( collect @ A
% 0.31/0.47 @ ^ [Uu: A] :
% 0.31/0.47 ? [X: B] :
% 0.31/0.47 ( ( Uu
% 0.31/0.47 = ( F2 @ X ) )
% 0.31/0.47 & ( member @ B @ X @ A3 ) ) )
% 0.31/0.47 = ( image @ B @ A @ F2 @ A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % Setcompr_eq_image
% 0.31/0.47 thf(fact_227_setcompr__eq__image,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,F2: B > A,P: B > $o] :
% 0.31/0.47 ( ( collect @ A
% 0.31/0.47 @ ^ [Uu: A] :
% 0.31/0.47 ? [X: B] :
% 0.31/0.47 ( ( Uu
% 0.31/0.47 = ( F2 @ X ) )
% 0.31/0.47 & ( P @ X ) ) )
% 0.31/0.47 = ( image @ B @ A @ F2 @ ( collect @ B @ P ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % setcompr_eq_image
% 0.31/0.47 thf(fact_228_lnull__ldropn,axiom,
% 0.31/0.47 ! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( coindu335574135_lnull @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
% 0.31/0.47 = ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( extended_enat2 @ N3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % lnull_ldropn
% 0.31/0.47 thf(fact_229_llength__def,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( coindu1018505716length @ A )
% 0.31/0.47 = ( coindu1491768222unfold @ ( coindu1593790203_llist @ A ) @ ( coindu335574135_lnull @ A ) @ ( coindu1047225960le_ltl @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % llength_def
% 0.31/0.47 thf(fact_230_add__le__cancel__right,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,C: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.31/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_le_cancel_right
% 0.31/0.47 thf(fact_231_add__le__cancel__left,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [C: A,A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.31/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_le_cancel_left
% 0.31/0.47 thf(fact_232_inf_Obounded__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
% 0.31/0.47 = ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 & ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf.bounded_iff
% 0.31/0.47 thf(fact_233_le__inf__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X3: A,Y2: A,Z: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) )
% 0.31/0.47 = ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.31/0.47 & ( ord_less_eq @ A @ X3 @ Z ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_inf_iff
% 0.31/0.47 thf(fact_234_lnull__ldrop,axiom,
% 0.31/0.47 ! [A: $tType,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.31/0.47 ( ( coindu335574135_lnull @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) )
% 0.31/0.47 = ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % lnull_ldrop
% 0.31/0.47 thf(fact_235_le__numeral__extra_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_numeral_extra(4)
% 0.31/0.47 thf(fact_236_enat__less__imp__le,axiom,
% 0.31/0.47 ! [N3: extended_enat,M: extended_enat] :
% 0.31/0.47 ( ! [K2: nat] :
% 0.31/0.47 ( ( ord_less @ extended_enat @ N3 @ ( extended_enat2 @ K2 ) )
% 0.31/0.47 => ( ord_less @ extended_enat @ M @ ( extended_enat2 @ K2 ) ) )
% 0.31/0.47 => ( ord_less_eq @ extended_enat @ M @ N3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % enat_less_imp_le
% 0.31/0.47 thf(fact_237_llength__ltakeWhile__le,axiom,
% 0.31/0.47 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] : ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) @ ( coindu1018505716length @ A @ Xs ) ) ).
% 0.31/0.47
% 0.31/0.47 % llength_ltakeWhile_le
% 0.31/0.47 thf(fact_238_inf__sup__ord_I2_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Y2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_ord(2)
% 0.31/0.47 thf(fact_239_inf__sup__ord_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ X3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_sup_ord(1)
% 0.31/0.47 thf(fact_240_inf__le1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ X3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_le1
% 0.31/0.47 thf(fact_241_inf__le2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Y2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_le2
% 0.31/0.47 thf(fact_242_le__infE,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X3: A,A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) )
% 0.31/0.47 => ~ ( ( ord_less_eq @ A @ X3 @ A2 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ X3 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_infE
% 0.31/0.47 thf(fact_243_le__infI,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X3: A,A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X3 @ A2 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ X3 @ B2 )
% 0.31/0.47 => ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_infI
% 0.31/0.47 thf(fact_244_inf__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,C: A,B2: A,D: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ C )
% 0.31/0.47 => ( ( ord_less_eq @ A @ B2 @ D )
% 0.31/0.47 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C @ D ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_mono
% 0.31/0.47 thf(fact_245_le__infI1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,X3: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ X3 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_infI1
% 0.31/0.47 thf(fact_246_le__infI2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,X3: A,A2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ B2 @ X3 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_infI2
% 0.31/0.47 thf(fact_247_inf_OorderE,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( A2
% 0.31/0.47 = ( inf_inf @ A @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf.orderE
% 0.31/0.47 thf(fact_248_inf_OorderI,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( inf_inf @ A @ A2 @ B2 ) )
% 0.31/0.47 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf.orderI
% 0.31/0.47 thf(fact_249_inf__unique,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [F2: A > A > A,X3: A,Y2: A] :
% 0.31/0.47 ( ! [X2: A,Y5: A] : ( ord_less_eq @ A @ ( F2 @ X2 @ Y5 ) @ X2 )
% 0.31/0.47 => ( ! [X2: A,Y5: A] : ( ord_less_eq @ A @ ( F2 @ X2 @ Y5 ) @ Y5 )
% 0.31/0.47 => ( ! [X2: A,Y5: A,Z2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X2 @ Y5 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ X2 @ Z2 )
% 0.31/0.47 => ( ord_less_eq @ A @ X2 @ ( F2 @ Y5 @ Z2 ) ) ) )
% 0.31/0.47 => ( ( inf_inf @ A @ X3 @ Y2 )
% 0.31/0.47 = ( F2 @ X3 @ Y2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf_unique
% 0.31/0.47 thf(fact_250_le__iff__inf,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ( ord_less_eq @ A )
% 0.31/0.47 = ( ^ [X: A,Y4: A] :
% 0.31/0.47 ( ( inf_inf @ A @ X @ Y4 )
% 0.31/0.47 = X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_iff_inf
% 0.31/0.47 thf(fact_251_inf_Oabsorb1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( inf_inf @ A @ A2 @ B2 )
% 0.31/0.47 = A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf.absorb1
% 0.31/0.47 thf(fact_252_inf_Oabsorb2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,A2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.31/0.47 => ( ( inf_inf @ A @ A2 @ B2 )
% 0.31/0.47 = B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inf.absorb2
% 0.31/0.47
% 0.31/0.47 %----Type constructors (27)
% 0.31/0.47 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.31/0.47 ! [A8: $tType,A9: $tType] :
% 0.31/0.47 ( ( semilattice_inf @ A9 @ ( type2 @ A9 ) )
% 0.31/0.47 => ( semilattice_inf @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.31/0.47 ! [A8: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.31/0.47 & ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
% 0.31/0.47 => ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.31/0.47 ! [A8: $tType,A9: $tType] :
% 0.31/0.47 ( ( lattice @ A9 @ ( type2 @ A9 ) )
% 0.31/0.47 => ( lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 0.31/0.47 strict2144017051up_add @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.31/0.47 ordere223160158up_add @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.31/0.47 ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.31/0.47 ordere779506340up_add @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.31/0.47 cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 0.31/0.47 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_1,axiom,
% 0.31/0.47 semilattice_inf @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.31/0.47 ab_semigroup_add @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.31/0.47 semigroup_add @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Lattices_Olattice_2,axiom,
% 0.31/0.47 lattice @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 0.31/0.47 one @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_3,axiom,
% 0.31/0.47 ! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Finite__Set_Ofinite_4,axiom,
% 0.31/0.47 ! [A8: $tType] :
% 0.31/0.47 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.31/0.47 => ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Lattices_Olattice_5,axiom,
% 0.31/0.47 ! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_6,axiom,
% 0.31/0.47 semilattice_inf @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Finite__Set_Ofinite_7,axiom,
% 0.31/0.47 finite_finite @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Lattices_Olattice_8,axiom,
% 0.31/0.47 lattice @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_9,axiom,
% 0.31/0.47 strict2144017051up_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_10,axiom,
% 0.31/0.47 ordere779506340up_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_11,axiom,
% 0.31/0.47 semilattice_inf @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_12,axiom,
% 0.31/0.47 ab_semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_13,axiom,
% 0.31/0.47 semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_14,axiom,
% 0.31/0.47 lattice @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Extended__Nat_Oenat___Groups_Oone_15,axiom,
% 0.31/0.47 one @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.31/0.47
% 0.31/0.47 %----Conjectures (1)
% 0.31/0.47 thf(conj_0,conjecture,
% 0.31/0.47 ( finite_finite2 @ nat
% 0.31/0.47 @ ( image @ nat @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ n @ ( one_one @ nat ) ) )
% 0.31/0.47 @ ( inf_inf @ ( set @ nat )
% 0.31/0.47 @ ( collect @ nat
% 0.31/0.47 @ ^ [N: nat] :
% 0.31/0.47 ( ord_less @ extended_enat @ ( extended_enat2 @ N )
% 0.31/0.47 @ ( coindu1018505716length @ a
% 0.31/0.47 @ ( coindu1047225960le_ltl @ a
% 0.31/0.47 @ ( coindu438612276pWhile @ a
% 0.31/0.47 @ ^ [X: a] :
% 0.31/0.47 ~ ( p @ X )
% 0.31/0.47 @ xsa ) ) ) ) )
% 0.31/0.47 @ ( collect @ nat
% 0.31/0.47 @ ^ [N: nat] :
% 0.31/0.47 ( p
% 0.31/0.47 @ ( coindu749330388e_lnth @ a
% 0.31/0.47 @ ( coindu1047225960le_ltl @ a
% 0.31/0.47 @ ( coindu438612276pWhile @ a
% 0.31/0.47 @ ^ [X: a] :
% 0.31/0.47 ~ ( p @ X )
% 0.31/0.47 @ xsa ) )
% 0.31/0.47 @ N ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 ------------------------
% 2.54/1.48 ---- Embedded in THF ---
% 2.54/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.54/1.48 %%% Generated on Tue Feb 24 06:07:39 EST 2026
% 2.54/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.54/1.48 %%% Logic specification used:
% 2.54/1.48 %%% thf(spec, logic, $$dhol).
% 2.54/1.48
% 2.54/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.WzKKhsWUic/DTF2DTF9660.p
% 2.54/1.48 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist, type, coindu1593790203_llist: ($tType > $tType)).
% 2.54/1.48 thf(ty_t_Extended__Nat_Oenat, type, extended_enat: $tType).
% 2.54/1.48 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.54/1.48 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.54/1.48 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.54/1.48 thf(ty_tf_a, type, a: $tType).
% 2.54/1.48 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Lattices_Osemilattice__inf, type, semilattice_inf: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add, type, ordere223160158up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add, type, strict2144017051up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__llength, type, coindu493225792length: (!> [A:$tType]: ((nat > ((coindu1593790203_llist @ A) > extended_enat))))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldrop, type, coindu191418589_ldrop: (!> [A:$tType]: ((extended_enat > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OldropWhile, type, coindu438612276pWhile: (!> [A:$tType]: (((A > $o) > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldropn, type, coindu531130065ldropn: (!> [A:$tType]: ((nat > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfilter, type, coindu1889960678filter: (!> [A:$tType]: (((A > $o) > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite, type, coindu1213758845finite: (!> [A:$tType]: (((coindu1593790203_llist @ A) > $o)))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollast, type, coindu2000965700_llast: (!> [A:$tType]: (((coindu1593790203_llist @ A) > A)))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength, type, coindu1018505716length: (!> [A:$tType]: (((coindu1593790203_llist @ A) > extended_enat)))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olhd, type, coindu1046438764le_lhd: (!> [A:$tType]: (((coindu1593790203_llist @ A) > A)))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull, type, coindu335574135_lnull: (!> [A:$tType]: (((coindu1593790203_llist @ A) > $o)))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olset, type, coindu1112613586e_lset: (!> [A:$tType]: (((coindu1593790203_llist @ A) > (set @ A))))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Oltl, type, coindu1047225960le_ltl: (!> [A:$tType]: (((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A))))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olnth, type, coindu749330388e_lnth: (!> [A:$tType]: (((coindu1593790203_llist @ A) > (nat > A))))).
% 2.54/1.48 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OltakeWhile, type, coindu721411036eWhile: (!> [A:$tType]: (((A > $o) > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 2.54/1.48 thf(sy_c_Coinductive__Nat_Oenat__unfold, type, coindu1491768222unfold: (!> [A:$tType]: (((A > $o) > ((A > A) > (A > extended_enat)))))).
% 2.54/1.48 thf(sy_c_Extended__Nat_Oenat, type, extended_enat2: (nat > extended_enat)).
% 2.54/1.48 thf(sy_c_Extended__Nat_Oenat_Orec__enat, type, extended_rec_enat: (!> [T:$tType]: (((nat > T) > (T > (extended_enat > T)))))).
% 2.54/1.48 thf(sy_c_Extended__Nat_Othe__enat, type, extended_the_enat: (extended_enat > nat)).
% 2.54/1.48 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.54/1.48 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 2.54/1.48 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.54/1.48 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 2.54/1.48 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.54/1.48 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.54/1.48 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.54/1.48 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.54/1.48 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.54/1.48 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.54/1.48 thf(sy_v_P, type, p: (a > $o)).
% 2.54/1.48 thf(sy_v_n____, type, n: nat).
% 2.54/1.48 thf(sy_v_xs, type, xs: (coindu1593790203_llist @ a)).
% 2.54/1.48 thf(sy_v_xsa____, type, xsa: (coindu1593790203_llist @ a)).
% 2.54/1.48 thf(fact_0__092_060open_062finite_A_I_123n_O_Aenat_An_A_060_Allength_A_Iltl_A_IldropWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_J_125_A_092_060inter_062_A_123n_O_AP_A_Ilnth_A_Iltl_A_IldropWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_J_An_J_125_J_092_060close_062, axiom, ((finite_finite2 @ nat) @ (((inf_inf @ (set @ nat)) @ ((collect @ nat) @ (^ [N:nat]: ((((ord_less @ extended_enat) @ (extended_enat2 @ N)) @ ((coindu1018505716length @ a) @ ((coindu1047225960le_ltl @ a) @ (((coindu438612276pWhile @ a) @ (^ [X:a]: ((~ ((p @ X)))))) @ xsa)))))))) @ ((collect @ nat) @ (^ [N:nat]: ((p @ (((coindu749330388e_lnth @ a) @ ((coindu1047225960le_ltl @ a) @ (((coindu438612276pWhile @ a) @ (^ [X:a]: ((~ ((p @ X)))))) @ xsa))) @ N)))))))).
% 2.54/1.48 thf(fact_1_LCons_Oprems, axiom, (~ (((coindu1213758845finite @ a) @ xsa)))).
% 2.54/1.48 thf(fact_2_n, axiom, (((coindu1018505716length @ a) @ (((coindu721411036eWhile @ a) @ (^ [X:a]: ((~ ((p @ X)))))) @ xsa)) = (extended_enat2 @ n))).
% 2.54/1.48 thf(fact_3_finite__Int, axiom, (! [A:$tType,F:(set @ A),G:(set @ A)]: (((((finite_finite2 @ A) @ F) | ((finite_finite2 @ A) @ G)) => ((finite_finite2 @ A) @ (((inf_inf @ (set @ A)) @ F) @ G)))))).
% 2.54/1.48 thf(fact_4_finite__imageI, axiom, (! [B:$tType,A:$tType,F:(set @ A),H:(A > B)]: ((((finite_finite2 @ A) @ F) => ((finite_finite2 @ B) @ ((((image @ A) @ B) @ H) @ F)))))).
% 2.54/1.48 thf(fact_5_add__less__cancel__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [C:A,A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) = (((ord_less @ A) @ A2) @ B2)))))))).
% 2.54/1.48 thf(fact_6_add__less__cancel__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [A2:A,C:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) = (((ord_less @ A) @ A2) @ B2)))))))).
% 2.54/1.48 thf(fact_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_Allength_A_IltakeWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_A_061_Aenat_An_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [N2:nat]: ((((coindu1018505716length @ a) @ (((coindu721411036eWhile @ a) @ (^ [X:a]: ((~ ((p @ X)))))) @ xsa)) != (extended_enat2 @ N2))))))).
% 2.54/1.48 thf(fact_8_finite__Collect__conjI, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((finite_finite2 @ A) @ ((collect @ A) @ P)) | ((finite_finite2 @ A) @ ((collect @ A) @ Q))) => ((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: (((P @ X) & (Q @ X)))))))))).
% 2.54/1.48 thf(fact_9_finite__Collect__disjI, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: (((P @ X) | (Q @ X)))))) = (((finite_finite2 @ A) @ ((collect @ A) @ P)) & ((finite_finite2 @ A) @ ((collect @ A) @ Q))))))).
% 2.54/1.48 thf(fact_10_image__ident, axiom, (! [A:$tType,Y:(set @ A)]: ((((((image @ A) @ A) @ (^ [X:A]: (X))) @ Y) = Y)))).
% 2.54/1.48 thf(fact_11_less__add__one, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ A2) @ (one_one @ A))))))))).
% 2.54/1.48 thf(fact_12_exP, axiom, (? [X2:a]: (((((member @ a) @ X2) @ ((coindu1112613586e_lset @ a) @ xsa)) & (p @ X2))))).
% 2.54/1.48 thf(fact_13_enat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((extended_enat2 @ Nat) = (extended_enat2 @ Nat2)) = (Nat = Nat2))))).
% 2.54/1.48 thf(fact_14_LCons_Ohyps_I3_J, axiom, (! [Xs:(coindu1593790203_llist @ a)]: (((((coindu1047225960le_ltl @ a) @ (((coindu1889960678filter @ a) @ p) @ xsa)) = (((coindu1889960678filter @ a) @ p) @ Xs)) => ((~ (((coindu1213758845finite @ a) @ Xs))) => ((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N:nat]: (((((ord_less @ extended_enat) @ (extended_enat2 @ N)) @ ((coindu1018505716length @ a) @ Xs)) & (p @ (((coindu749330388e_lnth @ a) @ Xs) @ N)))))))))))).
% 2.54/1.48 thf(fact_15__092_060open_062_092_060not_062_Alfinite_Axs_092_060close_062, axiom, (~ (((coindu1213758845finite @ a) @ xs)))).
% 2.54/1.48 thf(fact_16_LCons_Ohyps_I1_J, axiom, ((coindu1213758845finite @ a) @ (((coindu1889960678filter @ a) @ p) @ xsa))).
% 2.54/1.48 thf(fact_17__092_060open_062lfinite_A_Ilfilter_AP_Axs_J_092_060close_062, axiom, ((coindu1213758845finite @ a) @ (((coindu1889960678filter @ a) @ p) @ xs))).
% 2.54/1.48 thf(fact_18_add__right__cancel, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B2 = C)))))))).
% 2.54/1.48 thf(fact_19_add__left__cancel, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C)) = (B2 = C)))))))).
% 2.54/1.48 thf(fact_20_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F2:(B > A),X3:B,A3:(set @ B)]: (((B2 = (F2 @ X3)) => ((((member @ B) @ X3) @ A3) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F2) @ A3))))))).
% 2.54/1.48 thf(fact_21_finite__code, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => ((finite_finite2 @ A) = (^ [A4:(set @ A)]: ($true))))))).
% 2.54/1.48 thf(fact_22_Int__iff, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) = ((((member @ A) @ C) @ A3) & (((member @ A) @ C) @ B3)))))).
% 2.54/1.48 thf(fact_23_IntI, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ A3) => ((((member @ A) @ C) @ B3) => (((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B3))))))).
% 2.54/1.48 thf(fact_24_lfilter__idem, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: (((((coindu1889960678filter @ A) @ P) @ (((coindu1889960678filter @ A) @ P) @ Xs)) = (((coindu1889960678filter @ A) @ P) @ Xs))))).
% 2.54/1.48 thf(fact_25_ltakeWhile__K__True, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: (((((coindu721411036eWhile @ A) @ (^ [Uu:A]: ($true))) @ Xs) = Xs)))).
% 2.54/1.48 thf(fact_26_finite__Collect__less__nat, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N:nat]: ((((ord_less @ nat) @ N) @ K)))))))).
% 2.54/1.48 thf(fact_27_lfilter__K__True, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: (((((coindu1889960678filter @ A) @ (^ [Uu:A]: ($true))) @ Xs) = Xs)))).
% 2.54/1.48 thf(fact_28_plus__enat__simps_I1_J, axiom, (! [M:nat,N3:nat]: (((((plus_plus @ extended_enat) @ (extended_enat2 @ M)) @ (extended_enat2 @ N3)) = (extended_enat2 @ (((plus_plus @ nat) @ M) @ N3)))))).
% 2.54/1.48 thf(fact_29_lfinite__ltl, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ ((coindu1047225960le_ltl @ A) @ Xs)) = ((coindu1213758845finite @ A) @ Xs))))).
% 2.54/1.48 thf(fact_30_enat__ord__simps_I2_J, axiom, (! [M:nat,N3:nat]: (((((ord_less @ extended_enat) @ (extended_enat2 @ M)) @ (extended_enat2 @ N3)) = (((ord_less @ nat) @ M) @ N3))))).
% 2.54/1.48 thf(fact_31_lset__lfilter, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: ((((coindu1112613586e_lset @ A) @ (((coindu1889960678filter @ A) @ P) @ Xs)) = ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ ((coindu1112613586e_lset @ A) @ Xs)) & (P @ X))))))))).
% 2.54/1.48 thf(fact_32_LCons_Ohyps_I2_J, axiom, (~ (((coindu335574135_lnull @ a) @ (((coindu1889960678filter @ a) @ p) @ xsa))))).
% 2.54/1.48 thf(fact_33_lfilter__cong, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X2:A]: (((((member @ A) @ X2) @ ((coindu1112613586e_lset @ A) @ Ys)) => ((P @ X2) = (Q @ X2))))) => ((((coindu1889960678filter @ A) @ P) @ Xs) = (((coindu1889960678filter @ A) @ Q) @ Ys))))))).
% 2.54/1.48 thf(fact_34_ltakeWhile__all, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ ((coindu1112613586e_lset @ A) @ Xs)) => (P @ X2)))) => ((((coindu721411036eWhile @ A) @ P) @ Xs) = Xs))))).
% 2.54/1.48 thf(fact_35_ltakeWhile__cong, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X2:A]: (((((member @ A) @ X2) @ ((coindu1112613586e_lset @ A) @ Ys)) => ((P @ X2) = (Q @ X2))))) => ((((coindu721411036eWhile @ A) @ P) @ Xs) = (((coindu721411036eWhile @ A) @ Q) @ Ys))))))).
% 2.54/1.48 thf(fact_36_lfinite__lfilterI, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: ((((coindu1213758845finite @ A) @ Xs) => ((coindu1213758845finite @ A) @ (((coindu1889960678filter @ A) @ P) @ Xs)))))).
% 2.54/1.48 thf(fact_37_lset__ltakeWhileD, axiom, (! [A:$tType,X3:A,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: (((((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs))) => ((((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ Xs)) & (P @ X3)))))).
% 2.54/1.48 thf(fact_38_lfinite__ltakeWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs)) = (((coindu1213758845finite @ A) @ Xs) | (? [X:A]: (((((member @ A) @ X) @ ((coindu1112613586e_lset @ A) @ Xs)) & (~ ((P @ X))))))))))).
% 2.54/1.48 thf(fact_39_lfinite__imp__finite__lset, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ Xs) => ((finite_finite2 @ A) @ ((coindu1112613586e_lset @ A) @ Xs)))))).
% 2.54/1.48 thf(fact_40_lfinite__ldropWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ (((coindu438612276pWhile @ A) @ P) @ Xs)) = ((? [X:A]: (((((member @ A) @ X) @ ((coindu1112613586e_lset @ A) @ Xs)) & (~ ((P @ X)))))) => ((coindu1213758845finite @ A) @ Xs)))))).
% 2.54/1.48 thf(fact_41_finite__psubset__induct, axiom, (! [A:$tType,A3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A3) => ((! [A5:(set @ A)]: ((((finite_finite2 @ A) @ A5) => ((! [B4:(set @ A)]: (((((ord_less @ (set @ A)) @ B4) @ A5) => (P @ B4)))) => (P @ A5))))) => (P @ A3)))))).
% 2.54/1.48 thf(fact_42_lfilter__lfilter, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o),Xs:(coindu1593790203_llist @ A)]: (((((coindu1889960678filter @ A) @ P) @ (((coindu1889960678filter @ A) @ Q) @ Xs)) = (((coindu1889960678filter @ A) @ (^ [X:A]: (((P @ X) & (Q @ X))))) @ Xs))))).
% 2.54/1.48 thf(fact_43_llength__ltakeWhile__lt__iff, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: (((((ord_less @ extended_enat) @ ((coindu1018505716length @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs))) @ ((coindu1018505716length @ A) @ Xs)) = (? [X:A]: (((((member @ A) @ X) @ ((coindu1112613586e_lset @ A) @ Xs)) & (~ ((P @ X)))))))))).
% 2.54/1.48 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.54/1.48 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A3)))) = A3)))).
% 2.54/1.48 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.54/1.48 thf(fact_47_ext, axiom, (! [B:$tType,A:$tType,F2:(A > B),G2:(A > B)]: (((! [X2:A]: (((F2 @ X2) = (G2 @ X2)))) => (F2 = G2))))).
% 2.54/1.48 thf(fact_48_in__lset__ltlD, axiom, (! [A:$tType,X3:A,Xs:(coindu1593790203_llist @ A)]: (((((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ ((coindu1047225960le_ltl @ A) @ Xs))) => (((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ Xs)))))).
% 2.54/1.48 thf(fact_49_llength__ltakeWhile__all, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: (((((coindu1018505716length @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs)) = ((coindu1018505716length @ A) @ Xs)) = ((((coindu721411036eWhile @ A) @ P) @ Xs) = Xs))))).
% 2.54/1.48 thf(fact_50_in__lset__ldropWhileD, axiom, (! [A:$tType,X3:A,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: (((((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ (((coindu438612276pWhile @ A) @ P) @ Xs))) => (((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ Xs)))))).
% 2.54/1.48 thf(fact_51_ldropWhile__cong, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X2:A]: (((((member @ A) @ X2) @ ((coindu1112613586e_lset @ A) @ Ys)) => ((P @ X2) = (Q @ X2))))) => ((((coindu438612276pWhile @ A) @ P) @ Xs) = (((coindu438612276pWhile @ A) @ Q) @ Ys))))))).
% 2.54/1.48 thf(fact_52_one__enat__def, axiom, ((one_one @ extended_enat) = (extended_enat2 @ (one_one @ nat)))).
% 2.54/1.48 thf(fact_53_enat__1__iff_I1_J, axiom, (! [X3:nat]: ((((extended_enat2 @ X3) = (one_one @ extended_enat)) = (X3 = (one_one @ nat)))))).
% 2.54/1.48 thf(fact_54_enat__1__iff_I2_J, axiom, (! [X3:nat]: ((((one_one @ extended_enat) = (extended_enat2 @ X3)) = (X3 = (one_one @ nat)))))).
% 2.54/1.48 thf(fact_55_less__enatE, axiom, (! [N3:extended_enat,M:nat]: (((((ord_less @ extended_enat) @ N3) @ (extended_enat2 @ M)) => (~ ((! [K2:nat]: (((N3 = (extended_enat2 @ K2)) => (~ ((((ord_less @ nat) @ K2) @ M)))))))))))).
% 2.54/1.48 thf(fact_56_finite__conv__nat__seg__image, axiom, (! [A:$tType]: (((finite_finite2 @ A) = (^ [A4:(set @ A)]: ((? [N:nat,F3:(nat > A)]: ((A4 = ((((image @ nat) @ A) @ F3) @ ((collect @ nat) @ (^ [I:nat]: ((((ord_less @ nat) @ I) @ N)))))))))))))).
% 2.54/1.48 thf(fact_57_nat__seg__image__imp__finite, axiom, (! [A:$tType,A3:(set @ A),F2:(nat > A),N3:nat]: (((A3 = ((((image @ nat) @ A) @ F2) @ ((collect @ nat) @ (^ [I:nat]: ((((ord_less @ nat) @ I) @ N3)))))) => ((finite_finite2 @ A) @ A3))))).
% 2.54/1.48 thf(fact_58_lfinite__conv__llength__enat, axiom, (! [A:$tType]: (((coindu1213758845finite @ A) = (^ [Xs2:(coindu1593790203_llist @ A)]: ((? [N:nat]: ((((coindu1018505716length @ A) @ Xs2) = (extended_enat2 @ N)))))))))).
% 2.54/1.48 thf(fact_59_llength__eq__enat__lfiniteD, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),N3:nat]: (((((coindu1018505716length @ A) @ Xs) = (extended_enat2 @ N3)) => ((coindu1213758845finite @ A) @ Xs))))).
% 2.54/1.48 thf(fact_60_lfinite__llength__enat, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ Xs) => (? [N2:nat]: ((((coindu1018505716length @ A) @ Xs) = (extended_enat2 @ N2)))))))).
% 2.54/1.48 thf(fact_61_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((X3 != Y2) => ((~ ((((ord_less @ A) @ X3) @ Y2))) => (((ord_less @ A) @ Y2) @ X3))))))))).
% 2.54/1.48 thf(fact_62_add__right__imp__eq, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) => (B2 = C)))))))).
% 2.54/1.48 thf(fact_63_add__left__imp__eq, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C)) => (B2 = C)))))))).
% 2.54/1.48 thf(fact_64_add_Oleft__commute, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((plus_plus @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.54/1.48 thf(fact_65_add_Ocommute, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type2 @ A)) => ((plus_plus @ A) = (^ [A6:A,B5:A]: ((((plus_plus @ A) @ B5) @ A6)))))))).
