%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT121^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n012.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 08:44:50 AM UTC 2026
% Result : Theorem 2.88s 1.72s
% Output : Refutation 2.88s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : DAT121^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.07 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.08/0.25 % Computer : n012.cluster.edu
% 0.08/0.25 % Model : x86_64 x86_64
% 0.08/0.25 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.25 % Memory : 8042.1875MB
% 0.08/0.25 % OS : Linux 3.10.0-693.el7.x86_64
% 0.08/0.25 % CPULimit : 300
% 0.08/0.25 % WCLimit : 300
% 0.08/0.25 % DateTime : Tue Feb 24 06:07:15 EST 2026
% 0.08/0.25 % CPUTime :
% 0.08/0.25 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.08/0.35 ---- Original DTF file ---
% 0.08/0.35 thf(spec,logic,$$dhol).
% 0.08/0.35 %------------------------------------------------------------------------------
% 0.08/0.35 % File : DAT121^1 : TPTP v9.2.1. Released v7.0.0.
% 0.08/0.35 % Domain : Data Structures
% 0.08/0.35 % Problem : Coinductive list 1690
% 0.08/0.35 % Version : [Bla16] axioms : Especial.
% 0.08/0.35 % English :
% 0.08/0.35
% 0.08/0.35 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.08/0.35 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.08/0.35 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.08/0.35 % Source : [Bla16]
% 0.08/0.35 % Names : coinductive_list__1690.p [Bla16]
% 0.08/0.35
% 0.08/0.35 % Status : Theorem
% 0.08/0.35 % Rating : 0.33 v9.1.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.08/0.35 % Syntax : Number of formulae : 416 ( 179 unt; 70 typ; 0 def)
% 0.08/0.35 % Number of atoms : 784 ( 267 equ; 0 cnn)
% 0.08/0.35 % Maximal formula atoms : 13 ( 2 avg)
% 0.08/0.35 % Number of connectives : 3118 ( 67 ~; 20 |; 22 &;2713 @)
% 0.08/0.35 % ( 0 <=>; 296 =>; 0 <=; 0 <~>)
% 0.08/0.35 % Maximal formula depth : 19 ( 6 avg)
% 0.08/0.35 % Number of types : 8 ( 7 usr)
% 0.08/0.35 % Number of type conns : 151 ( 151 >; 0 *; 0 +; 0 <<)
% 0.08/0.35 % Number of symbols : 66 ( 63 usr; 4 con; 0-4 aty)
% 0.08/0.35 % Number of variables : 750 ( 23 ^; 666 !; 6 ?; 750 :)
% 0.08/0.35 % ( 55 !>; 0 ?*; 0 @-; 0 @+)
% 0.08/0.35 % SPC : TH1_THM_EQU_NAR
% 0.08/0.35
% 0.08/0.35 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.08/0.35 % 2016-07-13 14:51:55.110
% 0.08/0.35 %------------------------------------------------------------------------------
% 0.08/0.35 %----Could-be-implicit typings (9)
% 0.08/0.35 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
% 0.08/0.35 coindu1593790203_llist: $tType > $tType ).
% 0.08/0.35
% 0.08/0.35 thf(ty_t_Code__Numeral_Onatural,type,
% 0.08/0.35 code_natural: $tType ).
% 0.08/0.35
% 0.08/0.35 thf(ty_t_Extended__Nat_Oenat,type,
% 0.08/0.35 extended_enat: $tType ).
% 0.08/0.35
% 0.08/0.35 thf(ty_t_Real_Oreal,type,
% 0.08/0.35 real: $tType ).
% 0.08/0.35
% 0.08/0.35 thf(ty_t_Num_Onum,type,
% 0.08/0.35 num: $tType ).
% 0.08/0.35
% 0.08/0.35 thf(ty_t_Nat_Onat,type,
% 0.08/0.35 nat: $tType ).
% 0.08/0.35
% 0.08/0.35 thf(ty_t_Int_Oint,type,
% 0.08/0.35 int: $tType ).
% 0.08/0.35
% 0.08/0.35 thf(ty_t_itself,type,
% 0.08/0.35 itself: $tType > $tType ).
% 0.08/0.35
% 0.08/0.35 thf(ty_tf_a,type,
% 0.08/0.35 a: $tType ).
% 0.08/0.35
% 0.08/0.35 %----Explicit typings (61)
% 0.08/0.35 thf(sy_cl_HOL_Otype,type,
% 0.08/0.35 type:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Groups_Ozero,type,
% 0.08/0.35 zero:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Num_Onumeral,type,
% 0.08/0.35 numeral:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Groups_Ominus,type,
% 0.08/0.35 minus:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Rings_Oring__1,type,
% 0.08/0.35 ring_1:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Orderings_Oord,type,
% 0.08/0.35 ord:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Nat_Oring__char__0,type,
% 0.08/0.35 ring_char_0:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Num_Oneg__numeral,type,
% 0.08/0.35 neg_numeral:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Orderings_Oorder,type,
% 0.08/0.35 order:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Groups_Ogroup__add,type,
% 0.08/0.35 group_add:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Rings_Osemiring__1,type,
% 0.08/0.35 semiring_1:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Orderings_Olinorder,type,
% 0.08/0.35 linorder:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Orderings_Opreorder,type,
% 0.08/0.35 preorder:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.08/0.35 semiring_char_0:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.08/0.35 linordered_idom:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.08/0.35 comm_monoid_diff:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.08/0.35 linordered_semidom:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.08/0.35 ordered_ab_group_add:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.08/0.35 cancel1352612707id_add:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.08/0.35 cancel146912293up_add:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
% 0.08/0.35 archim1727834104eiling:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
% 0.08/0.35 ordere142940540dd_abs:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.08/0.35 comple187826305attice:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 0.08/0.35 archim1804426504_field:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.08/0.35 canoni770627133id_add:
% 0.08/0.35 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Archimedean__Field_Oceiling,type,
% 0.08/0.35 archimedean_ceiling:
% 0.08/0.35 !>[A: $tType] : ( A > int ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Archimedean__Field_Oround,type,
% 0.08/0.35 archimedean_round:
% 0.08/0.35 !>[A: $tType] : ( A > int ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Code__Numeral_OSuc,type,
% 0.08/0.35 code_Suc: code_natural > code_natural ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
% 0.08/0.35 code_case_natural:
% 0.08/0.35 !>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
% 0.08/0.35 code_rec_natural:
% 0.08/0.35 !>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Code__Numeral_Onatural_Osize__natural,type,
% 0.08/0.35 code_size_natural: code_natural > nat ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__llength,type,
% 0.08/0.35 coindu493225792length:
% 0.08/0.35 !>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend,type,
% 0.08/0.35 coindu268472904append:
% 0.08/0.35 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldrop,type,
% 0.08/0.35 coindu191418589_ldrop:
% 0.08/0.35 !>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
% 0.08/0.35 coindu1018505716length:
% 0.08/0.35 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil,type,
% 0.08/0.35 coindu1598213697e_LNil:
% 0.08/0.35 !>[A: $tType] : ( coindu1593790203_llist @ A ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olhd,type,
% 0.08/0.35 coindu1046438764le_lhd:
% 0.08/0.35 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
% 0.08/0.35 coindu335574135_lnull:
% 0.08/0.35 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olmember,type,
% 0.08/0.35 coindu567634248member:
% 0.08/0.35 !>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olprefix,type,
% 0.08/0.35 coindu1696667936prefix:
% 0.08/0.35 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olstrict__prefix,type,
% 0.08/0.35 coindu574146665prefix:
% 0.08/0.35 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oltake,type,
% 0.08/0.35 coindu1802687541_ltake:
% 0.08/0.35 !>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OltakeWhile,type,
% 0.08/0.35 coindu721411036eWhile:
% 0.08/0.35 !>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
% 0.08/0.35 extended_case_enat:
% 0.08/0.35 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Extended__Nat_Othe__enat,type,
% 0.08/0.35 extended_the_enat: extended_enat > nat ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 0.08/0.35 abs_abs:
% 0.08/0.35 !>[A: $tType] : ( A > A ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.08/0.35 minus_minus:
% 0.08/0.35 !>[A: $tType] : ( A > A > A ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.08/0.35 zero_zero:
% 0.08/0.35 !>[A: $tType] : A ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Hilbert__Choice_OLeastM,type,
% 0.08/0.35 hilbert_LeastM:
% 0.08/0.35 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_If,type,
% 0.08/0.35 if:
% 0.08/0.35 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Int_Onat,type,
% 0.08/0.35 nat2: int > nat ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 0.08/0.35 ring_1_of_int:
% 0.08/0.35 !>[A: $tType] : ( int > A ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.08/0.35 semiring_1_of_nat:
% 0.08/0.35 !>[A: $tType] : ( nat > A ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.08/0.35 size_size:
% 0.08/0.35 !>[A: $tType] : ( A > nat ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Nat__Transfer_Otsub,type,
% 0.08/0.35 nat_tsub: int > int > int ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 0.08/0.35 neg_numeral_dbl:
% 0.08/0.35 !>[A: $tType] : ( A > A ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 0.08/0.35 numeral_numeral:
% 0.08/0.35 !>[A: $tType] : ( num > A ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.08/0.35 ord_less_eq:
% 0.08/0.35 !>[A: $tType] : ( A > A > $o ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_c_Pure_Otype,type,
% 0.08/0.35 type2:
% 0.08/0.35 !>[A: $tType] : ( itself @ A ) ).
% 0.08/0.35
% 0.08/0.35 thf(sy_v_m,type,
% 0.08/0.35 m: extended_enat ).
% 0.08/0.35
% 0.08/0.35 thf(sy_v_xs,type,
% 0.08/0.35 xs: coindu1593790203_llist @ a ).
% 0.08/0.35
% 0.08/0.35 %----Relevant facts (250)
% 0.08/0.35 thf(fact_0_ldrop__LNil,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat] :
% 0.08/0.35 ( ( coindu191418589_ldrop @ A @ N @ ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) ) ).
% 0.08/0.35
% 0.08/0.35 % ldrop_LNil
% 0.08/0.35 thf(fact_1_ldrop__eq__LNil,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ( coindu191418589_ldrop @ A @ N @ Xs )
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 = ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N ) ) ).
% 0.08/0.35
% 0.08/0.35 % ldrop_eq_LNil
% 0.08/0.35 thf(fact_2_wlog__linorder__le,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( linorder @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [P: A > A > $o,B2: A,A2: A] :
% 0.08/0.35 ( ! [A3: A,B3: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.08/0.35 => ( P @ A3 @ B3 ) )
% 0.08/0.35 => ( ( ( P @ B2 @ A2 )
% 0.08/0.35 => ( P @ A2 @ B2 ) )
% 0.08/0.35 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % wlog_linorder_le
% 0.08/0.35 thf(fact_3_order__refl,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( preorder @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.08/0.35
% 0.08/0.35 % order_refl
% 0.08/0.35 thf(fact_4_lnull__ldrop,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ ( coindu191418589_ldrop @ A @ N @ Xs ) )
% 0.08/0.35 = ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N ) ) ).
% 0.08/0.35
% 0.08/0.35 % lnull_ldrop
% 0.08/0.35 thf(fact_5_lmember__code_I1_J,axiom,
% 0.08/0.35 ! [A: $tType,X: A] :
% 0.08/0.35 ~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% 0.08/0.35
% 0.08/0.35 % lmember_code(1)
% 0.08/0.35 thf(fact_6_llength__LNil,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( coindu1018505716length @ A @ ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 = ( zero_zero @ extended_enat ) ) ).
% 0.08/0.35
% 0.08/0.35 % llength_LNil
% 0.08/0.35 thf(fact_7_le__funD,axiom,
% 0.08/0.35 ! [B: $tType,A: $tType] :
% 0.08/0.35 ( ( ord @ B @ ( type2 @ B ) )
% 0.08/0.35 => ! [F: A > B,G: A > B,X: A] :
% 0.08/0.35 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.08/0.35 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_funD
% 0.08/0.35 thf(fact_8_le__funE,axiom,
% 0.08/0.35 ! [B: $tType,A: $tType] :
% 0.08/0.35 ( ( ord @ B @ ( type2 @ B ) )
% 0.08/0.35 => ! [F: A > B,G: A > B,X: A] :
% 0.08/0.35 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.08/0.35 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_funE
% 0.08/0.35 thf(fact_9_le__funI,axiom,
% 0.08/0.35 ! [B: $tType,A: $tType] :
% 0.08/0.35 ( ( ord @ B @ ( type2 @ B ) )
% 0.08/0.35 => ! [F: A > B,G: A > B] :
% 0.08/0.35 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.08/0.35 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_funI
% 0.08/0.35 thf(fact_10_le__fun__def,axiom,
% 0.08/0.35 ! [B: $tType,A: $tType] :
% 0.08/0.35 ( ( ord @ B @ ( type2 @ B ) )
% 0.08/0.35 => ( ( ord_less_eq @ ( A > B ) )
% 0.08/0.35 = ( ^ [F2: A > B,G2: A > B] :
% 0.08/0.35 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_fun_def
% 0.08/0.35 thf(fact_11_order__subst1,axiom,
% 0.08/0.35 ! [A: $tType,B: $tType] :
% 0.08/0.35 ( ( ( order @ B @ ( type2 @ B ) )
% 0.08/0.35 & ( order @ A @ ( type2 @ A ) ) )
% 0.08/0.35 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.08/0.35 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.08/0.35 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.08/0.35 => ( ! [X2: B,Y: B] :
% 0.08/0.35 ( ( ord_less_eq @ B @ X2 @ Y )
% 0.08/0.35 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.08/0.35 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % order_subst1
% 0.08/0.35 thf(fact_12_order__subst2,axiom,
% 0.08/0.35 ! [A: $tType,C2: $tType] :
% 0.08/0.35 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.08/0.35 & ( order @ A @ ( type2 @ A ) ) )
% 0.08/0.35 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.08/0.35 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.08/0.35 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 0.08/0.35 => ( ! [X2: A,Y: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.08/0.35 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.08/0.35 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % order_subst2
% 0.08/0.35 thf(fact_13_ord__eq__le__subst,axiom,
% 0.08/0.35 ! [A: $tType,B: $tType] :
% 0.08/0.35 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.08/0.35 & ( ord @ A @ ( type2 @ A ) ) )
% 0.08/0.35 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.08/0.35 ( ( A2
% 0.08/0.35 = ( F @ B2 ) )
% 0.08/0.35 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.08/0.35 => ( ! [X2: B,Y: B] :
% 0.08/0.35 ( ( ord_less_eq @ B @ X2 @ Y )
% 0.08/0.35 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.08/0.35 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ord_eq_le_subst
% 0.08/0.35 thf(fact_14_ord__le__eq__subst,axiom,
% 0.08/0.35 ! [A: $tType,B: $tType] :
% 0.08/0.35 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.08/0.35 & ( ord @ A @ ( type2 @ A ) ) )
% 0.08/0.35 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.08/0.35 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.08/0.35 => ( ( ( F @ B2 )
% 0.08/0.35 = C )
% 0.08/0.35 => ( ! [X2: A,Y: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.08/0.35 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.08/0.35 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ord_le_eq_subst
% 0.08/0.35 thf(fact_15_ldrop__0,axiom,
% 0.08/0.35 ! [B: $tType,Xs: coindu1593790203_llist @ B] :
% 0.08/0.35 ( ( coindu191418589_ldrop @ B @ ( zero_zero @ extended_enat ) @ Xs )
% 0.08/0.35 = Xs ) ).
% 0.08/0.35
% 0.08/0.35 % ldrop_0
% 0.08/0.35 thf(fact_16_llength__lnull,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ( ( coindu1018505716length @ A @ Xs )
% 0.08/0.35 = ( zero_zero @ extended_enat ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % llength_lnull
% 0.08/0.35 thf(fact_17_llength__eq__0,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ( coindu1018505716length @ A @ Xs )
% 0.08/0.35 = ( zero_zero @ extended_enat ) )
% 0.08/0.35 = ( coindu335574135_lnull @ A @ Xs ) ) ).
% 0.08/0.35
% 0.08/0.35 % llength_eq_0
% 0.08/0.35 thf(fact_18_lzip_Oexhaust,axiom,
% 0.08/0.35 ! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
% 0.08/0.35 ( ~ ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 | ( coindu335574135_lnull @ B @ Ys ) )
% 0.08/0.35 => ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lzip.exhaust
% 0.08/0.35 thf(fact_19_ltake_Oexhaust,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ~ ( ( N
% 0.08/0.35 = ( zero_zero @ extended_enat ) )
% 0.08/0.35 | ( coindu335574135_lnull @ A @ Xs ) )
% 0.08/0.35 => ~ ( ( N
% 0.08/0.35 != ( zero_zero @ extended_enat ) )
% 0.08/0.35 => ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake.exhaust
% 0.08/0.35 thf(fact_20_lappend_Oexhaust,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ~ ( coindu335574135_lnull @ A @ Ys ) )
% 0.08/0.35 => ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 | ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lappend.exhaust
% 0.08/0.35 thf(fact_21_lnull__def,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A )
% 0.08/0.35 = ( ^ [Llist: coindu1593790203_llist @ A] :
% 0.08/0.35 ( Llist
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lnull_def
% 0.08/0.35 thf(fact_22_llist_Ocollapse_I1_J,axiom,
% 0.08/0.35 ! [A: $tType,Llist2: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ Llist2 )
% 0.08/0.35 => ( Llist2
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % llist.collapse(1)
% 0.08/0.35 thf(fact_23_llist_OdiscI_I1_J,axiom,
% 0.08/0.35 ! [A: $tType,Llist2: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( Llist2
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 => ( coindu335574135_lnull @ A @ Llist2 ) ) ).
% 0.08/0.35
% 0.08/0.35 % llist.discI(1)
% 0.08/0.35 thf(fact_24_llist_Odisc_I1_J,axiom,
% 0.08/0.35 ! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% 0.08/0.35
% 0.08/0.35 % llist.disc(1)
% 0.08/0.35 thf(fact_25_ldrop__lnull,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,N: extended_enat] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ( ( coindu191418589_ldrop @ A @ N @ Xs )
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ldrop_lnull
% 0.08/0.35 thf(fact_26_dual__order_Oantisym,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( order @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [B2: A,A2: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.08/0.35 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.08/0.35 => ( A2 = B2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % dual_order.antisym
% 0.08/0.35 thf(fact_27_dual__order_Otrans,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( order @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [B2: A,A2: A,C: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.08/0.35 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.08/0.35 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % dual_order.trans
% 0.08/0.35 thf(fact_28_linorder__wlog,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( linorder @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.08/0.35 ( ! [A3: A,B3: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.08/0.35 => ( P @ A3 @ B3 ) )
% 0.08/0.35 => ( ! [A3: A,B3: A] :
% 0.08/0.35 ( ( P @ B3 @ A3 )
% 0.08/0.35 => ( P @ A3 @ B3 ) )
% 0.08/0.35 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % linorder_wlog
% 0.08/0.35 thf(fact_29_dual__order_Orefl,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( order @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.08/0.35
% 0.08/0.35 % dual_order.refl
% 0.08/0.35 thf(fact_30_order__trans,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( preorder @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A,Y2: A,Z: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.08/0.35 => ( ( ord_less_eq @ A @ Y2 @ Z )
% 0.08/0.35 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % order_trans
% 0.08/0.35 thf(fact_31_order__class_Oorder_Oantisym,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( order @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A,B2: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.08/0.35 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.08/0.35 => ( A2 = B2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % order_class.order.antisym
% 0.08/0.35 thf(fact_32_ord__le__eq__trans,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ord @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A,B2: A,C: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.08/0.35 => ( ( B2 = C )
% 0.08/0.35 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ord_le_eq_trans
% 0.08/0.35 thf(fact_33_ord__eq__le__trans,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ord @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A,B2: A,C: A] :
% 0.08/0.35 ( ( A2 = B2 )
% 0.08/0.35 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.08/0.35 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ord_eq_le_trans
% 0.08/0.35 thf(fact_34_antisym__conv,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( order @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [Y2: A,X: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ Y2 @ X )
% 0.08/0.35 => ( ( ord_less_eq @ A @ X @ Y2 )
% 0.08/0.35 = ( X = Y2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % antisym_conv
% 0.08/0.35 thf(fact_35_le__cases3,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( linorder @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A,Y2: A,Z: A] :
% 0.08/0.35 ( ( ( ord_less_eq @ A @ X @ Y2 )
% 0.08/0.35 => ~ ( ord_less_eq @ A @ Y2 @ Z ) )
% 0.08/0.35 => ( ( ( ord_less_eq @ A @ Y2 @ X )
% 0.08/0.35 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.08/0.35 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.08/0.35 => ~ ( ord_less_eq @ A @ Z @ Y2 ) )
% 0.08/0.35 => ( ( ( ord_less_eq @ A @ Z @ Y2 )
% 0.08/0.35 => ~ ( ord_less_eq @ A @ Y2 @ X ) )
% 0.08/0.35 => ( ( ( ord_less_eq @ A @ Y2 @ Z )
% 0.08/0.35 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.08/0.35 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.08/0.35 => ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_cases3
% 0.08/0.35 thf(fact_36_order_Otrans,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( order @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A,B2: A,C: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.08/0.35 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.08/0.35 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % order.trans
% 0.08/0.35 thf(fact_37_le__cases,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( linorder @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A,Y2: A] :
% 0.08/0.35 ( ~ ( ord_less_eq @ A @ X @ Y2 )
% 0.08/0.35 => ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_cases
% 0.08/0.35 thf(fact_38_eq__refl,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( preorder @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A,Y2: A] :
% 0.08/0.35 ( ( X = Y2 )
% 0.08/0.35 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % eq_refl
% 0.08/0.35 thf(fact_39_linear,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( linorder @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A,Y2: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.08/0.35 | ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % linear
% 0.08/0.35 thf(fact_40_antisym,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( order @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A,Y2: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.08/0.35 => ( ( ord_less_eq @ A @ Y2 @ X )
% 0.08/0.35 => ( X = Y2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % antisym
% 0.08/0.35 thf(fact_41_eq__iff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( order @ A @ ( type2 @ A ) )
% 0.08/0.35 => ( ( ^ [Y3: A,Z2: A] : ( Y3 = Z2 ) )
% 0.08/0.35 = ( ^ [X3: A,Y4: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.08/0.35 & ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % eq_iff
% 0.08/0.35 thf(fact_42_le__zero__eq,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [N: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.08/0.35 = ( N
% 0.08/0.35 = ( zero_zero @ A ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_zero_eq
% 0.08/0.35 thf(fact_43_ext,axiom,
% 0.08/0.35 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.08/0.35 ( ! [X2: A] :
% 0.08/0.35 ( ( F @ X2 )
% 0.08/0.35 = ( G @ X2 ) )
% 0.08/0.35 => ( F = G ) ) ).
% 0.08/0.35
% 0.08/0.35 % ext
% 0.08/0.35 thf(fact_44_ile0__eq,axiom,
% 0.08/0.35 ! [N: extended_enat] :
% 0.08/0.35 ( ( ord_less_eq @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
% 0.08/0.35 = ( N
% 0.08/0.35 = ( zero_zero @ extended_enat ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ile0_eq
% 0.08/0.35 thf(fact_45_i0__lb,axiom,
% 0.08/0.35 ! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% 0.08/0.35
% 0.08/0.35 % i0_lb
% 0.08/0.35 thf(fact_46_zero__le,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.08/0.35
% 0.08/0.35 % zero_le
% 0.08/0.35 thf(fact_47_le__numeral__extra_I3_J,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.08/0.35 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_numeral_extra(3)
% 0.08/0.35 thf(fact_48_ltake_Octr_I1_J,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ( N
% 0.08/0.35 = ( zero_zero @ extended_enat ) )
% 0.08/0.35 | ( coindu335574135_lnull @ A @ Xs ) )
% 0.08/0.35 => ( ( coindu1802687541_ltake @ A @ N @ Xs )
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake.ctr(1)
% 0.08/0.35 thf(fact_49_zero__reorient,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( zero @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A] :
% 0.08/0.35 ( ( ( zero_zero @ A )
% 0.08/0.35 = X )
% 0.08/0.35 = ( X
% 0.08/0.35 = ( zero_zero @ A ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % zero_reorient
% 0.08/0.35 thf(fact_50_lfp_Oleq__antisym,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A,Y2: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.08/0.35 => ( ( ord_less_eq @ A @ Y2 @ X )
% 0.08/0.35 => ( X = Y2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lfp.leq_antisym
% 0.08/0.35 thf(fact_51_ltake__LNil,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat] :
% 0.08/0.35 ( ( coindu1802687541_ltake @ A @ N @ ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake_LNil
% 0.08/0.35 thf(fact_52_lnull__ltake,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
% 0.08/0.35 = ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 | ( N
% 0.08/0.35 = ( zero_zero @ extended_enat ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lnull_ltake
% 0.08/0.35 thf(fact_53_ltake_Odisc__iff_I1_J,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
% 0.08/0.35 = ( ( N
% 0.08/0.35 = ( zero_zero @ extended_enat ) )
% 0.08/0.35 | ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake.disc_iff(1)
% 0.08/0.35 thf(fact_54_ltake_Odisc__iff_I2_J,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ~ ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) )
% 0.08/0.35 = ( ( N
% 0.08/0.35 != ( zero_zero @ extended_enat ) )
% 0.08/0.35 & ~ ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake.disc_iff(2)
% 0.08/0.35 thf(fact_55_ltake__0,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu1802687541_ltake @ A @ ( zero_zero @ extended_enat ) @ Xs )
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake_0
% 0.08/0.35 thf(fact_56_LNil__eq__ltake__iff,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ( coindu1598213697e_LNil @ A )
% 0.08/0.35 = ( coindu1802687541_ltake @ A @ N @ Xs ) )
% 0.08/0.35 = ( ( Xs
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 | ( N
% 0.08/0.35 = ( zero_zero @ extended_enat ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % LNil_eq_ltake_iff
% 0.08/0.35 thf(fact_57_ltake__eq__ltake__antimono,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,M: extended_enat] :
% 0.08/0.35 ( ( ( coindu1802687541_ltake @ A @ N @ Xs )
% 0.08/0.35 = ( coindu1802687541_ltake @ A @ N @ Ys ) )
% 0.08/0.35 => ( ( ord_less_eq @ extended_enat @ M @ N )
% 0.08/0.35 => ( ( coindu1802687541_ltake @ A @ M @ Xs )
% 0.08/0.35 = ( coindu1802687541_ltake @ A @ M @ Ys ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake_eq_ltake_antimono
% 0.08/0.35 thf(fact_58_ltake_Odisc_I1_J,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ( N
% 0.08/0.35 = ( zero_zero @ extended_enat ) )
% 0.08/0.35 | ( coindu335574135_lnull @ A @ Xs ) )
% 0.08/0.35 => ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake.disc(1)
% 0.08/0.35 thf(fact_59_ltake_Odisc_I2_J,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( N
% 0.08/0.35 != ( zero_zero @ extended_enat ) )
% 0.08/0.35 => ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ~ ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake.disc(2)
% 0.08/0.35 thf(fact_60_ltake__eq__LNil__iff,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ( coindu1802687541_ltake @ A @ N @ Xs )
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 = ( ( Xs
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 | ( N
% 0.08/0.35 = ( zero_zero @ extended_enat ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake_eq_LNil_iff
% 0.08/0.35 thf(fact_61_ltake__all,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,M: extended_enat] :
% 0.08/0.35 ( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M )
% 0.08/0.35 => ( ( coindu1802687541_ltake @ A @ M @ Xs )
% 0.08/0.35 = Xs ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake_all
% 0.08/0.35 thf(fact_62_lfp_Oleq__refl,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.08/0.35
% 0.08/0.35 % lfp.leq_refl
% 0.08/0.35 thf(fact_63_gfp_Oleq__trans,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [Y2: A,X: A,Z: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ Y2 @ X )
% 0.08/0.35 => ( ( ord_less_eq @ A @ Z @ Y2 )
% 0.08/0.35 => ( ord_less_eq @ A @ Z @ X ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % gfp.leq_trans
% 0.08/0.35 thf(fact_64_lfp_Oleq__trans,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A,Y2: A,Z: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.08/0.35 => ( ( ord_less_eq @ A @ Y2 @ Z )
% 0.08/0.35 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lfp.leq_trans
% 0.08/0.35 thf(fact_65_gfp_Oleq__antisym,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [Y2: A,X: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ Y2 @ X )
% 0.08/0.35 => ( ( ord_less_eq @ A @ X @ Y2 )
% 0.08/0.35 => ( X = Y2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % gfp.leq_antisym
% 0.08/0.35 thf(fact_66_the__enat__0,axiom,
% 0.08/0.35 ( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
% 0.08/0.35 = ( zero_zero @ nat ) ) ).
