%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT136^1 : 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 : n024.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 1.76s 1.18s
% Output : Refutation 1.76s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : DAT136^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.07 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.07/0.25 % Computer : n024.cluster.edu
% 0.07/0.25 % Model : x86_64 x86_64
% 0.07/0.25 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.25 % Memory : 8042.1875MB
% 0.07/0.25 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.25 % CPULimit : 300
% 0.07/0.25 % WCLimit : 300
% 0.07/0.25 % DateTime : Tue Feb 24 01:11:10 EST 2026
% 0.07/0.25 % CPUTime :
% 0.07/0.25 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.07/0.31 ---- Original DTF file ---
% 0.07/0.31 thf(spec,logic,$$dhol).
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 % File : DAT136^1 : TPTP v9.2.1. Released v7.0.0.
% 0.07/0.31 % Domain : Data Structures
% 0.07/0.31 % Problem : Coinductive list 4627
% 0.07/0.31 % Version : [Bla16] axioms : Especial.
% 0.07/0.31 % English :
% 0.07/0.31
% 0.07/0.31 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.07/0.31 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.07/0.31 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.07/0.31 % Source : [Bla16]
% 0.07/0.31 % Names : coinductive_list__4627.p [Bla16]
% 0.07/0.31
% 0.07/0.31 % Status : Theorem
% 0.07/0.31 % Rating : 0.33 v9.1.0, 0.00 v8.1.0, 0.25 v7.5.0, 0.00 v7.1.0
% 0.07/0.31 % Syntax : Number of formulae : 354 ( 148 unt; 62 typ; 0 def)
% 0.07/0.31 % Number of atoms : 662 ( 301 equ; 0 cnn)
% 0.07/0.31 % Maximal formula atoms : 12 ( 2 avg)
% 0.07/0.31 % Number of connectives : 2885 ( 84 ~; 10 |; 27 &;2512 @)
% 0.07/0.31 % ( 0 <=>; 252 =>; 0 <=; 0 <~>)
% 0.07/0.31 % Maximal formula depth : 21 ( 6 avg)
% 0.07/0.31 % Number of types : 4 ( 3 usr)
% 0.07/0.31 % Number of type conns : 107 ( 107 >; 0 *; 0 +; 0 <<)
% 0.07/0.31 % Number of symbols : 60 ( 59 usr; 10 con; 0-4 aty)
% 0.07/0.31 % Number of variables : 756 ( 26 ^; 657 !; 30 ?; 756 :)
% 0.07/0.31 % ( 43 !>; 0 ?*; 0 @-; 0 @+)
% 0.07/0.31 % SPC : TH1_THM_EQU_NAR
% 0.07/0.31
% 0.07/0.31 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.07/0.31 % 2016-07-13 15:08:08.226
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 %----Could-be-implicit typings (7)
% 0.07/0.31 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
% 0.07/0.31 coindu1593790203_llist: $tType > $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_t_Extended__Nat_Oenat,type,
% 0.07/0.31 extended_enat: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_t_List_Olist,type,
% 0.07/0.31 list: $tType > $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_t_Set_Oset,type,
% 0.07/0.31 set: $tType > $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_t_Nat_Onat,type,
% 0.07/0.31 nat: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_t_itself,type,
% 0.07/0.31 itself: $tType > $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_tf_a,type,
% 0.07/0.31 a: $tType ).
% 0.07/0.31
% 0.07/0.31 %----Explicit typings (55)
% 0.07/0.31 thf(sy_cl_HOL_Otype,type,
% 0.07/0.31 type:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ozero,type,
% 0.07/0.31 zero:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Orderings_Oord,type,
% 0.07/0.31 ord:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Orderings_Oorder,type,
% 0.07/0.31 order:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ogroup__add,type,
% 0.07/0.31 group_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Omonoid__add,type,
% 0.07/0.31 monoid_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Orderings_Olinorder,type,
% 0.07/0.31 linorder:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.07/0.31 semigroup_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.07/0.31 comm_semiring_1:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.07/0.31 comm_monoid_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.07/0.31 ab_semigroup_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.07/0.31 cancel_semigroup_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.07/0.31 cancel1352612707id_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.07/0.31 linord219039673up_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 0.07/0.31 ordere216010020id_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.07/0.31 ordere779506340up_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.07/0.31 canoni770627133id_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
% 0.07/0.31 strict797366125id_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.07/0.31 ordere236663937imp_le:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 0.07/0.31 ordere223160158up_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 0.07/0.31 strict2144017051up_add:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 0.07/0.31 ordere516151231imp_le:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
% 0.07/0.31 semiri456707255roduct:
% 0.07/0.31 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__llength,type,
% 0.07/0.31 coindu493225792length:
% 0.07/0.31 !>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olconcat,type,
% 0.07/0.31 coindu441856546concat:
% 0.07/0.31 !>[A: $tType] : ( ( coindu1593790203_llist @ ( coindu1593790203_llist @ A ) ) > ( coindu1593790203_llist @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldropn,type,
% 0.07/0.31 coindu531130065ldropn:
% 0.07/0.31 !>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite,type,
% 0.07/0.31 coindu1213758845finite:
% 0.07/0.31 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
% 0.07/0.31 coindu1018505716length:
% 0.07/0.31 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLCons,type,
% 0.07/0.31 coindu1121789889_LCons:
% 0.07/0.31 !>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil,type,
% 0.07/0.31 coindu1598213697e_LNil:
% 0.07/0.31 !>[A: $tType] : ( coindu1593790203_llist @ A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
% 0.07/0.31 coindu335574135_lnull:
% 0.07/0.31 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist__of,type,
% 0.07/0.31 coindu1079202755ist_of:
% 0.07/0.31 !>[A: $tType] : ( ( list @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olnth,type,
% 0.07/0.31 coindu749330388e_lnth:
% 0.07/0.31 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > nat > A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Extended__Nat_Oenat,type,
% 0.07/0.31 extended_enat2: nat > extended_enat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Extended__Nat_Oenat_Osize__enat,type,
% 0.07/0.31 extended_size_enat: extended_enat > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.07/0.31 plus_plus:
% 0.07/0.31 !>[A: $tType] : ( A > A > A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.07/0.31 zero_zero:
% 0.07/0.31 !>[A: $tType] : A ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osetsum,type,
% 0.07/0.31 groups15040474setsum:
% 0.07/0.31 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_OSuc,type,
% 0.07/0.31 suc: nat > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.07/0.31 size_size:
% 0.07/0.31 !>[A: $tType] : ( A > nat ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.07/0.31 ord_less:
% 0.07/0.31 !>[A: $tType] : ( A > A > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Pure_Otype,type,
% 0.07/0.31 type2:
% 0.07/0.31 !>[A: $tType] : ( itself @ A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_OCollect,type,
% 0.07/0.31 collect:
% 0.07/0.31 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
% 0.07/0.31 set_ord_lessThan:
% 0.07/0.31 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member,type,
% 0.07/0.31 member:
% 0.07/0.31 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_N____,type,
% 0.07/0.31 n: nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_m____,type,
% 0.07/0.31 m: nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_n,type,
% 0.07/0.31 n2: nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_n_H____,type,
% 0.07/0.31 n3: nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_na____,type,
% 0.07/0.31 na: nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_x____,type,
% 0.07/0.31 x: a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_xs____,type,
% 0.07/0.31 xs: coindu1593790203_llist @ a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_xss,type,
% 0.07/0.31 xss: coindu1593790203_llist @ ( coindu1593790203_llist @ a ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_xss_H____,type,
% 0.07/0.31 xss2: list @ ( coindu1593790203_llist @ a ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_xssa____,type,
% 0.07/0.31 xssa: coindu1593790203_llist @ ( coindu1593790203_llist @ a ) ).
% 0.07/0.31
% 0.07/0.31 %----Relevant facts (250)
% 0.07/0.31 thf(fact_0__092_060open_062enat_A_Ilength_Axss_H_J_A_060_Allength_Axss_092_060close_062,axiom,
% 0.07/0.31 ord_less @ extended_enat @ ( extended_enat2 @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ a ) @ xssa ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>enat (length xss') < llength xss\<close>
% 0.07/0.31 thf(fact_1_assms,axiom,
% 0.07/0.31 ord_less @ extended_enat @ ( extended_enat2 @ n2 ) @ ( coindu1018505716length @ a @ ( coindu441856546concat @ a @ xss ) ) ).
% 0.07/0.31
% 0.07/0.31 % assms
% 0.07/0.31 thf(fact_2__092_060open_062enat_A_ISuc_An_H_J_A_060_Allength_A_Ilnth_Axss_A_Ilength_Axss_H_J_J_092_060close_062,axiom,
% 0.07/0.31 ord_less @ extended_enat @ ( extended_enat2 @ ( suc @ n3 ) ) @ ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>enat (Suc n') < llength (lnth xss (length xss'))\<close>
% 0.07/0.31 thf(fact_3_Suc_Oprems,axiom,
% 0.07/0.31 ord_less @ extended_enat @ ( extended_enat2 @ ( suc @ na ) ) @ ( coindu1018505716length @ a @ ( coindu441856546concat @ a @ xssa ) ) ).
% 0.07/0.31
% 0.07/0.31 % Suc.prems
% 0.07/0.31 thf(fact_4__092_060open_062lnth_A_Ilconcat_Axss_J_A_ISuc_An_J_A_061_Alnth_A_Ilnth_Axss_A_Ilength_Axss_H_J_J_A_ISuc_An_H_J_092_060close_062,axiom,
% 0.07/0.31 ( ( coindu749330388e_lnth @ a @ ( coindu441856546concat @ a @ xssa ) @ ( suc @ na ) )
% 0.07/0.31 = ( coindu749330388e_lnth @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) @ ( suc @ n3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>lnth (lconcat xss) (Suc n) = lnth (lnth xss (length xss')) (Suc n')\<close>
% 0.07/0.31 thf(fact_5__092_060open_062enat_A_ISuc_An_J_A_061_A_I_092_060Sum_062i_060length_Axss_H_O_Allength_A_Ilnth_Axss_Ai_J_J_A_L_Aenat_A_ISuc_An_H_J_092_060close_062,axiom,
% 0.07/0.31 ( ( extended_enat2 @ ( suc @ na ) )
% 0.07/0.31 = ( plus_plus @ extended_enat
% 0.07/0.31 @ ( groups15040474setsum @ nat @ extended_enat
% 0.07/0.31 @ ^ [I: nat] : ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ I ) )
% 0.07/0.31 @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) )
% 0.07/0.31 @ ( extended_enat2 @ ( suc @ n3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>enat (Suc n) = (\<Sum>i<length xss'. llength (lnth xss i)) + enat (Suc n')\<close>
% 0.07/0.31 thf(fact_6_Suc_Ohyps,axiom,
% 0.07/0.31 ! [Xss: coindu1593790203_llist @ ( coindu1593790203_llist @ a )] :
% 0.07/0.31 ( ( ord_less @ extended_enat @ ( extended_enat2 @ na ) @ ( coindu1018505716length @ a @ ( coindu441856546concat @ a @ Xss ) ) )
% 0.07/0.31 => ? [M: nat,N: nat] :
% 0.07/0.31 ( ( ( coindu749330388e_lnth @ a @ ( coindu441856546concat @ a @ Xss ) @ na )
% 0.07/0.31 = ( coindu749330388e_lnth @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ Xss @ M ) @ N ) )
% 0.07/0.31 & ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ Xss @ M ) ) )
% 0.07/0.31 & ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ a ) @ Xss ) )
% 0.07/0.31 & ( ( extended_enat2 @ na )
% 0.07/0.31 = ( plus_plus @ extended_enat
% 0.07/0.31 @ ( groups15040474setsum @ nat @ extended_enat
% 0.07/0.31 @ ^ [I: nat] : ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ Xss @ I ) )
% 0.07/0.31 @ ( set_ord_lessThan @ nat @ M ) )
% 0.07/0.31 @ ( extended_enat2 @ N ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Suc.hyps
% 0.07/0.31 thf(fact_7_setsum__lessThan__Suc,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [F: nat > A,N2: nat] :
% 0.07/0.31 ( ( groups15040474setsum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
% 0.07/0.31 = ( plus_plus @ A @ ( groups15040474setsum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( F @ N2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % setsum_lessThan_Suc
% 0.07/0.31 thf(fact_8_enat__add__mono,axiom,
% 0.07/0.31 ! [X: nat,Y: extended_enat,Z: extended_enat] :
% 0.07/0.31 ( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
% 0.07/0.31 = ( ord_less @ extended_enat @ Y @ Z ) ) ).
% 0.07/0.31
% 0.07/0.31 % enat_add_mono
% 0.07/0.31 thf(fact_9_enat__add1__eq,axiom,
% 0.07/0.31 ! [X: nat,Y: extended_enat,Z: extended_enat] :
% 0.07/0.31 ( ( ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y )
% 0.07/0.31 = ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
% 0.07/0.31 = ( Y = Z ) ) ).
% 0.07/0.31
% 0.07/0.31 % enat_add1_eq
% 0.07/0.31 thf(fact_10_enat__add2__eq,axiom,
% 0.07/0.31 ! [Y: extended_enat,X: nat,Z: extended_enat] :
% 0.07/0.31 ( ( ( plus_plus @ extended_enat @ Y @ ( extended_enat2 @ X ) )
% 0.07/0.31 = ( plus_plus @ extended_enat @ Z @ ( extended_enat2 @ X ) ) )
% 0.07/0.31 = ( Y = Z ) ) ).
% 0.07/0.31
% 0.07/0.31 % enat_add2_eq
% 0.07/0.31 thf(fact_11_lessThan__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ord @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [I2: A,K: A] :
% 0.07/0.31 ( ( member @ A @ I2 @ ( set_ord_lessThan @ A @ K ) )
% 0.07/0.31 = ( ord_less @ A @ I2 @ K ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lessThan_iff
% 0.07/0.31 thf(fact_12_add__less__cancel__left,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [C: A,A2: A,B2: A] :
% 0.07/0.31 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.07/0.31 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_less_cancel_left
% 0.07/0.31 thf(fact_13_add__less__cancel__right,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,C: A,B2: A] :
% 0.07/0.31 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.07/0.31 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_less_cancel_right
% 0.07/0.31 thf(fact_14__092_060open_062_I_092_060Sum_062i_060length_Axss_H_O_Allength_A_Ilnth_Axss_Ai_J_J_A_061_A0_092_060close_062,axiom,
% 0.07/0.31 ( ( groups15040474setsum @ nat @ extended_enat
% 0.07/0.31 @ ^ [I: nat] : ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ I ) )
% 0.07/0.31 @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) )
% 0.07/0.31 = ( zero_zero @ extended_enat ) ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>(\<Sum>i<length xss'. llength (lnth xss i)) = 0\<close>
% 0.07/0.31 thf(fact_15_enat_Oinject,axiom,
% 0.07/0.31 ! [Nat: nat,Nat2: nat] :
% 0.07/0.31 ( ( ( extended_enat2 @ Nat )
% 0.07/0.31 = ( extended_enat2 @ Nat2 ) )
% 0.07/0.31 = ( Nat = Nat2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % enat.inject
% 0.07/0.31 thf(fact_16_lessThan__eq__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linorder @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [X: A,Y: A] :
% 0.07/0.31 ( ( ( set_ord_lessThan @ A @ X )
% 0.07/0.31 = ( set_ord_lessThan @ A @ Y ) )
% 0.07/0.31 = ( X = Y ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lessThan_eq_iff
% 0.07/0.31 thf(fact_17_nat_Oinject,axiom,
% 0.07/0.31 ! [X2: nat,Y2: nat] :
% 0.07/0.31 ( ( ( suc @ X2 )
% 0.07/0.31 = ( suc @ Y2 ) )
% 0.07/0.31 = ( X2 = Y2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % nat.inject
% 0.07/0.31 thf(fact_18_n,axiom,
% 0.07/0.31 ( na
% 0.07/0.31 = ( plus_plus @ nat @ n @ n3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % n
% 0.07/0.31 thf(fact_19_add__right__cancel,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [B2: A,A2: A,C: A] :
% 0.07/0.31 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.07/0.31 = ( plus_plus @ A @ C @ A2 ) )
% 0.07/0.31 = ( B2 = C ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_right_cancel
% 0.07/0.31 thf(fact_20_add__left__cancel,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A,C: A] :
% 0.07/0.31 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.07/0.31 = ( plus_plus @ A @ A2 @ C ) )
% 0.07/0.31 = ( B2 = C ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_left_cancel
% 0.07/0.31 thf(fact_21_add__Suc__right,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( plus_plus @ nat @ M2 @ ( suc @ N2 ) )
% 0.07/0.31 = ( suc @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_Suc_right
% 0.07/0.31 thf(fact_22_Suc__less__eq,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
% 0.07/0.31 = ( ord_less @ nat @ M2 @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % Suc_less_eq
% 0.07/0.31 thf(fact_23_Suc__mono,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ M2 @ N2 )
% 0.07/0.31 => ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Suc_mono
% 0.07/0.31 thf(fact_24_lessI,axiom,
% 0.07/0.31 ! [N2: nat] : ( ord_less @ nat @ N2 @ ( suc @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % lessI
% 0.07/0.31 thf(fact_25_old_Onat_Oinject,axiom,
% 0.07/0.31 ! [Nat: nat,Nat2: nat] :
% 0.07/0.31 ( ( ( suc @ Nat )
% 0.07/0.31 = ( suc @ Nat2 ) )
% 0.07/0.31 = ( Nat = Nat2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % old.nat.inject
% 0.07/0.31 thf(fact_26_not__gr__zero,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [N2: A] :
% 0.07/0.31 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
% 0.07/0.31 = ( N2
% 0.07/0.31 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % not_gr_zero
% 0.07/0.31 thf(fact_27_add__cancel__right__right,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A] :
% 0.07/0.31 ( ( A2
% 0.07/0.31 = ( plus_plus @ A @ A2 @ B2 ) )
% 0.07/0.31 = ( B2
% 0.07/0.31 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_cancel_right_right
% 0.07/0.31 thf(fact_28_add__cancel__right__left,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A] :
% 0.07/0.31 ( ( A2
% 0.07/0.31 = ( plus_plus @ A @ B2 @ A2 ) )
% 0.07/0.31 = ( B2
% 0.07/0.31 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_cancel_right_left
% 0.07/0.31 thf(fact_29_add__cancel__left__right,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A] :
% 0.07/0.31 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.07/0.31 = A2 )
% 0.07/0.31 = ( B2
% 0.07/0.31 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_cancel_left_right
% 0.07/0.31 thf(fact_30_add__cancel__left__left,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [B2: A,A2: A] :
% 0.07/0.31 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.07/0.31 = A2 )
% 0.07/0.31 = ( B2
% 0.07/0.31 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_cancel_left_left
% 0.07/0.31 thf(fact_31_double__zero__sym,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A] :
% 0.07/0.31 ( ( ( zero_zero @ A )
% 0.07/0.31 = ( plus_plus @ A @ A2 @ A2 ) )
% 0.07/0.31 = ( A2
% 0.07/0.31 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % double_zero_sym
% 0.07/0.31 thf(fact_32_double__zero,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A] :
% 0.07/0.31 ( ( ( plus_plus @ A @ A2 @ A2 )
% 0.07/0.31 = ( zero_zero @ A ) )
% 0.07/0.31 = ( A2
% 0.07/0.31 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % double_zero
% 0.07/0.31 thf(fact_33_add_Oright__neutral,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A] :
% 0.07/0.31 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.31 = A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % add.right_neutral
% 0.07/0.31 thf(fact_34_add_Oleft__neutral,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A] :
% 0.07/0.31 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.31 = A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % add.left_neutral
% 0.07/0.31 thf(fact_35_enat__ord__simps_I2_J,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
% 0.07/0.31 = ( ord_less @ nat @ M2 @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % enat_ord_simps(2)
% 0.07/0.31 thf(fact_36_i0__less,axiom,
% 0.07/0.31 ! [N2: extended_enat] :
% 0.07/0.31 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 )
% 0.07/0.31 = ( N2
% 0.07/0.31 != ( zero_zero @ extended_enat ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % i0_less
% 0.07/0.31 thf(fact_37_plus__enat__simps_I1_J,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( plus_plus @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
% 0.07/0.31 = ( extended_enat2 @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % plus_enat_simps(1)
% 0.07/0.31 thf(fact_38_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A] :
% 0.07/0.31 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.07/0.31 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % zero_less_double_add_iff_zero_less_single_add
% 0.07/0.31 thf(fact_39_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A] :
% 0.07/0.31 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.07/0.31 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % double_add_less_zero_iff_single_add_less_zero
% 0.07/0.31 thf(fact_40_less__add__same__cancel2,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A] :
% 0.07/0.31 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
% 0.07/0.31 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_add_same_cancel2
% 0.07/0.31 thf(fact_41_less__add__same__cancel1,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A] :
% 0.07/0.31 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.07/0.31 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_add_same_cancel1
% 0.07/0.31 thf(fact_42_add__less__same__cancel2,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A] :
% 0.07/0.31 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.07/0.31 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_less_same_cancel2
% 0.07/0.31 thf(fact_43_mem__Collect__eq,axiom,
% 0.07/0.31 ! [A: $tType,A2: A,P: A > $o] :
% 0.07/0.31 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.07/0.31 = ( P @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % mem_Collect_eq
% 0.07/0.31 thf(fact_44_Collect__mem__eq,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A] :
% 0.07/0.31 ( ( collect @ A
% 0.07/0.31 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.07/0.31 = A3 ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mem_eq
% 0.07/0.31 thf(fact_45_Collect__cong,axiom,
% 0.07/0.31 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.07/0.31 ( ! [X4: A] :
% 0.07/0.31 ( ( P @ X4 )
% 0.07/0.31 = ( Q @ X4 ) )
% 0.07/0.31 => ( ( collect @ A @ P )
% 0.07/0.31 = ( collect @ A @ Q ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_cong
% 0.07/0.31 thf(fact_46_ext,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.07/0.31 ( ! [X4: A] :
% 0.07/0.31 ( ( F @ X4 )
% 0.07/0.31 = ( G @ X4 ) )
% 0.07/0.31 => ( F = G ) ) ).
% 0.07/0.31
% 0.07/0.31 % ext
% 0.07/0.31 thf(fact_47_add__less__same__cancel1,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [B2: A,A2: A] :
% 0.07/0.31 ( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
% 0.07/0.31 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_less_same_cancel1
% 0.07/0.31 thf(fact_48_less__add__Suc1,axiom,
% 0.07/0.31 ! [I2: nat,M2: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ I2 @ M2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_add_Suc1
% 0.07/0.31 thf(fact_49_less__add__Suc2,axiom,
% 0.07/0.31 ! [I2: nat,M2: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ M2 @ I2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_add_Suc2
% 0.07/0.31 thf(fact_50_zero__reorient,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( zero @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [X: A] :
% 0.07/0.31 ( ( ( zero_zero @ A )
% 0.07/0.31 = X )
% 0.07/0.31 = ( X
% 0.07/0.31 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % zero_reorient
% 0.07/0.31 thf(fact_51_less__iff__Suc__add,axiom,
% 0.07/0.31 ( ( ord_less @ nat )
% 0.07/0.31 = ( ^ [M3: nat,N3: nat] :
% 0.07/0.31 ? [K2: nat] :
% 0.07/0.31 ( N3
% 0.07/0.31 = ( suc @ ( plus_plus @ nat @ M3 @ K2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_iff_Suc_add
% 0.07/0.31 thf(fact_52_less__imp__Suc__add,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ M2 @ N2 )
% 0.07/0.31 => ? [K3: nat] :
% 0.07/0.31 ( N2
% 0.07/0.31 = ( suc @ ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_imp_Suc_add
% 0.07/0.31 thf(fact_53_add__Suc,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( plus_plus @ nat @ ( suc @ M2 ) @ N2 )
% 0.07/0.31 = ( suc @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_Suc
% 0.07/0.31 thf(fact_54_add__Suc__shift,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( plus_plus @ nat @ ( suc @ M2 ) @ N2 )
% 0.07/0.31 = ( plus_plus @ nat @ M2 @ ( suc @ N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_Suc_shift
% 0.07/0.31 thf(fact_55_zero__less__iff__neq__zero,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [N2: A] :
% 0.07/0.31 ( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
% 0.07/0.31 = ( N2
% 0.07/0.31 != ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % zero_less_iff_neq_zero
% 0.07/0.31 thf(fact_56_gr__implies__not__zero,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [M2: A,N2: A] :
% 0.07/0.31 ( ( ord_less @ A @ M2 @ N2 )
% 0.07/0.31 => ( N2
% 0.07/0.31 != ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % gr_implies_not_zero
% 0.07/0.31 thf(fact_57_not__less__zero,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [N2: A] :
% 0.07/0.31 ~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % not_less_zero
% 0.07/0.31 thf(fact_58_gr__zeroI,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [N2: A] :
% 0.07/0.31 ( ( N2
% 0.07/0.31 != ( zero_zero @ A ) )
% 0.07/0.31 => ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % gr_zeroI
% 0.07/0.31 thf(fact_59_not__less__less__Suc__eq,axiom,
% 0.07/0.31 ! [N2: nat,M2: nat] :
% 0.07/0.31 ( ~ ( ord_less @ nat @ N2 @ M2 )
% 0.07/0.31 => ( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
% 0.07/0.31 = ( N2 = M2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % not_less_less_Suc_eq
% 0.07/0.31 thf(fact_60_strict__inc__induct,axiom,
% 0.07/0.31 ! [I2: nat,J: nat,P: nat > $o] :
% 0.07/0.31 ( ( ord_less @ nat @ I2 @ J )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( J
% 0.07/0.31 = ( suc @ I3 ) )
% 0.07/0.31 => ( P @ I3 ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ I3 @ J )
% 0.07/0.31 => ( ( P @ ( suc @ I3 ) )
% 0.07/0.31 => ( P @ I3 ) ) )
% 0.07/0.31 => ( P @ I2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % strict_inc_induct
% 0.07/0.31 thf(fact_61_less__Suc__induct,axiom,
% 0.07/0.31 ! [I2: nat,J: nat,P: nat > nat > $o] :
% 0.07/0.31 ( ( ord_less @ nat @ I2 @ J )
% 0.07/0.31 => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 0.07/0.31 => ( ! [I3: nat,J2: nat,K3: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ I3 @ J2 )
% 0.07/0.31 => ( ( ord_less @ nat @ J2 @ K3 )
% 0.07/0.31 => ( ( P @ I3 @ J2 )
% 0.07/0.31 => ( ( P @ J2 @ K3 )
% 0.07/0.31 => ( P @ I3 @ K3 ) ) ) ) )
% 0.07/0.31 => ( P @ I2 @ J ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_Suc_induct
% 0.07/0.31 thf(fact_62_less__trans__Suc,axiom,
% 0.07/0.31 ! [I2: nat,J: nat,K: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ I2 @ J )
% 0.07/0.31 => ( ( ord_less @ nat @ J @ K )
% 0.07/0.31 => ( ord_less @ nat @ ( suc @ I2 ) @ K ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_trans_Suc
% 0.07/0.31 thf(fact_63_Suc__less__SucD,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
% 0.07/0.31 => ( ord_less @ nat @ M2 @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % Suc_less_SucD
% 0.07/0.31 thf(fact_64_less__antisym,axiom,
% 0.07/0.31 ! [N2: nat,M2: nat] :
% 0.07/0.31 ( ~ ( ord_less @ nat @ N2 @ M2 )
% 0.07/0.31 => ( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
% 0.07/0.31 => ( M2 = N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_antisym
% 0.07/0.31 thf(fact_65_Suc__less__eq2,axiom,
% 0.07/0.31 ! [N2: nat,M2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ ( suc @ N2 ) @ M2 )
% 0.07/0.31 = ( ? [M4: nat] :
% 0.07/0.31 ( ( M2
% 0.07/0.31 = ( suc @ M4 ) )
% 0.07/0.31 & ( ord_less @ nat @ N2 @ M4 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Suc_less_eq2
% 0.07/0.31 thf(fact_66_not__less__eq,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ~ ( ord_less @ nat @ M2 @ N2 ) )
% 0.07/0.31 = ( ord_less @ nat @ N2 @ ( suc @ M2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % not_less_eq
% 0.07/0.31 thf(fact_67_less__Suc__eq,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
% 0.07/0.31 = ( ( ord_less @ nat @ M2 @ N2 )
% 0.07/0.31 | ( M2 = N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_Suc_eq
% 0.07/0.31 thf(fact_68_less__SucI,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ M2 @ N2 )
% 0.07/0.31 => ( ord_less @ nat @ M2 @ ( suc @ N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_SucI
% 0.07/0.31 thf(fact_69_less__SucE,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
% 0.07/0.31 => ( ~ ( ord_less @ nat @ M2 @ N2 )
% 0.07/0.31 => ( M2 = N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_SucE
% 0.07/0.31 thf(fact_70_Suc__lessI,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ M2 @ N2 )
% 0.07/0.31 => ( ( ( suc @ M2 )
% 0.07/0.31 != N2 )
% 0.07/0.31 => ( ord_less @ nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Suc_lessI
% 0.07/0.31 thf(fact_71_Suc__lessE,axiom,
% 0.07/0.31 ! [I2: nat,K: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ ( suc @ I2 ) @ K )
% 0.07/0.31 => ~ ! [J2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ I2 @ J2 )
% 0.07/0.31 => ( K
% 0.07/0.31 != ( suc @ J2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Suc_lessE
% 0.07/0.31 thf(fact_72_Suc__lessD,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ ( suc @ M2 ) @ N2 )
% 0.07/0.31 => ( ord_less @ nat @ M2 @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % Suc_lessD
% 0.07/0.31 thf(fact_73_lessE,axiom,
% 0.07/0.31 ! [I2: nat,K: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ I2 @ K )
% 0.07/0.31 => ( ( K
% 0.07/0.31 != ( suc @ I2 ) )
% 0.07/0.31 => ~ ! [J2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ I2 @ J2 )
% 0.07/0.31 => ( K
% 0.07/0.31 != ( suc @ J2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lessE
% 0.07/0.31 thf(fact_74_add__eq__0__iff__both__eq__0,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [X: A,Y: A] :
% 0.07/0.31 ( ( ( plus_plus @ A @ X @ Y )
% 0.07/0.31 = ( zero_zero @ A ) )
% 0.07/0.31 = ( ( X
% 0.07/0.31 = ( zero_zero @ A ) )
% 0.07/0.31 & ( Y
% 0.07/0.31 = ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_eq_0_iff_both_eq_0
% 0.07/0.31 thf(fact_75_add_Ogroup__left__neutral,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( group_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A] :
% 0.07/0.31 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.31 = A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % add.group_left_neutral
% 0.07/0.31 thf(fact_76_add_Ocomm__neutral,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A] :
% 0.07/0.31 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.31 = A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % add.comm_neutral
% 0.07/0.31 thf(fact_77_comm__monoid__add__class_Oadd__0,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A] :
% 0.07/0.31 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.31 = A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % comm_monoid_add_class.add_0
% 0.07/0.31 thf(fact_78_not__iless0,axiom,
% 0.07/0.31 ! [N2: extended_enat] :
% 0.07/0.31 ~ ( ord_less @ extended_enat @ N2 @ ( zero_zero @ extended_enat ) ) ).
% 0.07/0.31
% 0.07/0.31 % not_iless0
% 0.07/0.31 thf(fact_79_plus__enat__eq__0__conv,axiom,
% 0.07/0.31 ! [M2: extended_enat,N2: extended_enat] :
% 0.07/0.31 ( ( ( plus_plus @ extended_enat @ M2 @ N2 )
% 0.07/0.31 = ( zero_zero @ extended_enat ) )
% 0.07/0.31 = ( ( M2
% 0.07/0.31 = ( zero_zero @ extended_enat ) )
% 0.07/0.31 & ( N2
% 0.07/0.31 = ( zero_zero @ extended_enat ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % plus_enat_eq_0_conv
% 0.07/0.31 thf(fact_80_pos__add__strict,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( strict797366125id_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A,C: A] :
% 0.07/0.31 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.31 => ( ( ord_less @ A @ B2 @ C )
% 0.07/0.31 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % pos_add_strict
% 0.07/0.31 thf(fact_81_add__pos__pos,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A] :
% 0.07/0.31 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.31 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.07/0.31 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_pos_pos
% 0.07/0.31 thf(fact_82_add__neg__neg,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A] :
% 0.07/0.31 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.31 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.07/0.31 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_neg_neg
% 0.07/0.31 thf(fact_83_less__enatE,axiom,
% 0.07/0.31 ! [N2: extended_enat,M2: nat] :
% 0.07/0.31 ( ( ord_less @ extended_enat @ N2 @ ( extended_enat2 @ M2 ) )
% 0.07/0.31 => ~ ! [K3: nat] :
% 0.07/0.31 ( ( N2
% 0.07/0.31 = ( extended_enat2 @ K3 ) )
% 0.07/0.31 => ~ ( ord_less @ nat @ K3 @ M2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_enatE
% 0.07/0.31 thf(fact_84_enat__less__enat__plusI2,axiom,
% 0.07/0.31 ! [Y: nat,Z: extended_enat,X: nat] :
% 0.07/0.31 ( ( ord_less @ extended_enat @ ( extended_enat2 @ Y ) @ Z )
% 0.07/0.31 => ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ X @ Y ) ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % enat_less_enat_plusI2
% 0.07/0.31 thf(fact_85_enat__less__enat__plusI,axiom,
% 0.07/0.31 ! [X: nat,Y: nat,Z: extended_enat] :
% 0.07/0.31 ( ( ord_less @ nat @ X @ Y )
% 0.07/0.31 => ( ord_less @ extended_enat @ ( extended_enat2 @ X ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ Y ) @ Z ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % enat_less_enat_plusI
% 0.07/0.31 thf(fact_86_add__right__imp__eq,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [B2: A,A2: A,C: A] :
% 0.07/0.31 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.07/0.31 = ( plus_plus @ A @ C @ A2 ) )
% 0.07/0.31 => ( B2 = C ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_right_imp_eq
% 0.07/0.31 thf(fact_87_add__left__imp__eq,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A,C: A] :
% 0.07/0.31 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.07/0.31 = ( plus_plus @ A @ A2 @ C ) )
% 0.07/0.31 => ( B2 = C ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_left_imp_eq
% 0.07/0.31 thf(fact_88_add_Oleft__commute,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [B2: A,A2: A,C: A] :
% 0.07/0.31 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
% 0.07/0.31 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add.left_commute
% 0.07/0.31 thf(fact_89_add_Ocommute,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ( ( plus_plus @ A )
% 0.07/0.31 = ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add.commute
% 0.07/0.31 thf(fact_90_add_Oright__cancel,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( group_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [B2: A,A2: A,C: A] :
% 0.07/0.31 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.07/0.31 = ( plus_plus @ A @ C @ A2 ) )
% 0.07/0.31 = ( B2 = C ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add.right_cancel
% 0.07/0.31 thf(fact_91_add_Oleft__cancel,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( group_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A,C: A] :
% 0.07/0.31 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.07/0.31 = ( plus_plus @ A @ A2 @ C ) )
% 0.07/0.31 = ( B2 = C ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add.left_cancel
% 0.07/0.31 thf(fact_92_add_Oassoc,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semigroup_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A,C: A] :
% 0.07/0.31 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.07/0.31 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add.assoc
% 0.07/0.31 thf(fact_93_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [I2: A,J: A,K: A,L: A] :
% 0.07/0.31 ( ( ( I2 = J )
% 0.07/0.31 & ( K = L ) )
% 0.07/0.31 => ( ( plus_plus @ A @ I2 @ K )
% 0.07/0.31 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_mono_thms_linordered_semiring(4)
% 0.07/0.31 thf(fact_94_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A,C: A] :
% 0.07/0.31 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.07/0.31 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ab_semigroup_add_class.add_ac(1)
% 0.07/0.31 thf(fact_95_n__not__Suc__n,axiom,
% 0.07/0.31 ! [N2: nat] :
% 0.07/0.31 ( N2
% 0.07/0.31 != ( suc @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % n_not_Suc_n
% 0.07/0.31 thf(fact_96_Suc__inject,axiom,
% 0.07/0.31 ! [X: nat,Y: nat] :
% 0.07/0.31 ( ( ( suc @ X )
% 0.07/0.31 = ( suc @ Y ) )
% 0.07/0.31 => ( X = Y ) ) ).