% 2.54/1.48 thf(fact_66_add_Oright__cancel, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B2 = C)))))))).
% 2.54/1.48 thf(fact_67_add_Oleft__cancel, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C)) = (B2 = C)))))))).
% 2.54/1.48 thf(fact_68_add_Oassoc, axiom, (! [A:$tType]: ((((semigroup_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.54/1.48 thf(fact_69_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type2 @ A)) => (! [I2:A,J:A,K:A,L:A]: ((((I2 = J) & (K = L)) => ((((plus_plus @ A) @ I2) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 2.54/1.48 thf(fact_70_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.54/1.48 thf(fact_71_one__reorient, axiom, (! [A:$tType]: ((((one @ A) @ (type2 @ A)) => (! [X3:A]: ((((one_one @ A) = X3) = (X3 = (one_one @ A))))))))).
% 2.54/1.48 thf(fact_72_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X3:A,A3:(set @ A),B2:B,F2:(A > B)]: (((((member @ A) @ X3) @ A3) => ((B2 = (F2 @ X3)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F2) @ A3))))))).
% 2.54/1.48 thf(fact_73_ball__imageD, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ ((((image @ B) @ A) @ F2) @ A3)) => (P @ X2)))) => (! [X4:B]: (((((member @ B) @ X4) @ A3) => (P @ (F2 @ X4))))))))).
% 2.54/1.48 thf(fact_74_image__cong, axiom, (! [B:$tType,A:$tType,M2:(set @ A),N4:(set @ A),F2:(A > B),G2:(A > B)]: (((M2 = N4) => ((! [X2:A]: (((((member @ A) @ X2) @ N4) => ((F2 @ X2) = (G2 @ X2))))) => (((((image @ A) @ B) @ F2) @ M2) = ((((image @ A) @ B) @ G2) @ N4))))))).
% 2.54/1.48 thf(fact_75_bex__imageD, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F2) @ A3)) & (P @ X4)))) => (? [X2:B]: (((((member @ B) @ X2) @ A3) & (P @ (F2 @ X2))))))))).
% 2.54/1.48 thf(fact_76_image__iff, axiom, (! [A:$tType,B:$tType,Z:A,F2:(B > A),A3:(set @ B)]: (((((member @ A) @ Z) @ ((((image @ B) @ A) @ F2) @ A3)) = (? [X:B]: (((((member @ B) @ X) @ A3) & (Z = (F2 @ X))))))))).
% 2.54/1.48 thf(fact_77_imageI, axiom, (! [B:$tType,A:$tType,X3:A,A3:(set @ A),F2:(A > B)]: (((((member @ A) @ X3) @ A3) => (((member @ B) @ (F2 @ X3)) @ ((((image @ A) @ B) @ F2) @ A3)))))).
% 2.54/1.48 thf(fact_78_finite, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => (! [A3:(set @ A)]: (((finite_finite2 @ A) @ A3))))))).
% 2.54/1.48 thf(fact_79_finite__set__choice, axiom, (! [B:$tType,A:$tType,A3:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (? [X1:B]: (((P @ X2) @ X1)))))) => (? [F4:(A > B)]: ((! [X4:A]: (((((member @ A) @ X4) @ A3) => ((P @ X4) @ (F4 @ X4)))))))))))).
% 2.54/1.48 thf(fact_80_Int__left__commute, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ B3) @ C2)) = (((inf_inf @ (set @ A)) @ B3) @ (((inf_inf @ (set @ A)) @ A3) @ C2)))))).
% 2.54/1.48 thf(fact_81_Int__left__absorb, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) = (((inf_inf @ (set @ A)) @ A3) @ B3))))).
% 2.54/1.48 thf(fact_82_Int__commute, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A4:(set @ A),B6:(set @ A)]: ((((inf_inf @ (set @ A)) @ B6) @ A4))))))).
% 2.54/1.48 thf(fact_83_Int__absorb, axiom, (! [A:$tType,A3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ A3) = A3)))).
% 2.54/1.48 thf(fact_84_Int__assoc, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ C2) = (((inf_inf @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ B3) @ C2)))))).
% 2.54/1.48 thf(fact_85_IntD2, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) => (((member @ A) @ C) @ B3))))).
% 2.54/1.48 thf(fact_86_IntD1, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) => (((member @ A) @ C) @ A3))))).
% 2.54/1.48 thf(fact_87_IntE, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) => (~ (((((member @ A) @ C) @ A3) => (~ ((((member @ A) @ C) @ B3)))))))))).
% 2.54/1.48 thf(fact_88_enat__less__induct, axiom, (! [P:(extended_enat > $o),N3:extended_enat]: (((! [N2:extended_enat]: (((! [M3:extended_enat]: (((((ord_less @ extended_enat) @ M3) @ N2) => (P @ M3)))) => (P @ N2)))) => (P @ N3))))).
% 2.54/1.48 thf(fact_89_in__lset__conv__lnth, axiom, (! [A:$tType,X3:A,Xs:(coindu1593790203_llist @ A)]: (((((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ Xs)) = (? [N:nat]: (((((ord_less @ extended_enat) @ (extended_enat2 @ N)) @ ((coindu1018505716length @ A) @ Xs)) & ((((coindu749330388e_lnth @ A) @ Xs) @ N) = X3)))))))).
% 2.54/1.48 thf(fact_90_ltakeWhile__nth, axiom, (! [A:$tType,I2:nat,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: (((((ord_less @ extended_enat) @ (extended_enat2 @ I2)) @ ((coindu1018505716length @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs))) => ((((coindu749330388e_lnth @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs)) @ I2) = (((coindu749330388e_lnth @ A) @ Xs) @ I2)))))).
% 2.54/1.48 thf(fact_91_lfinite__finite__index, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ Xs) => ((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N:nat]: ((((ord_less @ extended_enat) @ (extended_enat2 @ N)) @ ((coindu1018505716length @ A) @ Xs)))))))))).
% 2.54/1.48 thf(fact_92_Compr__image__eq, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B),P:(A > $o)]: ((((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ ((((image @ B) @ A) @ F2) @ A3)) & (P @ X))))) = ((((image @ B) @ A) @ F2) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ A3) & (P @ (F2 @ X))))))))))).
% 2.54/1.48 thf(fact_93_image__image, axiom, (! [A:$tType,B:$tType,C3:$tType,F2:(B > A),G2:(C3 > B),A3:(set @ C3)]: ((((((image @ B) @ A) @ F2) @ ((((image @ C3) @ B) @ G2) @ A3)) = ((((image @ C3) @ A) @ (^ [X:C3]: ((F2 @ (G2 @ X))))) @ A3))))).
% 2.54/1.48 thf(fact_94_imageE, axiom, (! [A:$tType,B:$tType,B2:A,F2:(B > A),A3:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ F2) @ A3)) => (~ ((! [X2:B]: (((B2 = (F2 @ X2)) => (~ ((((member @ B) @ X2) @ A3)))))))))))).
% 2.54/1.48 thf(fact_95_pigeonhole__infinite__rel, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B),R:(A > (B > $o))]: (((~ (((finite_finite2 @ A) @ A3))) => (((finite_finite2 @ B) @ B3) => ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (? [Xa:B]: (((((member @ B) @ Xa) @ B3) & ((R @ X2) @ Xa))))))) => (? [X2:B]: (((((member @ B) @ X2) @ B3) & (~ (((finite_finite2 @ A) @ ((collect @ A) @ (^ [A6:A]: (((((member @ A) @ A6) @ A3) & ((R @ A6) @ X2))))))))))))))))).
% 2.54/1.48 thf(fact_96_not__finite__existsD, axiom, (! [A:$tType,P:(A > $o)]: (((~ (((finite_finite2 @ A) @ ((collect @ A) @ P)))) => (? [X12:A]: ((P @ X12))))))).
% 2.54/1.48 thf(fact_97_Collect__conj__eq, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((collect @ A) @ (^ [X:A]: (((P @ X) & (Q @ X))))) = (((inf_inf @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.54/1.48 thf(fact_98_Int__Collect, axiom, (! [A:$tType,X3:A,A3:(set @ A),P:(A > $o)]: (((((member @ A) @ X3) @ (((inf_inf @ (set @ A)) @ A3) @ ((collect @ A) @ P))) = ((((member @ A) @ X3) @ A3) & (P @ X3)))))).
% 2.54/1.48 thf(fact_99_Int__def, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A4:(set @ A),B6:(set @ A)]: (((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A4) & (((member @ A) @ X) @ B6))))))))))).
% 2.54/1.48 thf(fact_100_add__less__imp__less__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [A2:A,C:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) => (((ord_less @ A) @ A2) @ B2)))))))).
% 2.54/1.48 thf(fact_101_add__less__imp__less__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [C:A,A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) => (((ord_less @ A) @ A2) @ B2)))))))).
% 2.54/1.48 thf(fact_102_add__strict__right__mono, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.54/1.48 thf(fact_103_add__strict__left__mono, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2))))))))).
% 2.54/1.48 thf(fact_104_add__strict__mono, axiom, (! [A:$tType]: ((((strict2144017051up_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A,D:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ D)))))))))).
% 2.54/1.48 thf(fact_105_add__mono__thms__linordered__field_I1_J, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [I2:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I2) @ J) & (K = L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I2) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.54/1.48 thf(fact_106_add__mono__thms__linordered__field_I2_J, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [I2:A,J:A,K:A,L:A]: ((((I2 = J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I2) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.54/1.48 thf(fact_107_add__mono__thms__linordered__field_I5_J, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [I2:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I2) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I2) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.54/1.48 thf(fact_108_enat__iless, axiom, (! [N3:extended_enat,M:nat]: (((((ord_less @ extended_enat) @ N3) @ (extended_enat2 @ M)) => (? [K2:nat]: ((N3 = (extended_enat2 @ K2)))))))).
% 2.54/1.48 thf(fact_109_chain__incr, axiom, (! [A:$tType,Y:(A > extended_enat),K:nat]: (((! [I3:A]: ((? [J2:A]: ((((ord_less @ extended_enat) @ (Y @ I3)) @ (Y @ J2)))))) => (? [J3:A]: ((((ord_less @ extended_enat) @ (extended_enat2 @ K)) @ (Y @ J3)))))))).
% 2.54/1.48 thf(fact_110_pigeonhole__infinite, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F2:(A > B)]: (((~ (((finite_finite2 @ A) @ A3))) => (((finite_finite2 @ B) @ ((((image @ A) @ B) @ F2) @ A3)) => (? [X2:A]: (((((member @ A) @ X2) @ A3) & (~ (((finite_finite2 @ A) @ ((collect @ A) @ (^ [A6:A]: (((((member @ A) @ A6) @ A3) & ((F2 @ A6) = (F2 @ X2))))))))))))))))).
% 2.54/1.48 thf(fact_111_enat__add__mono, axiom, (! [X3:nat,Y2:extended_enat,Z:extended_enat]: (((((ord_less @ extended_enat) @ (((plus_plus @ extended_enat) @ (extended_enat2 @ X3)) @ Y2)) @ (((plus_plus @ extended_enat) @ (extended_enat2 @ X3)) @ Z)) = (((ord_less @ extended_enat) @ Y2) @ Z))))).
% 2.54/1.48 thf(fact_112_enat__less__enat__plusI2, axiom, (! [Y2:nat,Z:extended_enat,X3:nat]: (((((ord_less @ extended_enat) @ (extended_enat2 @ Y2)) @ Z) => (((ord_less @ extended_enat) @ (extended_enat2 @ (((plus_plus @ nat) @ X3) @ Y2))) @ (((plus_plus @ extended_enat) @ (extended_enat2 @ X3)) @ Z)))))).
% 2.54/1.48 thf(fact_113_enat__add2__eq, axiom, (! [Y2:extended_enat,X3:nat,Z:extended_enat]: ((((((plus_plus @ extended_enat) @ Y2) @ (extended_enat2 @ X3)) = (((plus_plus @ extended_enat) @ Z) @ (extended_enat2 @ X3))) = (Y2 = Z))))).
% 2.54/1.48 thf(fact_114_enat__add1__eq, axiom, (! [X3:nat,Y2:extended_enat,Z:extended_enat]: ((((((plus_plus @ extended_enat) @ (extended_enat2 @ X3)) @ Y2) = (((plus_plus @ extended_enat) @ (extended_enat2 @ X3)) @ Z)) = (Y2 = Z))))).
% 2.54/1.48 thf(fact_115_nat__add__left__cancel__less, axiom, (! [K:nat,M:nat,N3:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N3)) = (((ord_less @ nat) @ M) @ N3))))).
% 2.54/1.48 thf(fact_116_inf__right__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X3) @ Y2)) @ Y2) = (((inf_inf @ A) @ X3) @ Y2)))))))).
% 2.54/1.48 thf(fact_117_inf_Oright__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ B2) = (((inf_inf @ A) @ A2) @ B2)))))))).
% 2.54/1.48 thf(fact_118_inf__left__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((inf_inf @ A) @ X3) @ (((inf_inf @ A) @ X3) @ Y2)) = (((inf_inf @ A) @ X3) @ Y2)))))))).
% 2.54/1.48 thf(fact_119_inf_Oleft__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ A2) @ B2)) = (((inf_inf @ A) @ A2) @ B2)))))))).
% 2.54/1.48 thf(fact_120_inf__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X3:A]: (((((inf_inf @ A) @ X3) @ X3) = X3))))))).
% 2.54/1.48 thf(fact_121_inf__apply, axiom, (! [B:$tType,A:$tType]: ((((semilattice_inf @ B) @ (type2 @ B)) => ((inf_inf @ (A > B)) = (^ [F3:(A > B),G3:(A > B),X:A]: ((((inf_inf @ B) @ (F3 @ X)) @ (G3 @ X))))))))).
% 2.54/1.48 thf(fact_122_inf_Oidem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A]: (((((inf_inf @ A) @ A2) @ A2) = A2))))))).
% 2.54/1.48 thf(fact_123_lnull__lfilter, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ (((coindu1889960678filter @ A) @ P) @ Xs)) = (! [X:A]: (((((member @ A) @ X) @ ((coindu1112613586e_lset @ A) @ Xs)) => (~ ((P @ X)))))))))).
% 2.54/1.48 thf(fact_124_lnull__ldropWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ (((coindu438612276pWhile @ A) @ P) @ Xs)) = (! [X:A]: (((((member @ A) @ X) @ ((coindu1112613586e_lset @ A) @ Xs)) => (P @ X)))))))).
% 2.54/1.48 thf(fact_125_psubsetD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C:A]: (((((ord_less @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B3)))))).
% 2.54/1.48 thf(fact_126_nat__neq__iff, axiom, (! [M:nat,N3:nat]: (((M != N3) = ((((ord_less @ nat) @ M) @ N3) | (((ord_less @ nat) @ N3) @ M)))))).
% 2.54/1.48 thf(fact_127_less__set__def, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A4:(set @ A),B6:(set @ A)]: ((((ord_less @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ A4)))) @ (^ [X:A]: ((((member @ A) @ X) @ B6)))))))))).
% 2.54/1.48 thf(fact_128_less__not__refl, axiom, (! [N3:nat]: ((~ ((((ord_less @ nat) @ N3) @ N3)))))).
% 2.54/1.48 thf(fact_129_psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) => ((((ord_less @ (set @ A)) @ B3) @ C2) => (((ord_less @ (set @ A)) @ A3) @ C2)))))).
% 2.54/1.48 thf(fact_130_less__not__refl2, axiom, (! [N3:nat,M:nat]: (((((ord_less @ nat) @ N3) @ M) => (M != N3))))).
% 2.54/1.48 thf(fact_131_less__not__refl3, axiom, (! [S:nat,T2:nat]: (((((ord_less @ nat) @ S) @ T2) => (S != T2))))).
% 2.54/1.48 thf(fact_132_measure__induct, axiom, (! [A:$tType,F2:(A > nat),P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y3:A]: (((((ord_less @ nat) @ (F2 @ Y3)) @ (F2 @ X2)) => (P @ Y3)))) => (P @ X2)))) => (P @ A2))))).
% 2.54/1.48 thf(fact_133_less__irrefl__nat, axiom, (! [N3:nat]: ((~ ((((ord_less @ nat) @ N3) @ N3)))))).
% 2.54/1.48 thf(fact_134_nat__less__induct, axiom, (! [P:(nat > $o),N3:nat]: (((! [N2:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N2) => (P @ M3)))) => (P @ N2)))) => (P @ N3))))).
% 2.54/1.48 thf(fact_135_infinite__descent, axiom, (! [P:(nat > $o),N3:nat]: (((! [N2:nat]: (((~ ((P @ N2))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N2) & (~ ((P @ M3))))))))) => (P @ N3))))).
% 2.54/1.48 thf(fact_136_linorder__neqE__nat, axiom, (! [X3:nat,Y2:nat]: (((X3 != Y2) => ((~ ((((ord_less @ nat) @ X3) @ Y2))) => (((ord_less @ nat) @ Y2) @ X3)))))).
% 2.54/1.48 thf(fact_137_measure__induct__rule, axiom, (! [A:$tType,F2:(A > nat),P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y3:A]: (((((ord_less @ nat) @ (F2 @ Y3)) @ (F2 @ X2)) => (P @ Y3)))) => (P @ X2)))) => (P @ A2))))).
% 2.54/1.48 thf(fact_138_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X3:A]: (((! [X2:A]: (((~ ((P @ X2))) => (? [Y3:A]: (((((ord_less @ nat) @ (V @ Y3)) @ (V @ X2)) & (~ ((P @ Y3))))))))) => (P @ X3))))).
% 2.54/1.48 thf(fact_139_lappend_Oexhaust, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((((coindu335574135_lnull @ A) @ Xs) => (~ (((coindu335574135_lnull @ A) @ Ys)))) => ((~ (((coindu335574135_lnull @ A) @ Xs))) | (~ (((coindu335574135_lnull @ A) @ Ys)))))))).
% 2.54/1.48 thf(fact_140_lzip_Oexhaust, axiom, (! [A:$tType,B:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ B)]: (((~ ((((coindu335574135_lnull @ A) @ Xs) | ((coindu335574135_lnull @ B) @ Ys)))) => (~ (((~ (((coindu335574135_lnull @ A) @ Xs))) => ((coindu335574135_lnull @ B) @ Ys)))))))).
% 2.54/1.48 thf(fact_141_lnull__ltlI, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Xs) => ((coindu335574135_lnull @ A) @ ((coindu1047225960le_ltl @ A) @ Xs)))))).
% 2.54/1.48 thf(fact_142_lnull__imp__lfinite, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Xs) => ((coindu1213758845finite @ A) @ Xs))))).
% 2.54/1.48 thf(fact_143_inf__set__def, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A4:(set @ A),B6:(set @ A)]: (((collect @ A) @ (((inf_inf @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ A4)))) @ (^ [X:A]: ((((member @ A) @ X) @ B6))))))))))).
% 2.54/1.48 thf(fact_144_llist_Oset__sel_I2_J, axiom, (! [A:$tType,A2:(coindu1593790203_llist @ A),X3:A]: (((~ (((coindu335574135_lnull @ A) @ A2))) => ((((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ ((coindu1047225960le_ltl @ A) @ A2))) => (((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ A2))))))).
% 2.54/1.48 thf(fact_145_lfinite__induct, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:((coindu1593790203_llist @ A) > $o)]: ((((coindu1213758845finite @ A) @ Xs) => ((! [Xs3:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Xs3) => (P @ Xs3)))) => ((! [Xs3:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ Xs3) => ((~ (((coindu335574135_lnull @ A) @ Xs3))) => ((P @ ((coindu1047225960le_ltl @ A) @ Xs3)) => (P @ Xs3)))))) => (P @ Xs))))))).
% 2.54/1.48 thf(fact_146_inf__sup__aci_I4_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((inf_inf @ A) @ X3) @ (((inf_inf @ A) @ X3) @ Y2)) = (((inf_inf @ A) @ X3) @ Y2)))))))).
% 2.54/1.48 thf(fact_147_inf__sup__aci_I3_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X3:A,Y2:A,Z:A]: (((((inf_inf @ A) @ X3) @ (((inf_inf @ A) @ Y2) @ Z)) = (((inf_inf @ A) @ Y2) @ (((inf_inf @ A) @ X3) @ Z))))))))).
% 2.54/1.48 thf(fact_148_inf__sup__aci_I2_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X3:A,Y2:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X3) @ Y2)) @ Z) = (((inf_inf @ A) @ X3) @ (((inf_inf @ A) @ Y2) @ Z))))))))).
% 2.54/1.48 thf(fact_149_inf__sup__aci_I1_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => ((inf_inf @ A) = (^ [X:A,Y4:A]: ((((inf_inf @ A) @ Y4) @ X)))))))).
% 2.54/1.48 thf(fact_150_inf__fun__def, axiom, (! [B:$tType,A:$tType]: ((((semilattice_inf @ B) @ (type2 @ B)) => ((inf_inf @ (A > B)) = (^ [F3:(A > B),G3:(A > B),X:A]: ((((inf_inf @ B) @ (F3 @ X)) @ (G3 @ X))))))))).
% 2.54/1.48 thf(fact_151_inf_Oassoc, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C))))))))).
% 2.54/1.48 thf(fact_152_inf__assoc, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X3:A,Y2:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X3) @ Y2)) @ Z) = (((inf_inf @ A) @ X3) @ (((inf_inf @ A) @ Y2) @ Z))))))))).
% 2.54/1.48 thf(fact_153_inf_Ocommute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => ((inf_inf @ A) = (^ [A6:A,B5:A]: ((((inf_inf @ A) @ B5) @ A6)))))))).
% 2.54/1.48 thf(fact_154_inf__commute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => ((inf_inf @ A) = (^ [X:A,Y4:A]: ((((inf_inf @ A) @ Y4) @ X)))))))).
% 2.54/1.48 thf(fact_155_inf_Oleft__commute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((inf_inf @ A) @ B2) @ (((inf_inf @ A) @ A2) @ C)) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C))))))))).
% 2.54/1.48 thf(fact_156_inf__left__commute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X3:A,Y2:A,Z:A]: (((((inf_inf @ A) @ X3) @ (((inf_inf @ A) @ Y2) @ Z)) = (((inf_inf @ A) @ Y2) @ (((inf_inf @ A) @ X3) @ Z))))))))).
% 2.54/1.48 thf(fact_157_nat__add__left__cancel, axiom, (! [K:nat,M:nat,N3:nat]: ((((((plus_plus @ nat) @ K) @ M) = (((plus_plus @ nat) @ K) @ N3)) = (M = N3))))).
% 2.54/1.48 thf(fact_158_nat__add__right__cancel, axiom, (! [M:nat,K:nat,N3:nat]: ((((((plus_plus @ nat) @ M) @ K) = (((plus_plus @ nat) @ N3) @ K)) = (M = N3))))).
% 2.54/1.48 thf(fact_159_add__lessD1, axiom, (! [I2:nat,J:nat,K:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I2) @ J)) @ K) => (((ord_less @ nat) @ I2) @ K))))).
% 2.54/1.48 thf(fact_160_add__less__mono, axiom, (! [I2:nat,J:nat,K:nat,L:nat]: (((((ord_less @ nat) @ I2) @ J) => ((((ord_less @ nat) @ K) @ L) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I2) @ K)) @ (((plus_plus @ nat) @ J) @ L))))))).
% 2.54/1.48 thf(fact_161_not__add__less1, axiom, (! [I2:nat,J:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ I2) @ J)) @ I2)))))).
% 2.54/1.48 thf(fact_162_not__add__less2, axiom, (! [J:nat,I2:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ J) @ I2)) @ I2)))))).
% 2.54/1.48 thf(fact_163_add__less__mono1, axiom, (! [I2:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I2) @ J) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I2) @ K)) @ (((plus_plus @ nat) @ J) @ K)))))).
% 2.54/1.48 thf(fact_164_trans__less__add1, axiom, (! [I2:nat,J:nat,M:nat]: (((((ord_less @ nat) @ I2) @ J) => (((ord_less @ nat) @ I2) @ (((plus_plus @ nat) @ J) @ M)))))).
% 2.54/1.48 thf(fact_165_trans__less__add2, axiom, (! [I2:nat,J:nat,M:nat]: (((((ord_less @ nat) @ I2) @ J) => (((ord_less @ nat) @ I2) @ (((plus_plus @ nat) @ M) @ J)))))).
% 2.54/1.48 thf(fact_166_less__add__eq__less, axiom, (! [K:nat,L:nat,M:nat,N3:nat]: (((((ord_less @ nat) @ K) @ L) => (((((plus_plus @ nat) @ M) @ L) = (((plus_plus @ nat) @ K) @ N3)) => (((ord_less @ nat) @ M) @ N3)))))).
% 2.54/1.48 thf(fact_167_less__infI1, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,X3:A,B2:A]: (((((ord_less @ A) @ A2) @ X3) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ X3)))))))).
% 2.54/1.48 thf(fact_168_less__infI2, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [B2:A,X3:A,A2:A]: (((((ord_less @ A) @ B2) @ X3) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ X3)))))))).
% 2.54/1.48 thf(fact_169_inf_Ostrict__boundedE, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C)) => (~ (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ A2) @ C))))))))))))).
% 2.54/1.48 thf(fact_170_inf_Ostrict__order__iff, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A6:A,B5:A]: (((A6 = (((inf_inf @ A) @ A6) @ B5)) & (A6 != B5))))))))).
% 2.54/1.48 thf(fact_171_inf_Ostrict__coboundedI1, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,C:A,B2:A]: (((((ord_less @ A) @ A2) @ C) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C)))))))).
% 2.54/1.48 thf(fact_172_inf_Ostrict__coboundedI2, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [B2:A,C:A,A2:A]: (((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C)))))))).
% 2.54/1.48 thf(fact_173_enat__less__enat__plusI, axiom, (! [X3:nat,Y2:nat,Z:extended_enat]: (((((ord_less @ nat) @ X3) @ Y2) => (((ord_less @ extended_enat) @ (extended_enat2 @ X3)) @ (((plus_plus @ extended_enat) @ (extended_enat2 @ Y2)) @ Z)))))).
% 2.54/1.48 thf(fact_174_fold__atLeastAtMost__nat_Oinduct, axiom, (! [A:$tType,P:((nat > (A > A)) > (nat > (nat > (A > $o)))),A0:(nat > (A > A)),A1:nat,A22:nat,A32:A]: (((! [F4:(nat > (A > A)),A7:nat,B7:nat,Acc:A]: ((((~ ((((ord_less @ nat) @ B7) @ A7))) => ((((P @ F4) @ (((plus_plus @ nat) @ A7) @ (one_one @ nat))) @ B7) @ ((F4 @ A7) @ Acc))) => ((((P @ F4) @ A7) @ B7) @ Acc)))) => ((((P @ A0) @ A1) @ A22) @ A32))))).
% 2.54/1.48 thf(fact_175_finite__M__bounded__by__nat, axiom, (! [P:(nat > $o),I2:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((P @ K3) & (((ord_less @ nat) @ K3) @ I2))))))))).
% 2.54/1.48 thf(fact_176_gen__llength__def, axiom, (! [A:$tType]: (((coindu493225792length @ A) = (^ [N:nat,Xs2:(coindu1593790203_llist @ A)]: ((((plus_plus @ extended_enat) @ (extended_enat2 @ N)) @ ((coindu1018505716length @ A) @ Xs2)))))))).
% 2.54/1.48 thf(fact_177_unbounded__k__infinite, axiom, (! [K:nat,S2:(set @ nat)]: (((! [M4:nat]: (((((ord_less @ nat) @ K) @ M4) => (? [N5:nat]: (((((ord_less @ nat) @ M4) @ N5) & (((member @ nat) @ N5) @ S2))))))) => (~ (((finite_finite2 @ nat) @ S2))))))).
% 2.54/1.48 thf(fact_178_inf1I, axiom, (! [A:$tType,A3:(A > $o),X3:A,B3:(A > $o)]: (((A3 @ X3) => ((B3 @ X3) => ((((inf_inf @ (A > $o)) @ A3) @ B3) @ X3)))))).
% 2.54/1.48 thf(fact_179_inf1E, axiom, (! [A:$tType,A3:(A > $o),B3:(A > $o),X3:A]: ((((((inf_inf @ (A > $o)) @ A3) @ B3) @ X3) => (~ (((A3 @ X3) => (~ ((B3 @ X3)))))))))).
% 2.54/1.48 thf(fact_180_inf1D1, axiom, (! [A:$tType,A3:(A > $o),B3:(A > $o),X3:A]: ((((((inf_inf @ (A > $o)) @ A3) @ B3) @ X3) => (A3 @ X3))))).
% 2.54/1.48 thf(fact_181_inf1D2, axiom, (! [A:$tType,A3:(A > $o),B3:(A > $o),X3:A]: ((((((inf_inf @ (A > $o)) @ A3) @ B3) @ X3) => (B3 @ X3))))).
% 2.54/1.48 thf(fact_182_finite__nat__set__iff__bounded, axiom, ((finite_finite2 @ nat) = (^ [N6:(set @ nat)]: ((? [M5:nat]: ((! [X:nat]: (((((member @ nat) @ X) @ N6) => (((ord_less @ nat) @ X) @ M5)))))))))).
% 2.54/1.48 thf(fact_183_infinite__nat__iff__unbounded, axiom, (! [S2:(set @ nat)]: (((~ (((finite_finite2 @ nat) @ S2))) = (! [M5:nat]: ((? [N:nat]: (((((ord_less @ nat) @ M5) @ N) & (((member @ nat) @ N) @ S2)))))))))).
% 2.54/1.48 thf(fact_184_bounded__nat__set__is__finite, axiom, (! [N4:(set @ nat),N3:nat]: (((! [X2:nat]: (((((member @ nat) @ X2) @ N4) => (((ord_less @ nat) @ X2) @ N3)))) => ((finite_finite2 @ nat) @ N4))))).