% 0.08/0.35
% 0.08/0.35 % the_enat_0
% 0.08/0.35 thf(fact_67_case__enat__0,axiom,
% 0.08/0.35 ! [A: $tType,F: nat > A,I: A] :
% 0.08/0.35 ( ( extended_case_enat @ A @ F @ I @ ( zero_zero @ extended_enat ) )
% 0.08/0.35 = ( F @ ( zero_zero @ nat ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % case_enat_0
% 0.08/0.35 thf(fact_68_lprefix__ltake__same,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,M: extended_enat] :
% 0.08/0.35 ( ( coindu1696667936prefix @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ A @ M @ Xs ) )
% 0.08/0.35 = ( ( ord_less_eq @ extended_enat @ N @ M )
% 0.08/0.35 | ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lprefix_ltake_same
% 0.08/0.35 thf(fact_69_ltake__lappend1,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ord_less_eq @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) )
% 0.08/0.35 => ( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 0.08/0.35 = ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake_lappend1
% 0.08/0.35 thf(fact_70_llength__code,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( coindu1018505716length @ A )
% 0.08/0.35 = ( coindu493225792length @ A @ ( zero_zero @ nat ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % llength_code
% 0.08/0.35 thf(fact_71_dbl__simps_I2_J,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( neg_numeral @ A @ ( type2 @ A ) )
% 0.08/0.35 => ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
% 0.08/0.35 = ( zero_zero @ A ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % dbl_simps(2)
% 0.08/0.35 thf(fact_72_ltake_Osimps_I3_J,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( N
% 0.08/0.35 != ( zero_zero @ extended_enat ) )
% 0.08/0.35 => ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ( ( coindu1046438764le_lhd @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
% 0.08/0.35 = ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake.simps(3)
% 0.08/0.35 thf(fact_73_lprefix__refl,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ Xs ) ).
% 0.08/0.35
% 0.08/0.35 % lprefix_refl
% 0.08/0.35 thf(fact_74_llist_Oleq__refl,axiom,
% 0.08/0.35 ! [A: $tType,X: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X @ X ) ).
% 0.08/0.35
% 0.08/0.35 % llist.leq_refl
% 0.08/0.35 thf(fact_75_lprefix__code_I1_J,axiom,
% 0.08/0.35 ! [A: $tType,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% 0.08/0.35
% 0.08/0.35 % lprefix_code(1)
% 0.08/0.35 thf(fact_76_ltake__is__lprefix,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ Xs ) ).
% 0.08/0.35
% 0.08/0.35 % ltake_is_lprefix
% 0.08/0.35 thf(fact_77_lappend_Odisc__iff_I2_J,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) )
% 0.08/0.35 = ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 | ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lappend.disc_iff(2)
% 0.08/0.35 thf(fact_78_lnull__lappend,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 0.08/0.35 = ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 & ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lnull_lappend
% 0.08/0.35 thf(fact_79_lappend__LNil2,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 = Xs ) ).
% 0.08/0.35
% 0.08/0.35 % lappend_LNil2
% 0.08/0.35 thf(fact_80_lappend__code_I1_J,axiom,
% 0.08/0.35 ! [A: $tType,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
% 0.08/0.35 = Ys ) ).
% 0.08/0.35
% 0.08/0.35 % lappend_code(1)
% 0.08/0.35 thf(fact_81_lprefix__LNil,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu1696667936prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 = ( coindu335574135_lnull @ A @ Xs ) ) ).
% 0.08/0.35
% 0.08/0.35 % lprefix_LNil
% 0.08/0.35 thf(fact_82_lhd__lappend,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 0.08/0.35 = ( coindu1046438764le_lhd @ A @ Ys ) ) )
% 0.08/0.35 & ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 0.08/0.35 = ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lhd_lappend
% 0.08/0.35 thf(fact_83_lhd__ltake,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( N
% 0.08/0.35 != ( zero_zero @ extended_enat ) )
% 0.08/0.35 => ( ( coindu1046438764le_lhd @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
% 0.08/0.35 = ( coindu1046438764le_lhd @ A @ Xs ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lhd_ltake
% 0.08/0.35 thf(fact_84_lprefix__lhdD,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 0.08/0.35 => ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ( ( coindu1046438764le_lhd @ A @ Xs )
% 0.08/0.35 = ( coindu1046438764le_lhd @ A @ Ys ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lprefix_lhdD
% 0.08/0.35 thf(fact_85_lappend__assoc,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
% 0.08/0.35 = ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lappend_assoc
% 0.08/0.35 thf(fact_86_lprefix__trans,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 0.08/0.35 => ( ( coindu1696667936prefix @ A @ Ys @ Zs )
% 0.08/0.35 => ( coindu1696667936prefix @ A @ Xs @ Zs ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lprefix_trans
% 0.08/0.35 thf(fact_87_llist_Oleq__trans,axiom,
% 0.08/0.35 ! [A: $tType,X: coindu1593790203_llist @ A,Y2: coindu1593790203_llist @ A,Z: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu1696667936prefix @ A @ X @ Y2 )
% 0.08/0.35 => ( ( coindu1696667936prefix @ A @ Y2 @ Z )
% 0.08/0.35 => ( coindu1696667936prefix @ A @ X @ Z ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % llist.leq_trans
% 0.08/0.35 thf(fact_88_lprefix__antisym,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 0.08/0.35 => ( ( coindu1696667936prefix @ A @ Ys @ Xs )
% 0.08/0.35 => ( Xs = Ys ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lprefix_antisym
% 0.08/0.35 thf(fact_89_lprefix__lappend,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% 0.08/0.35
% 0.08/0.35 % lprefix_lappend
% 0.08/0.35 thf(fact_90_llist_Oleq__antisym,axiom,
% 0.08/0.35 ! [A: $tType,X: coindu1593790203_llist @ A,Y2: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu1696667936prefix @ A @ X @ Y2 )
% 0.08/0.35 => ( ( coindu1696667936prefix @ A @ Y2 @ X )
% 0.08/0.35 => ( X = Y2 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % llist.leq_antisym
% 0.08/0.35 thf(fact_91_lprefix__down__linear,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu1696667936prefix @ A @ Xs @ Zs )
% 0.08/0.35 => ( ( coindu1696667936prefix @ A @ Ys @ Zs )
% 0.08/0.35 => ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 0.08/0.35 | ( coindu1696667936prefix @ A @ Ys @ Xs ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lprefix_down_linear
% 0.08/0.35 thf(fact_92_lprefix__lappend__sameI,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 0.08/0.35 => ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Zs @ Xs ) @ ( coindu268472904append @ A @ Zs @ Ys ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lprefix_lappend_sameI
% 0.08/0.35 thf(fact_93_ltakeWhile_Oexhaust,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.08/0.35 ( ~ ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
% 0.08/0.35 => ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltakeWhile.exhaust
% 0.08/0.35 thf(fact_94_lnull__lprefix,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ).
% 0.08/0.35
% 0.08/0.35 % lnull_lprefix
% 0.08/0.35 thf(fact_95_lprefix__lnull,axiom,
% 0.08/0.35 ! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ Ys )
% 0.08/0.35 => ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 0.08/0.35 = ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lprefix_lnull
% 0.08/0.35 thf(fact_96_lprefix__lnullD,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 0.08/0.35 => ( ( coindu335574135_lnull @ A @ Ys )
% 0.08/0.35 => ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lprefix_lnullD
% 0.08/0.35 thf(fact_97_lprefix__not__lnullD,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 0.08/0.35 => ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lprefix_not_lnullD
% 0.08/0.35 thf(fact_98_lappend_Odisc_I2_J,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 | ~ ( coindu335574135_lnull @ A @ Ys ) )
% 0.08/0.35 => ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lappend.disc(2)
% 0.08/0.35 thf(fact_99_lappend_Odisc_I1_J,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ( ( coindu335574135_lnull @ A @ Ys )
% 0.08/0.35 => ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lappend.disc(1)
% 0.08/0.35 thf(fact_100_lappend__lnull1,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ( ( coindu268472904append @ A @ Xs @ Ys )
% 0.08/0.35 = Ys ) ) ).
% 0.08/0.35
% 0.08/0.35 % lappend_lnull1
% 0.08/0.35 thf(fact_101_lappend__lnull2,axiom,
% 0.08/0.35 ! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ Ys )
% 0.08/0.35 => ( ( coindu268472904append @ A @ Xs @ Ys )
% 0.08/0.35 = Xs ) ) ).
% 0.08/0.35
% 0.08/0.35 % lappend_lnull2
% 0.08/0.35 thf(fact_102_LNil__lprefix,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% 0.08/0.35
% 0.08/0.35 % LNil_lprefix
% 0.08/0.35 thf(fact_103_lappend__eq__LNil__iff,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ( coindu268472904append @ A @ Xs @ Ys )
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 = ( ( Xs
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 & ( Ys
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lappend_eq_LNil_iff
% 0.08/0.35 thf(fact_104_LNil__eq__lappend__iff,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ( coindu1598213697e_LNil @ A )
% 0.08/0.35 = ( coindu268472904append @ A @ Xs @ Ys ) )
% 0.08/0.35 = ( ( Xs
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 & ( Ys
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % LNil_eq_lappend_iff
% 0.08/0.35 thf(fact_105_lappend__LNil__LNil,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) ) ).
% 0.08/0.35
% 0.08/0.35 % lappend_LNil_LNil
% 0.08/0.35 thf(fact_106_lappend_Octr_I1_J,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ( ( coindu335574135_lnull @ A @ Ys )
% 0.08/0.35 => ( ( coindu268472904append @ A @ Xs @ Ys )
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lappend.ctr(1)
% 0.08/0.35 thf(fact_107_le0,axiom,
% 0.08/0.35 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.08/0.35
% 0.08/0.35 % le0
% 0.08/0.35 thf(fact_108_lstrict__prefix__def,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( coindu574146665prefix @ A )
% 0.08/0.35 = ( ^ [Xs2: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu1696667936prefix @ A @ Xs2 @ Ys2 )
% 0.08/0.35 & ( Xs2 != Ys2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % lstrict_prefix_def
% 0.08/0.35 thf(fact_109_ltakeWhile_Octr_I1_J,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.08/0.35 ( ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
% 0.08/0.35 => ( ( coindu721411036eWhile @ A @ P @ Xs )
% 0.08/0.35 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltakeWhile.ctr(1)
% 0.08/0.35 thf(fact_110_ltakeWhile_Odisc__iff_I1_J,axiom,
% 0.08/0.35 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 0.08/0.35 = ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltakeWhile.disc_iff(1)
% 0.08/0.35 thf(fact_111_ltakeWhile_Odisc__iff_I2_J,axiom,
% 0.08/0.35 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
% 0.08/0.35 = ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 & ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltakeWhile.disc_iff(2)
% 0.08/0.35 thf(fact_112_ltake__lappend2,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,N: extended_enat,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N )
% 0.08/0.35 => ( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 0.08/0.35 = ( coindu268472904append @ A @ Xs @ ( coindu1802687541_ltake @ A @ ( minus_minus @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) ) @ Ys ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake_lappend2
% 0.08/0.35 thf(fact_113_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A] :
% 0.08/0.35 ( ( minus_minus @ A @ A2 @ A2 )
% 0.08/0.35 = ( zero_zero @ A ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % cancel_comm_monoid_add_class.diff_cancel
% 0.08/0.35 thf(fact_114_diff__zero,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A] :
% 0.08/0.35 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.08/0.35 = A2 ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_zero
% 0.08/0.35 thf(fact_115_zero__diff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( comm_monoid_diff @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A] :
% 0.08/0.35 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.08/0.35 = ( zero_zero @ A ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % zero_diff
% 0.08/0.35 thf(fact_116_diff__0__right,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( group_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A] :
% 0.08/0.35 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.08/0.35 = A2 ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_0_right
% 0.08/0.35 thf(fact_117_diff__self,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( group_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A] :
% 0.08/0.35 ( ( minus_minus @ A @ A2 @ A2 )
% 0.08/0.35 = ( zero_zero @ A ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_self
% 0.08/0.35 thf(fact_118_idiff__0__right,axiom,
% 0.08/0.35 ! [N: extended_enat] :
% 0.08/0.35 ( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
% 0.08/0.35 = N ) ).
% 0.08/0.35
% 0.08/0.35 % idiff_0_right
% 0.08/0.35 thf(fact_119_idiff__0,axiom,
% 0.08/0.35 ! [N: extended_enat] :
% 0.08/0.35 ( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
% 0.08/0.35 = ( zero_zero @ extended_enat ) ) ).
% 0.08/0.35
% 0.08/0.35 % idiff_0
% 0.08/0.35 thf(fact_120_diff__ge__0__iff__ge,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A,B2: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
% 0.08/0.35 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_ge_0_iff_ge
% 0.08/0.35 thf(fact_121_le__refl,axiom,
% 0.08/0.35 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.08/0.35
% 0.08/0.35 % le_refl
% 0.08/0.35 thf(fact_122_le__trans,axiom,
% 0.08/0.35 ! [I: nat,J: nat,K: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ I @ J )
% 0.08/0.35 => ( ( ord_less_eq @ nat @ J @ K )
% 0.08/0.35 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_trans
% 0.08/0.35 thf(fact_123_eq__imp__le,axiom,
% 0.08/0.35 ! [M: nat,N: nat] :
% 0.08/0.35 ( ( M = N )
% 0.08/0.35 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.08/0.35
% 0.08/0.35 % eq_imp_le
% 0.08/0.35 thf(fact_124_le__antisym,axiom,
% 0.08/0.35 ! [M: nat,N: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ M @ N )
% 0.08/0.35 => ( ( ord_less_eq @ nat @ N @ M )
% 0.08/0.35 => ( M = N ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_antisym
% 0.08/0.35 thf(fact_125_nat__le__linear,axiom,
% 0.08/0.35 ! [M: nat,N: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ M @ N )
% 0.08/0.35 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.08/0.35
% 0.08/0.35 % nat_le_linear
% 0.08/0.35 thf(fact_126_diff__eq__diff__eq,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( group_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A,B2: A,C: A,D: A] :
% 0.08/0.35 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.08/0.35 = ( minus_minus @ A @ C @ D ) )
% 0.08/0.35 => ( ( A2 = B2 )
% 0.08/0.35 = ( C = D ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_eq_diff_eq
% 0.08/0.35 thf(fact_127_diff__right__commute,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( cancel146912293up_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A,C: A,B2: A] :
% 0.08/0.35 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 )
% 0.08/0.35 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_right_commute
% 0.08/0.35 thf(fact_128_diff__mono,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A,B2: A,D: A,C: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.08/0.35 => ( ( ord_less_eq @ A @ D @ C )
% 0.08/0.35 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_mono
% 0.08/0.35 thf(fact_129_diff__left__mono,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [B2: A,A2: A,C: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.08/0.35 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_left_mono
% 0.08/0.35 thf(fact_130_diff__right__mono,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A,B2: A,C: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.08/0.35 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_right_mono
% 0.08/0.35 thf(fact_131_diff__eq__diff__less__eq,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [A2: A,B2: A,C: A,D: A] :
% 0.08/0.35 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.08/0.35 = ( minus_minus @ A @ C @ D ) )
% 0.08/0.35 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.08/0.35 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_eq_diff_less_eq
% 0.08/0.35 thf(fact_132_eq__iff__diff__eq__0,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( group_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ( ( ^ [Y3: A,Z2: A] : ( Y3 = Z2 ) )
% 0.08/0.35 = ( ^ [A4: A,B4: A] :
% 0.08/0.35 ( ( minus_minus @ A @ A4 @ B4 )
% 0.08/0.35 = ( zero_zero @ A ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % eq_iff_diff_eq_0
% 0.08/0.35 thf(fact_133_enat__minus__mono1,axiom,
% 0.08/0.35 ! [X: extended_enat,Y2: extended_enat,M: extended_enat] :
% 0.08/0.35 ( ( ord_less_eq @ extended_enat @ X @ Y2 )
% 0.08/0.35 => ( ord_less_eq @ extended_enat @ ( minus_minus @ extended_enat @ X @ M ) @ ( minus_minus @ extended_enat @ Y2 @ M ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % enat_minus_mono1
% 0.08/0.35 thf(fact_134_enat__diff__cancel__left,axiom,
% 0.08/0.35 ! [M: extended_enat,X: extended_enat,Y2: extended_enat] :
% 0.08/0.35 ( ( ord_less_eq @ extended_enat @ M @ X )
% 0.08/0.35 => ( ( ord_less_eq @ extended_enat @ M @ Y2 )
% 0.08/0.35 => ( ( ( minus_minus @ extended_enat @ X @ M )
% 0.08/0.35 = ( minus_minus @ extended_enat @ Y2 @ M ) )
% 0.08/0.35 = ( X = Y2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % enat_diff_cancel_left
% 0.08/0.35 thf(fact_135_le__iff__diff__le__0,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.08/0.35 => ( ( ord_less_eq @ A )
% 0.08/0.35 = ( ^ [A4: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_iff_diff_le_0
% 0.08/0.35 thf(fact_136_le__0__eq,axiom,
% 0.08/0.35 ! [N: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.08/0.35 = ( N
% 0.08/0.35 = ( zero_zero @ nat ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_0_eq
% 0.08/0.35 thf(fact_137_less__eq__nat_Osimps_I1_J,axiom,
% 0.08/0.35 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.08/0.35
% 0.08/0.35 % less_eq_nat.simps(1)
% 0.08/0.35 thf(fact_138_ltake__lappend,axiom,
% 0.08/0.35 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.08/0.35 ( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 0.08/0.35 = ( coindu268472904append @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ A @ ( minus_minus @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) ) @ Ys ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltake_lappend
% 0.08/0.35 thf(fact_139_ltakeWhile_Odisc_I2_J,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.08/0.35 ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
% 0.08/0.35 => ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltakeWhile.disc(2)
% 0.08/0.35 thf(fact_140_ltakeWhile_Odisc_I1_J,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.08/0.35 ( ( ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
% 0.08/0.35 => ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltakeWhile.disc(1)
% 0.08/0.35 thf(fact_141_ltakeWhile_Osimps_I3_J,axiom,
% 0.08/0.35 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 0.08/0.35 ( ~ ( coindu335574135_lnull @ A @ Xs )
% 0.08/0.35 => ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
% 0.08/0.35 => ( ( coindu1046438764le_lhd @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 0.08/0.35 = ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ltakeWhile.simps(3)
% 0.08/0.35 thf(fact_142_minus__apply,axiom,
% 0.08/0.35 ! [B: $tType,A: $tType] :
% 0.08/0.35 ( ( minus @ B @ ( type2 @ B ) )
% 0.08/0.35 => ( ( minus_minus @ ( A > B ) )
% 0.08/0.35 = ( ^ [A5: A > B,B5: A > B,X3: A] : ( minus_minus @ B @ ( A5 @ X3 ) @ ( B5 @ X3 ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % minus_apply
% 0.08/0.35 thf(fact_143_of__nat__le__0__iff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [M: nat] :
% 0.08/0.35 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 0.08/0.35 = ( M
% 0.08/0.35 = ( zero_zero @ nat ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_nat_le_0_iff
% 0.08/0.35 thf(fact_144_ex__has__least__nat,axiom,
% 0.08/0.35 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 0.08/0.35 ( ( P @ K )
% 0.08/0.35 => ? [X2: A] :
% 0.08/0.35 ( ( P @ X2 )
% 0.08/0.35 & ! [Y5: A] :
% 0.08/0.35 ( ( P @ Y5 )
% 0.08/0.35 => ( ord_less_eq @ nat @ ( M @ X2 ) @ ( M @ Y5 ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ex_has_least_nat
% 0.08/0.35 thf(fact_145_zero__natural_Orsp,axiom,
% 0.08/0.35 ( ( zero_zero @ nat )
% 0.08/0.35 = ( zero_zero @ nat ) ) ).
% 0.08/0.35
% 0.08/0.35 % zero_natural.rsp
% 0.08/0.35 thf(fact_146_diff__self__eq__0,axiom,
% 0.08/0.35 ! [M: nat] :
% 0.08/0.35 ( ( minus_minus @ nat @ M @ M )
% 0.08/0.35 = ( zero_zero @ nat ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_self_eq_0
% 0.08/0.35 thf(fact_147_diff__0__eq__0,axiom,
% 0.08/0.35 ! [N: nat] :
% 0.08/0.35 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 0.08/0.35 = ( zero_zero @ nat ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_0_eq_0
% 0.08/0.35 thf(fact_148_of__nat__eq__iff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [M: nat,N: nat] :
% 0.08/0.35 ( ( ( semiring_1_of_nat @ A @ M )
% 0.08/0.35 = ( semiring_1_of_nat @ A @ N ) )
% 0.08/0.35 = ( M = N ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_nat_eq_iff
% 0.08/0.35 thf(fact_149_diff__diff__cancel,axiom,
% 0.08/0.35 ! [I: nat,N: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ I @ N )
% 0.08/0.35 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 0.08/0.35 = I ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_diff_cancel
% 0.08/0.35 thf(fact_150_diff__is__0__eq_H,axiom,
% 0.08/0.35 ! [M: nat,N: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ M @ N )
% 0.08/0.35 => ( ( minus_minus @ nat @ M @ N )
% 0.08/0.35 = ( zero_zero @ nat ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_is_0_eq'
% 0.08/0.35 thf(fact_151_diff__is__0__eq,axiom,
% 0.08/0.35 ! [M: nat,N: nat] :
% 0.08/0.35 ( ( ( minus_minus @ nat @ M @ N )
% 0.08/0.35 = ( zero_zero @ nat ) )
% 0.08/0.35 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_is_0_eq
% 0.08/0.35 thf(fact_152_of__nat__eq__0__iff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [M: nat] :
% 0.08/0.35 ( ( ( semiring_1_of_nat @ A @ M )
% 0.08/0.35 = ( zero_zero @ A ) )
% 0.08/0.35 = ( M
% 0.08/0.35 = ( zero_zero @ nat ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_nat_eq_0_iff
% 0.08/0.35 thf(fact_153_of__nat__0__eq__iff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [N: nat] :
% 0.08/0.35 ( ( ( zero_zero @ A )
% 0.08/0.35 = ( semiring_1_of_nat @ A @ N ) )
% 0.08/0.35 = ( ( zero_zero @ nat )
% 0.08/0.35 = N ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_nat_0_eq_iff
% 0.08/0.35 thf(fact_154_of__nat__0,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 0.08/0.35 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 0.08/0.35 = ( zero_zero @ A ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_nat_0
% 0.08/0.35 thf(fact_155_of__nat__le__iff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [M: nat,N: nat] :
% 0.08/0.35 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.08/0.35 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_nat_le_iff
% 0.08/0.35 thf(fact_156_diffs0__imp__equal,axiom,
% 0.08/0.35 ! [M: nat,N: nat] :
% 0.08/0.35 ( ( ( minus_minus @ nat @ M @ N )
% 0.08/0.35 = ( zero_zero @ nat ) )
% 0.08/0.35 => ( ( ( minus_minus @ nat @ N @ M )
% 0.08/0.35 = ( zero_zero @ nat ) )
% 0.08/0.35 => ( M = N ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % diffs0_imp_equal
% 0.08/0.35 thf(fact_157_minus__nat_Odiff__0,axiom,
% 0.08/0.35 ! [M: nat] :
% 0.08/0.35 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 0.08/0.35 = M ) ).
% 0.08/0.35
% 0.08/0.35 % minus_nat.diff_0
% 0.08/0.35 thf(fact_158_eq__diff__iff,axiom,
% 0.08/0.35 ! [K: nat,M: nat,N: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ K @ M )
% 0.08/0.35 => ( ( ord_less_eq @ nat @ K @ N )
% 0.08/0.35 => ( ( ( minus_minus @ nat @ M @ K )
% 0.08/0.35 = ( minus_minus @ nat @ N @ K ) )
% 0.08/0.35 = ( M = N ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % eq_diff_iff
% 0.08/0.35 thf(fact_159_le__diff__iff,axiom,
% 0.08/0.35 ! [K: nat,M: nat,N: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ K @ M )
% 0.08/0.35 => ( ( ord_less_eq @ nat @ K @ N )
% 0.08/0.35 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.08/0.35 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_diff_iff
% 0.08/0.35 thf(fact_160_Nat_Odiff__diff__eq,axiom,
% 0.08/0.35 ! [K: nat,M: nat,N: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ K @ M )
% 0.08/0.35 => ( ( ord_less_eq @ nat @ K @ N )
% 0.08/0.35 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.08/0.35 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % Nat.diff_diff_eq
% 0.08/0.35 thf(fact_161_diff__le__mono,axiom,
% 0.08/0.35 ! [M: nat,N: nat,L: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ M @ N )
% 0.08/0.35 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_le_mono
% 0.08/0.35 thf(fact_162_diff__le__self,axiom,
% 0.08/0.35 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 0.08/0.35
% 0.08/0.35 % diff_le_self
% 0.08/0.35 thf(fact_163_le__diff__iff_H,axiom,
% 0.08/0.35 ! [A2: nat,C: nat,B2: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ A2 @ C )
% 0.08/0.35 => ( ( ord_less_eq @ nat @ B2 @ C )
% 0.08/0.35 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B2 ) )
% 0.08/0.35 = ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_diff_iff'
% 0.08/0.35 thf(fact_164_diff__le__mono2,axiom,
% 0.08/0.35 ! [M: nat,N: nat,L: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ M @ N )
% 0.08/0.35 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_le_mono2
% 0.08/0.35 thf(fact_165_of__nat__0__le__iff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_nat_0_le_iff
% 0.08/0.35 thf(fact_166_of__nat__diff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ring_1 @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [N: nat,M: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ N @ M )
% 0.08/0.35 => ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N ) )
% 0.08/0.35 = ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_nat_diff
% 0.08/0.35 thf(fact_167_fun__diff__def,axiom,
% 0.08/0.35 ! [B: $tType,A: $tType] :
% 0.08/0.35 ( ( minus @ B @ ( type2 @ B ) )
% 0.08/0.35 => ( ( minus_minus @ ( A > B ) )
% 0.08/0.35 = ( ^ [A5: A > B,B5: A > B,X3: A] : ( minus_minus @ B @ ( A5 @ X3 ) @ ( B5 @ X3 ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % fun_diff_def
% 0.08/0.35 thf(fact_168_real__arch__simple,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A] :
% 0.08/0.35 ? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % real_arch_simple
% 0.08/0.35 thf(fact_169_natural_Osize_I1_J,axiom,
% 0.08/0.35 ( ( code_size_natural @ ( zero_zero @ code_natural ) )
% 0.08/0.35 = ( zero_zero @ nat ) ) ).