% 0.07/0.31
% 0.07/0.31 % Suc_inject
% 0.07/0.31 thf(fact_97_enat__less__induct,axiom,
% 0.07/0.31 ! [P: extended_enat > $o,N2: extended_enat] :
% 0.07/0.31 ( ! [N4: extended_enat] :
% 0.07/0.31 ( ! [M5: extended_enat] :
% 0.07/0.31 ( ( ord_less @ extended_enat @ M5 @ N4 )
% 0.07/0.31 => ( P @ M5 ) )
% 0.07/0.31 => ( P @ N4 ) )
% 0.07/0.31 => ( P @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % enat_less_induct
% 0.07/0.31 thf(fact_98_add__less__imp__less__right,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,C: A,B2: A] :
% 0.07/0.31 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.07/0.31 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_less_imp_less_right
% 0.07/0.31 thf(fact_99_add__less__imp__less__left,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [C: A,A2: A,B2: A] :
% 0.07/0.31 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.07/0.31 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_less_imp_less_left
% 0.07/0.31 thf(fact_100_add__strict__right__mono,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A,C: A] :
% 0.07/0.31 ( ( ord_less @ A @ A2 @ B2 )
% 0.07/0.31 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_strict_right_mono
% 0.07/0.31 thf(fact_101_add__strict__left__mono,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A,C: A] :
% 0.07/0.31 ( ( ord_less @ A @ A2 @ B2 )
% 0.07/0.31 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_strict_left_mono
% 0.07/0.31 thf(fact_102_add__strict__mono,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( strict2144017051up_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A2: A,B2: A,C: A,D: A] :
% 0.07/0.31 ( ( ord_less @ A @ A2 @ B2 )
% 0.07/0.31 => ( ( ord_less @ A @ C @ D )
% 0.07/0.31 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_strict_mono
% 0.07/0.31 thf(fact_103_add__mono__thms__linordered__field_I1_J,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [I2: A,J: A,K: A,L: A] :
% 0.07/0.31 ( ( ( ord_less @ A @ I2 @ J )
% 0.07/0.31 & ( K = L ) )
% 0.07/0.31 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_mono_thms_linordered_field(1)
% 0.07/0.31 thf(fact_104_add__mono__thms__linordered__field_I2_J,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [I2: A,J: A,K: A,L: A] :
% 0.07/0.31 ( ( ( I2 = J )
% 0.07/0.31 & ( ord_less @ A @ K @ L ) )
% 0.07/0.31 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_mono_thms_linordered_field(2)
% 0.07/0.31 thf(fact_105_add__mono__thms__linordered__field_I5_J,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [I2: A,J: A,K: A,L: A] :
% 0.07/0.31 ( ( ( ord_less @ A @ I2 @ J )
% 0.07/0.31 & ( ord_less @ A @ K @ L ) )
% 0.07/0.31 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_mono_thms_linordered_field(5)
% 0.07/0.31 thf(fact_106_lift__Suc__mono__less__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( order @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [F: nat > A,N2: nat,M2: nat] :
% 0.07/0.31 ( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.07/0.31 => ( ( ord_less @ A @ ( F @ N2 ) @ ( F @ M2 ) )
% 0.07/0.31 = ( ord_less @ nat @ N2 @ M2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lift_Suc_mono_less_iff
% 0.07/0.31 thf(fact_107_lift__Suc__mono__less,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( order @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [F: nat > A,N2: nat,N5: nat] :
% 0.07/0.31 ( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.07/0.31 => ( ( ord_less @ nat @ N2 @ N5 )
% 0.07/0.31 => ( ord_less @ A @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lift_Suc_mono_less
% 0.07/0.31 thf(fact_108_lessThan__strict__subset__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linorder @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [M2: A,N2: A] :
% 0.07/0.31 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M2 ) @ ( set_ord_lessThan @ A @ N2 ) )
% 0.07/0.31 = ( ord_less @ A @ M2 @ N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lessThan_strict_subset_iff
% 0.07/0.31 thf(fact_109_enat__iless,axiom,
% 0.07/0.31 ! [N2: extended_enat,M2: nat] :
% 0.07/0.31 ( ( ord_less @ extended_enat @ N2 @ ( extended_enat2 @ M2 ) )
% 0.07/0.31 => ? [K3: nat] :
% 0.07/0.31 ( N2
% 0.07/0.31 = ( extended_enat2 @ K3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % enat_iless
% 0.07/0.31 thf(fact_110_chain__incr,axiom,
% 0.07/0.31 ! [A: $tType,Y3: A > extended_enat,K: nat] :
% 0.07/0.31 ( ! [I3: A] :
% 0.07/0.31 ? [J3: A] : ( ord_less @ extended_enat @ ( Y3 @ I3 ) @ ( Y3 @ J3 ) )
% 0.07/0.31 => ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K ) @ ( Y3 @ J2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % chain_incr
% 0.07/0.31 thf(fact_111_lessThan__def,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ord @ A @ ( type2 @ A ) )
% 0.07/0.31 => ( ( set_ord_lessThan @ A )
% 0.07/0.31 = ( ^ [U: A] :
% 0.07/0.31 ( collect @ A
% 0.07/0.31 @ ^ [X3: A] : ( ord_less @ A @ X3 @ U ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lessThan_def
% 0.07/0.31 thf(fact_112_setsum_Oneutral__const,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType] :
% 0.07/0.31 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A3: set @ B] :
% 0.07/0.31 ( ( groups15040474setsum @ B @ A
% 0.07/0.31 @ ^ [Uu: B] : ( zero_zero @ A )
% 0.07/0.31 @ A3 )
% 0.07/0.31 = ( zero_zero @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % setsum.neutral_const
% 0.07/0.31 thf(fact_113_lnth__prefix,axiom,
% 0.07/0.31 ! [I4: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) )
% 0.07/0.31 => ( ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ I4 )
% 0.07/0.31 = ( coindu1598213697e_LNil @ a ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lnth_prefix
% 0.07/0.31 thf(fact_114_llength__lconcat__lfinite__conv__sum,axiom,
% 0.07/0.31 ! [A: $tType,Xss: coindu1593790203_llist @ ( coindu1593790203_llist @ A )] :
% 0.07/0.31 ( ( coindu1213758845finite @ ( coindu1593790203_llist @ A ) @ Xss )
% 0.07/0.31 => ( ( coindu1018505716length @ A @ ( coindu441856546concat @ A @ Xss ) )
% 0.07/0.31 = ( groups15040474setsum @ nat @ extended_enat
% 0.07/0.31 @ ^ [I: nat] : ( coindu1018505716length @ A @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ A ) @ Xss @ I ) )
% 0.07/0.31 @ ( collect @ nat
% 0.07/0.31 @ ^ [I: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ I ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ A ) @ Xss ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % llength_lconcat_lfinite_conv_sum
% 0.07/0.31 thf(fact_115_setsum_Odistrib,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType] :
% 0.07/0.31 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [G: B > A,H: B > A,A3: set @ B] :
% 0.07/0.31 ( ( groups15040474setsum @ B @ A
% 0.07/0.31 @ ^ [X3: B] : ( plus_plus @ A @ ( G @ X3 ) @ ( H @ X3 ) )
% 0.07/0.31 @ A3 )
% 0.07/0.31 = ( plus_plus @ A @ ( groups15040474setsum @ B @ A @ G @ A3 ) @ ( groups15040474setsum @ B @ A @ H @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % setsum.distrib
% 0.07/0.31 thf(fact_116__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Axs_O_Alconcat_Axss_A_061_ALCons_Ax_Axs_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.07/0.31 ~ ! [X4: a,Xs: coindu1593790203_llist @ a] :
% 0.07/0.31 ( ( coindu441856546concat @ a @ xssa )
% 0.07/0.31 != ( coindu1121789889_LCons @ a @ X4 @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>\<And>thesis. (\<And>x xs. lconcat xss = LCons x xs \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.07/0.31 thf(fact_117__092_060open_062_092_060not_062_Alnull_A_Ilconcat_Axss_J_092_060close_062,axiom,
% 0.07/0.31 ~ ( coindu335574135_lnull @ a @ ( coindu441856546concat @ a @ xssa ) ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>\<not> lnull (lconcat xss)\<close>
% 0.07/0.31 thf(fact_118_gen__llength__def,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( coindu493225792length @ A )
% 0.07/0.31 = ( ^ [N3: nat,Xs2: coindu1593790203_llist @ A] : ( plus_plus @ extended_enat @ ( extended_enat2 @ N3 ) @ ( coindu1018505716length @ A @ Xs2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % gen_llength_def
% 0.07/0.31 thf(fact_119_setsum_Onot__neutral__contains__not__neutral,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType] :
% 0.07/0.31 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [G: B > A,A3: set @ B] :
% 0.07/0.31 ( ( ( groups15040474setsum @ B @ A @ G @ A3 )
% 0.07/0.31 != ( zero_zero @ A ) )
% 0.07/0.31 => ~ ! [A5: B] :
% 0.07/0.31 ( ( member @ B @ A5 @ A3 )
% 0.07/0.31 => ( ( G @ A5 )
% 0.07/0.31 = ( zero_zero @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % setsum.not_neutral_contains_not_neutral
% 0.07/0.31 thf(fact_120_setsum_Oneutral,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType] :
% 0.07/0.31 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [A3: set @ B,G: B > A] :
% 0.07/0.31 ( ! [X4: B] :
% 0.07/0.31 ( ( member @ B @ X4 @ A3 )
% 0.07/0.31 => ( ( G @ X4 )
% 0.07/0.31 = ( zero_zero @ A ) ) )
% 0.07/0.31 => ( ( groups15040474setsum @ B @ A @ G @ A3 )
% 0.07/0.31 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % setsum.neutral
% 0.07/0.31 thf(fact_121_add__0__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semiri456707255roduct @ A @ ( type2 @ A ) )
% 0.07/0.31 => ! [B2: A,A2: A] :
% 0.07/0.31 ( ( B2
% 0.07/0.31 = ( plus_plus @ A @ B2 @ A2 ) )
% 0.07/0.31 = ( A2
% 0.07/0.31 = ( zero_zero @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_0_iff
% 0.07/0.31 thf(fact_122_False,axiom,
% 0.07/0.31 ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ m ) ).
% 0.07/0.31
% 0.07/0.31 % False
% 0.07/0.31 thf(fact_123_neq0__conv,axiom,
% 0.07/0.31 ! [N2: nat] :
% 0.07/0.31 ( ( N2
% 0.07/0.31 != ( zero_zero @ nat ) )
% 0.07/0.31 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % neq0_conv
% 0.07/0.31 thf(fact_124_add__is__0,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ( plus_plus @ nat @ M2 @ N2 )
% 0.07/0.31 = ( zero_zero @ nat ) )
% 0.07/0.31 = ( ( M2
% 0.07/0.31 = ( zero_zero @ nat ) )
% 0.07/0.31 & ( N2
% 0.07/0.31 = ( zero_zero @ nat ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_is_0
% 0.07/0.31 thf(fact_125_Nat_Oadd__0__right,axiom,
% 0.07/0.31 ! [M2: nat] :
% 0.07/0.31 ( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
% 0.07/0.31 = M2 ) ).
% 0.07/0.31
% 0.07/0.31 % Nat.add_0_right
% 0.07/0.31 thf(fact_126_llist_Oinject,axiom,
% 0.07/0.31 ! [A: $tType,X21: A,X22: coindu1593790203_llist @ A,Y21: A,Y22: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( ( coindu1121789889_LCons @ A @ X21 @ X22 )
% 0.07/0.31 = ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) )
% 0.07/0.31 = ( ( X21 = Y21 )
% 0.07/0.31 & ( X22 = Y22 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % llist.inject
% 0.07/0.31 thf(fact_127_zero__less__Suc,axiom,
% 0.07/0.31 ! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % zero_less_Suc
% 0.07/0.31 thf(fact_128_less__Suc0,axiom,
% 0.07/0.31 ! [N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
% 0.07/0.31 = ( N2
% 0.07/0.31 = ( zero_zero @ nat ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_Suc0
% 0.07/0.31 thf(fact_129_add__gr__0,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M2 @ N2 ) )
% 0.07/0.31 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 0.07/0.31 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_gr_0
% 0.07/0.31 thf(fact_130_nat__add__left__cancel__less,axiom,
% 0.07/0.31 ! [K: nat,M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N2 ) )
% 0.07/0.31 = ( ord_less @ nat @ M2 @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % nat_add_left_cancel_less
% 0.07/0.31 thf(fact_131_lfinite__LCons,axiom,
% 0.07/0.31 ! [A: $tType,X: A,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) )
% 0.07/0.31 = ( coindu1213758845finite @ A @ Xs3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % lfinite_LCons
% 0.07/0.31 thf(fact_132_lfinite__code_I2_J,axiom,
% 0.07/0.31 ! [B: $tType,X: B,Xs3: coindu1593790203_llist @ B] :
% 0.07/0.31 ( ( coindu1213758845finite @ B @ ( coindu1121789889_LCons @ B @ X @ Xs3 ) )
% 0.07/0.31 = ( coindu1213758845finite @ B @ Xs3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % lfinite_code(2)
% 0.07/0.31 thf(fact_133_lfinite__code_I1_J,axiom,
% 0.07/0.31 ! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 % lfinite_code(1)
% 0.07/0.31 thf(fact_134_lnth__0,axiom,
% 0.07/0.31 ! [A: $tType,X: A,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( coindu749330388e_lnth @ A @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) @ ( zero_zero @ nat ) )
% 0.07/0.31 = X ) ).
% 0.07/0.31
% 0.07/0.31 % lnth_0
% 0.07/0.31 thf(fact_135_lconcat__LNil,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( coindu441856546concat @ A @ ( coindu1598213697e_LNil @ ( coindu1593790203_llist @ A ) ) )
% 0.07/0.31 = ( coindu1598213697e_LNil @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 % lconcat_LNil
% 0.07/0.31 thf(fact_136_lnth__Suc__LCons,axiom,
% 0.07/0.31 ! [A: $tType,X: A,Xs3: coindu1593790203_llist @ A,N2: nat] :
% 0.07/0.31 ( ( coindu749330388e_lnth @ A @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) @ ( suc @ N2 ) )
% 0.07/0.31 = ( coindu749330388e_lnth @ A @ Xs3 @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % lnth_Suc_LCons
% 0.07/0.31 thf(fact_137_llength__eq__0,axiom,
% 0.07/0.31 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( ( coindu1018505716length @ A @ Xs3 )
% 0.07/0.31 = ( zero_zero @ extended_enat ) )
% 0.07/0.31 = ( coindu335574135_lnull @ A @ Xs3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % llength_eq_0
% 0.07/0.31 thf(fact_138_llength__lnull,axiom,
% 0.07/0.31 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( coindu335574135_lnull @ A @ Xs3 )
% 0.07/0.31 => ( ( coindu1018505716length @ A @ Xs3 )
% 0.07/0.31 = ( zero_zero @ extended_enat ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % llength_lnull
% 0.07/0.31 thf(fact_139_llength__LNil,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( coindu1018505716length @ A @ ( coindu1598213697e_LNil @ A ) )
% 0.07/0.31 = ( zero_zero @ extended_enat ) ) ).
% 0.07/0.31
% 0.07/0.31 % llength_LNil
% 0.07/0.31 thf(fact_140_concat__xss,axiom,
% 0.07/0.31 ( ( coindu441856546concat @ a @ xssa )
% 0.07/0.31 = ( coindu1121789889_LCons @ a @ x @ xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % concat_xss
% 0.07/0.31 thf(fact_141_gr0I,axiom,
% 0.07/0.31 ! [N2: nat] :
% 0.07/0.31 ( ( N2
% 0.07/0.31 != ( zero_zero @ nat ) )
% 0.07/0.31 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % gr0I
% 0.07/0.31 thf(fact_142_not__gr0,axiom,
% 0.07/0.31 ! [N2: nat] :
% 0.07/0.31 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
% 0.07/0.31 = ( N2
% 0.07/0.31 = ( zero_zero @ nat ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % not_gr0
% 0.07/0.31 thf(fact_143_not__less0,axiom,
% 0.07/0.31 ! [N2: nat] :
% 0.07/0.31 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 0.07/0.31
% 0.07/0.31 % not_less0
% 0.07/0.31 thf(fact_144_add__lessD1,axiom,
% 0.07/0.31 ! [I2: nat,J: nat,K: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
% 0.07/0.31 => ( ord_less @ nat @ I2 @ K ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_lessD1
% 0.07/0.31 thf(fact_145_less__zeroE,axiom,
% 0.07/0.31 ! [N2: nat] :
% 0.07/0.31 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_zeroE
% 0.07/0.31 thf(fact_146_nat__neq__iff,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( M2 != N2 )
% 0.07/0.31 = ( ( ord_less @ nat @ M2 @ N2 )
% 0.07/0.31 | ( ord_less @ nat @ N2 @ M2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % nat_neq_iff
% 0.07/0.31 thf(fact_147_add__less__mono,axiom,
% 0.07/0.31 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ I2 @ J )
% 0.07/0.31 => ( ( ord_less @ nat @ K @ L )
% 0.07/0.31 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_less_mono
% 0.07/0.31 thf(fact_148_less__not__refl,axiom,
% 0.07/0.31 ! [N2: nat] :
% 0.07/0.31 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 0.07/0.31
% 0.07/0.31 % less_not_refl
% 0.07/0.31 thf(fact_149_not__add__less1,axiom,
% 0.07/0.31 ! [I2: nat,J: nat] :
% 0.07/0.31 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ I2 ) ).
% 0.07/0.31
% 0.07/0.31 % not_add_less1
% 0.07/0.31 thf(fact_150_not__add__less2,axiom,
% 0.07/0.31 ! [J: nat,I2: nat] :
% 0.07/0.31 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I2 ) @ I2 ) ).
% 0.07/0.31
% 0.07/0.31 % not_add_less2
% 0.07/0.31 thf(fact_151_add__less__mono1,axiom,
% 0.07/0.31 ! [I2: nat,J: nat,K: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ I2 @ J )
% 0.07/0.31 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_less_mono1
% 0.07/0.31 thf(fact_152_less__not__refl2,axiom,
% 0.07/0.31 ! [N2: nat,M2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ N2 @ M2 )
% 0.07/0.31 => ( M2 != N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_not_refl2
% 0.07/0.31 thf(fact_153_less__not__refl3,axiom,
% 0.07/0.31 ! [S: nat,T: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ S @ T )
% 0.07/0.31 => ( S != T ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_not_refl3
% 0.07/0.31 thf(fact_154_measure__induct,axiom,
% 0.07/0.31 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.07/0.31 ( ! [X4: A] :
% 0.07/0.31 ( ! [Y4: A] :
% 0.07/0.31 ( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
% 0.07/0.31 => ( P @ Y4 ) )
% 0.07/0.31 => ( P @ X4 ) )
% 0.07/0.31 => ( P @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % measure_induct
% 0.07/0.31 thf(fact_155_plus__nat_Oadd__0,axiom,
% 0.07/0.31 ! [N2: nat] :
% 0.07/0.31 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.07/0.31 = N2 ) ).
% 0.07/0.31
% 0.07/0.31 % plus_nat.add_0
% 0.07/0.31 thf(fact_156_gr__implies__not0,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ M2 @ N2 )
% 0.07/0.31 => ( N2
% 0.07/0.31 != ( zero_zero @ nat ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % gr_implies_not0
% 0.07/0.31 thf(fact_157_less__irrefl__nat,axiom,
% 0.07/0.31 ! [N2: nat] :
% 0.07/0.31 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 0.07/0.31
% 0.07/0.31 % less_irrefl_nat
% 0.07/0.31 thf(fact_158_nat__less__induct,axiom,
% 0.07/0.31 ! [P: nat > $o,N2: nat] :
% 0.07/0.31 ( ! [N4: nat] :
% 0.07/0.31 ( ! [M5: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ M5 @ N4 )
% 0.07/0.31 => ( P @ M5 ) )
% 0.07/0.31 => ( P @ N4 ) )
% 0.07/0.31 => ( P @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % nat_less_induct
% 0.07/0.31 thf(fact_159_trans__less__add1,axiom,
% 0.07/0.31 ! [I2: nat,J: nat,M2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ I2 @ J )
% 0.07/0.31 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % trans_less_add1
% 0.07/0.31 thf(fact_160_trans__less__add2,axiom,
% 0.07/0.31 ! [I2: nat,J: nat,M2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ I2 @ J )
% 0.07/0.31 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % trans_less_add2
% 0.07/0.31 thf(fact_161_add__eq__self__zero,axiom,
% 0.07/0.31 ! [M2: nat,N2: nat] :
% 0.07/0.31 ( ( ( plus_plus @ nat @ M2 @ N2 )
% 0.07/0.31 = M2 )
% 0.07/0.31 => ( N2
% 0.07/0.31 = ( zero_zero @ nat ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % add_eq_self_zero
% 0.07/0.31 thf(fact_162_infinite__descent,axiom,
% 0.07/0.31 ! [P: nat > $o,N2: nat] :
% 0.07/0.31 ( ! [N4: nat] :
% 0.07/0.31 ( ~ ( P @ N4 )
% 0.07/0.31 => ? [M5: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ M5 @ N4 )
% 0.07/0.31 & ~ ( P @ M5 ) ) )
% 0.07/0.31 => ( P @ N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % infinite_descent
% 0.07/0.31 thf(fact_163_less__add__eq__less,axiom,
% 0.07/0.31 ! [K: nat,L: nat,M2: nat,N2: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ K @ L )
% 0.07/0.31 => ( ( ( plus_plus @ nat @ M2 @ L )
% 0.07/0.31 = ( plus_plus @ nat @ K @ N2 ) )
% 0.07/0.31 => ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_add_eq_less
% 0.07/0.31 thf(fact_164_infinite__descent0,axiom,
% 0.07/0.31 ! [P: nat > $o,N2: nat] :
% 0.07/0.31 ( ( P @ ( zero_zero @ nat ) )
% 0.07/0.31 => ( ! [N4: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 0.07/0.31 => ( ~ ( P @ N4 )
% 0.07/0.31 => ? [M5: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ M5 @ N4 )
% 0.07/0.31 & ~ ( P @ M5 ) ) ) )
% 0.07/0.31 => ( P @ N2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % infinite_descent0
% 0.07/0.31 thf(fact_165_linorder__neqE__nat,axiom,
% 0.07/0.31 ! [X: nat,Y: nat] :
% 0.07/0.31 ( ( X != Y )
% 0.07/0.31 => ( ~ ( ord_less @ nat @ X @ Y )
% 0.07/0.31 => ( ord_less @ nat @ Y @ X ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % linorder_neqE_nat
% 0.07/0.31 thf(fact_166_setsum__SucD,axiom,
% 0.07/0.31 ! [A: $tType,F: A > nat,A3: set @ A,N2: nat] :
% 0.07/0.31 ( ( ( groups15040474setsum @ A @ nat @ F @ A3 )
% 0.07/0.31 = ( suc @ N2 ) )
% 0.07/0.31 => ? [X4: A] :
% 0.07/0.31 ( ( member @ A @ X4 @ A3 )
% 0.07/0.31 & ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X4 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % setsum_SucD
% 0.07/0.31 thf(fact_167_less__nat__zero__code,axiom,
% 0.07/0.31 ! [N2: nat] :
% 0.07/0.31 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_nat_zero_code
% 0.07/0.31 thf(fact_168_measure__induct__rule,axiom,
% 0.07/0.31 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.07/0.31 ( ! [X4: A] :
% 0.07/0.31 ( ! [Y4: A] :
% 0.07/0.31 ( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
% 0.07/0.31 => ( P @ Y4 ) )
% 0.07/0.31 => ( P @ X4 ) )
% 0.07/0.31 => ( P @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % measure_induct_rule
% 0.07/0.31 thf(fact_169_nat__add__left__cancel,axiom,
% 0.07/0.31 ! [K: nat,M2: nat,N2: nat] :
% 0.07/0.31 ( ( ( plus_plus @ nat @ K @ M2 )
% 0.07/0.31 = ( plus_plus @ nat @ K @ N2 ) )
% 0.07/0.31 = ( M2 = N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % nat_add_left_cancel
% 0.07/0.31 thf(fact_170_nat__add__right__cancel,axiom,
% 0.07/0.31 ! [M2: nat,K: nat,N2: nat] :
% 0.07/0.31 ( ( ( plus_plus @ nat @ M2 @ K )
% 0.07/0.31 = ( plus_plus @ nat @ N2 @ K ) )
% 0.07/0.31 = ( M2 = N2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % nat_add_right_cancel
% 0.07/0.31 thf(fact_171_less__imp__add__positive,axiom,
% 0.07/0.31 ! [I2: nat,J: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ I2 @ J )
% 0.07/0.31 => ? [K3: nat] :
% 0.07/0.31 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
% 0.07/0.31 & ( ( plus_plus @ nat @ I2 @ K3 )
% 0.07/0.31 = J ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_imp_add_positive
% 0.07/0.31 thf(fact_172_infinite__descent__measure,axiom,
% 0.07/0.31 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 0.07/0.31 ( ! [X4: A] :
% 0.07/0.31 ( ~ ( P @ X4 )
% 0.07/0.31 => ? [Y4: A] :
% 0.07/0.31 ( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X4 ) )
% 0.07/0.31 & ~ ( P @ Y4 ) ) )
% 0.07/0.31 => ( P @ X ) ) ).
% 0.07/0.31
% 0.07/0.31 % infinite_descent_measure
% 0.07/0.31 thf(fact_173_infinite__descent0__measure,axiom,
% 0.07/0.31 ! [A: $tType,V: A > nat,P: A > $o,X: A] :
% 0.07/0.31 ( ! [X4: A] :
% 0.07/0.31 ( ( ( V @ X4 )
% 0.07/0.31 = ( zero_zero @ nat ) )
% 0.07/0.31 => ( P @ X4 ) )
% 0.07/0.31 => ( ! [X4: A] :
% 0.07/0.31 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X4 ) )
% 0.07/0.31 => ( ~ ( P @ X4 )
% 0.07/0.31 => ? [Y4: A] :
% 0.07/0.31 ( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X4 ) )
% 0.07/0.31 & ~ ( P @ Y4 ) ) ) )
% 0.07/0.31 => ( P @ X ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % infinite_descent0_measure
% 0.07/0.31 thf(fact_174_lfinite__LConsI,axiom,
% 0.07/0.31 ! [A: $tType,Xs3: coindu1593790203_llist @ A,X: A] :
% 0.07/0.31 ( ( coindu1213758845finite @ A @ Xs3 )
% 0.07/0.31 => ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lfinite_LConsI
% 0.07/0.31 thf(fact_175_lfinite__LNil,axiom,
% 0.07/0.31 ! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 % lfinite_LNil
% 0.07/0.31 thf(fact_176_lnull__imp__lfinite,axiom,
% 0.07/0.31 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( coindu335574135_lnull @ A @ Xs3 )
% 0.07/0.31 => ( coindu1213758845finite @ A @ Xs3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % lnull_imp_lfinite
% 0.07/0.31 thf(fact_177_lnull__def,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( coindu335574135_lnull @ A )
% 0.07/0.31 = ( ^ [Llist: coindu1593790203_llist @ A] :
% 0.07/0.31 ( Llist
% 0.07/0.31 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lnull_def
% 0.07/0.31 thf(fact_178_lfinite_Oinducts,axiom,
% 0.07/0.31 ! [A: $tType,X: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
% 0.07/0.31 ( ( coindu1213758845finite @ A @ X )
% 0.07/0.31 => ( ( P @ ( coindu1598213697e_LNil @ A ) )
% 0.07/0.31 => ( ! [Xs: coindu1593790203_llist @ A,X4: A] :
% 0.07/0.31 ( ( coindu1213758845finite @ A @ Xs )
% 0.07/0.31 => ( ( P @ Xs )
% 0.07/0.31 => ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs ) ) ) )
% 0.07/0.31 => ( P @ X ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lfinite.inducts
% 0.07/0.31 thf(fact_179_lappend_Oexhaust,axiom,
% 0.07/0.31 ! [A: $tType,Xs3: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( ( coindu335574135_lnull @ A @ Xs3 )
% 0.07/0.31 => ~ ( coindu335574135_lnull @ A @ Ys ) )
% 0.07/0.31 => ( ~ ( coindu335574135_lnull @ A @ Xs3 )
% 0.07/0.31 | ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lappend.exhaust
% 0.07/0.31 thf(fact_180_not__lnull__conv,axiom,
% 0.07/0.31 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( ~ ( coindu335574135_lnull @ A @ Xs3 ) )
% 0.07/0.31 = ( ? [X3: A,Xs4: coindu1593790203_llist @ A] :
% 0.07/0.31 ( Xs3
% 0.07/0.31 = ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % not_lnull_conv
% 0.07/0.31 thf(fact_181_neq__LNil__conv,axiom,
% 0.07/0.31 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( Xs3
% 0.07/0.31 != ( coindu1598213697e_LNil @ A ) )
% 0.07/0.31 = ( ? [X3: A,Xs4: coindu1593790203_llist @ A] :
% 0.07/0.31 ( Xs3
% 0.07/0.31 = ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % neq_LNil_conv
% 0.07/0.31 thf(fact_182_llist_Oexhaust,axiom,
% 0.07/0.31 ! [A: $tType,Y: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( Y
% 0.07/0.31 != ( coindu1598213697e_LNil @ A ) )
% 0.07/0.31 => ~ ! [X212: A,X222: coindu1593790203_llist @ A] :
% 0.07/0.31 ( Y
% 0.07/0.31 != ( coindu1121789889_LCons @ A @ X212 @ X222 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % llist.exhaust
% 0.07/0.31 thf(fact_183_lfinite_Osimps,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( coindu1213758845finite @ A )
% 0.07/0.31 = ( ^ [A4: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( A4
% 0.07/0.31 = ( coindu1598213697e_LNil @ A ) )
% 0.07/0.31 | ? [Xs2: coindu1593790203_llist @ A,X3: A] :
% 0.07/0.31 ( ( A4
% 0.07/0.31 = ( coindu1121789889_LCons @ A @ X3 @ Xs2 ) )
% 0.07/0.31 & ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lfinite.simps
% 0.07/0.31 thf(fact_184_lfinite_Ocases,axiom,
% 0.07/0.31 ! [A: $tType,A2: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( coindu1213758845finite @ A @ A2 )
% 0.07/0.31 => ( ( A2
% 0.07/0.31 != ( coindu1598213697e_LNil @ A ) )
% 0.07/0.31 => ~ ! [Xs: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ? [X4: A] :
% 0.07/0.31 ( A2
% 0.07/0.31 = ( coindu1121789889_LCons @ A @ X4 @ Xs ) )
% 0.07/0.31 => ~ ( coindu1213758845finite @ A @ Xs ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lfinite.cases
% 0.07/0.31 thf(fact_185_lzip_Oexhaust,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,Xs3: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
% 0.07/0.31 ( ~ ( ( coindu335574135_lnull @ A @ Xs3 )
% 0.07/0.31 | ( coindu335574135_lnull @ B @ Ys ) )
% 0.07/0.31 => ~ ( ~ ( coindu335574135_lnull @ A @ Xs3 )
% 0.07/0.31 => ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % lzip.exhaust
% 0.07/0.31 thf(fact_186_llist_Odistinct_I1_J,axiom,
% 0.07/0.31 ! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( coindu1598213697e_LNil @ A )
% 0.07/0.31 != ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% 0.07/0.31
% 0.07/0.31 % llist.distinct(1)
% 0.07/0.31 thf(fact_187_llist_Ocollapse_I1_J,axiom,
% 0.07/0.31 ! [A: $tType,Llist2: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( coindu335574135_lnull @ A @ Llist2 )
% 0.07/0.31 => ( Llist2
% 0.07/0.31 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % llist.collapse(1)
% 0.07/0.31 thf(fact_188_llist_OdiscI_I1_J,axiom,
% 0.07/0.31 ! [A: $tType,Llist2: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( Llist2
% 0.07/0.31 = ( coindu1598213697e_LNil @ A ) )
% 0.07/0.31 => ( coindu335574135_lnull @ A @ Llist2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % llist.discI(1)
% 0.07/0.31 thf(fact_189_llist_OdiscI_I2_J,axiom,
% 0.07/0.31 ! [A: $tType,Llist2: coindu1593790203_llist @ A,X21: A,X22: coindu1593790203_llist @ A] :
% 0.07/0.31 ( ( Llist2
% 0.07/0.31 = ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
% 0.07/0.31 => ~ ( coindu335574135_lnull @ A @ Llist2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % llist.discI(2)
% 0.07/0.31 thf(fact_190_llist_Odisc_I1_J,axiom,
% 0.07/0.31 ! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 % llist.disc(1)
% 0.07/0.32 thf(fact_191_llist_Odisc_I2_J,axiom,
% 0.07/0.32 ! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
% 0.07/0.32 ~ ( coindu335574135_lnull @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% 0.07/0.32
% 0.07/0.32 % llist.disc(2)
% 0.07/0.32 thf(fact_192_gen__llength__code_I1_J,axiom,
% 0.07/0.32 ! [A: $tType,N2: nat] :
% 0.07/0.32 ( ( coindu493225792length @ A @ N2 @ ( coindu1598213697e_LNil @ A ) )
% 0.07/0.32 = ( extended_enat2 @ N2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % gen_llength_code(1)
% 0.07/0.32 thf(fact_193_llength__code,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( coindu1018505716length @ A )
% 0.07/0.32 = ( coindu493225792length @ A @ ( zero_zero @ nat ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % llength_code
% 0.07/0.32 thf(fact_194_ltake_Oexhaust,axiom,
% 0.07/0.32 ! [A: $tType,N2: extended_enat,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.32 ( ~ ( ( N2
% 0.07/0.32 = ( zero_zero @ extended_enat ) )
% 0.07/0.32 | ( coindu335574135_lnull @ A @ Xs3 ) )
% 0.07/0.32 => ~ ( ( N2
% 0.07/0.32 != ( zero_zero @ extended_enat ) )
% 0.07/0.32 => ( coindu335574135_lnull @ A @ Xs3 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ltake.exhaust
% 0.07/0.32 thf(fact_195_nat_Odistinct_I1_J,axiom,
% 0.07/0.32 ! [X2: nat] :
% 0.07/0.32 ( ( zero_zero @ nat )
% 0.07/0.32 != ( suc @ X2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % nat.distinct(1)
% 0.07/0.32 thf(fact_196_old_Onat_Odistinct_I2_J,axiom,
% 0.07/0.32 ! [Nat3: nat] :
% 0.07/0.32 ( ( suc @ Nat3 )
% 0.07/0.32 != ( zero_zero @ nat ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.nat.distinct(2)
% 0.07/0.32 thf(fact_197_old_Onat_Odistinct_I1_J,axiom,
% 0.07/0.32 ! [Nat2: nat] :
% 0.07/0.32 ( ( zero_zero @ nat )
% 0.07/0.32 != ( suc @ Nat2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.nat.distinct(1)
% 0.07/0.32 thf(fact_198_nat_OdiscI,axiom,
% 0.07/0.32 ! [Nat: nat,X2: nat] :
% 0.07/0.32 ( ( Nat
% 0.07/0.32 = ( suc @ X2 ) )
% 0.07/0.32 => ( Nat
% 0.07/0.32 != ( zero_zero @ nat ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % nat.discI
% 0.07/0.32 thf(fact_199_nat__induct,axiom,
% 0.07/0.32 ! [P: nat > $o,N2: nat] :
% 0.07/0.32 ( ( P @ ( zero_zero @ nat ) )
% 0.07/0.32 => ( ! [N4: nat] :
% 0.07/0.32 ( ( P @ N4 )
% 0.07/0.32 => ( P @ ( suc @ N4 ) ) )
% 0.07/0.32 => ( P @ N2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % nat_induct
% 0.07/0.32 thf(fact_200_diff__induct,axiom,
% 0.07/0.32 ! [P: nat > nat > $o,M2: nat,N2: nat] :
% 0.07/0.32 ( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
% 0.07/0.32 => ( ! [Y5: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y5 ) )
% 0.07/0.32 => ( ! [X4: nat,Y5: nat] :
% 0.07/0.32 ( ( P @ X4 @ Y5 )
% 0.07/0.32 => ( P @ ( suc @ X4 ) @ ( suc @ Y5 ) ) )
% 0.07/0.32 => ( P @ M2 @ N2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % diff_induct
% 0.07/0.32 thf(fact_201_zero__induct,axiom,
% 0.07/0.32 ! [P: nat > $o,K: nat] :
% 0.07/0.32 ( ( P @ K )
% 0.07/0.32 => ( ! [N4: nat] :
% 0.07/0.32 ( ( P @ ( suc @ N4 ) )
% 0.07/0.32 => ( P @ N4 ) )
% 0.07/0.32 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_induct
% 0.07/0.32 thf(fact_202_Suc__neq__Zero,axiom,
% 0.07/0.32 ! [M2: nat] :
% 0.07/0.32 ( ( suc @ M2 )
% 0.07/0.32 != ( zero_zero @ nat ) ) ).