% 2.54/1.48 thf(fact_185_inf__Int__eq, axiom, (! [A:$tType,R:(set @ A),S2:(set @ A)]: (((((inf_inf @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ R)))) @ (^ [X:A]: ((((member @ A) @ X) @ S2)))) = (^ [X:A]: ((((member @ A) @ X) @ (((inf_inf @ (set @ A)) @ R) @ S2)))))))).
% 2.54/1.48 thf(fact_186_lnth__llength__ltakeWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: (((((ord_less @ extended_enat) @ ((coindu1018505716length @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs))) @ ((coindu1018505716length @ A) @ Xs)) => (~ ((P @ (((coindu749330388e_lnth @ A) @ Xs) @ (extended_the_enat @ ((coindu1018505716length @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs))))))))))).
% 2.54/1.48 thf(fact_187_lnth__ldropn, axiom, (! [A:$tType,N3:nat,M:nat,Xs:(coindu1593790203_llist @ A)]: (((((ord_less @ extended_enat) @ (extended_enat2 @ (((plus_plus @ nat) @ N3) @ M))) @ ((coindu1018505716length @ A) @ Xs)) => ((((coindu749330388e_lnth @ A) @ (((coindu531130065ldropn @ A) @ N3) @ Xs)) @ M) = (((coindu749330388e_lnth @ A) @ Xs) @ (((plus_plus @ nat) @ M) @ N3))))))).
% 2.54/1.48 thf(fact_188_ldropn__ldropn, axiom, (! [A:$tType,N3:nat,M:nat,Xs:(coindu1593790203_llist @ A)]: (((((coindu531130065ldropn @ A) @ N3) @ (((coindu531130065ldropn @ A) @ M) @ Xs)) = (((coindu531130065ldropn @ A) @ (((plus_plus @ nat) @ N3) @ M)) @ Xs))))).
% 2.54/1.48 thf(fact_189_lfinite__ldropn, axiom, (! [A:$tType,N3:nat,Xs:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ (((coindu531130065ldropn @ A) @ N3) @ Xs)) = ((coindu1213758845finite @ A) @ Xs))))).
% 2.54/1.48 thf(fact_190_the__enat_Osimps, axiom, (! [N3:nat]: (((extended_the_enat @ (extended_enat2 @ N3)) = N3)))).
% 2.54/1.48 thf(fact_191_ltl__ldropn, axiom, (! [A:$tType,N3:nat,Xs:(coindu1593790203_llist @ A)]: ((((coindu1047225960le_ltl @ A) @ (((coindu531130065ldropn @ A) @ N3) @ Xs)) = (((coindu531130065ldropn @ A) @ N3) @ ((coindu1047225960le_ltl @ A) @ Xs)))))).
% 2.54/1.48 thf(fact_192_in__lset__ldropnD, axiom, (! [A:$tType,X3:A,N3:nat,Xs:(coindu1593790203_llist @ A)]: (((((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ (((coindu531130065ldropn @ A) @ N3) @ Xs))) => (((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ Xs)))))).
% 2.54/1.48 thf(fact_193_lnth__ldrop, axiom, (! [A:$tType,N3:extended_enat,M:nat,Xs:(coindu1593790203_llist @ A)]: (((((ord_less @ extended_enat) @ (((plus_plus @ extended_enat) @ N3) @ (extended_enat2 @ M))) @ ((coindu1018505716length @ A) @ Xs)) => ((((coindu749330388e_lnth @ A) @ (((coindu191418589_ldrop @ A) @ N3) @ Xs)) @ M) = (((coindu749330388e_lnth @ A) @ Xs) @ (((plus_plus @ nat) @ M) @ (extended_the_enat @ N3)))))))).
% 2.54/1.48 thf(fact_194_less__numeral__extra_I4_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (one_one @ A)))))))).
% 2.54/1.48 thf(fact_195_ldrop__ldrop, axiom, (! [A:$tType,N3:extended_enat,M:extended_enat,Xs:(coindu1593790203_llist @ A)]: (((((coindu191418589_ldrop @ A) @ N3) @ (((coindu191418589_ldrop @ A) @ M) @ Xs)) = (((coindu191418589_ldrop @ A) @ (((plus_plus @ extended_enat) @ N3) @ M)) @ Xs))))).
% 2.54/1.48 thf(fact_196_in__lset__ldropD, axiom, (! [A:$tType,X3:A,N3:extended_enat,Xs:(coindu1593790203_llist @ A)]: (((((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ (((coindu191418589_ldrop @ A) @ N3) @ Xs))) => (((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ Xs)))))).
% 2.54/1.48 thf(fact_197_ltl__ldrop, axiom, (! [A:$tType,N3:extended_enat,Xs:(coindu1593790203_llist @ A)]: ((((coindu1047225960le_ltl @ A) @ (((coindu191418589_ldrop @ A) @ N3) @ Xs)) = (((coindu191418589_ldrop @ A) @ N3) @ ((coindu1047225960le_ltl @ A) @ Xs)))))).
% 2.54/1.48 thf(fact_198_ldrop__enat, axiom, (! [A:$tType,N3:nat,Xs:(coindu1593790203_llist @ A)]: (((((coindu191418589_ldrop @ A) @ (extended_enat2 @ N3)) @ Xs) = (((coindu531130065ldropn @ A) @ N3) @ Xs))))).
% 2.54/1.48 thf(fact_199_is__num__normalize_I1_J, axiom, (! [A:$tType]: ((((neg_numeral @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.54/1.48 thf(fact_200_ldropWhile__eq__ldrop, axiom, (! [A:$tType]: (((coindu438612276pWhile @ A) = (^ [P2:(A > $o),Xs2:(coindu1593790203_llist @ A)]: ((((coindu191418589_ldrop @ A) @ ((coindu1018505716length @ A) @ (((coindu721411036eWhile @ A) @ P2) @ Xs2))) @ Xs2))))))).
% 2.54/1.48 thf(fact_201_lhd__ldropWhile__conv__lnth, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((coindu1112613586e_lset @ A) @ Xs)) & (~ ((P @ X4)))))) => (((coindu1046438764le_lhd @ A) @ (((coindu438612276pWhile @ A) @ P) @ Xs)) = (((coindu749330388e_lnth @ A) @ Xs) @ (extended_the_enat @ ((coindu1018505716length @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs))))))))).
% 2.54/1.48 thf(fact_202_llast__ldropn, axiom, (! [A:$tType,N3:nat,Xs:(coindu1593790203_llist @ A)]: (((((ord_less @ extended_enat) @ (extended_enat2 @ N3)) @ ((coindu1018505716length @ A) @ Xs)) => (((coindu2000965700_llast @ A) @ (((coindu531130065ldropn @ A) @ N3) @ Xs)) = ((coindu2000965700_llast @ A) @ Xs)))))).
% 2.54/1.48 thf(fact_203_lnull__ltakeWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs)) = ((~ (((coindu335574135_lnull @ A) @ Xs))) => (~ ((P @ ((coindu1046438764le_lhd @ A) @ Xs))))))))).
% 2.54/1.48 thf(fact_204_ltakeWhile_Odisc__iff_I1_J, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs)) = (((coindu335574135_lnull @ A) @ Xs) | (~ ((P @ ((coindu1046438764le_lhd @ A) @ Xs))))))))).
% 2.54/1.48 thf(fact_205_ltakeWhile_Odisc__iff_I2_J, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: (((~ (((coindu335574135_lnull @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs)))) = ((~ (((coindu335574135_lnull @ A) @ Xs))) & (P @ ((coindu1046438764le_lhd @ A) @ Xs))))))).
% 2.54/1.48 thf(fact_206_ltakeWhile_Oexhaust, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((~ ((((coindu335574135_lnull @ A) @ Xs) | (~ ((P @ ((coindu1046438764le_lhd @ A) @ Xs))))))) => (~ (((~ (((coindu335574135_lnull @ A) @ Xs))) => (~ ((P @ ((coindu1046438764le_lhd @ A) @ Xs))))))))))).
% 2.54/1.48 thf(fact_207_llist_Oset__sel_I1_J, axiom, (! [A:$tType,A2:(coindu1593790203_llist @ A)]: (((~ (((coindu335574135_lnull @ A) @ A2))) => (((member @ A) @ ((coindu1046438764le_lhd @ A) @ A2)) @ ((coindu1112613586e_lset @ A) @ A2)))))).
% 2.54/1.48 thf(fact_208_llist_Ocoinduct__strong, axiom, (! [A:$tType,R:((coindu1593790203_llist @ A) > ((coindu1593790203_llist @ A) > $o)),Llist:(coindu1593790203_llist @ A),Llist2:(coindu1593790203_llist @ A)]: ((((R @ Llist) @ Llist2) => ((! [Llist3:(coindu1593790203_llist @ A),Llist4:(coindu1593790203_llist @ A)]: ((((R @ Llist3) @ Llist4) => ((((coindu335574135_lnull @ A) @ Llist3) = ((coindu335574135_lnull @ A) @ Llist4)) & ((~ (((coindu335574135_lnull @ A) @ Llist3))) => ((~ (((coindu335574135_lnull @ A) @ Llist4))) => ((((coindu1046438764le_lhd @ A) @ Llist3) = ((coindu1046438764le_lhd @ A) @ Llist4)) & (((R @ ((coindu1047225960le_ltl @ A) @ Llist3)) @ ((coindu1047225960le_ltl @ A) @ Llist4)) | (((coindu1047225960le_ltl @ A) @ Llist3) = ((coindu1047225960le_ltl @ A) @ Llist4)))))))))) => (Llist = Llist2)))))).
% 2.54/1.48 thf(fact_209_llist_Ocoinduct, axiom, (! [A:$tType,R:((coindu1593790203_llist @ A) > ((coindu1593790203_llist @ A) > $o)),Llist:(coindu1593790203_llist @ A),Llist2:(coindu1593790203_llist @ A)]: ((((R @ Llist) @ Llist2) => ((! [Llist3:(coindu1593790203_llist @ A),Llist4:(coindu1593790203_llist @ A)]: ((((R @ Llist3) @ Llist4) => ((((coindu335574135_lnull @ A) @ Llist3) = ((coindu335574135_lnull @ A) @ Llist4)) & ((~ (((coindu335574135_lnull @ A) @ Llist3))) => ((~ (((coindu335574135_lnull @ A) @ Llist4))) => ((((coindu1046438764le_lhd @ A) @ Llist3) = ((coindu1046438764le_lhd @ A) @ Llist4)) & ((R @ ((coindu1047225960le_ltl @ A) @ Llist3)) @ ((coindu1047225960le_ltl @ A) @ Llist4))))))))) => (Llist = Llist2)))))).
% 2.54/1.48 thf(fact_210_llist_Oexpand, axiom, (! [A:$tType,Llist:(coindu1593790203_llist @ A),Llist2:(coindu1593790203_llist @ A)]: (((((coindu335574135_lnull @ A) @ Llist) = ((coindu335574135_lnull @ A) @ Llist2)) => (((~ (((coindu335574135_lnull @ A) @ Llist))) => ((~ (((coindu335574135_lnull @ A) @ Llist2))) => ((((coindu1046438764le_lhd @ A) @ Llist) = ((coindu1046438764le_lhd @ A) @ Llist2)) & (((coindu1047225960le_ltl @ A) @ Llist) = ((coindu1047225960le_ltl @ A) @ Llist2))))) => (Llist = Llist2)))))).
% 2.54/1.48 thf(fact_211_ltakeWhile_Odisc_I2_J, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((~ (((coindu335574135_lnull @ A) @ Xs))) => ((P @ ((coindu1046438764le_lhd @ A) @ Xs)) => (~ (((coindu335574135_lnull @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs))))))))).
% 2.54/1.48 thf(fact_212_ltakeWhile_Odisc_I1_J, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((((coindu335574135_lnull @ A) @ Xs) | (~ ((P @ ((coindu1046438764le_lhd @ A) @ Xs))))) => ((coindu335574135_lnull @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs)))))).
% 2.54/1.48 thf(fact_213_lhd__ltakeWhile, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((~ (((coindu335574135_lnull @ A) @ Xs))) => ((P @ ((coindu1046438764le_lhd @ A) @ Xs)) => (((coindu1046438764le_lhd @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs)) = ((coindu1046438764le_lhd @ A) @ Xs))))))).
% 2.54/1.48 thf(fact_214_lhd__ldropWhile__in__lset, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((coindu1112613586e_lset @ A) @ Xs)) & (~ ((P @ X4)))))) => (((member @ A) @ ((coindu1046438764le_lhd @ A) @ (((coindu438612276pWhile @ A) @ P) @ Xs))) @ ((coindu1112613586e_lset @ A) @ Xs)))))).
% 2.54/1.48 thf(fact_215_lhd__ldropWhile, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((coindu1112613586e_lset @ A) @ Xs)) & (~ ((P @ X4)))))) => (~ ((P @ ((coindu1046438764le_lhd @ A) @ (((coindu438612276pWhile @ A) @ P) @ Xs))))))))).
% 2.54/1.48 thf(fact_216_llist__set__induct, axiom, (! [A:$tType,X3:A,Xs:(coindu1593790203_llist @ A),P:(A > ((coindu1593790203_llist @ A) > $o))]: (((((member @ A) @ X3) @ ((coindu1112613586e_lset @ A) @ Xs)) => ((! [Xs3:(coindu1593790203_llist @ A)]: (((~ (((coindu335574135_lnull @ A) @ Xs3))) => ((P @ ((coindu1046438764le_lhd @ A) @ Xs3)) @ Xs3)))) => ((! [Xs3:(coindu1593790203_llist @ A),Y5:A]: (((~ (((coindu335574135_lnull @ A) @ Xs3))) => ((((member @ A) @ Y5) @ ((coindu1112613586e_lset @ A) @ ((coindu1047225960le_ltl @ A) @ Xs3))) => (((P @ Y5) @ ((coindu1047225960le_ltl @ A) @ Xs3)) => ((P @ Y5) @ Xs3)))))) => ((P @ X3) @ Xs))))))).
% 2.54/1.48 thf(fact_217_ltakeWhile_Osimps_I4_J, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((~ (((coindu335574135_lnull @ A) @ Xs))) => ((P @ ((coindu1046438764le_lhd @ A) @ Xs)) => (((coindu1047225960le_ltl @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs)) = (((coindu721411036eWhile @ A) @ P) @ ((coindu1047225960le_ltl @ A) @ Xs)))))))).
% 2.54/1.48 thf(fact_218_llast__ldrop, axiom, (! [A:$tType,N3:extended_enat,Xs:(coindu1593790203_llist @ A)]: (((((ord_less @ extended_enat) @ N3) @ ((coindu1018505716length @ A) @ Xs)) => (((coindu2000965700_llast @ A) @ (((coindu191418589_ldrop @ A) @ N3) @ Xs)) = ((coindu2000965700_llast @ A) @ Xs)))))).
% 2.54/1.48 thf(fact_219_lhd__ldropn, axiom, (! [A:$tType,N3:nat,Xs:(coindu1593790203_llist @ A)]: (((((ord_less @ extended_enat) @ (extended_enat2 @ N3)) @ ((coindu1018505716length @ A) @ Xs)) => (((coindu1046438764le_lhd @ A) @ (((coindu531130065ldropn @ A) @ N3) @ Xs)) = (((coindu749330388e_lnth @ A) @ Xs) @ N3)))))).
% 2.54/1.48 thf(fact_220_lhd__ldrop, axiom, (! [A:$tType,N3:extended_enat,Xs:(coindu1593790203_llist @ A)]: (((((ord_less @ extended_enat) @ N3) @ ((coindu1018505716length @ A) @ Xs)) => (((coindu1046438764le_lhd @ A) @ (((coindu191418589_ldrop @ A) @ N3) @ Xs)) = (((coindu749330388e_lnth @ A) @ Xs) @ (extended_the_enat @ N3))))))).
% 2.54/1.48 thf(fact_221_enat_Osimps_I6_J, axiom, (! [T:$tType,F1:(nat > T),F22:T,Nat:nat]: ((((((extended_rec_enat @ T) @ F1) @ F22) @ (extended_enat2 @ Nat)) = (F1 @ Nat))))).
% 2.54/1.48 thf(fact_222_lset__conv__lnth, axiom, (! [A:$tType]: (((coindu1112613586e_lset @ A) = (^ [Xs2:(coindu1593790203_llist @ A)]: (((collect @ A) @ (^ [Uu:A]: ((? [N:nat]: (((Uu = (((coindu749330388e_lnth @ A) @ Xs2) @ N)) & (((ord_less @ extended_enat) @ (extended_enat2 @ N)) @ ((coindu1018505716length @ A) @ Xs2)))))))))))))).
% 2.54/1.48 thf(fact_223_finite__Collect__bounded__ex, axiom, (! [B:$tType,A:$tType,P:(A > $o),Q:(B > (A > $o))]: ((((finite_finite2 @ A) @ ((collect @ A) @ P)) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [X:B]: ((? [Y4:A]: (((P @ Y4) & ((Q @ X) @ Y4)))))))) = (! [Y4:A]: (((P @ Y4) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [X:B]: (((Q @ X) @ Y4))))))))))))).
% 2.54/1.48 thf(fact_224_finite__image__set, axiom, (! [A:$tType,B:$tType,P:(A > $o),F2:(A > B)]: ((((finite_finite2 @ A) @ ((collect @ A) @ P)) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [Uu:B]: ((? [X:A]: (((Uu = (F2 @ X)) & (P @ X)))))))))))).
% 2.54/1.48 thf(fact_225_finite__image__set2, axiom, (! [A:$tType,B:$tType,C3:$tType,P:(A > $o),Q:(B > $o),F2:(A > (B > C3))]: ((((finite_finite2 @ A) @ ((collect @ A) @ P)) => (((finite_finite2 @ B) @ ((collect @ B) @ Q)) => ((finite_finite2 @ C3) @ ((collect @ C3) @ (^ [Uu:C3]: ((? [X:A,Y4:B]: (((Uu = ((F2 @ X) @ Y4)) & ((P @ X) & (Q @ Y4)))))))))))))).
% 2.54/1.48 thf(fact_226_Setcompr__eq__image, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B)]: ((((collect @ A) @ (^ [Uu:A]: ((? [X:B]: (((Uu = (F2 @ X)) & (((member @ B) @ X) @ A3))))))) = ((((image @ B) @ A) @ F2) @ A3))))).
% 2.54/1.48 thf(fact_227_setcompr__eq__image, axiom, (! [A:$tType,B:$tType,F2:(B > A),P:(B > $o)]: ((((collect @ A) @ (^ [Uu:A]: ((? [X:B]: (((Uu = (F2 @ X)) & (P @ X))))))) = ((((image @ B) @ A) @ F2) @ ((collect @ B) @ P)))))).
% 2.54/1.48 thf(fact_228_lnull__ldropn, axiom, (! [A:$tType,N3:nat,Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ (((coindu531130065ldropn @ A) @ N3) @ Xs)) = (((ord_less_eq @ extended_enat) @ ((coindu1018505716length @ A) @ Xs)) @ (extended_enat2 @ N3)))))).
% 2.54/1.48 thf(fact_229_llength__def, axiom, (! [A:$tType]: (((coindu1018505716length @ A) = (((coindu1491768222unfold @ (coindu1593790203_llist @ A)) @ (coindu335574135_lnull @ A)) @ (coindu1047225960le_ltl @ A)))))).
% 2.54/1.48 thf(fact_230_add__le__cancel__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [A2:A,C:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.54/1.48 thf(fact_231_add__le__cancel__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [C:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.54/1.48 thf(fact_232_inf_Obounded__iff, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C)) = ((((ord_less_eq @ A) @ A2) @ B2) & (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.54/1.48 thf(fact_233_le__inf__iff, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X3:A,Y2:A,Z:A]: (((((ord_less_eq @ A) @ X3) @ (((inf_inf @ A) @ Y2) @ Z)) = ((((ord_less_eq @ A) @ X3) @ Y2) & (((ord_less_eq @ A) @ X3) @ Z))))))))).
% 2.54/1.48 thf(fact_234_lnull__ldrop, axiom, (! [A:$tType,N3:extended_enat,Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ (((coindu191418589_ldrop @ A) @ N3) @ Xs)) = (((ord_less_eq @ extended_enat) @ ((coindu1018505716length @ A) @ Xs)) @ N3))))).
% 2.54/1.48 thf(fact_235_le__numeral__extra_I4_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))).
% 2.54/1.48 thf(fact_236_enat__less__imp__le, axiom, (! [N3:extended_enat,M:extended_enat]: (((! [K2:nat]: (((((ord_less @ extended_enat) @ N3) @ (extended_enat2 @ K2)) => (((ord_less @ extended_enat) @ M) @ (extended_enat2 @ K2))))) => (((ord_less_eq @ extended_enat) @ M) @ N3))))).
% 2.54/1.48 thf(fact_237_llength__ltakeWhile__le, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: ((((ord_less_eq @ extended_enat) @ ((coindu1018505716length @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs))) @ ((coindu1018505716length @ A) @ Xs))))).
% 2.54/1.48 thf(fact_238_inf__sup__ord_I2_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X3) @ Y2)) @ Y2))))))).
% 2.54/1.48 thf(fact_239_inf__sup__ord_I1_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X3) @ Y2)) @ X3))))))).
% 2.54/1.48 thf(fact_240_inf__le1, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X3) @ Y2)) @ X3))))))).
% 2.54/1.48 thf(fact_241_inf__le2, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X3) @ Y2)) @ Y2))))))).
% 2.54/1.48 thf(fact_242_le__infE, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X3:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ X3) @ (((inf_inf @ A) @ A2) @ B2)) => (~ (((((ord_less_eq @ A) @ X3) @ A2) => (~ ((((ord_less_eq @ A) @ X3) @ B2))))))))))))).
% 2.54/1.48 thf(fact_243_le__infI, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X3:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ X3) @ A2) => ((((ord_less_eq @ A) @ X3) @ B2) => (((ord_less_eq @ A) @ X3) @ (((inf_inf @ A) @ A2) @ B2)))))))))).
% 2.54/1.48 thf(fact_244_inf__mono, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,C:A,B2:A,D:A]: (((((ord_less_eq @ A) @ A2) @ C) => ((((ord_less_eq @ A) @ B2) @ D) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ (((inf_inf @ A) @ C) @ D)))))))))).
% 2.54/1.48 thf(fact_245_le__infI1, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,X3:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ X3) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ X3)))))))).
% 2.54/1.48 thf(fact_246_le__infI2, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [B2:A,X3:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ X3) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ X3)))))))).
% 2.54/1.48 thf(fact_247_inf_OorderE, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (A2 = (((inf_inf @ A) @ A2) @ B2))))))))).
% 2.54/1.48 thf(fact_248_inf_OorderI, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((A2 = (((inf_inf @ A) @ A2) @ B2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.54/1.48 thf(fact_249_inf__unique, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [F2:(A > (A > A)),X3:A,Y2:A]: (((! [X2:A,Y5:A]: ((((ord_less_eq @ A) @ ((F2 @ X2) @ Y5)) @ X2))) => ((! [X2:A,Y5:A]: ((((ord_less_eq @ A) @ ((F2 @ X2) @ Y5)) @ Y5))) => ((! [X2:A,Y5:A,Z2:A]: (((((ord_less_eq @ A) @ X2) @ Y5) => ((((ord_less_eq @ A) @ X2) @ Z2) => (((ord_less_eq @ A) @ X2) @ ((F2 @ Y5) @ Z2)))))) => ((((inf_inf @ A) @ X3) @ Y2) = ((F2 @ X3) @ Y2))))))))))).
% 2.54/1.48 thf(fact_250_le__iff__inf, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X:A,Y4:A]: (((((inf_inf @ A) @ X) @ Y4) = X)))))))).
% 2.54/1.48 thf(fact_251_inf_Oabsorb1, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((inf_inf @ A) @ A2) @ B2) = A2)))))))).
% 2.54/1.48 thf(fact_252_inf_Oabsorb2, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((inf_inf @ A) @ A2) @ B2) = B2)))))))).
% 2.54/1.48 thf(tcon_fun___Lattices_Osemilattice__inf, axiom, (! [A8:$tType,A9:$tType]: ((((semilattice_inf @ A9) @ (type2 @ A9)) => ((semilattice_inf @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.54/1.48 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))))))).
% 2.54/1.48 thf(tcon_fun___Lattices_Olattice, axiom, (! [A8:$tType,A9:$tType]: ((((lattice @ A9) @ (type2 @ A9)) => ((lattice @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.54/1.48 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add, axiom, ((strict2144017051up_add @ nat) @ (type2 @ nat))).
% 2.54/1.48 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add, axiom, ((ordere223160158up_add @ nat) @ (type2 @ nat))).
% 2.54/1.48 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, axiom, ((ordere236663937imp_le @ nat) @ (type2 @ nat))).
% 2.54/1.48 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, axiom, ((ordere779506340up_add @ nat) @ (type2 @ nat))).
% 2.54/1.48 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add, axiom, ((cancel_semigroup_add @ nat) @ (type2 @ nat))).
% 2.54/1.48 thf(tcon_Nat_Onat___Rings_Olinordered__semidom, axiom, ((linordered_semidom @ nat) @ (type2 @ nat))).
% 2.54/1.48 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_1, axiom, ((semilattice_inf @ nat) @ (type2 @ nat))).
% 2.54/1.48 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add, axiom, ((ab_semigroup_add @ nat) @ (type2 @ nat))).
% 2.54/1.48 thf(tcon_Nat_Onat___Groups_Osemigroup__add, axiom, ((semigroup_add @ nat) @ (type2 @ nat))).
% 2.54/1.48 thf(tcon_Nat_Onat___Lattices_Olattice_2, axiom, ((lattice @ nat) @ (type2 @ nat))).
% 2.54/1.48 thf(tcon_Nat_Onat___Groups_Oone, axiom, ((one @ nat) @ (type2 @ nat))).
% 2.54/1.48 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_3, axiom, (! [A8:$tType]: (((semilattice_inf @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.54/1.48 thf(tcon_Set_Oset___Finite__Set_Ofinite_4, axiom, (! [A8:$tType]: ((((finite_finite @ A8) @ (type2 @ A8)) => ((finite_finite @ (set @ A8)) @ (type2 @ (set @ A8))))))).
% 2.54/1.48 thf(tcon_Set_Oset___Lattices_Olattice_5, axiom, (! [A8:$tType]: (((lattice @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.54/1.48 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_6, axiom, ((semilattice_inf @ $o) @ (type2 @ $o))).
% 2.54/1.48 thf(tcon_HOL_Obool___Finite__Set_Ofinite_7, axiom, ((finite_finite @ $o) @ (type2 @ $o))).
% 2.54/1.48 thf(tcon_HOL_Obool___Lattices_Olattice_8, axiom, ((lattice @ $o) @ (type2 @ $o))).
% 2.54/1.48 thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_9, axiom, ((strict2144017051up_add @ extended_enat) @ (type2 @ extended_enat))).
% 2.54/1.48 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_10, axiom, ((ordere779506340up_add @ extended_enat) @ (type2 @ extended_enat))).
% 2.54/1.48 thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_11, axiom, ((semilattice_inf @ extended_enat) @ (type2 @ extended_enat))).
% 2.54/1.48 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_12, axiom, ((ab_semigroup_add @ extended_enat) @ (type2 @ extended_enat))).
% 2.54/1.48 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_13, axiom, ((semigroup_add @ extended_enat) @ (type2 @ extended_enat))).
% 2.54/1.48 thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_14, axiom, ((lattice @ extended_enat) @ (type2 @ extended_enat))).
% 2.54/1.48 thf(tcon_Extended__Nat_Oenat___Groups_Oone_15, axiom, ((one @ extended_enat) @ (type2 @ extended_enat))).
% 2.54/1.48 thf(conj_0, conjecture, ((finite_finite2 @ nat) @ ((((image @ nat) @ nat) @ ((plus_plus @ nat) @ (((plus_plus @ nat) @ n) @ (one_one @ nat)))) @ (((inf_inf @ (set @ nat)) @ ((collect @ nat) @ (^ [N:nat]: ((((ord_less @ extended_enat) @ (extended_enat2 @ N)) @ ((coindu1018505716length @ a) @ ((coindu1047225960le_ltl @ a) @ (((coindu438612276pWhile @ a) @ (^ [X:a]: ((~ ((p @ X)))))) @ xsa)))))))) @ ((collect @ nat) @ (^ [N:nat]: ((p @ (((coindu749330388e_lnth @ a) @ ((coindu1047225960le_ltl @ a) @ (((coindu438612276pWhile @ a) @ (^ [X:a]: ((~ ((p @ X)))))) @ xsa))) @ N))))))))).
% 2.54/1.48 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.WzKKhsWUic/DTF2DTF9660.p
% 2.54/1.48 ------------------------
% 2.82/1.53 ---- Cleaned THF ---
% 2.82/1.53 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
% 2.82/1.53 coindu1593790203_llist: $tType > $tType ).
% 2.82/1.53
% 2.82/1.53 thf(ty_t_Extended__Nat_Oenat,type,
% 2.82/1.53 extended_enat: $tType ).
% 2.82/1.53
% 2.82/1.53 thf(ty_t_Set_Oset,type,
% 2.82/1.53 set: $tType > $tType ).
% 2.82/1.53
% 2.82/1.53 thf(ty_t_Nat_Onat,type,
% 2.82/1.53 nat: $tType ).
% 2.82/1.53
% 2.82/1.53 thf(ty_t_itself,type,
% 2.82/1.53 itself: $tType > $tType ).