% 0.08/0.35
% 0.08/0.35 % natural.size(1)
% 0.08/0.35 thf(fact_170_natural_Osize_I3_J,axiom,
% 0.08/0.35 ( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
% 0.08/0.35 = ( zero_zero @ nat ) ) ).
% 0.08/0.35
% 0.08/0.35 % natural.size(3)
% 0.08/0.35 thf(fact_171_diff__commute,axiom,
% 0.08/0.35 ! [I: nat,J: nat,K: nat] :
% 0.08/0.35 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 0.08/0.35 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 0.08/0.35
% 0.08/0.35 % diff_commute
% 0.08/0.35 thf(fact_172_natural_Osimps_I4_J,axiom,
% 0.08/0.35 ! [T: $tType,F1: T,F22: code_natural > T] :
% 0.08/0.35 ( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 0.08/0.35 = F1 ) ).
% 0.08/0.35
% 0.08/0.35 % natural.simps(4)
% 0.08/0.35 thf(fact_173_natural_Osimps_I6_J,axiom,
% 0.08/0.35 ! [T: $tType,F1: T,F22: code_natural > T > T] :
% 0.08/0.35 ( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 0.08/0.35 = F1 ) ).
% 0.08/0.35
% 0.08/0.35 % natural.simps(6)
% 0.08/0.35 thf(fact_174_LeastMI2,axiom,
% 0.08/0.35 ! [B: $tType,A: $tType] :
% 0.08/0.35 ( ( ord @ B @ ( type2 @ B ) )
% 0.08/0.35 => ! [P: A > $o,X: A,M: A > B,Q: A > $o] :
% 0.08/0.35 ( ( P @ X )
% 0.08/0.35 => ( ! [Y: A] :
% 0.08/0.35 ( ( P @ Y )
% 0.08/0.35 => ( ord_less_eq @ B @ ( M @ X ) @ ( M @ Y ) ) )
% 0.08/0.35 => ( ! [X2: A] :
% 0.08/0.35 ( ( P @ X2 )
% 0.08/0.35 => ( ! [Y5: A] :
% 0.08/0.35 ( ( P @ Y5 )
% 0.08/0.35 => ( ord_less_eq @ B @ ( M @ X2 ) @ ( M @ Y5 ) ) )
% 0.08/0.35 => ( Q @ X2 ) ) )
% 0.08/0.35 => ( Q @ ( hilbert_LeastM @ A @ B @ M @ P ) ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % LeastMI2
% 0.08/0.35 thf(fact_175_LeastM__nat__le,axiom,
% 0.08/0.35 ! [A: $tType,P: A > $o,X: A,M: A > nat] :
% 0.08/0.35 ( ( P @ X )
% 0.08/0.35 => ( ord_less_eq @ nat @ ( M @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) @ ( M @ X ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % LeastM_nat_le
% 0.08/0.35 thf(fact_176_LeastM__nat__lemma,axiom,
% 0.08/0.35 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 0.08/0.35 ( ( P @ K )
% 0.08/0.35 => ( ( P @ ( hilbert_LeastM @ A @ nat @ M @ P ) )
% 0.08/0.35 & ! [Y5: A] :
% 0.08/0.35 ( ( P @ Y5 )
% 0.08/0.35 => ( ord_less_eq @ nat @ ( M @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) @ ( M @ Y5 ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % LeastM_nat_lemma
% 0.08/0.35 thf(fact_177_natural_Osimps_I5_J,axiom,
% 0.08/0.35 ! [T: $tType,F1: T,F22: code_natural > T,Natural: code_natural] :
% 0.08/0.35 ( ( code_case_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
% 0.08/0.35 = ( F22 @ Natural ) ) ).
% 0.08/0.35
% 0.08/0.35 % natural.simps(5)
% 0.08/0.35 thf(fact_178_transfer__int__nat__relations_I3_J,axiom,
% 0.08/0.35 ! [X: nat,Y2: nat] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) )
% 0.08/0.35 = ( ord_less_eq @ nat @ X @ Y2 ) ) ).
% 0.08/0.35
% 0.08/0.35 % transfer_int_nat_relations(3)
% 0.08/0.35 thf(fact_179_zle__int,axiom,
% 0.08/0.35 ! [M: nat,N: nat] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 0.08/0.35 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.08/0.35
% 0.08/0.35 % zle_int
% 0.08/0.35 thf(fact_180_natural_Oinject,axiom,
% 0.08/0.35 ! [Natural: code_natural,Natural2: code_natural] :
% 0.08/0.35 ( ( ( code_Suc @ Natural )
% 0.08/0.35 = ( code_Suc @ Natural2 ) )
% 0.08/0.35 = ( Natural = Natural2 ) ) ).
% 0.08/0.35
% 0.08/0.35 % natural.inject
% 0.08/0.35 thf(fact_181_natural_Osimps_I7_J,axiom,
% 0.08/0.35 ! [T: $tType,F1: T,F22: code_natural > T > T,Natural: code_natural] :
% 0.08/0.35 ( ( code_rec_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
% 0.08/0.35 = ( F22 @ Natural @ ( code_rec_natural @ T @ F1 @ F22 @ Natural ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % natural.simps(7)
% 0.08/0.35 thf(fact_182_LeastM__natI,axiom,
% 0.08/0.35 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 0.08/0.35 ( ( P @ K )
% 0.08/0.35 => ( P @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % LeastM_natI
% 0.08/0.35 thf(fact_183_transfer__int__nat__relations_I1_J,axiom,
% 0.08/0.35 ! [X: nat,Y2: nat] :
% 0.08/0.35 ( ( ( semiring_1_of_nat @ int @ X )
% 0.08/0.35 = ( semiring_1_of_nat @ int @ Y2 ) )
% 0.08/0.35 = ( X = Y2 ) ) ).
% 0.08/0.35
% 0.08/0.35 % transfer_int_nat_relations(1)
% 0.08/0.35 thf(fact_184_int__int__eq,axiom,
% 0.08/0.35 ! [M: nat,N: nat] :
% 0.08/0.35 ( ( ( semiring_1_of_nat @ int @ M )
% 0.08/0.35 = ( semiring_1_of_nat @ int @ N ) )
% 0.08/0.35 = ( M = N ) ) ).
% 0.08/0.35
% 0.08/0.35 % int_int_eq
% 0.08/0.35 thf(fact_185_int__if__cong,axiom,
% 0.08/0.35 ! [P: $o,X: nat,Y2: nat] :
% 0.08/0.35 ( ( P
% 0.08/0.35 => ( ( semiring_1_of_nat @ int @ X )
% 0.08/0.35 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y2 ) ) ) )
% 0.08/0.35 & ( ~ P
% 0.08/0.35 => ( ( semiring_1_of_nat @ int @ Y2 )
% 0.08/0.35 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y2 ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % int_if_cong
% 0.08/0.35 thf(fact_186_natural_Odistinct_I2_J,axiom,
% 0.08/0.35 ! [Natural3: code_natural] :
% 0.08/0.35 ( ( code_Suc @ Natural3 )
% 0.08/0.35 != ( zero_zero @ code_natural ) ) ).
% 0.08/0.35
% 0.08/0.35 % natural.distinct(2)
% 0.08/0.35 thf(fact_187_natural_Odistinct_I1_J,axiom,
% 0.08/0.35 ! [Natural2: code_natural] :
% 0.08/0.35 ( ( zero_zero @ code_natural )
% 0.08/0.35 != ( code_Suc @ Natural2 ) ) ).
% 0.08/0.35
% 0.08/0.35 % natural.distinct(1)
% 0.08/0.35 thf(fact_188_natural_Oexhaust,axiom,
% 0.08/0.35 ! [Y2: code_natural] :
% 0.08/0.35 ( ( Y2
% 0.08/0.35 != ( zero_zero @ code_natural ) )
% 0.08/0.35 => ~ ! [Natural4: code_natural] :
% 0.08/0.35 ( Y2
% 0.08/0.35 != ( code_Suc @ Natural4 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % natural.exhaust
% 0.08/0.35 thf(fact_189_natural_Oinducts,axiom,
% 0.08/0.35 ! [P: code_natural > $o,Natural: code_natural] :
% 0.08/0.35 ( ( P @ ( zero_zero @ code_natural ) )
% 0.08/0.35 => ( ! [Natural4: code_natural] :
% 0.08/0.35 ( ( P @ Natural4 )
% 0.08/0.35 => ( P @ ( code_Suc @ Natural4 ) ) )
% 0.08/0.35 => ( P @ Natural ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % natural.inducts
% 0.08/0.35 thf(fact_190_int__diff__cases,axiom,
% 0.08/0.35 ! [Z: int] :
% 0.08/0.35 ~ ! [M2: nat,N2: nat] :
% 0.08/0.35 ( Z
% 0.08/0.35 != ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % int_diff_cases
% 0.08/0.35 thf(fact_191_zero__le__imp__eq__int,axiom,
% 0.08/0.35 ! [K: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.08/0.35 => ? [N2: nat] :
% 0.08/0.35 ( K
% 0.08/0.35 = ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % zero_le_imp_eq_int
% 0.08/0.35 thf(fact_192_nonneg__int__cases,axiom,
% 0.08/0.35 ! [K: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.08/0.35 => ~ ! [N2: nat] :
% 0.08/0.35 ( K
% 0.08/0.35 != ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % nonneg_int_cases
% 0.08/0.35 thf(fact_193_nonneg__eq__int,axiom,
% 0.08/0.35 ! [Z: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.08/0.35 => ~ ! [M2: nat] :
% 0.08/0.35 ( Z
% 0.08/0.35 != ( semiring_1_of_nat @ int @ M2 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % nonneg_eq_int
% 0.08/0.35 thf(fact_194_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
% 0.08/0.35 ! [Z: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ Z ) ) ).
% 0.08/0.35
% 0.08/0.35 % Nat_Transfer.transfer_nat_int_function_closures(9)
% 0.08/0.35 thf(fact_195_transfer__int__nat__quantifiers_I1_J,axiom,
% 0.08/0.35 ! [P: int > $o] :
% 0.08/0.35 ( ( ! [X3: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
% 0.08/0.35 => ( P @ X3 ) ) )
% 0.08/0.35 = ( ! [X3: nat] : ( P @ ( semiring_1_of_nat @ int @ X3 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % transfer_int_nat_quantifiers(1)
% 0.08/0.35 thf(fact_196_transfer__int__nat__quantifiers_I2_J,axiom,
% 0.08/0.35 ! [P: int > $o] :
% 0.08/0.35 ( ( ? [X3: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
% 0.08/0.35 & ( P @ X3 ) ) )
% 0.08/0.35 = ( ? [X3: nat] : ( P @ ( semiring_1_of_nat @ int @ X3 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % transfer_int_nat_quantifiers(2)
% 0.08/0.35 thf(fact_197_transfer__int__nat__numerals_I1_J,axiom,
% 0.08/0.35 ( ( zero_zero @ int )
% 0.08/0.35 = ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % transfer_int_nat_numerals(1)
% 0.08/0.35 thf(fact_198_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
% 0.08/0.35 ! [X: nat,Y2: nat] :
% 0.08/0.35 ( ( nat_tsub @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) )
% 0.08/0.35 = ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y2 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % Nat_Transfer.transfer_int_nat_functions(3)
% 0.08/0.35 thf(fact_199_round__of__nat,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [N: nat] :
% 0.08/0.35 ( ( archimedean_round @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.08/0.35 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % round_of_nat
% 0.08/0.35 thf(fact_200_round__0,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.08/0.35 => ( ( archimedean_round @ A @ ( zero_zero @ A ) )
% 0.08/0.35 = ( zero_zero @ int ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % round_0
% 0.08/0.35 thf(fact_201_minus__int__code_I1_J,axiom,
% 0.08/0.35 ! [K: int] :
% 0.08/0.35 ( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
% 0.08/0.35 = K ) ).
% 0.08/0.35
% 0.08/0.35 % minus_int_code(1)
% 0.08/0.35 thf(fact_202_zero__integer_Orsp,axiom,
% 0.08/0.35 ( ( zero_zero @ int )
% 0.08/0.35 = ( zero_zero @ int ) ) ).
% 0.08/0.35
% 0.08/0.35 % zero_integer.rsp
% 0.08/0.35 thf(fact_203_tsub__def,axiom,
% 0.08/0.35 ( nat_tsub
% 0.08/0.35 = ( ^ [X3: int,Y4: int] : ( if @ int @ ( ord_less_eq @ int @ Y4 @ X3 ) @ ( minus_minus @ int @ X3 @ Y4 ) @ ( zero_zero @ int ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % tsub_def
% 0.08/0.35 thf(fact_204_tsub__eq,axiom,
% 0.08/0.35 ! [Y2: int,X: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ Y2 @ X )
% 0.08/0.35 => ( ( nat_tsub @ X @ Y2 )
% 0.08/0.35 = ( minus_minus @ int @ X @ Y2 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % tsub_eq
% 0.08/0.35 thf(fact_205_round__mono,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A,Y2: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.08/0.35 => ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % round_mono
% 0.08/0.35 thf(fact_206_ceiling__le__zero,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
% 0.08/0.35 = ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ceiling_le_zero
% 0.08/0.35 thf(fact_207_of__int__0__le__iff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [Z: int] :
% 0.08/0.35 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 0.08/0.35 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_int_0_le_iff
% 0.08/0.35 thf(fact_208_of__int__0,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ring_1 @ A @ ( type2 @ A ) )
% 0.08/0.35 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
% 0.08/0.35 = ( zero_zero @ A ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_int_0
% 0.08/0.35 thf(fact_209_of__int__0__eq__iff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ring_char_0 @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [Z: int] :
% 0.08/0.35 ( ( ( zero_zero @ A )
% 0.08/0.35 = ( ring_1_of_int @ A @ Z ) )
% 0.08/0.35 = ( Z
% 0.08/0.35 = ( zero_zero @ int ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_int_0_eq_iff
% 0.08/0.35 thf(fact_210_of__int__eq__0__iff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ring_char_0 @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [Z: int] :
% 0.08/0.35 ( ( ( ring_1_of_int @ A @ Z )
% 0.08/0.35 = ( zero_zero @ A ) )
% 0.08/0.35 = ( Z
% 0.08/0.35 = ( zero_zero @ int ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_int_eq_0_iff
% 0.08/0.35 thf(fact_211_of__int__le__iff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [W: int,Z: int] :
% 0.08/0.35 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
% 0.08/0.35 = ( ord_less_eq @ int @ W @ Z ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_int_le_iff
% 0.08/0.35 thf(fact_212_ceiling__zero,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.08/0.35 => ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
% 0.08/0.35 = ( zero_zero @ int ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ceiling_zero
% 0.08/0.35 thf(fact_213_of__int__diff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ring_1 @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [W: int,Z: int] :
% 0.08/0.35 ( ( ring_1_of_int @ A @ ( minus_minus @ int @ W @ Z ) )
% 0.08/0.35 = ( minus_minus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_int_diff
% 0.08/0.35 thf(fact_214_of__int__of__nat__eq,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ring_1 @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [N: nat] :
% 0.08/0.35 ( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N ) )
% 0.08/0.35 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_int_of_nat_eq
% 0.08/0.35 thf(fact_215_ceiling__of__nat,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [N: nat] :
% 0.08/0.35 ( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.08/0.35 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ceiling_of_nat
% 0.08/0.35 thf(fact_216_ceiling__diff__of__int,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A,Z: int] :
% 0.08/0.35 ( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) )
% 0.08/0.35 = ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ Z ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ceiling_diff_of_int
% 0.08/0.35 thf(fact_217_of__int__le__0__iff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [Z: int] :
% 0.08/0.35 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
% 0.08/0.35 = ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_int_le_0_iff
% 0.08/0.35 thf(fact_218_ex__le__of__int,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A] :
% 0.08/0.35 ? [Z3: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ex_le_of_int
% 0.08/0.35 thf(fact_219_ceiling__mono,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [Y2: A,X: A] :
% 0.08/0.35 ( ( ord_less_eq @ A @ Y2 @ X )
% 0.08/0.35 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y2 ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ceiling_mono
% 0.08/0.35 thf(fact_220_le__of__int__ceiling,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_of_int_ceiling
% 0.08/0.35 thf(fact_221_ceiling__le__iff,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A,Z: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
% 0.08/0.35 = ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ceiling_le_iff
% 0.08/0.35 thf(fact_222_of__int__nonneg,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [Z: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.08/0.35 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_int_nonneg
% 0.08/0.35 thf(fact_223_ceiling__le,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [X: A,A2: int] :
% 0.08/0.35 ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) )
% 0.08/0.35 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A2 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % ceiling_le
% 0.08/0.35 thf(fact_224_of__nat__nat,axiom,
% 0.08/0.35 ! [A: $tType] :
% 0.08/0.35 ( ( ring_1 @ A @ ( type2 @ A ) )
% 0.08/0.35 => ! [Z: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.08/0.35 => ( ( semiring_1_of_nat @ A @ ( nat2 @ Z ) )
% 0.08/0.35 = ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % of_nat_nat
% 0.08/0.35 thf(fact_225_nat__ceiling__le__eq,axiom,
% 0.08/0.35 ! [X: real,A2: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A2 )
% 0.08/0.35 = ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % nat_ceiling_le_eq
% 0.08/0.35 thf(fact_226_nat__int,axiom,
% 0.08/0.35 ! [N: nat] :
% 0.08/0.35 ( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
% 0.08/0.35 = N ) ).
% 0.08/0.35
% 0.08/0.35 % nat_int
% 0.08/0.35 thf(fact_227_nat__0,axiom,
% 0.08/0.35 ( ( nat2 @ ( zero_zero @ int ) )
% 0.08/0.35 = ( zero_zero @ nat ) ) ).
% 0.08/0.35
% 0.08/0.35 % nat_0
% 0.08/0.35 thf(fact_228_nat__0__iff,axiom,
% 0.08/0.35 ! [I: int] :
% 0.08/0.35 ( ( ( nat2 @ I )
% 0.08/0.35 = ( zero_zero @ nat ) )
% 0.08/0.35 = ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % nat_0_iff
% 0.08/0.35 thf(fact_229_nat__le__0,axiom,
% 0.08/0.35 ! [Z: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
% 0.08/0.35 => ( ( nat2 @ Z )
% 0.08/0.35 = ( zero_zero @ nat ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % nat_le_0
% 0.08/0.35 thf(fact_230_int__nat__eq,axiom,
% 0.08/0.35 ! [Z: int] :
% 0.08/0.35 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.08/0.35 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 0.08/0.35 = Z ) )
% 0.08/0.35 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.08/0.35 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 0.08/0.35 = ( zero_zero @ int ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % int_nat_eq
% 0.08/0.35 thf(fact_231_real__nat__ceiling__ge,axiom,
% 0.08/0.35 ! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % real_nat_ceiling_ge
% 0.08/0.35 thf(fact_232_nat__mono,axiom,
% 0.08/0.35 ! [X: int,Y2: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ X @ Y2 )
% 0.08/0.35 => ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % nat_mono
% 0.08/0.35 thf(fact_233_transfer__nat__int__relations_I3_J,axiom,
% 0.08/0.35 ! [X: int,Y2: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.08/0.35 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
% 0.08/0.35 => ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) )
% 0.08/0.35 = ( ord_less_eq @ int @ X @ Y2 ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % transfer_nat_int_relations(3)
% 0.08/0.35 thf(fact_234_nat__le__iff,axiom,
% 0.08/0.35 ! [X: int,N: nat] :
% 0.08/0.35 ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
% 0.08/0.35 = ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % nat_le_iff
% 0.08/0.35 thf(fact_235_int__eq__iff,axiom,
% 0.08/0.35 ! [M: nat,Z: int] :
% 0.08/0.35 ( ( ( semiring_1_of_nat @ int @ M )
% 0.08/0.35 = Z )
% 0.08/0.35 = ( ( M
% 0.08/0.35 = ( nat2 @ Z ) )
% 0.08/0.35 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % int_eq_iff
% 0.08/0.35 thf(fact_236_nat__0__le,axiom,
% 0.08/0.35 ! [Z: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.08/0.35 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 0.08/0.35 = Z ) ) ).
% 0.08/0.35
% 0.08/0.35 % nat_0_le
% 0.08/0.35 thf(fact_237_nat__eq__iff,axiom,
% 0.08/0.35 ! [W: int,M: nat] :
% 0.08/0.35 ( ( ( nat2 @ W )
% 0.08/0.35 = M )
% 0.08/0.35 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.08/0.35 => ( W
% 0.08/0.35 = ( semiring_1_of_nat @ int @ M ) ) )
% 0.08/0.35 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.08/0.35 => ( M
% 0.08/0.35 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % nat_eq_iff
% 0.08/0.35 thf(fact_238_nat__eq__iff2,axiom,
% 0.08/0.35 ! [M: nat,W: int] :
% 0.08/0.35 ( ( M
% 0.08/0.35 = ( nat2 @ W ) )
% 0.08/0.35 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.08/0.35 => ( W
% 0.08/0.35 = ( semiring_1_of_nat @ int @ M ) ) )
% 0.08/0.35 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.08/0.35 => ( M
% 0.08/0.35 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % nat_eq_iff2
% 0.08/0.35 thf(fact_239_le__nat__iff,axiom,
% 0.08/0.35 ! [K: int,N: nat] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.08/0.35 => ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
% 0.08/0.35 = ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % le_nat_iff
% 0.08/0.35 thf(fact_240_nat__diff__distrib_H,axiom,
% 0.08/0.35 ! [X: int,Y2: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.08/0.35 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
% 0.08/0.35 => ( ( nat2 @ ( minus_minus @ int @ X @ Y2 ) )
% 0.08/0.35 = ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % nat_diff_distrib'
% 0.08/0.35 thf(fact_241_nat__diff__distrib,axiom,
% 0.08/0.35 ! [Z4: int,Z: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z4 )
% 0.08/0.35 => ( ( ord_less_eq @ int @ Z4 @ Z )
% 0.08/0.35 => ( ( nat2 @ ( minus_minus @ int @ Z @ Z4 ) )
% 0.08/0.35 = ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % nat_diff_distrib
% 0.08/0.35 thf(fact_242_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,
% 0.08/0.35 ! [X: int,Y2: int] :
% 0.08/0.35 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.08/0.35 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
% 0.08/0.35 => ( ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) )
% 0.08/0.35 = ( nat2 @ ( nat_tsub @ X @ Y2 ) ) ) ) ) ).
% 0.08/0.35
% 0.08/0.35 % Nat_Transfer.transfer_nat_int_functions(3)
% 0.08/0.35 thf(fact_243_int__eq__iff__numeral,axiom,
% 0.08/0.35 ! [M: nat,V: num] :
% 0.08/0.35 ( ( ( semiring_1_of_nat @ int @ M )
% 0.08/0.35 = ( numeral_numeral @ int @ V ) )
% 0.18/0.35 = ( ( M
% 0.18/0.35 = ( nat2 @ ( numeral_numeral @ int @ V ) ) )
% 0.18/0.35 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 0.18/0.35
% 0.18/0.35 % int_eq_iff_numeral
% 0.18/0.35 thf(fact_244_nat__abs__int__diff,axiom,
% 0.18/0.35 ! [A2: nat,B2: nat] :
% 0.18/0.35 ( ( ( ord_less_eq @ nat @ A2 @ B2 )
% 0.18/0.35 => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
% 0.18/0.35 = ( minus_minus @ nat @ B2 @ A2 ) ) )
% 0.18/0.35 & ( ~ ( ord_less_eq @ nat @ A2 @ B2 )
% 0.18/0.35 => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
% 0.18/0.35 = ( minus_minus @ nat @ A2 @ B2 ) ) ) ) ).
% 0.18/0.35
% 0.18/0.35 % nat_abs_int_diff
% 0.18/0.35 thf(fact_245_numeral__eq__iff,axiom,
% 0.18/0.35 ! [A: $tType] :
% 0.18/0.35 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.18/0.35 => ! [M: num,N: num] :
% 0.18/0.35 ( ( ( numeral_numeral @ A @ M )
% 0.18/0.35 = ( numeral_numeral @ A @ N ) )
% 0.18/0.35 = ( M = N ) ) ) ).
% 0.18/0.35
% 0.18/0.35 % numeral_eq_iff
% 0.18/0.35 thf(fact_246_abs__idempotent,axiom,
% 0.18/0.35 ! [A: $tType] :
% 0.18/0.35 ( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
% 0.18/0.35 => ! [A2: A] :
% 0.18/0.35 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 0.18/0.35 = ( abs_abs @ A @ A2 ) ) ) ).
% 0.18/0.35
% 0.18/0.35 % abs_idempotent
% 0.18/0.35 thf(fact_247_numeral__le__iff,axiom,
% 0.18/0.35 ! [A: $tType] :
% 0.18/0.35 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.18/0.35 => ! [M: num,N: num] :
% 0.18/0.35 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.18/0.35 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 0.18/0.35
% 0.18/0.35 % numeral_le_iff
% 0.18/0.35 thf(fact_248_abs__0__eq,axiom,
% 0.18/0.35 ! [A: $tType] :
% 0.18/0.35 ( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
% 0.18/0.35 => ! [A2: A] :
% 0.18/0.35 ( ( ( zero_zero @ A )
% 0.18/0.35 = ( abs_abs @ A @ A2 ) )
% 0.18/0.35 = ( A2
% 0.18/0.35 = ( zero_zero @ A ) ) ) ) ).
% 0.18/0.35
% 0.18/0.35 % abs_0_eq
% 0.18/0.35 thf(fact_249_abs__eq__0,axiom,
% 0.18/0.35 ! [A: $tType] :
% 0.18/0.35 ( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
% 0.18/0.35 => ! [A2: A] :
% 0.18/0.35 ( ( ( abs_abs @ A @ A2 )
% 0.18/0.35 = ( zero_zero @ A ) )
% 0.18/0.35 = ( A2
% 0.18/0.35 = ( zero_zero @ A ) ) ) ) ).