% 0.07/0.32
% 0.07/0.32 % Suc_neq_Zero
% 0.07/0.32 thf(fact_203_Zero__neq__Suc,axiom,
% 0.07/0.32 ! [M2: nat] :
% 0.07/0.32 ( ( zero_zero @ nat )
% 0.07/0.32 != ( suc @ M2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % Zero_neq_Suc
% 0.07/0.32 thf(fact_204_Zero__not__Suc,axiom,
% 0.07/0.32 ! [M2: nat] :
% 0.07/0.32 ( ( zero_zero @ nat )
% 0.07/0.32 != ( suc @ M2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % Zero_not_Suc
% 0.07/0.32 thf(fact_205_old_Onat_Oexhaust,axiom,
% 0.07/0.32 ! [Y: nat] :
% 0.07/0.32 ( ( Y
% 0.07/0.32 != ( zero_zero @ nat ) )
% 0.07/0.32 => ~ ! [Nat4: nat] :
% 0.07/0.32 ( Y
% 0.07/0.32 != ( suc @ Nat4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.nat.exhaust
% 0.07/0.32 thf(fact_206_old_Onat_Oinducts,axiom,
% 0.07/0.32 ! [P: nat > $o,Nat: nat] :
% 0.07/0.32 ( ( P @ ( zero_zero @ nat ) )
% 0.07/0.32 => ( ! [Nat4: nat] :
% 0.07/0.32 ( ( P @ Nat4 )
% 0.07/0.32 => ( P @ ( suc @ Nat4 ) ) )
% 0.07/0.32 => ( P @ Nat ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.nat.inducts
% 0.07/0.32 thf(fact_207_not0__implies__Suc,axiom,
% 0.07/0.32 ! [N2: nat] :
% 0.07/0.32 ( ( N2
% 0.07/0.32 != ( zero_zero @ nat ) )
% 0.07/0.32 => ? [M: nat] :
% 0.07/0.32 ( N2
% 0.07/0.32 = ( suc @ M ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % not0_implies_Suc
% 0.07/0.32 thf(fact_208_less__Suc__eq__0__disj,axiom,
% 0.07/0.32 ! [M2: nat,N2: nat] :
% 0.07/0.32 ( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
% 0.07/0.32 = ( ( M2
% 0.07/0.32 = ( zero_zero @ nat ) )
% 0.07/0.32 | ? [J4: nat] :
% 0.07/0.32 ( ( M2
% 0.07/0.32 = ( suc @ J4 ) )
% 0.07/0.32 & ( ord_less @ nat @ J4 @ N2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_Suc_eq_0_disj
% 0.07/0.32 thf(fact_209_gr0__implies__Suc,axiom,
% 0.07/0.32 ! [N2: nat] :
% 0.07/0.32 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.07/0.32 => ? [M: nat] :
% 0.07/0.32 ( N2
% 0.07/0.32 = ( suc @ M ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % gr0_implies_Suc
% 0.07/0.32 thf(fact_210_gr0__conv__Suc,axiom,
% 0.07/0.32 ! [N2: nat] :
% 0.07/0.32 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.07/0.32 = ( ? [M3: nat] :
% 0.07/0.32 ( N2
% 0.07/0.32 = ( suc @ M3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % gr0_conv_Suc
% 0.07/0.32 thf(fact_211_one__is__add,axiom,
% 0.07/0.32 ! [M2: nat,N2: nat] :
% 0.07/0.32 ( ( ( suc @ ( zero_zero @ nat ) )
% 0.07/0.32 = ( plus_plus @ nat @ M2 @ N2 ) )
% 0.07/0.32 = ( ( ( M2
% 0.07/0.32 = ( suc @ ( zero_zero @ nat ) ) )
% 0.07/0.32 & ( N2
% 0.07/0.32 = ( zero_zero @ nat ) ) )
% 0.07/0.32 | ( ( M2
% 0.07/0.32 = ( zero_zero @ nat ) )
% 0.07/0.32 & ( N2
% 0.07/0.32 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % one_is_add
% 0.07/0.32 thf(fact_212_add__is__1,axiom,
% 0.07/0.32 ! [M2: nat,N2: nat] :
% 0.07/0.32 ( ( ( plus_plus @ nat @ M2 @ N2 )
% 0.07/0.32 = ( suc @ ( zero_zero @ nat ) ) )
% 0.07/0.32 = ( ( ( M2
% 0.07/0.32 = ( suc @ ( zero_zero @ nat ) ) )
% 0.07/0.32 & ( N2
% 0.07/0.32 = ( zero_zero @ nat ) ) )
% 0.07/0.32 | ( ( M2
% 0.07/0.32 = ( zero_zero @ nat ) )
% 0.07/0.32 & ( N2
% 0.07/0.32 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_is_1
% 0.07/0.32 thf(fact_213_setsum__cong__Suc,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.32 => ! [A3: set @ nat,F: nat > A,G: nat > A] :
% 0.07/0.32 ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 )
% 0.07/0.32 => ( ! [X4: nat] :
% 0.07/0.32 ( ( member @ nat @ ( suc @ X4 ) @ A3 )
% 0.07/0.32 => ( ( F @ ( suc @ X4 ) )
% 0.07/0.32 = ( G @ ( suc @ X4 ) ) ) )
% 0.07/0.32 => ( ( groups15040474setsum @ nat @ A @ F @ A3 )
% 0.07/0.32 = ( groups15040474setsum @ nat @ A @ G @ A3 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % setsum_cong_Suc
% 0.07/0.32 thf(fact_214_lfinite__llength__enat,axiom,
% 0.07/0.32 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.32 ( ( coindu1213758845finite @ A @ Xs3 )
% 0.07/0.32 => ? [N4: nat] :
% 0.07/0.32 ( ( coindu1018505716length @ A @ Xs3 )
% 0.07/0.32 = ( extended_enat2 @ N4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % lfinite_llength_enat
% 0.07/0.32 thf(fact_215_llength__eq__enat__lfiniteD,axiom,
% 0.07/0.32 ! [A: $tType,Xs3: coindu1593790203_llist @ A,N2: nat] :
% 0.07/0.32 ( ( ( coindu1018505716length @ A @ Xs3 )
% 0.07/0.32 = ( extended_enat2 @ N2 ) )
% 0.07/0.32 => ( coindu1213758845finite @ A @ Xs3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % llength_eq_enat_lfiniteD
% 0.07/0.32 thf(fact_216_lfinite__conv__llength__enat,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( coindu1213758845finite @ A )
% 0.07/0.32 = ( ^ [Xs2: coindu1593790203_llist @ A] :
% 0.07/0.32 ? [N3: nat] :
% 0.07/0.32 ( ( coindu1018505716length @ A @ Xs2 )
% 0.07/0.32 = ( extended_enat2 @ N3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % lfinite_conv_llength_enat
% 0.07/0.32 thf(fact_217_zero__enat__def,axiom,
% 0.07/0.32 ( ( zero_zero @ extended_enat )
% 0.07/0.32 = ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_enat_def
% 0.07/0.32 thf(fact_218_enat__0__iff_I1_J,axiom,
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ( extended_enat2 @ X )
% 0.07/0.32 = ( zero_zero @ extended_enat ) )
% 0.07/0.32 = ( X
% 0.07/0.32 = ( zero_zero @ nat ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % enat_0_iff(1)
% 0.07/0.32 thf(fact_219_enat__0__iff_I2_J,axiom,
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ( zero_zero @ extended_enat )
% 0.07/0.32 = ( extended_enat2 @ X ) )
% 0.07/0.32 = ( X
% 0.07/0.32 = ( zero_zero @ nat ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % enat_0_iff(2)
% 0.07/0.32 thf(fact_220_semiring__normalization__rules_I25_J,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.07/0.32 => ! [A2: A,C: A,D: A] :
% 0.07/0.32 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
% 0.07/0.32 = ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ D ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % semiring_normalization_rules(25)
% 0.07/0.32 thf(fact_221_semiring__normalization__rules_I24_J,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.07/0.32 => ( ( plus_plus @ A )
% 0.07/0.32 = ( ^ [A4: A,C2: A] : ( plus_plus @ A @ C2 @ A4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % semiring_normalization_rules(24)
% 0.07/0.32 thf(fact_222_semiring__normalization__rules_I23_J,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.07/0.32 => ! [A2: A,B2: A,C: A] :
% 0.07/0.32 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.07/0.32 = ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % semiring_normalization_rules(23)
% 0.07/0.32 thf(fact_223_semiring__normalization__rules_I22_J,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.07/0.32 => ! [A2: A,C: A,D: A] :
% 0.07/0.32 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
% 0.07/0.32 = ( plus_plus @ A @ C @ ( plus_plus @ A @ A2 @ D ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % semiring_normalization_rules(22)
% 0.07/0.32 thf(fact_224_semiring__normalization__rules_I21_J,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.07/0.32 => ! [A2: A,B2: A,C: A] :
% 0.07/0.32 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.07/0.32 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % semiring_normalization_rules(21)
% 0.07/0.32 thf(fact_225_semiring__normalization__rules_I20_J,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.07/0.32 => ! [A2: A,B2: A,C: A,D: A] :
% 0.07/0.32 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C @ D ) )
% 0.07/0.32 = ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % semiring_normalization_rules(20)
% 0.07/0.32 thf(fact_226_setsum_Ocong,axiom,
% 0.07/0.32 ! [A: $tType,B: $tType] :
% 0.07/0.32 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.32 => ! [A3: set @ B,B4: set @ B,G: B > A,H: B > A] :
% 0.07/0.32 ( ( A3 = B4 )
% 0.07/0.32 => ( ! [X4: B] :
% 0.07/0.32 ( ( member @ B @ X4 @ B4 )
% 0.07/0.32 => ( ( G @ X4 )
% 0.07/0.32 = ( H @ X4 ) ) )
% 0.07/0.32 => ( ( groups15040474setsum @ B @ A @ G @ A3 )
% 0.07/0.32 = ( groups15040474setsum @ B @ A @ H @ B4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % setsum.cong
% 0.07/0.32 thf(fact_227_setsum_Oreindex__bij__witness,axiom,
% 0.07/0.32 ! [B: $tType,A: $tType,C3: $tType] :
% 0.07/0.32 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.32 => ! [S2: set @ B,I2: C3 > B,J: B > C3,T2: set @ C3,H: C3 > A,G: B > A] :
% 0.07/0.32 ( ! [A5: B] :
% 0.07/0.32 ( ( member @ B @ A5 @ S2 )
% 0.07/0.32 => ( ( I2 @ ( J @ A5 ) )
% 0.07/0.32 = A5 ) )
% 0.07/0.32 => ( ! [A5: B] :
% 0.07/0.32 ( ( member @ B @ A5 @ S2 )
% 0.07/0.32 => ( member @ C3 @ ( J @ A5 ) @ T2 ) )
% 0.07/0.32 => ( ! [B5: C3] :
% 0.07/0.32 ( ( member @ C3 @ B5 @ T2 )
% 0.07/0.32 => ( ( J @ ( I2 @ B5 ) )
% 0.07/0.32 = B5 ) )
% 0.07/0.32 => ( ! [B5: C3] :
% 0.07/0.32 ( ( member @ C3 @ B5 @ T2 )
% 0.07/0.32 => ( member @ B @ ( I2 @ B5 ) @ S2 ) )
% 0.07/0.32 => ( ! [A5: B] :
% 0.07/0.32 ( ( member @ B @ A5 @ S2 )
% 0.07/0.32 => ( ( H @ ( J @ A5 ) )
% 0.07/0.32 = ( G @ A5 ) ) )
% 0.07/0.32 => ( ( groups15040474setsum @ B @ A @ G @ S2 )
% 0.07/0.32 = ( groups15040474setsum @ C3 @ A @ H @ T2 ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % setsum.reindex_bij_witness
% 0.07/0.32 thf(fact_228_setsum_Ocommute,axiom,
% 0.07/0.32 ! [A: $tType,B: $tType,C3: $tType] :
% 0.07/0.32 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.32 => ! [G: B > C3 > A,B4: set @ C3,A3: set @ B] :
% 0.07/0.32 ( ( groups15040474setsum @ B @ A
% 0.07/0.32 @ ^ [I: B] : ( groups15040474setsum @ C3 @ A @ ( G @ I ) @ B4 )
% 0.07/0.32 @ A3 )
% 0.07/0.32 = ( groups15040474setsum @ C3 @ A
% 0.07/0.32 @ ^ [J4: C3] :
% 0.07/0.32 ( groups15040474setsum @ B @ A
% 0.07/0.32 @ ^ [I: B] : ( G @ I @ J4 )
% 0.07/0.32 @ A3 )
% 0.07/0.32 @ B4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % setsum.commute
% 0.07/0.32 thf(fact_229_setsum__lessThan__Suc__shift,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 0.07/0.32 => ! [F: nat > A,N2: nat] :
% 0.07/0.32 ( ( groups15040474setsum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
% 0.07/0.32 = ( plus_plus @ A @ ( F @ ( zero_zero @ nat ) )
% 0.07/0.32 @ ( groups15040474setsum @ nat @ A
% 0.07/0.32 @ ^ [I: nat] : ( F @ ( suc @ I ) )
% 0.07/0.32 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % setsum_lessThan_Suc_shift
% 0.07/0.32 thf(fact_230_semiring__normalization__rules_I6_J,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.07/0.32 = A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % semiring_normalization_rules(6)
% 0.07/0.32 thf(fact_231_semiring__normalization__rules_I5_J,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.07/0.32 => ! [A2: A] :
% 0.07/0.32 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.07/0.32 = A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % semiring_normalization_rules(5)
% 0.07/0.32 thf(fact_232_concat__xss_H,axiom,
% 0.07/0.32 coindu335574135_lnull @ a @ ( coindu441856546concat @ a @ ( coindu1079202755ist_of @ ( coindu1593790203_llist @ a ) @ xss2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % concat_xss'
% 0.07/0.32 thf(fact_233_enat_Osize_I1_J,axiom,
% 0.07/0.32 ! [Nat: nat] :
% 0.07/0.32 ( ( extended_size_enat @ ( extended_enat2 @ Nat ) )
% 0.07/0.32 = ( zero_zero @ nat ) ) ).
% 0.07/0.32
% 0.07/0.32 % enat.size(1)
% 0.07/0.32 thf(fact_234_length__induct,axiom,
% 0.07/0.32 ! [A: $tType,P: ( list @ A ) > $o,Xs3: list @ A] :
% 0.07/0.32 ( ! [Xs: list @ A] :
% 0.07/0.32 ( ! [Ys2: list @ A] :
% 0.07/0.32 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.07/0.32 => ( P @ Ys2 ) )
% 0.07/0.32 => ( P @ Xs ) )
% 0.07/0.32 => ( P @ Xs3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % length_induct
% 0.07/0.32 thf(fact_235_lnth__ldropn,axiom,
% 0.07/0.32 ! [A: $tType,N2: nat,M2: nat,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.32 ( ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ N2 @ M2 ) ) @ ( coindu1018505716length @ A @ Xs3 ) )
% 0.07/0.32 => ( ( coindu749330388e_lnth @ A @ ( coindu531130065ldropn @ A @ N2 @ Xs3 ) @ M2 )
% 0.07/0.32 = ( coindu749330388e_lnth @ A @ Xs3 @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % lnth_ldropn
% 0.07/0.32 thf(fact_236_llist__of__inject,axiom,
% 0.07/0.32 ! [A: $tType,Xs3: list @ A,Ys: list @ A] :
% 0.07/0.32 ( ( ( coindu1079202755ist_of @ A @ Xs3 )
% 0.07/0.32 = ( coindu1079202755ist_of @ A @ Ys ) )
% 0.07/0.32 = ( Xs3 = Ys ) ) ).
% 0.07/0.32
% 0.07/0.32 % llist_of_inject
% 0.07/0.32 thf(fact_237_ldropn__0,axiom,
% 0.07/0.32 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.32 ( ( coindu531130065ldropn @ A @ ( zero_zero @ nat ) @ Xs3 )
% 0.07/0.32 = Xs3 ) ).
% 0.07/0.32
% 0.07/0.32 % ldropn_0
% 0.07/0.32 thf(fact_238_ldropn__ldropn,axiom,
% 0.07/0.32 ! [A: $tType,N2: nat,M2: nat,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.32 ( ( coindu531130065ldropn @ A @ N2 @ ( coindu531130065ldropn @ A @ M2 @ Xs3 ) )
% 0.07/0.32 = ( coindu531130065ldropn @ A @ ( plus_plus @ nat @ N2 @ M2 ) @ Xs3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % ldropn_ldropn
% 0.07/0.32 thf(fact_239_ldropn__LNil,axiom,
% 0.07/0.32 ! [A: $tType,N2: nat] :
% 0.07/0.32 ( ( coindu531130065ldropn @ A @ N2 @ ( coindu1598213697e_LNil @ A ) )
% 0.07/0.32 = ( coindu1598213697e_LNil @ A ) ) ).
% 0.07/0.32
% 0.07/0.32 % ldropn_LNil
% 0.07/0.32 thf(fact_240_lfinite__ldropn,axiom,
% 0.07/0.32 ! [A: $tType,N2: nat,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.32 ( ( coindu1213758845finite @ A @ ( coindu531130065ldropn @ A @ N2 @ Xs3 ) )
% 0.07/0.32 = ( coindu1213758845finite @ A @ Xs3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % lfinite_ldropn
% 0.07/0.32 thf(fact_241_ldropn__Suc__LCons,axiom,
% 0.07/0.32 ! [A: $tType,N2: nat,X: A,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.32 ( ( coindu531130065ldropn @ A @ ( suc @ N2 ) @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) )
% 0.07/0.32 = ( coindu531130065ldropn @ A @ N2 @ Xs3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % ldropn_Suc_LCons
% 0.07/0.32 thf(fact_242__092_060open_062m_A_061_A0_092_060close_062,axiom,
% 0.07/0.32 ( m
% 0.07/0.32 = ( zero_zero @ nat ) ) ).
% 0.07/0.32
% 0.07/0.32 % \<open>m = 0\<close>
% 0.07/0.32 thf(fact_243_llength__llist__of,axiom,
% 0.07/0.32 ! [A: $tType,Xs3: list @ A] :
% 0.07/0.32 ( ( coindu1018505716length @ A @ ( coindu1079202755ist_of @ A @ Xs3 ) )
% 0.07/0.32 = ( extended_enat2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % llength_llist_of
% 0.07/0.32 thf(fact_244_lfinite__llist__of,axiom,
% 0.07/0.32 ! [A: $tType,Xs3: list @ A] : ( coindu1213758845finite @ A @ ( coindu1079202755ist_of @ A @ Xs3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % lfinite_llist_of
% 0.07/0.32 thf(fact_245_ldropn__lnull,axiom,
% 0.07/0.32 ! [A: $tType,Xs3: coindu1593790203_llist @ A,N2: nat] :
% 0.07/0.32 ( ( coindu335574135_lnull @ A @ Xs3 )
% 0.07/0.32 => ( ( coindu531130065ldropn @ A @ N2 @ Xs3 )
% 0.07/0.32 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ldropn_lnull
% 0.07/0.32 thf(fact_246_Ex__list__of__length,axiom,
% 0.07/0.32 ! [A: $tType,N2: nat] :
% 0.07/0.32 ? [Xs: list @ A] :
% 0.07/0.32 ( ( size_size @ ( list @ A ) @ Xs )
% 0.07/0.32 = N2 ) ).
% 0.07/0.32
% 0.07/0.32 % Ex_list_of_length
% 0.07/0.32 thf(fact_247_neq__if__length__neq,axiom,
% 0.07/0.32 ! [A: $tType,Xs3: list @ A,Ys: list @ A] :
% 0.07/0.32 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.07/0.32 != ( size_size @ ( list @ A ) @ Ys ) )
% 0.07/0.32 => ( Xs3 != Ys ) ) ).
% 0.07/0.32
% 0.07/0.32 % neq_if_length_neq
% 0.07/0.32 thf(fact_248_ldropn__eq__LConsD,axiom,
% 0.07/0.32 ! [A: $tType,N2: nat,Xs3: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
% 0.07/0.32 ( ( ( coindu531130065ldropn @ A @ N2 @ Xs3 )
% 0.07/0.32 = ( coindu1121789889_LCons @ A @ Y @ Ys ) )
% 0.07/0.32 => ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs3 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ldropn_eq_LConsD
% 0.07/0.32 thf(fact_249_ldropn__Suc__conv__ldropn,axiom,
% 0.07/0.32 ! [A: $tType,N2: nat,Xs3: coindu1593790203_llist @ A] :
% 0.07/0.32 ( ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs3 ) )
% 0.07/0.32 => ( ( coindu1121789889_LCons @ A @ ( coindu749330388e_lnth @ A @ Xs3 @ N2 ) @ ( coindu531130065ldropn @ A @ ( suc @ N2 ) @ Xs3 ) )
% 0.07/0.32 = ( coindu531130065ldropn @ A @ N2 @ Xs3 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ldropn_Suc_conv_ldropn
% 0.07/0.32
% 0.07/0.32 %----Type constructors (41)
% 0.07/0.32 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.07/0.32 ! [A6: $tType,A7: $tType] :
% 0.07/0.32 ( ( order @ A7 @ ( type2 @ A7 ) )
% 0.07/0.32 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_fun___Orderings_Oord,axiom,
% 0.07/0.32 ! [A6: $tType,A7: $tType] :
% 0.07/0.32 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 0.07/0.32 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
% 0.07/0.32 semiri456707255roduct @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.07/0.32 ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 0.07/0.32 strict2144017051up_add @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.07/0.32 ordere223160158up_add @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.07/0.32 ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 0.07/0.32 strict797366125id_add @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.07/0.32 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.07/0.32 ordere779506340up_add @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
% 0.07/0.32 ordere216010020id_add @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.07/0.32 cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.07/0.32 cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.07/0.32 ab_semigroup_add @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.07/0.32 comm_monoid_add @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
% 0.07/0.32 comm_semiring_1 @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.07/0.32 semigroup_add @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.07/0.32 linorder @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 0.07/0.32 monoid_add @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
% 0.07/0.32 order @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Orderings_Oord_2,axiom,
% 0.07/0.32 ord @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.07/0.32 zero @ nat @ ( type2 @ nat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 0.07/0.32 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
% 0.07/0.32 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Orderings_Olinorder_5,axiom,
% 0.07/0.32 linorder @ $o @ ( type2 @ $o ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Orderings_Oorder_6,axiom,
% 0.07/0.32 order @ $o @ ( type2 @ $o ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Orderings_Oord_7,axiom,
% 0.07/0.32 ord @ $o @ ( type2 @ $o ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_8,axiom,
% 0.07/0.32 strict2144017051up_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_9,axiom,
% 0.07/0.32 strict797366125id_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_10,axiom,
% 0.07/0.32 canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_11,axiom,
% 0.07/0.32 ordere779506340up_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_12,axiom,
% 0.07/0.32 ordere216010020id_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_13,axiom,
% 0.07/0.32 ab_semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_14,axiom,
% 0.07/0.32 comm_monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_15,axiom,
% 0.07/0.32 comm_semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_16,axiom,
% 0.07/0.32 semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_17,axiom,
% 0.07/0.32 linorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_18,axiom,
% 0.07/0.32 monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_19,axiom,
% 0.07/0.32 order @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_20,axiom,
% 0.07/0.32 ord @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_21,axiom,
% 0.07/0.32 zero @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.07/0.32
% 0.07/0.32 %----Conjectures (1)
% 0.07/0.32 thf(conj_0,conjecture,
% 0.07/0.32 ? [M5: nat,N6: nat] :
% 0.07/0.32 ( ( ( coindu749330388e_lnth @ a @ ( coindu441856546concat @ a @ xssa ) @ ( suc @ na ) )
% 0.07/0.32 = ( coindu749330388e_lnth @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ M5 ) @ N6 ) )
% 0.07/0.32 & ( ord_less @ extended_enat @ ( extended_enat2 @ N6 ) @ ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ M5 ) ) )
% 0.07/0.32 & ( ord_less @ extended_enat @ ( extended_enat2 @ M5 ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ a ) @ xssa ) )
% 0.07/0.32 & ( ( extended_enat2 @ ( suc @ na ) )
% 0.07/0.32 = ( plus_plus @ extended_enat
% 0.07/0.32 @ ( groups15040474setsum @ nat @ extended_enat
% 0.07/0.32 @ ^ [I: nat] : ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ I ) )
% 0.07/0.32 @ ( set_ord_lessThan @ nat @ M5 ) )
% 0.07/0.32 @ ( extended_enat2 @ N6 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 %------------------------------------------------------------------------------
% 0.07/0.32 ------------------------
% 1.64/1.06 ---- Embedded in THF ---
% 1.64/1.06 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.64/1.06 %%% Generated on Tue Feb 24 01:11:11 EST 2026
% 1.64/1.06 %%% using '$$dhol' embedding, version 1.3.0.
% 1.64/1.06 %%% Logic specification used:
% 1.64/1.06 %%% thf(spec, logic, $$dhol).
% 1.64/1.06
% 1.64/1.06 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Bd5MKNrlUu/DTF2DTF19363.p
% 1.64/1.06 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist, type, coindu1593790203_llist: ($tType > $tType)).
% 1.64/1.06 thf(ty_t_Extended__Nat_Oenat, type, extended_enat: $tType).
% 1.64/1.06 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 1.64/1.06 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 1.64/1.06 thf(ty_t_Nat_Onat, type, nat: $tType).
% 1.64/1.06 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 1.64/1.06 thf(ty_tf_a, type, a: $tType).
% 1.64/1.06 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Omonoid__add, type, monoid_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Rings_Ocomm__semiring__1, type, comm_semiring_1: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Oordered__comm__monoid__add, type, ordere216010020id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add, type, strict797366125id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add, type, ordere223160158up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add, type, strict2144017051up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le, type, ordere516151231imp_le: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, type, semiri456707255roduct: (!> [A:$tType]: (((itself @ A) > $o)))).
% 1.64/1.06 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__llength, type, coindu493225792length: (!> [A:$tType]: ((nat > ((coindu1593790203_llist @ A) > extended_enat))))).
% 1.64/1.06 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olconcat, type, coindu441856546concat: (!> [A:$tType]: (((coindu1593790203_llist @ (coindu1593790203_llist @ A)) > (coindu1593790203_llist @ A))))).
% 1.64/1.06 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldropn, type, coindu531130065ldropn: (!> [A:$tType]: ((nat > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 1.64/1.06 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite, type, coindu1213758845finite: (!> [A:$tType]: (((coindu1593790203_llist @ A) > $o)))).
% 1.64/1.06 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength, type, coindu1018505716length: (!> [A:$tType]: (((coindu1593790203_llist @ A) > extended_enat)))).
% 1.64/1.06 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLCons, type, coindu1121789889_LCons: (!> [A:$tType]: ((A > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 1.64/1.06 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil, type, coindu1598213697e_LNil: (!> [A:$tType]: ((coindu1593790203_llist @ A)))).
% 1.64/1.06 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull, type, coindu335574135_lnull: (!> [A:$tType]: (((coindu1593790203_llist @ A) > $o)))).
% 1.64/1.06 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist__of, type, coindu1079202755ist_of: (!> [A:$tType]: (((list @ A) > (coindu1593790203_llist @ A))))).
% 1.64/1.06 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olnth, type, coindu749330388e_lnth: (!> [A:$tType]: (((coindu1593790203_llist @ A) > (nat > A))))).
% 1.64/1.06 thf(sy_c_Extended__Nat_Oenat, type, extended_enat2: (nat > extended_enat)).
% 1.64/1.06 thf(sy_c_Extended__Nat_Oenat_Osize__enat, type, extended_size_enat: (extended_enat > nat)).
% 1.64/1.06 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 1.64/1.06 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 1.64/1.06 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osetsum, type, groups15040474setsum: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > A))))).
% 1.64/1.06 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 1.64/1.06 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 1.64/1.06 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 1.64/1.06 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 1.64/1.06 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 1.64/1.06 thf(sy_c_Set__Interval_Oord__class_OlessThan, type, set_ord_lessThan: (!> [A:$tType]: ((A > (set @ A))))).
% 1.64/1.06 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 1.64/1.06 thf(sy_v_N____, type, n: nat).
% 1.64/1.06 thf(sy_v_m____, type, m: nat).
% 1.64/1.06 thf(sy_v_n, type, n2: nat).
% 1.64/1.06 thf(sy_v_n_H____, type, n3: nat).
% 1.64/1.06 thf(sy_v_na____, type, na: nat).
% 1.64/1.06 thf(sy_v_x____, type, x: a).
% 1.64/1.06 thf(sy_v_xs____, type, xs: (coindu1593790203_llist @ a)).
% 1.64/1.06 thf(sy_v_xss, type, xss: (coindu1593790203_llist @ (coindu1593790203_llist @ a))).
% 1.64/1.06 thf(sy_v_xss_H____, type, xss2: (list @ (coindu1593790203_llist @ a))).
% 1.64/1.06 thf(sy_v_xssa____, type, xssa: (coindu1593790203_llist @ (coindu1593790203_llist @ a))).
% 1.64/1.06 thf(fact_0__092_060open_062enat_A_Ilength_Axss_H_J_A_060_Allength_Axss_092_060close_062, axiom, (((ord_less @ extended_enat) @ (extended_enat2 @ ((size_size @ (list @ (coindu1593790203_llist @ a))) @ xss2))) @ ((coindu1018505716length @ (coindu1593790203_llist @ a)) @ xssa))).
% 1.64/1.06 thf(fact_1_assms, axiom, (((ord_less @ extended_enat) @ (extended_enat2 @ n2)) @ ((coindu1018505716length @ a) @ ((coindu441856546concat @ a) @ xss)))).
% 1.64/1.06 thf(fact_2__092_060open_062enat_A_ISuc_An_H_J_A_060_Allength_A_Ilnth_Axss_A_Ilength_Axss_H_J_J_092_060close_062, axiom, (((ord_less @ extended_enat) @ (extended_enat2 @ (suc @ n3))) @ ((coindu1018505716length @ a) @ (((coindu749330388e_lnth @ (coindu1593790203_llist @ a)) @ xssa) @ ((size_size @ (list @ (coindu1593790203_llist @ a))) @ xss2))))).
% 1.64/1.06 thf(fact_3_Suc_Oprems, axiom, (((ord_less @ extended_enat) @ (extended_enat2 @ (suc @ na))) @ ((coindu1018505716length @ a) @ ((coindu441856546concat @ a) @ xssa)))).
% 1.64/1.06 thf(fact_4__092_060open_062lnth_A_Ilconcat_Axss_J_A_ISuc_An_J_A_061_Alnth_A_Ilnth_Axss_A_Ilength_Axss_H_J_J_A_ISuc_An_H_J_092_060close_062, axiom, ((((coindu749330388e_lnth @ a) @ ((coindu441856546concat @ a) @ xssa)) @ (suc @ na)) = (((coindu749330388e_lnth @ a) @ (((coindu749330388e_lnth @ (coindu1593790203_llist @ a)) @ xssa) @ ((size_size @ (list @ (coindu1593790203_llist @ a))) @ xss2))) @ (suc @ n3)))).
% 1.64/1.06 thf(fact_5__092_060open_062enat_A_ISuc_An_J_A_061_A_I_092_060Sum_062i_060length_Axss_H_O_Allength_A_Ilnth_Axss_Ai_J_J_A_L_Aenat_A_ISuc_An_H_J_092_060close_062, axiom, ((extended_enat2 @ (suc @ na)) = (((plus_plus @ extended_enat) @ ((((groups15040474setsum @ nat) @ extended_enat) @ (^ [I:nat]: (((coindu1018505716length @ a) @ (((coindu749330388e_lnth @ (coindu1593790203_llist @ a)) @ xssa) @ I))))) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ (coindu1593790203_llist @ a))) @ xss2)))) @ (extended_enat2 @ (suc @ n3))))).
% 1.64/1.06 thf(fact_6_Suc_Ohyps, axiom, (! [Xss:(coindu1593790203_llist @ (coindu1593790203_llist @ a))]: (((((ord_less @ extended_enat) @ (extended_enat2 @ na)) @ ((coindu1018505716length @ a) @ ((coindu441856546concat @ a) @ Xss))) => (? [M:nat,N:nat]: ((((((coindu749330388e_lnth @ a) @ ((coindu441856546concat @ a) @ Xss)) @ na) = (((coindu749330388e_lnth @ a) @ (((coindu749330388e_lnth @ (coindu1593790203_llist @ a)) @ Xss) @ M)) @ N)) & ((((ord_less @ extended_enat) @ (extended_enat2 @ N)) @ ((coindu1018505716length @ a) @ (((coindu749330388e_lnth @ (coindu1593790203_llist @ a)) @ Xss) @ M))) & ((((ord_less @ extended_enat) @ (extended_enat2 @ M)) @ ((coindu1018505716length @ (coindu1593790203_llist @ a)) @ Xss)) & ((extended_enat2 @ na) = (((plus_plus @ extended_enat) @ ((((groups15040474setsum @ nat) @ extended_enat) @ (^ [I:nat]: (((coindu1018505716length @ a) @ (((coindu749330388e_lnth @ (coindu1593790203_llist @ a)) @ Xss) @ I))))) @ ((set_ord_lessThan @ nat) @ M))) @ (extended_enat2 @ N)))))))))))).
% 1.64/1.06 thf(fact_7_setsum__lessThan__Suc, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [F:(nat > A),N2:nat]: ((((((groups15040474setsum @ nat) @ A) @ F) @ ((set_ord_lessThan @ nat) @ (suc @ N2))) = (((plus_plus @ A) @ ((((groups15040474setsum @ nat) @ A) @ F) @ ((set_ord_lessThan @ nat) @ N2))) @ (F @ N2))))))))).
% 1.64/1.06 thf(fact_8_enat__add__mono, axiom, (! [X:nat,Y:extended_enat,Z:extended_enat]: (((((ord_less @ extended_enat) @ (((plus_plus @ extended_enat) @ (extended_enat2 @ X)) @ Y)) @ (((plus_plus @ extended_enat) @ (extended_enat2 @ X)) @ Z)) = (((ord_less @ extended_enat) @ Y) @ Z))))).
% 1.64/1.06 thf(fact_9_enat__add1__eq, axiom, (! [X:nat,Y:extended_enat,Z:extended_enat]: ((((((plus_plus @ extended_enat) @ (extended_enat2 @ X)) @ Y) = (((plus_plus @ extended_enat) @ (extended_enat2 @ X)) @ Z)) = (Y = Z))))).
% 1.64/1.06 thf(fact_10_enat__add2__eq, axiom, (! [Y:extended_enat,X:nat,Z:extended_enat]: ((((((plus_plus @ extended_enat) @ Y) @ (extended_enat2 @ X)) = (((plus_plus @ extended_enat) @ Z) @ (extended_enat2 @ X))) = (Y = Z))))).
% 1.64/1.06 thf(fact_11_lessThan__iff, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [I2:A,K:A]: (((((member @ A) @ I2) @ ((set_ord_lessThan @ A) @ K)) = (((ord_less @ A) @ I2) @ K)))))))).
% 1.64/1.06 thf(fact_12_add__less__cancel__left, axiom, (! [A:$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)))))))).
% 1.64/1.06 thf(fact_13_add__less__cancel__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [A2:A,C:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) = (((ord_less @ A) @ A2) @ B2)))))))).
% 1.64/1.06 thf(fact_14__092_060open_062_I_092_060Sum_062i_060length_Axss_H_O_Allength_A_Ilnth_Axss_Ai_J_J_A_061_A0_092_060close_062, axiom, (((((groups15040474setsum @ nat) @ extended_enat) @ (^ [I:nat]: (((coindu1018505716length @ a) @ (((coindu749330388e_lnth @ (coindu1593790203_llist @ a)) @ xssa) @ I))))) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ (coindu1593790203_llist @ a))) @ xss2))) = (zero_zero @ extended_enat))).