% 2.82/1.53
% 2.82/1.53 thf(ty_tf_a,type,
% 2.82/1.53 a: $tType ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_HOL_Otype,type,
% 2.82/1.53 type:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Groups_Oone,type,
% 2.82/1.53 one:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Num_Oneg__numeral,type,
% 2.82/1.53 neg_numeral:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Groups_Ogroup__add,type,
% 2.82/1.53 group_add:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Lattices_Olattice,type,
% 2.82/1.53 lattice:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.82/1.53 finite_finite:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.82/1.53 semigroup_add:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.82/1.53 linordered_idom:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.82/1.53 ab_semigroup_add:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 2.82/1.53 semilattice_inf:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.82/1.53 linordered_semidom:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.82/1.53 cancel_semigroup_add:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.82/1.53 ordere779506340up_add:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.82/1.53 ordere236663937imp_le:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 2.82/1.53 ordere223160158up_add:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 2.82/1.53 strict2144017051up_add:
% 2.82/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__llength,type,
% 2.82/1.53 coindu493225792length:
% 2.82/1.53 !>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldrop,type,
% 2.82/1.53 coindu191418589_ldrop:
% 2.82/1.53 !>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OldropWhile,type,
% 2.82/1.53 coindu438612276pWhile:
% 2.82/1.53 !>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldropn,type,
% 2.82/1.53 coindu531130065ldropn:
% 2.82/1.53 !>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfilter,type,
% 2.82/1.53 coindu1889960678filter:
% 2.82/1.53 !>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite,type,
% 2.82/1.53 coindu1213758845finite:
% 2.82/1.53 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollast,type,
% 2.82/1.53 coindu2000965700_llast:
% 2.82/1.53 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
% 2.82/1.53 coindu1018505716length:
% 2.82/1.53 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olhd,type,
% 2.82/1.53 coindu1046438764le_lhd:
% 2.82/1.53 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
% 2.82/1.53 coindu335574135_lnull:
% 2.82/1.53 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olset,type,
% 2.82/1.53 coindu1112613586e_lset:
% 2.82/1.53 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Oltl,type,
% 2.82/1.53 coindu1047225960le_ltl:
% 2.82/1.53 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olnth,type,
% 2.82/1.53 coindu749330388e_lnth:
% 2.82/1.53 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > nat > A ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OltakeWhile,type,
% 2.82/1.53 coindu721411036eWhile:
% 2.82/1.53 !>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Coinductive__Nat_Oenat__unfold,type,
% 2.82/1.53 coindu1491768222unfold:
% 2.82/1.53 !>[A: $tType] : ( ( A > $o ) > ( A > A ) > A > extended_enat ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Extended__Nat_Oenat,type,
% 2.82/1.53 extended_enat2: nat > extended_enat ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
% 2.82/1.53 extended_rec_enat:
% 2.82/1.53 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Extended__Nat_Othe__enat,type,
% 2.82/1.53 extended_the_enat: extended_enat > nat ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Finite__Set_Ofinite,type,
% 2.82/1.53 finite_finite2:
% 2.82/1.53 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.82/1.53 one_one:
% 2.82/1.53 !>[A: $tType] : A ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.82/1.53 plus_plus:
% 2.82/1.53 !>[A: $tType] : ( A > A > A ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 2.82/1.53 inf_inf:
% 2.82/1.53 !>[A: $tType] : ( A > A > A ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.82/1.53 ord_less:
% 2.82/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.82/1.53 ord_less_eq:
% 2.82/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Pure_Otype,type,
% 2.82/1.53 type2:
% 2.82/1.53 !>[A: $tType] : ( itself @ A ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Set_OCollect,type,
% 2.82/1.53 collect:
% 2.82/1.53 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_Set_Oimage,type,
% 2.82/1.53 image:
% 2.82/1.53 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_c_member,type,
% 2.82/1.53 member:
% 2.82/1.53 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.82/1.53
% 2.82/1.53 thf(sy_v_P,type,
% 2.82/1.53 p: a > $o ).
% 2.82/1.53
% 2.82/1.53 thf(sy_v_n____,type,
% 2.82/1.53 n: nat ).
% 2.82/1.53
% 2.82/1.53 thf(sy_v_xs,type,
% 2.82/1.53 xs: coindu1593790203_llist @ a ).
% 2.82/1.53
% 2.82/1.53 thf(sy_v_xsa____,type,
% 2.82/1.53 xsa: coindu1593790203_llist @ a ).
% 2.82/1.53
% 2.82/1.53 thf(fact_0__092_060open_062finite_A_I_123n_O_Aenat_An_A_060_Allength_A_Iltl_A_IldropWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_J_125_A_092_060inter_062_A_123n_O_AP_A_Ilnth_A_Iltl_A_IldropWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_J_An_J_125_J_092_060close_062,axiom,
% 2.82/1.53 ( finite_finite2 @ nat
% 2.82/1.53 @ ( inf_inf @ ( set @ nat )
% 2.82/1.53 @ ( collect @ nat
% 2.82/1.53 @ ^ [N: nat] :
% 2.82/1.53 ( ord_less @ extended_enat @ ( extended_enat2 @ N )
% 2.82/1.53 @ ( coindu1018505716length @ a
% 2.82/1.53 @ ( coindu1047225960le_ltl @ a
% 2.82/1.53 @ ( coindu438612276pWhile @ a
% 2.82/1.53 @ ^ [X: a] :
% 2.82/1.53 ~ ( p @ X )
% 2.82/1.53 @ xsa ) ) ) ) )
% 2.82/1.53 @ ( collect @ nat
% 2.82/1.53 @ ^ [N: nat] :
% 2.82/1.53 ( p
% 2.82/1.53 @ ( coindu749330388e_lnth @ a
% 2.82/1.53 @ ( coindu1047225960le_ltl @ a
% 2.82/1.53 @ ( coindu438612276pWhile @ a
% 2.82/1.53 @ ^ [X: a] :
% 2.82/1.53 ~ ( p @ X )
% 2.82/1.53 @ xsa ) )
% 2.82/1.53 @ N ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_1_LCons_Oprems,axiom,
% 2.82/1.53 ~ ( coindu1213758845finite @ a @ xsa ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_2_n,axiom,
% 2.82/1.53 ( ( coindu1018505716length @ a
% 2.82/1.53 @ ( coindu721411036eWhile @ a
% 2.82/1.53 @ ^ [X: a] :
% 2.82/1.53 ~ ( p @ X )
% 2.82/1.53 @ xsa ) )
% 2.82/1.53 = ( extended_enat2 @ n ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_3_finite__Int,axiom,
% 2.82/1.53 ! [A: $tType,F: set @ A,G: set @ A] :
% 2.82/1.53 ( ( ( finite_finite2 @ A @ F )
% 2.82/1.53 | ( finite_finite2 @ A @ G ) )
% 2.82/1.53 => ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F @ G ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_4_finite__imageI,axiom,
% 2.82/1.53 ! [B: $tType,A: $tType,F: set @ A,H: A > B] :
% 2.82/1.53 ( ( finite_finite2 @ A @ F )
% 2.82/1.53 => ( finite_finite2 @ B @ ( image @ A @ B @ H @ F ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_5_add__less__cancel__left,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [C: A,A2: A,B2: A] :
% 2.82/1.53 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 2.82/1.53 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_6_add__less__cancel__right,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,C: A,B2: A] :
% 2.82/1.53 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 2.82/1.53 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_Allength_A_IltakeWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_A_061_Aenat_An_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.82/1.53 ~ ! [N2: nat] :
% 2.82/1.53 ( ( coindu1018505716length @ a
% 2.82/1.53 @ ( coindu721411036eWhile @ a
% 2.82/1.53 @ ^ [X: a] :
% 2.82/1.53 ~ ( p @ X )
% 2.82/1.53 @ xsa ) )
% 2.82/1.53 != ( extended_enat2 @ N2 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_8_finite__Collect__conjI,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.82/1.53 ( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.82/1.53 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
% 2.82/1.53 => ( finite_finite2 @ A
% 2.82/1.53 @ ( collect @ A
% 2.82/1.53 @ ^ [X: A] :
% 2.82/1.53 ( ( P @ X )
% 2.82/1.53 & ( Q @ X ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_9_finite__Collect__disjI,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.82/1.53 ( ( finite_finite2 @ A
% 2.82/1.53 @ ( collect @ A
% 2.82/1.53 @ ^ [X: A] :
% 2.82/1.53 ( ( P @ X )
% 2.82/1.53 | ( Q @ X ) ) ) )
% 2.82/1.53 = ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.82/1.53 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_10_image__ident,axiom,
% 2.82/1.53 ! [A: $tType,Y: set @ A] :
% 2.82/1.53 ( ( image @ A @ A
% 2.82/1.53 @ ^ [X: A] : X
% 2.82/1.53 @ Y )
% 2.82/1.53 = Y ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_11_less__add__one,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_12_exP,axiom,
% 2.82/1.53 ? [X2: a] :
% 2.82/1.53 ( ( member @ a @ X2 @ ( coindu1112613586e_lset @ a @ xsa ) )
% 2.82/1.53 & ( p @ X2 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_13_enat_Oinject,axiom,
% 2.82/1.53 ! [Nat: nat,Nat2: nat] :
% 2.82/1.53 ( ( ( extended_enat2 @ Nat )
% 2.82/1.53 = ( extended_enat2 @ Nat2 ) )
% 2.82/1.53 = ( Nat = Nat2 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_14_LCons_Ohyps_I3_J,axiom,
% 2.82/1.53 ! [Xs: coindu1593790203_llist @ a] :
% 2.82/1.53 ( ( ( coindu1047225960le_ltl @ a @ ( coindu1889960678filter @ a @ p @ xsa ) )
% 2.82/1.53 = ( coindu1889960678filter @ a @ p @ Xs ) )
% 2.82/1.53 => ( ~ ( coindu1213758845finite @ a @ Xs )
% 2.82/1.53 => ( finite_finite2 @ nat
% 2.82/1.53 @ ( collect @ nat
% 2.82/1.53 @ ^ [N: nat] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ a @ Xs ) )
% 2.82/1.53 & ( p @ ( coindu749330388e_lnth @ a @ Xs @ N ) ) ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_15__092_060open_062_092_060not_062_Alfinite_Axs_092_060close_062,axiom,
% 2.82/1.53 ~ ( coindu1213758845finite @ a @ xs ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_16_LCons_Ohyps_I1_J,axiom,
% 2.82/1.53 coindu1213758845finite @ a @ ( coindu1889960678filter @ a @ p @ xsa ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_17__092_060open_062lfinite_A_Ilfilter_AP_Axs_J_092_060close_062,axiom,
% 2.82/1.53 coindu1213758845finite @ a @ ( coindu1889960678filter @ a @ p @ xs ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_18_add__right__cancel,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [B2: A,A2: A,C: A] :
% 2.82/1.53 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.82/1.53 = ( plus_plus @ A @ C @ A2 ) )
% 2.82/1.53 = ( B2 = C ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_19_add__left__cancel,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A,C: A] :
% 2.82/1.53 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.82/1.53 = ( plus_plus @ A @ A2 @ C ) )
% 2.82/1.53 = ( B2 = C ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_20_image__eqI,axiom,
% 2.82/1.53 ! [A: $tType,B: $tType,B2: A,F2: B > A,X3: B,A3: set @ B] :
% 2.82/1.53 ( ( B2
% 2.82/1.53 = ( F2 @ X3 ) )
% 2.82/1.53 => ( ( member @ B @ X3 @ A3 )
% 2.82/1.53 => ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_21_finite__code,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( finite_finite @ A @ ( type2 @ A ) )
% 2.82/1.53 => ( ( finite_finite2 @ A )
% 2.82/1.53 = ( ^ [A4: set @ A] : $true ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_22_Int__iff,axiom,
% 2.82/1.53 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.82/1.53 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 2.82/1.53 = ( ( member @ A @ C @ A3 )
% 2.82/1.53 & ( member @ A @ C @ B3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_23_IntI,axiom,
% 2.82/1.53 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.82/1.53 ( ( member @ A @ C @ A3 )
% 2.82/1.53 => ( ( member @ A @ C @ B3 )
% 2.82/1.53 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_24_lfilter__idem,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1889960678filter @ A @ P @ ( coindu1889960678filter @ A @ P @ Xs ) )
% 2.82/1.53 = ( coindu1889960678filter @ A @ P @ Xs ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_25_ltakeWhile__K__True,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu721411036eWhile @ A
% 2.82/1.53 @ ^ [Uu: A] : $true
% 2.82/1.53 @ Xs )
% 2.82/1.53 = Xs ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_26_finite__Collect__less__nat,axiom,
% 2.82/1.53 ! [K: nat] :
% 2.82/1.53 ( finite_finite2 @ nat
% 2.82/1.53 @ ( collect @ nat
% 2.82/1.53 @ ^ [N: nat] : ( ord_less @ nat @ N @ K ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_27_lfilter__K__True,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1889960678filter @ A
% 2.82/1.53 @ ^ [Uu: A] : $true
% 2.82/1.53 @ Xs )
% 2.82/1.53 = Xs ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_28_plus__enat__simps_I1_J,axiom,
% 2.82/1.53 ! [M: nat,N3: nat] :
% 2.82/1.53 ( ( plus_plus @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N3 ) )
% 2.82/1.53 = ( extended_enat2 @ ( plus_plus @ nat @ M @ N3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_29_lfinite__ltl,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1213758845finite @ A @ ( coindu1047225960le_ltl @ A @ Xs ) )
% 2.82/1.53 = ( coindu1213758845finite @ A @ Xs ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_30_enat__ord__simps_I2_J,axiom,
% 2.82/1.53 ! [M: nat,N3: nat] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N3 ) )
% 2.82/1.53 = ( ord_less @ nat @ M @ N3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_31_lset__lfilter,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1112613586e_lset @ A @ ( coindu1889960678filter @ A @ P @ Xs ) )
% 2.82/1.53 = ( collect @ A
% 2.82/1.53 @ ^ [X: A] :
% 2.82/1.53 ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
% 2.82/1.53 & ( P @ X ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_32_LCons_Ohyps_I2_J,axiom,
% 2.82/1.53 ~ ( coindu335574135_lnull @ a @ ( coindu1889960678filter @ a @ p @ xsa ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_33_lfilter__cong,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,P: A > $o,Q: A > $o] :
% 2.82/1.53 ( ( Xs = Ys )
% 2.82/1.53 => ( ! [X2: A] :
% 2.82/1.53 ( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Ys ) )
% 2.82/1.53 => ( ( P @ X2 )
% 2.82/1.53 = ( Q @ X2 ) ) )
% 2.82/1.53 => ( ( coindu1889960678filter @ A @ P @ Xs )
% 2.82/1.53 = ( coindu1889960678filter @ A @ Q @ Ys ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_34_ltakeWhile__all,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.82/1.53 ( ! [X2: A] :
% 2.82/1.53 ( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 2.82/1.53 => ( P @ X2 ) )
% 2.82/1.53 => ( ( coindu721411036eWhile @ A @ P @ Xs )
% 2.82/1.53 = Xs ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_35_ltakeWhile__cong,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,P: A > $o,Q: A > $o] :
% 2.82/1.53 ( ( Xs = Ys )
% 2.82/1.53 => ( ! [X2: A] :
% 2.82/1.53 ( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Ys ) )
% 2.82/1.53 => ( ( P @ X2 )
% 2.82/1.53 = ( Q @ X2 ) ) )
% 2.82/1.53 => ( ( coindu721411036eWhile @ A @ P @ Xs )
% 2.82/1.53 = ( coindu721411036eWhile @ A @ Q @ Ys ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_36_lfinite__lfilterI,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.82/1.53 ( ( coindu1213758845finite @ A @ Xs )
% 2.82/1.53 => ( coindu1213758845finite @ A @ ( coindu1889960678filter @ A @ P @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_37_lset__ltakeWhileD,axiom,
% 2.82/1.53 ! [A: $tType,X3: A,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
% 2.82/1.53 => ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 2.82/1.53 & ( P @ X3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_38_lfinite__ltakeWhile,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1213758845finite @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 2.82/1.53 = ( ( coindu1213758845finite @ A @ Xs )
% 2.82/1.53 | ? [X: A] :
% 2.82/1.53 ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
% 2.82/1.53 & ~ ( P @ X ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_39_lfinite__imp__finite__lset,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1213758845finite @ A @ Xs )
% 2.82/1.53 => ( finite_finite2 @ A @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_40_lfinite__ldropWhile,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1213758845finite @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) )
% 2.82/1.53 = ( ? [X: A] :
% 2.82/1.53 ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
% 2.82/1.53 & ~ ( P @ X ) )
% 2.82/1.53 => ( coindu1213758845finite @ A @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_41_finite__psubset__induct,axiom,
% 2.82/1.53 ! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
% 2.82/1.53 ( ( finite_finite2 @ A @ A3 )
% 2.82/1.53 => ( ! [A5: set @ A] :
% 2.82/1.53 ( ( finite_finite2 @ A @ A5 )
% 2.82/1.53 => ( ! [B4: set @ A] :
% 2.82/1.53 ( ( ord_less @ ( set @ A ) @ B4 @ A5 )
% 2.82/1.53 => ( P @ B4 ) )
% 2.82/1.53 => ( P @ A5 ) ) )
% 2.82/1.53 => ( P @ A3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_42_lfilter__lfilter,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Q: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1889960678filter @ A @ P @ ( coindu1889960678filter @ A @ Q @ Xs ) )
% 2.82/1.53 = ( coindu1889960678filter @ A
% 2.82/1.53 @ ^ [X: A] :
% 2.82/1.53 ( ( P @ X )
% 2.82/1.53 & ( Q @ X ) )
% 2.82/1.53 @ Xs ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_43_llength__ltakeWhile__lt__iff,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) @ ( coindu1018505716length @ A @ Xs ) )
% 2.82/1.53 = ( ? [X: A] :
% 2.82/1.53 ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
% 2.82/1.53 & ~ ( P @ X ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_44_mem__Collect__eq,axiom,
% 2.82/1.53 ! [A: $tType,A2: A,P: A > $o] :
% 2.82/1.53 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.82/1.53 = ( P @ A2 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_45_Collect__mem__eq,axiom,
% 2.82/1.53 ! [A: $tType,A3: set @ A] :
% 2.82/1.53 ( ( collect @ A
% 2.82/1.53 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 2.82/1.53 = A3 ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_46_Collect__cong,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.82/1.53 ( ! [X2: A] :
% 2.82/1.53 ( ( P @ X2 )
% 2.82/1.53 = ( Q @ X2 ) )
% 2.82/1.53 => ( ( collect @ A @ P )
% 2.82/1.53 = ( collect @ A @ Q ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_47_ext,axiom,
% 2.82/1.53 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 2.82/1.53 ( ! [X2: A] :
% 2.82/1.53 ( ( F2 @ X2 )
% 2.82/1.53 = ( G2 @ X2 ) )
% 2.82/1.53 => ( F2 = G2 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_48_in__lset__ltlD,axiom,
% 2.82/1.53 ! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) )
% 2.82/1.53 => ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_49_llength__ltakeWhile__all,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 2.82/1.53 = ( coindu1018505716length @ A @ Xs ) )
% 2.82/1.53 = ( ( coindu721411036eWhile @ A @ P @ Xs )
% 2.82/1.53 = Xs ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_50_in__lset__ldropWhileD,axiom,
% 2.82/1.53 ! [A: $tType,X3: A,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) ) )
% 2.82/1.53 => ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_51_ldropWhile__cong,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,P: A > $o,Q: A > $o] :
% 2.82/1.53 ( ( Xs = Ys )
% 2.82/1.53 => ( ! [X2: A] :
% 2.82/1.53 ( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Ys ) )
% 2.82/1.53 => ( ( P @ X2 )
% 2.82/1.53 = ( Q @ X2 ) ) )
% 2.82/1.53 => ( ( coindu438612276pWhile @ A @ P @ Xs )
% 2.82/1.53 = ( coindu438612276pWhile @ A @ Q @ Ys ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_52_one__enat__def,axiom,
% 2.82/1.53 ( ( one_one @ extended_enat )
% 2.82/1.53 = ( extended_enat2 @ ( one_one @ nat ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_53_enat__1__iff_I1_J,axiom,
% 2.82/1.53 ! [X3: nat] :
% 2.82/1.53 ( ( ( extended_enat2 @ X3 )
% 2.82/1.53 = ( one_one @ extended_enat ) )
% 2.82/1.53 = ( X3
% 2.82/1.53 = ( one_one @ nat ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_54_enat__1__iff_I2_J,axiom,
% 2.82/1.53 ! [X3: nat] :
% 2.82/1.53 ( ( ( one_one @ extended_enat )
% 2.82/1.53 = ( extended_enat2 @ X3 ) )
% 2.82/1.53 = ( X3
% 2.82/1.53 = ( one_one @ nat ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_55_less__enatE,axiom,
% 2.82/1.53 ! [N3: extended_enat,M: nat] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ N3 @ ( extended_enat2 @ M ) )
% 2.82/1.53 => ~ ! [K2: nat] :
% 2.82/1.53 ( ( N3
% 2.82/1.53 = ( extended_enat2 @ K2 ) )
% 2.82/1.53 => ~ ( ord_less @ nat @ K2 @ M ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_56_finite__conv__nat__seg__image,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( finite_finite2 @ A )
% 2.82/1.53 = ( ^ [A4: set @ A] :
% 2.82/1.53 ? [N: nat,F3: nat > A] :
% 2.82/1.53 ( A4
% 2.82/1.53 = ( image @ nat @ A @ F3
% 2.82/1.53 @ ( collect @ nat
% 2.82/1.53 @ ^ [I: nat] : ( ord_less @ nat @ I @ N ) ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_57_nat__seg__image__imp__finite,axiom,
% 2.82/1.53 ! [A: $tType,A3: set @ A,F2: nat > A,N3: nat] :
% 2.82/1.53 ( ( A3
% 2.82/1.53 = ( image @ nat @ A @ F2
% 2.82/1.53 @ ( collect @ nat
% 2.82/1.53 @ ^ [I: nat] : ( ord_less @ nat @ I @ N3 ) ) ) )
% 2.82/1.53 => ( finite_finite2 @ A @ A3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_58_lfinite__conv__llength__enat,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( coindu1213758845finite @ A )
% 2.82/1.53 = ( ^ [Xs2: coindu1593790203_llist @ A] :
% 2.82/1.53 ? [N: nat] :
% 2.82/1.53 ( ( coindu1018505716length @ A @ Xs2 )
% 2.82/1.53 = ( extended_enat2 @ N ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_59_llength__eq__enat__lfiniteD,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,N3: nat] :
% 2.82/1.53 ( ( ( coindu1018505716length @ A @ Xs )
% 2.82/1.53 = ( extended_enat2 @ N3 ) )
% 2.82/1.53 => ( coindu1213758845finite @ A @ Xs ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_60_lfinite__llength__enat,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1213758845finite @ A @ Xs )
% 2.82/1.53 => ? [N2: nat] :
% 2.82/1.53 ( ( coindu1018505716length @ A @ Xs )
% 2.82/1.53 = ( extended_enat2 @ N2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_61_linorder__neqE__linordered__idom,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,Y2: A] :
% 2.82/1.53 ( ( X3 != Y2 )
% 2.82/1.53 => ( ~ ( ord_less @ A @ X3 @ Y2 )
% 2.82/1.53 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_62_add__right__imp__eq,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [B2: A,A2: A,C: A] :
% 2.82/1.53 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.82/1.53 = ( plus_plus @ A @ C @ A2 ) )
% 2.82/1.53 => ( B2 = C ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_63_add__left__imp__eq,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A,C: A] :
% 2.82/1.53 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.82/1.53 = ( plus_plus @ A @ A2 @ C ) )
% 2.82/1.53 => ( B2 = C ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_64_add_Oleft__commute,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [B2: A,A2: A,C: A] :
% 2.82/1.53 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
% 2.82/1.53 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_65_add_Ocommute,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ( ( plus_plus @ A )
% 2.82/1.53 = ( ^ [A6: A,B5: A] : ( plus_plus @ A @ B5 @ A6 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_66_add_Oright__cancel,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( group_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [B2: A,A2: A,C: A] :
% 2.82/1.53 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.82/1.53 = ( plus_plus @ A @ C @ A2 ) )
% 2.82/1.53 = ( B2 = C ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_67_add_Oleft__cancel,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( group_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A,C: A] :
% 2.82/1.53 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.82/1.53 = ( plus_plus @ A @ A2 @ C ) )
% 2.82/1.53 = ( B2 = C ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_68_add_Oassoc,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semigroup_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A,C: A] :
% 2.82/1.53 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 2.82/1.53 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_69_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [I2: A,J: A,K: A,L: A] :
% 2.82/1.53 ( ( ( I2 = J )
% 2.82/1.53 & ( K = L ) )
% 2.82/1.53 => ( ( plus_plus @ A @ I2 @ K )
% 2.82/1.53 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_70_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A,C: A] :
% 2.82/1.53 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 2.82/1.53 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_71_one__reorient,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( one @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A] :
% 2.82/1.53 ( ( ( one_one @ A )
% 2.82/1.53 = X3 )
% 2.82/1.53 = ( X3
% 2.82/1.53 = ( one_one @ A ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_72_rev__image__eqI,axiom,
% 2.82/1.53 ! [B: $tType,A: $tType,X3: A,A3: set @ A,B2: B,F2: A > B] :
% 2.82/1.53 ( ( member @ A @ X3 @ A3 )
% 2.82/1.53 => ( ( B2
% 2.82/1.53 = ( F2 @ X3 ) )
% 2.82/1.53 => ( member @ B @ B2 @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_73_ball__imageD,axiom,
% 2.82/1.53 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
% 2.82/1.53 ( ! [X2: A] :
% 2.82/1.53 ( ( member @ A @ X2 @ ( image @ B @ A @ F2 @ A3 ) )
% 2.82/1.53 => ( P @ X2 ) )
% 2.82/1.53 => ! [X4: B] :
% 2.82/1.53 ( ( member @ B @ X4 @ A3 )
% 2.82/1.53 => ( P @ ( F2 @ X4 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_74_image__cong,axiom,
% 2.82/1.53 ! [B: $tType,A: $tType,M2: set @ A,N4: set @ A,F2: A > B,G2: A > B] :
% 2.82/1.53 ( ( M2 = N4 )
% 2.82/1.53 => ( ! [X2: A] :
% 2.82/1.53 ( ( member @ A @ X2 @ N4 )
% 2.82/1.53 => ( ( F2 @ X2 )
% 2.82/1.53 = ( G2 @ X2 ) ) )
% 2.82/1.53 => ( ( image @ A @ B @ F2 @ M2 )
% 2.82/1.53 = ( image @ A @ B @ G2 @ N4 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_75_bex__imageD,axiom,
% 2.82/1.53 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
% 2.82/1.53 ( ? [X4: A] :
% 2.82/1.53 ( ( member @ A @ X4 @ ( image @ B @ A @ F2 @ A3 ) )
% 2.82/1.53 & ( P @ X4 ) )
% 2.82/1.53 => ? [X2: B] :
% 2.82/1.53 ( ( member @ B @ X2 @ A3 )
% 2.82/1.53 & ( P @ ( F2 @ X2 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_76_image__iff,axiom,
% 2.82/1.53 ! [A: $tType,B: $tType,Z: A,F2: B > A,A3: set @ B] :
% 2.82/1.53 ( ( member @ A @ Z @ ( image @ B @ A @ F2 @ A3 ) )
% 2.82/1.53 = ( ? [X: B] :
% 2.82/1.53 ( ( member @ B @ X @ A3 )
% 2.82/1.53 & ( Z
% 2.82/1.53 = ( F2 @ X ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_77_imageI,axiom,
% 2.82/1.53 ! [B: $tType,A: $tType,X3: A,A3: set @ A,F2: A > B] :
% 2.82/1.53 ( ( member @ A @ X3 @ A3 )
% 2.82/1.53 => ( member @ B @ ( F2 @ X3 ) @ ( image @ A @ B @ F2 @ A3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_78_finite,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( finite_finite @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_79_finite__set__choice,axiom,
% 2.82/1.53 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 2.82/1.53 ( ( finite_finite2 @ A @ A3 )
% 2.82/1.53 => ( ! [X2: A] :
% 2.82/1.53 ( ( member @ A @ X2 @ A3 )
% 2.82/1.53 => ? [X1: B] : ( P @ X2 @ X1 ) )
% 2.82/1.53 => ? [F4: A > B] :
% 2.82/1.53 ! [X4: A] :
% 2.82/1.53 ( ( member @ A @ X4 @ A3 )
% 2.82/1.53 => ( P @ X4 @ ( F4 @ X4 ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_80_Int__left__commute,axiom,
% 2.82/1.53 ! [A: $tType,A3: set @ A,B3: set @ A,C2: set @ A] :
% 2.82/1.53 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) )
% 2.82/1.53 = ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_81_Int__left__absorb,axiom,
% 2.82/1.53 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.82/1.53 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 2.82/1.53 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_82_Int__commute,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( inf_inf @ ( set @ A ) )
% 2.82/1.53 = ( ^ [A4: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A4 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_83_Int__absorb,axiom,
% 2.82/1.53 ! [A: $tType,A3: set @ A] :
% 2.82/1.53 ( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
% 2.82/1.53 = A3 ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_84_Int__assoc,axiom,
% 2.82/1.53 ! [A: $tType,A3: set @ A,B3: set @ A,C2: set @ A] :
% 2.82/1.53 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C2 )
% 2.82/1.53 = ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_85_IntD2,axiom,
% 2.82/1.53 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.82/1.53 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 2.82/1.53 => ( member @ A @ C @ B3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_86_IntD1,axiom,
% 2.82/1.53 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.82/1.53 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 2.82/1.53 => ( member @ A @ C @ A3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_87_IntE,axiom,