% 0.18/0.35
% 0.18/0.35 % abs_eq_0
% 0.18/0.35
% 0.18/0.35 %----Type constructors (91)
% 0.18/0.35 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.18/0.35 ! [A6: $tType,A7: $tType] :
% 0.18/0.35 ( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
% 0.18/0.35 => ( comple187826305attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.18/0.35 ! [A6: $tType,A7: $tType] :
% 0.18/0.35 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 0.18/0.35 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.18/0.35 ! [A6: $tType,A7: $tType] :
% 0.18/0.35 ( ( order @ A7 @ ( type2 @ A7 ) )
% 0.18/0.35 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_fun___Orderings_Oord,axiom,
% 0.18/0.35 ! [A6: $tType,A7: $tType] :
% 0.18/0.35 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 0.18/0.35 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_fun___Groups_Ominus,axiom,
% 0.18/0.35 ! [A6: $tType,A7: $tType] :
% 0.18/0.35 ( ( minus @ A7 @ ( type2 @ A7 ) )
% 0.18/0.35 => ( minus @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
% 0.18/0.35 ordere142940540dd_abs @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.18/0.35 cancel146912293up_add @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 0.18/0.35 cancel1352612707id_add @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 0.18/0.35 ordered_ab_group_add @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.18/0.35 linordered_semidom @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.18/0.35 linordered_idom @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.18/0.35 semiring_char_0 @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 0.18/0.35 preorder @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.18/0.35 linorder @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.18/0.35 semiring_1 @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 0.18/0.35 group_add @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 0.18/0.35 order @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
% 0.18/0.35 neg_numeral @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 0.18/0.35 ring_char_0 @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 0.18/0.35 ord @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Rings_Oring__1,axiom,
% 0.18/0.35 ring_1 @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Groups_Ominus_4,axiom,
% 0.18/0.35 minus @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Num_Onumeral,axiom,
% 0.18/0.35 numeral @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.18/0.35 zero @ int @ ( type2 @ int ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.18/0.35 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_5,axiom,
% 0.18/0.35 cancel146912293up_add @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_6,axiom,
% 0.18/0.35 cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_7,axiom,
% 0.18/0.35 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 0.18/0.35 comm_monoid_diff @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_8,axiom,
% 0.18/0.35 semiring_char_0 @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Orderings_Opreorder_9,axiom,
% 0.18/0.35 preorder @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Orderings_Olinorder_10,axiom,
% 0.18/0.35 linorder @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Rings_Osemiring__1_11,axiom,
% 0.18/0.35 semiring_1 @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Orderings_Oorder_12,axiom,
% 0.18/0.35 order @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Orderings_Oord_13,axiom,
% 0.18/0.35 ord @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Groups_Ominus_14,axiom,
% 0.18/0.35 minus @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Num_Onumeral_15,axiom,
% 0.18/0.35 numeral @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Nat_Onat___Groups_Ozero_16,axiom,
% 0.18/0.35 zero @ nat @ ( type2 @ nat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Num_Onum___Orderings_Opreorder_17,axiom,
% 0.18/0.35 preorder @ num @ ( type2 @ num ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Num_Onum___Orderings_Olinorder_18,axiom,
% 0.18/0.35 linorder @ num @ ( type2 @ num ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Num_Onum___Orderings_Oorder_19,axiom,
% 0.18/0.35 order @ num @ ( type2 @ num ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Num_Onum___Orderings_Oord_20,axiom,
% 0.18/0.35 ord @ num @ ( type2 @ num ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_21,axiom,
% 0.18/0.35 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_HOL_Obool___Orderings_Opreorder_22,axiom,
% 0.18/0.35 preorder @ $o @ ( type2 @ $o ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_HOL_Obool___Orderings_Olinorder_23,axiom,
% 0.18/0.35 linorder @ $o @ ( type2 @ $o ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_HOL_Obool___Orderings_Oorder_24,axiom,
% 0.18/0.35 order @ $o @ ( type2 @ $o ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_HOL_Obool___Orderings_Oord_25,axiom,
% 0.18/0.35 ord @ $o @ ( type2 @ $o ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_HOL_Obool___Groups_Ominus_26,axiom,
% 0.18/0.35 minus @ $o @ ( type2 @ $o ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 0.18/0.35 archim1804426504_field @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_27,axiom,
% 0.18/0.35 ordere142940540dd_abs @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling,axiom,
% 0.18/0.35 archim1727834104eiling @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_28,axiom,
% 0.18/0.35 cancel146912293up_add @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_29,axiom,
% 0.18/0.35 cancel1352612707id_add @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_30,axiom,
% 0.18/0.35 ordered_ab_group_add @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_31,axiom,
% 0.18/0.35 linordered_semidom @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Rings_Olinordered__idom_32,axiom,
% 0.18/0.35 linordered_idom @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_33,axiom,
% 0.18/0.35 semiring_char_0 @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Orderings_Opreorder_34,axiom,
% 0.18/0.35 preorder @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Orderings_Olinorder_35,axiom,
% 0.18/0.35 linorder @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Rings_Osemiring__1_36,axiom,
% 0.18/0.35 semiring_1 @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Groups_Ogroup__add_37,axiom,
% 0.18/0.35 group_add @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Orderings_Oorder_38,axiom,
% 0.18/0.35 order @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Num_Oneg__numeral_39,axiom,
% 0.18/0.35 neg_numeral @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Nat_Oring__char__0_40,axiom,
% 0.18/0.35 ring_char_0 @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Orderings_Oord_41,axiom,
% 0.18/0.35 ord @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Rings_Oring__1_42,axiom,
% 0.18/0.35 ring_1 @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Groups_Ominus_43,axiom,
% 0.18/0.35 minus @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Num_Onumeral_44,axiom,
% 0.18/0.35 numeral @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Real_Oreal___Groups_Ozero_45,axiom,
% 0.18/0.35 zero @ real @ ( type2 @ real ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_46,axiom,
% 0.18/0.35 canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_47,axiom,
% 0.18/0.35 comple187826305attice @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_48,axiom,
% 0.18/0.35 semiring_char_0 @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_49,axiom,
% 0.18/0.35 preorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_50,axiom,
% 0.18/0.35 linorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_51,axiom,
% 0.18/0.35 semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_52,axiom,
% 0.18/0.35 order @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_53,axiom,
% 0.18/0.35 ord @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Extended__Nat_Oenat___Groups_Ominus_54,axiom,
% 0.18/0.35 minus @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Extended__Nat_Oenat___Num_Onumeral_55,axiom,
% 0.18/0.35 numeral @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_56,axiom,
% 0.18/0.35 zero @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_57,axiom,
% 0.18/0.35 cancel146912293up_add @ code_natural @ ( type2 @ code_natural ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_58,axiom,
% 0.18/0.35 cancel1352612707id_add @ code_natural @ ( type2 @ code_natural ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_59,axiom,
% 0.18/0.35 comm_monoid_diff @ code_natural @ ( type2 @ code_natural ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_60,axiom,
% 0.18/0.35 preorder @ code_natural @ ( type2 @ code_natural ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_61,axiom,
% 0.18/0.35 linorder @ code_natural @ ( type2 @ code_natural ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_62,axiom,
% 0.18/0.35 semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_63,axiom,
% 0.18/0.35 order @ code_natural @ ( type2 @ code_natural ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Code__Numeral_Onatural___Orderings_Oord_64,axiom,
% 0.18/0.35 ord @ code_natural @ ( type2 @ code_natural ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Code__Numeral_Onatural___Groups_Ominus_65,axiom,
% 0.18/0.35 minus @ code_natural @ ( type2 @ code_natural ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Code__Numeral_Onatural___Num_Onumeral_66,axiom,
% 0.18/0.35 numeral @ code_natural @ ( type2 @ code_natural ) ).
% 0.18/0.35
% 0.18/0.35 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_67,axiom,
% 0.18/0.35 zero @ code_natural @ ( type2 @ code_natural ) ).
% 0.18/0.35
% 0.18/0.35 %----Helper facts (3)
% 0.18/0.35 thf(help_If_3_1_T,axiom,
% 0.18/0.35 ! [P: $o] :
% 0.18/0.35 ( ( P = $true )
% 0.18/0.35 | ( P = $false ) ) ).
% 0.18/0.35
% 0.18/0.35 thf(help_If_2_1_T,axiom,
% 0.18/0.35 ! [A: $tType,X: A,Y2: A] :
% 0.18/0.35 ( ( if @ A @ $false @ X @ Y2 )
% 0.18/0.35 = Y2 ) ).
% 0.18/0.35
% 0.18/0.35 thf(help_If_1_1_T,axiom,
% 0.18/0.35 ! [A: $tType,X: A,Y2: A] :
% 0.18/0.35 ( ( if @ A @ $true @ X @ Y2 )
% 0.18/0.35 = X ) ).
% 0.18/0.35
% 0.18/0.35 %----Conjectures (2)
% 0.18/0.35 thf(conj_0,hypothesis,
% 0.18/0.35 ord_less_eq @ extended_enat @ ( coindu1018505716length @ a @ xs ) @ m ).
% 0.18/0.35
% 0.18/0.35 thf(conj_1,conjecture,
% 0.18/0.35 ( ( coindu191418589_ldrop @ a @ m @ xs )
% 0.18/0.35 = ( coindu1598213697e_LNil @ a ) ) ).
% 0.18/0.35
% 0.18/0.35 %------------------------------------------------------------------------------
% 0.18/0.35 ------------------------
% 2.83/1.61 ---- Embedded in THF ---
% 2.83/1.62 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.83/1.62 %%% Generated on Tue Feb 24 06:07:17 EST 2026
% 2.83/1.62 %%% using '$$dhol' embedding, version 1.3.0.
% 2.83/1.62 %%% Logic specification used:
% 2.83/1.62 %%% thf(spec, logic, $$dhol).
% 2.83/1.62
% 2.83/1.62 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.bImcdsmqd4/DTF2DTF17343.p
% 2.83/1.62 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist, type, coindu1593790203_llist: ($tType > $tType)).
% 2.83/1.62 thf(ty_t_Code__Numeral_Onatural, type, code_natural: $tType).
% 2.83/1.62 thf(ty_t_Extended__Nat_Oenat, type, extended_enat: $tType).
% 2.83/1.62 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.83/1.62 thf(ty_t_Num_Onum, type, num: $tType).
% 2.83/1.62 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.83/1.62 thf(ty_t_Int_Oint, type, int: $tType).
% 2.83/1.62 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.83/1.62 thf(ty_tf_a, type, a: $tType).
% 2.83/1.62 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Num_Onumeral, type, numeral: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Rings_Oring__1, type, ring_1: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Nat_Oring__char__0, type, ring_char_0: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Archimedean__Field_Ofloor__ceiling, type, archim1727834104eiling: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Groups_Oordered__ab__group__add__abs, type, ordere142940540dd_abs: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Archimedean__Field_Oarchimedean__field, type, archim1804426504_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.83/1.62 thf(sy_c_Archimedean__Field_Oceiling, type, archimedean_ceiling: (!> [A:$tType]: ((A > int)))).
% 2.83/1.62 thf(sy_c_Archimedean__Field_Oround, type, archimedean_round: (!> [A:$tType]: ((A > int)))).
% 2.83/1.62 thf(sy_c_Code__Numeral_OSuc, type, code_Suc: (code_natural > code_natural)).
% 2.83/1.62 thf(sy_c_Code__Numeral_Onatural_Ocase__natural, type, code_case_natural: (!> [T:$tType]: ((T > ((code_natural > T) > (code_natural > T)))))).
% 2.83/1.62 thf(sy_c_Code__Numeral_Onatural_Orec__natural, type, code_rec_natural: (!> [T:$tType]: ((T > ((code_natural > (T > T)) > (code_natural > T)))))).
% 2.83/1.62 thf(sy_c_Code__Numeral_Onatural_Osize__natural, type, code_size_natural: (code_natural > nat)).
% 2.83/1.62 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__llength, type, coindu493225792length: (!> [A:$tType]: ((nat > ((coindu1593790203_llist @ A) > extended_enat))))).
% 2.83/1.62 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend, type, coindu268472904append: (!> [A:$tType]: (((coindu1593790203_llist @ A) > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 2.83/1.62 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldrop, type, coindu191418589_ldrop: (!> [A:$tType]: ((extended_enat > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 2.83/1.62 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength, type, coindu1018505716length: (!> [A:$tType]: (((coindu1593790203_llist @ A) > extended_enat)))).
% 2.83/1.62 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil, type, coindu1598213697e_LNil: (!> [A:$tType]: ((coindu1593790203_llist @ A)))).
% 2.83/1.62 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olhd, type, coindu1046438764le_lhd: (!> [A:$tType]: (((coindu1593790203_llist @ A) > A)))).
% 2.83/1.62 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull, type, coindu335574135_lnull: (!> [A:$tType]: (((coindu1593790203_llist @ A) > $o)))).
% 2.83/1.62 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olmember, type, coindu567634248member: (!> [A:$tType]: ((A > ((coindu1593790203_llist @ A) > $o))))).
% 2.83/1.62 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olprefix, type, coindu1696667936prefix: (!> [A:$tType]: (((coindu1593790203_llist @ A) > ((coindu1593790203_llist @ A) > $o))))).
% 2.83/1.62 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olstrict__prefix, type, coindu574146665prefix: (!> [A:$tType]: (((coindu1593790203_llist @ A) > ((coindu1593790203_llist @ A) > $o))))).
% 2.83/1.62 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oltake, type, coindu1802687541_ltake: (!> [A:$tType]: ((extended_enat > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 2.83/1.62 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OltakeWhile, type, coindu721411036eWhile: (!> [A:$tType]: (((A > $o) > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 2.83/1.62 thf(sy_c_Extended__Nat_Oenat_Ocase__enat, type, extended_case_enat: (!> [T:$tType]: (((nat > T) > (T > (extended_enat > T)))))).
% 2.83/1.62 thf(sy_c_Extended__Nat_Othe__enat, type, extended_the_enat: (extended_enat > nat)).
% 2.83/1.62 thf(sy_c_Groups_Oabs__class_Oabs, type, abs_abs: (!> [A:$tType]: ((A > A)))).
% 2.83/1.62 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.83/1.62 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.83/1.62 thf(sy_c_Hilbert__Choice_OLeastM, type, hilbert_LeastM: (!> [A:$tType,B:$tType]: (((A > B) > ((A > $o) > A))))).
% 2.83/1.62 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.83/1.62 thf(sy_c_Int_Onat, type, nat2: (int > nat)).
% 2.83/1.62 thf(sy_c_Int_Oring__1__class_Oof__int, type, ring_1_of_int: (!> [A:$tType]: ((int > A)))).
% 2.83/1.62 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 2.83/1.62 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.83/1.62 thf(sy_c_Nat__Transfer_Otsub, type, nat_tsub: (int > (int > int))).
% 2.83/1.62 thf(sy_c_Num_Oneg__numeral__class_Odbl, type, neg_numeral_dbl: (!> [A:$tType]: ((A > A)))).
% 2.83/1.62 thf(sy_c_Num_Onumeral__class_Onumeral, type, numeral_numeral: (!> [A:$tType]: ((num > A)))).
% 2.83/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.83/1.62 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.83/1.62 thf(sy_v_m, type, m: extended_enat).
% 2.83/1.62 thf(sy_v_xs, type, xs: (coindu1593790203_llist @ a)).
% 2.83/1.62 thf(fact_0_ldrop__LNil, axiom, (! [A:$tType,N:extended_enat]: (((((coindu191418589_ldrop @ A) @ N) @ (coindu1598213697e_LNil @ A)) = (coindu1598213697e_LNil @ A))))).
% 2.83/1.62 thf(fact_1_ldrop__eq__LNil, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: ((((((coindu191418589_ldrop @ A) @ N) @ Xs) = (coindu1598213697e_LNil @ A)) = (((ord_less_eq @ extended_enat) @ ((coindu1018505716length @ A) @ Xs)) @ N))))).
% 2.83/1.62 thf(fact_2_wlog__linorder__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),B2:A,A2:A]: (((! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((P @ A3) @ B3)))) => ((((P @ B2) @ A2) => ((P @ A2) @ B2)) => ((P @ A2) @ B2))))))))).
% 2.83/1.62 thf(fact_3_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.83/1.62 thf(fact_4_lnull__ldrop, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ (((coindu191418589_ldrop @ A) @ N) @ Xs)) = (((ord_less_eq @ extended_enat) @ ((coindu1018505716length @ A) @ Xs)) @ N))))).
% 2.83/1.62 thf(fact_5_lmember__code_I1_J, axiom, (! [A:$tType,X:A]: ((~ ((((coindu567634248member @ A) @ X) @ (coindu1598213697e_LNil @ A))))))).
% 2.83/1.62 thf(fact_6_llength__LNil, axiom, (! [A:$tType]: ((((coindu1018505716length @ A) @ (coindu1598213697e_LNil @ A)) = (zero_zero @ extended_enat))))).
% 2.83/1.62 thf(fact_7_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.83/1.62 thf(fact_8_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.83/1.62 thf(fact_9_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.83/1.62 thf(fact_10_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F2 @ X3)) @ (G2 @ X3))))))))))).
% 2.83/1.62 thf(fact_11_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X2:B,Y:B]: (((((ord_less_eq @ B) @ X2) @ Y) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.83/1.62 thf(fact_12_order__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ C2) @ (F @ A2)) @ C)))))))))).
% 2.83/1.62 thf(fact_13_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X2:B,Y:B]: (((((ord_less_eq @ B) @ X2) @ Y) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.83/1.62 thf(fact_14_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C)))))))))).
% 2.83/1.62 thf(fact_15_ldrop__0, axiom, (! [B:$tType,Xs:(coindu1593790203_llist @ B)]: (((((coindu191418589_ldrop @ B) @ (zero_zero @ extended_enat)) @ Xs) = Xs)))).
% 2.83/1.62 thf(fact_16_llength__lnull, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Xs) => (((coindu1018505716length @ A) @ Xs) = (zero_zero @ extended_enat)))))).
% 2.83/1.62 thf(fact_17_llength__eq__0, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: (((((coindu1018505716length @ A) @ Xs) = (zero_zero @ extended_enat)) = ((coindu335574135_lnull @ A) @ Xs))))).
% 2.83/1.62 thf(fact_18_lzip_Oexhaust, axiom, (! [A:$tType,B:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ B)]: (((~ ((((coindu335574135_lnull @ A) @ Xs) | ((coindu335574135_lnull @ B) @ Ys)))) => (~ (((~ (((coindu335574135_lnull @ A) @ Xs))) => ((coindu335574135_lnull @ B) @ Ys)))))))).
% 2.83/1.62 thf(fact_19_ltake_Oexhaust, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: (((~ (((N = (zero_zero @ extended_enat)) | ((coindu335574135_lnull @ A) @ Xs)))) => (~ (((N != (zero_zero @ extended_enat)) => ((coindu335574135_lnull @ A) @ Xs)))))))).
% 2.83/1.62 thf(fact_20_lappend_Oexhaust, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((((coindu335574135_lnull @ A) @ Xs) => (~ (((coindu335574135_lnull @ A) @ Ys)))) => ((~ (((coindu335574135_lnull @ A) @ Xs))) | (~ (((coindu335574135_lnull @ A) @ Ys)))))))).
% 2.83/1.62 thf(fact_21_lnull__def, axiom, (! [A:$tType]: (((coindu335574135_lnull @ A) = (^ [Llist:(coindu1593790203_llist @ A)]: ((Llist = (coindu1598213697e_LNil @ A)))))))).
% 2.83/1.62 thf(fact_22_llist_Ocollapse_I1_J, axiom, (! [A:$tType,Llist2:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Llist2) => (Llist2 = (coindu1598213697e_LNil @ A)))))).
% 2.83/1.62 thf(fact_23_llist_OdiscI_I1_J, axiom, (! [A:$tType,Llist2:(coindu1593790203_llist @ A)]: (((Llist2 = (coindu1598213697e_LNil @ A)) => ((coindu335574135_lnull @ A) @ Llist2))))).
% 2.83/1.62 thf(fact_24_llist_Odisc_I1_J, axiom, (! [A:$tType]: (((coindu335574135_lnull @ A) @ (coindu1598213697e_LNil @ A))))).
% 2.83/1.62 thf(fact_25_ldrop__lnull, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),N:extended_enat]: ((((coindu335574135_lnull @ A) @ Xs) => ((((coindu191418589_ldrop @ A) @ N) @ Xs) = (coindu1598213697e_LNil @ A)))))).
% 2.83/1.62 thf(fact_26_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.83/1.62 thf(fact_27_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.83/1.62 thf(fact_28_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((P @ A3) @ B3)))) => ((! [A3:A,B3:A]: ((((P @ B3) @ A3) => ((P @ A3) @ B3)))) => ((P @ A2) @ B2))))))))).
% 2.83/1.62 thf(fact_29_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.83/1.62 thf(fact_30_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.83/1.62 thf(fact_31_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.83/1.62 thf(fact_32_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.83/1.62 thf(fact_33_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.83/1.62 thf(fact_34_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y2:A,X:A]: (((((ord_less_eq @ A) @ Y2) @ X) => ((((ord_less_eq @ A) @ X) @ Y2) = (X = Y2))))))))).
% 2.83/1.62 thf(fact_35_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z:A]: ((((((ord_less_eq @ A) @ X) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z)))) => (((((ord_less_eq @ A) @ Y2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z)))) => (((((ord_less_eq @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y2)))) => (((((ord_less_eq @ A) @ Z) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X)))) => (((((ord_less_eq @ A) @ Y2) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X)))) => (~ (((((ord_less_eq @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y2))))))))))))))))).
% 2.83/1.62 thf(fact_36_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.83/1.62 thf(fact_37_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X)))))))).
% 2.83/1.62 thf(fact_38_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((X = Y2) => (((ord_less_eq @ A) @ X) @ Y2)))))))).
% 2.83/1.62 thf(fact_39_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X)))))))).
% 2.83/1.62 thf(fact_40_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X) => (X = Y2))))))))).
% 2.83/1.62 thf(fact_41_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y3:A,Z2:A]: ((Y3 = Z2))) = (^ [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X3))))))))).
% 2.83/1.62 thf(fact_42_le__zero__eq, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.83/1.62 thf(fact_43_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 2.83/1.62 thf(fact_44_ile0__eq, axiom, (! [N:extended_enat]: (((((ord_less_eq @ extended_enat) @ N) @ (zero_zero @ extended_enat)) = (N = (zero_zero @ extended_enat)))))).
% 2.83/1.62 thf(fact_45_i0__lb, axiom, (! [N:extended_enat]: ((((ord_less_eq @ extended_enat) @ (zero_zero @ extended_enat)) @ N)))).
% 2.83/1.62 thf(fact_46_zero__le, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.83/1.62 thf(fact_47_le__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.83/1.62 thf(fact_48_ltake_Octr_I1_J, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: ((((N = (zero_zero @ extended_enat)) | ((coindu335574135_lnull @ A) @ Xs)) => ((((coindu1802687541_ltake @ A) @ N) @ Xs) = (coindu1598213697e_LNil @ A)))))).
% 2.83/1.62 thf(fact_49_zero__reorient, axiom, (! [A:$tType]: ((((zero @ A) @ (type2 @ A)) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 2.83/1.62 thf(fact_50_lfp_Oleq__antisym, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X) => (X = Y2))))))))).
% 2.83/1.62 thf(fact_51_ltake__LNil, axiom, (! [A:$tType,N:extended_enat]: (((((coindu1802687541_ltake @ A) @ N) @ (coindu1598213697e_LNil @ A)) = (coindu1598213697e_LNil @ A))))).
% 2.83/1.62 thf(fact_52_lnull__ltake, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ (((coindu1802687541_ltake @ A) @ N) @ Xs)) = (((coindu335574135_lnull @ A) @ Xs) | (N = (zero_zero @ extended_enat))))))).
% 2.83/1.62 thf(fact_53_ltake_Odisc__iff_I1_J, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ (((coindu1802687541_ltake @ A) @ N) @ Xs)) = ((N = (zero_zero @ extended_enat)) | ((coindu335574135_lnull @ A) @ Xs)))))).
% 2.83/1.62 thf(fact_54_ltake_Odisc__iff_I2_J, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: (((~ (((coindu335574135_lnull @ A) @ (((coindu1802687541_ltake @ A) @ N) @ Xs)))) = ((N != (zero_zero @ extended_enat)) & (~ (((coindu335574135_lnull @ A) @ Xs)))))))).
% 2.83/1.62 thf(fact_55_ltake__0, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: (((((coindu1802687541_ltake @ A) @ (zero_zero @ extended_enat)) @ Xs) = (coindu1598213697e_LNil @ A))))).
% 2.83/1.62 thf(fact_56_LNil__eq__ltake__iff, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: ((((coindu1598213697e_LNil @ A) = (((coindu1802687541_ltake @ A) @ N) @ Xs)) = ((Xs = (coindu1598213697e_LNil @ A)) | (N = (zero_zero @ extended_enat))))))).
% 2.83/1.62 thf(fact_57_ltake__eq__ltake__antimono, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A),M:extended_enat]: ((((((coindu1802687541_ltake @ A) @ N) @ Xs) = (((coindu1802687541_ltake @ A) @ N) @ Ys)) => ((((ord_less_eq @ extended_enat) @ M) @ N) => ((((coindu1802687541_ltake @ A) @ M) @ Xs) = (((coindu1802687541_ltake @ A) @ M) @ Ys))))))).
% 2.83/1.62 thf(fact_58_ltake_Odisc_I1_J, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: ((((N = (zero_zero @ extended_enat)) | ((coindu335574135_lnull @ A) @ Xs)) => ((coindu335574135_lnull @ A) @ (((coindu1802687541_ltake @ A) @ N) @ Xs)))))).
% 2.83/1.62 thf(fact_59_ltake_Odisc_I2_J, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: (((N != (zero_zero @ extended_enat)) => ((~ (((coindu335574135_lnull @ A) @ Xs))) => (~ (((coindu335574135_lnull @ A) @ (((coindu1802687541_ltake @ A) @ N) @ Xs))))))))).
% 2.83/1.62 thf(fact_60_ltake__eq__LNil__iff, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: ((((((coindu1802687541_ltake @ A) @ N) @ Xs) = (coindu1598213697e_LNil @ A)) = ((Xs = (coindu1598213697e_LNil @ A)) | (N = (zero_zero @ extended_enat))))))).
% 2.83/1.62 thf(fact_61_ltake__all, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),M:extended_enat]: (((((ord_less_eq @ extended_enat) @ ((coindu1018505716length @ A) @ Xs)) @ M) => ((((coindu1802687541_ltake @ A) @ M) @ Xs) = Xs))))).
% 2.83/1.62 thf(fact_62_lfp_Oleq__refl, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.83/1.62 thf(fact_63_gfp_Oleq__trans, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [Y2:A,X:A,Z:A]: (((((ord_less_eq @ A) @ Y2) @ X) => ((((ord_less_eq @ A) @ Z) @ Y2) => (((ord_less_eq @ A) @ Z) @ X))))))))).
% 2.83/1.62 thf(fact_64_lfp_Oleq__trans, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.83/1.62 thf(fact_65_gfp_Oleq__antisym, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [Y2:A,X:A]: (((((ord_less_eq @ A) @ Y2) @ X) => ((((ord_less_eq @ A) @ X) @ Y2) => (X = Y2))))))))).
% 2.83/1.62 thf(fact_66_the__enat__0, axiom, ((extended_the_enat @ (zero_zero @ extended_enat)) = (zero_zero @ nat))).
% 2.83/1.62 thf(fact_67_case__enat__0, axiom, (! [A:$tType,F:(nat > A),I:A]: ((((((extended_case_enat @ A) @ F) @ I) @ (zero_zero @ extended_enat)) = (F @ (zero_zero @ nat)))))).
% 2.83/1.62 thf(fact_68_lprefix__ltake__same, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A),M:extended_enat]: (((((coindu1696667936prefix @ A) @ (((coindu1802687541_ltake @ A) @ N) @ Xs)) @ (((coindu1802687541_ltake @ A) @ M) @ Xs)) = ((((ord_less_eq @ extended_enat) @ N) @ M) | (((ord_less_eq @ extended_enat) @ ((coindu1018505716length @ A) @ Xs)) @ M)))))).
% 2.83/1.62 thf(fact_69_ltake__lappend1, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((((ord_less_eq @ extended_enat) @ N) @ ((coindu1018505716length @ A) @ Xs)) => ((((coindu1802687541_ltake @ A) @ N) @ (((coindu268472904append @ A) @ Xs) @ Ys)) = (((coindu1802687541_ltake @ A) @ N) @ Xs)))))).