% 1.64/1.06 thf(fact_15_enat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((extended_enat2 @ Nat) = (extended_enat2 @ Nat2)) = (Nat = Nat2))))).
% 1.64/1.06 thf(fact_16_lessThan__eq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((set_ord_lessThan @ A) @ X) = ((set_ord_lessThan @ A) @ Y)) = (X = Y)))))))).
% 1.64/1.06 thf(fact_17_nat_Oinject, axiom, (! [X2:nat,Y2:nat]: ((((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))).
% 1.64/1.06 thf(fact_18_n, axiom, (na = (((plus_plus @ nat) @ n) @ n3))).
% 1.64/1.06 thf(fact_19_add__right__cancel, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B2 = C)))))))).
% 1.64/1.06 thf(fact_20_add__left__cancel, axiom, (! [A:$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)))))))).
% 1.64/1.06 thf(fact_21_add__Suc__right, axiom, (! [M2:nat,N2:nat]: (((((plus_plus @ nat) @ M2) @ (suc @ N2)) = (suc @ (((plus_plus @ nat) @ M2) @ N2)))))).
% 1.64/1.06 thf(fact_22_Suc__less__eq, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ (suc @ M2)) @ (suc @ N2)) = (((ord_less @ nat) @ M2) @ N2))))).
% 1.64/1.06 thf(fact_23_Suc__mono, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (((ord_less @ nat) @ (suc @ M2)) @ (suc @ N2)))))).
% 1.64/1.06 thf(fact_24_lessI, axiom, (! [N2:nat]: ((((ord_less @ nat) @ N2) @ (suc @ N2))))).
% 1.64/1.06 thf(fact_25_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 1.64/1.06 thf(fact_26_not__gr__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N2:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N2))) = (N2 = (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_27_add__cancel__right__right, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((A2 = (((plus_plus @ A) @ A2) @ B2)) = (B2 = (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_28_add__cancel__right__left, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((A2 = (((plus_plus @ A) @ B2) @ A2)) = (B2 = (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_29_add__cancel__left__right, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: ((((((plus_plus @ A) @ A2) @ B2) = A2) = (B2 = (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_30_add__cancel__left__left, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: ((((((plus_plus @ A) @ B2) @ A2) = A2) = (B2 = (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_31_double__zero__sym, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ A2) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_32_double__zero, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: ((((((plus_plus @ A) @ A2) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_33_add_Oright__neutral, axiom, (! [A:$tType]: ((((monoid_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 1.64/1.06 thf(fact_34_add_Oleft__neutral, axiom, (! [A:$tType]: ((((monoid_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 1.64/1.06 thf(fact_35_enat__ord__simps_I2_J, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ extended_enat) @ (extended_enat2 @ M2)) @ (extended_enat2 @ N2)) = (((ord_less @ nat) @ M2) @ N2))))).
% 1.64/1.06 thf(fact_36_i0__less, axiom, (! [N2:extended_enat]: (((((ord_less @ extended_enat) @ (zero_zero @ extended_enat)) @ N2) = (N2 != (zero_zero @ extended_enat)))))).
% 1.64/1.06 thf(fact_37_plus__enat__simps_I1_J, axiom, (! [M2:nat,N2:nat]: (((((plus_plus @ extended_enat) @ (extended_enat2 @ M2)) @ (extended_enat2 @ N2)) = (extended_enat2 @ (((plus_plus @ nat) @ M2) @ N2)))))).
% 1.64/1.06 thf(fact_38_zero__less__double__add__iff__zero__less__single__add, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 1.64/1.06 thf(fact_39_double__add__less__zero__iff__single__add__less__zero, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_40_less__add__same__cancel2, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ B2) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))).
% 1.64/1.06 thf(fact_41_less__add__same__cancel1, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B2)) = (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))).
% 1.64/1.06 thf(fact_42_add__less__same__cancel2, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_43_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 1.64/1.06 thf(fact_44_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A3)))) = A3)))).
% 1.64/1.06 thf(fact_45_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 1.64/1.06 thf(fact_46_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 1.64/1.06 thf(fact_47_add__less__same__cancel1, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ B2) @ A2)) @ B2) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_48_less__add__Suc1, axiom, (! [I2:nat,M2:nat]: ((((ord_less @ nat) @ I2) @ (suc @ (((plus_plus @ nat) @ I2) @ M2)))))).
% 1.64/1.06 thf(fact_49_less__add__Suc2, axiom, (! [I2:nat,M2:nat]: ((((ord_less @ nat) @ I2) @ (suc @ (((plus_plus @ nat) @ M2) @ I2)))))).
% 1.64/1.06 thf(fact_50_zero__reorient, axiom, (! [A:$tType]: ((((zero @ A) @ (type2 @ A)) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_51_less__iff__Suc__add, axiom, ((ord_less @ nat) = (^ [M3:nat,N3:nat]: ((? [K2:nat]: ((N3 = (suc @ (((plus_plus @ nat) @ M3) @ K2))))))))).
% 1.64/1.06 thf(fact_52_less__imp__Suc__add, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (? [K3:nat]: ((N2 = (suc @ (((plus_plus @ nat) @ M2) @ K3))))))))).
% 1.64/1.06 thf(fact_53_add__Suc, axiom, (! [M2:nat,N2:nat]: (((((plus_plus @ nat) @ (suc @ M2)) @ N2) = (suc @ (((plus_plus @ nat) @ M2) @ N2)))))).
% 1.64/1.06 thf(fact_54_add__Suc__shift, axiom, (! [M2:nat,N2:nat]: (((((plus_plus @ nat) @ (suc @ M2)) @ N2) = (((plus_plus @ nat) @ M2) @ (suc @ N2)))))).
% 1.64/1.06 thf(fact_55_zero__less__iff__neq__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N2) = (N2 != (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_56_gr__implies__not__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [M2:A,N2:A]: (((((ord_less @ A) @ M2) @ N2) => (N2 != (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_57_not__less__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N2:A]: ((~ ((((ord_less @ A) @ N2) @ (zero_zero @ A)))))))))).
% 1.64/1.06 thf(fact_58_gr__zeroI, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N2:A]: (((N2 != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N2)))))))).
% 1.64/1.06 thf(fact_59_not__less__less__Suc__eq, axiom, (! [N2:nat,M2:nat]: (((~ ((((ord_less @ nat) @ N2) @ M2))) => ((((ord_less @ nat) @ N2) @ (suc @ M2)) = (N2 = M2)))))).
% 1.64/1.06 thf(fact_60_strict__inc__induct, axiom, (! [I2:nat,J:nat,P:(nat > $o)]: (((((ord_less @ nat) @ I2) @ J) => ((! [I3:nat]: (((J = (suc @ I3)) => (P @ I3)))) => ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ J) => ((P @ (suc @ I3)) => (P @ I3))))) => (P @ I2))))))).
% 1.64/1.06 thf(fact_61_less__Suc__induct, axiom, (! [I2:nat,J:nat,P:(nat > (nat > $o))]: (((((ord_less @ nat) @ I2) @ J) => ((! [I3:nat]: (((P @ I3) @ (suc @ I3)))) => ((! [I3:nat,J2:nat,K3:nat]: (((((ord_less @ nat) @ I3) @ J2) => ((((ord_less @ nat) @ J2) @ K3) => (((P @ I3) @ J2) => (((P @ J2) @ K3) => ((P @ I3) @ K3))))))) => ((P @ I2) @ J))))))).
% 1.64/1.06 thf(fact_62_less__trans__Suc, axiom, (! [I2:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I2) @ J) => ((((ord_less @ nat) @ J) @ K) => (((ord_less @ nat) @ (suc @ I2)) @ K)))))).
% 1.64/1.06 thf(fact_63_Suc__less__SucD, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ (suc @ M2)) @ (suc @ N2)) => (((ord_less @ nat) @ M2) @ N2))))).
% 1.64/1.06 thf(fact_64_less__antisym, axiom, (! [N2:nat,M2:nat]: (((~ ((((ord_less @ nat) @ N2) @ M2))) => ((((ord_less @ nat) @ N2) @ (suc @ M2)) => (M2 = N2)))))).
% 1.64/1.06 thf(fact_65_Suc__less__eq2, axiom, (! [N2:nat,M2:nat]: (((((ord_less @ nat) @ (suc @ N2)) @ M2) = (? [M4:nat]: (((M2 = (suc @ M4)) & (((ord_less @ nat) @ N2) @ M4)))))))).
% 1.64/1.06 thf(fact_66_not__less__eq, axiom, (! [M2:nat,N2:nat]: (((~ ((((ord_less @ nat) @ M2) @ N2))) = (((ord_less @ nat) @ N2) @ (suc @ M2)))))).
% 1.64/1.06 thf(fact_67_less__Suc__eq, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ (suc @ N2)) = ((((ord_less @ nat) @ M2) @ N2) | (M2 = N2)))))).
% 1.64/1.06 thf(fact_68_less__SucI, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (((ord_less @ nat) @ M2) @ (suc @ N2)))))).
% 1.64/1.06 thf(fact_69_less__SucE, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ (suc @ N2)) => ((~ ((((ord_less @ nat) @ M2) @ N2))) => (M2 = N2)))))).
% 1.64/1.06 thf(fact_70_Suc__lessI, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (((suc @ M2) != N2) => (((ord_less @ nat) @ (suc @ M2)) @ N2)))))).
% 1.64/1.06 thf(fact_71_Suc__lessE, axiom, (! [I2:nat,K:nat]: (((((ord_less @ nat) @ (suc @ I2)) @ K) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (K != (suc @ J2))))))))))).
% 1.64/1.06 thf(fact_72_Suc__lessD, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ (suc @ M2)) @ N2) => (((ord_less @ nat) @ M2) @ N2))))).
% 1.64/1.06 thf(fact_73_lessE, axiom, (! [I2:nat,K:nat]: (((((ord_less @ nat) @ I2) @ K) => ((K != (suc @ I2)) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (K != (suc @ J2)))))))))))).
% 1.64/1.06 thf(fact_74_add__eq__0__iff__both__eq__0, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((((plus_plus @ A) @ X) @ Y) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))).
% 1.64/1.06 thf(fact_75_add_Ogroup__left__neutral, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 1.64/1.06 thf(fact_76_add_Ocomm__neutral, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 1.64/1.06 thf(fact_77_comm__monoid__add__class_Oadd__0, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 1.64/1.06 thf(fact_78_not__iless0, axiom, (! [N2:extended_enat]: ((~ ((((ord_less @ extended_enat) @ N2) @ (zero_zero @ extended_enat))))))).
% 1.64/1.06 thf(fact_79_plus__enat__eq__0__conv, axiom, (! [M2:extended_enat,N2:extended_enat]: ((((((plus_plus @ extended_enat) @ M2) @ N2) = (zero_zero @ extended_enat)) = ((M2 = (zero_zero @ extended_enat)) & (N2 = (zero_zero @ extended_enat))))))).
% 1.64/1.06 thf(fact_80_pos__add__strict, axiom, (! [A:$tType]: ((((strict797366125id_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 1.64/1.06 thf(fact_81_add__pos__pos, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 1.64/1.06 thf(fact_82_add__neg__neg, axiom, (! [A:$tType]: ((((ordere216010020id_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))).
% 1.64/1.06 thf(fact_83_less__enatE, axiom, (! [N2:extended_enat,M2:nat]: (((((ord_less @ extended_enat) @ N2) @ (extended_enat2 @ M2)) => (~ ((! [K3:nat]: (((N2 = (extended_enat2 @ K3)) => (~ ((((ord_less @ nat) @ K3) @ M2)))))))))))).
% 1.64/1.06 thf(fact_84_enat__less__enat__plusI2, axiom, (! [Y:nat,Z:extended_enat,X:nat]: (((((ord_less @ extended_enat) @ (extended_enat2 @ Y)) @ Z) => (((ord_less @ extended_enat) @ (extended_enat2 @ (((plus_plus @ nat) @ X) @ Y))) @ (((plus_plus @ extended_enat) @ (extended_enat2 @ X)) @ Z)))))).
% 1.64/1.06 thf(fact_85_enat__less__enat__plusI, axiom, (! [X:nat,Y:nat,Z:extended_enat]: (((((ord_less @ nat) @ X) @ Y) => (((ord_less @ extended_enat) @ (extended_enat2 @ X)) @ (((plus_plus @ extended_enat) @ (extended_enat2 @ Y)) @ Z)))))).
% 1.64/1.06 thf(fact_86_add__right__imp__eq, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) => (B2 = C)))))))).
% 1.64/1.06 thf(fact_87_add__left__imp__eq, axiom, (! [A:$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)))))))).
% 1.64/1.06 thf(fact_88_add_Oleft__commute, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((plus_plus @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 1.64/1.06 thf(fact_89_add_Ocommute, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type2 @ A)) => ((plus_plus @ A) = (^ [A4:A,B3:A]: ((((plus_plus @ A) @ B3) @ A4)))))))).
% 1.64/1.06 thf(fact_90_add_Oright__cancel, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B2 = C)))))))).
% 1.64/1.06 thf(fact_91_add_Oleft__cancel, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C)) = (B2 = C)))))))).
% 1.64/1.06 thf(fact_92_add_Oassoc, axiom, (! [A:$tType]: ((((semigroup_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 1.64/1.06 thf(fact_93_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$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))))))))).
% 1.64/1.06 thf(fact_94_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))))))))).
% 1.64/1.06 thf(fact_95_n__not__Suc__n, axiom, (! [N2:nat]: ((N2 != (suc @ N2))))).
% 1.64/1.06 thf(fact_96_Suc__inject, axiom, (! [X:nat,Y:nat]: ((((suc @ X) = (suc @ Y)) => (X = Y))))).
% 1.64/1.06 thf(fact_97_enat__less__induct, axiom, (! [P:(extended_enat > $o),N2:extended_enat]: (((! [N4:extended_enat]: (((! [M5:extended_enat]: (((((ord_less @ extended_enat) @ M5) @ N4) => (P @ M5)))) => (P @ N4)))) => (P @ N2))))).
% 1.64/1.06 thf(fact_98_add__less__imp__less__right, axiom, (! [A:$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)))))))).
% 1.64/1.06 thf(fact_99_add__less__imp__less__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [C:A,A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) => (((ord_less @ A) @ A2) @ B2)))))))).
% 1.64/1.06 thf(fact_100_add__strict__right__mono, axiom, (! [A:$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))))))))).
% 1.64/1.06 thf(fact_101_add__strict__left__mono, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2))))))))).
% 1.64/1.06 thf(fact_102_add__strict__mono, axiom, (! [A:$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)))))))))).
% 1.64/1.06 thf(fact_103_add__mono__thms__linordered__field_I1_J, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [I2:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I2) @ J) & (K = L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I2) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 1.64/1.06 thf(fact_104_add__mono__thms__linordered__field_I2_J, axiom, (! [A:$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))))))))).
% 1.64/1.06 thf(fact_105_add__mono__thms__linordered__field_I5_J, axiom, (! [A:$tType]: ((((ordere223160158up_add @ A) @ (type2 @ A)) => (! [I2:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I2) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I2) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 1.64/1.06 thf(fact_106_lift__Suc__mono__less__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F:(nat > A),N2:nat,M2:nat]: (((! [N4:nat]: ((((ord_less @ A) @ (F @ N4)) @ (F @ (suc @ N4))))) => ((((ord_less @ A) @ (F @ N2)) @ (F @ M2)) = (((ord_less @ nat) @ N2) @ M2))))))))).
% 1.64/1.06 thf(fact_107_lift__Suc__mono__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F:(nat > A),N2:nat,N5:nat]: (((! [N4:nat]: ((((ord_less @ A) @ (F @ N4)) @ (F @ (suc @ N4))))) => ((((ord_less @ nat) @ N2) @ N5) => (((ord_less @ A) @ (F @ N2)) @ (F @ N5)))))))))).
% 1.64/1.06 thf(fact_108_lessThan__strict__subset__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [M2:A,N2:A]: (((((ord_less @ (set @ A)) @ ((set_ord_lessThan @ A) @ M2)) @ ((set_ord_lessThan @ A) @ N2)) = (((ord_less @ A) @ M2) @ N2)))))))).
% 1.64/1.06 thf(fact_109_enat__iless, axiom, (! [N2:extended_enat,M2:nat]: (((((ord_less @ extended_enat) @ N2) @ (extended_enat2 @ M2)) => (? [K3:nat]: ((N2 = (extended_enat2 @ K3)))))))).
% 1.64/1.06 thf(fact_110_chain__incr, axiom, (! [A:$tType,Y3:(A > extended_enat),K:nat]: (((! [I3:A]: ((? [J3:A]: ((((ord_less @ extended_enat) @ (Y3 @ I3)) @ (Y3 @ J3)))))) => (? [J2:A]: ((((ord_less @ extended_enat) @ (extended_enat2 @ K)) @ (Y3 @ J2)))))))).
% 1.64/1.06 thf(fact_111_lessThan__def, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => ((set_ord_lessThan @ A) = (^ [U:A]: (((collect @ A) @ (^ [X3:A]: ((((ord_less @ A) @ X3) @ U))))))))))).
% 1.64/1.06 thf(fact_112_setsum_Oneutral__const, axiom, (! [B:$tType,A:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [A3:(set @ B)]: ((((((groups15040474setsum @ B) @ A) @ (^ [Uu:B]: ((zero_zero @ A)))) @ A3) = (zero_zero @ A)))))))).
% 1.64/1.06 thf(fact_113_lnth__prefix, axiom, (! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ (coindu1593790203_llist @ a))) @ xss2)) => ((((coindu749330388e_lnth @ (coindu1593790203_llist @ a)) @ xssa) @ I4) = (coindu1598213697e_LNil @ a)))))).
% 1.64/1.06 thf(fact_114_llength__lconcat__lfinite__conv__sum, axiom, (! [A:$tType,Xss:(coindu1593790203_llist @ (coindu1593790203_llist @ A))]: ((((coindu1213758845finite @ (coindu1593790203_llist @ A)) @ Xss) => (((coindu1018505716length @ A) @ ((coindu441856546concat @ A) @ Xss)) = ((((groups15040474setsum @ nat) @ extended_enat) @ (^ [I:nat]: (((coindu1018505716length @ A) @ (((coindu749330388e_lnth @ (coindu1593790203_llist @ A)) @ Xss) @ I))))) @ ((collect @ nat) @ (^ [I:nat]: ((((ord_less @ extended_enat) @ (extended_enat2 @ I)) @ ((coindu1018505716length @ (coindu1593790203_llist @ A)) @ Xss))))))))))).
% 1.64/1.06 thf(fact_115_setsum_Odistrib, axiom, (! [A:$tType,B:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [G:(B > A),H:(B > A),A3:(set @ B)]: ((((((groups15040474setsum @ B) @ A) @ (^ [X3:B]: ((((plus_plus @ A) @ (G @ X3)) @ (H @ X3))))) @ A3) = (((plus_plus @ A) @ ((((groups15040474setsum @ B) @ A) @ G) @ A3)) @ ((((groups15040474setsum @ B) @ A) @ H) @ A3))))))))).
% 1.64/1.06 thf(fact_116__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Axs_O_Alconcat_Axss_A_061_ALCons_Ax_Axs_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [X4:a,Xs:(coindu1593790203_llist @ a)]: ((((coindu441856546concat @ a) @ xssa) != (((coindu1121789889_LCons @ a) @ X4) @ Xs))))))).
% 1.64/1.06 thf(fact_117__092_060open_062_092_060not_062_Alnull_A_Ilconcat_Axss_J_092_060close_062, axiom, (~ (((coindu335574135_lnull @ a) @ ((coindu441856546concat @ a) @ xssa))))).
% 1.64/1.06 thf(fact_118_gen__llength__def, axiom, (! [A:$tType]: (((coindu493225792length @ A) = (^ [N3:nat,Xs2:(coindu1593790203_llist @ A)]: ((((plus_plus @ extended_enat) @ (extended_enat2 @ N3)) @ ((coindu1018505716length @ A) @ Xs2)))))))).
% 1.64/1.06 thf(fact_119_setsum_Onot__neutral__contains__not__neutral, axiom, (! [B:$tType,A:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [G:(B > A),A3:(set @ B)]: (((((((groups15040474setsum @ B) @ A) @ G) @ A3) != (zero_zero @ A)) => (~ ((! [A5:B]: (((((member @ B) @ A5) @ A3) => ((G @ A5) = (zero_zero @ A)))))))))))))).
% 1.64/1.06 thf(fact_120_setsum_Oneutral, axiom, (! [B:$tType,A:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [A3:(set @ B),G:(B > A)]: (((! [X4:B]: (((((member @ B) @ X4) @ A3) => ((G @ X4) = (zero_zero @ A))))) => (((((groups15040474setsum @ B) @ A) @ G) @ A3) = (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_121_add__0__iff, axiom, (! [A:$tType]: ((((semiri456707255roduct @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((B2 = (((plus_plus @ A) @ B2) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 1.64/1.06 thf(fact_122_False, axiom, (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ m)))).
% 1.64/1.06 thf(fact_123_neq0__conv, axiom, (! [N2:nat]: (((N2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))).
% 1.64/1.06 thf(fact_124_add__is__0, axiom, (! [M2:nat,N2:nat]: ((((((plus_plus @ nat) @ M2) @ N2) = (zero_zero @ nat)) = ((M2 = (zero_zero @ nat)) & (N2 = (zero_zero @ nat))))))).
% 1.64/1.06 thf(fact_125_Nat_Oadd__0__right, axiom, (! [M2:nat]: (((((plus_plus @ nat) @ M2) @ (zero_zero @ nat)) = M2)))).
% 1.64/1.06 thf(fact_126_llist_Oinject, axiom, (! [A:$tType,X21:A,X22:(coindu1593790203_llist @ A),Y21:A,Y22:(coindu1593790203_llist @ A)]: ((((((coindu1121789889_LCons @ A) @ X21) @ X22) = (((coindu1121789889_LCons @ A) @ Y21) @ Y22)) = ((X21 = Y21) & (X22 = Y22)))))).
% 1.64/1.06 thf(fact_127_zero__less__Suc, axiom, (! [N2:nat]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (suc @ N2))))).
% 1.64/1.06 thf(fact_128_less__Suc0, axiom, (! [N2:nat]: (((((ord_less @ nat) @ N2) @ (suc @ (zero_zero @ nat))) = (N2 = (zero_zero @ nat)))))).
% 1.64/1.06 thf(fact_129_add__gr__0, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ M2) @ N2)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ M2) | (((ord_less @ nat) @ (zero_zero @ nat)) @ N2)))))).
% 1.64/1.06 thf(fact_130_nat__add__left__cancel__less, axiom, (! [K:nat,M2:nat,N2:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ K) @ M2)) @ (((plus_plus @ nat) @ K) @ N2)) = (((ord_less @ nat) @ M2) @ N2))))).
% 1.64/1.06 thf(fact_131_lfinite__LCons, axiom, (! [A:$tType,X:A,Xs3:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ (((coindu1121789889_LCons @ A) @ X) @ Xs3)) = ((coindu1213758845finite @ A) @ Xs3))))).
% 1.64/1.06 thf(fact_132_lfinite__code_I2_J, axiom, (! [B:$tType,X:B,Xs3:(coindu1593790203_llist @ B)]: ((((coindu1213758845finite @ B) @ (((coindu1121789889_LCons @ B) @ X) @ Xs3)) = ((coindu1213758845finite @ B) @ Xs3))))).
% 1.64/1.06 thf(fact_133_lfinite__code_I1_J, axiom, (! [A:$tType]: (((coindu1213758845finite @ A) @ (coindu1598213697e_LNil @ A))))).
% 1.64/1.06 thf(fact_134_lnth__0, axiom, (! [A:$tType,X:A,Xs3:(coindu1593790203_llist @ A)]: (((((coindu749330388e_lnth @ A) @ (((coindu1121789889_LCons @ A) @ X) @ Xs3)) @ (zero_zero @ nat)) = X)))).
% 1.64/1.06 thf(fact_135_lconcat__LNil, axiom, (! [A:$tType]: ((((coindu441856546concat @ A) @ (coindu1598213697e_LNil @ (coindu1593790203_llist @ A))) = (coindu1598213697e_LNil @ A))))).
% 1.64/1.06 thf(fact_136_lnth__Suc__LCons, axiom, (! [A:$tType,X:A,Xs3:(coindu1593790203_llist @ A),N2:nat]: (((((coindu749330388e_lnth @ A) @ (((coindu1121789889_LCons @ A) @ X) @ Xs3)) @ (suc @ N2)) = (((coindu749330388e_lnth @ A) @ Xs3) @ N2))))).
% 1.64/1.06 thf(fact_137_llength__eq__0, axiom, (! [A:$tType,Xs3:(coindu1593790203_llist @ A)]: (((((coindu1018505716length @ A) @ Xs3) = (zero_zero @ extended_enat)) = ((coindu335574135_lnull @ A) @ Xs3))))).
% 1.64/1.06 thf(fact_138_llength__lnull, axiom, (! [A:$tType,Xs3:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Xs3) => (((coindu1018505716length @ A) @ Xs3) = (zero_zero @ extended_enat)))))).
% 1.64/1.06 thf(fact_139_llength__LNil, axiom, (! [A:$tType]: ((((coindu1018505716length @ A) @ (coindu1598213697e_LNil @ A)) = (zero_zero @ extended_enat))))).
% 1.64/1.06 thf(fact_140_concat__xss, axiom, (((coindu441856546concat @ a) @ xssa) = (((coindu1121789889_LCons @ a) @ x) @ xs))).
% 1.64/1.06 thf(fact_141_gr0I, axiom, (! [N2:nat]: (((N2 != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))).
% 1.64/1.06 thf(fact_142_not__gr0, axiom, (! [N2:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2))) = (N2 = (zero_zero @ nat)))))).
% 1.64/1.06 thf(fact_143_not__less0, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ (zero_zero @ nat))))))).
% 1.64/1.06 thf(fact_144_add__lessD1, axiom, (! [I2:nat,J:nat,K:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I2) @ J)) @ K) => (((ord_less @ nat) @ I2) @ K))))).
% 1.64/1.06 thf(fact_145_less__zeroE, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ (zero_zero @ nat))))))).
% 1.64/1.06 thf(fact_146_nat__neq__iff, axiom, (! [M2:nat,N2:nat]: (((M2 != N2) = ((((ord_less @ nat) @ M2) @ N2) | (((ord_less @ nat) @ N2) @ M2)))))).
% 1.64/1.06 thf(fact_147_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))))))).
% 1.64/1.06 thf(fact_148_less__not__refl, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ N2)))))).
% 1.64/1.06 thf(fact_149_not__add__less1, axiom, (! [I2:nat,J:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ I2) @ J)) @ I2)))))).
% 1.64/1.06 thf(fact_150_not__add__less2, axiom, (! [J:nat,I2:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ J) @ I2)) @ I2)))))).
% 1.64/1.06 thf(fact_151_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)))))).
% 1.64/1.06 thf(fact_152_less__not__refl2, axiom, (! [N2:nat,M2:nat]: (((((ord_less @ nat) @ N2) @ M2) => (M2 != N2))))).
% 1.64/1.06 thf(fact_153_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 1.64/1.06 thf(fact_154_measure__induct, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X4:A]: (((! [Y4:A]: (((((ord_less @ nat) @ (F @ Y4)) @ (F @ X4)) => (P @ Y4)))) => (P @ X4)))) => (P @ A2))))).
% 1.64/1.06 thf(fact_155_plus__nat_Oadd__0, axiom, (! [N2:nat]: (((((plus_plus @ nat) @ (zero_zero @ nat)) @ N2) = N2)))).
% 1.64/1.06 thf(fact_156_gr__implies__not0, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (N2 != (zero_zero @ nat)))))).
% 1.64/1.06 thf(fact_157_less__irrefl__nat, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ N2)))))).
% 1.64/1.06 thf(fact_158_nat__less__induct, axiom, (! [P:(nat > $o),N2:nat]: (((! [N4:nat]: (((! [M5:nat]: (((((ord_less @ nat) @ M5) @ N4) => (P @ M5)))) => (P @ N4)))) => (P @ N2))))).
% 1.64/1.06 thf(fact_159_trans__less__add1, axiom, (! [I2:nat,J:nat,M2:nat]: (((((ord_less @ nat) @ I2) @ J) => (((ord_less @ nat) @ I2) @ (((plus_plus @ nat) @ J) @ M2)))))).
% 1.64/1.06 thf(fact_160_trans__less__add2, axiom, (! [I2:nat,J:nat,M2:nat]: (((((ord_less @ nat) @ I2) @ J) => (((ord_less @ nat) @ I2) @ (((plus_plus @ nat) @ M2) @ J)))))).
% 1.64/1.06 thf(fact_161_add__eq__self__zero, axiom, (! [M2:nat,N2:nat]: ((((((plus_plus @ nat) @ M2) @ N2) = M2) => (N2 = (zero_zero @ nat)))))).
% 1.64/1.06 thf(fact_162_infinite__descent, axiom, (! [P:(nat > $o),N2:nat]: (((! [N4:nat]: (((~ ((P @ N4))) => (? [M5:nat]: (((((ord_less @ nat) @ M5) @ N4) & (~ ((P @ M5))))))))) => (P @ N2))))).
% 1.64/1.06 thf(fact_163_less__add__eq__less, axiom, (! [K:nat,L:nat,M2:nat,N2:nat]: (((((ord_less @ nat) @ K) @ L) => (((((plus_plus @ nat) @ M2) @ L) = (((plus_plus @ nat) @ K) @ N2)) => (((ord_less @ nat) @ M2) @ N2)))))).
% 1.64/1.06 thf(fact_164_infinite__descent0, axiom, (! [P:(nat > $o),N2:nat]: (((P @ (zero_zero @ nat)) => ((! [N4:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N4) => ((~ ((P @ N4))) => (? [M5:nat]: (((((ord_less @ nat) @ M5) @ N4) & (~ ((P @ M5)))))))))) => (P @ N2)))))).
% 1.64/1.06 thf(fact_165_linorder__neqE__nat, axiom, (! [X:nat,Y:nat]: (((X != Y) => ((~ ((((ord_less @ nat) @ X) @ Y))) => (((ord_less @ nat) @ Y) @ X)))))).
% 1.64/1.06 thf(fact_166_setsum__SucD, axiom, (! [A:$tType,F:(A > nat),A3:(set @ A),N2:nat]: (((((((groups15040474setsum @ A) @ nat) @ F) @ A3) = (suc @ N2)) => (? [X4:A]: (((((member @ A) @ X4) @ A3) & (((ord_less @ nat) @ (zero_zero @ nat)) @ (F @ X4))))))))).
% 1.64/1.06 thf(fact_167_less__nat__zero__code, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ (zero_zero @ nat))))))).
% 1.64/1.06 thf(fact_168_measure__induct__rule, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X4:A]: (((! [Y4:A]: (((((ord_less @ nat) @ (F @ Y4)) @ (F @ X4)) => (P @ Y4)))) => (P @ X4)))) => (P @ A2))))).
% 1.64/1.06 thf(fact_169_nat__add__left__cancel, axiom, (! [K:nat,M2:nat,N2:nat]: ((((((plus_plus @ nat) @ K) @ M2) = (((plus_plus @ nat) @ K) @ N2)) = (M2 = N2))))).
% 1.64/1.06 thf(fact_170_nat__add__right__cancel, axiom, (! [M2:nat,K:nat,N2:nat]: ((((((plus_plus @ nat) @ M2) @ K) = (((plus_plus @ nat) @ N2) @ K)) = (M2 = N2))))).
% 1.64/1.06 thf(fact_171_less__imp__add__positive, axiom, (! [I2:nat,J:nat]: (((((ord_less @ nat) @ I2) @ J) => (? [K3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K3) & ((((plus_plus @ nat) @ I2) @ K3) = J)))))))).
% 1.64/1.06 thf(fact_172_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X:A]: (((! [X4:A]: (((~ ((P @ X4))) => (? [Y4:A]: (((((ord_less @ nat) @ (V @ Y4)) @ (V @ X4)) & (~ ((P @ Y4))))))))) => (P @ X))))).
% 1.64/1.06 thf(fact_173_infinite__descent0__measure, axiom, (! [A:$tType,V:(A > nat),P:(A > $o),X:A]: (((! [X4:A]: ((((V @ X4) = (zero_zero @ nat)) => (P @ X4)))) => ((! [X4:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V @ X4)) => ((~ ((P @ X4))) => (? [Y4:A]: (((((ord_less @ nat) @ (V @ Y4)) @ (V @ X4)) & (~ ((P @ Y4)))))))))) => (P @ X)))))).
% 1.64/1.06 thf(fact_174_lfinite__LConsI, axiom, (! [A:$tType,Xs3:(coindu1593790203_llist @ A),X:A]: ((((coindu1213758845finite @ A) @ Xs3) => ((coindu1213758845finite @ A) @ (((coindu1121789889_LCons @ A) @ X) @ Xs3)))))).
% 1.64/1.06 thf(fact_175_lfinite__LNil, axiom, (! [A:$tType]: (((coindu1213758845finite @ A) @ (coindu1598213697e_LNil @ A))))).
% 1.64/1.06 thf(fact_176_lnull__imp__lfinite, axiom, (! [A:$tType,Xs3:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Xs3) => ((coindu1213758845finite @ A) @ Xs3))))).
% 1.64/1.06 thf(fact_177_lnull__def, axiom, (! [A:$tType]: (((coindu335574135_lnull @ A) = (^ [Llist:(coindu1593790203_llist @ A)]: ((Llist = (coindu1598213697e_LNil @ A)))))))).
% 1.64/1.06 thf(fact_178_lfinite_Oinducts, axiom, (! [A:$tType,X:(coindu1593790203_llist @ A),P:((coindu1593790203_llist @ A) > $o)]: ((((coindu1213758845finite @ A) @ X) => ((P @ (coindu1598213697e_LNil @ A)) => ((! [Xs:(coindu1593790203_llist @ A),X4:A]: ((((coindu1213758845finite @ A) @ Xs) => ((P @ Xs) => (P @ (((coindu1121789889_LCons @ A) @ X4) @ Xs)))))) => (P @ X))))))).
% 1.64/1.06 thf(fact_179_lappend_Oexhaust, axiom, (! [A:$tType,Xs3:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((((coindu335574135_lnull @ A) @ Xs3) => (~ (((coindu335574135_lnull @ A) @ Ys)))) => ((~ (((coindu335574135_lnull @ A) @ Xs3))) | (~ (((coindu335574135_lnull @ A) @ Ys)))))))).
% 1.64/1.06 thf(fact_180_not__lnull__conv, axiom, (! [A:$tType,Xs3:(coindu1593790203_llist @ A)]: (((~ (((coindu335574135_lnull @ A) @ Xs3))) = (? [X3:A,Xs4:(coindu1593790203_llist @ A)]: ((Xs3 = (((coindu1121789889_LCons @ A) @ X3) @ Xs4)))))))).
% 1.64/1.06 thf(fact_181_neq__LNil__conv, axiom, (! [A:$tType,Xs3:(coindu1593790203_llist @ A)]: (((Xs3 != (coindu1598213697e_LNil @ A)) = (? [X3:A,Xs4:(coindu1593790203_llist @ A)]: ((Xs3 = (((coindu1121789889_LCons @ A) @ X3) @ Xs4)))))))).
% 1.64/1.06 thf(fact_182_llist_Oexhaust, axiom, (! [A:$tType,Y:(coindu1593790203_llist @ A)]: (((Y != (coindu1598213697e_LNil @ A)) => (~ ((! [X212:A,X222:(coindu1593790203_llist @ A)]: ((Y != (((coindu1121789889_LCons @ A) @ X212) @ X222)))))))))).
% 1.64/1.06 thf(fact_183_lfinite_Osimps, axiom, (! [A:$tType]: (((coindu1213758845finite @ A) = (^ [A4:(coindu1593790203_llist @ A)]: (((A4 = (coindu1598213697e_LNil @ A)) | (? [Xs2:(coindu1593790203_llist @ A),X3:A]: (((A4 = (((coindu1121789889_LCons @ A) @ X3) @ Xs2)) & ((coindu1213758845finite @ A) @ Xs2))))))))))).
% 1.64/1.06 thf(fact_184_lfinite_Ocases, axiom, (! [A:$tType,A2:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ A2) => ((A2 != (coindu1598213697e_LNil @ A)) => (~ ((! [Xs:(coindu1593790203_llist @ A)]: (((? [X4:A]: ((A2 = (((coindu1121789889_LCons @ A) @ X4) @ Xs)))) => (~ (((coindu1213758845finite @ A) @ Xs))))))))))))).