% 2.82/1.53 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.82/1.53 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 2.82/1.53 => ~ ( ( member @ A @ C @ A3 )
% 2.82/1.53 => ~ ( member @ A @ C @ B3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_88_enat__less__induct,axiom,
% 2.82/1.53 ! [P: extended_enat > $o,N3: extended_enat] :
% 2.82/1.53 ( ! [N2: extended_enat] :
% 2.82/1.53 ( ! [M3: extended_enat] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ M3 @ N2 )
% 2.82/1.53 => ( P @ M3 ) )
% 2.82/1.53 => ( P @ N2 ) )
% 2.82/1.53 => ( P @ N3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_89_in__lset__conv__lnth,axiom,
% 2.82/1.53 ! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 2.82/1.53 = ( ? [N: nat] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs ) )
% 2.82/1.53 & ( ( coindu749330388e_lnth @ A @ Xs @ N )
% 2.82/1.53 = X3 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_90_ltakeWhile__nth,axiom,
% 2.82/1.53 ! [A: $tType,I2: nat,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ ( extended_enat2 @ I2 ) @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
% 2.82/1.53 => ( ( coindu749330388e_lnth @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) @ I2 )
% 2.82/1.53 = ( coindu749330388e_lnth @ A @ Xs @ I2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_91_lfinite__finite__index,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1213758845finite @ A @ Xs )
% 2.82/1.53 => ( finite_finite2 @ nat
% 2.82/1.53 @ ( collect @ nat
% 2.82/1.53 @ ^ [N: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_92_Compr__image__eq,axiom,
% 2.82/1.53 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
% 2.82/1.53 ( ( collect @ A
% 2.82/1.53 @ ^ [X: A] :
% 2.82/1.53 ( ( member @ A @ X @ ( image @ B @ A @ F2 @ A3 ) )
% 2.82/1.53 & ( P @ X ) ) )
% 2.82/1.53 = ( image @ B @ A @ F2
% 2.82/1.53 @ ( collect @ B
% 2.82/1.53 @ ^ [X: B] :
% 2.82/1.53 ( ( member @ B @ X @ A3 )
% 2.82/1.53 & ( P @ ( F2 @ X ) ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_93_image__image,axiom,
% 2.82/1.53 ! [A: $tType,B: $tType,C3: $tType,F2: B > A,G2: C3 > B,A3: set @ C3] :
% 2.82/1.53 ( ( image @ B @ A @ F2 @ ( image @ C3 @ B @ G2 @ A3 ) )
% 2.82/1.53 = ( image @ C3 @ A
% 2.82/1.53 @ ^ [X: C3] : ( F2 @ ( G2 @ X ) )
% 2.82/1.53 @ A3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_94_imageE,axiom,
% 2.82/1.53 ! [A: $tType,B: $tType,B2: A,F2: B > A,A3: set @ B] :
% 2.82/1.53 ( ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) )
% 2.82/1.53 => ~ ! [X2: B] :
% 2.82/1.53 ( ( B2
% 2.82/1.53 = ( F2 @ X2 ) )
% 2.82/1.53 => ~ ( member @ B @ X2 @ A3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_95_pigeonhole__infinite__rel,axiom,
% 2.82/1.53 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R: A > B > $o] :
% 2.82/1.53 ( ~ ( finite_finite2 @ A @ A3 )
% 2.82/1.53 => ( ( finite_finite2 @ B @ B3 )
% 2.82/1.53 => ( ! [X2: A] :
% 2.82/1.53 ( ( member @ A @ X2 @ A3 )
% 2.82/1.53 => ? [Xa: B] :
% 2.82/1.53 ( ( member @ B @ Xa @ B3 )
% 2.82/1.53 & ( R @ X2 @ Xa ) ) )
% 2.82/1.53 => ? [X2: B] :
% 2.82/1.53 ( ( member @ B @ X2 @ B3 )
% 2.82/1.53 & ~ ( finite_finite2 @ A
% 2.82/1.53 @ ( collect @ A
% 2.82/1.53 @ ^ [A6: A] :
% 2.82/1.53 ( ( member @ A @ A6 @ A3 )
% 2.82/1.53 & ( R @ A6 @ X2 ) ) ) ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_96_not__finite__existsD,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o] :
% 2.82/1.53 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.82/1.53 => ? [X12: A] : ( P @ X12 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_97_Collect__conj__eq,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.82/1.53 ( ( collect @ A
% 2.82/1.53 @ ^ [X: A] :
% 2.82/1.53 ( ( P @ X )
% 2.82/1.53 & ( Q @ X ) ) )
% 2.82/1.53 = ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_98_Int__Collect,axiom,
% 2.82/1.53 ! [A: $tType,X3: A,A3: set @ A,P: A > $o] :
% 2.82/1.53 ( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) )
% 2.82/1.53 = ( ( member @ A @ X3 @ A3 )
% 2.82/1.53 & ( P @ X3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_99_Int__def,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( inf_inf @ ( set @ A ) )
% 2.82/1.53 = ( ^ [A4: set @ A,B6: set @ A] :
% 2.82/1.53 ( collect @ A
% 2.82/1.53 @ ^ [X: A] :
% 2.82/1.53 ( ( member @ A @ X @ A4 )
% 2.82/1.53 & ( member @ A @ X @ B6 ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_100_add__less__imp__less__right,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,C: A,B2: A] :
% 2.82/1.53 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 2.82/1.53 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_101_add__less__imp__less__left,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [C: A,A2: A,B2: A] :
% 2.82/1.53 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 2.82/1.53 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_102_add__strict__right__mono,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A,C: A] :
% 2.82/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 2.82/1.53 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_103_add__strict__left__mono,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A,C: A] :
% 2.82/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 2.82/1.53 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_104_add__strict__mono,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( strict2144017051up_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A,C: A,D: A] :
% 2.82/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 2.82/1.53 => ( ( ord_less @ A @ C @ D )
% 2.82/1.53 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_105_add__mono__thms__linordered__field_I1_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [I2: A,J: A,K: A,L: A] :
% 2.82/1.53 ( ( ( ord_less @ A @ I2 @ J )
% 2.82/1.53 & ( K = L ) )
% 2.82/1.53 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_106_add__mono__thms__linordered__field_I2_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [I2: A,J: A,K: A,L: A] :
% 2.82/1.53 ( ( ( I2 = J )
% 2.82/1.53 & ( ord_less @ A @ K @ L ) )
% 2.82/1.53 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_107_add__mono__thms__linordered__field_I5_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [I2: A,J: A,K: A,L: A] :
% 2.82/1.53 ( ( ( ord_less @ A @ I2 @ J )
% 2.82/1.53 & ( ord_less @ A @ K @ L ) )
% 2.82/1.53 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_108_enat__iless,axiom,
% 2.82/1.53 ! [N3: extended_enat,M: nat] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ N3 @ ( extended_enat2 @ M ) )
% 2.82/1.53 => ? [K2: nat] :
% 2.82/1.53 ( N3
% 2.82/1.53 = ( extended_enat2 @ K2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_109_chain__incr,axiom,
% 2.82/1.53 ! [A: $tType,Y: A > extended_enat,K: nat] :
% 2.82/1.53 ( ! [I3: A] :
% 2.82/1.53 ? [J2: A] : ( ord_less @ extended_enat @ ( Y @ I3 ) @ ( Y @ J2 ) )
% 2.82/1.53 => ? [J3: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K ) @ ( Y @ J3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_110_pigeonhole__infinite,axiom,
% 2.82/1.53 ! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
% 2.82/1.53 ( ~ ( finite_finite2 @ A @ A3 )
% 2.82/1.53 => ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A3 ) )
% 2.82/1.53 => ? [X2: A] :
% 2.82/1.53 ( ( member @ A @ X2 @ A3 )
% 2.82/1.53 & ~ ( finite_finite2 @ A
% 2.82/1.53 @ ( collect @ A
% 2.82/1.53 @ ^ [A6: A] :
% 2.82/1.53 ( ( member @ A @ A6 @ A3 )
% 2.82/1.53 & ( ( F2 @ A6 )
% 2.82/1.53 = ( F2 @ X2 ) ) ) ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_111_enat__add__mono,axiom,
% 2.82/1.53 ! [X3: nat,Y2: extended_enat,Z: extended_enat] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Y2 ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Z ) )
% 2.82/1.53 = ( ord_less @ extended_enat @ Y2 @ Z ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_112_enat__less__enat__plusI2,axiom,
% 2.82/1.53 ! [Y2: nat,Z: extended_enat,X3: nat] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ ( extended_enat2 @ Y2 ) @ Z )
% 2.82/1.53 => ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ X3 @ Y2 ) ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Z ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_113_enat__add2__eq,axiom,
% 2.82/1.53 ! [Y2: extended_enat,X3: nat,Z: extended_enat] :
% 2.82/1.53 ( ( ( plus_plus @ extended_enat @ Y2 @ ( extended_enat2 @ X3 ) )
% 2.82/1.53 = ( plus_plus @ extended_enat @ Z @ ( extended_enat2 @ X3 ) ) )
% 2.82/1.53 = ( Y2 = Z ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_114_enat__add1__eq,axiom,
% 2.82/1.53 ! [X3: nat,Y2: extended_enat,Z: extended_enat] :
% 2.82/1.53 ( ( ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Y2 )
% 2.82/1.53 = ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Z ) )
% 2.82/1.53 = ( Y2 = Z ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_115_nat__add__left__cancel__less,axiom,
% 2.82/1.53 ! [K: nat,M: nat,N3: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N3 ) )
% 2.82/1.53 = ( ord_less @ nat @ M @ N3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_116_inf__right__idem,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,Y2: A] :
% 2.82/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Y2 )
% 2.82/1.53 = ( inf_inf @ A @ X3 @ Y2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_117_inf_Oright__idem,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A] :
% 2.82/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
% 2.82/1.53 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_118_inf__left__idem,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,Y2: A] :
% 2.82/1.53 ( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ X3 @ Y2 ) )
% 2.82/1.53 = ( inf_inf @ A @ X3 @ Y2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_119_inf_Oleft__idem,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A] :
% 2.82/1.53 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
% 2.82/1.53 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_120_inf__idem,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A] :
% 2.82/1.53 ( ( inf_inf @ A @ X3 @ X3 )
% 2.82/1.53 = X3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_121_inf__apply,axiom,
% 2.82/1.53 ! [B: $tType,A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 2.82/1.53 => ( ( inf_inf @ ( A > B ) )
% 2.82/1.53 = ( ^ [F3: A > B,G3: A > B,X: A] : ( inf_inf @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_122_inf_Oidem,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A] :
% 2.82/1.53 ( ( inf_inf @ A @ A2 @ A2 )
% 2.82/1.53 = A2 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_123_lnull__lfilter,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu335574135_lnull @ A @ ( coindu1889960678filter @ A @ P @ Xs ) )
% 2.82/1.53 = ( ! [X: A] :
% 2.82/1.53 ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
% 2.82/1.53 => ~ ( P @ X ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_124_lnull__ldropWhile,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu335574135_lnull @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) )
% 2.82/1.53 = ( ! [X: A] :
% 2.82/1.53 ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
% 2.82/1.53 => ( P @ X ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_125_psubsetD,axiom,
% 2.82/1.53 ! [A: $tType,A3: set @ A,B3: set @ A,C: A] :
% 2.82/1.53 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.82/1.53 => ( ( member @ A @ C @ A3 )
% 2.82/1.53 => ( member @ A @ C @ B3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_126_nat__neq__iff,axiom,
% 2.82/1.53 ! [M: nat,N3: nat] :
% 2.82/1.53 ( ( M != N3 )
% 2.82/1.53 = ( ( ord_less @ nat @ M @ N3 )
% 2.82/1.53 | ( ord_less @ nat @ N3 @ M ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_127_less__set__def,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ord_less @ ( set @ A ) )
% 2.82/1.53 = ( ^ [A4: set @ A,B6: set @ A] :
% 2.82/1.53 ( ord_less @ ( A > $o )
% 2.82/1.53 @ ^ [X: A] : ( member @ A @ X @ A4 )
% 2.82/1.53 @ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_128_less__not__refl,axiom,
% 2.82/1.53 ! [N3: nat] :
% 2.82/1.53 ~ ( ord_less @ nat @ N3 @ N3 ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_129_psubset__trans,axiom,
% 2.82/1.53 ! [A: $tType,A3: set @ A,B3: set @ A,C2: set @ A] :
% 2.82/1.53 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.82/1.53 => ( ( ord_less @ ( set @ A ) @ B3 @ C2 )
% 2.82/1.53 => ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_130_less__not__refl2,axiom,
% 2.82/1.53 ! [N3: nat,M: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ N3 @ M )
% 2.82/1.53 => ( M != N3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_131_less__not__refl3,axiom,
% 2.82/1.53 ! [S: nat,T2: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ S @ T2 )
% 2.82/1.53 => ( S != T2 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_132_measure__induct,axiom,
% 2.82/1.53 ! [A: $tType,F2: A > nat,P: A > $o,A2: A] :
% 2.82/1.53 ( ! [X2: A] :
% 2.82/1.53 ( ! [Y3: A] :
% 2.82/1.53 ( ( ord_less @ nat @ ( F2 @ Y3 ) @ ( F2 @ X2 ) )
% 2.82/1.53 => ( P @ Y3 ) )
% 2.82/1.53 => ( P @ X2 ) )
% 2.82/1.53 => ( P @ A2 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_133_less__irrefl__nat,axiom,
% 2.82/1.53 ! [N3: nat] :
% 2.82/1.53 ~ ( ord_less @ nat @ N3 @ N3 ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_134_nat__less__induct,axiom,
% 2.82/1.53 ! [P: nat > $o,N3: nat] :
% 2.82/1.53 ( ! [N2: nat] :
% 2.82/1.53 ( ! [M3: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ M3 @ N2 )
% 2.82/1.53 => ( P @ M3 ) )
% 2.82/1.53 => ( P @ N2 ) )
% 2.82/1.53 => ( P @ N3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_135_infinite__descent,axiom,
% 2.82/1.53 ! [P: nat > $o,N3: nat] :
% 2.82/1.53 ( ! [N2: nat] :
% 2.82/1.53 ( ~ ( P @ N2 )
% 2.82/1.53 => ? [M3: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ M3 @ N2 )
% 2.82/1.53 & ~ ( P @ M3 ) ) )
% 2.82/1.53 => ( P @ N3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_136_linorder__neqE__nat,axiom,
% 2.82/1.53 ! [X3: nat,Y2: nat] :
% 2.82/1.53 ( ( X3 != Y2 )
% 2.82/1.53 => ( ~ ( ord_less @ nat @ X3 @ Y2 )
% 2.82/1.53 => ( ord_less @ nat @ Y2 @ X3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_137_measure__induct__rule,axiom,
% 2.82/1.53 ! [A: $tType,F2: A > nat,P: A > $o,A2: A] :
% 2.82/1.53 ( ! [X2: A] :
% 2.82/1.53 ( ! [Y3: A] :
% 2.82/1.53 ( ( ord_less @ nat @ ( F2 @ Y3 ) @ ( F2 @ X2 ) )
% 2.82/1.53 => ( P @ Y3 ) )
% 2.82/1.53 => ( P @ X2 ) )
% 2.82/1.53 => ( P @ A2 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_138_infinite__descent__measure,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,V: A > nat,X3: A] :
% 2.82/1.53 ( ! [X2: A] :
% 2.82/1.53 ( ~ ( P @ X2 )
% 2.82/1.53 => ? [Y3: A] :
% 2.82/1.53 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
% 2.82/1.53 & ~ ( P @ Y3 ) ) )
% 2.82/1.53 => ( P @ X3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_139_lappend_Oexhaust,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 => ~ ( coindu335574135_lnull @ A @ Ys ) )
% 2.82/1.53 => ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 | ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_140_lzip_Oexhaust,axiom,
% 2.82/1.53 ! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
% 2.82/1.53 ( ~ ( ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 | ( coindu335574135_lnull @ B @ Ys ) )
% 2.82/1.53 => ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 => ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_141_lnull__ltlI,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 => ( coindu335574135_lnull @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_142_lnull__imp__lfinite,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 => ( coindu1213758845finite @ A @ Xs ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_143_inf__set__def,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( inf_inf @ ( set @ A ) )
% 2.82/1.53 = ( ^ [A4: set @ A,B6: set @ A] :
% 2.82/1.53 ( collect @ A
% 2.82/1.53 @ ( inf_inf @ ( A > $o )
% 2.82/1.53 @ ^ [X: A] : ( member @ A @ X @ A4 )
% 2.82/1.53 @ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_144_llist_Oset__sel_I2_J,axiom,
% 2.82/1.53 ! [A: $tType,A2: coindu1593790203_llist @ A,X3: A] :
% 2.82/1.53 ( ~ ( coindu335574135_lnull @ A @ A2 )
% 2.82/1.53 => ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ A2 ) ) )
% 2.82/1.53 => ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ A2 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_145_lfinite__induct,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
% 2.82/1.53 ( ( coindu1213758845finite @ A @ Xs )
% 2.82/1.53 => ( ! [Xs3: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu335574135_lnull @ A @ Xs3 )
% 2.82/1.53 => ( P @ Xs3 ) )
% 2.82/1.53 => ( ! [Xs3: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1213758845finite @ A @ Xs3 )
% 2.82/1.53 => ( ~ ( coindu335574135_lnull @ A @ Xs3 )
% 2.82/1.53 => ( ( P @ ( coindu1047225960le_ltl @ A @ Xs3 ) )
% 2.82/1.53 => ( P @ Xs3 ) ) ) )
% 2.82/1.53 => ( P @ Xs ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_146_inf__sup__aci_I4_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,Y2: A] :
% 2.82/1.53 ( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ X3 @ Y2 ) )
% 2.82/1.53 = ( inf_inf @ A @ X3 @ Y2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_147_inf__sup__aci_I3_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,Y2: A,Z: A] :
% 2.82/1.53 ( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) )
% 2.82/1.53 = ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X3 @ Z ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_148_inf__sup__aci_I2_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,Y2: A,Z: A] :
% 2.82/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Z )
% 2.82/1.53 = ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_149_inf__sup__aci_I1_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.82/1.53 => ( ( inf_inf @ A )
% 2.82/1.53 = ( ^ [X: A,Y4: A] : ( inf_inf @ A @ Y4 @ X ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_150_inf__fun__def,axiom,
% 2.82/1.53 ! [B: $tType,A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 2.82/1.53 => ( ( inf_inf @ ( A > B ) )
% 2.82/1.53 = ( ^ [F3: A > B,G3: A > B,X: A] : ( inf_inf @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_151_inf_Oassoc,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A,C: A] :
% 2.82/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C )
% 2.82/1.53 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_152_inf__assoc,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,Y2: A,Z: A] :
% 2.82/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Z )
% 2.82/1.53 = ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_153_inf_Ocommute,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ( ( inf_inf @ A )
% 2.82/1.53 = ( ^ [A6: A,B5: A] : ( inf_inf @ A @ B5 @ A6 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_154_inf__commute,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ( ( inf_inf @ A )
% 2.82/1.53 = ( ^ [X: A,Y4: A] : ( inf_inf @ A @ Y4 @ X ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_155_inf_Oleft__commute,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [B2: A,A2: A,C: A] :
% 2.82/1.53 ( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C ) )
% 2.82/1.53 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_156_inf__left__commute,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,Y2: A,Z: A] :
% 2.82/1.53 ( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) )
% 2.82/1.53 = ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X3 @ Z ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_157_nat__add__left__cancel,axiom,
% 2.82/1.53 ! [K: nat,M: nat,N3: nat] :
% 2.82/1.53 ( ( ( plus_plus @ nat @ K @ M )
% 2.82/1.53 = ( plus_plus @ nat @ K @ N3 ) )
% 2.82/1.53 = ( M = N3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_158_nat__add__right__cancel,axiom,
% 2.82/1.53 ! [M: nat,K: nat,N3: nat] :
% 2.82/1.53 ( ( ( plus_plus @ nat @ M @ K )
% 2.82/1.53 = ( plus_plus @ nat @ N3 @ K ) )
% 2.82/1.53 = ( M = N3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_159_add__lessD1,axiom,
% 2.82/1.53 ! [I2: nat,J: nat,K: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
% 2.82/1.53 => ( ord_less @ nat @ I2 @ K ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_160_add__less__mono,axiom,
% 2.82/1.53 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ I2 @ J )
% 2.82/1.53 => ( ( ord_less @ nat @ K @ L )
% 2.82/1.53 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_161_not__add__less1,axiom,
% 2.82/1.53 ! [I2: nat,J: nat] :
% 2.82/1.53 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ I2 ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_162_not__add__less2,axiom,
% 2.82/1.53 ! [J: nat,I2: nat] :
% 2.82/1.53 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I2 ) @ I2 ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_163_add__less__mono1,axiom,
% 2.82/1.53 ! [I2: nat,J: nat,K: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ I2 @ J )
% 2.82/1.53 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_164_trans__less__add1,axiom,
% 2.82/1.53 ! [I2: nat,J: nat,M: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ I2 @ J )
% 2.82/1.53 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_165_trans__less__add2,axiom,
% 2.82/1.53 ! [I2: nat,J: nat,M: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ I2 @ J )
% 2.82/1.53 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_166_less__add__eq__less,axiom,
% 2.82/1.53 ! [K: nat,L: nat,M: nat,N3: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ K @ L )
% 2.82/1.53 => ( ( ( plus_plus @ nat @ M @ L )
% 2.82/1.53 = ( plus_plus @ nat @ K @ N3 ) )
% 2.82/1.53 => ( ord_less @ nat @ M @ N3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_167_less__infI1,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,X3: A,B2: A] :
% 2.82/1.53 ( ( ord_less @ A @ A2 @ X3 )
% 2.82/1.53 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_168_less__infI2,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [B2: A,X3: A,A2: A] :
% 2.82/1.53 ( ( ord_less @ A @ B2 @ X3 )
% 2.82/1.53 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_169_inf_Ostrict__boundedE,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A,C: A] :
% 2.82/1.53 ( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
% 2.82/1.53 => ~ ( ( ord_less @ A @ A2 @ B2 )
% 2.82/1.53 => ~ ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_170_inf_Ostrict__order__iff,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ( ( ord_less @ A )
% 2.82/1.53 = ( ^ [A6: A,B5: A] :
% 2.82/1.53 ( ( A6
% 2.82/1.53 = ( inf_inf @ A @ A6 @ B5 ) )
% 2.82/1.53 & ( A6 != B5 ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_171_inf_Ostrict__coboundedI1,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,C: A,B2: A] :
% 2.82/1.53 ( ( ord_less @ A @ A2 @ C )
% 2.82/1.53 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_172_inf_Ostrict__coboundedI2,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [B2: A,C: A,A2: A] :
% 2.82/1.53 ( ( ord_less @ A @ B2 @ C )
% 2.82/1.53 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_173_enat__less__enat__plusI,axiom,
% 2.82/1.53 ! [X3: nat,Y2: nat,Z: extended_enat] :
% 2.82/1.53 ( ( ord_less @ nat @ X3 @ Y2 )
% 2.82/1.53 => ( ord_less @ extended_enat @ ( extended_enat2 @ X3 ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ Y2 ) @ Z ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_174_fold__atLeastAtMost__nat_Oinduct,axiom,
% 2.82/1.53 ! [A: $tType,P: ( nat > A > A ) > nat > nat > A > $o,A0: nat > A > A,A1: nat,A22: nat,A32: A] :
% 2.82/1.53 ( ! [F4: nat > A > A,A7: nat,B7: nat,Acc: A] :
% 2.82/1.53 ( ( ~ ( ord_less @ nat @ B7 @ A7 )
% 2.82/1.53 => ( P @ F4 @ ( plus_plus @ nat @ A7 @ ( one_one @ nat ) ) @ B7 @ ( F4 @ A7 @ Acc ) ) )
% 2.82/1.53 => ( P @ F4 @ A7 @ B7 @ Acc ) )
% 2.82/1.53 => ( P @ A0 @ A1 @ A22 @ A32 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_175_finite__M__bounded__by__nat,axiom,
% 2.82/1.53 ! [P: nat > $o,I2: nat] :
% 2.82/1.53 ( finite_finite2 @ nat
% 2.82/1.53 @ ( collect @ nat
% 2.82/1.53 @ ^ [K3: nat] :
% 2.82/1.53 ( ( P @ K3 )
% 2.82/1.53 & ( ord_less @ nat @ K3 @ I2 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_176_gen__llength__def,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( coindu493225792length @ A )
% 2.82/1.53 = ( ^ [N: nat,Xs2: coindu1593790203_llist @ A] : ( plus_plus @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs2 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_177_unbounded__k__infinite,axiom,
% 2.82/1.53 ! [K: nat,S2: set @ nat] :
% 2.82/1.53 ( ! [M4: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ K @ M4 )
% 2.82/1.53 => ? [N5: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ M4 @ N5 )
% 2.82/1.53 & ( member @ nat @ N5 @ S2 ) ) )
% 2.82/1.53 => ~ ( finite_finite2 @ nat @ S2 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_178_inf1I,axiom,
% 2.82/1.53 ! [A: $tType,A3: A > $o,X3: A,B3: A > $o] :
% 2.82/1.53 ( ( A3 @ X3 )
% 2.82/1.53 => ( ( B3 @ X3 )
% 2.82/1.53 => ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_179_inf1E,axiom,
% 2.82/1.53 ! [A: $tType,A3: A > $o,B3: A > $o,X3: A] :
% 2.82/1.53 ( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X3 )
% 2.82/1.53 => ~ ( ( A3 @ X3 )
% 2.82/1.53 => ~ ( B3 @ X3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_180_inf1D1,axiom,
% 2.82/1.53 ! [A: $tType,A3: A > $o,B3: A > $o,X3: A] :
% 2.82/1.53 ( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X3 )
% 2.82/1.53 => ( A3 @ X3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_181_inf1D2,axiom,
% 2.82/1.53 ! [A: $tType,A3: A > $o,B3: A > $o,X3: A] :
% 2.82/1.53 ( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X3 )
% 2.82/1.53 => ( B3 @ X3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_182_finite__nat__set__iff__bounded,axiom,
% 2.82/1.53 ( ( finite_finite2 @ nat )
% 2.82/1.53 = ( ^ [N6: set @ nat] :
% 2.82/1.53 ? [M5: nat] :
% 2.82/1.53 ! [X: nat] :
% 2.82/1.53 ( ( member @ nat @ X @ N6 )
% 2.82/1.53 => ( ord_less @ nat @ X @ M5 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_183_infinite__nat__iff__unbounded,axiom,
% 2.82/1.53 ! [S2: set @ nat] :
% 2.82/1.53 ( ( ~ ( finite_finite2 @ nat @ S2 ) )
% 2.82/1.53 = ( ! [M5: nat] :
% 2.82/1.53 ? [N: nat] :
% 2.82/1.53 ( ( ord_less @ nat @ M5 @ N )
% 2.82/1.53 & ( member @ nat @ N @ S2 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_184_bounded__nat__set__is__finite,axiom,
% 2.82/1.53 ! [N4: set @ nat,N3: nat] :
% 2.82/1.53 ( ! [X2: nat] :
% 2.82/1.53 ( ( member @ nat @ X2 @ N4 )
% 2.82/1.53 => ( ord_less @ nat @ X2 @ N3 ) )
% 2.82/1.53 => ( finite_finite2 @ nat @ N4 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_185_inf__Int__eq,axiom,
% 2.82/1.53 ! [A: $tType,R: set @ A,S2: set @ A] :
% 2.82/1.53 ( ( inf_inf @ ( A > $o )
% 2.82/1.53 @ ^ [X: A] : ( member @ A @ X @ R )
% 2.82/1.53 @ ^ [X: A] : ( member @ A @ X @ S2 ) )
% 2.82/1.53 = ( ^ [X: A] : ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ R @ S2 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_186_lnth__llength__ltakeWhile,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) @ ( coindu1018505716length @ A @ Xs ) )
% 2.82/1.53 => ~ ( P @ ( coindu749330388e_lnth @ A @ Xs @ ( extended_the_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_187_lnth__ldropn,axiom,
% 2.82/1.53 ! [A: $tType,N3: nat,M: nat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ N3 @ M ) ) @ ( coindu1018505716length @ A @ Xs ) )
% 2.82/1.53 => ( ( coindu749330388e_lnth @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) @ M )
% 2.82/1.53 = ( coindu749330388e_lnth @ A @ Xs @ ( plus_plus @ nat @ M @ N3 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_188_ldropn__ldropn,axiom,
% 2.82/1.53 ! [A: $tType,N3: nat,M: nat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu531130065ldropn @ A @ N3 @ ( coindu531130065ldropn @ A @ M @ Xs ) )
% 2.82/1.53 = ( coindu531130065ldropn @ A @ ( plus_plus @ nat @ N3 @ M ) @ Xs ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_189_lfinite__ldropn,axiom,
% 2.82/1.53 ! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1213758845finite @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
% 2.82/1.53 = ( coindu1213758845finite @ A @ Xs ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_190_the__enat_Osimps,axiom,
% 2.82/1.53 ! [N3: nat] :
% 2.82/1.53 ( ( extended_the_enat @ ( extended_enat2 @ N3 ) )
% 2.82/1.53 = N3 ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_191_ltl__ldropn,axiom,
% 2.82/1.53 ! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1047225960le_ltl @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
% 2.82/1.53 = ( coindu531130065ldropn @ A @ N3 @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_192_in__lset__ldropnD,axiom,
% 2.82/1.53 ! [A: $tType,X3: A,N3: nat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) ) )
% 2.82/1.53 => ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_193_lnth__ldrop,axiom,
% 2.82/1.53 ! [A: $tType,N3: extended_enat,M: nat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ N3 @ ( extended_enat2 @ M ) ) @ ( coindu1018505716length @ A @ Xs ) )
% 2.82/1.53 => ( ( coindu749330388e_lnth @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) @ M )
% 2.82/1.53 = ( coindu749330388e_lnth @ A @ Xs @ ( plus_plus @ nat @ M @ ( extended_the_enat @ N3 ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_194_less__numeral__extra_I4_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.82/1.53 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_195_ldrop__ldrop,axiom,