% 2.83/1.62 thf(fact_70_llength__code, axiom, (! [A:$tType]: (((coindu1018505716length @ A) = ((coindu493225792length @ A) @ (zero_zero @ nat)))))).
% 2.83/1.62 thf(fact_71_dbl__simps_I2_J, axiom, (! [A:$tType]: ((((neg_numeral @ A) @ (type2 @ A)) => (((neg_numeral_dbl @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.83/1.62 thf(fact_72_ltake_Osimps_I3_J, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: (((N != (zero_zero @ extended_enat)) => ((~ (((coindu335574135_lnull @ A) @ Xs))) => (((coindu1046438764le_lhd @ A) @ (((coindu1802687541_ltake @ A) @ N) @ Xs)) = ((coindu1046438764le_lhd @ A) @ Xs))))))).
% 2.83/1.62 thf(fact_73_lprefix__refl, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: ((((coindu1696667936prefix @ A) @ Xs) @ Xs)))).
% 2.83/1.62 thf(fact_74_llist_Oleq__refl, axiom, (! [A:$tType,X:(coindu1593790203_llist @ A)]: ((((coindu1696667936prefix @ A) @ X) @ X)))).
% 2.83/1.62 thf(fact_75_lprefix__code_I1_J, axiom, (! [A:$tType,Ys:(coindu1593790203_llist @ A)]: ((((coindu1696667936prefix @ A) @ (coindu1598213697e_LNil @ A)) @ Ys)))).
% 2.83/1.62 thf(fact_76_ltake__is__lprefix, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: ((((coindu1696667936prefix @ A) @ (((coindu1802687541_ltake @ A) @ N) @ Xs)) @ Xs)))).
% 2.83/1.62 thf(fact_77_lappend_Odisc__iff_I2_J, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((~ (((coindu335574135_lnull @ A) @ (((coindu268472904append @ A) @ Xs) @ Ys)))) = ((~ (((coindu335574135_lnull @ A) @ Xs))) | (~ (((coindu335574135_lnull @ A) @ Ys)))))))).
% 2.83/1.62 thf(fact_78_lnull__lappend, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ (((coindu268472904append @ A) @ Xs) @ Ys)) = (((coindu335574135_lnull @ A) @ Xs) & ((coindu335574135_lnull @ A) @ Ys)))))).
% 2.83/1.62 thf(fact_79_lappend__LNil2, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: (((((coindu268472904append @ A) @ Xs) @ (coindu1598213697e_LNil @ A)) = Xs)))).
% 2.83/1.62 thf(fact_80_lappend__code_I1_J, axiom, (! [A:$tType,Ys:(coindu1593790203_llist @ A)]: (((((coindu268472904append @ A) @ (coindu1598213697e_LNil @ A)) @ Ys) = Ys)))).
% 2.83/1.62 thf(fact_81_lprefix__LNil, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: (((((coindu1696667936prefix @ A) @ Xs) @ (coindu1598213697e_LNil @ A)) = ((coindu335574135_lnull @ A) @ Xs))))).
% 2.83/1.62 thf(fact_82_lhd__lappend, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((((coindu335574135_lnull @ A) @ Xs) => (((coindu1046438764le_lhd @ A) @ (((coindu268472904append @ A) @ Xs) @ Ys)) = ((coindu1046438764le_lhd @ A) @ Ys))) & ((~ (((coindu335574135_lnull @ A) @ Xs))) => (((coindu1046438764le_lhd @ A) @ (((coindu268472904append @ A) @ Xs) @ Ys)) = ((coindu1046438764le_lhd @ A) @ Xs))))))).
% 2.83/1.62 thf(fact_83_lhd__ltake, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A)]: (((N != (zero_zero @ extended_enat)) => (((coindu1046438764le_lhd @ A) @ (((coindu1802687541_ltake @ A) @ N) @ Xs)) = ((coindu1046438764le_lhd @ A) @ Xs)))))).
% 2.83/1.62 thf(fact_84_lprefix__lhdD, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((((coindu1696667936prefix @ A) @ Xs) @ Ys) => ((~ (((coindu335574135_lnull @ A) @ Xs))) => (((coindu1046438764le_lhd @ A) @ Xs) = ((coindu1046438764le_lhd @ A) @ Ys))))))).
% 2.83/1.62 thf(fact_85_lappend__assoc, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A),Zs:(coindu1593790203_llist @ A)]: (((((coindu268472904append @ A) @ (((coindu268472904append @ A) @ Xs) @ Ys)) @ Zs) = (((coindu268472904append @ A) @ Xs) @ (((coindu268472904append @ A) @ Ys) @ Zs)))))).
% 2.83/1.62 thf(fact_86_lprefix__trans, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A),Zs:(coindu1593790203_llist @ A)]: (((((coindu1696667936prefix @ A) @ Xs) @ Ys) => ((((coindu1696667936prefix @ A) @ Ys) @ Zs) => (((coindu1696667936prefix @ A) @ Xs) @ Zs)))))).
% 2.83/1.62 thf(fact_87_llist_Oleq__trans, axiom, (! [A:$tType,X:(coindu1593790203_llist @ A),Y2:(coindu1593790203_llist @ A),Z:(coindu1593790203_llist @ A)]: (((((coindu1696667936prefix @ A) @ X) @ Y2) => ((((coindu1696667936prefix @ A) @ Y2) @ Z) => (((coindu1696667936prefix @ A) @ X) @ Z)))))).
% 2.83/1.62 thf(fact_88_lprefix__antisym, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((((coindu1696667936prefix @ A) @ Xs) @ Ys) => ((((coindu1696667936prefix @ A) @ Ys) @ Xs) => (Xs = Ys)))))).
% 2.83/1.62 thf(fact_89_lprefix__lappend, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: ((((coindu1696667936prefix @ A) @ Xs) @ (((coindu268472904append @ A) @ Xs) @ Ys))))).
% 2.83/1.62 thf(fact_90_llist_Oleq__antisym, axiom, (! [A:$tType,X:(coindu1593790203_llist @ A),Y2:(coindu1593790203_llist @ A)]: (((((coindu1696667936prefix @ A) @ X) @ Y2) => ((((coindu1696667936prefix @ A) @ Y2) @ X) => (X = Y2)))))).
% 2.83/1.62 thf(fact_91_lprefix__down__linear, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Zs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((((coindu1696667936prefix @ A) @ Xs) @ Zs) => ((((coindu1696667936prefix @ A) @ Ys) @ Zs) => ((((coindu1696667936prefix @ A) @ Xs) @ Ys) | (((coindu1696667936prefix @ A) @ Ys) @ Xs))))))).
% 2.83/1.62 thf(fact_92_lprefix__lappend__sameI, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A),Zs:(coindu1593790203_llist @ A)]: (((((coindu1696667936prefix @ A) @ Xs) @ Ys) => (((coindu1696667936prefix @ A) @ (((coindu268472904append @ A) @ Zs) @ Xs)) @ (((coindu268472904append @ A) @ Zs) @ Ys)))))).
% 2.83/1.62 thf(fact_93_ltakeWhile_Oexhaust, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((~ ((((coindu335574135_lnull @ A) @ Xs) | (~ ((P @ ((coindu1046438764le_lhd @ A) @ Xs))))))) => (~ (((~ (((coindu335574135_lnull @ A) @ Xs))) => (~ ((P @ ((coindu1046438764le_lhd @ A) @ Xs))))))))))).
% 2.83/1.62 thf(fact_94_lnull__lprefix, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Xs) => (((coindu1696667936prefix @ A) @ Xs) @ Ys))))).
% 2.83/1.62 thf(fact_95_lprefix__lnull, axiom, (! [A:$tType,Ys:(coindu1593790203_llist @ A),Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Ys) => ((((coindu1696667936prefix @ A) @ Xs) @ Ys) = ((coindu335574135_lnull @ A) @ Xs)))))).
% 2.83/1.62 thf(fact_96_lprefix__lnullD, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((((coindu1696667936prefix @ A) @ Xs) @ Ys) => (((coindu335574135_lnull @ A) @ Ys) => ((coindu335574135_lnull @ A) @ Xs)))))).
% 2.83/1.62 thf(fact_97_lprefix__not__lnullD, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((((coindu1696667936prefix @ A) @ Xs) @ Ys) => ((~ (((coindu335574135_lnull @ A) @ Xs))) => (~ (((coindu335574135_lnull @ A) @ Ys)))))))).
% 2.83/1.62 thf(fact_98_lappend_Odisc_I2_J, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: ((((~ (((coindu335574135_lnull @ A) @ Xs))) | (~ (((coindu335574135_lnull @ A) @ Ys)))) => (~ (((coindu335574135_lnull @ A) @ (((coindu268472904append @ A) @ Xs) @ Ys)))))))).
% 2.83/1.62 thf(fact_99_lappend_Odisc_I1_J, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Xs) => (((coindu335574135_lnull @ A) @ Ys) => ((coindu335574135_lnull @ A) @ (((coindu268472904append @ A) @ Xs) @ Ys))))))).
% 2.83/1.62 thf(fact_100_lappend__lnull1, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Xs) => ((((coindu268472904append @ A) @ Xs) @ Ys) = Ys))))).
% 2.83/1.62 thf(fact_101_lappend__lnull2, axiom, (! [A:$tType,Ys:(coindu1593790203_llist @ A),Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Ys) => ((((coindu268472904append @ A) @ Xs) @ Ys) = Xs))))).
% 2.83/1.62 thf(fact_102_LNil__lprefix, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: ((((coindu1696667936prefix @ A) @ (coindu1598213697e_LNil @ A)) @ Xs)))).
% 2.83/1.62 thf(fact_103_lappend__eq__LNil__iff, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: ((((((coindu268472904append @ A) @ Xs) @ Ys) = (coindu1598213697e_LNil @ A)) = ((Xs = (coindu1598213697e_LNil @ A)) & (Ys = (coindu1598213697e_LNil @ A))))))).
% 2.83/1.62 thf(fact_104_LNil__eq__lappend__iff, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: ((((coindu1598213697e_LNil @ A) = (((coindu268472904append @ A) @ Xs) @ Ys)) = ((Xs = (coindu1598213697e_LNil @ A)) & (Ys = (coindu1598213697e_LNil @ A))))))).
% 2.83/1.62 thf(fact_105_lappend__LNil__LNil, axiom, (! [A:$tType]: (((((coindu268472904append @ A) @ (coindu1598213697e_LNil @ A)) @ (coindu1598213697e_LNil @ A)) = (coindu1598213697e_LNil @ A))))).
% 2.83/1.62 thf(fact_106_lappend_Octr_I1_J, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ Xs) => (((coindu335574135_lnull @ A) @ Ys) => ((((coindu268472904append @ A) @ Xs) @ Ys) = (coindu1598213697e_LNil @ A))))))).
% 2.83/1.62 thf(fact_107_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.83/1.62 thf(fact_108_lstrict__prefix__def, axiom, (! [A:$tType]: (((coindu574146665prefix @ A) = (^ [Xs2:(coindu1593790203_llist @ A),Ys2:(coindu1593790203_llist @ A)]: (((((coindu1696667936prefix @ A) @ Xs2) @ Ys2) & (Xs2 != Ys2)))))))).
% 2.83/1.62 thf(fact_109_ltakeWhile_Octr_I1_J, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((((coindu335574135_lnull @ A) @ Xs) | (~ ((P @ ((coindu1046438764le_lhd @ A) @ Xs))))) => ((((coindu721411036eWhile @ A) @ P) @ Xs) = (coindu1598213697e_LNil @ A)))))).
% 2.83/1.62 thf(fact_110_ltakeWhile_Odisc__iff_I1_J, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: ((((coindu335574135_lnull @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs)) = (((coindu335574135_lnull @ A) @ Xs) | (~ ((P @ ((coindu1046438764le_lhd @ A) @ Xs))))))))).
% 2.83/1.62 thf(fact_111_ltakeWhile_Odisc__iff_I2_J, axiom, (! [A:$tType,P:(A > $o),Xs:(coindu1593790203_llist @ A)]: (((~ (((coindu335574135_lnull @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs)))) = ((~ (((coindu335574135_lnull @ A) @ Xs))) & (P @ ((coindu1046438764le_lhd @ A) @ Xs))))))).
% 2.83/1.62 thf(fact_112_ltake__lappend2, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),N:extended_enat,Ys:(coindu1593790203_llist @ A)]: (((((ord_less_eq @ extended_enat) @ ((coindu1018505716length @ A) @ Xs)) @ N) => ((((coindu1802687541_ltake @ A) @ N) @ (((coindu268472904append @ A) @ Xs) @ Ys)) = (((coindu268472904append @ A) @ Xs) @ (((coindu1802687541_ltake @ A) @ (((minus_minus @ extended_enat) @ N) @ ((coindu1018505716length @ A) @ Xs))) @ Ys))))))).
% 2.83/1.62 thf(fact_113_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.83/1.62 thf(fact_114_diff__zero, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.83/1.62 thf(fact_115_zero__diff, axiom, (! [A:$tType]: ((((comm_monoid_diff @ A) @ (type2 @ A)) => (! [A2:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.83/1.62 thf(fact_116_diff__0__right, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.83/1.62 thf(fact_117_diff__self, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.83/1.62 thf(fact_118_idiff__0__right, axiom, (! [N:extended_enat]: (((((minus_minus @ extended_enat) @ N) @ (zero_zero @ extended_enat)) = N)))).
% 2.83/1.62 thf(fact_119_idiff__0, axiom, (! [N:extended_enat]: (((((minus_minus @ extended_enat) @ (zero_zero @ extended_enat)) @ N) = (zero_zero @ extended_enat))))).
% 2.83/1.62 thf(fact_120_diff__ge__0__iff__ge, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B2)) = (((ord_less_eq @ A) @ B2) @ A2)))))))).
% 2.83/1.62 thf(fact_121_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.83/1.62 thf(fact_122_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.83/1.62 thf(fact_123_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.83/1.62 thf(fact_124_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.83/1.62 thf(fact_125_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.83/1.62 thf(fact_126_diff__eq__diff__eq, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A,D:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C) @ D)) => ((A2 = B2) = (C = D))))))))).
% 2.83/1.62 thf(fact_127_diff__right__commute, axiom, (! [A:$tType]: ((((cancel146912293up_add @ A) @ (type2 @ A)) => (! [A2:A,C:A,B2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C)) @ B2) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C)))))))).
% 2.83/1.62 thf(fact_128_diff__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,D:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ D) @ C) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B2) @ D)))))))))).
% 2.83/1.62 thf(fact_129_diff__left__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C) @ A2)) @ (((minus_minus @ A) @ C) @ B2))))))))).
% 2.83/1.62 thf(fact_130_diff__right__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B2) @ C))))))))).
% 2.83/1.62 thf(fact_131_diff__eq__diff__less__eq, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A,D:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C) @ D)) => ((((ord_less_eq @ A) @ A2) @ B2) = (((ord_less_eq @ A) @ C) @ D))))))))).
% 2.83/1.62 thf(fact_132_eq__iff__diff__eq__0, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => ((^ [Y3:A,Z2:A]: ((Y3 = Z2))) = (^ [A4:A,B4:A]: (((((minus_minus @ A) @ A4) @ B4) = (zero_zero @ A))))))))).
% 2.83/1.62 thf(fact_133_enat__minus__mono1, axiom, (! [X:extended_enat,Y2:extended_enat,M:extended_enat]: (((((ord_less_eq @ extended_enat) @ X) @ Y2) => (((ord_less_eq @ extended_enat) @ (((minus_minus @ extended_enat) @ X) @ M)) @ (((minus_minus @ extended_enat) @ Y2) @ M)))))).
% 2.83/1.62 thf(fact_134_enat__diff__cancel__left, axiom, (! [M:extended_enat,X:extended_enat,Y2:extended_enat]: (((((ord_less_eq @ extended_enat) @ M) @ X) => ((((ord_less_eq @ extended_enat) @ M) @ Y2) => (((((minus_minus @ extended_enat) @ X) @ M) = (((minus_minus @ extended_enat) @ Y2) @ M)) = (X = Y2))))))).
% 2.83/1.62 thf(fact_135_le__iff__diff__le__0, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ A4) @ B4)) @ (zero_zero @ A))))))))).
% 2.83/1.62 thf(fact_136_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.83/1.62 thf(fact_137_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.83/1.62 thf(fact_138_ltake__lappend, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((((coindu1802687541_ltake @ A) @ N) @ (((coindu268472904append @ A) @ Xs) @ Ys)) = (((coindu268472904append @ A) @ (((coindu1802687541_ltake @ A) @ N) @ Xs)) @ (((coindu1802687541_ltake @ A) @ (((minus_minus @ extended_enat) @ N) @ ((coindu1018505716length @ A) @ Xs))) @ Ys)))))).
% 2.83/1.62 thf(fact_139_ltakeWhile_Odisc_I2_J, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((~ (((coindu335574135_lnull @ A) @ Xs))) => ((P @ ((coindu1046438764le_lhd @ A) @ Xs)) => (~ (((coindu335574135_lnull @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs))))))))).
% 2.83/1.62 thf(fact_140_ltakeWhile_Odisc_I1_J, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((((coindu335574135_lnull @ A) @ Xs) | (~ ((P @ ((coindu1046438764le_lhd @ A) @ Xs))))) => ((coindu335574135_lnull @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs)))))).
% 2.83/1.62 thf(fact_141_ltakeWhile_Osimps_I3_J, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:(A > $o)]: (((~ (((coindu335574135_lnull @ A) @ Xs))) => ((P @ ((coindu1046438764le_lhd @ A) @ Xs)) => (((coindu1046438764le_lhd @ A) @ (((coindu721411036eWhile @ A) @ P) @ Xs)) = ((coindu1046438764le_lhd @ A) @ Xs))))))).
% 2.83/1.62 thf(fact_142_minus__apply, axiom, (! [B:$tType,A:$tType]: ((((minus @ B) @ (type2 @ B)) => ((minus_minus @ (A > B)) = (^ [A5:(A > B),B5:(A > B),X3:A]: ((((minus_minus @ B) @ (A5 @ X3)) @ (B5 @ X3))))))))).
% 2.83/1.62 thf(fact_143_of__nat__le__0__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.83/1.62 thf(fact_144_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M:(A > nat)]: (((P @ K) => (? [X2:A]: (((P @ X2) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M @ X2)) @ (M @ Y5)))))))))))).
% 2.83/1.62 thf(fact_145_zero__natural_Orsp, axiom, ((zero_zero @ nat) = (zero_zero @ nat))).
% 2.83/1.62 thf(fact_146_diff__self__eq__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ M) = (zero_zero @ nat))))).
% 2.83/1.62 thf(fact_147_diff__0__eq__0, axiom, (! [N:nat]: (((((minus_minus @ nat) @ (zero_zero @ nat)) @ N) = (zero_zero @ nat))))).
% 2.83/1.62 thf(fact_148_of__nat__eq__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (! [M:nat,N:nat]: (((((semiring_1_of_nat @ A) @ M) = ((semiring_1_of_nat @ A) @ N)) = (M = N)))))))).
% 2.83/1.62 thf(fact_149_diff__diff__cancel, axiom, (! [I:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ N) => ((((minus_minus @ nat) @ N) @ (((minus_minus @ nat) @ N) @ I)) = I))))).
% 2.83/1.62 thf(fact_150_diff__is__0__eq_H, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)))))).
% 2.83/1.62 thf(fact_151_diff__is__0__eq, axiom, (! [M:nat,N:nat]: ((((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.83/1.62 thf(fact_152_of__nat__eq__0__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (! [M:nat]: (((((semiring_1_of_nat @ A) @ M) = (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.83/1.62 thf(fact_153_of__nat__0__eq__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (! [N:nat]: ((((zero_zero @ A) = ((semiring_1_of_nat @ A) @ N)) = ((zero_zero @ nat) = N)))))))).
% 2.83/1.62 thf(fact_154_of__nat__0, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type2 @ A)) => (((semiring_1_of_nat @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))).
% 2.83/1.62 thf(fact_155_of__nat__le__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M:nat,N:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less_eq @ nat) @ M) @ N)))))))).
% 2.83/1.62 thf(fact_156_diffs0__imp__equal, axiom, (! [M:nat,N:nat]: ((((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)) => (((((minus_minus @ nat) @ N) @ M) = (zero_zero @ nat)) => (M = N)))))).
% 2.83/1.62 thf(fact_157_minus__nat_Odiff__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ (zero_zero @ nat)) = M)))).
% 2.83/1.62 thf(fact_158_eq__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => (((((minus_minus @ nat) @ M) @ K) = (((minus_minus @ nat) @ N) @ K)) = (M = N))))))).
% 2.83/1.62 thf(fact_159_le__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((ord_less_eq @ nat) @ M) @ N))))))).
% 2.83/1.62 thf(fact_160_Nat_Odiff__diff__eq, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((minus_minus @ nat) @ M) @ N))))))).
% 2.83/1.62 thf(fact_161_diff__le__mono, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ L)) @ (((minus_minus @ nat) @ N) @ L)))))).
% 2.83/1.62 thf(fact_162_diff__le__self, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ M)))).
% 2.83/1.62 thf(fact_163_le__diff__iff_H, axiom, (! [A2:nat,C:nat,B2:nat]: (((((ord_less_eq @ nat) @ A2) @ C) => ((((ord_less_eq @ nat) @ B2) @ C) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ C) @ A2)) @ (((minus_minus @ nat) @ C) @ B2)) = (((ord_less_eq @ nat) @ B2) @ A2))))))).
% 2.83/1.62 thf(fact_164_diff__le__mono2, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ L) @ N)) @ (((minus_minus @ nat) @ L) @ M)))))).
% 2.83/1.62 thf(fact_165_of__nat__0__le__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [N:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)))))))).
% 2.83/1.62 thf(fact_166_of__nat__diff, axiom, (! [A:$tType]: ((((ring_1 @ A) @ (type2 @ A)) => (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ N) @ M) => (((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ M) @ N)) = (((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)))))))))).
% 2.83/1.62 thf(fact_167_fun__diff__def, axiom, (! [B:$tType,A:$tType]: ((((minus @ B) @ (type2 @ B)) => ((minus_minus @ (A > B)) = (^ [A5:(A > B),B5:(A > B),X3:A]: ((((minus_minus @ B) @ (A5 @ X3)) @ (B5 @ X3))))))))).
% 2.83/1.62 thf(fact_168_real__arch__simple, axiom, (! [A:$tType]: ((((archim1804426504_field @ A) @ (type2 @ A)) => (! [X:A]: ((? [N2:nat]: ((((ord_less_eq @ A) @ X) @ ((semiring_1_of_nat @ A) @ N2)))))))))).
% 2.83/1.62 thf(fact_169_natural_Osize_I1_J, axiom, ((code_size_natural @ (zero_zero @ code_natural)) = (zero_zero @ nat))).
% 2.83/1.62 thf(fact_170_natural_Osize_I3_J, axiom, (((size_size @ code_natural) @ (zero_zero @ code_natural)) = (zero_zero @ nat))).
% 2.83/1.62 thf(fact_171_diff__commute, axiom, (! [I:nat,J:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ K) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ K)) @ J))))).
% 2.83/1.62 thf(fact_172_natural_Osimps_I4_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > T)]: ((((((code_case_natural @ T) @ F1) @ F22) @ (zero_zero @ code_natural)) = F1)))).
% 2.83/1.62 thf(fact_173_natural_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > (T > T))]: ((((((code_rec_natural @ T) @ F1) @ F22) @ (zero_zero @ code_natural)) = F1)))).
% 2.83/1.62 thf(fact_174_LeastMI2, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [P:(A > $o),X:A,M:(A > B),Q:(A > $o)]: (((P @ X) => ((! [Y:A]: (((P @ Y) => (((ord_less_eq @ B) @ (M @ X)) @ (M @ Y))))) => ((! [X2:A]: (((P @ X2) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ B) @ (M @ X2)) @ (M @ Y5))))) => (Q @ X2))))) => (Q @ ((((hilbert_LeastM @ A) @ B) @ M) @ P))))))))))).
% 2.83/1.62 thf(fact_175_LeastM__nat__le, axiom, (! [A:$tType,P:(A > $o),X:A,M:(A > nat)]: (((P @ X) => (((ord_less_eq @ nat) @ (M @ ((((hilbert_LeastM @ A) @ nat) @ M) @ P))) @ (M @ X)))))).
% 2.83/1.62 thf(fact_176_LeastM__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,M:(A > nat)]: (((P @ K) => ((P @ ((((hilbert_LeastM @ A) @ nat) @ M) @ P)) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M @ ((((hilbert_LeastM @ A) @ nat) @ M) @ P))) @ (M @ Y5)))))))))).
% 2.83/1.62 thf(fact_177_natural_Osimps_I5_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > T),Natural:code_natural]: ((((((code_case_natural @ T) @ F1) @ F22) @ (code_Suc @ Natural)) = (F22 @ Natural))))).
% 2.83/1.62 thf(fact_178_transfer__int__nat__relations_I3_J, axiom, (! [X:nat,Y2:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ X)) @ ((semiring_1_of_nat @ int) @ Y2)) = (((ord_less_eq @ nat) @ X) @ Y2))))).
% 2.83/1.62 thf(fact_179_zle__int, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.83/1.62 thf(fact_180_natural_Oinject, axiom, (! [Natural:code_natural,Natural2:code_natural]: ((((code_Suc @ Natural) = (code_Suc @ Natural2)) = (Natural = Natural2))))).
% 2.83/1.62 thf(fact_181_natural_Osimps_I7_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > (T > T)),Natural:code_natural]: ((((((code_rec_natural @ T) @ F1) @ F22) @ (code_Suc @ Natural)) = ((F22 @ Natural) @ ((((code_rec_natural @ T) @ F1) @ F22) @ Natural)))))).
% 2.83/1.62 thf(fact_182_LeastM__natI, axiom, (! [A:$tType,P:(A > $o),K:A,M:(A > nat)]: (((P @ K) => (P @ ((((hilbert_LeastM @ A) @ nat) @ M) @ P)))))).
% 2.83/1.62 thf(fact_183_transfer__int__nat__relations_I1_J, axiom, (! [X:nat,Y2:nat]: (((((semiring_1_of_nat @ int) @ X) = ((semiring_1_of_nat @ int) @ Y2)) = (X = Y2))))).
% 2.83/1.62 thf(fact_184_int__int__eq, axiom, (! [M:nat,N:nat]: (((((semiring_1_of_nat @ int) @ M) = ((semiring_1_of_nat @ int) @ N)) = (M = N))))).
% 2.83/1.62 thf(fact_185_int__if__cong, axiom, (! [P:$o,X:nat,Y2:nat]: (((P => (((semiring_1_of_nat @ int) @ X) = ((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ X) @ Y2)))) & ((~ (P)) => (((semiring_1_of_nat @ int) @ Y2) = ((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ X) @ Y2)))))))).
% 2.83/1.62 thf(fact_186_natural_Odistinct_I2_J, axiom, (! [Natural3:code_natural]: (((code_Suc @ Natural3) != (zero_zero @ code_natural))))).
% 2.83/1.62 thf(fact_187_natural_Odistinct_I1_J, axiom, (! [Natural2:code_natural]: (((zero_zero @ code_natural) != (code_Suc @ Natural2))))).
% 2.83/1.62 thf(fact_188_natural_Oexhaust, axiom, (! [Y2:code_natural]: (((Y2 != (zero_zero @ code_natural)) => (~ ((! [Natural4:code_natural]: ((Y2 != (code_Suc @ Natural4)))))))))).