% 1.64/1.06 thf(fact_185_lzip_Oexhaust, axiom, (! [A:$tType,B:$tType,Xs3:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ B)]: (((~ ((((coindu335574135_lnull @ A) @ Xs3) | ((coindu335574135_lnull @ B) @ Ys)))) => (~ (((~ (((coindu335574135_lnull @ A) @ Xs3))) => ((coindu335574135_lnull @ B) @ Ys)))))))).
% 1.64/1.06 thf(fact_186_llist_Odistinct_I1_J, axiom, (! [A:$tType,X21:A,X22:(coindu1593790203_llist @ A)]: (((coindu1598213697e_LNil @ A) != (((coindu1121789889_LCons @ A) @ X21) @ X22))))).
% 1.64/1.06 thf(fact_187_llist_Ocollapse_I1_J, axiom, (! [A:$tType,Llist2:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Llist2) => (Llist2 = (coindu1598213697e_LNil @ A)))))).
% 1.64/1.06 thf(fact_188_llist_OdiscI_I1_J, axiom, (! [A:$tType,Llist2:(coindu1593790203_llist @ A)]: (((Llist2 = (coindu1598213697e_LNil @ A)) => ((coindu335574135_lnull @ A) @ Llist2))))).
% 1.64/1.06 thf(fact_189_llist_OdiscI_I2_J, axiom, (! [A:$tType,Llist2:(coindu1593790203_llist @ A),X21:A,X22:(coindu1593790203_llist @ A)]: (((Llist2 = (((coindu1121789889_LCons @ A) @ X21) @ X22)) => (~ (((coindu335574135_lnull @ A) @ Llist2))))))).
% 1.64/1.06 thf(fact_190_llist_Odisc_I1_J, axiom, (! [A:$tType]: (((coindu335574135_lnull @ A) @ (coindu1598213697e_LNil @ A))))).
% 1.64/1.06 thf(fact_191_llist_Odisc_I2_J, axiom, (! [A:$tType,X21:A,X22:(coindu1593790203_llist @ A)]: ((~ (((coindu335574135_lnull @ A) @ (((coindu1121789889_LCons @ A) @ X21) @ X22))))))).
% 1.64/1.06 thf(fact_192_gen__llength__code_I1_J, axiom, (! [A:$tType,N2:nat]: (((((coindu493225792length @ A) @ N2) @ (coindu1598213697e_LNil @ A)) = (extended_enat2 @ N2))))).
% 1.64/1.06 thf(fact_193_llength__code, axiom, (! [A:$tType]: (((coindu1018505716length @ A) = ((coindu493225792length @ A) @ (zero_zero @ nat)))))).
% 1.64/1.06 thf(fact_194_ltake_Oexhaust, axiom, (! [A:$tType,N2:extended_enat,Xs3:(coindu1593790203_llist @ A)]: (((~ (((N2 = (zero_zero @ extended_enat)) | ((coindu335574135_lnull @ A) @ Xs3)))) => (~ (((N2 != (zero_zero @ extended_enat)) => ((coindu335574135_lnull @ A) @ Xs3)))))))).
% 1.64/1.06 thf(fact_195_nat_Odistinct_I1_J, axiom, (! [X2:nat]: (((zero_zero @ nat) != (suc @ X2))))).
% 1.64/1.06 thf(fact_196_old_Onat_Odistinct_I2_J, axiom, (! [Nat3:nat]: (((suc @ Nat3) != (zero_zero @ nat))))).
% 1.64/1.06 thf(fact_197_old_Onat_Odistinct_I1_J, axiom, (! [Nat2:nat]: (((zero_zero @ nat) != (suc @ Nat2))))).
% 1.64/1.06 thf(fact_198_nat_OdiscI, axiom, (! [Nat:nat,X2:nat]: (((Nat = (suc @ X2)) => (Nat != (zero_zero @ nat)))))).
% 1.64/1.06 thf(fact_199_nat__induct, axiom, (! [P:(nat > $o),N2:nat]: (((P @ (zero_zero @ nat)) => ((! [N4:nat]: (((P @ N4) => (P @ (suc @ N4))))) => (P @ N2)))))).
% 1.64/1.06 thf(fact_200_diff__induct, axiom, (! [P:(nat > (nat > $o)),M2:nat,N2:nat]: (((! [X4:nat]: (((P @ X4) @ (zero_zero @ nat)))) => ((! [Y5:nat]: (((P @ (zero_zero @ nat)) @ (suc @ Y5)))) => ((! [X4:nat,Y5:nat]: ((((P @ X4) @ Y5) => ((P @ (suc @ X4)) @ (suc @ Y5))))) => ((P @ M2) @ N2))))))).
% 1.64/1.06 thf(fact_201_zero__induct, axiom, (! [P:(nat > $o),K:nat]: (((P @ K) => ((! [N4:nat]: (((P @ (suc @ N4)) => (P @ N4)))) => (P @ (zero_zero @ nat))))))).
% 1.64/1.06 thf(fact_202_Suc__neq__Zero, axiom, (! [M2:nat]: (((suc @ M2) != (zero_zero @ nat))))).
% 1.64/1.06 thf(fact_203_Zero__neq__Suc, axiom, (! [M2:nat]: (((zero_zero @ nat) != (suc @ M2))))).
% 1.64/1.06 thf(fact_204_Zero__not__Suc, axiom, (! [M2:nat]: (((zero_zero @ nat) != (suc @ M2))))).
% 1.64/1.06 thf(fact_205_old_Onat_Oexhaust, axiom, (! [Y:nat]: (((Y != (zero_zero @ nat)) => (~ ((! [Nat4:nat]: ((Y != (suc @ Nat4)))))))))).
% 1.64/1.06 thf(fact_206_old_Onat_Oinducts, axiom, (! [P:(nat > $o),Nat:nat]: (((P @ (zero_zero @ nat)) => ((! [Nat4:nat]: (((P @ Nat4) => (P @ (suc @ Nat4))))) => (P @ Nat)))))).
% 1.64/1.06 thf(fact_207_not0__implies__Suc, axiom, (! [N2:nat]: (((N2 != (zero_zero @ nat)) => (? [M:nat]: ((N2 = (suc @ M)))))))).
% 1.64/1.06 thf(fact_208_less__Suc__eq__0__disj, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ (suc @ N2)) = ((M2 = (zero_zero @ nat)) | (? [J4:nat]: (((M2 = (suc @ J4)) & (((ord_less @ nat) @ J4) @ N2))))))))).
% 1.64/1.06 thf(fact_209_gr0__implies__Suc, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (? [M:nat]: ((N2 = (suc @ M)))))))).
% 1.64/1.06 thf(fact_210_gr0__conv__Suc, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) = (? [M3:nat]: ((N2 = (suc @ M3)))))))).
% 1.64/1.06 thf(fact_211_one__is__add, axiom, (! [M2:nat,N2:nat]: ((((suc @ (zero_zero @ nat)) = (((plus_plus @ nat) @ M2) @ N2)) = (((M2 = (suc @ (zero_zero @ nat))) & (N2 = (zero_zero @ nat))) | ((M2 = (zero_zero @ nat)) & (N2 = (suc @ (zero_zero @ nat))))))))).
% 1.64/1.06 thf(fact_212_add__is__1, axiom, (! [M2:nat,N2:nat]: ((((((plus_plus @ nat) @ M2) @ N2) = (suc @ (zero_zero @ nat))) = (((M2 = (suc @ (zero_zero @ nat))) & (N2 = (zero_zero @ nat))) | ((M2 = (zero_zero @ nat)) & (N2 = (suc @ (zero_zero @ nat))))))))).
% 1.64/1.06 thf(fact_213_setsum__cong__Suc, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [A3:(set @ nat),F:(nat > A),G:(nat > A)]: (((~ ((((member @ nat) @ (zero_zero @ nat)) @ A3))) => ((! [X4:nat]: (((((member @ nat) @ (suc @ X4)) @ A3) => ((F @ (suc @ X4)) = (G @ (suc @ X4)))))) => (((((groups15040474setsum @ nat) @ A) @ F) @ A3) = ((((groups15040474setsum @ nat) @ A) @ G) @ A3)))))))))).
% 1.64/1.06 thf(fact_214_lfinite__llength__enat, axiom, (! [A:$tType,Xs3:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ Xs3) => (? [N4:nat]: ((((coindu1018505716length @ A) @ Xs3) = (extended_enat2 @ N4)))))))).
% 1.64/1.06 thf(fact_215_llength__eq__enat__lfiniteD, axiom, (! [A:$tType,Xs3:(coindu1593790203_llist @ A),N2:nat]: (((((coindu1018505716length @ A) @ Xs3) = (extended_enat2 @ N2)) => ((coindu1213758845finite @ A) @ Xs3))))).
% 1.64/1.06 thf(fact_216_lfinite__conv__llength__enat, axiom, (! [A:$tType]: (((coindu1213758845finite @ A) = (^ [Xs2:(coindu1593790203_llist @ A)]: ((? [N3:nat]: ((((coindu1018505716length @ A) @ Xs2) = (extended_enat2 @ N3)))))))))).
% 1.64/1.06 thf(fact_217_zero__enat__def, axiom, ((zero_zero @ extended_enat) = (extended_enat2 @ (zero_zero @ nat)))).
% 1.64/1.06 thf(fact_218_enat__0__iff_I1_J, axiom, (! [X:nat]: ((((extended_enat2 @ X) = (zero_zero @ extended_enat)) = (X = (zero_zero @ nat)))))).
% 1.64/1.06 thf(fact_219_enat__0__iff_I2_J, axiom, (! [X:nat]: ((((zero_zero @ extended_enat) = (extended_enat2 @ X)) = (X = (zero_zero @ nat)))))).
% 1.64/1.06 thf(fact_220_semiring__normalization__rules_I25_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [A2:A,C:A,D:A]: (((((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ C) @ D)) = (((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ C)) @ D)))))))).
% 1.64/1.06 thf(fact_221_semiring__normalization__rules_I24_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => ((plus_plus @ A) = (^ [A4:A,C2:A]: ((((plus_plus @ A) @ C2) @ A4)))))))).
% 1.64/1.06 thf(fact_222_semiring__normalization__rules_I23_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C) = (((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ C)) @ B2)))))))).
% 1.64/1.06 thf(fact_223_semiring__normalization__rules_I22_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [A2:A,C:A,D:A]: (((((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ C) @ D)) = (((plus_plus @ A) @ C) @ (((plus_plus @ A) @ A2) @ D))))))))).
% 1.64/1.06 thf(fact_224_semiring__normalization__rules_I21_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ 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))))))))).
% 1.64/1.06 thf(fact_225_semiring__normalization__rules_I20_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A,D:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (((plus_plus @ A) @ C) @ D)) = (((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ D))))))))).
% 1.64/1.06 thf(fact_226_setsum_Ocong, axiom, (! [A:$tType,B:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [A3:(set @ B),B4:(set @ B),G:(B > A),H:(B > A)]: (((A3 = B4) => ((! [X4:B]: (((((member @ B) @ X4) @ B4) => ((G @ X4) = (H @ X4))))) => (((((groups15040474setsum @ B) @ A) @ G) @ A3) = ((((groups15040474setsum @ B) @ A) @ H) @ B4)))))))))).
% 1.64/1.06 thf(fact_227_setsum_Oreindex__bij__witness, axiom, (! [B:$tType,A:$tType,C3:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [S2:(set @ B),I2:(C3 > B),J:(B > C3),T2:(set @ C3),H:(C3 > A),G:(B > A)]: (((! [A5:B]: (((((member @ B) @ A5) @ S2) => ((I2 @ (J @ A5)) = A5)))) => ((! [A5:B]: (((((member @ B) @ A5) @ S2) => (((member @ C3) @ (J @ A5)) @ T2)))) => ((! [B5:C3]: (((((member @ C3) @ B5) @ T2) => ((J @ (I2 @ B5)) = B5)))) => ((! [B5:C3]: (((((member @ C3) @ B5) @ T2) => (((member @ B) @ (I2 @ B5)) @ S2)))) => ((! [A5:B]: (((((member @ B) @ A5) @ S2) => ((H @ (J @ A5)) = (G @ A5))))) => (((((groups15040474setsum @ B) @ A) @ G) @ S2) = ((((groups15040474setsum @ C3) @ A) @ H) @ T2))))))))))))).
% 1.64/1.06 thf(fact_228_setsum_Ocommute, axiom, (! [A:$tType,B:$tType,C3:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [G:(B > (C3 > A)),B4:(set @ C3),A3:(set @ B)]: ((((((groups15040474setsum @ B) @ A) @ (^ [I:B]: (((((groups15040474setsum @ C3) @ A) @ (G @ I)) @ B4)))) @ A3) = ((((groups15040474setsum @ C3) @ A) @ (^ [J4:C3]: (((((groups15040474setsum @ B) @ A) @ (^ [I:B]: (((G @ I) @ J4)))) @ A3)))) @ B4)))))))).
% 1.64/1.06 thf(fact_229_setsum__lessThan__Suc__shift, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) @ (type2 @ A)) => (! [F:(nat > A),N2:nat]: ((((((groups15040474setsum @ nat) @ A) @ F) @ ((set_ord_lessThan @ nat) @ (suc @ N2))) = (((plus_plus @ A) @ (F @ (zero_zero @ nat))) @ ((((groups15040474setsum @ nat) @ A) @ (^ [I:nat]: ((F @ (suc @ I))))) @ ((set_ord_lessThan @ nat) @ N2)))))))))).
% 1.64/1.06 thf(fact_230_semiring__normalization__rules_I6_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 1.64/1.06 thf(fact_231_semiring__normalization__rules_I5_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 1.64/1.06 thf(fact_232_concat__xss_H, axiom, ((coindu335574135_lnull @ a) @ ((coindu441856546concat @ a) @ ((coindu1079202755ist_of @ (coindu1593790203_llist @ a)) @ xss2)))).
% 1.64/1.06 thf(fact_233_enat_Osize_I1_J, axiom, (! [Nat:nat]: (((extended_size_enat @ (extended_enat2 @ Nat)) = (zero_zero @ nat))))).
% 1.64/1.06 thf(fact_234_length__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs3:(list @ A)]: (((! [Xs:(list @ A)]: (((! [Ys2:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys2)) @ ((size_size @ (list @ A)) @ Xs)) => (P @ Ys2)))) => (P @ Xs)))) => (P @ Xs3))))).
% 1.64/1.06 thf(fact_235_lnth__ldropn, axiom, (! [A:$tType,N2:nat,M2:nat,Xs3:(coindu1593790203_llist @ A)]: (((((ord_less @ extended_enat) @ (extended_enat2 @ (((plus_plus @ nat) @ N2) @ M2))) @ ((coindu1018505716length @ A) @ Xs3)) => ((((coindu749330388e_lnth @ A) @ (((coindu531130065ldropn @ A) @ N2) @ Xs3)) @ M2) = (((coindu749330388e_lnth @ A) @ Xs3) @ (((plus_plus @ nat) @ M2) @ N2))))))).
% 1.64/1.06 thf(fact_236_llist__of__inject, axiom, (! [A:$tType,Xs3:(list @ A),Ys:(list @ A)]: (((((coindu1079202755ist_of @ A) @ Xs3) = ((coindu1079202755ist_of @ A) @ Ys)) = (Xs3 = Ys))))).
% 1.64/1.06 thf(fact_237_ldropn__0, axiom, (! [A:$tType,Xs3:(coindu1593790203_llist @ A)]: (((((coindu531130065ldropn @ A) @ (zero_zero @ nat)) @ Xs3) = Xs3)))).
% 1.64/1.06 thf(fact_238_ldropn__ldropn, axiom, (! [A:$tType,N2:nat,M2:nat,Xs3:(coindu1593790203_llist @ A)]: (((((coindu531130065ldropn @ A) @ N2) @ (((coindu531130065ldropn @ A) @ M2) @ Xs3)) = (((coindu531130065ldropn @ A) @ (((plus_plus @ nat) @ N2) @ M2)) @ Xs3))))).
% 1.64/1.06 thf(fact_239_ldropn__LNil, axiom, (! [A:$tType,N2:nat]: (((((coindu531130065ldropn @ A) @ N2) @ (coindu1598213697e_LNil @ A)) = (coindu1598213697e_LNil @ A))))).
% 1.64/1.06 thf(fact_240_lfinite__ldropn, axiom, (! [A:$tType,N2:nat,Xs3:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ (((coindu531130065ldropn @ A) @ N2) @ Xs3)) = ((coindu1213758845finite @ A) @ Xs3))))).
% 1.64/1.06 thf(fact_241_ldropn__Suc__LCons, axiom, (! [A:$tType,N2:nat,X:A,Xs3:(coindu1593790203_llist @ A)]: (((((coindu531130065ldropn @ A) @ (suc @ N2)) @ (((coindu1121789889_LCons @ A) @ X) @ Xs3)) = (((coindu531130065ldropn @ A) @ N2) @ Xs3))))).
% 1.64/1.06 thf(fact_242__092_060open_062m_A_061_A0_092_060close_062, axiom, (m = (zero_zero @ nat))).
% 1.64/1.06 thf(fact_243_llength__llist__of, axiom, (! [A:$tType,Xs3:(list @ A)]: ((((coindu1018505716length @ A) @ ((coindu1079202755ist_of @ A) @ Xs3)) = (extended_enat2 @ ((size_size @ (list @ A)) @ Xs3)))))).
% 1.64/1.06 thf(fact_244_lfinite__llist__of, axiom, (! [A:$tType,Xs3:(list @ A)]: (((coindu1213758845finite @ A) @ ((coindu1079202755ist_of @ A) @ Xs3))))).
% 1.64/1.06 thf(fact_245_ldropn__lnull, axiom, (! [A:$tType,Xs3:(coindu1593790203_llist @ A),N2:nat]: ((((coindu335574135_lnull @ A) @ Xs3) => ((((coindu531130065ldropn @ A) @ N2) @ Xs3) = (coindu1598213697e_LNil @ A)))))).
% 1.64/1.06 thf(fact_246_Ex__list__of__length, axiom, (! [A:$tType,N2:nat]: ((? [Xs:(list @ A)]: ((((size_size @ (list @ A)) @ Xs) = N2)))))).
% 1.64/1.06 thf(fact_247_neq__if__length__neq, axiom, (! [A:$tType,Xs3:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) != ((size_size @ (list @ A)) @ Ys)) => (Xs3 != Ys))))).
% 1.64/1.06 thf(fact_248_ldropn__eq__LConsD, axiom, (! [A:$tType,N2:nat,Xs3:(coindu1593790203_llist @ A),Y:A,Ys:(coindu1593790203_llist @ A)]: ((((((coindu531130065ldropn @ A) @ N2) @ Xs3) = (((coindu1121789889_LCons @ A) @ Y) @ Ys)) => (((ord_less @ extended_enat) @ (extended_enat2 @ N2)) @ ((coindu1018505716length @ A) @ Xs3)))))).
% 1.64/1.06 thf(fact_249_ldropn__Suc__conv__ldropn, axiom, (! [A:$tType,N2:nat,Xs3:(coindu1593790203_llist @ A)]: (((((ord_less @ extended_enat) @ (extended_enat2 @ N2)) @ ((coindu1018505716length @ A) @ Xs3)) => ((((coindu1121789889_LCons @ A) @ (((coindu749330388e_lnth @ A) @ Xs3) @ N2)) @ (((coindu531130065ldropn @ A) @ (suc @ N2)) @ Xs3)) = (((coindu531130065ldropn @ A) @ N2) @ Xs3)))))).
% 1.64/1.06 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: ((((order @ A7) @ (type2 @ A7)) => ((order @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 1.64/1.06 thf(tcon_fun___Orderings_Oord, axiom, (! [A6:$tType,A7:$tType]: ((((ord @ A7) @ (type2 @ A7)) => ((ord @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 1.64/1.06 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, axiom, ((semiri456707255roduct @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le, axiom, ((ordere516151231imp_le @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add, axiom, ((strict2144017051up_add @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add, axiom, ((ordere223160158up_add @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, axiom, ((ordere236663937imp_le @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add, axiom, ((strict797366125id_add @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, axiom, ((ordere779506340up_add @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add, axiom, ((ordere216010020id_add @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, axiom, ((cancel1352612707id_add @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add, axiom, ((cancel_semigroup_add @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add, axiom, ((ab_semigroup_add @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add, axiom, ((comm_monoid_add @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1, axiom, ((comm_semiring_1 @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Osemigroup__add, axiom, ((semigroup_add @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Omonoid__add, axiom, ((monoid_add @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Orderings_Oorder_1, axiom, ((order @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Orderings_Oord_2, axiom, ((ord @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Nat_Onat___Groups_Ozero, axiom, ((zero @ nat) @ (type2 @ nat))).
% 1.64/1.06 thf(tcon_Set_Oset___Orderings_Oorder_3, axiom, (! [A6:$tType]: (((order @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 1.64/1.06 thf(tcon_Set_Oset___Orderings_Oord_4, axiom, (! [A6:$tType]: (((ord @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 1.64/1.06 thf(tcon_HOL_Obool___Orderings_Olinorder_5, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 1.64/1.06 thf(tcon_HOL_Obool___Orderings_Oorder_6, axiom, ((order @ $o) @ (type2 @ $o))).
% 1.64/1.06 thf(tcon_HOL_Obool___Orderings_Oord_7, axiom, ((ord @ $o) @ (type2 @ $o))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_8, axiom, ((strict2144017051up_add @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_9, axiom, ((strict797366125id_add @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_10, axiom, ((canoni770627133id_add @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_11, axiom, ((ordere779506340up_add @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_12, axiom, ((ordere216010020id_add @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_13, axiom, ((ab_semigroup_add @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_14, axiom, ((comm_monoid_add @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_15, axiom, ((comm_semiring_1 @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_16, axiom, ((semigroup_add @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_17, axiom, ((linorder @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_18, axiom, ((monoid_add @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_19, axiom, ((order @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_20, axiom, ((ord @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_21, axiom, ((zero @ extended_enat) @ (type2 @ extended_enat))).
% 1.64/1.06 thf(conj_0, conjecture, (? [M5:nat,N6:nat]: ((((((coindu749330388e_lnth @ a) @ ((coindu441856546concat @ a) @ xssa)) @ (suc @ na)) = (((coindu749330388e_lnth @ a) @ (((coindu749330388e_lnth @ (coindu1593790203_llist @ a)) @ xssa) @ M5)) @ N6)) & ((((ord_less @ extended_enat) @ (extended_enat2 @ N6)) @ ((coindu1018505716length @ a) @ (((coindu749330388e_lnth @ (coindu1593790203_llist @ a)) @ xssa) @ M5))) & ((((ord_less @ extended_enat) @ (extended_enat2 @ M5)) @ ((coindu1018505716length @ (coindu1593790203_llist @ a)) @ xssa)) & ((extended_enat2 @ (suc @ na)) = (((plus_plus @ extended_enat) @ ((((groups15040474setsum @ nat) @ extended_enat) @ (^ [I:nat]: (((coindu1018505716length @ a) @ (((coindu749330388e_lnth @ (coindu1593790203_llist @ a)) @ xssa) @ I))))) @ ((set_ord_lessThan @ nat) @ M5))) @ (extended_enat2 @ N6))))))))).
% 1.64/1.06 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Bd5MKNrlUu/DTF2DTF19363.p
% 1.64/1.06 ------------------------
% 1.64/1.08 ---- Cleaned THF ---
% 1.64/1.08 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
% 1.64/1.08 coindu1593790203_llist: $tType > $tType ).
% 1.64/1.08
% 1.64/1.08 thf(ty_t_Extended__Nat_Oenat,type,
% 1.64/1.08 extended_enat: $tType ).
% 1.64/1.08
% 1.64/1.08 thf(ty_t_List_Olist,type,
% 1.64/1.08 list: $tType > $tType ).
% 1.64/1.08
% 1.64/1.08 thf(ty_t_Set_Oset,type,
% 1.64/1.08 set: $tType > $tType ).
% 1.64/1.08
% 1.64/1.08 thf(ty_t_Nat_Onat,type,
% 1.64/1.08 nat: $tType ).
% 1.64/1.08
% 1.64/1.08 thf(ty_t_itself,type,
% 1.64/1.08 itself: $tType > $tType ).
% 1.64/1.08
% 1.64/1.08 thf(ty_tf_a,type,
% 1.64/1.08 a: $tType ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_HOL_Otype,type,
% 1.64/1.08 type:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Ozero,type,
% 1.64/1.08 zero:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Orderings_Oord,type,
% 1.64/1.08 ord:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Orderings_Oorder,type,
% 1.64/1.08 order:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Ogroup__add,type,
% 1.64/1.08 group_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Omonoid__add,type,
% 1.64/1.08 monoid_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Orderings_Olinorder,type,
% 1.64/1.08 linorder:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Osemigroup__add,type,
% 1.64/1.08 semigroup_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 1.64/1.08 comm_semiring_1:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 1.64/1.08 comm_monoid_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 1.64/1.08 ab_semigroup_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 1.64/1.08 cancel_semigroup_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 1.64/1.08 cancel1352612707id_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 1.64/1.08 linord219039673up_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 1.64/1.08 ordere216010020id_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 1.64/1.08 ordere779506340up_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 1.64/1.08 canoni770627133id_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
% 1.64/1.08 strict797366125id_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 1.64/1.08 ordere236663937imp_le:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 1.64/1.08 ordere223160158up_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 1.64/1.08 strict2144017051up_add:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 1.64/1.08 ordere516151231imp_le:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
% 1.64/1.08 semiri456707255roduct:
% 1.64/1.08 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__llength,type,
% 1.64/1.08 coindu493225792length:
% 1.64/1.08 !>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olconcat,type,
% 1.64/1.08 coindu441856546concat:
% 1.64/1.08 !>[A: $tType] : ( ( coindu1593790203_llist @ ( coindu1593790203_llist @ A ) ) > ( coindu1593790203_llist @ A ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldropn,type,
% 1.64/1.08 coindu531130065ldropn:
% 1.64/1.08 !>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite,type,
% 1.64/1.08 coindu1213758845finite:
% 1.64/1.08 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
% 1.64/1.08 coindu1018505716length:
% 1.64/1.08 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLCons,type,
% 1.64/1.08 coindu1121789889_LCons:
% 1.64/1.08 !>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil,type,
% 1.64/1.08 coindu1598213697e_LNil:
% 1.64/1.08 !>[A: $tType] : ( coindu1593790203_llist @ A ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
% 1.64/1.08 coindu335574135_lnull:
% 1.64/1.08 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist__of,type,
% 1.64/1.08 coindu1079202755ist_of:
% 1.64/1.08 !>[A: $tType] : ( ( list @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olnth,type,
% 1.64/1.08 coindu749330388e_lnth:
% 1.64/1.08 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > nat > A ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Extended__Nat_Oenat,type,
% 1.64/1.08 extended_enat2: nat > extended_enat ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Extended__Nat_Oenat_Osize__enat,type,
% 1.64/1.08 extended_size_enat: extended_enat > nat ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 1.64/1.08 plus_plus:
% 1.64/1.08 !>[A: $tType] : ( A > A > A ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 1.64/1.08 zero_zero:
% 1.64/1.08 !>[A: $tType] : A ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osetsum,type,
% 1.64/1.08 groups15040474setsum:
% 1.64/1.08 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Nat_OSuc,type,
% 1.64/1.08 suc: nat > nat ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Nat_Osize__class_Osize,type,
% 1.64/1.08 size_size:
% 1.64/1.08 !>[A: $tType] : ( A > nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Orderings_Oord__class_Oless,type,
% 1.64/1.08 ord_less:
% 1.64/1.08 !>[A: $tType] : ( A > A > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Pure_Otype,type,
% 1.64/1.08 type2:
% 1.64/1.08 !>[A: $tType] : ( itself @ A ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Set_OCollect,type,
% 1.64/1.08 collect:
% 1.64/1.08 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
% 1.64/1.08 set_ord_lessThan:
% 1.64/1.08 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_c_member,type,
% 1.64/1.08 member:
% 1.64/1.08 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_v_N____,type,
% 1.64/1.08 n: nat ).
% 1.64/1.08
% 1.64/1.08 thf(sy_v_m____,type,
% 1.64/1.08 m: nat ).
% 1.64/1.08
% 1.64/1.08 thf(sy_v_n,type,
% 1.64/1.08 n2: nat ).
% 1.64/1.08
% 1.64/1.08 thf(sy_v_n_H____,type,
% 1.64/1.08 n3: nat ).
% 1.64/1.08
% 1.64/1.08 thf(sy_v_na____,type,
% 1.64/1.08 na: nat ).
% 1.64/1.08
% 1.64/1.08 thf(sy_v_x____,type,
% 1.64/1.08 x: a ).
% 1.64/1.08
% 1.64/1.08 thf(sy_v_xs____,type,
% 1.64/1.08 xs: coindu1593790203_llist @ a ).
% 1.64/1.08
% 1.64/1.08 thf(sy_v_xss,type,
% 1.64/1.08 xss: coindu1593790203_llist @ ( coindu1593790203_llist @ a ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_v_xss_H____,type,
% 1.64/1.08 xss2: list @ ( coindu1593790203_llist @ a ) ).