% 2.82/1.53 ! [A: $tType,N3: extended_enat,M: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu191418589_ldrop @ A @ N3 @ ( coindu191418589_ldrop @ A @ M @ Xs ) )
% 2.82/1.53 = ( coindu191418589_ldrop @ A @ ( plus_plus @ extended_enat @ N3 @ M ) @ Xs ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_196_in__lset__ldropD,axiom,
% 2.82/1.53 ! [A: $tType,X3: A,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) ) )
% 2.82/1.53 => ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_197_ltl__ldrop,axiom,
% 2.82/1.53 ! [A: $tType,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu1047225960le_ltl @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) )
% 2.82/1.53 = ( coindu191418589_ldrop @ A @ N3 @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_198_ldrop__enat,axiom,
% 2.82/1.53 ! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu191418589_ldrop @ A @ ( extended_enat2 @ N3 ) @ Xs )
% 2.82/1.53 = ( coindu531130065ldropn @ A @ N3 @ Xs ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_199_is__num__normalize_I1_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( neg_numeral @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A,C: A] :
% 2.82/1.53 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 2.82/1.53 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_200_ldropWhile__eq__ldrop,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( coindu438612276pWhile @ A )
% 2.82/1.53 = ( ^ [P2: A > $o,Xs2: coindu1593790203_llist @ A] : ( coindu191418589_ldrop @ A @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P2 @ Xs2 ) ) @ Xs2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_201_lhd__ldropWhile__conv__lnth,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.82/1.53 ( ? [X4: A] :
% 2.82/1.53 ( ( member @ A @ X4 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 2.82/1.53 & ~ ( P @ X4 ) )
% 2.82/1.53 => ( ( coindu1046438764le_lhd @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) )
% 2.82/1.53 = ( coindu749330388e_lnth @ A @ Xs @ ( extended_the_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_202_llast__ldropn,axiom,
% 2.82/1.53 ! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ ( extended_enat2 @ N3 ) @ ( coindu1018505716length @ A @ Xs ) )
% 2.82/1.53 => ( ( coindu2000965700_llast @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
% 2.82/1.53 = ( coindu2000965700_llast @ A @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_203_lnull__ltakeWhile,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 2.82/1.53 = ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 => ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_204_ltakeWhile_Odisc__iff_I1_J,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 2.82/1.53 = ( ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_205_ltakeWhile_Odisc__iff_I2_J,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
% 2.82/1.53 = ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 & ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_206_ltakeWhile_Oexhaust,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.82/1.53 ( ~ ( ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
% 2.82/1.53 => ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 => ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_207_llist_Oset__sel_I1_J,axiom,
% 2.82/1.53 ! [A: $tType,A2: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ~ ( coindu335574135_lnull @ A @ A2 )
% 2.82/1.53 => ( member @ A @ ( coindu1046438764le_lhd @ A @ A2 ) @ ( coindu1112613586e_lset @ A @ A2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_208_llist_Ocoinduct__strong,axiom,
% 2.82/1.53 ! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist2: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( R @ Llist @ Llist2 )
% 2.82/1.53 => ( ! [Llist3: coindu1593790203_llist @ A,Llist4: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( R @ Llist3 @ Llist4 )
% 2.82/1.53 => ( ( ( coindu335574135_lnull @ A @ Llist3 )
% 2.82/1.53 = ( coindu335574135_lnull @ A @ Llist4 ) )
% 2.82/1.53 & ( ~ ( coindu335574135_lnull @ A @ Llist3 )
% 2.82/1.53 => ( ~ ( coindu335574135_lnull @ A @ Llist4 )
% 2.82/1.53 => ( ( ( coindu1046438764le_lhd @ A @ Llist3 )
% 2.82/1.53 = ( coindu1046438764le_lhd @ A @ Llist4 ) )
% 2.82/1.53 & ( ( R @ ( coindu1047225960le_ltl @ A @ Llist3 ) @ ( coindu1047225960le_ltl @ A @ Llist4 ) )
% 2.82/1.53 | ( ( coindu1047225960le_ltl @ A @ Llist3 )
% 2.82/1.53 = ( coindu1047225960le_ltl @ A @ Llist4 ) ) ) ) ) ) ) )
% 2.82/1.53 => ( Llist = Llist2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_209_llist_Ocoinduct,axiom,
% 2.82/1.53 ! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist2: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( R @ Llist @ Llist2 )
% 2.82/1.53 => ( ! [Llist3: coindu1593790203_llist @ A,Llist4: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( R @ Llist3 @ Llist4 )
% 2.82/1.53 => ( ( ( coindu335574135_lnull @ A @ Llist3 )
% 2.82/1.53 = ( coindu335574135_lnull @ A @ Llist4 ) )
% 2.82/1.53 & ( ~ ( coindu335574135_lnull @ A @ Llist3 )
% 2.82/1.53 => ( ~ ( coindu335574135_lnull @ A @ Llist4 )
% 2.82/1.53 => ( ( ( coindu1046438764le_lhd @ A @ Llist3 )
% 2.82/1.53 = ( coindu1046438764le_lhd @ A @ Llist4 ) )
% 2.82/1.53 & ( R @ ( coindu1047225960le_ltl @ A @ Llist3 ) @ ( coindu1047225960le_ltl @ A @ Llist4 ) ) ) ) ) ) )
% 2.82/1.53 => ( Llist = Llist2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_210_llist_Oexpand,axiom,
% 2.82/1.53 ! [A: $tType,Llist: coindu1593790203_llist @ A,Llist2: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( ( coindu335574135_lnull @ A @ Llist )
% 2.82/1.53 = ( coindu335574135_lnull @ A @ Llist2 ) )
% 2.82/1.53 => ( ( ~ ( coindu335574135_lnull @ A @ Llist )
% 2.82/1.53 => ( ~ ( coindu335574135_lnull @ A @ Llist2 )
% 2.82/1.53 => ( ( ( coindu1046438764le_lhd @ A @ Llist )
% 2.82/1.53 = ( coindu1046438764le_lhd @ A @ Llist2 ) )
% 2.82/1.53 & ( ( coindu1047225960le_ltl @ A @ Llist )
% 2.82/1.53 = ( coindu1047225960le_ltl @ A @ Llist2 ) ) ) ) )
% 2.82/1.53 => ( Llist = Llist2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_211_ltakeWhile_Odisc_I2_J,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.82/1.53 ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 => ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
% 2.82/1.53 => ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_212_ltakeWhile_Odisc_I1_J,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.82/1.53 ( ( ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
% 2.82/1.53 => ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_213_lhd__ltakeWhile,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.82/1.53 ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 => ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
% 2.82/1.53 => ( ( coindu1046438764le_lhd @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 2.82/1.53 = ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_214_lhd__ldropWhile__in__lset,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.82/1.53 ( ? [X4: A] :
% 2.82/1.53 ( ( member @ A @ X4 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 2.82/1.53 & ~ ( P @ X4 ) )
% 2.82/1.53 => ( member @ A @ ( coindu1046438764le_lhd @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) ) @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_215_lhd__ldropWhile,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.82/1.53 ( ? [X4: A] :
% 2.82/1.53 ( ( member @ A @ X4 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 2.82/1.53 & ~ ( P @ X4 ) )
% 2.82/1.53 => ~ ( P @ ( coindu1046438764le_lhd @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_216_llist__set__induct,axiom,
% 2.82/1.53 ! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A,P: A > ( coindu1593790203_llist @ A ) > $o] :
% 2.82/1.53 ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
% 2.82/1.53 => ( ! [Xs3: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ~ ( coindu335574135_lnull @ A @ Xs3 )
% 2.82/1.53 => ( P @ ( coindu1046438764le_lhd @ A @ Xs3 ) @ Xs3 ) )
% 2.82/1.53 => ( ! [Xs3: coindu1593790203_llist @ A,Y5: A] :
% 2.82/1.53 ( ~ ( coindu335574135_lnull @ A @ Xs3 )
% 2.82/1.53 => ( ( member @ A @ Y5 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs3 ) ) )
% 2.82/1.53 => ( ( P @ Y5 @ ( coindu1047225960le_ltl @ A @ Xs3 ) )
% 2.82/1.53 => ( P @ Y5 @ Xs3 ) ) ) )
% 2.82/1.53 => ( P @ X3 @ Xs ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_217_ltakeWhile_Osimps_I4_J,axiom,
% 2.82/1.53 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.82/1.53 ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.82/1.53 => ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
% 2.82/1.53 => ( ( coindu1047225960le_ltl @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 2.82/1.53 = ( coindu721411036eWhile @ A @ P @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_218_llast__ldrop,axiom,
% 2.82/1.53 ! [A: $tType,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ N3 @ ( coindu1018505716length @ A @ Xs ) )
% 2.82/1.53 => ( ( coindu2000965700_llast @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) )
% 2.82/1.53 = ( coindu2000965700_llast @ A @ Xs ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_219_lhd__ldropn,axiom,
% 2.82/1.53 ! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ ( extended_enat2 @ N3 ) @ ( coindu1018505716length @ A @ Xs ) )
% 2.82/1.53 => ( ( coindu1046438764le_lhd @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
% 2.82/1.53 = ( coindu749330388e_lnth @ A @ Xs @ N3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_220_lhd__ldrop,axiom,
% 2.82/1.53 ! [A: $tType,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ N3 @ ( coindu1018505716length @ A @ Xs ) )
% 2.82/1.53 => ( ( coindu1046438764le_lhd @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) )
% 2.82/1.53 = ( coindu749330388e_lnth @ A @ Xs @ ( extended_the_enat @ N3 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_221_enat_Osimps_I6_J,axiom,
% 2.82/1.53 ! [T: $tType,F1: nat > T,F22: T,Nat: nat] :
% 2.82/1.53 ( ( extended_rec_enat @ T @ F1 @ F22 @ ( extended_enat2 @ Nat ) )
% 2.82/1.53 = ( F1 @ Nat ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_222_lset__conv__lnth,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( coindu1112613586e_lset @ A )
% 2.82/1.53 = ( ^ [Xs2: coindu1593790203_llist @ A] :
% 2.82/1.53 ( collect @ A
% 2.82/1.53 @ ^ [Uu: A] :
% 2.82/1.53 ? [N: nat] :
% 2.82/1.53 ( ( Uu
% 2.82/1.53 = ( coindu749330388e_lnth @ A @ Xs2 @ N ) )
% 2.82/1.53 & ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs2 ) ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_223_finite__Collect__bounded__ex,axiom,
% 2.82/1.53 ! [B: $tType,A: $tType,P: A > $o,Q: B > A > $o] :
% 2.82/1.53 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.82/1.53 => ( ( finite_finite2 @ B
% 2.82/1.53 @ ( collect @ B
% 2.82/1.53 @ ^ [X: B] :
% 2.82/1.53 ? [Y4: A] :
% 2.82/1.53 ( ( P @ Y4 )
% 2.82/1.53 & ( Q @ X @ Y4 ) ) ) )
% 2.82/1.53 = ( ! [Y4: A] :
% 2.82/1.53 ( ( P @ Y4 )
% 2.82/1.53 => ( finite_finite2 @ B
% 2.82/1.53 @ ( collect @ B
% 2.82/1.53 @ ^ [X: B] : ( Q @ X @ Y4 ) ) ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_224_finite__image__set,axiom,
% 2.82/1.53 ! [A: $tType,B: $tType,P: A > $o,F2: A > B] :
% 2.82/1.53 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.82/1.53 => ( finite_finite2 @ B
% 2.82/1.53 @ ( collect @ B
% 2.82/1.53 @ ^ [Uu: B] :
% 2.82/1.53 ? [X: A] :
% 2.82/1.53 ( ( Uu
% 2.82/1.53 = ( F2 @ X ) )
% 2.82/1.53 & ( P @ X ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_225_finite__image__set2,axiom,
% 2.82/1.53 ! [A: $tType,B: $tType,C3: $tType,P: A > $o,Q: B > $o,F2: A > B > C3] :
% 2.82/1.53 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.82/1.53 => ( ( finite_finite2 @ B @ ( collect @ B @ Q ) )
% 2.82/1.53 => ( finite_finite2 @ C3
% 2.82/1.53 @ ( collect @ C3
% 2.82/1.53 @ ^ [Uu: C3] :
% 2.82/1.53 ? [X: A,Y4: B] :
% 2.82/1.53 ( ( Uu
% 2.82/1.53 = ( F2 @ X @ Y4 ) )
% 2.82/1.53 & ( P @ X )
% 2.82/1.53 & ( Q @ Y4 ) ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_226_Setcompr__eq__image,axiom,
% 2.82/1.53 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
% 2.82/1.53 ( ( collect @ A
% 2.82/1.53 @ ^ [Uu: A] :
% 2.82/1.53 ? [X: B] :
% 2.82/1.53 ( ( Uu
% 2.82/1.53 = ( F2 @ X ) )
% 2.82/1.53 & ( member @ B @ X @ A3 ) ) )
% 2.82/1.53 = ( image @ B @ A @ F2 @ A3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_227_setcompr__eq__image,axiom,
% 2.82/1.53 ! [A: $tType,B: $tType,F2: B > A,P: B > $o] :
% 2.82/1.53 ( ( collect @ A
% 2.82/1.53 @ ^ [Uu: A] :
% 2.82/1.53 ? [X: B] :
% 2.82/1.53 ( ( Uu
% 2.82/1.53 = ( F2 @ X ) )
% 2.82/1.53 & ( P @ X ) ) )
% 2.82/1.53 = ( image @ B @ A @ F2 @ ( collect @ B @ P ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_228_lnull__ldropn,axiom,
% 2.82/1.53 ! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu335574135_lnull @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
% 2.82/1.53 = ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( extended_enat2 @ N3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_229_llength__def,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( coindu1018505716length @ A )
% 2.82/1.53 = ( coindu1491768222unfold @ ( coindu1593790203_llist @ A ) @ ( coindu335574135_lnull @ A ) @ ( coindu1047225960le_ltl @ A ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_230_add__le__cancel__right,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,C: A,B2: A] :
% 2.82/1.53 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 2.82/1.53 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_231_add__le__cancel__left,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [C: A,A2: A,B2: A] :
% 2.82/1.53 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 2.82/1.53 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_232_inf_Obounded__iff,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A,C: A] :
% 2.82/1.53 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
% 2.82/1.53 = ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.82/1.53 & ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_233_le__inf__iff,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,Y2: A,Z: A] :
% 2.82/1.53 ( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) )
% 2.82/1.53 = ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.82/1.53 & ( ord_less_eq @ A @ X3 @ Z ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_234_lnull__ldrop,axiom,
% 2.82/1.53 ! [A: $tType,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.82/1.53 ( ( coindu335574135_lnull @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) )
% 2.82/1.53 = ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_235_le__numeral__extra_I4_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.82/1.53 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_236_enat__less__imp__le,axiom,
% 2.82/1.53 ! [N3: extended_enat,M: extended_enat] :
% 2.82/1.53 ( ! [K2: nat] :
% 2.82/1.53 ( ( ord_less @ extended_enat @ N3 @ ( extended_enat2 @ K2 ) )
% 2.82/1.53 => ( ord_less @ extended_enat @ M @ ( extended_enat2 @ K2 ) ) )
% 2.82/1.53 => ( ord_less_eq @ extended_enat @ M @ N3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_237_llength__ltakeWhile__le,axiom,
% 2.82/1.53 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] : ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) @ ( coindu1018505716length @ A @ Xs ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_238_inf__sup__ord_I2_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Y2 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_239_inf__sup__ord_I1_J,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ X3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_240_inf__le1,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ X3 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_241_inf__le2,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Y2 ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_242_le__infE,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,A2: A,B2: A] :
% 2.82/1.53 ( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) )
% 2.82/1.53 => ~ ( ( ord_less_eq @ A @ X3 @ A2 )
% 2.82/1.53 => ~ ( ord_less_eq @ A @ X3 @ B2 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_243_le__infI,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [X3: A,A2: A,B2: A] :
% 2.82/1.53 ( ( ord_less_eq @ A @ X3 @ A2 )
% 2.82/1.53 => ( ( ord_less_eq @ A @ X3 @ B2 )
% 2.82/1.53 => ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_244_inf__mono,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,C: A,B2: A,D: A] :
% 2.82/1.53 ( ( ord_less_eq @ A @ A2 @ C )
% 2.82/1.53 => ( ( ord_less_eq @ A @ B2 @ D )
% 2.82/1.53 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C @ D ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_245_le__infI1,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,X3: A,B2: A] :
% 2.82/1.53 ( ( ord_less_eq @ A @ A2 @ X3 )
% 2.82/1.53 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_246_le__infI2,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [B2: A,X3: A,A2: A] :
% 2.82/1.53 ( ( ord_less_eq @ A @ B2 @ X3 )
% 2.82/1.53 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_247_inf_OorderE,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A] :
% 2.82/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.82/1.53 => ( A2
% 2.82/1.53 = ( inf_inf @ A @ A2 @ B2 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_248_inf_OorderI,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A] :
% 2.82/1.53 ( ( A2
% 2.82/1.53 = ( inf_inf @ A @ A2 @ B2 ) )
% 2.82/1.53 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_249_inf__unique,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [F2: A > A > A,X3: A,Y2: A] :
% 2.82/1.53 ( ! [X2: A,Y5: A] : ( ord_less_eq @ A @ ( F2 @ X2 @ Y5 ) @ X2 )
% 2.82/1.53 => ( ! [X2: A,Y5: A] : ( ord_less_eq @ A @ ( F2 @ X2 @ Y5 ) @ Y5 )
% 2.82/1.53 => ( ! [X2: A,Y5: A,Z2: A] :
% 2.82/1.53 ( ( ord_less_eq @ A @ X2 @ Y5 )
% 2.82/1.53 => ( ( ord_less_eq @ A @ X2 @ Z2 )
% 2.82/1.53 => ( ord_less_eq @ A @ X2 @ ( F2 @ Y5 @ Z2 ) ) ) )
% 2.82/1.53 => ( ( inf_inf @ A @ X3 @ Y2 )
% 2.82/1.53 = ( F2 @ X3 @ Y2 ) ) ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_250_le__iff__inf,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ( ( ord_less_eq @ A )
% 2.82/1.53 = ( ^ [X: A,Y4: A] :
% 2.82/1.53 ( ( inf_inf @ A @ X @ Y4 )
% 2.82/1.53 = X ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_251_inf_Oabsorb1,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [A2: A,B2: A] :
% 2.82/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.82/1.53 => ( ( inf_inf @ A @ A2 @ B2 )
% 2.82/1.53 = A2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(fact_252_inf_Oabsorb2,axiom,
% 2.82/1.53 ! [A: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.82/1.53 => ! [B2: A,A2: A] :
% 2.82/1.53 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.82/1.53 => ( ( inf_inf @ A @ A2 @ B2 )
% 2.82/1.53 = B2 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 2.82/1.53 ! [A8: $tType,A9: $tType] :
% 2.82/1.53 ( ( semilattice_inf @ A9 @ ( type2 @ A9 ) )
% 2.82/1.53 => ( semilattice_inf @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.82/1.53 ! [A8: $tType,A9: $tType] :
% 2.82/1.53 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 2.82/1.53 & ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
% 2.82/1.53 => ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.82/1.53 ! [A8: $tType,A9: $tType] :
% 2.82/1.53 ( ( lattice @ A9 @ ( type2 @ A9 ) )
% 2.82/1.53 => ( lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 2.82/1.53 strict2144017051up_add @ nat @ ( type2 @ nat ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 2.82/1.53 ordere223160158up_add @ nat @ ( type2 @ nat ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.82/1.53 ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 2.82/1.53 ordere779506340up_add @ nat @ ( type2 @ nat ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 2.82/1.53 cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 2.82/1.53 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_1,axiom,
% 2.82/1.53 semilattice_inf @ nat @ ( type2 @ nat ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 2.82/1.53 ab_semigroup_add @ nat @ ( type2 @ nat ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 2.82/1.53 semigroup_add @ nat @ ( type2 @ nat ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Nat_Onat___Lattices_Olattice_2,axiom,
% 2.82/1.53 lattice @ nat @ ( type2 @ nat ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 2.82/1.53 one @ nat @ ( type2 @ nat ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_3,axiom,
% 2.82/1.53 ! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Set_Oset___Finite__Set_Ofinite_4,axiom,
% 2.82/1.53 ! [A8: $tType] :
% 2.82/1.53 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 2.82/1.53 => ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_Set_Oset___Lattices_Olattice_5,axiom,
% 2.82/1.53 ! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.82/1.53
% 2.82/1.53 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_6,axiom,
% 2.82/1.54 semilattice_inf @ $o @ ( type2 @ $o ) ).
% 2.82/1.54
% 2.82/1.54 thf(tcon_HOL_Obool___Finite__Set_Ofinite_7,axiom,
% 2.82/1.54 finite_finite @ $o @ ( type2 @ $o ) ).
% 2.82/1.54
% 2.82/1.54 thf(tcon_HOL_Obool___Lattices_Olattice_8,axiom,
% 2.82/1.54 lattice @ $o @ ( type2 @ $o ) ).
% 2.82/1.54
% 2.82/1.54 thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_9,axiom,
% 2.82/1.54 strict2144017051up_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.82/1.54
% 2.82/1.54 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_10,axiom,
% 2.82/1.54 ordere779506340up_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.82/1.54
% 2.82/1.54 thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_11,axiom,
% 2.82/1.54 semilattice_inf @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.82/1.54
% 2.82/1.54 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_12,axiom,
% 2.82/1.54 ab_semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.82/1.54
% 2.82/1.54 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_13,axiom,
% 2.82/1.54 semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.82/1.54
% 2.82/1.54 thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_14,axiom,
% 2.82/1.54 lattice @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.82/1.54
% 2.82/1.54 thf(tcon_Extended__Nat_Oenat___Groups_Oone_15,axiom,
% 2.82/1.54 one @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.82/1.54
% 2.82/1.54 thf(conj_0,conjecture,
% 2.82/1.54 ( finite_finite2 @ nat
% 2.82/1.54 @ ( image @ nat @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ n @ ( one_one @ nat ) ) )
% 2.82/1.54 @ ( inf_inf @ ( set @ nat )
% 2.82/1.54 @ ( collect @ nat
% 2.82/1.54 @ ^ [N: nat] :
% 2.82/1.54 ( ord_less @ extended_enat @ ( extended_enat2 @ N )
% 2.82/1.54 @ ( coindu1018505716length @ a
% 2.82/1.54 @ ( coindu1047225960le_ltl @ a
% 2.82/1.54 @ ( coindu438612276pWhile @ a
% 2.82/1.54 @ ^ [X: a] :
% 2.82/1.54 ~ ( p @ X )
% 2.82/1.54 @ xsa ) ) ) ) )
% 2.82/1.54 @ ( collect @ nat
% 2.82/1.54 @ ^ [N: nat] :
% 2.82/1.54 ( p
% 2.82/1.54 @ ( coindu749330388e_lnth @ a
% 2.82/1.54 @ ( coindu1047225960le_ltl @ a
% 2.82/1.54 @ ( coindu438612276pWhile @ a
% 2.82/1.54 @ ^ [X: a] :
% 2.82/1.54 ~ ( p @ X )
% 2.82/1.54 @ xsa ) )
% 2.82/1.54 @ N ) ) ) ) ) ) ).
% 2.82/1.54 ------------------------
% 2.82/1.55 % (9957)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/2Mi)
% 2.82/1.55 % (9958)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_9660 for (2999ds/2Mi)
% 2.82/1.55 % (9960)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/18Mi)
% 2.82/1.56 % (9958)Instruction limit reached!
% 2.82/1.56 % (9958)------------------------------
% 2.82/1.56 % (9958)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.56 % (9958)Termination reason: Unknown
% 2.82/1.56 % (9958)Termination phase: shuffling
% 2.82/1.56
% 2.82/1.56 % (9958)Memory used [KB]: 1279
% 2.82/1.56 % (9958)Time elapsed: 0.002 s
% 2.82/1.56 % (9958)Instructions burned: 2 (million)
% 2.82/1.56 % (9958)------------------------------
% 2.82/1.56 % (9958)------------------------------
% 2.82/1.56 % (9957)Instruction limit reached!
% 2.82/1.56 % (9957)------------------------------
% 2.82/1.56 % (9957)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.56 % (9957)Termination reason: Unknown
% 2.82/1.56 % (9957)Termination phase: shuffling
% 2.82/1.56
% 2.82/1.56 % (9957)Memory used [KB]: 1407
% 2.82/1.56 % (9957)Time elapsed: 0.003 s
% 2.82/1.56 % (9957)Instructions burned: 4 (million)
% 2.82/1.56 % (9957)------------------------------
% 2.82/1.56 % (9957)------------------------------
% 2.82/1.56 % (9960)Instruction limit reached!
% 2.82/1.56 % (9960)------------------------------
% 2.82/1.56 % (9960)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.56 % (9960)Termination reason: Unknown
% 2.82/1.56 % (9960)Termination phase: shuffling
% 2.82/1.56
% 2.82/1.56 % (9960)Memory used [KB]: 1791
% 2.82/1.56 % (9960)Time elapsed: 0.008 s
% 2.82/1.56 % (9960)Instructions burned: 18 (million)
% 2.82/1.56 % (9960)------------------------------
% 2.82/1.56 % (9960)------------------------------
% 2.82/1.56 % (9954)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/183Mi)
% 2.82/1.56 % (9955)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_9660 for (2999ds/4Mi)
% 2.82/1.56 % (9956)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_9660 for (2999ds/27Mi)
% 2.82/1.56 % (9959)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_9660 for (2999ds/275Mi)
% 2.82/1.56 % (9955)Instruction limit reached!
% 2.82/1.56 % (9955)------------------------------
% 2.82/1.56 % (9955)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.56 % (9955)Termination reason: Unknown
% 2.82/1.56 % (9955)Termination phase: shuffling
% 2.82/1.56
% 2.82/1.56 % (9955)Memory used [KB]: 1407
% 2.82/1.56 % (9955)Time elapsed: 0.005 s
% 2.82/1.56 % (9955)Instructions burned: 5 (million)
% 2.82/1.56 % (9955)------------------------------
% 2.82/1.56 % (9955)------------------------------
% 2.82/1.57 % (9961)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/3Mi)
% 2.82/1.57 % (9962)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_9660 for (2999ds/37Mi)
% 2.82/1.57 % (9961)Instruction limit reached!
% 2.82/1.57 % (9961)------------------------------
% 2.82/1.57 % (9961)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.57 % (9961)Termination reason: Unknown
% 2.82/1.57 % (9961)Termination phase: shuffling
% 2.82/1.57
% 2.82/1.57 % (9961)Memory used [KB]: 1407
% 2.82/1.57 % (9961)Time elapsed: 0.003 s
% 2.82/1.57 % (9961)Instructions burned: 4 (million)
% 2.82/1.57 % (9961)------------------------------
% 2.82/1.57 % (9961)------------------------------
% 2.82/1.57 % (9963)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_9660 for (2999ds/15Mi)
% 2.82/1.58 % (9965)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_9660 for (2999ds/1041Mi)
% 2.82/1.58 % (9962)Instruction limit reached!
% 2.82/1.58 % (9962)------------------------------
% 2.82/1.58 % (9962)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.58 % (9962)Termination reason: Unknown
% 2.82/1.58 % (9962)Termination phase: shuffling
% 2.82/1.58
% 2.82/1.58 % (9962)Memory used [KB]: 2174
% 2.82/1.58 % (9962)Time elapsed: 0.012 s
% 2.82/1.58 % (9962)Instructions burned: 37 (million)
% 2.82/1.58 % (9962)------------------------------
% 2.82/1.58 % (9962)------------------------------
% 2.82/1.58 % (9963)Instruction limit reached!
% 2.82/1.58 % (9963)------------------------------
% 2.82/1.58 % (9963)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.58 % (9963)Termination reason: Unknown
% 2.82/1.58 % (9963)Termination phase: shuffling
% 2.82/1.58
% 2.82/1.58 % (9963)Memory used [KB]: 1663
% 2.82/1.58 % (9963)Time elapsed: 0.007 s
% 2.82/1.58 % (9963)Instructions burned: 16 (million)
% 2.82/1.58 % (9963)------------------------------
% 2.82/1.58 % (9963)------------------------------
% 2.82/1.58 % (9956)Instruction limit reached!
% 2.82/1.58 % (9956)------------------------------
% 2.82/1.58 % (9956)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.58 % (9956)Termination reason: Unknown
% 2.82/1.58 % (9956)Termination phase: Property scanning
% 2.82/1.58
% 2.82/1.58 % (9956)Memory used [KB]: 1791
% 2.82/1.58 % (9956)Time elapsed: 0.019 s
% 2.82/1.58 % (9956)Instructions burned: 27 (million)
% 2.82/1.58 % (9956)------------------------------
% 2.82/1.58 % (9956)------------------------------
% 2.82/1.58 % (9964)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_9660 for (2999ds/3Mi)
% 2.82/1.58 % (9964)Instruction limit reached!
% 2.82/1.58 % (9964)------------------------------
% 2.82/1.58 % (9964)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.58 % (9964)Termination reason: Unknown
% 2.82/1.58 % (9964)Termination phase: shuffling
% 2.82/1.58
% 2.82/1.58 % (9964)Memory used [KB]: 1407
% 2.82/1.58 % (9964)Time elapsed: 0.004 s
% 2.82/1.58 % (9964)Instructions burned: 3 (million)
% 2.82/1.58 % (9964)------------------------------
% 2.82/1.58 % (9964)------------------------------
% 2.82/1.59 % (9966)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/7Mi)
% 2.82/1.59 % (9967)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/16Mi)
% 2.82/1.59 % (9966)Instruction limit reached!