% 2.83/1.62 thf(fact_189_natural_Oinducts, axiom, (! [P:(code_natural > $o),Natural:code_natural]: (((P @ (zero_zero @ code_natural)) => ((! [Natural4:code_natural]: (((P @ Natural4) => (P @ (code_Suc @ Natural4))))) => (P @ Natural)))))).
% 2.83/1.62 thf(fact_190_int__diff__cases, axiom, (! [Z:int]: ((~ ((! [M2:nat,N2:nat]: ((Z != (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ M2)) @ ((semiring_1_of_nat @ int) @ N2)))))))))).
% 2.83/1.62 thf(fact_191_zero__le__imp__eq__int, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (? [N2:nat]: ((K = ((semiring_1_of_nat @ int) @ N2)))))))).
% 2.83/1.62 thf(fact_192_nonneg__int__cases, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N2:nat]: ((K != ((semiring_1_of_nat @ int) @ N2)))))))))).
% 2.83/1.62 thf(fact_193_nonneg__eq__int, axiom, (! [Z:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => (~ ((! [M2:nat]: ((Z != ((semiring_1_of_nat @ int) @ M2)))))))))).
% 2.83/1.62 thf(fact_194_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J, axiom, (! [Z:nat]: ((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((semiring_1_of_nat @ int) @ Z))))).
% 2.83/1.62 thf(fact_195_transfer__int__nat__quantifiers_I1_J, axiom, (! [P:(int > $o)]: (((! [X3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X3) => (P @ X3)))) = (! [X3:nat]: ((P @ ((semiring_1_of_nat @ int) @ X3)))))))).
% 2.83/1.62 thf(fact_196_transfer__int__nat__quantifiers_I2_J, axiom, (! [P:(int > $o)]: (((? [X3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X3) & (P @ X3)))) = (? [X3:nat]: ((P @ ((semiring_1_of_nat @ int) @ X3)))))))).
% 2.83/1.62 thf(fact_197_transfer__int__nat__numerals_I1_J, axiom, ((zero_zero @ int) = ((semiring_1_of_nat @ int) @ (zero_zero @ nat)))).
% 2.83/1.62 thf(fact_198_Nat__Transfer_Otransfer__int__nat__functions_I3_J, axiom, (! [X:nat,Y2:nat]: ((((nat_tsub @ ((semiring_1_of_nat @ int) @ X)) @ ((semiring_1_of_nat @ int) @ Y2)) = ((semiring_1_of_nat @ int) @ (((minus_minus @ nat) @ X) @ Y2)))))).
% 2.83/1.62 thf(fact_199_round__of__nat, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [N:nat]: ((((archimedean_round @ A) @ ((semiring_1_of_nat @ A) @ N)) = ((semiring_1_of_nat @ int) @ N)))))))).
% 2.83/1.62 thf(fact_200_round__0, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (((archimedean_round @ A) @ (zero_zero @ A)) = (zero_zero @ int)))))).
% 2.83/1.62 thf(fact_201_minus__int__code_I1_J, axiom, (! [K:int]: (((((minus_minus @ int) @ K) @ (zero_zero @ int)) = K)))).
% 2.83/1.62 thf(fact_202_zero__integer_Orsp, axiom, ((zero_zero @ int) = (zero_zero @ int))).
% 2.83/1.62 thf(fact_203_tsub__def, axiom, (nat_tsub = (^ [X3:int,Y4:int]: (((((if @ int) @ (((ord_less_eq @ int) @ Y4) @ X3)) @ (((minus_minus @ int) @ X3) @ Y4)) @ (zero_zero @ int)))))).
% 2.83/1.62 thf(fact_204_tsub__eq, axiom, (! [Y2:int,X:int]: (((((ord_less_eq @ int) @ Y2) @ X) => (((nat_tsub @ X) @ Y2) = (((minus_minus @ int) @ X) @ Y2)))))).
% 2.83/1.62 thf(fact_205_round__mono, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => (((ord_less_eq @ int) @ ((archimedean_round @ A) @ X)) @ ((archimedean_round @ A) @ Y2))))))))).
% 2.83/1.62 thf(fact_206_ceiling__le__zero, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ (zero_zero @ int)) = (((ord_less_eq @ A) @ X) @ (zero_zero @ A))))))))).
% 2.83/1.62 thf(fact_207_of__int__0__le__iff, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [Z:int]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z)))))))).
% 2.83/1.62 thf(fact_208_of__int__0, axiom, (! [A:$tType]: ((((ring_1 @ A) @ (type2 @ A)) => (((ring_1_of_int @ A) @ (zero_zero @ int)) = (zero_zero @ A)))))).
% 2.83/1.62 thf(fact_209_of__int__0__eq__iff, axiom, (! [A:$tType]: ((((ring_char_0 @ A) @ (type2 @ A)) => (! [Z:int]: ((((zero_zero @ A) = ((ring_1_of_int @ A) @ Z)) = (Z = (zero_zero @ int))))))))).
% 2.83/1.62 thf(fact_210_of__int__eq__0__iff, axiom, (! [A:$tType]: ((((ring_char_0 @ A) @ (type2 @ A)) => (! [Z:int]: (((((ring_1_of_int @ A) @ Z) = (zero_zero @ A)) = (Z = (zero_zero @ int))))))))).
% 2.83/1.62 thf(fact_211_of__int__le__iff, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [W:int,Z:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ W)) @ ((ring_1_of_int @ A) @ Z)) = (((ord_less_eq @ int) @ W) @ Z)))))))).
% 2.83/1.62 thf(fact_212_ceiling__zero, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (((archimedean_ceiling @ A) @ (zero_zero @ A)) = (zero_zero @ int)))))).
% 2.83/1.62 thf(fact_213_of__int__diff, axiom, (! [A:$tType]: ((((ring_1 @ A) @ (type2 @ A)) => (! [W:int,Z:int]: ((((ring_1_of_int @ A) @ (((minus_minus @ int) @ W) @ Z)) = (((minus_minus @ A) @ ((ring_1_of_int @ A) @ W)) @ ((ring_1_of_int @ A) @ Z))))))))).
% 2.83/1.62 thf(fact_214_of__int__of__nat__eq, axiom, (! [A:$tType]: ((((ring_1 @ A) @ (type2 @ A)) => (! [N:nat]: ((((ring_1_of_int @ A) @ ((semiring_1_of_nat @ int) @ N)) = ((semiring_1_of_nat @ A) @ N)))))))).
% 2.83/1.62 thf(fact_215_ceiling__of__nat, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [N:nat]: ((((archimedean_ceiling @ A) @ ((semiring_1_of_nat @ A) @ N)) = ((semiring_1_of_nat @ int) @ N)))))))).
% 2.83/1.62 thf(fact_216_ceiling__diff__of__int, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A,Z:int]: ((((archimedean_ceiling @ A) @ (((minus_minus @ A) @ X) @ ((ring_1_of_int @ A) @ Z))) = (((minus_minus @ int) @ ((archimedean_ceiling @ A) @ X)) @ Z)))))))).
% 2.83/1.62 thf(fact_217_of__int__le__0__iff, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [Z:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z)) @ (zero_zero @ A)) = (((ord_less_eq @ int) @ Z) @ (zero_zero @ int))))))))).
% 2.83/1.62 thf(fact_218_ex__le__of__int, axiom, (! [A:$tType]: ((((archim1804426504_field @ A) @ (type2 @ A)) => (! [X:A]: ((? [Z3:int]: ((((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ Z3)))))))))).
% 2.83/1.62 thf(fact_219_ceiling__mono, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [Y2:A,X:A]: (((((ord_less_eq @ A) @ Y2) @ X) => (((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ Y2)) @ ((archimedean_ceiling @ A) @ X))))))))).
% 2.83/1.62 thf(fact_220_le__of__int__ceiling, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ X))))))))).
% 2.83/1.62 thf(fact_221_ceiling__le__iff, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A,Z:int]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ Z) = (((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ Z))))))))).
% 2.83/1.62 thf(fact_222_of__int__nonneg, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [Z:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z))))))))).
% 2.83/1.62 thf(fact_223_ceiling__le, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A,A2:int]: (((((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ A2)) => (((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ A2)))))))).
% 2.83/1.62 thf(fact_224_of__nat__nat, axiom, (! [A:$tType]: ((((ring_1 @ A) @ (type2 @ A)) => (! [Z:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => (((semiring_1_of_nat @ A) @ (nat2 @ Z)) = ((ring_1_of_int @ A) @ Z))))))))).
% 2.83/1.62 thf(fact_225_nat__ceiling__le__eq, axiom, (! [X:real,A2:nat]: (((((ord_less_eq @ nat) @ (nat2 @ ((archimedean_ceiling @ real) @ X))) @ A2) = (((ord_less_eq @ real) @ X) @ ((semiring_1_of_nat @ real) @ A2)))))).
% 2.83/1.62 thf(fact_226_nat__int, axiom, (! [N:nat]: (((nat2 @ ((semiring_1_of_nat @ int) @ N)) = N)))).
% 2.83/1.62 thf(fact_227_nat__0, axiom, ((nat2 @ (zero_zero @ int)) = (zero_zero @ nat))).
% 2.83/1.62 thf(fact_228_nat__0__iff, axiom, (! [I:int]: ((((nat2 @ I) = (zero_zero @ nat)) = (((ord_less_eq @ int) @ I) @ (zero_zero @ int)))))).
% 2.83/1.62 thf(fact_229_nat__le__0, axiom, (! [Z:int]: (((((ord_less_eq @ int) @ Z) @ (zero_zero @ int)) => ((nat2 @ Z) = (zero_zero @ nat)))))).
% 2.83/1.62 thf(fact_230_int__nat__eq, axiom, (! [Z:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => (((semiring_1_of_nat @ int) @ (nat2 @ Z)) = Z)) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z))) => (((semiring_1_of_nat @ int) @ (nat2 @ Z)) = (zero_zero @ int))))))).
% 2.83/1.62 thf(fact_231_real__nat__ceiling__ge, axiom, (! [X:real]: ((((ord_less_eq @ real) @ X) @ ((semiring_1_of_nat @ real) @ (nat2 @ ((archimedean_ceiling @ real) @ X))))))).
% 2.83/1.62 thf(fact_232_nat__mono, axiom, (! [X:int,Y2:int]: (((((ord_less_eq @ int) @ X) @ Y2) => (((ord_less_eq @ nat) @ (nat2 @ X)) @ (nat2 @ Y2)))))).
% 2.83/1.62 thf(fact_233_transfer__nat__int__relations_I3_J, axiom, (! [X:int,Y2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y2) => ((((ord_less_eq @ nat) @ (nat2 @ X)) @ (nat2 @ Y2)) = (((ord_less_eq @ int) @ X) @ Y2))))))).
% 2.83/1.62 thf(fact_234_nat__le__iff, axiom, (! [X:int,N:nat]: (((((ord_less_eq @ nat) @ (nat2 @ X)) @ N) = (((ord_less_eq @ int) @ X) @ ((semiring_1_of_nat @ int) @ N)))))).
% 2.83/1.62 thf(fact_235_int__eq__iff, axiom, (! [M:nat,Z:int]: (((((semiring_1_of_nat @ int) @ M) = Z) = ((M = (nat2 @ Z)) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z)))))).
% 2.83/1.62 thf(fact_236_nat__0__le, axiom, (! [Z:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => (((semiring_1_of_nat @ int) @ (nat2 @ Z)) = Z))))).
% 2.83/1.62 thf(fact_237_nat__eq__iff, axiom, (! [W:int,M:nat]: ((((nat2 @ W) = M) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W) => (W = ((semiring_1_of_nat @ int) @ M))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ W))) => (M = (zero_zero @ nat)))))))).
% 2.83/1.62 thf(fact_238_nat__eq__iff2, axiom, (! [M:nat,W:int]: (((M = (nat2 @ W)) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W) => (W = ((semiring_1_of_nat @ int) @ M))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ W))) => (M = (zero_zero @ nat)))))))).
% 2.83/1.62 thf(fact_239_le__nat__iff, axiom, (! [K:int,N:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less_eq @ nat) @ N) @ (nat2 @ K)) = (((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ N)) @ K)))))).
% 2.83/1.62 thf(fact_240_nat__diff__distrib_H, axiom, (! [X:int,Y2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y2) => ((nat2 @ (((minus_minus @ int) @ X) @ Y2)) = (((minus_minus @ nat) @ (nat2 @ X)) @ (nat2 @ Y2)))))))).
% 2.83/1.62 thf(fact_241_nat__diff__distrib, axiom, (! [Z4:int,Z:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z4) => ((((ord_less_eq @ int) @ Z4) @ Z) => ((nat2 @ (((minus_minus @ int) @ Z) @ Z4)) = (((minus_minus @ nat) @ (nat2 @ Z)) @ (nat2 @ Z4)))))))).
% 2.83/1.62 thf(fact_242_Nat__Transfer_Otransfer__nat__int__functions_I3_J, axiom, (! [X:int,Y2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y2) => ((((minus_minus @ nat) @ (nat2 @ X)) @ (nat2 @ Y2)) = (nat2 @ ((nat_tsub @ X) @ Y2)))))))).
% 2.83/1.62 thf(fact_243_int__eq__iff__numeral, axiom, (! [M:nat,V:num]: (((((semiring_1_of_nat @ int) @ M) = ((numeral_numeral @ int) @ V)) = ((M = (nat2 @ ((numeral_numeral @ int) @ V))) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ ((numeral_numeral @ int) @ V))))))).
% 2.83/1.62 thf(fact_244_nat__abs__int__diff, axiom, (! [A2:nat,B2:nat]: ((((((ord_less_eq @ nat) @ A2) @ B2) => ((nat2 @ ((abs_abs @ int) @ (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ A2)) @ ((semiring_1_of_nat @ int) @ B2)))) = (((minus_minus @ nat) @ B2) @ A2))) & ((~ ((((ord_less_eq @ nat) @ A2) @ B2))) => ((nat2 @ ((abs_abs @ int) @ (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ A2)) @ ((semiring_1_of_nat @ int) @ B2)))) = (((minus_minus @ nat) @ A2) @ B2))))))).
% 2.83/1.62 thf(fact_245_numeral__eq__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (! [M:num,N:num]: (((((numeral_numeral @ A) @ M) = ((numeral_numeral @ A) @ N)) = (M = N)))))))).
% 2.83/1.62 thf(fact_246_abs__idempotent, axiom, (! [A:$tType]: ((((ordere142940540dd_abs @ A) @ (type2 @ A)) => (! [A2:A]: ((((abs_abs @ A) @ ((abs_abs @ A) @ A2)) = ((abs_abs @ A) @ A2)))))))).
% 2.83/1.62 thf(fact_247_numeral__le__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M:num,N:num]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = (((ord_less_eq @ num) @ M) @ N)))))))).
% 2.83/1.62 thf(fact_248_abs__0__eq, axiom, (! [A:$tType]: ((((ordere142940540dd_abs @ A) @ (type2 @ A)) => (! [A2:A]: ((((zero_zero @ A) = ((abs_abs @ A) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 2.83/1.62 thf(fact_249_abs__eq__0, axiom, (! [A:$tType]: ((((ordere142940540dd_abs @ A) @ (type2 @ A)) => (! [A2:A]: (((((abs_abs @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.83/1.62 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A6:$tType,A7:$tType]: ((((comple187826305attice @ A7) @ (type2 @ A7)) => ((comple187826305attice @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.83/1.62 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A6:$tType,A7:$tType]: ((((preorder @ A7) @ (type2 @ A7)) => ((preorder @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.83/1.62 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: ((((order @ A7) @ (type2 @ A7)) => ((order @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.83/1.62 thf(tcon_fun___Orderings_Oord, axiom, (! [A6:$tType,A7:$tType]: ((((ord @ A7) @ (type2 @ A7)) => ((ord @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.83/1.62 thf(tcon_fun___Groups_Ominus, axiom, (! [A6:$tType,A7:$tType]: ((((minus @ A7) @ (type2 @ A7)) => ((minus @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.83/1.62 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs, axiom, ((ordere142940540dd_abs @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add, axiom, ((cancel146912293up_add @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, axiom, ((cancel1352612707id_add @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add, axiom, ((ordered_ab_group_add @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, ((linordered_semidom @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, ((linordered_idom @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, ((semiring_char_0 @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Orderings_Opreorder_1, axiom, ((preorder @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, ((linorder @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, ((semiring_1 @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, ((group_add @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Orderings_Oorder_2, axiom, ((order @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Num_Oneg__numeral, axiom, ((neg_numeral @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Nat_Oring__char__0, axiom, ((ring_char_0 @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Orderings_Oord_3, axiom, ((ord @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Rings_Oring__1, axiom, ((ring_1 @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Groups_Ominus_4, axiom, ((minus @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Num_Onumeral, axiom, ((numeral @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Int_Oint___Groups_Ozero, axiom, ((zero @ int) @ (type2 @ int))).
% 2.83/1.62 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_5, axiom, ((cancel146912293up_add @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_6, axiom, ((cancel1352612707id_add @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_7, axiom, ((linordered_semidom @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff, axiom, ((comm_monoid_diff @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_8, axiom, ((semiring_char_0 @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Nat_Onat___Orderings_Opreorder_9, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Nat_Onat___Orderings_Olinorder_10, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Nat_Onat___Rings_Osemiring__1_11, axiom, ((semiring_1 @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Nat_Onat___Orderings_Oorder_12, axiom, ((order @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Nat_Onat___Orderings_Oord_13, axiom, ((ord @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Nat_Onat___Groups_Ominus_14, axiom, ((minus @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Nat_Onat___Num_Onumeral_15, axiom, ((numeral @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Nat_Onat___Groups_Ozero_16, axiom, ((zero @ nat) @ (type2 @ nat))).
% 2.83/1.62 thf(tcon_Num_Onum___Orderings_Opreorder_17, axiom, ((preorder @ num) @ (type2 @ num))).
% 2.83/1.62 thf(tcon_Num_Onum___Orderings_Olinorder_18, axiom, ((linorder @ num) @ (type2 @ num))).
% 2.83/1.62 thf(tcon_Num_Onum___Orderings_Oorder_19, axiom, ((order @ num) @ (type2 @ num))).
% 2.83/1.62 thf(tcon_Num_Onum___Orderings_Oord_20, axiom, ((ord @ num) @ (type2 @ num))).
% 2.83/1.62 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_21, axiom, ((comple187826305attice @ $o) @ (type2 @ $o))).
% 2.83/1.62 thf(tcon_HOL_Obool___Orderings_Opreorder_22, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.83/1.62 thf(tcon_HOL_Obool___Orderings_Olinorder_23, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.83/1.62 thf(tcon_HOL_Obool___Orderings_Oorder_24, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.83/1.62 thf(tcon_HOL_Obool___Orderings_Oord_25, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.83/1.62 thf(tcon_HOL_Obool___Groups_Ominus_26, axiom, ((minus @ $o) @ (type2 @ $o))).
% 2.83/1.62 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field, axiom, ((archim1804426504_field @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_27, axiom, ((ordere142940540dd_abs @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling, axiom, ((archim1727834104eiling @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_28, axiom, ((cancel146912293up_add @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_29, axiom, ((cancel1352612707id_add @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_30, axiom, ((ordered_ab_group_add @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_31, axiom, ((linordered_semidom @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Rings_Olinordered__idom_32, axiom, ((linordered_idom @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_33, axiom, ((semiring_char_0 @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Orderings_Opreorder_34, axiom, ((preorder @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Orderings_Olinorder_35, axiom, ((linorder @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Rings_Osemiring__1_36, axiom, ((semiring_1 @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Groups_Ogroup__add_37, axiom, ((group_add @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Orderings_Oorder_38, axiom, ((order @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Num_Oneg__numeral_39, axiom, ((neg_numeral @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Nat_Oring__char__0_40, axiom, ((ring_char_0 @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Orderings_Oord_41, axiom, ((ord @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Rings_Oring__1_42, axiom, ((ring_1 @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Groups_Ominus_43, axiom, ((minus @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Num_Onumeral_44, axiom, ((numeral @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Real_Oreal___Groups_Ozero_45, axiom, ((zero @ real) @ (type2 @ real))).
% 2.83/1.62 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_46, axiom, ((canoni770627133id_add @ extended_enat) @ (type2 @ extended_enat))).
% 2.83/1.62 thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_47, axiom, ((comple187826305attice @ extended_enat) @ (type2 @ extended_enat))).
% 2.83/1.62 thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_48, axiom, ((semiring_char_0 @ extended_enat) @ (type2 @ extended_enat))).
% 2.83/1.62 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_49, axiom, ((preorder @ extended_enat) @ (type2 @ extended_enat))).
% 2.83/1.62 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_50, axiom, ((linorder @ extended_enat) @ (type2 @ extended_enat))).
% 2.83/1.62 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_51, axiom, ((semiring_1 @ extended_enat) @ (type2 @ extended_enat))).
% 2.83/1.62 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_52, axiom, ((order @ extended_enat) @ (type2 @ extended_enat))).
% 2.83/1.62 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_53, axiom, ((ord @ extended_enat) @ (type2 @ extended_enat))).
% 2.83/1.62 thf(tcon_Extended__Nat_Oenat___Groups_Ominus_54, axiom, ((minus @ extended_enat) @ (type2 @ extended_enat))).
% 2.83/1.62 thf(tcon_Extended__Nat_Oenat___Num_Onumeral_55, axiom, ((numeral @ extended_enat) @ (type2 @ extended_enat))).
% 2.83/1.62 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_56, axiom, ((zero @ extended_enat) @ (type2 @ extended_enat))).
% 2.83/1.62 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_57, axiom, ((cancel146912293up_add @ code_natural) @ (type2 @ code_natural))).
% 2.83/1.62 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_58, axiom, ((cancel1352612707id_add @ code_natural) @ (type2 @ code_natural))).
% 2.83/1.62 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_59, axiom, ((comm_monoid_diff @ code_natural) @ (type2 @ code_natural))).
% 2.83/1.62 thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_60, axiom, ((preorder @ code_natural) @ (type2 @ code_natural))).
% 2.83/1.62 thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_61, axiom, ((linorder @ code_natural) @ (type2 @ code_natural))).
% 2.83/1.62 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_62, axiom, ((semiring_1 @ code_natural) @ (type2 @ code_natural))).
% 2.83/1.62 thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_63, axiom, ((order @ code_natural) @ (type2 @ code_natural))).
% 2.83/1.62 thf(tcon_Code__Numeral_Onatural___Orderings_Oord_64, axiom, ((ord @ code_natural) @ (type2 @ code_natural))).
% 2.83/1.62 thf(tcon_Code__Numeral_Onatural___Groups_Ominus_65, axiom, ((minus @ code_natural) @ (type2 @ code_natural))).
% 2.83/1.62 thf(tcon_Code__Numeral_Onatural___Num_Onumeral_66, axiom, ((numeral @ code_natural) @ (type2 @ code_natural))).
% 2.83/1.62 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_67, axiom, ((zero @ code_natural) @ (type2 @ code_natural))).
% 2.83/1.62 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.83/1.62 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y2:A]: ((((((if @ A) @ $false) @ X) @ Y2) = Y2)))).
% 2.83/1.62 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y2:A]: ((((((if @ A) @ $true) @ X) @ Y2) = X)))).
% 2.83/1.62 thf(conj_0, hypothesis, (((ord_less_eq @ extended_enat) @ ((coindu1018505716length @ a) @ xs)) @ m)).
% 2.83/1.62 thf(conj_1, conjecture, ((((coindu191418589_ldrop @ a) @ m) @ xs) = (coindu1598213697e_LNil @ a))).
% 2.83/1.62 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.bImcdsmqd4/DTF2DTF17343.p
% 2.83/1.62 ------------------------
% 2.83/1.66 ---- Cleaned THF ---
% 2.83/1.66 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
% 2.83/1.66 coindu1593790203_llist: $tType > $tType ).
% 2.83/1.66
% 2.83/1.66 thf(ty_t_Code__Numeral_Onatural,type,
% 2.83/1.66 code_natural: $tType ).
% 2.83/1.66
% 2.83/1.66 thf(ty_t_Extended__Nat_Oenat,type,
% 2.83/1.66 extended_enat: $tType ).
% 2.83/1.66
% 2.83/1.66 thf(ty_t_Real_Oreal,type,
% 2.83/1.66 real: $tType ).
% 2.83/1.66
% 2.83/1.66 thf(ty_t_Num_Onum,type,
% 2.83/1.66 num: $tType ).
% 2.83/1.66
% 2.83/1.66 thf(ty_t_Nat_Onat,type,
% 2.83/1.66 nat: $tType ).
% 2.83/1.66
% 2.83/1.66 thf(ty_t_Int_Oint,type,
% 2.83/1.66 int: $tType ).
% 2.83/1.66
% 2.83/1.66 thf(ty_t_itself,type,
% 2.83/1.66 itself: $tType > $tType ).
% 2.83/1.66
% 2.83/1.66 thf(ty_tf_a,type,
% 2.83/1.66 a: $tType ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_HOL_Otype,type,
% 2.83/1.66 type:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Groups_Ozero,type,
% 2.83/1.66 zero:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Num_Onumeral,type,
% 2.83/1.66 numeral:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Groups_Ominus,type,
% 2.83/1.66 minus:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Rings_Oring__1,type,
% 2.83/1.66 ring_1:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Orderings_Oord,type,
% 2.83/1.66 ord:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Nat_Oring__char__0,type,
% 2.83/1.66 ring_char_0:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Num_Oneg__numeral,type,
% 2.83/1.66 neg_numeral:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Orderings_Oorder,type,
% 2.83/1.66 order:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Groups_Ogroup__add,type,
% 2.83/1.66 group_add:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Rings_Osemiring__1,type,
% 2.83/1.66 semiring_1:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Orderings_Olinorder,type,
% 2.83/1.66 linorder:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Orderings_Opreorder,type,
% 2.83/1.66 preorder:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Nat_Osemiring__char__0,type,
% 2.83/1.66 semiring_char_0:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.83/1.66 linordered_idom:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 2.83/1.66 comm_monoid_diff:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.83/1.66 linordered_semidom:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.83/1.66 ordered_ab_group_add:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.83/1.66 cancel1352612707id_add:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.83/1.66 cancel146912293up_add:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
% 2.83/1.66 archim1727834104eiling:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
% 2.83/1.66 ordere142940540dd_abs:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 2.83/1.66 comple187826305attice:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 2.83/1.66 archim1804426504_field:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.83/1.66 canoni770627133id_add:
% 2.83/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Archimedean__Field_Oceiling,type,
% 2.83/1.66 archimedean_ceiling:
% 2.83/1.66 !>[A: $tType] : ( A > int ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Archimedean__Field_Oround,type,
% 2.83/1.66 archimedean_round:
% 2.83/1.66 !>[A: $tType] : ( A > int ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Code__Numeral_OSuc,type,
% 2.83/1.66 code_Suc: code_natural > code_natural ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
% 2.83/1.66 code_case_natural:
% 2.83/1.66 !>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
% 2.83/1.66 code_rec_natural:
% 2.83/1.66 !>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Code__Numeral_Onatural_Osize__natural,type,
% 2.83/1.66 code_size_natural: code_natural > nat ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__llength,type,
% 2.83/1.66 coindu493225792length:
% 2.83/1.66 !>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend,type,
% 2.83/1.66 coindu268472904append:
% 2.83/1.66 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldrop,type,
% 2.83/1.66 coindu191418589_ldrop:
% 2.83/1.66 !>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
% 2.83/1.66 coindu1018505716length:
% 2.83/1.66 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil,type,
% 2.83/1.66 coindu1598213697e_LNil:
% 2.83/1.66 !>[A: $tType] : ( coindu1593790203_llist @ A ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olhd,type,
% 2.83/1.66 coindu1046438764le_lhd:
% 2.83/1.66 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
% 2.83/1.66 coindu335574135_lnull:
% 2.83/1.66 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olmember,type,
% 2.83/1.66 coindu567634248member:
% 2.83/1.66 !>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olprefix,type,
% 2.83/1.66 coindu1696667936prefix:
% 2.83/1.66 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olstrict__prefix,type,
% 2.83/1.66 coindu574146665prefix:
% 2.83/1.66 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oltake,type,
% 2.83/1.66 coindu1802687541_ltake:
% 2.83/1.66 !>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OltakeWhile,type,
% 2.83/1.66 coindu721411036eWhile:
% 2.83/1.66 !>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
% 2.83/1.66 extended_case_enat:
% 2.83/1.66 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Extended__Nat_Othe__enat,type,
% 2.83/1.66 extended_the_enat: extended_enat > nat ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 2.83/1.66 abs_abs:
% 2.83/1.66 !>[A: $tType] : ( A > A ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.83/1.66 minus_minus:
% 2.83/1.66 !>[A: $tType] : ( A > A > A ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.83/1.66 zero_zero:
% 2.83/1.66 !>[A: $tType] : A ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Hilbert__Choice_OLeastM,type,
% 2.83/1.66 hilbert_LeastM:
% 2.83/1.66 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_If,type,
% 2.83/1.66 if:
% 2.83/1.66 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Int_Onat,type,
% 2.83/1.66 nat2: int > nat ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 2.83/1.66 ring_1_of_int:
% 2.83/1.66 !>[A: $tType] : ( int > A ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 2.83/1.66 semiring_1_of_nat:
% 2.83/1.66 !>[A: $tType] : ( nat > A ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.83/1.66 size_size:
% 2.83/1.66 !>[A: $tType] : ( A > nat ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Nat__Transfer_Otsub,type,
% 2.83/1.66 nat_tsub: int > int > int ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 2.83/1.66 neg_numeral_dbl:
% 2.83/1.66 !>[A: $tType] : ( A > A ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 2.83/1.66 numeral_numeral:
% 2.83/1.66 !>[A: $tType] : ( num > A ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.83/1.66 ord_less_eq:
% 2.83/1.66 !>[A: $tType] : ( A > A > $o ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_c_Pure_Otype,type,
% 2.83/1.66 type2:
% 2.83/1.66 !>[A: $tType] : ( itself @ A ) ).