% 1.64/1.08
% 1.64/1.08 thf(sy_v_xssa____,type,
% 1.64/1.08 xssa: coindu1593790203_llist @ ( coindu1593790203_llist @ a ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_0__092_060open_062enat_A_Ilength_Axss_H_J_A_060_Allength_Axss_092_060close_062,axiom,
% 1.64/1.08 ord_less @ extended_enat @ ( extended_enat2 @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ a ) @ xssa ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_1_assms,axiom,
% 1.64/1.08 ord_less @ extended_enat @ ( extended_enat2 @ n2 ) @ ( coindu1018505716length @ a @ ( coindu441856546concat @ a @ xss ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_2__092_060open_062enat_A_ISuc_An_H_J_A_060_Allength_A_Ilnth_Axss_A_Ilength_Axss_H_J_J_092_060close_062,axiom,
% 1.64/1.08 ord_less @ extended_enat @ ( extended_enat2 @ ( suc @ n3 ) ) @ ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_3_Suc_Oprems,axiom,
% 1.64/1.08 ord_less @ extended_enat @ ( extended_enat2 @ ( suc @ na ) ) @ ( coindu1018505716length @ a @ ( coindu441856546concat @ a @ xssa ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_4__092_060open_062lnth_A_Ilconcat_Axss_J_A_ISuc_An_J_A_061_Alnth_A_Ilnth_Axss_A_Ilength_Axss_H_J_J_A_ISuc_An_H_J_092_060close_062,axiom,
% 1.64/1.08 ( ( coindu749330388e_lnth @ a @ ( coindu441856546concat @ a @ xssa ) @ ( suc @ na ) )
% 1.64/1.08 = ( coindu749330388e_lnth @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) @ ( suc @ n3 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_5__092_060open_062enat_A_ISuc_An_J_A_061_A_I_092_060Sum_062i_060length_Axss_H_O_Allength_A_Ilnth_Axss_Ai_J_J_A_L_Aenat_A_ISuc_An_H_J_092_060close_062,axiom,
% 1.64/1.08 ( ( extended_enat2 @ ( suc @ na ) )
% 1.64/1.08 = ( plus_plus @ extended_enat
% 1.64/1.08 @ ( groups15040474setsum @ nat @ extended_enat
% 1.64/1.08 @ ^ [I: nat] : ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ I ) )
% 1.64/1.08 @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) )
% 1.64/1.08 @ ( extended_enat2 @ ( suc @ n3 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_6_Suc_Ohyps,axiom,
% 1.64/1.08 ! [Xss: coindu1593790203_llist @ ( coindu1593790203_llist @ a )] :
% 1.64/1.08 ( ( ord_less @ extended_enat @ ( extended_enat2 @ na ) @ ( coindu1018505716length @ a @ ( coindu441856546concat @ a @ Xss ) ) )
% 1.64/1.08 => ? [M: nat,N: nat] :
% 1.64/1.08 ( ( ( coindu749330388e_lnth @ a @ ( coindu441856546concat @ a @ Xss ) @ na )
% 1.64/1.08 = ( coindu749330388e_lnth @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ Xss @ M ) @ N ) )
% 1.64/1.08 & ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ Xss @ M ) ) )
% 1.64/1.08 & ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ a ) @ Xss ) )
% 1.64/1.08 & ( ( extended_enat2 @ na )
% 1.64/1.08 = ( plus_plus @ extended_enat
% 1.64/1.08 @ ( groups15040474setsum @ nat @ extended_enat
% 1.64/1.08 @ ^ [I: nat] : ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ Xss @ I ) )
% 1.64/1.08 @ ( set_ord_lessThan @ nat @ M ) )
% 1.64/1.08 @ ( extended_enat2 @ N ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_7_setsum__lessThan__Suc,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [F: nat > A,N2: nat] :
% 1.64/1.08 ( ( groups15040474setsum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
% 1.64/1.08 = ( plus_plus @ A @ ( groups15040474setsum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( F @ N2 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_8_enat__add__mono,axiom,
% 1.64/1.08 ! [X: nat,Y: extended_enat,Z: extended_enat] :
% 1.64/1.08 ( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
% 1.64/1.08 = ( ord_less @ extended_enat @ Y @ Z ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_9_enat__add1__eq,axiom,
% 1.64/1.08 ! [X: nat,Y: extended_enat,Z: extended_enat] :
% 1.64/1.08 ( ( ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y )
% 1.64/1.08 = ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
% 1.64/1.08 = ( Y = Z ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_10_enat__add2__eq,axiom,
% 1.64/1.08 ! [Y: extended_enat,X: nat,Z: extended_enat] :
% 1.64/1.08 ( ( ( plus_plus @ extended_enat @ Y @ ( extended_enat2 @ X ) )
% 1.64/1.08 = ( plus_plus @ extended_enat @ Z @ ( extended_enat2 @ X ) ) )
% 1.64/1.08 = ( Y = Z ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_11_lessThan__iff,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ord @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [I2: A,K: A] :
% 1.64/1.08 ( ( member @ A @ I2 @ ( set_ord_lessThan @ A @ K ) )
% 1.64/1.08 = ( ord_less @ A @ I2 @ K ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_12_add__less__cancel__left,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [C: A,A2: A,B2: A] :
% 1.64/1.08 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 1.64/1.08 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_13_add__less__cancel__right,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,C: A,B2: A] :
% 1.64/1.08 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 1.64/1.08 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_14__092_060open_062_I_092_060Sum_062i_060length_Axss_H_O_Allength_A_Ilnth_Axss_Ai_J_J_A_061_A0_092_060close_062,axiom,
% 1.64/1.08 ( ( groups15040474setsum @ nat @ extended_enat
% 1.64/1.08 @ ^ [I: nat] : ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ I ) )
% 1.64/1.08 @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) )
% 1.64/1.08 = ( zero_zero @ extended_enat ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_15_enat_Oinject,axiom,
% 1.64/1.08 ! [Nat: nat,Nat2: nat] :
% 1.64/1.08 ( ( ( extended_enat2 @ Nat )
% 1.64/1.08 = ( extended_enat2 @ Nat2 ) )
% 1.64/1.08 = ( Nat = Nat2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_16_lessThan__eq__iff,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( linorder @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [X: A,Y: A] :
% 1.64/1.08 ( ( ( set_ord_lessThan @ A @ X )
% 1.64/1.08 = ( set_ord_lessThan @ A @ Y ) )
% 1.64/1.08 = ( X = Y ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_17_nat_Oinject,axiom,
% 1.64/1.08 ! [X2: nat,Y2: nat] :
% 1.64/1.08 ( ( ( suc @ X2 )
% 1.64/1.08 = ( suc @ Y2 ) )
% 1.64/1.08 = ( X2 = Y2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_18_n,axiom,
% 1.64/1.08 ( na
% 1.64/1.08 = ( plus_plus @ nat @ n @ n3 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_19_add__right__cancel,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [B2: A,A2: A,C: A] :
% 1.64/1.08 ( ( ( plus_plus @ A @ B2 @ A2 )
% 1.64/1.08 = ( plus_plus @ A @ C @ A2 ) )
% 1.64/1.08 = ( B2 = C ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_20_add__left__cancel,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A,C: A] :
% 1.64/1.08 ( ( ( plus_plus @ A @ A2 @ B2 )
% 1.64/1.08 = ( plus_plus @ A @ A2 @ C ) )
% 1.64/1.08 = ( B2 = C ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_21_add__Suc__right,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( plus_plus @ nat @ M2 @ ( suc @ N2 ) )
% 1.64/1.08 = ( suc @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_22_Suc__less__eq,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
% 1.64/1.08 = ( ord_less @ nat @ M2 @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_23_Suc__mono,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ M2 @ N2 )
% 1.64/1.08 => ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_24_lessI,axiom,
% 1.64/1.08 ! [N2: nat] : ( ord_less @ nat @ N2 @ ( suc @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_25_old_Onat_Oinject,axiom,
% 1.64/1.08 ! [Nat: nat,Nat2: nat] :
% 1.64/1.08 ( ( ( suc @ Nat )
% 1.64/1.08 = ( suc @ Nat2 ) )
% 1.64/1.08 = ( Nat = Nat2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_26_not__gr__zero,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [N2: A] :
% 1.64/1.08 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
% 1.64/1.08 = ( N2
% 1.64/1.08 = ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_27_add__cancel__right__right,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A] :
% 1.64/1.08 ( ( A2
% 1.64/1.08 = ( plus_plus @ A @ A2 @ B2 ) )
% 1.64/1.08 = ( B2
% 1.64/1.08 = ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_28_add__cancel__right__left,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A] :
% 1.64/1.08 ( ( A2
% 1.64/1.08 = ( plus_plus @ A @ B2 @ A2 ) )
% 1.64/1.08 = ( B2
% 1.64/1.08 = ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_29_add__cancel__left__right,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A] :
% 1.64/1.08 ( ( ( plus_plus @ A @ A2 @ B2 )
% 1.64/1.08 = A2 )
% 1.64/1.08 = ( B2
% 1.64/1.08 = ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_30_add__cancel__left__left,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [B2: A,A2: A] :
% 1.64/1.08 ( ( ( plus_plus @ A @ B2 @ A2 )
% 1.64/1.08 = A2 )
% 1.64/1.08 = ( B2
% 1.64/1.08 = ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_31_double__zero__sym,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A] :
% 1.64/1.08 ( ( ( zero_zero @ A )
% 1.64/1.08 = ( plus_plus @ A @ A2 @ A2 ) )
% 1.64/1.08 = ( A2
% 1.64/1.08 = ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_32_double__zero,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A] :
% 1.64/1.08 ( ( ( plus_plus @ A @ A2 @ A2 )
% 1.64/1.08 = ( zero_zero @ A ) )
% 1.64/1.08 = ( A2
% 1.64/1.08 = ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_33_add_Oright__neutral,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A] :
% 1.64/1.08 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 1.64/1.08 = A2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_34_add_Oleft__neutral,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A] :
% 1.64/1.08 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 1.64/1.08 = A2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_35_enat__ord__simps_I2_J,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
% 1.64/1.08 = ( ord_less @ nat @ M2 @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_36_i0__less,axiom,
% 1.64/1.08 ! [N2: extended_enat] :
% 1.64/1.08 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 )
% 1.64/1.08 = ( N2
% 1.64/1.08 != ( zero_zero @ extended_enat ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_37_plus__enat__simps_I1_J,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( plus_plus @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
% 1.64/1.08 = ( extended_enat2 @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_38_zero__less__double__add__iff__zero__less__single__add,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A] :
% 1.64/1.08 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 1.64/1.08 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_39_double__add__less__zero__iff__single__add__less__zero,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A] :
% 1.64/1.08 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 1.64/1.08 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_40_less__add__same__cancel2,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A] :
% 1.64/1.08 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
% 1.64/1.08 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_41_less__add__same__cancel1,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A] :
% 1.64/1.08 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 1.64/1.08 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_42_add__less__same__cancel2,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A] :
% 1.64/1.08 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 1.64/1.08 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_43_mem__Collect__eq,axiom,
% 1.64/1.08 ! [A: $tType,A2: A,P: A > $o] :
% 1.64/1.08 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 1.64/1.08 = ( P @ A2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_44_Collect__mem__eq,axiom,
% 1.64/1.08 ! [A: $tType,A3: set @ A] :
% 1.64/1.08 ( ( collect @ A
% 1.64/1.08 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 1.64/1.08 = A3 ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_45_Collect__cong,axiom,
% 1.64/1.08 ! [A: $tType,P: A > $o,Q: A > $o] :
% 1.64/1.08 ( ! [X4: A] :
% 1.64/1.08 ( ( P @ X4 )
% 1.64/1.08 = ( Q @ X4 ) )
% 1.64/1.08 => ( ( collect @ A @ P )
% 1.64/1.08 = ( collect @ A @ Q ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_46_ext,axiom,
% 1.64/1.08 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 1.64/1.08 ( ! [X4: A] :
% 1.64/1.08 ( ( F @ X4 )
% 1.64/1.08 = ( G @ X4 ) )
% 1.64/1.08 => ( F = G ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_47_add__less__same__cancel1,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [B2: A,A2: A] :
% 1.64/1.08 ( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
% 1.64/1.08 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_48_less__add__Suc1,axiom,
% 1.64/1.08 ! [I2: nat,M2: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ I2 @ M2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_49_less__add__Suc2,axiom,
% 1.64/1.08 ! [I2: nat,M2: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ M2 @ I2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_50_zero__reorient,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( zero @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [X: A] :
% 1.64/1.08 ( ( ( zero_zero @ A )
% 1.64/1.08 = X )
% 1.64/1.08 = ( X
% 1.64/1.08 = ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_51_less__iff__Suc__add,axiom,
% 1.64/1.08 ( ( ord_less @ nat )
% 1.64/1.08 = ( ^ [M3: nat,N3: nat] :
% 1.64/1.08 ? [K2: nat] :
% 1.64/1.08 ( N3
% 1.64/1.08 = ( suc @ ( plus_plus @ nat @ M3 @ K2 ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_52_less__imp__Suc__add,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ M2 @ N2 )
% 1.64/1.08 => ? [K3: nat] :
% 1.64/1.08 ( N2
% 1.64/1.08 = ( suc @ ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_53_add__Suc,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( plus_plus @ nat @ ( suc @ M2 ) @ N2 )
% 1.64/1.08 = ( suc @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_54_add__Suc__shift,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( plus_plus @ nat @ ( suc @ M2 ) @ N2 )
% 1.64/1.08 = ( plus_plus @ nat @ M2 @ ( suc @ N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_55_zero__less__iff__neq__zero,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [N2: A] :
% 1.64/1.08 ( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
% 1.64/1.08 = ( N2
% 1.64/1.08 != ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_56_gr__implies__not__zero,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [M2: A,N2: A] :
% 1.64/1.08 ( ( ord_less @ A @ M2 @ N2 )
% 1.64/1.08 => ( N2
% 1.64/1.08 != ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_57_not__less__zero,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [N2: A] :
% 1.64/1.08 ~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_58_gr__zeroI,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [N2: A] :
% 1.64/1.08 ( ( N2
% 1.64/1.08 != ( zero_zero @ A ) )
% 1.64/1.08 => ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_59_not__less__less__Suc__eq,axiom,
% 1.64/1.08 ! [N2: nat,M2: nat] :
% 1.64/1.08 ( ~ ( ord_less @ nat @ N2 @ M2 )
% 1.64/1.08 => ( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
% 1.64/1.08 = ( N2 = M2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_60_strict__inc__induct,axiom,
% 1.64/1.08 ! [I2: nat,J: nat,P: nat > $o] :
% 1.64/1.08 ( ( ord_less @ nat @ I2 @ J )
% 1.64/1.08 => ( ! [I3: nat] :
% 1.64/1.08 ( ( J
% 1.64/1.08 = ( suc @ I3 ) )
% 1.64/1.08 => ( P @ I3 ) )
% 1.64/1.08 => ( ! [I3: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ I3 @ J )
% 1.64/1.08 => ( ( P @ ( suc @ I3 ) )
% 1.64/1.08 => ( P @ I3 ) ) )
% 1.64/1.08 => ( P @ I2 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_61_less__Suc__induct,axiom,
% 1.64/1.08 ! [I2: nat,J: nat,P: nat > nat > $o] :
% 1.64/1.08 ( ( ord_less @ nat @ I2 @ J )
% 1.64/1.08 => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 1.64/1.08 => ( ! [I3: nat,J2: nat,K3: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ I3 @ J2 )
% 1.64/1.08 => ( ( ord_less @ nat @ J2 @ K3 )
% 1.64/1.08 => ( ( P @ I3 @ J2 )
% 1.64/1.08 => ( ( P @ J2 @ K3 )
% 1.64/1.08 => ( P @ I3 @ K3 ) ) ) ) )
% 1.64/1.08 => ( P @ I2 @ J ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_62_less__trans__Suc,axiom,
% 1.64/1.08 ! [I2: nat,J: nat,K: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ I2 @ J )
% 1.64/1.08 => ( ( ord_less @ nat @ J @ K )
% 1.64/1.08 => ( ord_less @ nat @ ( suc @ I2 ) @ K ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_63_Suc__less__SucD,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
% 1.64/1.08 => ( ord_less @ nat @ M2 @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_64_less__antisym,axiom,
% 1.64/1.08 ! [N2: nat,M2: nat] :
% 1.64/1.08 ( ~ ( ord_less @ nat @ N2 @ M2 )
% 1.64/1.08 => ( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
% 1.64/1.08 => ( M2 = N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_65_Suc__less__eq2,axiom,
% 1.64/1.08 ! [N2: nat,M2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ ( suc @ N2 ) @ M2 )
% 1.64/1.08 = ( ? [M4: nat] :
% 1.64/1.08 ( ( M2
% 1.64/1.08 = ( suc @ M4 ) )
% 1.64/1.08 & ( ord_less @ nat @ N2 @ M4 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_66_not__less__eq,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ~ ( ord_less @ nat @ M2 @ N2 ) )
% 1.64/1.08 = ( ord_less @ nat @ N2 @ ( suc @ M2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_67_less__Suc__eq,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
% 1.64/1.08 = ( ( ord_less @ nat @ M2 @ N2 )
% 1.64/1.08 | ( M2 = N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_68_less__SucI,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ M2 @ N2 )
% 1.64/1.08 => ( ord_less @ nat @ M2 @ ( suc @ N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_69_less__SucE,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
% 1.64/1.08 => ( ~ ( ord_less @ nat @ M2 @ N2 )
% 1.64/1.08 => ( M2 = N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_70_Suc__lessI,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ M2 @ N2 )
% 1.64/1.08 => ( ( ( suc @ M2 )
% 1.64/1.08 != N2 )
% 1.64/1.08 => ( ord_less @ nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_71_Suc__lessE,axiom,
% 1.64/1.08 ! [I2: nat,K: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ ( suc @ I2 ) @ K )
% 1.64/1.08 => ~ ! [J2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ I2 @ J2 )
% 1.64/1.08 => ( K
% 1.64/1.08 != ( suc @ J2 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_72_Suc__lessD,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ ( suc @ M2 ) @ N2 )
% 1.64/1.08 => ( ord_less @ nat @ M2 @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_73_lessE,axiom,
% 1.64/1.08 ! [I2: nat,K: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ I2 @ K )
% 1.64/1.08 => ( ( K
% 1.64/1.08 != ( suc @ I2 ) )
% 1.64/1.08 => ~ ! [J2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ I2 @ J2 )
% 1.64/1.08 => ( K
% 1.64/1.08 != ( suc @ J2 ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_74_add__eq__0__iff__both__eq__0,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [X: A,Y: A] :
% 1.64/1.08 ( ( ( plus_plus @ A @ X @ Y )
% 1.64/1.08 = ( zero_zero @ A ) )
% 1.64/1.08 = ( ( X
% 1.64/1.08 = ( zero_zero @ A ) )
% 1.64/1.08 & ( Y
% 1.64/1.08 = ( zero_zero @ A ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_75_add_Ogroup__left__neutral,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( group_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A] :
% 1.64/1.08 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 1.64/1.08 = A2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_76_add_Ocomm__neutral,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A] :
% 1.64/1.08 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 1.64/1.08 = A2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_77_comm__monoid__add__class_Oadd__0,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A] :
% 1.64/1.08 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 1.64/1.08 = A2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_78_not__iless0,axiom,
% 1.64/1.08 ! [N2: extended_enat] :
% 1.64/1.08 ~ ( ord_less @ extended_enat @ N2 @ ( zero_zero @ extended_enat ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_79_plus__enat__eq__0__conv,axiom,
% 1.64/1.08 ! [M2: extended_enat,N2: extended_enat] :
% 1.64/1.08 ( ( ( plus_plus @ extended_enat @ M2 @ N2 )
% 1.64/1.08 = ( zero_zero @ extended_enat ) )
% 1.64/1.08 = ( ( M2
% 1.64/1.08 = ( zero_zero @ extended_enat ) )
% 1.64/1.08 & ( N2
% 1.64/1.08 = ( zero_zero @ extended_enat ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_80_pos__add__strict,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( strict797366125id_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A,C: A] :
% 1.64/1.08 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 1.64/1.08 => ( ( ord_less @ A @ B2 @ C )
% 1.64/1.08 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_81_add__pos__pos,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A] :
% 1.64/1.08 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 1.64/1.08 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 1.64/1.08 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_82_add__neg__neg,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere216010020id_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A] :
% 1.64/1.08 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 1.64/1.08 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 1.64/1.08 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_83_less__enatE,axiom,
% 1.64/1.08 ! [N2: extended_enat,M2: nat] :
% 1.64/1.08 ( ( ord_less @ extended_enat @ N2 @ ( extended_enat2 @ M2 ) )
% 1.64/1.08 => ~ ! [K3: nat] :
% 1.64/1.08 ( ( N2
% 1.64/1.08 = ( extended_enat2 @ K3 ) )
% 1.64/1.08 => ~ ( ord_less @ nat @ K3 @ M2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_84_enat__less__enat__plusI2,axiom,
% 1.64/1.08 ! [Y: nat,Z: extended_enat,X: nat] :
% 1.64/1.08 ( ( ord_less @ extended_enat @ ( extended_enat2 @ Y ) @ Z )
% 1.64/1.08 => ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ X @ Y ) ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_85_enat__less__enat__plusI,axiom,
% 1.64/1.08 ! [X: nat,Y: nat,Z: extended_enat] :
% 1.64/1.08 ( ( ord_less @ nat @ X @ Y )
% 1.64/1.08 => ( ord_less @ extended_enat @ ( extended_enat2 @ X ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ Y ) @ Z ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_86_add__right__imp__eq,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [B2: A,A2: A,C: A] :
% 1.64/1.08 ( ( ( plus_plus @ A @ B2 @ A2 )
% 1.64/1.08 = ( plus_plus @ A @ C @ A2 ) )
% 1.64/1.08 => ( B2 = C ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_87_add__left__imp__eq,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A,C: A] :
% 1.64/1.08 ( ( ( plus_plus @ A @ A2 @ B2 )
% 1.64/1.08 = ( plus_plus @ A @ A2 @ C ) )
% 1.64/1.08 => ( B2 = C ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_88_add_Oleft__commute,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [B2: A,A2: A,C: A] :
% 1.64/1.08 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
% 1.64/1.08 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_89_add_Ocommute,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ( ( plus_plus @ A )
% 1.64/1.08 = ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_90_add_Oright__cancel,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( group_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [B2: A,A2: A,C: A] :
% 1.64/1.08 ( ( ( plus_plus @ A @ B2 @ A2 )
% 1.64/1.08 = ( plus_plus @ A @ C @ A2 ) )
% 1.64/1.08 = ( B2 = C ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_91_add_Oleft__cancel,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( group_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A,C: A] :
% 1.64/1.08 ( ( ( plus_plus @ A @ A2 @ B2 )
% 1.64/1.08 = ( plus_plus @ A @ A2 @ C ) )
% 1.64/1.08 = ( B2 = C ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_92_add_Oassoc,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( semigroup_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A,C: A] :
% 1.64/1.08 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 1.64/1.08 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_93_add__mono__thms__linordered__semiring_I4_J,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere779506340up_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [I2: A,J: A,K: A,L: A] :
% 1.64/1.08 ( ( ( I2 = J )
% 1.64/1.08 & ( K = L ) )
% 1.64/1.08 => ( ( plus_plus @ A @ I2 @ K )
% 1.64/1.08 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_94_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ab_semigroup_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A,C: A] :
% 1.64/1.08 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 1.64/1.08 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_95_n__not__Suc__n,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ( N2
% 1.64/1.08 != ( suc @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_96_Suc__inject,axiom,
% 1.64/1.08 ! [X: nat,Y: nat] :
% 1.64/1.08 ( ( ( suc @ X )
% 1.64/1.08 = ( suc @ Y ) )
% 1.64/1.08 => ( X = Y ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_97_enat__less__induct,axiom,
% 1.64/1.08 ! [P: extended_enat > $o,N2: extended_enat] :
% 1.64/1.08 ( ! [N4: extended_enat] :
% 1.64/1.08 ( ! [M5: extended_enat] :
% 1.64/1.08 ( ( ord_less @ extended_enat @ M5 @ N4 )
% 1.64/1.08 => ( P @ M5 ) )
% 1.64/1.08 => ( P @ N4 ) )
% 1.64/1.08 => ( P @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_98_add__less__imp__less__right,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,C: A,B2: A] :
% 1.64/1.08 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 1.64/1.08 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_99_add__less__imp__less__left,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [C: A,A2: A,B2: A] :
% 1.64/1.08 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 1.64/1.08 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_100_add__strict__right__mono,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A,C: A] :
% 1.64/1.08 ( ( ord_less @ A @ A2 @ B2 )
% 1.64/1.08 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_101_add__strict__left__mono,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A,C: A] :
% 1.64/1.08 ( ( ord_less @ A @ A2 @ B2 )
% 1.64/1.08 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_102_add__strict__mono,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( strict2144017051up_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A,C: A,D: A] :
% 1.64/1.08 ( ( ord_less @ A @ A2 @ B2 )
% 1.64/1.08 => ( ( ord_less @ A @ C @ D )
% 1.64/1.08 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_103_add__mono__thms__linordered__field_I1_J,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [I2: A,J: A,K: A,L: A] :
% 1.64/1.08 ( ( ( ord_less @ A @ I2 @ J )
% 1.64/1.08 & ( K = L ) )
% 1.64/1.08 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_104_add__mono__thms__linordered__field_I2_J,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [I2: A,J: A,K: A,L: A] :
% 1.64/1.08 ( ( ( I2 = J )
% 1.64/1.08 & ( ord_less @ A @ K @ L ) )
% 1.64/1.08 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_105_add__mono__thms__linordered__field_I5_J,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ordere223160158up_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [I2: A,J: A,K: A,L: A] :
% 1.64/1.08 ( ( ( ord_less @ A @ I2 @ J )
% 1.64/1.08 & ( ord_less @ A @ K @ L ) )
% 1.64/1.08 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_106_lift__Suc__mono__less__iff,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( order @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [F: nat > A,N2: nat,M2: nat] :
% 1.64/1.08 ( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 1.64/1.08 => ( ( ord_less @ A @ ( F @ N2 ) @ ( F @ M2 ) )
% 1.64/1.08 = ( ord_less @ nat @ N2 @ M2 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_107_lift__Suc__mono__less,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( order @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [F: nat > A,N2: nat,N5: nat] :
% 1.64/1.08 ( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 1.64/1.08 => ( ( ord_less @ nat @ N2 @ N5 )
% 1.64/1.08 => ( ord_less @ A @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_108_lessThan__strict__subset__iff,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( linorder @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [M2: A,N2: A] :
% 1.64/1.08 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M2 ) @ ( set_ord_lessThan @ A @ N2 ) )
% 1.64/1.08 = ( ord_less @ A @ M2 @ N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_109_enat__iless,axiom,
% 1.64/1.08 ! [N2: extended_enat,M2: nat] :
% 1.64/1.08 ( ( ord_less @ extended_enat @ N2 @ ( extended_enat2 @ M2 ) )
% 1.64/1.08 => ? [K3: nat] :
% 1.64/1.08 ( N2
% 1.64/1.08 = ( extended_enat2 @ K3 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_110_chain__incr,axiom,
% 1.64/1.08 ! [A: $tType,Y3: A > extended_enat,K: nat] :
% 1.64/1.08 ( ! [I3: A] :
% 1.64/1.08 ? [J3: A] : ( ord_less @ extended_enat @ ( Y3 @ I3 ) @ ( Y3 @ J3 ) )
% 1.64/1.08 => ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K ) @ ( Y3 @ J2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_111_lessThan__def,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( ord @ A @ ( type2 @ A ) )
% 1.64/1.08 => ( ( set_ord_lessThan @ A )
% 1.64/1.08 = ( ^ [U: A] :
% 1.64/1.08 ( collect @ A
% 1.64/1.08 @ ^ [X3: A] : ( ord_less @ A @ X3 @ U ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_112_setsum_Oneutral__const,axiom,
% 1.64/1.08 ! [B: $tType,A: $tType] :
% 1.64/1.08 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A3: set @ B] :
% 1.64/1.08 ( ( groups15040474setsum @ B @ A
% 1.64/1.08 @ ^ [Uu: B] : ( zero_zero @ A )
% 1.64/1.08 @ A3 )
% 1.64/1.08 = ( zero_zero @ A ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_113_lnth__prefix,axiom,
% 1.64/1.08 ! [I4: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) )
% 1.64/1.08 => ( ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ I4 )
% 1.64/1.08 = ( coindu1598213697e_LNil @ a ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_114_llength__lconcat__lfinite__conv__sum,axiom,
% 1.64/1.08 ! [A: $tType,Xss: coindu1593790203_llist @ ( coindu1593790203_llist @ A )] :
% 1.64/1.08 ( ( coindu1213758845finite @ ( coindu1593790203_llist @ A ) @ Xss )
% 1.64/1.08 => ( ( coindu1018505716length @ A @ ( coindu441856546concat @ A @ Xss ) )
% 1.64/1.08 = ( groups15040474setsum @ nat @ extended_enat
% 1.64/1.08 @ ^ [I: nat] : ( coindu1018505716length @ A @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ A ) @ Xss @ I ) )
% 1.64/1.08 @ ( collect @ nat
% 1.64/1.08 @ ^ [I: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ I ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ A ) @ Xss ) ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_115_setsum_Odistrib,axiom,
% 1.64/1.08 ! [A: $tType,B: $tType] :
% 1.64/1.08 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [G: B > A,H: B > A,A3: set @ B] :
% 1.64/1.08 ( ( groups15040474setsum @ B @ A
% 1.64/1.08 @ ^ [X3: B] : ( plus_plus @ A @ ( G @ X3 ) @ ( H @ X3 ) )
% 1.64/1.08 @ A3 )
% 1.64/1.08 = ( plus_plus @ A @ ( groups15040474setsum @ B @ A @ G @ A3 ) @ ( groups15040474setsum @ B @ A @ H @ A3 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_116__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Axs_O_Alconcat_Axss_A_061_ALCons_Ax_Axs_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 1.64/1.08 ~ ! [X4: a,Xs: coindu1593790203_llist @ a] :
% 1.64/1.08 ( ( coindu441856546concat @ a @ xssa )
% 1.64/1.08 != ( coindu1121789889_LCons @ a @ X4 @ Xs ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_117__092_060open_062_092_060not_062_Alnull_A_Ilconcat_Axss_J_092_060close_062,axiom,
% 1.64/1.08 ~ ( coindu335574135_lnull @ a @ ( coindu441856546concat @ a @ xssa ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_118_gen__llength__def,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( coindu493225792length @ A )
% 1.64/1.08 = ( ^ [N3: nat,Xs2: coindu1593790203_llist @ A] : ( plus_plus @ extended_enat @ ( extended_enat2 @ N3 ) @ ( coindu1018505716length @ A @ Xs2 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_119_setsum_Onot__neutral__contains__not__neutral,axiom,
% 1.64/1.08 ! [B: $tType,A: $tType] :
% 1.64/1.08 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [G: B > A,A3: set @ B] :
% 1.64/1.08 ( ( ( groups15040474setsum @ B @ A @ G @ A3 )
% 1.64/1.08 != ( zero_zero @ A ) )
% 1.64/1.08 => ~ ! [A5: B] :
% 1.64/1.08 ( ( member @ B @ A5 @ A3 )
% 1.64/1.08 => ( ( G @ A5 )
% 1.64/1.08 = ( zero_zero @ A ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_120_setsum_Oneutral,axiom,
% 1.64/1.08 ! [B: $tType,A: $tType] :
% 1.64/1.08 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A3: set @ B,G: B > A] :
% 1.64/1.08 ( ! [X4: B] :
% 1.64/1.08 ( ( member @ B @ X4 @ A3 )
% 1.64/1.08 => ( ( G @ X4 )
% 1.64/1.08 = ( zero_zero @ A ) ) )
% 1.64/1.08 => ( ( groups15040474setsum @ B @ A @ G @ A3 )
% 1.64/1.08 = ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_121_add__0__iff,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( semiri456707255roduct @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [B2: A,A2: A] :
% 1.64/1.08 ( ( B2
% 1.64/1.08 = ( plus_plus @ A @ B2 @ A2 ) )
% 1.64/1.08 = ( A2
% 1.64/1.08 = ( zero_zero @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_122_False,axiom,
% 1.64/1.08 ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ m ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_123_neq0__conv,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ( ( N2
% 1.64/1.08 != ( zero_zero @ nat ) )
% 1.64/1.08 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_124_add__is__0,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ( plus_plus @ nat @ M2 @ N2 )
% 1.64/1.08 = ( zero_zero @ nat ) )
% 1.64/1.08 = ( ( M2
% 1.64/1.08 = ( zero_zero @ nat ) )
% 1.64/1.08 & ( N2
% 1.64/1.08 = ( zero_zero @ nat ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_125_Nat_Oadd__0__right,axiom,
% 1.64/1.08 ! [M2: nat] :
% 1.64/1.08 ( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
% 1.64/1.08 = M2 ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_126_llist_Oinject,axiom,
% 1.64/1.08 ! [A: $tType,X21: A,X22: coindu1593790203_llist @ A,Y21: A,Y22: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( ( coindu1121789889_LCons @ A @ X21 @ X22 )
% 1.64/1.08 = ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) )
% 1.64/1.08 = ( ( X21 = Y21 )
% 1.64/1.08 & ( X22 = Y22 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_127_zero__less__Suc,axiom,
% 1.64/1.08 ! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_128_less__Suc0,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
% 1.64/1.08 = ( N2
% 1.64/1.08 = ( zero_zero @ nat ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_129_add__gr__0,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M2 @ N2 ) )
% 1.64/1.08 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 1.64/1.08 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_130_nat__add__left__cancel__less,axiom,
% 1.64/1.08 ! [K: nat,M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N2 ) )
% 1.64/1.08 = ( ord_less @ nat @ M2 @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_131_lfinite__LCons,axiom,
% 1.64/1.08 ! [A: $tType,X: A,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) )
% 1.64/1.08 = ( coindu1213758845finite @ A @ Xs3 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_132_lfinite__code_I2_J,axiom,
% 1.64/1.08 ! [B: $tType,X: B,Xs3: coindu1593790203_llist @ B] :
% 1.64/1.08 ( ( coindu1213758845finite @ B @ ( coindu1121789889_LCons @ B @ X @ Xs3 ) )
% 1.64/1.08 = ( coindu1213758845finite @ B @ Xs3 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_133_lfinite__code_I1_J,axiom,
% 1.64/1.08 ! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_134_lnth__0,axiom,
% 1.64/1.08 ! [A: $tType,X: A,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( coindu749330388e_lnth @ A @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) @ ( zero_zero @ nat ) )
% 1.64/1.08 = X ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_135_lconcat__LNil,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( coindu441856546concat @ A @ ( coindu1598213697e_LNil @ ( coindu1593790203_llist @ A ) ) )
% 1.64/1.08 = ( coindu1598213697e_LNil @ A ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_136_lnth__Suc__LCons,axiom,
% 1.64/1.08 ! [A: $tType,X: A,Xs3: coindu1593790203_llist @ A,N2: nat] :
% 1.64/1.08 ( ( coindu749330388e_lnth @ A @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) @ ( suc @ N2 ) )
% 1.64/1.08 = ( coindu749330388e_lnth @ A @ Xs3 @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_137_llength__eq__0,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( ( coindu1018505716length @ A @ Xs3 )
% 1.64/1.08 = ( zero_zero @ extended_enat ) )
% 1.64/1.08 = ( coindu335574135_lnull @ A @ Xs3 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_138_llength__lnull,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( coindu335574135_lnull @ A @ Xs3 )
% 1.64/1.08 => ( ( coindu1018505716length @ A @ Xs3 )
% 1.64/1.08 = ( zero_zero @ extended_enat ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_139_llength__LNil,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( coindu1018505716length @ A @ ( coindu1598213697e_LNil @ A ) )
% 1.64/1.08 = ( zero_zero @ extended_enat ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_140_concat__xss,axiom,
% 1.64/1.08 ( ( coindu441856546concat @ a @ xssa )
% 1.64/1.08 = ( coindu1121789889_LCons @ a @ x @ xs ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_141_gr0I,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ( ( N2
% 1.64/1.08 != ( zero_zero @ nat ) )
% 1.64/1.08 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_142_not__gr0,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
% 1.64/1.08 = ( N2
% 1.64/1.08 = ( zero_zero @ nat ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_143_not__less0,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_144_add__lessD1,axiom,
% 1.64/1.08 ! [I2: nat,J: nat,K: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
% 1.64/1.08 => ( ord_less @ nat @ I2 @ K ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_145_less__zeroE,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_146_nat__neq__iff,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( M2 != N2 )
% 1.64/1.08 = ( ( ord_less @ nat @ M2 @ N2 )
% 1.64/1.08 | ( ord_less @ nat @ N2 @ M2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_147_add__less__mono,axiom,
% 1.64/1.08 ! [I2: nat,J: nat,K: nat,L: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ I2 @ J )
% 1.64/1.08 => ( ( ord_less @ nat @ K @ L )
% 1.64/1.08 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_148_less__not__refl,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_149_not__add__less1,axiom,
% 1.64/1.08 ! [I2: nat,J: nat] :
% 1.64/1.08 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ I2 ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_150_not__add__less2,axiom,
% 1.64/1.08 ! [J: nat,I2: nat] :
% 1.64/1.08 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I2 ) @ I2 ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_151_add__less__mono1,axiom,