% 2.82/1.59 % (9966)------------------------------
% 2.82/1.59 % (9966)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.59 % (9966)Termination reason: Unknown
% 2.82/1.59 % (9966)Termination phase: shuffling
% 2.82/1.59
% 2.82/1.59 % (9966)Memory used [KB]: 1535
% 2.82/1.59 % (9966)Time elapsed: 0.004 s
% 2.82/1.59 % (9966)Instructions burned: 9 (million)
% 2.82/1.59 % (9966)------------------------------
% 2.82/1.59 % (9966)------------------------------
% 2.82/1.59 % (9967)Instruction limit reached!
% 2.82/1.59 % (9967)------------------------------
% 2.82/1.59 % (9967)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.59 % (9967)Termination reason: Unknown
% 2.82/1.59 % (9967)Termination phase: shuffling
% 2.82/1.59
% 2.82/1.59 % (9967)Memory used [KB]: 1663
% 2.82/1.59 % (9967)Time elapsed: 0.007 s
% 2.82/1.59 % (9967)Instructions burned: 17 (million)
% 2.82/1.59 % (9967)------------------------------
% 2.82/1.59 % (9967)------------------------------
% 2.82/1.60 % (9968)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_9660 for (2999ds/3Mi)
% 2.87/1.60 % (9968)Instruction limit reached!
% 2.87/1.60 % (9968)------------------------------
% 2.87/1.60 % (9968)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.60 % (9968)Termination reason: Unknown
% 2.87/1.60 % (9968)Termination phase: shuffling
% 2.87/1.60
% 2.87/1.60 % (9968)Memory used [KB]: 1407
% 2.87/1.60 % (9968)Time elapsed: 0.004 s
% 2.87/1.60 % (9968)Instructions burned: 3 (million)
% 2.87/1.60 % (9968)------------------------------
% 2.87/1.60 % (9968)------------------------------
% 2.87/1.60 % (9970)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/7Mi)
% 2.87/1.60 % (9970)Instruction limit reached!
% 2.87/1.60 % (9970)------------------------------
% 2.87/1.60 % (9970)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.60 % (9970)Termination reason: Unknown
% 2.87/1.60 % (9970)Termination phase: shuffling
% 2.87/1.60
% 2.87/1.60 % (9970)Memory used [KB]: 1535
% 2.87/1.60 % (9970)Time elapsed: 0.004 s
% 2.87/1.60 % (9970)Instructions burned: 9 (million)
% 2.87/1.60 % (9970)------------------------------
% 2.87/1.60 % (9970)------------------------------
% 2.87/1.60 % (9969)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_9660 for (2999ds/3Mi)
% 2.87/1.60 % (9969)Instruction limit reached!
% 2.87/1.60 % (9969)------------------------------
% 2.87/1.60 % (9969)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.60 % (9969)Termination reason: Unknown
% 2.87/1.60 % (9969)Termination phase: shuffling
% 2.87/1.60
% 2.87/1.60 % (9969)Memory used [KB]: 1407
% 2.87/1.60 % (9969)Time elapsed: 0.004 s
% 2.87/1.60 % (9969)Instructions burned: 3 (million)
% 2.87/1.60 % (9969)------------------------------
% 2.87/1.60 % (9969)------------------------------
% 2.87/1.60 % (9971)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/3Mi)
% 2.87/1.61 % (9971)Instruction limit reached!
% 2.87/1.61 % (9971)------------------------------
% 2.87/1.61 % (9971)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.61 % (9971)Termination reason: Unknown
% 2.87/1.61 % (9971)Termination phase: shuffling
% 2.87/1.61
% 2.87/1.61 % (9971)Memory used [KB]: 1407
% 2.87/1.61 % (9971)Time elapsed: 0.003 s
% 2.87/1.61 % (9971)Instructions burned: 4 (million)
% 2.87/1.61 % (9971)------------------------------
% 2.87/1.61 % (9971)------------------------------
% 2.87/1.61 % (9973)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/18Mi)
% 2.87/1.62 % (9972)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/4Mi)
% 2.87/1.62 % (9975)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/6Mi)
% 2.87/1.62 % (9973)Instruction limit reached!
% 2.87/1.62 % (9973)------------------------------
% 2.87/1.62 % (9973)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.62 % (9973)Termination reason: Unknown
% 2.87/1.62 % (9973)Termination phase: shuffling
% 2.87/1.62
% 2.87/1.62 % (9973)Memory used [KB]: 1791
% 2.87/1.62 % (9973)Time elapsed: 0.008 s
% 2.87/1.62 % (9973)Instructions burned: 19 (million)
% 2.87/1.62 % (9973)------------------------------
% 2.87/1.62 % (9973)------------------------------
% 2.87/1.62 % (9975)Instruction limit reached!
% 2.87/1.62 % (9975)------------------------------
% 2.87/1.62 % (9975)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.62 % (9975)Termination reason: Unknown
% 2.87/1.62 % (9975)Termination phase: shuffling
% 2.87/1.62
% 2.87/1.62 % (9975)Memory used [KB]: 1407
% 2.87/1.62 % (9975)Time elapsed: 0.004 s
% 2.87/1.62 % (9975)Instructions burned: 6 (million)
% 2.87/1.62 % (9975)------------------------------
% 2.87/1.62 % (9975)------------------------------
% 2.87/1.62 % (9972)Instruction limit reached!
% 2.87/1.62 % (9972)------------------------------
% 2.87/1.62 % (9972)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.62 % (9972)Termination reason: Unknown
% 2.87/1.62 % (9972)Termination phase: shuffling
% 2.87/1.62
% 2.87/1.62 % (9972)Memory used [KB]: 1407
% 2.87/1.62 % (9972)Time elapsed: 0.005 s
% 2.87/1.62 % (9972)Instructions burned: 5 (million)
% 2.87/1.62 % (9972)------------------------------
% 2.87/1.62 % (9972)------------------------------
% 2.87/1.62 % (9974)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/710Mi)
% 2.87/1.63 % (9977)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_9660 for (2999ds/21Mi)
% 2.87/1.63 % (9976)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_9660 for (2999ds/902Mi)
% 2.87/1.64 % (9978)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/5Mi)
% 2.87/1.64 % (9977)Instruction limit reached!
% 2.87/1.64 % (9977)------------------------------
% 2.87/1.64 % (9977)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.64 % (9977)Termination reason: Unknown
% 2.87/1.64 % (9977)Termination phase: shuffling
% 2.87/1.64
% 2.87/1.64 % (9977)Memory used [KB]: 1791
% 2.87/1.64 % (9977)Time elapsed: 0.010 s
% 2.87/1.64 % (9977)Instructions burned: 23 (million)
% 2.87/1.64 % (9977)------------------------------
% 2.87/1.64 % (9977)------------------------------
% 2.87/1.64 % (9978)Instruction limit reached!
% 2.87/1.64 % (9978)------------------------------
% 2.87/1.64 % (9978)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.64 % (9978)Termination reason: Unknown
% 2.87/1.64 % (9978)Termination phase: shuffling
% 2.87/1.64
% 2.87/1.64 % (9978)Memory used [KB]: 1407
% 2.87/1.64 % (9978)Time elapsed: 0.005 s
% 2.87/1.64 % (9978)Instructions burned: 5 (million)
% 2.87/1.64 % (9978)------------------------------
% 2.87/1.64 % (9978)------------------------------
% 2.87/1.65 % (9979)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_9660 for (2999ds/6Mi)
% 2.87/1.65 % (9979)Instruction limit reached!
% 2.87/1.65 % (9979)------------------------------
% 2.87/1.65 % (9979)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.65 % (9979)Termination reason: Unknown
% 2.87/1.65 % (9979)Termination phase: shuffling
% 2.87/1.65
% 2.87/1.65 % (9979)Memory used [KB]: 1407
% 2.87/1.65 % (9979)Time elapsed: 0.004 s
% 2.87/1.65 % (9979)Instructions burned: 7 (million)
% 2.87/1.65 % (9979)------------------------------
% 2.87/1.65 % (9979)------------------------------
% 2.87/1.65 % (9954)Instruction limit reached!
% 2.87/1.65 % (9954)------------------------------
% 2.87/1.65 % (9954)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.65 % (9954)Termination reason: Unknown
% 2.87/1.65 % (9954)Termination phase: Saturation
% 2.87/1.65
% 2.87/1.65 % (9954)Memory used [KB]: 7164
% 2.87/1.65 % (9954)Time elapsed: 0.091 s
% 2.87/1.65 % (9954)Instructions burned: 184 (million)
% 2.87/1.65 % (9954)------------------------------
% 2.87/1.65 % (9954)------------------------------
% 2.87/1.66 % (9980)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_9660 for (2999ds/377Mi)
% 2.87/1.66 % (9981)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_9660 for (2998ds/779Mi)
% 2.87/1.66 % (9982)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_9660 for (2998ds/19Mi)
% 2.87/1.67 % (9959)Instruction limit reached!
% 2.87/1.67 % (9959)------------------------------
% 2.87/1.67 % (9959)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.67 % (9959)Termination reason: Unknown
% 2.87/1.67 % (9959)Termination phase: Saturation
% 2.87/1.67
% 2.87/1.67 % (9959)Memory used [KB]: 7291
% 2.87/1.67 % (9959)Time elapsed: 0.106 s
% 2.87/1.67 % (9959)Instructions burned: 277 (million)
% 2.87/1.67 % (9959)------------------------------
% 2.87/1.67 % (9959)------------------------------
% 2.87/1.67 % (9982)Instruction limit reached!
% 2.87/1.67 % (9982)------------------------------
% 2.87/1.67 % (9982)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.67 % (9982)Termination reason: Unknown
% 2.87/1.67 % (9982)Termination phase: shuffling
% 2.87/1.67
% 2.87/1.67 % (9982)Memory used [KB]: 1791
% 2.87/1.67 % (9982)Time elapsed: 0.008 s
% 2.87/1.67 % (9982)Instructions burned: 20 (million)
% 2.87/1.67 % (9982)------------------------------
% 2.87/1.67 % (9982)------------------------------
% 2.87/1.67 % (9983)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_9660 for (2998ds/879Mi)
% 2.87/1.68 % (9984)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_9660 for (2998ds/17Mi)
% 2.87/1.68 % (9984)Instruction limit reached!
% 2.87/1.68 % (9984)------------------------------
% 2.87/1.68 % (9984)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.68 % (9984)Termination reason: Unknown
% 2.87/1.68 % (9984)Termination phase: shuffling
% 2.87/1.68
% 2.87/1.68 % (9984)Memory used [KB]: 1663
% 2.87/1.68 % (9984)Time elapsed: 0.008 s
% 2.87/1.68 % (9984)Instructions burned: 17 (million)
% 2.87/1.68 % (9984)------------------------------
% 2.87/1.68 % (9984)------------------------------
% 2.87/1.69 % (9985)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_9660 for (2998ds/3Mi)
% 2.87/1.69 % (9985)Instruction limit reached!
% 2.87/1.69 % (9985)------------------------------
% 2.87/1.69 % (9985)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.69 % (9985)Termination reason: Unknown
% 2.87/1.69 % (9985)Termination phase: shuffling
% 2.87/1.69
% 2.87/1.69 % (9985)Memory used [KB]: 1407
% 2.87/1.69 % (9985)Time elapsed: 0.003 s
% 2.87/1.69 % (9985)Instructions burned: 3 (million)
% 2.87/1.69 % (9985)------------------------------
% 2.87/1.69 % (9985)------------------------------
% 2.87/1.70 % (9986)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_9660 for (2998ds/30Mi)
% 2.87/1.72 % (9986)Instruction limit reached!
% 2.87/1.72 % (9986)------------------------------
% 2.87/1.72 % (9986)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.72 % (9986)Termination reason: Unknown
% 2.87/1.72 % (9986)Termination phase: Property scanning
% 2.87/1.72
% 2.87/1.72 % (9986)Memory used [KB]: 1791
% 2.87/1.72 % (9986)Time elapsed: 0.012 s
% 2.87/1.72 % (9986)Instructions burned: 32 (million)
% 2.87/1.72 % (9986)------------------------------
% 2.87/1.72 % (9986)------------------------------
% 2.87/1.73 % (9987)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_9660 for (2998ds/127Mi)
% 2.87/1.77 % (9987)Instruction limit reached!
% 2.87/1.77 % (9987)------------------------------
% 2.87/1.77 % (9987)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.77 % (9987)Termination reason: Unknown
% 2.87/1.77 % (9987)Termination phase: Saturation
% 2.87/1.77
% 2.87/1.77 % (9987)Memory used [KB]: 6908
% 2.87/1.77 % (9987)Time elapsed: 0.042 s
% 2.87/1.77 % (9987)Instructions burned: 129 (million)
% 2.87/1.77 % (9987)------------------------------
% 2.87/1.77 % (9987)------------------------------
% 2.87/1.78 % (9988)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_9660 for (2997ds/100Mi)
% 2.87/1.79 % (9980)Instruction limit reached!
% 2.87/1.79 % (9980)------------------------------
% 2.87/1.79 % (9980)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.79 % (9980)Termination reason: Unknown
% 2.87/1.79 % (9980)Termination phase: Saturation
% 2.87/1.79
% 2.87/1.79 % (9980)Memory used [KB]: 8059
% 2.87/1.79 % (9980)Time elapsed: 0.140 s
% 2.87/1.79 % (9980)Instructions burned: 379 (million)
% 2.87/1.79 % (9980)------------------------------
% 2.87/1.79 % (9980)------------------------------
% 2.87/1.80 % (9989)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_9660 for (2997ds/3Mi)
% 2.87/1.81 % (9989)Instruction limit reached!
% 2.87/1.81 % (9989)------------------------------
% 2.87/1.81 % (9989)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.81 % (9989)Termination reason: Unknown
% 2.87/1.81 % (9989)Termination phase: shuffling
% 2.87/1.81
% 2.87/1.81 % (9989)Memory used [KB]: 1407
% 2.87/1.81 % (9989)Time elapsed: 0.003 s
% 2.87/1.81 % (9989)Instructions burned: 4 (million)
% 2.87/1.81 % (9989)------------------------------
% 2.87/1.81 % (9989)------------------------------
% 2.87/1.81 % (9988)Instruction limit reached!
% 2.87/1.81 % (9988)------------------------------
% 2.87/1.81 % (9988)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.81 % (9988)Termination reason: Unknown
% 2.87/1.81 % (9988)Termination phase: Saturation
% 2.87/1.81
% 2.87/1.81 % (9988)Memory used [KB]: 6780
% 2.87/1.81 % (9988)Time elapsed: 0.032 s
% 2.87/1.81 % (9988)Instructions burned: 103 (million)
% 2.87/1.81 % (9988)------------------------------
% 2.87/1.81 % (9988)------------------------------
% 2.87/1.82 % (9990)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_9660 for (2997ds/20Mi)
% 2.87/1.82 % (9991)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_9660 for (2997ds/86Mi)
% 2.87/1.82 % (9990)Instruction limit reached!
% 2.87/1.82 % (9990)------------------------------
% 2.87/1.82 % (9990)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.82 % (9990)Termination reason: Unknown
% 2.87/1.82 % (9990)Termination phase: shuffling
% 2.87/1.82
% 2.87/1.82 % (9990)Memory used [KB]: 1791
% 2.87/1.82 % (9990)Time elapsed: 0.009 s
% 2.87/1.82 % (9990)Instructions burned: 20 (million)
% 2.87/1.82 % (9990)------------------------------
% 2.87/1.82 % (9990)------------------------------
% 2.87/1.83 % (9992)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_9660 for (2997ds/107Mi)
% 2.87/1.84 % (9991)Instruction limit reached!
% 2.87/1.84 % (9991)------------------------------
% 2.87/1.84 % (9991)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.84 % (9991)Termination reason: Unknown
% 2.87/1.84 % (9991)Termination phase: Property scanning
% 2.87/1.84
% 2.87/1.84 % (9991)Memory used [KB]: 2174
% 2.87/1.84 % (9991)Time elapsed: 0.029 s
% 2.87/1.84 % (9991)Instructions burned: 87 (million)
% 2.87/1.84 % (9991)------------------------------
% 2.87/1.84 % (9991)------------------------------
% 2.87/1.85 % (9993)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_9660 for (2996ds/39Mi)
% 2.87/1.87 % (9993)Instruction limit reached!
% 2.87/1.87 % (9993)------------------------------
% 2.87/1.87 % (9993)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.87 % (9993)Termination reason: Unknown
% 2.87/1.87 % (9993)Termination phase: Property scanning
% 2.87/1.87
% 2.87/1.87 % (9993)Memory used [KB]: 2174
% 2.87/1.87 % (9993)Time elapsed: 0.014 s
% 2.87/1.87 % (9993)Instructions burned: 41 (million)
% 2.87/1.87 % (9993)------------------------------
% 2.87/1.87 % (9993)------------------------------
% 2.87/1.87 % (9974)Instruction limit reached!
% 2.87/1.87 % (9974)------------------------------
% 2.87/1.87 % (9974)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.87 % (9974)Termination reason: Unknown
% 2.87/1.87 % (9974)Termination phase: Saturation
% 2.87/1.87
% 2.87/1.87 % (9992)Instruction limit reached!
% 2.87/1.87 % (9992)------------------------------
% 2.87/1.87 % (9992)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.87 % (9974)Memory used [KB]: 8443
% 2.87/1.87 % (9992)Termination reason: Unknown
% 2.87/1.87 % (9974)Time elapsed: 0.249 s
% 2.87/1.87 % (9992)Termination phase: Saturation
% 2.87/1.87 % (9974)Instructions burned: 712 (million)
% 2.87/1.87
% 2.87/1.87 % (9974)------------------------------
% 2.87/1.87 % (9974)------------------------------
% 2.87/1.87 % (9992)Memory used [KB]: 6652
% 2.87/1.87 % (9992)Time elapsed: 0.036 s
% 2.87/1.87 % (9992)Instructions burned: 109 (million)
% 2.87/1.87 % (9992)------------------------------
% 2.87/1.87 % (9992)------------------------------
% 2.87/1.88 % (9994)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_9660 for (2996ds/448Mi)
% 2.87/1.88 % (9995)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_9660 for (2996ds/46Mi)
% 2.87/1.88 % (9996)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_9660 for (2996ds/98Mi)
% 2.87/1.89 % (9976)Instruction limit reached!
% 2.87/1.89 % (9976)------------------------------
% 2.87/1.89 % (9976)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.89 % (9976)Termination reason: Unknown
% 2.87/1.89 % (9976)Termination phase: Saturation
% 2.87/1.89
% 2.87/1.89 % (9976)Memory used [KB]: 7547
% 2.87/1.89 % (9976)Time elapsed: 0.261 s
% 2.87/1.89 % (9976)Instructions burned: 904 (million)
% 2.87/1.89 % (9976)------------------------------
% 2.87/1.89 % (9976)------------------------------
% 2.87/1.89 % (9995)Instruction limit reached!
% 2.87/1.89 % (9995)------------------------------
% 2.87/1.89 % (9995)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.89 % (9995)Termination reason: Unknown
% 2.87/1.89 % (9995)Termination phase: Preprocessing 3
% 2.87/1.89
% 2.87/1.89 % (9995)Memory used [KB]: 1918
% 2.87/1.89 % (9995)Time elapsed: 0.018 s
% 2.87/1.89 % (9995)Instructions burned: 48 (million)
% 2.87/1.89 % (9995)------------------------------
% 2.87/1.89 % (9995)------------------------------
% 2.87/1.90 % (9965)Instruction limit reached!
% 2.87/1.90 % (9965)------------------------------
% 2.87/1.90 % (9965)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.90 % (9965)Termination reason: Unknown
% 2.87/1.90 % (9965)Termination phase: Saturation
% 2.87/1.90
% 2.87/1.90 % (9965)Memory used [KB]: 10234
% 2.87/1.90 % (9965)Time elapsed: 0.321 s
% 2.87/1.90 % (9965)Instructions burned: 1041 (million)
% 2.87/1.90 % (9965)------------------------------
% 2.87/1.90 % (9965)------------------------------
% 2.87/1.90 % (9997)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_9660 for (2996ds/507Mi)
% 2.87/1.90 % (9981)Instruction limit reached!
% 2.87/1.90 % (9981)------------------------------
% 2.87/1.90 % (9981)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.90 % (9981)Termination reason: Unknown
% 2.87/1.90 % (9981)Termination phase: Saturation
% 2.87/1.90
% 2.87/1.90 % (9981)Memory used [KB]: 8187
% 2.87/1.90 % (9981)Time elapsed: 0.242 s
% 2.87/1.90 % (9981)Instructions burned: 781 (million)
% 2.87/1.90 % (9981)------------------------------
% 2.87/1.90 % (9981)------------------------------
% 2.87/1.90 % (9998)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_9660 for (2996ds/149Mi)
% 2.87/1.91 % (9999)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_9660 for (2996ds/39Mi)
% 2.87/1.91 % (9996)Instruction limit reached!
% 2.87/1.91 % (9996)------------------------------
% 2.87/1.91 % (9996)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.91 % (9996)Termination reason: Unknown
% 2.87/1.91 % (9996)Termination phase: Saturation
% 2.87/1.91
% 2.87/1.91 % (9996)Memory used [KB]: 2302
% 2.87/1.91 % (9996)Time elapsed: 0.030 s
% 2.87/1.91 % (9996)Instructions burned: 99 (million)
% 2.87/1.91 % (9996)------------------------------
% 2.87/1.91 % (9996)------------------------------
% 2.87/1.91 % (10000)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_9660 for (2996ds/6Mi)
% 2.87/1.91 % (10000)Instruction limit reached!
% 2.87/1.91 % (10000)------------------------------
% 2.87/1.91 % (10000)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.91 % (10000)Termination reason: Unknown
% 2.87/1.91 % (10000)Termination phase: shuffling
% 2.87/1.91
% 2.87/1.91 % (10000)Memory used [KB]: 1407
% 2.87/1.91 % (10000)Time elapsed: 0.003 s
% 2.87/1.91 % (10000)Instructions burned: 7 (million)
% 2.87/1.91 % (10000)------------------------------
% 2.87/1.91 % (10000)------------------------------
% 2.87/1.92 % (10001)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_9660 for (2996ds/1087Mi)
% 2.87/1.92 % (9999)Instruction limit reached!
% 2.87/1.92 % (9999)------------------------------
% 2.87/1.92 % (9999)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.92 % (9999)Termination reason: Unknown
% 2.87/1.92 % (9999)Termination phase: Preprocessing 3
% 2.87/1.92
% 2.87/1.92 % (9999)Memory used [KB]: 1918
% 2.87/1.92 % (9999)Time elapsed: 0.014 s
% 2.87/1.92 % (9999)Instructions burned: 41 (million)
% 2.87/1.92 % (9999)------------------------------
% 2.87/1.92 % (9999)------------------------------
% 2.87/1.92 % (10002)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_9660 for (2996ds/47Mi)
% 2.87/1.93 % (10003)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_9660 for (2996ds/44Mi)
% 2.87/1.93 % (10002)Instruction limit reached!
% 2.87/1.93 % (10002)------------------------------
% 2.87/1.93 % (10002)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.93 % (10002)Termination reason: Unknown
% 2.87/1.93 % (10002)Termination phase: Property scanning
% 2.87/1.93
% 2.87/1.93 % (10002)Memory used [KB]: 2046
% 2.87/1.93 % (10002)Time elapsed: 0.016 s
% 2.87/1.93 % (10002)Instructions burned: 48 (million)
% 2.87/1.93 % (10002)------------------------------
% 2.87/1.93 % (10002)------------------------------
% 2.87/1.94 % (10003)Instruction limit reached!
% 2.87/1.94 % (10003)------------------------------
% 2.87/1.94 % (10003)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.94 % (10003)Termination reason: Unknown
% 2.87/1.94 % (10003)Termination phase: Property scanning
% 2.87/1.94
% 2.87/1.94 % (10003)Memory used [KB]: 2174
% 2.87/1.94 % (10003)Time elapsed: 0.013 s
% 2.87/1.94 % (10003)Instructions burned: 44 (million)
% 2.87/1.94 % (10003)------------------------------
% 2.87/1.94 % (10003)------------------------------
% 2.87/1.94 % (9997)First to succeed.
% 2.87/1.94 % (10004)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_9660 for (2995ds/432Mi)
% 2.87/1.95 % (9997)Refutation found. Thanks to Tanya!
% 2.87/1.95 % SZS status Theorem for DTF2THF_9660
% 2.87/1.95 % SZS output start Proof for DTF2THF_9660
% 2.87/1.95 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.87/1.95 thf(type_def_7, type, nat: $tType).
% 2.87/1.95 thf(type_def_8, type, coindu1593790203_llist: ($tType) > $tType).
% 2.87/1.95 thf(type_def_9, type, extended_enat: $tType).
% 2.87/1.95 thf(type_def_10, type, set: ($tType) > $tType).
% 2.87/1.95 thf(type_def_11, type, a: $tType).
% 2.87/1.95 thf(func_def_0, type, coindu1593790203_llist: ($tType) > $tType).
% 2.87/1.95 thf(func_def_1, type, extended_enat: $tType).
% 2.87/1.95 thf(func_def_2, type, set: ($tType) > $tType).
% 2.87/1.95 thf(func_def_3, type, nat: $tType).
% 2.87/1.95 thf(func_def_4, type, itself: ($tType) > $tType).
% 2.87/1.95 thf(func_def_5, type, a: $tType).
% 2.87/1.95 thf(func_def_6, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_7, type, one: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_8, type, neg_numeral: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_9, type, group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_10, type, lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_11, type, finite_finite: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_12, type, semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_13, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_14, type, ab_semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_15, type, semilattice_inf: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_16, type, linordered_semidom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_17, type, cancel_semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_18, type, ordere779506340up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_19, type, ordere236663937imp_le: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_20, type, ordere223160158up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_21, type, strict2144017051up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.95 thf(func_def_22, type, coindu493225792length: !>[X0: $tType]:(nat > coindu1593790203_llist @ X0 > extended_enat)).
% 2.87/1.95 thf(func_def_23, type, coindu191418589_ldrop: !>[X0: $tType]:(extended_enat > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_24, type, coindu438612276pWhile: !>[X0: $tType]:((X0 > $o) > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_25, type, coindu531130065ldropn: !>[X0: $tType]:(nat > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_26, type, coindu1889960678filter: !>[X0: $tType]:((X0 > $o) > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_27, type, coindu1213758845finite: !>[X0: $tType]:(coindu1593790203_llist @ X0 > $o)).
% 2.87/1.95 thf(func_def_28, type, coindu2000965700_llast: !>[X0: $tType]:(coindu1593790203_llist @ X0 > X0)).
% 2.87/1.95 thf(func_def_29, type, coindu1018505716length: !>[X0: $tType]:(coindu1593790203_llist @ X0 > extended_enat)).
% 2.87/1.95 thf(func_def_30, type, coindu1046438764le_lhd: !>[X0: $tType]:(coindu1593790203_llist @ X0 > X0)).
% 2.87/1.95 thf(func_def_31, type, coindu335574135_lnull: !>[X0: $tType]:(coindu1593790203_llist @ X0 > $o)).
% 2.87/1.95 thf(func_def_32, type, coindu1112613586e_lset: !>[X0: $tType]:(coindu1593790203_llist @ X0 > set @ X0)).
% 2.87/1.95 thf(func_def_33, type, coindu1047225960le_ltl: !>[X0: $tType]:(coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_34, type, coindu749330388e_lnth: !>[X0: $tType]:(coindu1593790203_llist @ X0 > nat > X0)).
% 2.87/1.95 thf(func_def_35, type, coindu721411036eWhile: !>[X0: $tType]:((X0 > $o) > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_36, type, coindu1491768222unfold: !>[X0: $tType]:((X0 > $o) > (X0 > X0) > X0 > extended_enat)).
% 2.87/1.95 thf(func_def_37, type, extended_enat2: nat > extended_enat).
% 2.87/1.95 thf(func_def_38, type, extended_rec_enat: !>[X0: $tType]:((nat > X0) > X0 > extended_enat > X0)).
% 2.87/1.95 thf(func_def_39, type, extended_the_enat: extended_enat > nat).
% 2.87/1.95 thf(func_def_40, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 2.87/1.95 thf(func_def_41, type, one_one: !>[X0: $tType]:(X0)).
% 2.87/1.95 thf(func_def_42, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.87/1.95 thf(func_def_43, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.87/1.95 thf(func_def_44, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.87/1.95 thf(func_def_45, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.87/1.95 thf(func_def_46, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.87/1.95 thf(func_def_47, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.87/1.95 thf(func_def_48, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 2.87/1.95 thf(func_def_49, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.87/1.95 thf(func_def_50, type, p: a > $o).
% 2.87/1.95 thf(func_def_51, type, n: nat).
% 2.87/1.95 thf(func_def_52, type, xs: coindu1593790203_llist @ a).
% 2.87/1.95 thf(func_def_53, type, xsa: coindu1593790203_llist @ a).
% 2.87/1.95 thf(func_def_68, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.87/1.95 thf(func_def_69, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.87/1.95 thf(func_def_70, type, sK2: !>[X0: $tType]:((X0 > coindu1593790203_llist @ X0 > $o) > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_71, type, sK3: !>[X0: $tType]:((X0 > coindu1593790203_llist @ X0 > $o) > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_72, type, sK4: !>[X0: $tType]:((X0 > coindu1593790203_llist @ X0 > $o) > X0)).