% 2.83/1.66
% 2.83/1.66 thf(sy_v_m,type,
% 2.83/1.66 m: extended_enat ).
% 2.83/1.66
% 2.83/1.66 thf(sy_v_xs,type,
% 2.83/1.66 xs: coindu1593790203_llist @ a ).
% 2.83/1.66
% 2.83/1.66 thf(fact_0_ldrop__LNil,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat] :
% 2.83/1.66 ( ( coindu191418589_ldrop @ A @ N @ ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_1_ldrop__eq__LNil,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ( coindu191418589_ldrop @ A @ N @ Xs )
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 = ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_2_wlog__linorder__le,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linorder @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [P: A > A > $o,B2: A,A2: A] :
% 2.83/1.66 ( ! [A3: A,B3: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.83/1.66 => ( P @ A3 @ B3 ) )
% 2.83/1.66 => ( ( ( P @ B2 @ A2 )
% 2.83/1.66 => ( P @ A2 @ B2 ) )
% 2.83/1.66 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_3_order__refl,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( preorder @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_4_lnull__ldrop,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ ( coindu191418589_ldrop @ A @ N @ Xs ) )
% 2.83/1.66 = ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_5_lmember__code_I1_J,axiom,
% 2.83/1.66 ! [A: $tType,X: A] :
% 2.83/1.66 ~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_6_llength__LNil,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( coindu1018505716length @ A @ ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 = ( zero_zero @ extended_enat ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_7_le__funD,axiom,
% 2.83/1.66 ! [B: $tType,A: $tType] :
% 2.83/1.66 ( ( ord @ B @ ( type2 @ B ) )
% 2.83/1.66 => ! [F: A > B,G: A > B,X: A] :
% 2.83/1.66 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.83/1.66 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_8_le__funE,axiom,
% 2.83/1.66 ! [B: $tType,A: $tType] :
% 2.83/1.66 ( ( ord @ B @ ( type2 @ B ) )
% 2.83/1.66 => ! [F: A > B,G: A > B,X: A] :
% 2.83/1.66 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.83/1.66 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_9_le__funI,axiom,
% 2.83/1.66 ! [B: $tType,A: $tType] :
% 2.83/1.66 ( ( ord @ B @ ( type2 @ B ) )
% 2.83/1.66 => ! [F: A > B,G: A > B] :
% 2.83/1.66 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 2.83/1.66 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_10_le__fun__def,axiom,
% 2.83/1.66 ! [B: $tType,A: $tType] :
% 2.83/1.66 ( ( ord @ B @ ( type2 @ B ) )
% 2.83/1.66 => ( ( ord_less_eq @ ( A > B ) )
% 2.83/1.66 = ( ^ [F2: A > B,G2: A > B] :
% 2.83/1.66 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_11_order__subst1,axiom,
% 2.83/1.66 ! [A: $tType,B: $tType] :
% 2.83/1.66 ( ( ( order @ B @ ( type2 @ B ) )
% 2.83/1.66 & ( order @ A @ ( type2 @ A ) ) )
% 2.83/1.66 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.83/1.66 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.83/1.66 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.83/1.66 => ( ! [X2: B,Y: B] :
% 2.83/1.66 ( ( ord_less_eq @ B @ X2 @ Y )
% 2.83/1.66 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.83/1.66 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_12_order__subst2,axiom,
% 2.83/1.66 ! [A: $tType,C2: $tType] :
% 2.83/1.66 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.83/1.66 & ( order @ A @ ( type2 @ A ) ) )
% 2.83/1.66 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.83/1.66 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.66 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 2.83/1.66 => ( ! [X2: A,Y: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.83/1.66 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.83/1.66 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_13_ord__eq__le__subst,axiom,
% 2.83/1.66 ! [A: $tType,B: $tType] :
% 2.83/1.66 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.83/1.66 & ( ord @ A @ ( type2 @ A ) ) )
% 2.83/1.66 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.83/1.66 ( ( A2
% 2.83/1.66 = ( F @ B2 ) )
% 2.83/1.66 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.83/1.66 => ( ! [X2: B,Y: B] :
% 2.83/1.66 ( ( ord_less_eq @ B @ X2 @ Y )
% 2.83/1.66 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.83/1.66 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_14_ord__le__eq__subst,axiom,
% 2.83/1.66 ! [A: $tType,B: $tType] :
% 2.83/1.66 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.83/1.66 & ( ord @ A @ ( type2 @ A ) ) )
% 2.83/1.66 => ! [A2: A,B2: A,F: A > B,C: B] :
% 2.83/1.66 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.66 => ( ( ( F @ B2 )
% 2.83/1.66 = C )
% 2.83/1.66 => ( ! [X2: A,Y: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.83/1.66 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.83/1.66 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_15_ldrop__0,axiom,
% 2.83/1.66 ! [B: $tType,Xs: coindu1593790203_llist @ B] :
% 2.83/1.66 ( ( coindu191418589_ldrop @ B @ ( zero_zero @ extended_enat ) @ Xs )
% 2.83/1.66 = Xs ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_16_llength__lnull,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ( ( coindu1018505716length @ A @ Xs )
% 2.83/1.66 = ( zero_zero @ extended_enat ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_17_llength__eq__0,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ( coindu1018505716length @ A @ Xs )
% 2.83/1.66 = ( zero_zero @ extended_enat ) )
% 2.83/1.66 = ( coindu335574135_lnull @ A @ Xs ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_18_lzip_Oexhaust,axiom,
% 2.83/1.66 ! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
% 2.83/1.66 ( ~ ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 | ( coindu335574135_lnull @ B @ Ys ) )
% 2.83/1.66 => ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_19_ltake_Oexhaust,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ~ ( ( N
% 2.83/1.66 = ( zero_zero @ extended_enat ) )
% 2.83/1.66 | ( coindu335574135_lnull @ A @ Xs ) )
% 2.83/1.66 => ~ ( ( N
% 2.83/1.66 != ( zero_zero @ extended_enat ) )
% 2.83/1.66 => ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_20_lappend_Oexhaust,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ~ ( coindu335574135_lnull @ A @ Ys ) )
% 2.83/1.66 => ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 | ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_21_lnull__def,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A )
% 2.83/1.66 = ( ^ [Llist: coindu1593790203_llist @ A] :
% 2.83/1.66 ( Llist
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_22_llist_Ocollapse_I1_J,axiom,
% 2.83/1.66 ! [A: $tType,Llist2: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ Llist2 )
% 2.83/1.66 => ( Llist2
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_23_llist_OdiscI_I1_J,axiom,
% 2.83/1.66 ! [A: $tType,Llist2: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( Llist2
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 => ( coindu335574135_lnull @ A @ Llist2 ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_24_llist_Odisc_I1_J,axiom,
% 2.83/1.66 ! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_25_ldrop__lnull,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,N: extended_enat] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ( ( coindu191418589_ldrop @ A @ N @ Xs )
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_26_dual__order_Oantisym,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( order @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [B2: A,A2: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.83/1.66 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.66 => ( A2 = B2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_27_dual__order_Otrans,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( order @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [B2: A,A2: A,C: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.83/1.66 => ( ( ord_less_eq @ A @ C @ B2 )
% 2.83/1.66 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_28_linorder__wlog,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linorder @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.83/1.66 ( ! [A3: A,B3: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.83/1.66 => ( P @ A3 @ B3 ) )
% 2.83/1.66 => ( ! [A3: A,B3: A] :
% 2.83/1.66 ( ( P @ B3 @ A3 )
% 2.83/1.66 => ( P @ A3 @ B3 ) )
% 2.83/1.66 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_29_dual__order_Orefl,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( order @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_30_order__trans,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( preorder @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A,Y2: A,Z: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.83/1.66 => ( ( ord_less_eq @ A @ Y2 @ Z )
% 2.83/1.66 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_31_order__class_Oorder_Oantisym,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( order @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A,B2: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.66 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.83/1.66 => ( A2 = B2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_32_ord__le__eq__trans,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ord @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A,B2: A,C: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.66 => ( ( B2 = C )
% 2.83/1.66 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_33_ord__eq__le__trans,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ord @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A,B2: A,C: A] :
% 2.83/1.66 ( ( A2 = B2 )
% 2.83/1.66 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.83/1.66 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_34_antisym__conv,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( order @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [Y2: A,X: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ Y2 @ X )
% 2.83/1.66 => ( ( ord_less_eq @ A @ X @ Y2 )
% 2.83/1.66 = ( X = Y2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_35_le__cases3,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linorder @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A,Y2: A,Z: A] :
% 2.83/1.66 ( ( ( ord_less_eq @ A @ X @ Y2 )
% 2.83/1.66 => ~ ( ord_less_eq @ A @ Y2 @ Z ) )
% 2.83/1.66 => ( ( ( ord_less_eq @ A @ Y2 @ X )
% 2.83/1.66 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 2.83/1.66 => ( ( ( ord_less_eq @ A @ X @ Z )
% 2.83/1.66 => ~ ( ord_less_eq @ A @ Z @ Y2 ) )
% 2.83/1.66 => ( ( ( ord_less_eq @ A @ Z @ Y2 )
% 2.83/1.66 => ~ ( ord_less_eq @ A @ Y2 @ X ) )
% 2.83/1.66 => ( ( ( ord_less_eq @ A @ Y2 @ Z )
% 2.83/1.66 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 2.83/1.66 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 2.83/1.66 => ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_36_order_Otrans,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( order @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A,B2: A,C: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.66 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.83/1.66 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_37_le__cases,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linorder @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A,Y2: A] :
% 2.83/1.66 ( ~ ( ord_less_eq @ A @ X @ Y2 )
% 2.83/1.66 => ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_38_eq__refl,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( preorder @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A,Y2: A] :
% 2.83/1.66 ( ( X = Y2 )
% 2.83/1.66 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_39_linear,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linorder @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A,Y2: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.83/1.66 | ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_40_antisym,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( order @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A,Y2: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.83/1.66 => ( ( ord_less_eq @ A @ Y2 @ X )
% 2.83/1.66 => ( X = Y2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_41_eq__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( order @ A @ ( type2 @ A ) )
% 2.83/1.66 => ( ( ^ [Y3: A,Z2: A] : ( Y3 = Z2 ) )
% 2.83/1.66 = ( ^ [X3: A,Y4: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.83/1.66 & ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_42_le__zero__eq,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [N: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.83/1.66 = ( N
% 2.83/1.66 = ( zero_zero @ A ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_43_ext,axiom,
% 2.83/1.66 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.83/1.66 ( ! [X2: A] :
% 2.83/1.66 ( ( F @ X2 )
% 2.83/1.66 = ( G @ X2 ) )
% 2.83/1.66 => ( F = G ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_44_ile0__eq,axiom,
% 2.83/1.66 ! [N: extended_enat] :
% 2.83/1.66 ( ( ord_less_eq @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
% 2.83/1.66 = ( N
% 2.83/1.66 = ( zero_zero @ extended_enat ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_45_i0__lb,axiom,
% 2.83/1.66 ! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_46_zero__le,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_47_le__numeral__extra_I3_J,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.83/1.66 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_48_ltake_Octr_I1_J,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ( N
% 2.83/1.66 = ( zero_zero @ extended_enat ) )
% 2.83/1.66 | ( coindu335574135_lnull @ A @ Xs ) )
% 2.83/1.66 => ( ( coindu1802687541_ltake @ A @ N @ Xs )
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_49_zero__reorient,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( zero @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A] :
% 2.83/1.66 ( ( ( zero_zero @ A )
% 2.83/1.66 = X )
% 2.83/1.66 = ( X
% 2.83/1.66 = ( zero_zero @ A ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_50_lfp_Oleq__antisym,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A,Y2: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.83/1.66 => ( ( ord_less_eq @ A @ Y2 @ X )
% 2.83/1.66 => ( X = Y2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_51_ltake__LNil,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat] :
% 2.83/1.66 ( ( coindu1802687541_ltake @ A @ N @ ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_52_lnull__ltake,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
% 2.83/1.66 = ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 | ( N
% 2.83/1.66 = ( zero_zero @ extended_enat ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_53_ltake_Odisc__iff_I1_J,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
% 2.83/1.66 = ( ( N
% 2.83/1.66 = ( zero_zero @ extended_enat ) )
% 2.83/1.66 | ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_54_ltake_Odisc__iff_I2_J,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ~ ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) )
% 2.83/1.66 = ( ( N
% 2.83/1.66 != ( zero_zero @ extended_enat ) )
% 2.83/1.66 & ~ ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_55_ltake__0,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu1802687541_ltake @ A @ ( zero_zero @ extended_enat ) @ Xs )
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_56_LNil__eq__ltake__iff,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ( coindu1598213697e_LNil @ A )
% 2.83/1.66 = ( coindu1802687541_ltake @ A @ N @ Xs ) )
% 2.83/1.66 = ( ( Xs
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 | ( N
% 2.83/1.66 = ( zero_zero @ extended_enat ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_57_ltake__eq__ltake__antimono,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,M: extended_enat] :
% 2.83/1.66 ( ( ( coindu1802687541_ltake @ A @ N @ Xs )
% 2.83/1.66 = ( coindu1802687541_ltake @ A @ N @ Ys ) )
% 2.83/1.66 => ( ( ord_less_eq @ extended_enat @ M @ N )
% 2.83/1.66 => ( ( coindu1802687541_ltake @ A @ M @ Xs )
% 2.83/1.66 = ( coindu1802687541_ltake @ A @ M @ Ys ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_58_ltake_Odisc_I1_J,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ( N
% 2.83/1.66 = ( zero_zero @ extended_enat ) )
% 2.83/1.66 | ( coindu335574135_lnull @ A @ Xs ) )
% 2.83/1.66 => ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_59_ltake_Odisc_I2_J,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( N
% 2.83/1.66 != ( zero_zero @ extended_enat ) )
% 2.83/1.66 => ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ~ ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_60_ltake__eq__LNil__iff,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ( coindu1802687541_ltake @ A @ N @ Xs )
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 = ( ( Xs
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 | ( N
% 2.83/1.66 = ( zero_zero @ extended_enat ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_61_ltake__all,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,M: extended_enat] :
% 2.83/1.66 ( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M )
% 2.83/1.66 => ( ( coindu1802687541_ltake @ A @ M @ Xs )
% 2.83/1.66 = Xs ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_62_lfp_Oleq__refl,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_63_gfp_Oleq__trans,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [Y2: A,X: A,Z: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ Y2 @ X )
% 2.83/1.66 => ( ( ord_less_eq @ A @ Z @ Y2 )
% 2.83/1.66 => ( ord_less_eq @ A @ Z @ X ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_64_lfp_Oleq__trans,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A,Y2: A,Z: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.83/1.66 => ( ( ord_less_eq @ A @ Y2 @ Z )
% 2.83/1.66 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_65_gfp_Oleq__antisym,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [Y2: A,X: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ Y2 @ X )
% 2.83/1.66 => ( ( ord_less_eq @ A @ X @ Y2 )
% 2.83/1.66 => ( X = Y2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_66_the__enat__0,axiom,
% 2.83/1.66 ( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
% 2.83/1.66 = ( zero_zero @ nat ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_67_case__enat__0,axiom,
% 2.83/1.66 ! [A: $tType,F: nat > A,I: A] :
% 2.83/1.66 ( ( extended_case_enat @ A @ F @ I @ ( zero_zero @ extended_enat ) )
% 2.83/1.66 = ( F @ ( zero_zero @ nat ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_68_lprefix__ltake__same,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,M: extended_enat] :
% 2.83/1.66 ( ( coindu1696667936prefix @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ A @ M @ Xs ) )
% 2.83/1.66 = ( ( ord_less_eq @ extended_enat @ N @ M )
% 2.83/1.66 | ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_69_ltake__lappend1,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ord_less_eq @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) )
% 2.83/1.66 => ( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 2.83/1.66 = ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_70_llength__code,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( coindu1018505716length @ A )
% 2.83/1.66 = ( coindu493225792length @ A @ ( zero_zero @ nat ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_71_dbl__simps_I2_J,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( neg_numeral @ A @ ( type2 @ A ) )
% 2.83/1.66 => ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
% 2.83/1.66 = ( zero_zero @ A ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_72_ltake_Osimps_I3_J,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( N
% 2.83/1.66 != ( zero_zero @ extended_enat ) )
% 2.83/1.66 => ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ( ( coindu1046438764le_lhd @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
% 2.83/1.66 = ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_73_lprefix__refl,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ Xs ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_74_llist_Oleq__refl,axiom,
% 2.83/1.66 ! [A: $tType,X: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X @ X ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_75_lprefix__code_I1_J,axiom,
% 2.83/1.66 ! [A: $tType,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_76_ltake__is__lprefix,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ Xs ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_77_lappend_Odisc__iff_I2_J,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) )
% 2.83/1.66 = ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 | ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_78_lnull__lappend,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 2.83/1.66 = ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 & ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_79_lappend__LNil2,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 = Xs ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_80_lappend__code_I1_J,axiom,
% 2.83/1.66 ! [A: $tType,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
% 2.83/1.66 = Ys ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_81_lprefix__LNil,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu1696667936prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 = ( coindu335574135_lnull @ A @ Xs ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_82_lhd__lappend,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 2.83/1.66 = ( coindu1046438764le_lhd @ A @ Ys ) ) )
% 2.83/1.66 & ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 2.83/1.66 = ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_83_lhd__ltake,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( N
% 2.83/1.66 != ( zero_zero @ extended_enat ) )
% 2.83/1.66 => ( ( coindu1046438764le_lhd @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
% 2.83/1.66 = ( coindu1046438764le_lhd @ A @ Xs ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_84_lprefix__lhdD,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 2.83/1.66 => ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ( ( coindu1046438764le_lhd @ A @ Xs )
% 2.83/1.66 = ( coindu1046438764le_lhd @ A @ Ys ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_85_lappend__assoc,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
% 2.83/1.66 = ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_86_lprefix__trans,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 2.83/1.66 => ( ( coindu1696667936prefix @ A @ Ys @ Zs )
% 2.83/1.66 => ( coindu1696667936prefix @ A @ Xs @ Zs ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_87_llist_Oleq__trans,axiom,
% 2.83/1.66 ! [A: $tType,X: coindu1593790203_llist @ A,Y2: coindu1593790203_llist @ A,Z: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu1696667936prefix @ A @ X @ Y2 )
% 2.83/1.66 => ( ( coindu1696667936prefix @ A @ Y2 @ Z )
% 2.83/1.66 => ( coindu1696667936prefix @ A @ X @ Z ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_88_lprefix__antisym,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 2.83/1.66 => ( ( coindu1696667936prefix @ A @ Ys @ Xs )
% 2.83/1.66 => ( Xs = Ys ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_89_lprefix__lappend,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_90_llist_Oleq__antisym,axiom,
% 2.83/1.66 ! [A: $tType,X: coindu1593790203_llist @ A,Y2: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu1696667936prefix @ A @ X @ Y2 )
% 2.83/1.66 => ( ( coindu1696667936prefix @ A @ Y2 @ X )
% 2.83/1.66 => ( X = Y2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_91_lprefix__down__linear,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu1696667936prefix @ A @ Xs @ Zs )
% 2.83/1.66 => ( ( coindu1696667936prefix @ A @ Ys @ Zs )
% 2.83/1.66 => ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 2.83/1.66 | ( coindu1696667936prefix @ A @ Ys @ Xs ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_92_lprefix__lappend__sameI,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 2.83/1.66 => ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Zs @ Xs ) @ ( coindu268472904append @ A @ Zs @ Ys ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_93_ltakeWhile_Oexhaust,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.83/1.66 ( ~ ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
% 2.83/1.66 => ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_94_lnull__lprefix,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_95_lprefix__lnull,axiom,
% 2.83/1.66 ! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ Ys )
% 2.83/1.66 => ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 2.83/1.66 = ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_96_lprefix__lnullD,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 2.83/1.66 => ( ( coindu335574135_lnull @ A @ Ys )
% 2.83/1.66 => ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_97_lprefix__not__lnullD,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu1696667936prefix @ A @ Xs @ Ys )
% 2.83/1.66 => ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_98_lappend_Odisc_I2_J,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 | ~ ( coindu335574135_lnull @ A @ Ys ) )
% 2.83/1.66 => ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_99_lappend_Odisc_I1_J,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ( ( coindu335574135_lnull @ A @ Ys )
% 2.83/1.66 => ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_100_lappend__lnull1,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ( ( coindu268472904append @ A @ Xs @ Ys )
% 2.83/1.66 = Ys ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_101_lappend__lnull2,axiom,
% 2.83/1.66 ! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ Ys )
% 2.83/1.66 => ( ( coindu268472904append @ A @ Xs @ Ys )
% 2.83/1.66 = Xs ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_102_LNil__lprefix,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_103_lappend__eq__LNil__iff,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ( coindu268472904append @ A @ Xs @ Ys )
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 = ( ( Xs
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 & ( Ys
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_104_LNil__eq__lappend__iff,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ( coindu1598213697e_LNil @ A )
% 2.83/1.66 = ( coindu268472904append @ A @ Xs @ Ys ) )
% 2.83/1.66 = ( ( Xs
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 & ( Ys
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_105_lappend__LNil__LNil,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_106_lappend_Octr_I1_J,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ( ( coindu335574135_lnull @ A @ Ys )
% 2.83/1.66 => ( ( coindu268472904append @ A @ Xs @ Ys )
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_107_le0,axiom,
% 2.83/1.66 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_108_lstrict__prefix__def,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( coindu574146665prefix @ A )
% 2.83/1.66 = ( ^ [Xs2: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu1696667936prefix @ A @ Xs2 @ Ys2 )
% 2.83/1.66 & ( Xs2 != Ys2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_109_ltakeWhile_Octr_I1_J,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.83/1.66 ( ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
% 2.83/1.66 => ( ( coindu721411036eWhile @ A @ P @ Xs )
% 2.83/1.66 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_110_ltakeWhile_Odisc__iff_I1_J,axiom,
% 2.83/1.66 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 2.83/1.66 = ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_111_ltakeWhile_Odisc__iff_I2_J,axiom,
% 2.83/1.66 ! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
% 2.83/1.66 = ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 & ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_112_ltake__lappend2,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,N: extended_enat,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N )
% 2.83/1.66 => ( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 2.83/1.66 = ( coindu268472904append @ A @ Xs @ ( coindu1802687541_ltake @ A @ ( minus_minus @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) ) @ Ys ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_113_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A] :
% 2.83/1.66 ( ( minus_minus @ A @ A2 @ A2 )
% 2.83/1.66 = ( zero_zero @ A ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_114_diff__zero,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A] :
% 2.83/1.66 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.83/1.66 = A2 ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_115_zero__diff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( comm_monoid_diff @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A] :
% 2.83/1.66 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 2.83/1.66 = ( zero_zero @ A ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_116_diff__0__right,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( group_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A] :
% 2.83/1.66 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.83/1.66 = A2 ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_117_diff__self,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( group_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A] :
% 2.83/1.66 ( ( minus_minus @ A @ A2 @ A2 )
% 2.83/1.66 = ( zero_zero @ A ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_118_idiff__0__right,axiom,
% 2.83/1.66 ! [N: extended_enat] :
% 2.83/1.66 ( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
% 2.83/1.66 = N ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_119_idiff__0,axiom,
% 2.83/1.66 ! [N: extended_enat] :
% 2.83/1.66 ( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
% 2.83/1.66 = ( zero_zero @ extended_enat ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_120_diff__ge__0__iff__ge,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A,B2: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
% 2.83/1.66 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_121_le__refl,axiom,
% 2.83/1.66 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_122_le__trans,axiom,
% 2.83/1.66 ! [I: nat,J: nat,K: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ I @ J )
% 2.83/1.66 => ( ( ord_less_eq @ nat @ J @ K )
% 2.83/1.66 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_123_eq__imp__le,axiom,
% 2.83/1.66 ! [M: nat,N: nat] :
% 2.83/1.66 ( ( M = N )
% 2.83/1.66 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_124_le__antisym,axiom,
% 2.83/1.66 ! [M: nat,N: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ M @ N )