% 1.64/1.08 ! [I2: nat,J: nat,K: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ I2 @ J )
% 1.64/1.08 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_152_less__not__refl2,axiom,
% 1.64/1.08 ! [N2: nat,M2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ N2 @ M2 )
% 1.64/1.08 => ( M2 != N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_153_less__not__refl3,axiom,
% 1.64/1.08 ! [S: nat,T: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ S @ T )
% 1.64/1.08 => ( S != T ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_154_measure__induct,axiom,
% 1.64/1.08 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 1.64/1.08 ( ! [X4: A] :
% 1.64/1.08 ( ! [Y4: A] :
% 1.64/1.08 ( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
% 1.64/1.08 => ( P @ Y4 ) )
% 1.64/1.08 => ( P @ X4 ) )
% 1.64/1.08 => ( P @ A2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_155_plus__nat_Oadd__0,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N2 )
% 1.64/1.08 = N2 ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_156_gr__implies__not0,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ M2 @ N2 )
% 1.64/1.08 => ( N2
% 1.64/1.08 != ( zero_zero @ nat ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_157_less__irrefl__nat,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_158_nat__less__induct,axiom,
% 1.64/1.08 ! [P: nat > $o,N2: nat] :
% 1.64/1.08 ( ! [N4: nat] :
% 1.64/1.08 ( ! [M5: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ M5 @ N4 )
% 1.64/1.08 => ( P @ M5 ) )
% 1.64/1.08 => ( P @ N4 ) )
% 1.64/1.08 => ( P @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_159_trans__less__add1,axiom,
% 1.64/1.08 ! [I2: nat,J: nat,M2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ I2 @ J )
% 1.64/1.08 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_160_trans__less__add2,axiom,
% 1.64/1.08 ! [I2: nat,J: nat,M2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ I2 @ J )
% 1.64/1.08 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_161_add__eq__self__zero,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ( plus_plus @ nat @ M2 @ N2 )
% 1.64/1.08 = M2 )
% 1.64/1.08 => ( N2
% 1.64/1.08 = ( zero_zero @ nat ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_162_infinite__descent,axiom,
% 1.64/1.08 ! [P: nat > $o,N2: nat] :
% 1.64/1.08 ( ! [N4: nat] :
% 1.64/1.08 ( ~ ( P @ N4 )
% 1.64/1.08 => ? [M5: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ M5 @ N4 )
% 1.64/1.08 & ~ ( P @ M5 ) ) )
% 1.64/1.08 => ( P @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_163_less__add__eq__less,axiom,
% 1.64/1.08 ! [K: nat,L: nat,M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ K @ L )
% 1.64/1.08 => ( ( ( plus_plus @ nat @ M2 @ L )
% 1.64/1.08 = ( plus_plus @ nat @ K @ N2 ) )
% 1.64/1.08 => ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_164_infinite__descent0,axiom,
% 1.64/1.08 ! [P: nat > $o,N2: nat] :
% 1.64/1.08 ( ( P @ ( zero_zero @ nat ) )
% 1.64/1.08 => ( ! [N4: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 1.64/1.08 => ( ~ ( P @ N4 )
% 1.64/1.08 => ? [M5: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ M5 @ N4 )
% 1.64/1.08 & ~ ( P @ M5 ) ) ) )
% 1.64/1.08 => ( P @ N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_165_linorder__neqE__nat,axiom,
% 1.64/1.08 ! [X: nat,Y: nat] :
% 1.64/1.08 ( ( X != Y )
% 1.64/1.08 => ( ~ ( ord_less @ nat @ X @ Y )
% 1.64/1.08 => ( ord_less @ nat @ Y @ X ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_166_setsum__SucD,axiom,
% 1.64/1.08 ! [A: $tType,F: A > nat,A3: set @ A,N2: nat] :
% 1.64/1.08 ( ( ( groups15040474setsum @ A @ nat @ F @ A3 )
% 1.64/1.08 = ( suc @ N2 ) )
% 1.64/1.08 => ? [X4: A] :
% 1.64/1.08 ( ( member @ A @ X4 @ A3 )
% 1.64/1.08 & ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X4 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_167_less__nat__zero__code,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_168_measure__induct__rule,axiom,
% 1.64/1.08 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 1.64/1.08 ( ! [X4: A] :
% 1.64/1.08 ( ! [Y4: A] :
% 1.64/1.08 ( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
% 1.64/1.08 => ( P @ Y4 ) )
% 1.64/1.08 => ( P @ X4 ) )
% 1.64/1.08 => ( P @ A2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_169_nat__add__left__cancel,axiom,
% 1.64/1.08 ! [K: nat,M2: nat,N2: nat] :
% 1.64/1.08 ( ( ( plus_plus @ nat @ K @ M2 )
% 1.64/1.08 = ( plus_plus @ nat @ K @ N2 ) )
% 1.64/1.08 = ( M2 = N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_170_nat__add__right__cancel,axiom,
% 1.64/1.08 ! [M2: nat,K: nat,N2: nat] :
% 1.64/1.08 ( ( ( plus_plus @ nat @ M2 @ K )
% 1.64/1.08 = ( plus_plus @ nat @ N2 @ K ) )
% 1.64/1.08 = ( M2 = N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_171_less__imp__add__positive,axiom,
% 1.64/1.08 ! [I2: nat,J: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ I2 @ J )
% 1.64/1.08 => ? [K3: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
% 1.64/1.08 & ( ( plus_plus @ nat @ I2 @ K3 )
% 1.64/1.08 = J ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_172_infinite__descent__measure,axiom,
% 1.64/1.08 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 1.64/1.08 ( ! [X4: A] :
% 1.64/1.08 ( ~ ( P @ X4 )
% 1.64/1.08 => ? [Y4: A] :
% 1.64/1.08 ( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X4 ) )
% 1.64/1.08 & ~ ( P @ Y4 ) ) )
% 1.64/1.08 => ( P @ X ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_173_infinite__descent0__measure,axiom,
% 1.64/1.08 ! [A: $tType,V: A > nat,P: A > $o,X: A] :
% 1.64/1.08 ( ! [X4: A] :
% 1.64/1.08 ( ( ( V @ X4 )
% 1.64/1.08 = ( zero_zero @ nat ) )
% 1.64/1.08 => ( P @ X4 ) )
% 1.64/1.08 => ( ! [X4: A] :
% 1.64/1.08 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X4 ) )
% 1.64/1.08 => ( ~ ( P @ X4 )
% 1.64/1.08 => ? [Y4: A] :
% 1.64/1.08 ( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X4 ) )
% 1.64/1.08 & ~ ( P @ Y4 ) ) ) )
% 1.64/1.08 => ( P @ X ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_174_lfinite__LConsI,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: coindu1593790203_llist @ A,X: A] :
% 1.64/1.08 ( ( coindu1213758845finite @ A @ Xs3 )
% 1.64/1.08 => ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_175_lfinite__LNil,axiom,
% 1.64/1.08 ! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_176_lnull__imp__lfinite,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( coindu335574135_lnull @ A @ Xs3 )
% 1.64/1.08 => ( coindu1213758845finite @ A @ Xs3 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_177_lnull__def,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( coindu335574135_lnull @ A )
% 1.64/1.08 = ( ^ [Llist: coindu1593790203_llist @ A] :
% 1.64/1.08 ( Llist
% 1.64/1.08 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_178_lfinite_Oinducts,axiom,
% 1.64/1.08 ! [A: $tType,X: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
% 1.64/1.08 ( ( coindu1213758845finite @ A @ X )
% 1.64/1.08 => ( ( P @ ( coindu1598213697e_LNil @ A ) )
% 1.64/1.08 => ( ! [Xs: coindu1593790203_llist @ A,X4: A] :
% 1.64/1.08 ( ( coindu1213758845finite @ A @ Xs )
% 1.64/1.08 => ( ( P @ Xs )
% 1.64/1.08 => ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs ) ) ) )
% 1.64/1.08 => ( P @ X ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_179_lappend_Oexhaust,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( ( coindu335574135_lnull @ A @ Xs3 )
% 1.64/1.08 => ~ ( coindu335574135_lnull @ A @ Ys ) )
% 1.64/1.08 => ( ~ ( coindu335574135_lnull @ A @ Xs3 )
% 1.64/1.08 | ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_180_not__lnull__conv,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( ~ ( coindu335574135_lnull @ A @ Xs3 ) )
% 1.64/1.08 = ( ? [X3: A,Xs4: coindu1593790203_llist @ A] :
% 1.64/1.08 ( Xs3
% 1.64/1.08 = ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_181_neq__LNil__conv,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( Xs3
% 1.64/1.08 != ( coindu1598213697e_LNil @ A ) )
% 1.64/1.08 = ( ? [X3: A,Xs4: coindu1593790203_llist @ A] :
% 1.64/1.08 ( Xs3
% 1.64/1.08 = ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_182_llist_Oexhaust,axiom,
% 1.64/1.08 ! [A: $tType,Y: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( Y
% 1.64/1.08 != ( coindu1598213697e_LNil @ A ) )
% 1.64/1.08 => ~ ! [X212: A,X222: coindu1593790203_llist @ A] :
% 1.64/1.08 ( Y
% 1.64/1.08 != ( coindu1121789889_LCons @ A @ X212 @ X222 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_183_lfinite_Osimps,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( coindu1213758845finite @ A )
% 1.64/1.08 = ( ^ [A4: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( A4
% 1.64/1.08 = ( coindu1598213697e_LNil @ A ) )
% 1.64/1.08 | ? [Xs2: coindu1593790203_llist @ A,X3: A] :
% 1.64/1.08 ( ( A4
% 1.64/1.08 = ( coindu1121789889_LCons @ A @ X3 @ Xs2 ) )
% 1.64/1.08 & ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_184_lfinite_Ocases,axiom,
% 1.64/1.08 ! [A: $tType,A2: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( coindu1213758845finite @ A @ A2 )
% 1.64/1.08 => ( ( A2
% 1.64/1.08 != ( coindu1598213697e_LNil @ A ) )
% 1.64/1.08 => ~ ! [Xs: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ? [X4: A] :
% 1.64/1.08 ( A2
% 1.64/1.08 = ( coindu1121789889_LCons @ A @ X4 @ Xs ) )
% 1.64/1.08 => ~ ( coindu1213758845finite @ A @ Xs ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_185_lzip_Oexhaust,axiom,
% 1.64/1.08 ! [A: $tType,B: $tType,Xs3: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
% 1.64/1.08 ( ~ ( ( coindu335574135_lnull @ A @ Xs3 )
% 1.64/1.08 | ( coindu335574135_lnull @ B @ Ys ) )
% 1.64/1.08 => ~ ( ~ ( coindu335574135_lnull @ A @ Xs3 )
% 1.64/1.08 => ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_186_llist_Odistinct_I1_J,axiom,
% 1.64/1.08 ! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( coindu1598213697e_LNil @ A )
% 1.64/1.08 != ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_187_llist_Ocollapse_I1_J,axiom,
% 1.64/1.08 ! [A: $tType,Llist2: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( coindu335574135_lnull @ A @ Llist2 )
% 1.64/1.08 => ( Llist2
% 1.64/1.08 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_188_llist_OdiscI_I1_J,axiom,
% 1.64/1.08 ! [A: $tType,Llist2: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( Llist2
% 1.64/1.08 = ( coindu1598213697e_LNil @ A ) )
% 1.64/1.08 => ( coindu335574135_lnull @ A @ Llist2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_189_llist_OdiscI_I2_J,axiom,
% 1.64/1.08 ! [A: $tType,Llist2: coindu1593790203_llist @ A,X21: A,X22: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( Llist2
% 1.64/1.08 = ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
% 1.64/1.08 => ~ ( coindu335574135_lnull @ A @ Llist2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_190_llist_Odisc_I1_J,axiom,
% 1.64/1.08 ! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_191_llist_Odisc_I2_J,axiom,
% 1.64/1.08 ! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
% 1.64/1.08 ~ ( coindu335574135_lnull @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_192_gen__llength__code_I1_J,axiom,
% 1.64/1.08 ! [A: $tType,N2: nat] :
% 1.64/1.08 ( ( coindu493225792length @ A @ N2 @ ( coindu1598213697e_LNil @ A ) )
% 1.64/1.08 = ( extended_enat2 @ N2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_193_llength__code,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( coindu1018505716length @ A )
% 1.64/1.08 = ( coindu493225792length @ A @ ( zero_zero @ nat ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_194_ltake_Oexhaust,axiom,
% 1.64/1.08 ! [A: $tType,N2: extended_enat,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ~ ( ( N2
% 1.64/1.08 = ( zero_zero @ extended_enat ) )
% 1.64/1.08 | ( coindu335574135_lnull @ A @ Xs3 ) )
% 1.64/1.08 => ~ ( ( N2
% 1.64/1.08 != ( zero_zero @ extended_enat ) )
% 1.64/1.08 => ( coindu335574135_lnull @ A @ Xs3 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_195_nat_Odistinct_I1_J,axiom,
% 1.64/1.08 ! [X2: nat] :
% 1.64/1.08 ( ( zero_zero @ nat )
% 1.64/1.08 != ( suc @ X2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_196_old_Onat_Odistinct_I2_J,axiom,
% 1.64/1.08 ! [Nat3: nat] :
% 1.64/1.08 ( ( suc @ Nat3 )
% 1.64/1.08 != ( zero_zero @ nat ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_197_old_Onat_Odistinct_I1_J,axiom,
% 1.64/1.08 ! [Nat2: nat] :
% 1.64/1.08 ( ( zero_zero @ nat )
% 1.64/1.08 != ( suc @ Nat2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_198_nat_OdiscI,axiom,
% 1.64/1.08 ! [Nat: nat,X2: nat] :
% 1.64/1.08 ( ( Nat
% 1.64/1.08 = ( suc @ X2 ) )
% 1.64/1.08 => ( Nat
% 1.64/1.08 != ( zero_zero @ nat ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_199_nat__induct,axiom,
% 1.64/1.08 ! [P: nat > $o,N2: nat] :
% 1.64/1.08 ( ( P @ ( zero_zero @ nat ) )
% 1.64/1.08 => ( ! [N4: nat] :
% 1.64/1.08 ( ( P @ N4 )
% 1.64/1.08 => ( P @ ( suc @ N4 ) ) )
% 1.64/1.08 => ( P @ N2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_200_diff__induct,axiom,
% 1.64/1.08 ! [P: nat > nat > $o,M2: nat,N2: nat] :
% 1.64/1.08 ( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
% 1.64/1.08 => ( ! [Y5: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y5 ) )
% 1.64/1.08 => ( ! [X4: nat,Y5: nat] :
% 1.64/1.08 ( ( P @ X4 @ Y5 )
% 1.64/1.08 => ( P @ ( suc @ X4 ) @ ( suc @ Y5 ) ) )
% 1.64/1.08 => ( P @ M2 @ N2 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_201_zero__induct,axiom,
% 1.64/1.08 ! [P: nat > $o,K: nat] :
% 1.64/1.08 ( ( P @ K )
% 1.64/1.08 => ( ! [N4: nat] :
% 1.64/1.08 ( ( P @ ( suc @ N4 ) )
% 1.64/1.08 => ( P @ N4 ) )
% 1.64/1.08 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_202_Suc__neq__Zero,axiom,
% 1.64/1.08 ! [M2: nat] :
% 1.64/1.08 ( ( suc @ M2 )
% 1.64/1.08 != ( zero_zero @ nat ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_203_Zero__neq__Suc,axiom,
% 1.64/1.08 ! [M2: nat] :
% 1.64/1.08 ( ( zero_zero @ nat )
% 1.64/1.08 != ( suc @ M2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_204_Zero__not__Suc,axiom,
% 1.64/1.08 ! [M2: nat] :
% 1.64/1.08 ( ( zero_zero @ nat )
% 1.64/1.08 != ( suc @ M2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_205_old_Onat_Oexhaust,axiom,
% 1.64/1.08 ! [Y: nat] :
% 1.64/1.08 ( ( Y
% 1.64/1.08 != ( zero_zero @ nat ) )
% 1.64/1.08 => ~ ! [Nat4: nat] :
% 1.64/1.08 ( Y
% 1.64/1.08 != ( suc @ Nat4 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_206_old_Onat_Oinducts,axiom,
% 1.64/1.08 ! [P: nat > $o,Nat: nat] :
% 1.64/1.08 ( ( P @ ( zero_zero @ nat ) )
% 1.64/1.08 => ( ! [Nat4: nat] :
% 1.64/1.08 ( ( P @ Nat4 )
% 1.64/1.08 => ( P @ ( suc @ Nat4 ) ) )
% 1.64/1.08 => ( P @ Nat ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_207_not0__implies__Suc,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ( ( N2
% 1.64/1.08 != ( zero_zero @ nat ) )
% 1.64/1.08 => ? [M: nat] :
% 1.64/1.08 ( N2
% 1.64/1.08 = ( suc @ M ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_208_less__Suc__eq__0__disj,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
% 1.64/1.08 = ( ( M2
% 1.64/1.08 = ( zero_zero @ nat ) )
% 1.64/1.08 | ? [J4: nat] :
% 1.64/1.08 ( ( M2
% 1.64/1.08 = ( suc @ J4 ) )
% 1.64/1.08 & ( ord_less @ nat @ J4 @ N2 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_209_gr0__implies__Suc,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 1.64/1.08 => ? [M: nat] :
% 1.64/1.08 ( N2
% 1.64/1.08 = ( suc @ M ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_210_gr0__conv__Suc,axiom,
% 1.64/1.08 ! [N2: nat] :
% 1.64/1.08 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 1.64/1.08 = ( ? [M3: nat] :
% 1.64/1.08 ( N2
% 1.64/1.08 = ( suc @ M3 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_211_one__is__add,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ( suc @ ( zero_zero @ nat ) )
% 1.64/1.08 = ( plus_plus @ nat @ M2 @ N2 ) )
% 1.64/1.08 = ( ( ( M2
% 1.64/1.08 = ( suc @ ( zero_zero @ nat ) ) )
% 1.64/1.08 & ( N2
% 1.64/1.08 = ( zero_zero @ nat ) ) )
% 1.64/1.08 | ( ( M2
% 1.64/1.08 = ( zero_zero @ nat ) )
% 1.64/1.08 & ( N2
% 1.64/1.08 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_212_add__is__1,axiom,
% 1.64/1.08 ! [M2: nat,N2: nat] :
% 1.64/1.08 ( ( ( plus_plus @ nat @ M2 @ N2 )
% 1.64/1.08 = ( suc @ ( zero_zero @ nat ) ) )
% 1.64/1.08 = ( ( ( M2
% 1.64/1.08 = ( suc @ ( zero_zero @ nat ) ) )
% 1.64/1.08 & ( N2
% 1.64/1.08 = ( zero_zero @ nat ) ) )
% 1.64/1.08 | ( ( M2
% 1.64/1.08 = ( zero_zero @ nat ) )
% 1.64/1.08 & ( N2
% 1.64/1.08 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_213_setsum__cong__Suc,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A3: set @ nat,F: nat > A,G: nat > A] :
% 1.64/1.08 ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 )
% 1.64/1.08 => ( ! [X4: nat] :
% 1.64/1.08 ( ( member @ nat @ ( suc @ X4 ) @ A3 )
% 1.64/1.08 => ( ( F @ ( suc @ X4 ) )
% 1.64/1.08 = ( G @ ( suc @ X4 ) ) ) )
% 1.64/1.08 => ( ( groups15040474setsum @ nat @ A @ F @ A3 )
% 1.64/1.08 = ( groups15040474setsum @ nat @ A @ G @ A3 ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_214_lfinite__llength__enat,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( coindu1213758845finite @ A @ Xs3 )
% 1.64/1.08 => ? [N4: nat] :
% 1.64/1.08 ( ( coindu1018505716length @ A @ Xs3 )
% 1.64/1.08 = ( extended_enat2 @ N4 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_215_llength__eq__enat__lfiniteD,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: coindu1593790203_llist @ A,N2: nat] :
% 1.64/1.08 ( ( ( coindu1018505716length @ A @ Xs3 )
% 1.64/1.08 = ( extended_enat2 @ N2 ) )
% 1.64/1.08 => ( coindu1213758845finite @ A @ Xs3 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_216_lfinite__conv__llength__enat,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( coindu1213758845finite @ A )
% 1.64/1.08 = ( ^ [Xs2: coindu1593790203_llist @ A] :
% 1.64/1.08 ? [N3: nat] :
% 1.64/1.08 ( ( coindu1018505716length @ A @ Xs2 )
% 1.64/1.08 = ( extended_enat2 @ N3 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_217_zero__enat__def,axiom,
% 1.64/1.08 ( ( zero_zero @ extended_enat )
% 1.64/1.08 = ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_218_enat__0__iff_I1_J,axiom,
% 1.64/1.08 ! [X: nat] :
% 1.64/1.08 ( ( ( extended_enat2 @ X )
% 1.64/1.08 = ( zero_zero @ extended_enat ) )
% 1.64/1.08 = ( X
% 1.64/1.08 = ( zero_zero @ nat ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_219_enat__0__iff_I2_J,axiom,
% 1.64/1.08 ! [X: nat] :
% 1.64/1.08 ( ( ( zero_zero @ extended_enat )
% 1.64/1.08 = ( extended_enat2 @ X ) )
% 1.64/1.08 = ( X
% 1.64/1.08 = ( zero_zero @ nat ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_220_semiring__normalization__rules_I25_J,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,C: A,D: A] :
% 1.64/1.08 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
% 1.64/1.08 = ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ D ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_221_semiring__normalization__rules_I24_J,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 1.64/1.08 => ( ( plus_plus @ A )
% 1.64/1.08 = ( ^ [A4: A,C2: A] : ( plus_plus @ A @ C2 @ A4 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_222_semiring__normalization__rules_I23_J,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A,C: A] :
% 1.64/1.08 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 1.64/1.08 = ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_223_semiring__normalization__rules_I22_J,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,C: A,D: A] :
% 1.64/1.08 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
% 1.64/1.08 = ( plus_plus @ A @ C @ ( plus_plus @ A @ A2 @ D ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_224_semiring__normalization__rules_I21_J,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A,C: A] :
% 1.64/1.08 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 1.64/1.08 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_225_semiring__normalization__rules_I20_J,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A,B2: A,C: A,D: A] :
% 1.64/1.08 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C @ D ) )
% 1.64/1.08 = ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_226_setsum_Ocong,axiom,
% 1.64/1.08 ! [A: $tType,B: $tType] :
% 1.64/1.08 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A3: set @ B,B4: set @ B,G: B > A,H: B > A] :
% 1.64/1.08 ( ( A3 = B4 )
% 1.64/1.08 => ( ! [X4: B] :
% 1.64/1.08 ( ( member @ B @ X4 @ B4 )
% 1.64/1.08 => ( ( G @ X4 )
% 1.64/1.08 = ( H @ X4 ) ) )
% 1.64/1.08 => ( ( groups15040474setsum @ B @ A @ G @ A3 )
% 1.64/1.08 = ( groups15040474setsum @ B @ A @ H @ B4 ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_227_setsum_Oreindex__bij__witness,axiom,
% 1.64/1.08 ! [B: $tType,A: $tType,C3: $tType] :
% 1.64/1.08 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [S2: set @ B,I2: C3 > B,J: B > C3,T2: set @ C3,H: C3 > A,G: B > A] :
% 1.64/1.08 ( ! [A5: B] :
% 1.64/1.08 ( ( member @ B @ A5 @ S2 )
% 1.64/1.08 => ( ( I2 @ ( J @ A5 ) )
% 1.64/1.08 = A5 ) )
% 1.64/1.08 => ( ! [A5: B] :
% 1.64/1.08 ( ( member @ B @ A5 @ S2 )
% 1.64/1.08 => ( member @ C3 @ ( J @ A5 ) @ T2 ) )
% 1.64/1.08 => ( ! [B5: C3] :
% 1.64/1.08 ( ( member @ C3 @ B5 @ T2 )
% 1.64/1.08 => ( ( J @ ( I2 @ B5 ) )
% 1.64/1.08 = B5 ) )
% 1.64/1.08 => ( ! [B5: C3] :
% 1.64/1.08 ( ( member @ C3 @ B5 @ T2 )
% 1.64/1.08 => ( member @ B @ ( I2 @ B5 ) @ S2 ) )
% 1.64/1.08 => ( ! [A5: B] :
% 1.64/1.08 ( ( member @ B @ A5 @ S2 )
% 1.64/1.08 => ( ( H @ ( J @ A5 ) )
% 1.64/1.08 = ( G @ A5 ) ) )
% 1.64/1.08 => ( ( groups15040474setsum @ B @ A @ G @ S2 )
% 1.64/1.08 = ( groups15040474setsum @ C3 @ A @ H @ T2 ) ) ) ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_228_setsum_Ocommute,axiom,
% 1.64/1.08 ! [A: $tType,B: $tType,C3: $tType] :
% 1.64/1.08 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [G: B > C3 > A,B4: set @ C3,A3: set @ B] :
% 1.64/1.08 ( ( groups15040474setsum @ B @ A
% 1.64/1.08 @ ^ [I: B] : ( groups15040474setsum @ C3 @ A @ ( G @ I ) @ B4 )
% 1.64/1.08 @ A3 )
% 1.64/1.08 = ( groups15040474setsum @ C3 @ A
% 1.64/1.08 @ ^ [J4: C3] :
% 1.64/1.08 ( groups15040474setsum @ B @ A
% 1.64/1.08 @ ^ [I: B] : ( G @ I @ J4 )
% 1.64/1.08 @ A3 )
% 1.64/1.08 @ B4 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_229_setsum__lessThan__Suc__shift,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( comm_monoid_add @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [F: nat > A,N2: nat] :
% 1.64/1.08 ( ( groups15040474setsum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
% 1.64/1.08 = ( plus_plus @ A @ ( F @ ( zero_zero @ nat ) )
% 1.64/1.08 @ ( groups15040474setsum @ nat @ A
% 1.64/1.08 @ ^ [I: nat] : ( F @ ( suc @ I ) )
% 1.64/1.08 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_230_semiring__normalization__rules_I6_J,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A] :
% 1.64/1.08 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 1.64/1.08 = A2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_231_semiring__normalization__rules_I5_J,axiom,
% 1.64/1.08 ! [A: $tType] :
% 1.64/1.08 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 1.64/1.08 => ! [A2: A] :
% 1.64/1.08 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 1.64/1.08 = A2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_232_concat__xss_H,axiom,
% 1.64/1.08 coindu335574135_lnull @ a @ ( coindu441856546concat @ a @ ( coindu1079202755ist_of @ ( coindu1593790203_llist @ a ) @ xss2 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_233_enat_Osize_I1_J,axiom,
% 1.64/1.08 ! [Nat: nat] :
% 1.64/1.08 ( ( extended_size_enat @ ( extended_enat2 @ Nat ) )
% 1.64/1.08 = ( zero_zero @ nat ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_234_length__induct,axiom,
% 1.64/1.08 ! [A: $tType,P: ( list @ A ) > $o,Xs3: list @ A] :
% 1.64/1.08 ( ! [Xs: list @ A] :
% 1.64/1.08 ( ! [Ys2: list @ A] :
% 1.64/1.08 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 1.64/1.08 => ( P @ Ys2 ) )
% 1.64/1.08 => ( P @ Xs ) )
% 1.64/1.08 => ( P @ Xs3 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_235_lnth__ldropn,axiom,
% 1.64/1.08 ! [A: $tType,N2: nat,M2: nat,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ N2 @ M2 ) ) @ ( coindu1018505716length @ A @ Xs3 ) )
% 1.64/1.08 => ( ( coindu749330388e_lnth @ A @ ( coindu531130065ldropn @ A @ N2 @ Xs3 ) @ M2 )
% 1.64/1.08 = ( coindu749330388e_lnth @ A @ Xs3 @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_236_llist__of__inject,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: list @ A,Ys: list @ A] :
% 1.64/1.08 ( ( ( coindu1079202755ist_of @ A @ Xs3 )
% 1.64/1.08 = ( coindu1079202755ist_of @ A @ Ys ) )
% 1.64/1.08 = ( Xs3 = Ys ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_237_ldropn__0,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( coindu531130065ldropn @ A @ ( zero_zero @ nat ) @ Xs3 )
% 1.64/1.08 = Xs3 ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_238_ldropn__ldropn,axiom,
% 1.64/1.08 ! [A: $tType,N2: nat,M2: nat,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( coindu531130065ldropn @ A @ N2 @ ( coindu531130065ldropn @ A @ M2 @ Xs3 ) )
% 1.64/1.08 = ( coindu531130065ldropn @ A @ ( plus_plus @ nat @ N2 @ M2 ) @ Xs3 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_239_ldropn__LNil,axiom,
% 1.64/1.08 ! [A: $tType,N2: nat] :
% 1.64/1.08 ( ( coindu531130065ldropn @ A @ N2 @ ( coindu1598213697e_LNil @ A ) )
% 1.64/1.08 = ( coindu1598213697e_LNil @ A ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_240_lfinite__ldropn,axiom,
% 1.64/1.08 ! [A: $tType,N2: nat,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( coindu1213758845finite @ A @ ( coindu531130065ldropn @ A @ N2 @ Xs3 ) )
% 1.64/1.08 = ( coindu1213758845finite @ A @ Xs3 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_241_ldropn__Suc__LCons,axiom,
% 1.64/1.08 ! [A: $tType,N2: nat,X: A,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( coindu531130065ldropn @ A @ ( suc @ N2 ) @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) )
% 1.64/1.08 = ( coindu531130065ldropn @ A @ N2 @ Xs3 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_242__092_060open_062m_A_061_A0_092_060close_062,axiom,
% 1.64/1.08 ( m
% 1.64/1.08 = ( zero_zero @ nat ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_243_llength__llist__of,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: list @ A] :
% 1.64/1.08 ( ( coindu1018505716length @ A @ ( coindu1079202755ist_of @ A @ Xs3 ) )
% 1.64/1.08 = ( extended_enat2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_244_lfinite__llist__of,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: list @ A] : ( coindu1213758845finite @ A @ ( coindu1079202755ist_of @ A @ Xs3 ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_245_ldropn__lnull,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: coindu1593790203_llist @ A,N2: nat] :
% 1.64/1.08 ( ( coindu335574135_lnull @ A @ Xs3 )
% 1.64/1.08 => ( ( coindu531130065ldropn @ A @ N2 @ Xs3 )
% 1.64/1.08 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_246_Ex__list__of__length,axiom,
% 1.64/1.08 ! [A: $tType,N2: nat] :
% 1.64/1.08 ? [Xs: list @ A] :
% 1.64/1.08 ( ( size_size @ ( list @ A ) @ Xs )
% 1.64/1.08 = N2 ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_247_neq__if__length__neq,axiom,
% 1.64/1.08 ! [A: $tType,Xs3: list @ A,Ys: list @ A] :
% 1.64/1.08 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 1.64/1.08 != ( size_size @ ( list @ A ) @ Ys ) )
% 1.64/1.08 => ( Xs3 != Ys ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_248_ldropn__eq__LConsD,axiom,
% 1.64/1.08 ! [A: $tType,N2: nat,Xs3: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( ( coindu531130065ldropn @ A @ N2 @ Xs3 )
% 1.64/1.08 = ( coindu1121789889_LCons @ A @ Y @ Ys ) )
% 1.64/1.08 => ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs3 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(fact_249_ldropn__Suc__conv__ldropn,axiom,
% 1.64/1.08 ! [A: $tType,N2: nat,Xs3: coindu1593790203_llist @ A] :
% 1.64/1.08 ( ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs3 ) )
% 1.64/1.08 => ( ( coindu1121789889_LCons @ A @ ( coindu749330388e_lnth @ A @ Xs3 @ N2 ) @ ( coindu531130065ldropn @ A @ ( suc @ N2 ) @ Xs3 ) )
% 1.64/1.08 = ( coindu531130065ldropn @ A @ N2 @ Xs3 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_fun___Orderings_Oorder,axiom,
% 1.64/1.08 ! [A6: $tType,A7: $tType] :
% 1.64/1.08 ( ( order @ A7 @ ( type2 @ A7 ) )
% 1.64/1.08 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_fun___Orderings_Oord,axiom,
% 1.64/1.08 ! [A6: $tType,A7: $tType] :
% 1.64/1.08 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 1.64/1.08 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
% 1.64/1.08 semiri456707255roduct @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 1.64/1.08 ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 1.64/1.08 strict2144017051up_add @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 1.64/1.08 ordere223160158up_add @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 1.64/1.08 ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 1.64/1.08 strict797366125id_add @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 1.64/1.08 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 1.64/1.08 ordere779506340up_add @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
% 1.64/1.08 ordere216010020id_add @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 1.64/1.08 cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 1.64/1.08 cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 1.64/1.08 ab_semigroup_add @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 1.64/1.08 comm_monoid_add @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
% 1.64/1.08 comm_semiring_1 @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 1.64/1.08 semigroup_add @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 1.64/1.08 linorder @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 1.64/1.08 monoid_add @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
% 1.64/1.08 order @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Orderings_Oord_2,axiom,
% 1.64/1.08 ord @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 1.64/1.08 zero @ nat @ ( type2 @ nat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 1.64/1.08 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
% 1.64/1.08 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_HOL_Obool___Orderings_Olinorder_5,axiom,
% 1.64/1.08 linorder @ $o @ ( type2 @ $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_HOL_Obool___Orderings_Oorder_6,axiom,
% 1.64/1.08 order @ $o @ ( type2 @ $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_HOL_Obool___Orderings_Oord_7,axiom,
% 1.64/1.08 ord @ $o @ ( type2 @ $o ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_8,axiom,
% 1.64/1.08 strict2144017051up_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_9,axiom,
% 1.64/1.08 strict797366125id_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_10,axiom,
% 1.64/1.08 canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_11,axiom,
% 1.64/1.08 ordere779506340up_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_12,axiom,
% 1.64/1.08 ordere216010020id_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_13,axiom,
% 1.64/1.08 ab_semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_14,axiom,
% 1.64/1.08 comm_monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_15,axiom,
% 1.64/1.08 comm_semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_16,axiom,
% 1.64/1.08 semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_17,axiom,
% 1.64/1.08 linorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_18,axiom,
% 1.64/1.08 monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_19,axiom,
% 1.64/1.08 order @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_20,axiom,
% 1.64/1.08 ord @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_21,axiom,
% 1.64/1.08 zero @ extended_enat @ ( type2 @ extended_enat ) ).
% 1.64/1.08
% 1.64/1.08 thf(conj_0,conjecture,
% 1.64/1.08 ? [M5: nat,N6: nat] :
% 1.64/1.08 ( ( ( coindu749330388e_lnth @ a @ ( coindu441856546concat @ a @ xssa ) @ ( suc @ na ) )
% 1.64/1.08 = ( coindu749330388e_lnth @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ M5 ) @ N6 ) )
% 1.64/1.08 & ( ord_less @ extended_enat @ ( extended_enat2 @ N6 ) @ ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ M5 ) ) )
% 1.64/1.08 & ( ord_less @ extended_enat @ ( extended_enat2 @ M5 ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ a ) @ xssa ) )
% 1.64/1.08 & ( ( extended_enat2 @ ( suc @ na ) )
% 1.64/1.08 = ( plus_plus @ extended_enat
% 1.64/1.08 @ ( groups15040474setsum @ nat @ extended_enat
% 1.64/1.08 @ ^ [I: nat] : ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ I ) )
% 1.64/1.08 @ ( set_ord_lessThan @ nat @ M5 ) )
% 1.64/1.08 @ ( extended_enat2 @ N6 ) ) ) ) ).
% 1.64/1.08 ------------------------
% 1.64/1.10 % (19692)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/183Mi)
% 1.64/1.10 % (19696)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_19363 for (2999ds/2Mi)
% 1.64/1.10 % (19693)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_19363 for (2999ds/4Mi)
% 1.64/1.10 % (19697)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_19363 for (2999ds/275Mi)
% 1.64/1.10 % (19698)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/18Mi)
% 1.64/1.10 % (19695)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/2Mi)
% 1.64/1.10 % (19694)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_19363 for (2999ds/27Mi)
% 1.64/1.10 % (19695)Instruction limit reached!
% 1.64/1.10 % (19695)------------------------------
% 1.64/1.10 % (19695)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.64/1.10 % (19695)Termination reason: Unknown
% 1.64/1.10 % (19695)Termination phase: shuffling
% 1.64/1.10
% 1.64/1.10 % (19695)Memory used [KB]: 1279
% 1.64/1.10 % (19695)Time elapsed: 0.002 s
% 1.64/1.10 % (19695)Instructions burned: 2 (million)
% 1.64/1.10 % (19695)------------------------------
% 1.64/1.10 % (19695)------------------------------
% 1.64/1.10 % (19696)Instruction limit reached!
% 1.64/1.10 % (19696)------------------------------
% 1.64/1.10 % (19696)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.64/1.10 % (19696)Termination reason: Unknown
% 1.64/1.10 % (19696)Termination phase: shuffling
% 1.64/1.10
% 1.64/1.10 % (19696)Memory used [KB]: 1279
% 1.64/1.10 % (19696)Time elapsed: 0.002 s
% 1.64/1.10 % (19696)Instructions burned: 2 (million)
% 1.64/1.10 % (19696)------------------------------
% 1.64/1.10 % (19696)------------------------------
% 1.64/1.10 % (19693)Instruction limit reached!
% 1.64/1.10 % (19693)------------------------------
% 1.64/1.10 % (19693)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.64/1.10 % (19693)Termination reason: Unknown
% 1.64/1.10 % (19693)Termination phase: shuffling
% 1.64/1.10
% 1.64/1.10 % (19693)Memory used [KB]: 1407
% 1.64/1.10 % (19693)Time elapsed: 0.003 s
% 1.64/1.10 % (19693)Instructions burned: 6 (million)
% 1.64/1.10 % (19693)------------------------------
% 1.64/1.10 % (19693)------------------------------
% 1.64/1.11 % (19698)Instruction limit reached!
% 1.64/1.11 % (19698)------------------------------
% 1.64/1.11 % (19698)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.64/1.11 % (19698)Termination reason: Unknown
% 1.64/1.11 % (19698)Termination phase: Property scanning
% 1.64/1.11
% 1.64/1.11 % (19698)Memory used [KB]: 1663
% 1.64/1.11 % (19698)Time elapsed: 0.007 s
% 1.64/1.11 % (19698)Instructions burned: 18 (million)
% 1.64/1.11 % (19698)------------------------------
% 1.64/1.11 % (19698)------------------------------
% 1.64/1.11 % (19694)Instruction limit reached!
% 1.64/1.11 % (19694)------------------------------
% 1.64/1.11 % (19694)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.64/1.11 % (19694)Termination reason: Unknown
% 1.64/1.11 % (19694)Termination phase: Preprocessing 1
% 1.64/1.11
% 1.64/1.11 % (19694)Memory used [KB]: 1663
% 1.64/1.11 % (19694)Time elapsed: 0.010 s
% 1.64/1.11 % (19694)Instructions burned: 28 (million)
% 1.64/1.11 % (19694)------------------------------
% 1.64/1.11 % (19694)------------------------------
% 1.76/1.11 % (19699)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/3Mi)
% 1.76/1.11 % (19700)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_19363 for (2999ds/37Mi)
% 1.76/1.11 % (19701)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_19363 for (2999ds/15Mi)
% 1.76/1.11 % (19699)Instruction limit reached!
% 1.76/1.11 % (19699)------------------------------
% 1.76/1.11 % (19699)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.11 % (19699)Termination reason: Unknown
% 1.76/1.11 % (19699)Termination phase: shuffling
% 1.76/1.11
% 1.76/1.11 % (19699)Memory used [KB]: 1279
% 1.76/1.11 % (19699)Time elapsed: 0.002 s
% 1.76/1.11 % (19699)Instructions burned: 3 (million)
% 1.76/1.11 % (19699)------------------------------
% 1.76/1.11 % (19699)------------------------------
% 1.76/1.12 % (19702)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_19363 for (2999ds/3Mi)
% 1.76/1.12 % (19701)Instruction limit reached!
% 1.76/1.12 % (19701)------------------------------
% 1.76/1.12 % (19701)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.12 % (19701)Termination reason: Unknown
% 1.76/1.12 % (19701)Termination phase: shuffling
% 1.76/1.12
% 1.76/1.12 % (19701)Memory used [KB]: 1535
% 1.76/1.12 % (19701)Time elapsed: 0.006 s
% 1.76/1.12 % (19701)Instructions burned: 15 (million)
% 1.76/1.12 % (19701)------------------------------
% 1.76/1.12 % (19701)------------------------------
% 1.76/1.12 % (19702)Instruction limit reached!
% 1.76/1.12 % (19702)------------------------------
% 1.76/1.12 % (19702)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.12 % (19702)Termination reason: Unknown
% 1.76/1.12 % (19702)Termination phase: shuffling
% 1.76/1.12
% 1.76/1.12 % (19702)Memory used [KB]: 1279
% 1.76/1.12 % (19702)Time elapsed: 0.002 s
% 1.76/1.12 % (19702)Instructions burned: 3 (million)
% 1.76/1.12 % (19702)------------------------------
% 1.76/1.12 % (19702)------------------------------
% 1.76/1.12 % (19703)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_19363 for (2999ds/1041Mi)
% 1.76/1.12 % (19704)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/7Mi)
% 1.76/1.12 % (19700)Instruction limit reached!
% 1.76/1.12 % (19700)------------------------------
% 1.76/1.12 % (19700)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.12 % (19700)Termination reason: Unknown
% 1.76/1.12 % (19700)Termination phase: Property scanning
% 1.76/1.12
% 1.76/1.12 % (19700)Memory used [KB]: 2046
% 1.76/1.12 % (19700)Time elapsed: 0.012 s
% 1.76/1.12 % (19700)Instructions burned: 40 (million)
% 1.76/1.12 % (19700)------------------------------
% 1.76/1.12 % (19700)------------------------------
% 1.76/1.12 % (19697)Refutation not found, incomplete strategy
% 1.76/1.12 % (19697)------------------------------
% 1.76/1.12 % (19697)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.12 % (19697)Termination reason: Refutation not found, incomplete strategy
% 1.76/1.12
% 1.76/1.12
% 1.76/1.12 % (19697)Memory used [KB]: 6652
% 1.76/1.12 % (19697)Time elapsed: 0.024 s
% 1.76/1.12 % (19697)Instructions burned: 72 (million)
% 1.76/1.12 % (19697)------------------------------
% 1.76/1.12 % (19697)------------------------------
% 1.76/1.12 % (19704)Instruction limit reached!