% 2.87/1.95 thf(func_def_73, type, sK5: !>[X0: $tType]:(nat > (X0 > extended_enat) > X0)).
% 2.87/1.95 thf(func_def_74, type, sK6: !>[X0: $tType]:((X0 > extended_enat) > X0)).
% 2.87/1.95 thf(func_def_75, type, sK7: (nat > $o) > nat).
% 2.87/1.95 thf(func_def_76, type, sK8: !>[X0: $tType]:((coindu1593790203_llist @ X0 > $o) > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_77, type, sK9: !>[X0: $tType]:((coindu1593790203_llist @ X0 > $o) > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_78, type, sK10: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 2.87/1.95 thf(func_def_79, type, sK11: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1)).
% 2.87/1.95 thf(func_def_80, type, sK12: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1 > X0)).
% 2.87/1.95 thf(func_def_81, type, sK13: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.87/1.95 thf(func_def_82, type, sK14: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > coindu1593790203_llist @ X0 > X0)).
% 2.87/1.95 thf(func_def_83, type, sK15: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X0)).
% 2.87/1.95 thf(func_def_84, type, sK16: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.87/1.95 thf(func_def_85, type, sK17: (nat > $o) > nat).
% 2.87/1.95 thf(func_def_86, type, sK18: (extended_enat > $o) > extended_enat).
% 2.87/1.95 thf(func_def_87, type, sK19: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.87/1.95 thf(func_def_88, type, sK20: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > coindu1593790203_llist @ X0 > X0)).
% 2.87/1.95 thf(func_def_89, type, sK21: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.87/1.95 thf(func_def_90, type, sK22: extended_enat > nat).
% 2.87/1.95 thf(func_def_91, type, sK23: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.87/1.95 thf(func_def_92, type, sK24: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.87/1.95 thf(func_def_93, type, sK25: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X1)).
% 2.87/1.95 thf(func_def_94, type, sK26: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X1)).
% 2.87/1.95 thf(func_def_95, type, sK27: nat > set @ nat > nat).
% 2.87/1.95 thf(func_def_96, type, sK28: set @ nat > nat > nat).
% 2.87/1.95 thf(func_def_97, type, sK29: !>[X0: $tType]:(((nat > X0 > X0) > nat > nat > X0 > $o) > nat > X0 > X0)).
% 2.87/1.95 thf(func_def_98, type, sK30: !>[X0: $tType]:(((nat > X0 > X0) > nat > nat > X0 > $o) > nat)).
% 2.87/1.95 thf(func_def_99, type, sK31: !>[X0: $tType]:(((nat > X0 > X0) > nat > nat > X0 > $o) > nat)).
% 2.87/1.95 thf(func_def_100, type, sK32: !>[X0: $tType]:(((nat > X0 > X0) > nat > nat > X0 > $o) > X0)).
% 2.87/1.95 thf(func_def_101, type, sK33: nat).
% 2.87/1.95 thf(func_def_102, type, sK34: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 2.87/1.95 thf(func_def_103, type, sK35: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X0 > set @ X1 > X1)).
% 2.87/1.95 thf(func_def_104, type, sK36: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X0 > set @ X1 > X0)).
% 2.87/1.95 thf(func_def_105, type, sK37: !>[X0: $tType]:((X0 > $o) > coindu1593790203_llist @ X0 > X0)).
% 2.87/1.95 thf(func_def_106, type, sK38: nat > extended_enat > nat).
% 2.87/1.95 thf(func_def_107, type, sK39: !>[X0: $tType]:(coindu1593790203_llist @ X0 > nat)).
% 2.87/1.95 thf(func_def_108, type, sK40: a).
% 2.87/1.95 thf(func_def_109, type, sK41: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.87/1.95 thf(func_def_110, type, sK42: !>[X0: $tType]:((coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0 > $o) > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_111, type, sK43: !>[X0: $tType]:((coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0 > $o) > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_112, type, sK44: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.87/1.95 thf(func_def_113, type, sK45: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.87/1.95 thf(func_def_114, type, sK46: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.87/1.95 thf(func_def_115, type, sK47: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.87/1.95 thf(func_def_116, type, sK48: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.87/1.95 thf(func_def_117, type, sK49: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.87/1.95 thf(func_def_118, type, sK50: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.87/1.95 thf(func_def_119, type, sK51: extended_enat > extended_enat > nat).
% 2.87/1.95 thf(func_def_120, type, sK52: !>[X0: $tType]:((coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0 > $o) > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_121, type, sK53: !>[X0: $tType]:((coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0 > $o) > coindu1593790203_llist @ X0)).
% 2.87/1.95 thf(func_def_122, type, sK54: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > coindu1593790203_llist @ X0 > X0)).
% 2.87/1.95 thf(func_def_124, type, ph56: !>[X0: $tType]:(X0)).
% 2.87/1.95 thf(f1768,plain,(
% 2.87/1.95 $false),
% 2.87/1.95 inference(subsumption_resolution,[status(thm)],[f1767,f1737])).
% 2.87/1.95 thf(f1737,plain,(
% 2.87/1.95 ($true = (?? @ (nat > nat) @ (^[Y0 : nat > nat]: (?? @ nat @ (^[Y1 : nat]: ((image @ nat @ nat @ Y0 @ (collect @ nat @ (^[Y2 : nat]: (ord_less @ nat @ Y2 @ Y1)))) = (inf_inf @ set @ nat @ (collect @ nat @ (^[Y2 : nat]: (ord_less @ extended_enat @ (extended_enat2 @ Y2) @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu1047225960le_ltl @ a @ (coindu191418589_ldrop @ a @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu721411036eWhile @ a @ (^[Y3 : a]: (~ (p @ Y3))) @ xsa)) @ xsa)))))) @ (collect @ nat @ (^[Y2 : nat]: (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu191418589_ldrop @ a @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu721411036eWhile @ a @ (^[Y3 : a]: (~ (p @ Y3))) @ xsa)) @ xsa)) @ Y2)))))))))))),
% 2.87/1.95 inference(beta_eta_normalization,[status(thm)],[f1598])).
% 2.87/1.95 thf(f1598,plain,(
% 2.87/1.95 ($true = ((^[Y0 : set @ nat]: (?? @ (nat > nat) @ (^[Y1 : nat > nat]: (?? @ nat @ (^[Y2 : nat]: ((image @ nat @ nat @ Y1 @ (collect @ nat @ (^[Y3 : nat]: (ord_less @ nat @ Y3 @ Y2)))) = Y0)))))) @ (inf_inf @ set @ nat @ (collect @ nat @ (^[Y0 : nat]: (ord_less @ extended_enat @ (extended_enat2 @ Y0) @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu1047225960le_ltl @ a @ ((^[Y1 : a > $o]: ((^[Y2 : coindu1593790203_llist @ a]: (coindu191418589_ldrop @ a @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu721411036eWhile @ a @ Y1 @ Y2)) @ Y2)))) @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)))))) @ (collect @ nat @ (^[Y0 : nat]: (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ ((^[Y1 : a > $o]: ((^[Y2 : coindu1593790203_llist @ a]: (coindu191418589_ldrop @ a @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu721411036eWhile @ a @ Y1 @ Y2)) @ Y2)))) @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)) @ Y0)))))))),
% 2.87/1.95 inference(definition_unfolding,[status(thm)],[f1360,f1245,f1337,f1548,f1548])).
% 2.87/1.95 thf(f1548,plain,(
% 2.87/1.95 ( ! [X0 : $tType] : ((coindu438612276pWhile @ X0 = (^[Y0 : X0 > $o]: ((^[Y1 : coindu1593790203_llist @ X0]: (coindu191418589_ldrop @ X0 @ (coindu1491768222unfold @ coindu1593790203_llist @ X0 @ coindu335574135_lnull @ X0 @ coindu1047225960le_ltl @ X0 @ (coindu721411036eWhile @ X0 @ Y0 @ Y1)) @ Y1)))))) )),
% 2.87/1.95 inference(definition_unfolding,[status(thm)],[f1520,f1337])).
% 2.87/1.95 thf(f1520,plain,(
% 2.87/1.95 ( ! [X0 : $tType] : ((coindu438612276pWhile @ X0 = (^[Y0 : X0 > $o]: ((^[Y1 : coindu1593790203_llist @ X0]: (coindu191418589_ldrop @ X0 @ (coindu1018505716length @ X0 @ (coindu721411036eWhile @ X0 @ Y0 @ Y1)) @ Y1)))))) )),
% 2.87/1.95 inference(cnf_transformation,[status(thm)],[f648])).
% 2.87/1.95 thf(f648,plain,(
% 2.87/1.95 ! [X0 : $tType] : (coindu438612276pWhile @ X0 = (^[Y0 : X0 > $o]: ((^[Y1 : coindu1593790203_llist @ X0]: (coindu191418589_ldrop @ X0 @ (coindu1018505716length @ X0 @ (coindu721411036eWhile @ X0 @ Y0 @ Y1)) @ Y1)))))),
% 2.87/1.95 inference(fool_elimination,[status(thm)],[f647])).
% 2.87/1.95 thf(f647,plain,(
% 2.87/1.95 ! [X0 : $tType] : ((^[X1 : X0 > $o, X2 : coindu1593790203_llist @ X0] : (coindu191418589_ldrop @ X0 @ (coindu1018505716length @ X0 @ (coindu721411036eWhile @ X0 @ X1 @ X2)) @ X2)) = coindu438612276pWhile @ X0)),
% 2.87/1.95 inference(rectify,[status(thm)],[f201])).
% 2.87/1.95 thf(f201,axiom,(
% 2.87/1.95 ! [X0 : $tType] : ((^[X79 : X0 > $o, X35 : coindu1593790203_llist @ X0] : (coindu191418589_ldrop @ X0 @ (coindu1018505716length @ X0 @ (coindu721411036eWhile @ X0 @ X79 @ X35)) @ X35)) = coindu438612276pWhile @ X0)),
% 2.87/1.95 file('/export/starexec/sandbox2/tmp/tmp.WzKKhsWUic/DTF2THF_9660.p',fact_200_ldropWhile__eq__ldrop)).
% 2.87/1.95 thf(f1337,plain,(
% 2.87/1.95 ( ! [X0 : $tType] : ((coindu1018505716length @ X0 = (coindu1491768222unfold @ coindu1593790203_llist @ X0 @ coindu335574135_lnull @ X0 @ coindu1047225960le_ltl @ X0))) )),
% 2.87/1.95 inference(cnf_transformation,[status(thm)],[f707])).
% 2.87/1.95 thf(f707,plain,(
% 2.87/1.95 ! [X0 : $tType] : (coindu1018505716length @ X0 = (coindu1491768222unfold @ coindu1593790203_llist @ X0 @ coindu335574135_lnull @ X0 @ coindu1047225960le_ltl @ X0))),
% 2.87/1.95 inference(fool_elimination,[status(thm)],[f230])).
% 2.87/1.95 thf(f230,axiom,(
% 2.87/1.95 ! [X0 : $tType] : (coindu1018505716length @ X0 = (coindu1491768222unfold @ coindu1593790203_llist @ X0 @ coindu335574135_lnull @ X0 @ coindu1047225960le_ltl @ X0))),
% 2.87/1.95 file('/export/starexec/sandbox2/tmp/tmp.WzKKhsWUic/DTF2THF_9660.p',fact_229_llength__def)).
% 2.87/1.95 thf(f1245,plain,(
% 2.87/1.95 ( ! [X0 : $tType] : ((finite_finite2 @ X0 = (^[Y0 : set @ X0]: (?? @ (nat > X0) @ (^[Y1 : nat > X0]: (?? @ nat @ (^[Y2 : nat]: ((image @ nat @ X0 @ Y1 @ (collect @ nat @ (^[Y3 : nat]: (ord_less @ nat @ Y3 @ Y2)))) = Y0)))))))) )),
% 2.87/1.95 inference(cnf_transformation,[status(thm)],[f610])).
% 2.87/1.95 thf(f610,plain,(
% 2.87/1.95 ! [X0 : $tType] : (finite_finite2 @ X0 = (^[Y0 : set @ X0]: (?? @ (nat > X0) @ (^[Y1 : nat > X0]: (?? @ nat @ (^[Y2 : nat]: ((image @ nat @ X0 @ Y1 @ (collect @ nat @ (^[Y3 : nat]: (ord_less @ nat @ Y3 @ Y2)))) = Y0)))))))),
% 2.87/1.95 inference(fool_elimination,[status(thm)],[f609])).
% 2.87/1.95 thf(f609,plain,(
% 2.87/1.95 ! [X0 : $tType] : ((^[X1 : set @ X0] : (? [X2 : nat,X3 : nat > X0] : ((image @ nat @ X0 @ X3 @ (collect @ nat @ (^[X4 : nat] : (ord_less @ nat @ X4 @ X2)))) = X1))) = finite_finite2 @ X0)),
% 2.87/1.95 inference(rectify,[status(thm)],[f57])).
% 2.87/1.95 thf(f57,axiom,(
% 2.87/1.95 ! [X0 : $tType] : ((^[X22 : set @ X0] : (? [X3 : nat,X33 : nat > X0] : ((image @ nat @ X0 @ X33 @ (collect @ nat @ (^[X34 : nat] : (ord_less @ nat @ X34 @ X3)))) = X22))) = finite_finite2 @ X0)),
% 2.87/1.95 file('/export/starexec/sandbox2/tmp/tmp.WzKKhsWUic/DTF2THF_9660.p',fact_56_finite__conv__nat__seg__image)).
% 2.87/1.95 thf(f1360,plain,(
% 2.87/1.95 ($true = (finite_finite2 @ nat @ (inf_inf @ set @ nat @ (collect @ nat @ (^[Y0 : nat]: (ord_less @ extended_enat @ (extended_enat2 @ Y0) @ (coindu1018505716length @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)))))) @ (collect @ nat @ (^[Y0 : nat]: (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)) @ Y0)))))))),
% 2.87/1.95 inference(cnf_transformation,[status(thm)],[f498])).
% 2.87/1.95 thf(f498,plain,(
% 2.87/1.95 ($true = (finite_finite2 @ nat @ (inf_inf @ set @ nat @ (collect @ nat @ (^[Y0 : nat]: (ord_less @ extended_enat @ (extended_enat2 @ Y0) @ (coindu1018505716length @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)))))) @ (collect @ nat @ (^[Y0 : nat]: (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)) @ Y0)))))))),
% 2.87/1.95 inference(fool_elimination,[status(thm)],[f497])).
% 2.87/1.95 thf(f497,plain,(
% 2.87/1.95 (finite_finite2 @ nat @ (inf_inf @ set @ nat @ (collect @ nat @ (^[X0 : nat] : (ord_less @ extended_enat @ (extended_enat2 @ X0) @ (coindu1018505716length @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[X1 : a] : (~(p @ X1))) @ xsa)))))) @ (collect @ nat @ (^[X2 : nat] : (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[X3 : a] : (~(p @ X3))) @ xsa)) @ X2))))))),
% 2.87/1.95 inference(rectify,[status(thm)],[f1])).
% 2.87/1.95 thf(f1,axiom,(
% 2.87/1.95 (finite_finite2 @ nat @ (inf_inf @ set @ nat @ (collect @ nat @ (^[X3 : nat] : (ord_less @ extended_enat @ (extended_enat2 @ X3) @ (coindu1018505716length @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[X4 : a] : (~(p @ X4))) @ xsa)))))) @ (collect @ nat @ (^[X3 : nat] : (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[X4 : a] : (~(p @ X4))) @ xsa)) @ X3))))))),
% 2.87/1.95 file('/export/starexec/sandbox2/tmp/tmp.WzKKhsWUic/DTF2THF_9660.p',fact_0__092_060open_062finite_A_I_123n_O_Aenat_An_A_060_Allength_A_Iltl_A_IldropWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_J_125_A_092_060inter_062_A_123n_O_AP_A_Ilnth_A_Iltl_A_IldropWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_J_An_J_125_J_092_060close_062)).
% 2.87/1.95 thf(f1767,plain,(
% 2.87/1.95 ($true != (?? @ (nat > nat) @ (^[Y0 : nat > nat]: (?? @ nat @ (^[Y1 : nat]: ((image @ nat @ nat @ Y0 @ (collect @ nat @ (^[Y2 : nat]: (ord_less @ nat @ Y2 @ Y1)))) = (inf_inf @ set @ nat @ (collect @ nat @ (^[Y2 : nat]: (ord_less @ extended_enat @ (extended_enat2 @ Y2) @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu1047225960le_ltl @ a @ (coindu191418589_ldrop @ a @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu721411036eWhile @ a @ (^[Y3 : a]: (~ (p @ Y3))) @ xsa)) @ xsa)))))) @ (collect @ nat @ (^[Y2 : nat]: (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu191418589_ldrop @ a @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu721411036eWhile @ a @ (^[Y3 : a]: (~ (p @ Y3))) @ xsa)) @ xsa)) @ Y2)))))))))))),
% 2.87/1.95 inference(trivial_inequality_removal,[status(thm)],[f1758])).
% 2.87/1.95 thf(f1758,plain,(
% 2.87/1.95 ($true != (?? @ (nat > nat) @ (^[Y0 : nat > nat]: (?? @ nat @ (^[Y1 : nat]: ((image @ nat @ nat @ Y0 @ (collect @ nat @ (^[Y2 : nat]: (ord_less @ nat @ Y2 @ Y1)))) = (inf_inf @ set @ nat @ (collect @ nat @ (^[Y2 : nat]: (ord_less @ extended_enat @ (extended_enat2 @ Y2) @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu1047225960le_ltl @ a @ (coindu191418589_ldrop @ a @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu721411036eWhile @ a @ (^[Y3 : a]: (~ (p @ Y3))) @ xsa)) @ xsa)))))) @ (collect @ nat @ (^[Y2 : nat]: (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu191418589_ldrop @ a @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu721411036eWhile @ a @ (^[Y3 : a]: (~ (p @ Y3))) @ xsa)) @ xsa)) @ Y2))))))))))) | ($true != $true)),
% 2.87/1.95 inference(superposition,[status(thm)],[f1752,f1710])).
% 2.87/1.95 thf(f1710,plain,(
% 2.87/1.95 ( ! [X1 : $tType,X0 : $tType,X2 : set @ X1,X3 : X1 > X0] : (($true = (?? @ (nat > X0) @ (^[Y0 : nat > X0]: (?? @ nat @ (^[Y1 : nat]: ((image @ nat @ X0 @ Y0 @ (collect @ nat @ (^[Y2 : nat]: (ord_less @ nat @ Y2 @ Y1)))) = (image @ X1 @ X0 @ X3 @ X2))))))) | ($true != (?? @ (nat > X1) @ (^[Y0 : nat > X1]: (?? @ nat @ (^[Y1 : nat]: ((image @ nat @ X1 @ Y0 @ (collect @ nat @ (^[Y2 : nat]: (ord_less @ nat @ Y2 @ Y1)))) = X2))))))) )),
% 2.87/1.95 inference(beta_eta_normalization,[status(thm)],[f1587])).
% 2.87/1.95 thf(f1587,plain,(
% 2.87/1.95 ( ! [X1 : $tType,X0 : $tType,X2 : set @ X1,X3 : X1 > X0] : ((((^[Y0 : set @ X1]: (?? @ (nat > X1) @ (^[Y1 : nat > X1]: (?? @ nat @ (^[Y2 : nat]: ((image @ nat @ X1 @ Y1 @ (collect @ nat @ (^[Y3 : nat]: (ord_less @ nat @ Y3 @ Y2)))) = Y0)))))) @ X2) != $true) | ($true = ((^[Y0 : set @ X0]: (?? @ (nat > X0) @ (^[Y1 : nat > X0]: (?? @ nat @ (^[Y2 : nat]: ((image @ nat @ X0 @ Y1 @ (collect @ nat @ (^[Y3 : nat]: (ord_less @ nat @ Y3 @ Y2)))) = Y0)))))) @ (image @ X1 @ X0 @ X3 @ X2)))) )),
% 2.87/1.95 inference(definition_unfolding,[status(thm)],[f1319,f1245,f1245])).
% 2.87/1.95 thf(f1319,plain,(
% 2.87/1.95 ( ! [X1 : $tType,X0 : $tType,X2 : set @ X1,X3 : X1 > X0] : (((finite_finite2 @ X0 @ (image @ X1 @ X0 @ X3 @ X2)) = $true) | ((finite_finite2 @ X1 @ X2) != $true)) )),
% 2.87/1.95 inference(cnf_transformation,[status(thm)],[f1014])).
% 2.87/1.95 thf(f1014,plain,(
% 2.87/1.95 ! [X0 : $tType,X1 : $tType,X2 : set @ X1,X3 : X1 > X0] : (((finite_finite2 @ X0 @ (image @ X1 @ X0 @ X3 @ X2)) = $true) | ((finite_finite2 @ X1 @ X2) != $true))),
% 2.87/1.95 inference(ennf_transformation,[status(thm)],[f289])).
% 2.87/1.95 thf(f289,plain,(
% 2.87/1.95 ! [X0 : $tType,X1 : $tType,X2 : set @ X1,X3 : X1 > X0] : (((finite_finite2 @ X1 @ X2) = $true) => ((finite_finite2 @ X0 @ (image @ X1 @ X0 @ X3 @ X2)) = $true))),
% 2.87/1.95 inference(fool_elimination,[status(thm)],[f288])).
% 2.87/1.95 thf(f288,plain,(
% 2.87/1.95 ! [X0 : $tType,X1 : $tType,X2 : set @ X1,X3 : X1 > X0] : ((finite_finite2 @ X1 @ X2) => (finite_finite2 @ X0 @ (image @ X1 @ X0 @ X3 @ X2)))),
% 2.87/1.95 inference(rectify,[status(thm)],[f5])).
% 2.87/1.95 thf(f5,axiom,(
% 2.87/1.95 ! [X2 : $tType,X0 : $tType,X5 : set @ X0,X7 : X0 > X2] : ((finite_finite2 @ X0 @ X5) => (finite_finite2 @ X2 @ (image @ X0 @ X2 @ X7 @ X5)))),
% 2.87/1.95 file('/export/starexec/sandbox2/tmp/tmp.WzKKhsWUic/DTF2THF_9660.p',fact_4_finite__imageI)).
% 2.87/1.95 thf(f1752,plain,(
% 2.87/1.95 ($true != (?? @ (nat > nat) @ (^[Y0 : nat > nat]: (?? @ nat @ (^[Y1 : nat]: ((image @ nat @ nat @ Y0 @ (collect @ nat @ (^[Y2 : nat]: (ord_less @ nat @ Y2 @ Y1)))) = (image @ nat @ nat @ (plus_plus @ nat @ (plus_plus @ nat @ n @ one_one @ nat)) @ (inf_inf @ set @ nat @ (collect @ nat @ (^[Y2 : nat]: (ord_less @ extended_enat @ (extended_enat2 @ Y2) @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu1047225960le_ltl @ a @ (coindu191418589_ldrop @ a @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu721411036eWhile @ a @ (^[Y3 : a]: (~ (p @ Y3))) @ xsa)) @ xsa)))))) @ (collect @ nat @ (^[Y2 : nat]: (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu191418589_ldrop @ a @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu721411036eWhile @ a @ (^[Y3 : a]: (~ (p @ Y3))) @ xsa)) @ xsa)) @ Y2))))))))))))),
% 2.87/1.95 inference(beta_eta_normalization,[status(thm)],[f1573])).
% 2.87/1.95 thf(f1573,plain,(
% 2.87/1.95 (((^[Y0 : set @ nat]: (?? @ (nat > nat) @ (^[Y1 : nat > nat]: (?? @ nat @ (^[Y2 : nat]: ((image @ nat @ nat @ Y1 @ (collect @ nat @ (^[Y3 : nat]: (ord_less @ nat @ Y3 @ Y2)))) = Y0)))))) @ (image @ nat @ nat @ (plus_plus @ nat @ (plus_plus @ nat @ n @ one_one @ nat)) @ (inf_inf @ set @ nat @ (collect @ nat @ (^[Y0 : nat]: (ord_less @ extended_enat @ (extended_enat2 @ Y0) @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu1047225960le_ltl @ a @ ((^[Y1 : a > $o]: ((^[Y2 : coindu1593790203_llist @ a]: (coindu191418589_ldrop @ a @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu721411036eWhile @ a @ Y1 @ Y2)) @ Y2)))) @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)))))) @ (collect @ nat @ (^[Y0 : nat]: (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ ((^[Y1 : a > $o]: ((^[Y2 : coindu1593790203_llist @ a]: (coindu191418589_ldrop @ a @ (coindu1491768222unfold @ coindu1593790203_llist @ a @ coindu335574135_lnull @ a @ coindu1047225960le_ltl @ a @ (coindu721411036eWhile @ a @ Y1 @ Y2)) @ Y2)))) @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)) @ Y0))))))) != $true)),
% 2.87/1.95 inference(definition_unfolding,[status(thm)],[f1249,f1245,f1337,f1548,f1548])).
% 2.87/1.95 thf(f1249,plain,(
% 2.87/1.95 ($true != (finite_finite2 @ nat @ (image @ nat @ nat @ (plus_plus @ nat @ (plus_plus @ nat @ n @ one_one @ nat)) @ (inf_inf @ set @ nat @ (collect @ nat @ (^[Y0 : nat]: (ord_less @ extended_enat @ (extended_enat2 @ Y0) @ (coindu1018505716length @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)))))) @ (collect @ nat @ (^[Y0 : nat]: (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)) @ Y0))))))))),
% 2.87/1.95 inference(cnf_transformation,[status(thm)],[f899])).
% 2.87/1.95 thf(f899,plain,(
% 2.87/1.95 ($true != (finite_finite2 @ nat @ (image @ nat @ nat @ (plus_plus @ nat @ (plus_plus @ nat @ n @ one_one @ nat)) @ (inf_inf @ set @ nat @ (collect @ nat @ (^[Y0 : nat]: (ord_less @ extended_enat @ (extended_enat2 @ Y0) @ (coindu1018505716length @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)))))) @ (collect @ nat @ (^[Y0 : nat]: (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)) @ Y0))))))))),
% 2.87/1.95 inference(flattening,[status(thm)],[f797])).
% 2.87/1.95 thf(f797,plain,(
% 2.87/1.95 ~($true = (finite_finite2 @ nat @ (image @ nat @ nat @ (plus_plus @ nat @ (plus_plus @ nat @ n @ one_one @ nat)) @ (inf_inf @ set @ nat @ (collect @ nat @ (^[Y0 : nat]: (ord_less @ extended_enat @ (extended_enat2 @ Y0) @ (coindu1018505716length @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)))))) @ (collect @ nat @ (^[Y0 : nat]: (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[Y1 : a]: (~ (p @ Y1))) @ xsa)) @ Y0))))))))),
% 2.87/1.95 inference(fool_elimination,[status(thm)],[f796])).
% 2.87/1.95 thf(f796,plain,(
% 2.87/1.95 ~(finite_finite2 @ nat @ (image @ nat @ nat @ (plus_plus @ nat @ (plus_plus @ nat @ n @ one_one @ nat)) @ (inf_inf @ set @ nat @ (collect @ nat @ (^[X0 : nat] : (ord_less @ extended_enat @ (extended_enat2 @ X0) @ (coindu1018505716length @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[X1 : a] : (~(p @ X1))) @ xsa)))))) @ (collect @ nat @ (^[X2 : nat] : (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[X3 : a] : (~(p @ X3))) @ xsa)) @ X2)))))))),
% 2.87/1.95 inference(rectify,[status(thm)],[f282])).
% 2.87/1.95 thf(f282,negated_conjecture,(
% 2.87/1.95 ~(finite_finite2 @ nat @ (image @ nat @ nat @ (plus_plus @ nat @ (plus_plus @ nat @ n @ one_one @ nat)) @ (inf_inf @ set @ nat @ (collect @ nat @ (^[X3 : nat] : (ord_less @ extended_enat @ (extended_enat2 @ X3) @ (coindu1018505716length @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[X4 : a] : (~(p @ X4))) @ xsa)))))) @ (collect @ nat @ (^[X3 : nat] : (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[X4 : a] : (~(p @ X4))) @ xsa)) @ X3)))))))),
% 2.87/1.95 inference(negated_conjecture,[status(cth)],[f281])).
% 2.87/1.95 thf(f281,conjecture,(
% 2.87/1.95 (finite_finite2 @ nat @ (image @ nat @ nat @ (plus_plus @ nat @ (plus_plus @ nat @ n @ one_one @ nat)) @ (inf_inf @ set @ nat @ (collect @ nat @ (^[X3 : nat] : (ord_less @ extended_enat @ (extended_enat2 @ X3) @ (coindu1018505716length @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[X4 : a] : (~(p @ X4))) @ xsa)))))) @ (collect @ nat @ (^[X3 : nat] : (p @ (coindu749330388e_lnth @ a @ (coindu1047225960le_ltl @ a @ (coindu438612276pWhile @ a @ (^[X4 : a] : (~(p @ X4))) @ xsa)) @ X3)))))))),
% 2.87/1.95 file('/export/starexec/sandbox2/tmp/tmp.WzKKhsWUic/DTF2THF_9660.p',conj_0)).
% 2.87/1.95 % SZS output end Proof for DTF2THF_9660
% 2.87/1.95 % (9997)------------------------------
% 2.87/1.95 % (9997)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.95 % (9997)Termination reason: Refutation
% 2.87/1.95
% 2.87/1.95 % (9997)Memory used [KB]: 7419
% 2.87/1.95 % (9997)Time elapsed: 0.047 s
% 2.87/1.95 % (9997)Instructions burned: 146 (million)
% 2.87/1.95 % (9997)------------------------------
% 2.87/1.95 % (9997)------------------------------
% 2.87/1.95 % (9953)Success in time 0.411 s
%------------------------------------------------------------------------------