% 1.76/1.12 % (19704)------------------------------
% 1.76/1.12 % (19704)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.12 % (19704)Termination reason: Unknown
% 1.76/1.12 % (19704)Termination phase: shuffling
% 1.76/1.12
% 1.76/1.12 % (19704)Memory used [KB]: 1407
% 1.76/1.12 % (19704)Time elapsed: 0.004 s
% 1.76/1.12 % (19704)Instructions burned: 8 (million)
% 1.76/1.12 % (19704)------------------------------
% 1.76/1.12 % (19704)------------------------------
% 1.76/1.13 % (19705)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/16Mi)
% 1.76/1.13 % (19706)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_19363 for (2999ds/3Mi)
% 1.76/1.13 % (19706)Instruction limit reached!
% 1.76/1.13 % (19706)------------------------------
% 1.76/1.13 % (19706)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.13 % (19706)Termination reason: Unknown
% 1.76/1.13 % (19706)Termination phase: shuffling
% 1.76/1.13
% 1.76/1.13 % (19706)Memory used [KB]: 1279
% 1.76/1.13 % (19706)Time elapsed: 0.002 s
% 1.76/1.13 % (19706)Instructions burned: 3 (million)
% 1.76/1.13 % (19706)------------------------------
% 1.76/1.13 % (19706)------------------------------
% 1.76/1.13 % (19707)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_19363 for (2999ds/3Mi)
% 1.76/1.13 % (19705)Instruction limit reached!
% 1.76/1.13 % (19705)------------------------------
% 1.76/1.13 % (19705)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.13 % (19705)Termination reason: Unknown
% 1.76/1.13 % (19705)Termination phase: Property scanning
% 1.76/1.13
% 1.76/1.13 % (19705)Memory used [KB]: 1663
% 1.76/1.13 % (19705)Time elapsed: 0.007 s
% 1.76/1.13 % (19705)Instructions burned: 18 (million)
% 1.76/1.13 % (19705)------------------------------
% 1.76/1.13 % (19705)------------------------------
% 1.76/1.13 % (19707)Instruction limit reached!
% 1.76/1.13 % (19707)------------------------------
% 1.76/1.13 % (19707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.13 % (19707)Termination reason: Unknown
% 1.76/1.13 % (19707)Termination phase: shuffling
% 1.76/1.13
% 1.76/1.13 % (19707)Memory used [KB]: 1279
% 1.76/1.13 % (19707)Time elapsed: 0.002 s
% 1.76/1.13 % (19707)Instructions burned: 3 (million)
% 1.76/1.13 % (19707)------------------------------
% 1.76/1.13 % (19707)------------------------------
% 1.76/1.13 % (19708)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/7Mi)
% 1.76/1.13 % (19709)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/3Mi)
% 1.76/1.14 % (19709)Instruction limit reached!
% 1.76/1.14 % (19709)------------------------------
% 1.76/1.14 % (19709)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.14 % (19709)Termination reason: Unknown
% 1.76/1.14 % (19709)Termination phase: shuffling
% 1.76/1.14
% 1.76/1.14 % (19709)Memory used [KB]: 1407
% 1.76/1.14 % (19709)Time elapsed: 0.002 s
% 1.76/1.14 % (19709)Instructions burned: 4 (million)
% 1.76/1.14 % (19709)------------------------------
% 1.76/1.14 % (19709)------------------------------
% 1.76/1.14 % (19710)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/4Mi)
% 1.76/1.14 % (19708)Instruction limit reached!
% 1.76/1.14 % (19708)------------------------------
% 1.76/1.14 % (19708)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.14 % (19708)Termination reason: Unknown
% 1.76/1.14 % (19708)Termination phase: shuffling
% 1.76/1.14
% 1.76/1.14 % (19708)Memory used [KB]: 1535
% 1.76/1.14 % (19708)Time elapsed: 0.004 s
% 1.76/1.14 % (19708)Instructions burned: 9 (million)
% 1.76/1.14 % (19708)------------------------------
% 1.76/1.14 % (19708)------------------------------
% 1.76/1.14 % (19710)Instruction limit reached!
% 1.76/1.14 % (19710)------------------------------
% 1.76/1.14 % (19710)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.14 % (19710)Termination reason: Unknown
% 1.76/1.14 % (19710)Termination phase: shuffling
% 1.76/1.14
% 1.76/1.14 % (19710)Memory used [KB]: 1407
% 1.76/1.14 % (19710)Time elapsed: 0.003 s
% 1.76/1.14 % (19710)Instructions burned: 6 (million)
% 1.76/1.14 % (19710)------------------------------
% 1.76/1.14 % (19710)------------------------------
% 1.76/1.14 % (19712)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/18Mi)
% 1.76/1.14 % (19713)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/710Mi)
% 1.76/1.14 % (19714)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/6Mi)
% 1.76/1.15 % (19715)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_19363 for (2999ds/902Mi)
% 1.76/1.15 % (19714)Instruction limit reached!
% 1.76/1.15 % (19714)------------------------------
% 1.76/1.15 % (19714)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.15 % (19714)Termination reason: Unknown
% 1.76/1.15 % (19714)Termination phase: shuffling
% 1.76/1.15
% 1.76/1.15 % (19714)Memory used [KB]: 1407
% 1.76/1.15 % (19714)Time elapsed: 0.003 s
% 1.76/1.15 % (19714)Instructions burned: 7 (million)
% 1.76/1.15 % (19714)------------------------------
% 1.76/1.15 % (19714)------------------------------
% 1.76/1.15 % (19712)Instruction limit reached!
% 1.76/1.15 % (19712)------------------------------
% 1.76/1.15 % (19712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.15 % (19712)Termination reason: Unknown
% 1.76/1.15 % (19712)Termination phase: Property scanning
% 1.76/1.15
% 1.76/1.15 % (19712)Memory used [KB]: 1663
% 1.76/1.15 % (19712)Time elapsed: 0.007 s
% 1.76/1.15 % (19712)Instructions burned: 19 (million)
% 1.76/1.15 % (19712)------------------------------
% 1.76/1.15 % (19712)------------------------------
% 1.76/1.15 % (19716)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_19363 for (2999ds/21Mi)
% 1.76/1.16 % (19716)Instruction limit reached!
% 1.76/1.16 % (19716)------------------------------
% 1.76/1.16 % (19716)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.16 % (19716)Termination reason: Unknown
% 1.76/1.16 % (19716)Termination phase: Property scanning
% 1.76/1.16
% 1.76/1.16 % (19716)Memory used [KB]: 1663
% 1.76/1.16 % (19716)Time elapsed: 0.008 s
% 1.76/1.16 % (19716)Instructions burned: 22 (million)
% 1.76/1.16 % (19716)------------------------------
% 1.76/1.16 % (19716)------------------------------
% 1.76/1.16 % (19717)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/5Mi)
% 1.76/1.16 % (19718)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_19363 for (2999ds/6Mi)
% 1.76/1.16 % (19717)Instruction limit reached!
% 1.76/1.16 % (19717)------------------------------
% 1.76/1.16 % (19717)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.16 % (19717)Termination reason: Unknown
% 1.76/1.16 % (19717)Termination phase: shuffling
% 1.76/1.16
% 1.76/1.16 % (19717)Memory used [KB]: 1407
% 1.76/1.16 % (19717)Time elapsed: 0.003 s
% 1.76/1.16 % (19717)Instructions burned: 7 (million)
% 1.76/1.16 % (19717)------------------------------
% 1.76/1.16 % (19717)------------------------------
% 1.76/1.16 % (19692)Instruction limit reached!
% 1.76/1.16 % (19692)------------------------------
% 1.76/1.16 % (19692)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.16 % (19692)Termination reason: Unknown
% 1.76/1.16 % (19692)Termination phase: Saturation
% 1.76/1.16
% 1.76/1.16 % (19692)Memory used [KB]: 7036
% 1.76/1.16 % (19692)Time elapsed: 0.058 s
% 1.76/1.16 % (19692)Instructions burned: 183 (million)
% 1.76/1.16 % (19692)------------------------------
% 1.76/1.16 % (19692)------------------------------
% 1.76/1.16 % (19718)Instruction limit reached!
% 1.76/1.16 % (19718)------------------------------
% 1.76/1.16 % (19718)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.16 % (19718)Termination reason: Unknown
% 1.76/1.16 % (19718)Termination phase: shuffling
% 1.76/1.16
% 1.76/1.16 % (19718)Memory used [KB]: 1407
% 1.76/1.16 % (19718)Time elapsed: 0.003 s
% 1.76/1.16 % (19718)Instructions burned: 7 (million)
% 1.76/1.16 % (19718)------------------------------
% 1.76/1.16 % (19718)------------------------------
% 1.76/1.16 % (19719)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/377Mi)
% 1.76/1.17 % (19720)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_19363 for (2999ds/779Mi)
% 1.76/1.17 % (19721)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_19363 for (2999ds/19Mi)
% 1.76/1.17 % (19722)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/879Mi)
% 1.76/1.17 % (19721)Instruction limit reached!
% 1.76/1.17 % (19721)------------------------------
% 1.76/1.17 % (19721)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.17 % (19721)Termination reason: Unknown
% 1.76/1.17 % (19721)Termination phase: shuffling
% 1.76/1.17
% 1.76/1.17 % (19721)Memory used [KB]: 1791
% 1.76/1.17 % (19721)Time elapsed: 0.007 s
% 1.76/1.17 % (19721)Instructions burned: 22 (million)
% 1.76/1.17 % (19721)------------------------------
% 1.76/1.17 % (19721)------------------------------
% 1.76/1.18 % (19715)First to succeed.
% 1.76/1.18 % (19723)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_19363 for (2999ds/17Mi)
% 1.76/1.18 % (19715)Refutation found. Thanks to Tanya!
% 1.76/1.18 % SZS status Theorem for DTF2THF_19363
% 1.76/1.18 % SZS output start Proof for DTF2THF_19363
% 1.76/1.18 thf(type_def_5, type, itself: ($tType) > $tType).
% 1.76/1.18 thf(type_def_7, type, nat: $tType).
% 1.76/1.18 thf(type_def_8, type, coindu1593790203_llist: ($tType) > $tType).
% 1.76/1.18 thf(type_def_9, type, extended_enat: $tType).
% 1.76/1.18 thf(type_def_10, type, list: ($tType) > $tType).
% 1.76/1.18 thf(type_def_11, type, set: ($tType) > $tType).
% 1.76/1.18 thf(type_def_12, type, a: $tType).
% 1.76/1.18 thf(func_def_0, type, coindu1593790203_llist: ($tType) > $tType).
% 1.76/1.18 thf(func_def_1, type, extended_enat: $tType).
% 1.76/1.18 thf(func_def_2, type, list: ($tType) > $tType).
% 1.76/1.18 thf(func_def_3, type, set: ($tType) > $tType).
% 1.76/1.18 thf(func_def_4, type, nat: $tType).
% 1.76/1.18 thf(func_def_5, type, itself: ($tType) > $tType).
% 1.76/1.18 thf(func_def_6, type, a: $tType).
% 1.76/1.18 thf(func_def_7, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_8, type, zero: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_9, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_10, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_11, type, group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_12, type, monoid_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_13, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_14, type, semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_15, type, comm_semiring_1: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_16, type, comm_monoid_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_17, type, ab_semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_18, type, cancel_semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_19, type, cancel1352612707id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_20, type, linord219039673up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_21, type, ordere216010020id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_22, type, ordere779506340up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_23, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_24, type, strict797366125id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_25, type, ordere236663937imp_le: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_26, type, ordere223160158up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_27, type, strict2144017051up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_28, type, ordere516151231imp_le: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_29, type, semiri456707255roduct: !>[X0: $tType]:(itself @ X0 > $o)).
% 1.76/1.18 thf(func_def_30, type, coindu493225792length: !>[X0: $tType]:(nat > coindu1593790203_llist @ X0 > extended_enat)).
% 1.76/1.18 thf(func_def_31, type, coindu441856546concat: !>[X0: $tType]:(coindu1593790203_llist @ coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 1.76/1.18 thf(func_def_32, type, coindu531130065ldropn: !>[X0: $tType]:(nat > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 1.76/1.18 thf(func_def_33, type, coindu1213758845finite: !>[X0: $tType]:(coindu1593790203_llist @ X0 > $o)).
% 1.76/1.18 thf(func_def_34, type, coindu1018505716length: !>[X0: $tType]:(coindu1593790203_llist @ X0 > extended_enat)).
% 1.76/1.18 thf(func_def_35, type, coindu1121789889_LCons: !>[X0: $tType]:(X0 > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 1.76/1.18 thf(func_def_36, type, coindu1598213697e_LNil: !>[X0: $tType]:(coindu1593790203_llist @ X0)).
% 1.76/1.18 thf(func_def_37, type, coindu335574135_lnull: !>[X0: $tType]:(coindu1593790203_llist @ X0 > $o)).
% 1.76/1.18 thf(func_def_38, type, coindu1079202755ist_of: !>[X0: $tType]:(list @ X0 > coindu1593790203_llist @ X0)).
% 1.76/1.18 thf(func_def_39, type, coindu749330388e_lnth: !>[X0: $tType]:(coindu1593790203_llist @ X0 > nat > X0)).
% 1.76/1.18 thf(func_def_40, type, extended_enat2: nat > extended_enat).
% 1.76/1.18 thf(func_def_41, type, extended_size_enat: extended_enat > nat).
% 1.76/1.18 thf(func_def_42, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 1.76/1.18 thf(func_def_43, type, zero_zero: !>[X0: $tType]:(X0)).
% 1.76/1.18 thf(func_def_44, type, groups15040474setsum: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 1.76/1.18 thf(func_def_45, type, suc: nat > nat).
% 1.76/1.18 thf(func_def_46, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 1.76/1.18 thf(func_def_47, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 1.76/1.18 thf(func_def_48, type, type2: !>[X0: $tType]:(itself @ X0)).
% 1.76/1.18 thf(func_def_49, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 1.76/1.18 thf(func_def_50, type, set_ord_lessThan: !>[X0: $tType]:(X0 > set @ X0)).
% 1.76/1.18 thf(func_def_51, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 1.76/1.18 thf(func_def_52, type, n: nat).
% 1.76/1.18 thf(func_def_53, type, m: nat).
% 1.76/1.18 thf(func_def_54, type, n2: nat).
% 1.76/1.18 thf(func_def_55, type, n3: nat).
% 1.76/1.18 thf(func_def_56, type, na: nat).
% 1.76/1.18 thf(func_def_57, type, x: a).
% 1.76/1.18 thf(func_def_58, type, xs: coindu1593790203_llist @ a).
% 1.76/1.18 thf(func_def_59, type, xss: coindu1593790203_llist @ coindu1593790203_llist @ a).
% 1.76/1.18 thf(func_def_60, type, xss2: list @ coindu1593790203_llist @ a).
% 1.76/1.18 thf(func_def_61, type, xssa: coindu1593790203_llist @ coindu1593790203_llist @ a).
% 1.76/1.18 thf(func_def_73, type, sP0: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > $o)).
% 1.76/1.18 thf(func_def_74, type, sP1: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > X1) > (X1 > X0) > $o)).
% 1.76/1.18 thf(func_def_75, type, sK2: coindu1593790203_llist @ coindu1593790203_llist @ a > nat).
% 1.76/1.18 thf(func_def_76, type, sK3: coindu1593790203_llist @ coindu1593790203_llist @ a > nat).
% 1.76/1.18 thf(func_def_77, type, sK4: nat > nat > nat).
% 1.76/1.18 thf(func_def_78, type, sK5: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X0)).
% 1.76/1.18 thf(func_def_79, type, sK6: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 1.76/1.18 thf(func_def_80, type, sK7: !>[X0: $tType]:(coindu1593790203_llist @ X0 > nat)).
% 1.76/1.18 thf(func_def_81, type, sK8: nat > nat).
% 1.76/1.18 thf(func_def_82, type, sK9: (nat > $o) > nat).
% 1.76/1.18 thf(func_def_83, type, sK10: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 1.76/1.18 thf(func_def_84, type, sK11: nat > nat).
% 1.76/1.18 thf(func_def_85, type, sK12: (nat > $o) > nat).
% 1.76/1.18 thf(func_def_86, type, sK13: !>[X0: $tType]:(coindu1593790203_llist @ X0 > X0)).
% 1.76/1.18 thf(func_def_87, type, sK14: !>[X0: $tType]:(coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 1.76/1.18 thf(func_def_88, type, sK15: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 1.76/1.18 thf(func_def_89, type, sK16: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 1.76/1.18 thf(func_def_90, type, sK17: !>[X0: $tType]:(set @ X0 > (X0 > nat) > X0)).
% 1.76/1.18 thf(func_def_91, type, sK18: (nat > $o) > nat).
% 1.76/1.18 thf(func_def_92, type, sK19: !>[X0: $tType]:((nat > X0) > nat)).
% 1.76/1.18 thf(func_def_93, type, sK20: !>[X0: $tType]:((nat > X0) > nat)).
% 1.76/1.18 thf(func_def_94, type, sK21: a).
% 1.76/1.18 thf(func_def_95, type, sK22: coindu1593790203_llist @ a).
% 1.76/1.18 thf(func_def_96, type, sK23: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X0)).
% 1.76/1.18 thf(func_def_97, type, sK24: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 1.76/1.18 thf(func_def_98, type, sK25: nat > nat > nat).
% 1.76/1.18 thf(func_def_99, type, sK26: nat > nat).
% 1.76/1.18 thf(func_def_100, type, sK27: (nat > nat > $o) > nat).
% 1.76/1.18 thf(func_def_101, type, sK28: (nat > nat > $o) > nat).
% 1.76/1.18 thf(func_def_102, type, sK29: (nat > nat > $o) > nat).
% 1.76/1.18 thf(func_def_103, type, sK30: (nat > nat > $o) > nat).
% 1.76/1.18 thf(func_def_104, type, sK31: !>[X0: $tType]:((nat > X0) > (nat > X0) > set @ nat > nat)).
% 1.76/1.18 thf(func_def_105, type, sK32: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 1.76/1.18 thf(func_def_106, type, sK33: !>[X0: $tType]:(coindu1593790203_llist @ X0 > X0)).
% 1.76/1.18 thf(func_def_107, type, sK34: !>[X0: $tType]:(coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 1.76/1.18 thf(func_def_108, type, sK35: (nat > $o) > nat > nat).
% 1.76/1.18 thf(func_def_109, type, sK36: (nat > $o) > nat > nat).
% 1.76/1.18 thf(func_def_110, type, sK37: !>[X0: $tType]:((coindu1593790203_llist @ X0 > $o) > coindu1593790203_llist @ X0)).
% 1.76/1.18 thf(func_def_111, type, sK38: !>[X0: $tType]:((coindu1593790203_llist @ X0 > $o) > X0)).
% 1.76/1.18 thf(func_def_112, type, sK39: (nat > $o) > nat).
% 1.76/1.18 thf(func_def_113, type, sK40: (nat > $o) > nat).
% 1.76/1.18 thf(func_def_114, type, sK41: nat > extended_enat > nat).
% 1.76/1.18 thf(func_def_115, type, sK42: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > X1) > (X1 > X0) > X1)).
% 1.76/1.18 thf(func_def_116, type, sK43: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > X1)).
% 1.76/1.18 thf(func_def_117, type, sK44: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > X1) > set @ X0 > X0)).
% 1.76/1.18 thf(func_def_118, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X1) > (X0 > X2) > set @ X0 > X0)).
% 1.76/1.18 thf(func_def_119, type, sK46: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X1 > X0) > set @ X0 > X0)).
% 1.76/1.18 thf(func_def_120, type, sK47: extended_enat > nat).
% 1.76/1.18 thf(func_def_121, type, sK48: nat > nat).
% 1.76/1.18 thf(func_def_122, type, sK49: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 1.76/1.18 thf(func_def_123, type, sK50: (extended_enat > $o) > extended_enat).
% 1.76/1.18 thf(func_def_124, type, sK51: nat > nat > nat).
% 1.76/1.18 thf(func_def_125, type, sK52: nat > nat > nat).
% 1.76/1.18 thf(func_def_126, type, sK53: nat > nat > nat).
% 1.76/1.18 thf(func_def_127, type, sK54: !>[X0: $tType]:(coindu1593790203_llist @ X0 > X0)).
% 1.76/1.18 thf(func_def_128, type, sK55: !>[X0: $tType]:(coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 1.76/1.18 thf(func_def_129, type, sK56: !>[X0: $tType]:(coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 1.76/1.18 thf(func_def_130, type, sK57: !>[X0: $tType]:(coindu1593790203_llist @ X0 > X0)).
% 1.76/1.18 thf(func_def_131, type, sK58: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 1.76/1.18 thf(func_def_132, type, sK59: (nat > $o) > nat).
% 1.76/1.18 thf(func_def_133, type, sK60: (nat > nat > $o) > nat).
% 1.76/1.18 thf(func_def_134, type, sK61: (nat > nat > $o) > nat).
% 1.76/1.18 thf(func_def_135, type, sK62: (nat > nat > $o) > nat).
% 1.76/1.18 thf(func_def_136, type, sK63: (nat > nat > $o) > nat).
% 1.76/1.18 thf(func_def_137, type, sK64: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 1.76/1.18 thf(func_def_138, type, sK65: !>[X0: $tType]:(nat > list @ X0)).
% 1.76/1.18 thf(func_def_139, type, sK66: !>[X0: $tType]:((X0 > extended_enat) > X0)).
% 1.76/1.18 thf(func_def_140, type, sK67: !>[X0: $tType]:(nat > (X0 > extended_enat) > X0)).
% 1.76/1.18 thf(func_def_141, type, sK68: nat > nat > nat).
% 1.76/1.18 thf(func_def_143, type, ph70: !>[X0: $tType]:(X0)).
% 1.76/1.18 thf(f1786,plain,(
% 1.76/1.18 $false),
% 1.76/1.18 inference(trivial_inequality_removal,[status(thm)],[f1785])).
% 1.76/1.18 thf(f1785,plain,(
% 1.76/1.18 ($true != $true)),
% 1.76/1.18 inference(superposition,[status(thm)],[f1755,f1469])).
% 1.76/1.18 thf(f1469,plain,(
% 1.76/1.18 ((ord_less @ extended_enat @ (extended_enat2 @ (suc @ n3)) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)))) = $true)),
% 1.76/1.18 inference(cnf_transformation,[status(thm)],[f470])).
% 1.76/1.18 thf(f470,plain,(
% 1.76/1.18 ((ord_less @ extended_enat @ (extended_enat2 @ (suc @ n3)) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)))) = $true)),
% 1.76/1.18 inference(fool_elimination,[status(thm)],[f469])).
% 1.76/1.18 thf(f469,plain,(
% 1.76/1.18 (ord_less @ extended_enat @ (extended_enat2 @ (suc @ n3)) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2))))),
% 1.76/1.18 inference(rectify,[status(thm)],[f3])).
% 1.76/1.18 thf(f3,axiom,(
% 1.76/1.18 (ord_less @ extended_enat @ (extended_enat2 @ (suc @ n3)) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2))))),
% 1.76/1.18 file('/export/starexec/sandbox2/tmp/tmp.Bd5MKNrlUu/DTF2THF_19363.p',fact_2__092_060open_062enat_A_ISuc_An_H_J_A_060_Allength_A_Ilnth_Axss_A_Ilength_Axss_H_J_J_092_060close_062)).
% 1.76/1.18 thf(f1755,plain,(
% 1.76/1.18 ((ord_less @ extended_enat @ (extended_enat2 @ (suc @ n3)) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)))) != $true)),
% 1.76/1.18 inference(trivial_inequality_removal,[status(thm)],[f1754])).
% 1.76/1.18 thf(f1754,plain,(
% 1.76/1.18 ((ord_less @ extended_enat @ (extended_enat2 @ (suc @ n3)) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)))) != $true) | ($true != $true)),
% 1.76/1.18 inference(forward_demodulation,[status(thm)],[f1753,f1563])).
% 1.76/1.18 thf(f1563,plain,(
% 1.76/1.18 ((ord_less @ extended_enat @ (extended_enat2 @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa)) = $true)),
% 1.76/1.18 inference(cnf_transformation,[status(thm)],[f542])).
% 1.76/1.18 thf(f542,plain,(
% 1.76/1.18 ((ord_less @ extended_enat @ (extended_enat2 @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa)) = $true)),
% 1.76/1.18 inference(fool_elimination,[status(thm)],[f541])).
% 1.76/1.18 thf(f541,plain,(
% 1.76/1.18 (ord_less @ extended_enat @ (extended_enat2 @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa))),
% 1.76/1.18 inference(rectify,[status(thm)],[f1])).
% 1.76/1.18 thf(f1,axiom,(
% 1.76/1.18 (ord_less @ extended_enat @ (extended_enat2 @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa))),
% 1.76/1.18 file('/export/starexec/sandbox2/tmp/tmp.Bd5MKNrlUu/DTF2THF_19363.p',fact_0__092_060open_062enat_A_Ilength_Axss_H_J_A_060_Allength_Axss_092_060close_062)).
% 1.76/1.18 thf(f1753,plain,(
% 1.76/1.18 ((ord_less @ extended_enat @ (extended_enat2 @ (suc @ n3)) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)))) != $true) | ((ord_less @ extended_enat @ (extended_enat2 @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa)) != $true)),
% 1.76/1.18 inference(trivial_inequality_removal,[status(thm)],[f1752])).
% 1.76/1.18 thf(f1752,plain,(
% 1.76/1.18 ((coindu749330388e_lnth @ a @ (coindu441856546concat @ a @ xssa) @ (suc @ na)) != (coindu749330388e_lnth @ a @ (coindu441856546concat @ a @ xssa) @ (suc @ na))) | ((ord_less @ extended_enat @ (extended_enat2 @ (suc @ n3)) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)))) != $true) | ((ord_less @ extended_enat @ (extended_enat2 @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa)) != $true)),
% 1.76/1.18 inference(forward_demodulation,[status(thm)],[f1711,f1419])).
% 1.76/1.18 thf(f1419,plain,(
% 1.76/1.18 ((coindu749330388e_lnth @ a @ (coindu441856546concat @ a @ xssa) @ (suc @ na)) = (coindu749330388e_lnth @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)) @ (suc @ n3)))),
% 1.76/1.18 inference(cnf_transformation,[status(thm)],[f5])).
% 1.76/1.18 thf(f5,axiom,(
% 1.76/1.18 ((coindu749330388e_lnth @ a @ (coindu441856546concat @ a @ xssa) @ (suc @ na)) = (coindu749330388e_lnth @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)) @ (suc @ n3)))),
% 1.76/1.18 file('/export/starexec/sandbox2/tmp/tmp.Bd5MKNrlUu/DTF2THF_19363.p',fact_4__092_060open_062lnth_A_Ilconcat_Axss_J_A_ISuc_An_J_A_061_Alnth_A_Ilnth_Axss_A_Ilength_Axss_H_J_J_A_ISuc_An_H_J_092_060close_062)).
% 1.76/1.18 thf(f1711,plain,(
% 1.76/1.18 ((ord_less @ extended_enat @ (extended_enat2 @ (suc @ n3)) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)))) != $true) | ((coindu749330388e_lnth @ a @ (coindu441856546concat @ a @ xssa) @ (suc @ na)) != (coindu749330388e_lnth @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)) @ (suc @ n3))) | ((ord_less @ extended_enat @ (extended_enat2 @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa)) != $true)),
% 1.76/1.18 inference(trivial_inequality_removal,[status(thm)],[f1710])).
% 1.76/1.18 thf(f1710,plain,(
% 1.76/1.18 ((extended_enat2 @ (suc @ na)) != (extended_enat2 @ (suc @ na))) | ((coindu749330388e_lnth @ a @ (coindu441856546concat @ a @ xssa) @ (suc @ na)) != (coindu749330388e_lnth @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)) @ (suc @ n3))) | ((ord_less @ extended_enat @ (extended_enat2 @ (suc @ n3)) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)))) != $true) | ((ord_less @ extended_enat @ (extended_enat2 @ (size_size @ list @ coindu1593790203_llist @ a @ xss2)) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa)) != $true)),
% 1.76/1.18 inference(superposition,[status(thm)],[f1615,f1277])).
% 1.76/1.18 thf(f1277,plain,(
% 1.76/1.18 ((extended_enat2 @ (suc @ na)) = (plus_plus @ extended_enat @ (groups15040474setsum @ nat @ extended_enat @ (^[Y0 : nat]: (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ Y0))) @ (set_ord_lessThan @ nat @ (size_size @ list @ coindu1593790203_llist @ a @ xss2))) @ (extended_enat2 @ (suc @ n3))))),
% 1.76/1.18 inference(cnf_transformation,[status(thm)],[f764])).
% 1.76/1.18 thf(f764,plain,(
% 1.76/1.18 ((extended_enat2 @ (suc @ na)) = (plus_plus @ extended_enat @ (groups15040474setsum @ nat @ extended_enat @ (^[Y0 : nat]: (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ Y0))) @ (set_ord_lessThan @ nat @ (size_size @ list @ coindu1593790203_llist @ a @ xss2))) @ (extended_enat2 @ (suc @ n3))))),
% 1.76/1.18 inference(fool_elimination,[status(thm)],[f763])).
% 1.76/1.18 thf(f763,plain,(
% 1.76/1.18 ((plus_plus @ extended_enat @ (groups15040474setsum @ nat @ extended_enat @ (^[X0 : nat] : (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X0))) @ (set_ord_lessThan @ nat @ (size_size @ list @ coindu1593790203_llist @ a @ xss2))) @ (extended_enat2 @ (suc @ n3))) = (extended_enat2 @ (suc @ na)))),
% 1.76/1.18 inference(rectify,[status(thm)],[f6])).
% 1.76/1.18 thf(f6,axiom,(
% 1.76/1.18 ((plus_plus @ extended_enat @ (groups15040474setsum @ nat @ extended_enat @ (^[X2 : nat] : (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X2))) @ (set_ord_lessThan @ nat @ (size_size @ list @ coindu1593790203_llist @ a @ xss2))) @ (extended_enat2 @ (suc @ n3))) = (extended_enat2 @ (suc @ na)))),
% 1.76/1.18 file('/export/starexec/sandbox2/tmp/tmp.Bd5MKNrlUu/DTF2THF_19363.p',fact_5__092_060open_062enat_A_ISuc_An_J_A_061_A_I_092_060Sum_062i_060length_Axss_H_O_Allength_A_Ilnth_Axss_Ai_J_J_A_L_Aenat_A_ISuc_An_H_J_092_060close_062)).
% 1.76/1.18 thf(f1615,plain,(
% 1.76/1.18 ( ! [X0 : nat,X1 : nat] : (((extended_enat2 @ (suc @ na)) != (plus_plus @ extended_enat @ (groups15040474setsum @ nat @ extended_enat @ (^[Y0 : nat]: (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ Y0))) @ (set_ord_lessThan @ nat @ X0)) @ (extended_enat2 @ X1))) | ((ord_less @ extended_enat @ (extended_enat2 @ X0) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa)) != $true) | ($true != (ord_less @ extended_enat @ (extended_enat2 @ X1) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X0)))) | ((coindu749330388e_lnth @ a @ (coindu441856546concat @ a @ xssa) @ (suc @ na)) != (coindu749330388e_lnth @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X0) @ X1))) )),
% 1.76/1.18 inference(cnf_transformation,[status(thm)],[f867])).
% 1.76/1.18 thf(f867,plain,(
% 1.76/1.18 ! [X0 : nat,X1 : nat] : (((ord_less @ extended_enat @ (extended_enat2 @ X0) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa)) != $true) | ($true != (ord_less @ extended_enat @ (extended_enat2 @ X1) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X0)))) | ((coindu749330388e_lnth @ a @ (coindu441856546concat @ a @ xssa) @ (suc @ na)) != (coindu749330388e_lnth @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X0) @ X1)) | ((extended_enat2 @ (suc @ na)) != (plus_plus @ extended_enat @ (groups15040474setsum @ nat @ extended_enat @ (^[Y0 : nat]: (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ Y0))) @ (set_ord_lessThan @ nat @ X0)) @ (extended_enat2 @ X1))))),
% 1.76/1.18 inference(ennf_transformation,[status(thm)],[f402])).
% 1.76/1.18 thf(f402,plain,(
% 1.76/1.18 ~? [X0 : nat,X1 : nat] : (((coindu749330388e_lnth @ a @ (coindu441856546concat @ a @ xssa) @ (suc @ na)) = (coindu749330388e_lnth @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X0) @ X1)) & ((extended_enat2 @ (suc @ na)) = (plus_plus @ extended_enat @ (groups15040474setsum @ nat @ extended_enat @ (^[Y0 : nat]: (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ Y0))) @ (set_ord_lessThan @ nat @ X0)) @ (extended_enat2 @ X1))) & ((ord_less @ extended_enat @ (extended_enat2 @ X0) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa)) = $true) & ($true = (ord_less @ extended_enat @ (extended_enat2 @ X1) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X0)))))),
% 1.76/1.18 inference(fool_elimination,[status(thm)],[f401])).
% 1.76/1.18 thf(f401,plain,(
% 1.76/1.18 ~? [X0 : nat,X1 : nat] : (((extended_enat2 @ (suc @ na)) = (plus_plus @ extended_enat @ (groups15040474setsum @ nat @ extended_enat @ (^[X2 : nat] : (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X2))) @ (set_ord_lessThan @ nat @ X0)) @ (extended_enat2 @ X1))) & ((coindu749330388e_lnth @ a @ (coindu441856546concat @ a @ xssa) @ (suc @ na)) = (coindu749330388e_lnth @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X0) @ X1)) & (ord_less @ extended_enat @ (extended_enat2 @ X0) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa)) & (ord_less @ extended_enat @ (extended_enat2 @ X1) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X0))))),
% 1.76/1.18 inference(rectify,[status(thm)],[f293])).
% 1.76/1.18 thf(f293,negated_conjecture,(
% 1.76/1.18 ~? [X39 : nat,X79 : nat] : (((extended_enat2 @ (suc @ na)) = (plus_plus @ extended_enat @ (groups15040474setsum @ nat @ extended_enat @ (^[X2 : nat] : (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X2))) @ (set_ord_lessThan @ nat @ X39)) @ (extended_enat2 @ X79))) & ((coindu749330388e_lnth @ a @ (coindu441856546concat @ a @ xssa) @ (suc @ na)) = (coindu749330388e_lnth @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X39) @ X79)) & (ord_less @ extended_enat @ (extended_enat2 @ X39) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa)) & (ord_less @ extended_enat @ (extended_enat2 @ X79) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X39))))),
% 1.76/1.18 inference(negated_conjecture,[status(cth)],[f292])).
% 1.76/1.18 thf(f292,conjecture,(
% 1.76/1.18 ? [X39 : nat,X79 : nat] : (((extended_enat2 @ (suc @ na)) = (plus_plus @ extended_enat @ (groups15040474setsum @ nat @ extended_enat @ (^[X2 : nat] : (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X2))) @ (set_ord_lessThan @ nat @ X39)) @ (extended_enat2 @ X79))) & ((coindu749330388e_lnth @ a @ (coindu441856546concat @ a @ xssa) @ (suc @ na)) = (coindu749330388e_lnth @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X39) @ X79)) & (ord_less @ extended_enat @ (extended_enat2 @ X39) @ (coindu1018505716length @ coindu1593790203_llist @ a @ xssa)) & (ord_less @ extended_enat @ (extended_enat2 @ X79) @ (coindu1018505716length @ a @ (coindu749330388e_lnth @ coindu1593790203_llist @ a @ xssa @ X39))))),
% 1.76/1.18 file('/export/starexec/sandbox2/tmp/tmp.Bd5MKNrlUu/DTF2THF_19363.p',conj_0)).
% 1.76/1.18 % SZS output end Proof for DTF2THF_19363
% 1.76/1.18 % (19715)------------------------------
% 1.76/1.18 % (19715)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.18 % (19715)Termination reason: Refutation
% 1.76/1.18
% 1.76/1.18 % (19715)Memory used [KB]: 6908
% 1.76/1.18 % (19715)Time elapsed: 0.037 s
% 1.76/1.18 % (19715)Instructions burned: 113 (million)
% 1.76/1.18 % (19715)------------------------------
% 1.76/1.18 % (19715)------------------------------
% 1.76/1.18 % (19691)Success in time 0.09 s
%------------------------------------------------------------------------------