% 2.83/1.66 => ( ( ord_less_eq @ nat @ N @ M )
% 2.83/1.66 => ( M = N ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_125_nat__le__linear,axiom,
% 2.83/1.66 ! [M: nat,N: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ M @ N )
% 2.83/1.66 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_126_diff__eq__diff__eq,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( group_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A,B2: A,C: A,D: A] :
% 2.83/1.66 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.83/1.66 = ( minus_minus @ A @ C @ D ) )
% 2.83/1.66 => ( ( A2 = B2 )
% 2.83/1.66 = ( C = D ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_127_diff__right__commute,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( cancel146912293up_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A,C: A,B2: A] :
% 2.83/1.66 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 )
% 2.83/1.66 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_128_diff__mono,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A,B2: A,D: A,C: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.66 => ( ( ord_less_eq @ A @ D @ C )
% 2.83/1.66 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_129_diff__left__mono,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [B2: A,A2: A,C: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.83/1.66 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_130_diff__right__mono,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A,B2: A,C: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.66 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_131_diff__eq__diff__less__eq,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A,B2: A,C: A,D: A] :
% 2.83/1.66 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.83/1.66 = ( minus_minus @ A @ C @ D ) )
% 2.83/1.66 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.66 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_132_eq__iff__diff__eq__0,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( group_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ( ( ^ [Y3: A,Z2: A] : ( Y3 = Z2 ) )
% 2.83/1.66 = ( ^ [A4: A,B4: A] :
% 2.83/1.66 ( ( minus_minus @ A @ A4 @ B4 )
% 2.83/1.66 = ( zero_zero @ A ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_133_enat__minus__mono1,axiom,
% 2.83/1.66 ! [X: extended_enat,Y2: extended_enat,M: extended_enat] :
% 2.83/1.66 ( ( ord_less_eq @ extended_enat @ X @ Y2 )
% 2.83/1.66 => ( ord_less_eq @ extended_enat @ ( minus_minus @ extended_enat @ X @ M ) @ ( minus_minus @ extended_enat @ Y2 @ M ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_134_enat__diff__cancel__left,axiom,
% 2.83/1.66 ! [M: extended_enat,X: extended_enat,Y2: extended_enat] :
% 2.83/1.66 ( ( ord_less_eq @ extended_enat @ M @ X )
% 2.83/1.66 => ( ( ord_less_eq @ extended_enat @ M @ Y2 )
% 2.83/1.66 => ( ( ( minus_minus @ extended_enat @ X @ M )
% 2.83/1.66 = ( minus_minus @ extended_enat @ Y2 @ M ) )
% 2.83/1.66 = ( X = Y2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_135_le__iff__diff__le__0,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.83/1.66 => ( ( ord_less_eq @ A )
% 2.83/1.66 = ( ^ [A4: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_136_le__0__eq,axiom,
% 2.83/1.66 ! [N: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.83/1.66 = ( N
% 2.83/1.66 = ( zero_zero @ nat ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_137_less__eq__nat_Osimps_I1_J,axiom,
% 2.83/1.66 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_138_ltake__lappend,axiom,
% 2.83/1.66 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 2.83/1.66 ( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 2.83/1.66 = ( coindu268472904append @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ A @ ( minus_minus @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) ) @ Ys ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_139_ltakeWhile_Odisc_I2_J,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.83/1.66 ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
% 2.83/1.66 => ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_140_ltakeWhile_Odisc_I1_J,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.83/1.66 ( ( ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 | ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
% 2.83/1.66 => ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_141_ltakeWhile_Osimps_I3_J,axiom,
% 2.83/1.66 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
% 2.83/1.66 ( ~ ( coindu335574135_lnull @ A @ Xs )
% 2.83/1.66 => ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
% 2.83/1.66 => ( ( coindu1046438764le_lhd @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
% 2.83/1.66 = ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_142_minus__apply,axiom,
% 2.83/1.66 ! [B: $tType,A: $tType] :
% 2.83/1.66 ( ( minus @ B @ ( type2 @ B ) )
% 2.83/1.66 => ( ( minus_minus @ ( A > B ) )
% 2.83/1.66 = ( ^ [A5: A > B,B5: A > B,X3: A] : ( minus_minus @ B @ ( A5 @ X3 ) @ ( B5 @ X3 ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_143_of__nat__le__0__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [M: nat] :
% 2.83/1.66 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 2.83/1.66 = ( M
% 2.83/1.66 = ( zero_zero @ nat ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_144_ex__has__least__nat,axiom,
% 2.83/1.66 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 2.83/1.66 ( ( P @ K )
% 2.83/1.66 => ? [X2: A] :
% 2.83/1.66 ( ( P @ X2 )
% 2.83/1.66 & ! [Y5: A] :
% 2.83/1.66 ( ( P @ Y5 )
% 2.83/1.66 => ( ord_less_eq @ nat @ ( M @ X2 ) @ ( M @ Y5 ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_145_zero__natural_Orsp,axiom,
% 2.83/1.66 ( ( zero_zero @ nat )
% 2.83/1.66 = ( zero_zero @ nat ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_146_diff__self__eq__0,axiom,
% 2.83/1.66 ! [M: nat] :
% 2.83/1.66 ( ( minus_minus @ nat @ M @ M )
% 2.83/1.66 = ( zero_zero @ nat ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_147_diff__0__eq__0,axiom,
% 2.83/1.66 ! [N: nat] :
% 2.83/1.66 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 2.83/1.66 = ( zero_zero @ nat ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_148_of__nat__eq__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [M: nat,N: nat] :
% 2.83/1.66 ( ( ( semiring_1_of_nat @ A @ M )
% 2.83/1.66 = ( semiring_1_of_nat @ A @ N ) )
% 2.83/1.66 = ( M = N ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_149_diff__diff__cancel,axiom,
% 2.83/1.66 ! [I: nat,N: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ I @ N )
% 2.83/1.66 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 2.83/1.66 = I ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_150_diff__is__0__eq_H,axiom,
% 2.83/1.66 ! [M: nat,N: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ M @ N )
% 2.83/1.66 => ( ( minus_minus @ nat @ M @ N )
% 2.83/1.66 = ( zero_zero @ nat ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_151_diff__is__0__eq,axiom,
% 2.83/1.66 ! [M: nat,N: nat] :
% 2.83/1.66 ( ( ( minus_minus @ nat @ M @ N )
% 2.83/1.66 = ( zero_zero @ nat ) )
% 2.83/1.66 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_152_of__nat__eq__0__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [M: nat] :
% 2.83/1.66 ( ( ( semiring_1_of_nat @ A @ M )
% 2.83/1.66 = ( zero_zero @ A ) )
% 2.83/1.66 = ( M
% 2.83/1.66 = ( zero_zero @ nat ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_153_of__nat__0__eq__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [N: nat] :
% 2.83/1.66 ( ( ( zero_zero @ A )
% 2.83/1.66 = ( semiring_1_of_nat @ A @ N ) )
% 2.83/1.66 = ( ( zero_zero @ nat )
% 2.83/1.66 = N ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_154_of__nat__0,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 2.83/1.66 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 2.83/1.66 = ( zero_zero @ A ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_155_of__nat__le__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [M: nat,N: nat] :
% 2.83/1.66 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.83/1.66 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_156_diffs0__imp__equal,axiom,
% 2.83/1.66 ! [M: nat,N: nat] :
% 2.83/1.66 ( ( ( minus_minus @ nat @ M @ N )
% 2.83/1.66 = ( zero_zero @ nat ) )
% 2.83/1.66 => ( ( ( minus_minus @ nat @ N @ M )
% 2.83/1.66 = ( zero_zero @ nat ) )
% 2.83/1.66 => ( M = N ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_157_minus__nat_Odiff__0,axiom,
% 2.83/1.66 ! [M: nat] :
% 2.83/1.66 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 2.83/1.66 = M ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_158_eq__diff__iff,axiom,
% 2.83/1.66 ! [K: nat,M: nat,N: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ K @ M )
% 2.83/1.66 => ( ( ord_less_eq @ nat @ K @ N )
% 2.83/1.66 => ( ( ( minus_minus @ nat @ M @ K )
% 2.83/1.66 = ( minus_minus @ nat @ N @ K ) )
% 2.83/1.66 = ( M = N ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_159_le__diff__iff,axiom,
% 2.83/1.66 ! [K: nat,M: nat,N: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ K @ M )
% 2.83/1.66 => ( ( ord_less_eq @ nat @ K @ N )
% 2.83/1.66 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.83/1.66 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_160_Nat_Odiff__diff__eq,axiom,
% 2.83/1.66 ! [K: nat,M: nat,N: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ K @ M )
% 2.83/1.66 => ( ( ord_less_eq @ nat @ K @ N )
% 2.83/1.66 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.83/1.66 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_161_diff__le__mono,axiom,
% 2.83/1.66 ! [M: nat,N: nat,L: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ M @ N )
% 2.83/1.66 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_162_diff__le__self,axiom,
% 2.83/1.66 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_163_le__diff__iff_H,axiom,
% 2.83/1.66 ! [A2: nat,C: nat,B2: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ A2 @ C )
% 2.83/1.66 => ( ( ord_less_eq @ nat @ B2 @ C )
% 2.83/1.66 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B2 ) )
% 2.83/1.66 = ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_164_diff__le__mono2,axiom,
% 2.83/1.66 ! [M: nat,N: nat,L: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ M @ N )
% 2.83/1.66 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_165_of__nat__0__le__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_166_of__nat__diff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ring_1 @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [N: nat,M: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ N @ M )
% 2.83/1.66 => ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N ) )
% 2.83/1.66 = ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_167_fun__diff__def,axiom,
% 2.83/1.66 ! [B: $tType,A: $tType] :
% 2.83/1.66 ( ( minus @ B @ ( type2 @ B ) )
% 2.83/1.66 => ( ( minus_minus @ ( A > B ) )
% 2.83/1.66 = ( ^ [A5: A > B,B5: A > B,X3: A] : ( minus_minus @ B @ ( A5 @ X3 ) @ ( B5 @ X3 ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_168_real__arch__simple,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A] :
% 2.83/1.66 ? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_169_natural_Osize_I1_J,axiom,
% 2.83/1.66 ( ( code_size_natural @ ( zero_zero @ code_natural ) )
% 2.83/1.66 = ( zero_zero @ nat ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_170_natural_Osize_I3_J,axiom,
% 2.83/1.66 ( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
% 2.83/1.66 = ( zero_zero @ nat ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_171_diff__commute,axiom,
% 2.83/1.66 ! [I: nat,J: nat,K: nat] :
% 2.83/1.66 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 2.83/1.66 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_172_natural_Osimps_I4_J,axiom,
% 2.83/1.66 ! [T: $tType,F1: T,F22: code_natural > T] :
% 2.83/1.66 ( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 2.83/1.66 = F1 ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_173_natural_Osimps_I6_J,axiom,
% 2.83/1.66 ! [T: $tType,F1: T,F22: code_natural > T > T] :
% 2.83/1.66 ( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 2.83/1.66 = F1 ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_174_LeastMI2,axiom,
% 2.83/1.66 ! [B: $tType,A: $tType] :
% 2.83/1.66 ( ( ord @ B @ ( type2 @ B ) )
% 2.83/1.66 => ! [P: A > $o,X: A,M: A > B,Q: A > $o] :
% 2.83/1.66 ( ( P @ X )
% 2.83/1.66 => ( ! [Y: A] :
% 2.83/1.66 ( ( P @ Y )
% 2.83/1.66 => ( ord_less_eq @ B @ ( M @ X ) @ ( M @ Y ) ) )
% 2.83/1.66 => ( ! [X2: A] :
% 2.83/1.66 ( ( P @ X2 )
% 2.83/1.66 => ( ! [Y5: A] :
% 2.83/1.66 ( ( P @ Y5 )
% 2.83/1.66 => ( ord_less_eq @ B @ ( M @ X2 ) @ ( M @ Y5 ) ) )
% 2.83/1.66 => ( Q @ X2 ) ) )
% 2.83/1.66 => ( Q @ ( hilbert_LeastM @ A @ B @ M @ P ) ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_175_LeastM__nat__le,axiom,
% 2.83/1.66 ! [A: $tType,P: A > $o,X: A,M: A > nat] :
% 2.83/1.66 ( ( P @ X )
% 2.83/1.66 => ( ord_less_eq @ nat @ ( M @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) @ ( M @ X ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_176_LeastM__nat__lemma,axiom,
% 2.83/1.66 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 2.83/1.66 ( ( P @ K )
% 2.83/1.66 => ( ( P @ ( hilbert_LeastM @ A @ nat @ M @ P ) )
% 2.83/1.66 & ! [Y5: A] :
% 2.83/1.66 ( ( P @ Y5 )
% 2.83/1.66 => ( ord_less_eq @ nat @ ( M @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) @ ( M @ Y5 ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_177_natural_Osimps_I5_J,axiom,
% 2.83/1.66 ! [T: $tType,F1: T,F22: code_natural > T,Natural: code_natural] :
% 2.83/1.66 ( ( code_case_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
% 2.83/1.66 = ( F22 @ Natural ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_178_transfer__int__nat__relations_I3_J,axiom,
% 2.83/1.66 ! [X: nat,Y2: nat] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) )
% 2.83/1.66 = ( ord_less_eq @ nat @ X @ Y2 ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_179_zle__int,axiom,
% 2.83/1.66 ! [M: nat,N: nat] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 2.83/1.66 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_180_natural_Oinject,axiom,
% 2.83/1.66 ! [Natural: code_natural,Natural2: code_natural] :
% 2.83/1.66 ( ( ( code_Suc @ Natural )
% 2.83/1.66 = ( code_Suc @ Natural2 ) )
% 2.83/1.66 = ( Natural = Natural2 ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_181_natural_Osimps_I7_J,axiom,
% 2.83/1.66 ! [T: $tType,F1: T,F22: code_natural > T > T,Natural: code_natural] :
% 2.83/1.66 ( ( code_rec_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
% 2.83/1.66 = ( F22 @ Natural @ ( code_rec_natural @ T @ F1 @ F22 @ Natural ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_182_LeastM__natI,axiom,
% 2.83/1.66 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 2.83/1.66 ( ( P @ K )
% 2.83/1.66 => ( P @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_183_transfer__int__nat__relations_I1_J,axiom,
% 2.83/1.66 ! [X: nat,Y2: nat] :
% 2.83/1.66 ( ( ( semiring_1_of_nat @ int @ X )
% 2.83/1.66 = ( semiring_1_of_nat @ int @ Y2 ) )
% 2.83/1.66 = ( X = Y2 ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_184_int__int__eq,axiom,
% 2.83/1.66 ! [M: nat,N: nat] :
% 2.83/1.66 ( ( ( semiring_1_of_nat @ int @ M )
% 2.83/1.66 = ( semiring_1_of_nat @ int @ N ) )
% 2.83/1.66 = ( M = N ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_185_int__if__cong,axiom,
% 2.83/1.66 ! [P: $o,X: nat,Y2: nat] :
% 2.83/1.66 ( ( P
% 2.83/1.66 => ( ( semiring_1_of_nat @ int @ X )
% 2.83/1.66 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y2 ) ) ) )
% 2.83/1.66 & ( ~ P
% 2.83/1.66 => ( ( semiring_1_of_nat @ int @ Y2 )
% 2.83/1.66 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y2 ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_186_natural_Odistinct_I2_J,axiom,
% 2.83/1.66 ! [Natural3: code_natural] :
% 2.83/1.66 ( ( code_Suc @ Natural3 )
% 2.83/1.66 != ( zero_zero @ code_natural ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_187_natural_Odistinct_I1_J,axiom,
% 2.83/1.66 ! [Natural2: code_natural] :
% 2.83/1.66 ( ( zero_zero @ code_natural )
% 2.83/1.66 != ( code_Suc @ Natural2 ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_188_natural_Oexhaust,axiom,
% 2.83/1.66 ! [Y2: code_natural] :
% 2.83/1.66 ( ( Y2
% 2.83/1.66 != ( zero_zero @ code_natural ) )
% 2.83/1.66 => ~ ! [Natural4: code_natural] :
% 2.83/1.66 ( Y2
% 2.83/1.66 != ( code_Suc @ Natural4 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_189_natural_Oinducts,axiom,
% 2.83/1.66 ! [P: code_natural > $o,Natural: code_natural] :
% 2.83/1.66 ( ( P @ ( zero_zero @ code_natural ) )
% 2.83/1.66 => ( ! [Natural4: code_natural] :
% 2.83/1.66 ( ( P @ Natural4 )
% 2.83/1.66 => ( P @ ( code_Suc @ Natural4 ) ) )
% 2.83/1.66 => ( P @ Natural ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_190_int__diff__cases,axiom,
% 2.83/1.66 ! [Z: int] :
% 2.83/1.66 ~ ! [M2: nat,N2: nat] :
% 2.83/1.66 ( Z
% 2.83/1.66 != ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_191_zero__le__imp__eq__int,axiom,
% 2.83/1.66 ! [K: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.83/1.66 => ? [N2: nat] :
% 2.83/1.66 ( K
% 2.83/1.66 = ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_192_nonneg__int__cases,axiom,
% 2.83/1.66 ! [K: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.83/1.66 => ~ ! [N2: nat] :
% 2.83/1.66 ( K
% 2.83/1.66 != ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_193_nonneg__eq__int,axiom,
% 2.83/1.66 ! [Z: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.83/1.66 => ~ ! [M2: nat] :
% 2.83/1.66 ( Z
% 2.83/1.66 != ( semiring_1_of_nat @ int @ M2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_194_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
% 2.83/1.66 ! [Z: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ Z ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_195_transfer__int__nat__quantifiers_I1_J,axiom,
% 2.83/1.66 ! [P: int > $o] :
% 2.83/1.66 ( ( ! [X3: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
% 2.83/1.66 => ( P @ X3 ) ) )
% 2.83/1.66 = ( ! [X3: nat] : ( P @ ( semiring_1_of_nat @ int @ X3 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_196_transfer__int__nat__quantifiers_I2_J,axiom,
% 2.83/1.66 ! [P: int > $o] :
% 2.83/1.66 ( ( ? [X3: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
% 2.83/1.66 & ( P @ X3 ) ) )
% 2.83/1.66 = ( ? [X3: nat] : ( P @ ( semiring_1_of_nat @ int @ X3 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_197_transfer__int__nat__numerals_I1_J,axiom,
% 2.83/1.66 ( ( zero_zero @ int )
% 2.83/1.66 = ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_198_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
% 2.83/1.66 ! [X: nat,Y2: nat] :
% 2.83/1.66 ( ( nat_tsub @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) )
% 2.83/1.66 = ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_199_round__of__nat,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [N: nat] :
% 2.83/1.66 ( ( archimedean_round @ A @ ( semiring_1_of_nat @ A @ N ) )
% 2.83/1.66 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_200_round__0,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.83/1.66 => ( ( archimedean_round @ A @ ( zero_zero @ A ) )
% 2.83/1.66 = ( zero_zero @ int ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_201_minus__int__code_I1_J,axiom,
% 2.83/1.66 ! [K: int] :
% 2.83/1.66 ( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
% 2.83/1.66 = K ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_202_zero__integer_Orsp,axiom,
% 2.83/1.66 ( ( zero_zero @ int )
% 2.83/1.66 = ( zero_zero @ int ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_203_tsub__def,axiom,
% 2.83/1.66 ( nat_tsub
% 2.83/1.66 = ( ^ [X3: int,Y4: int] : ( if @ int @ ( ord_less_eq @ int @ Y4 @ X3 ) @ ( minus_minus @ int @ X3 @ Y4 ) @ ( zero_zero @ int ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_204_tsub__eq,axiom,
% 2.83/1.66 ! [Y2: int,X: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ Y2 @ X )
% 2.83/1.66 => ( ( nat_tsub @ X @ Y2 )
% 2.83/1.66 = ( minus_minus @ int @ X @ Y2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_205_round__mono,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A,Y2: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.83/1.66 => ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_206_ceiling__le__zero,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
% 2.83/1.66 = ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_207_of__int__0__le__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [Z: int] :
% 2.83/1.66 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 2.83/1.66 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_208_of__int__0,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ring_1 @ A @ ( type2 @ A ) )
% 2.83/1.66 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
% 2.83/1.66 = ( zero_zero @ A ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_209_of__int__0__eq__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ring_char_0 @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [Z: int] :
% 2.83/1.66 ( ( ( zero_zero @ A )
% 2.83/1.66 = ( ring_1_of_int @ A @ Z ) )
% 2.83/1.66 = ( Z
% 2.83/1.66 = ( zero_zero @ int ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_210_of__int__eq__0__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ring_char_0 @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [Z: int] :
% 2.83/1.66 ( ( ( ring_1_of_int @ A @ Z )
% 2.83/1.66 = ( zero_zero @ A ) )
% 2.83/1.66 = ( Z
% 2.83/1.66 = ( zero_zero @ int ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_211_of__int__le__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [W: int,Z: int] :
% 2.83/1.66 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
% 2.83/1.66 = ( ord_less_eq @ int @ W @ Z ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_212_ceiling__zero,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.83/1.66 => ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
% 2.83/1.66 = ( zero_zero @ int ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_213_of__int__diff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ring_1 @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [W: int,Z: int] :
% 2.83/1.66 ( ( ring_1_of_int @ A @ ( minus_minus @ int @ W @ Z ) )
% 2.83/1.66 = ( minus_minus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_214_of__int__of__nat__eq,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ring_1 @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [N: nat] :
% 2.83/1.66 ( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N ) )
% 2.83/1.66 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_215_ceiling__of__nat,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [N: nat] :
% 2.83/1.66 ( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
% 2.83/1.66 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_216_ceiling__diff__of__int,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A,Z: int] :
% 2.83/1.66 ( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) )
% 2.83/1.66 = ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ Z ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_217_of__int__le__0__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [Z: int] :
% 2.83/1.66 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
% 2.83/1.66 = ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_218_ex__le__of__int,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A] :
% 2.83/1.66 ? [Z3: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_219_ceiling__mono,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [Y2: A,X: A] :
% 2.83/1.66 ( ( ord_less_eq @ A @ Y2 @ X )
% 2.83/1.66 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y2 ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_220_le__of__int__ceiling,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_221_ceiling__le__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A,Z: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
% 2.83/1.66 = ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_222_of__int__nonneg,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [Z: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.83/1.66 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_223_ceiling__le,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [X: A,A2: int] :
% 2.83/1.66 ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) )
% 2.83/1.66 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_224_of__nat__nat,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ring_1 @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [Z: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.83/1.66 => ( ( semiring_1_of_nat @ A @ ( nat2 @ Z ) )
% 2.83/1.66 = ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_225_nat__ceiling__le__eq,axiom,
% 2.83/1.66 ! [X: real,A2: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A2 )
% 2.83/1.66 = ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_226_nat__int,axiom,
% 2.83/1.66 ! [N: nat] :
% 2.83/1.66 ( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
% 2.83/1.66 = N ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_227_nat__0,axiom,
% 2.83/1.66 ( ( nat2 @ ( zero_zero @ int ) )
% 2.83/1.66 = ( zero_zero @ nat ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_228_nat__0__iff,axiom,
% 2.83/1.66 ! [I: int] :
% 2.83/1.66 ( ( ( nat2 @ I )
% 2.83/1.66 = ( zero_zero @ nat ) )
% 2.83/1.66 = ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_229_nat__le__0,axiom,
% 2.83/1.66 ! [Z: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
% 2.83/1.66 => ( ( nat2 @ Z )
% 2.83/1.66 = ( zero_zero @ nat ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_230_int__nat__eq,axiom,
% 2.83/1.66 ! [Z: int] :
% 2.83/1.66 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.83/1.66 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 2.83/1.66 = Z ) )
% 2.83/1.66 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.83/1.66 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 2.83/1.66 = ( zero_zero @ int ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_231_real__nat__ceiling__ge,axiom,
% 2.83/1.66 ! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_232_nat__mono,axiom,
% 2.83/1.66 ! [X: int,Y2: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ X @ Y2 )
% 2.83/1.66 => ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_233_transfer__nat__int__relations_I3_J,axiom,
% 2.83/1.66 ! [X: int,Y2: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.83/1.66 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
% 2.83/1.66 => ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) )
% 2.83/1.66 = ( ord_less_eq @ int @ X @ Y2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_234_nat__le__iff,axiom,
% 2.83/1.66 ! [X: int,N: nat] :
% 2.83/1.66 ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
% 2.83/1.66 = ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_235_int__eq__iff,axiom,
% 2.83/1.66 ! [M: nat,Z: int] :
% 2.83/1.66 ( ( ( semiring_1_of_nat @ int @ M )
% 2.83/1.66 = Z )
% 2.83/1.66 = ( ( M
% 2.83/1.66 = ( nat2 @ Z ) )
% 2.83/1.66 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_236_nat__0__le,axiom,
% 2.83/1.66 ! [Z: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.83/1.66 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 2.83/1.66 = Z ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_237_nat__eq__iff,axiom,
% 2.83/1.66 ! [W: int,M: nat] :
% 2.83/1.66 ( ( ( nat2 @ W )
% 2.83/1.66 = M )
% 2.83/1.66 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.83/1.66 => ( W
% 2.83/1.66 = ( semiring_1_of_nat @ int @ M ) ) )
% 2.83/1.66 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.83/1.66 => ( M
% 2.83/1.66 = ( zero_zero @ nat ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_238_nat__eq__iff2,axiom,
% 2.83/1.66 ! [M: nat,W: int] :
% 2.83/1.66 ( ( M
% 2.83/1.66 = ( nat2 @ W ) )
% 2.83/1.66 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.83/1.66 => ( W
% 2.83/1.66 = ( semiring_1_of_nat @ int @ M ) ) )
% 2.83/1.66 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.83/1.66 => ( M
% 2.83/1.66 = ( zero_zero @ nat ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_239_le__nat__iff,axiom,
% 2.83/1.66 ! [K: int,N: nat] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.83/1.66 => ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
% 2.83/1.66 = ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_240_nat__diff__distrib_H,axiom,
% 2.83/1.66 ! [X: int,Y2: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.83/1.66 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
% 2.83/1.66 => ( ( nat2 @ ( minus_minus @ int @ X @ Y2 ) )
% 2.83/1.66 = ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_241_nat__diff__distrib,axiom,
% 2.83/1.66 ! [Z4: int,Z: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z4 )
% 2.83/1.66 => ( ( ord_less_eq @ int @ Z4 @ Z )
% 2.83/1.66 => ( ( nat2 @ ( minus_minus @ int @ Z @ Z4 ) )
% 2.83/1.66 = ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_242_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,
% 2.83/1.66 ! [X: int,Y2: int] :
% 2.83/1.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.83/1.66 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
% 2.83/1.66 => ( ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) )
% 2.83/1.66 = ( nat2 @ ( nat_tsub @ X @ Y2 ) ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_243_int__eq__iff__numeral,axiom,
% 2.83/1.66 ! [M: nat,V: num] :
% 2.83/1.66 ( ( ( semiring_1_of_nat @ int @ M )
% 2.83/1.66 = ( numeral_numeral @ int @ V ) )
% 2.83/1.66 = ( ( M
% 2.83/1.66 = ( nat2 @ ( numeral_numeral @ int @ V ) ) )
% 2.83/1.66 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_244_nat__abs__int__diff,axiom,
% 2.83/1.66 ! [A2: nat,B2: nat] :
% 2.83/1.66 ( ( ( ord_less_eq @ nat @ A2 @ B2 )
% 2.83/1.66 => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
% 2.83/1.66 = ( minus_minus @ nat @ B2 @ A2 ) ) )
% 2.83/1.66 & ( ~ ( ord_less_eq @ nat @ A2 @ B2 )
% 2.83/1.66 => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
% 2.83/1.66 = ( minus_minus @ nat @ A2 @ B2 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_245_numeral__eq__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [M: num,N: num] :
% 2.83/1.66 ( ( ( numeral_numeral @ A @ M )
% 2.83/1.66 = ( numeral_numeral @ A @ N ) )
% 2.83/1.66 = ( M = N ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_246_abs__idempotent,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A] :
% 2.83/1.66 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 2.83/1.66 = ( abs_abs @ A @ A2 ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_247_numeral__le__iff,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [M: num,N: num] :
% 2.83/1.66 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 2.83/1.66 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_248_abs__0__eq,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A] :
% 2.83/1.66 ( ( ( zero_zero @ A )
% 2.83/1.66 = ( abs_abs @ A @ A2 ) )
% 2.83/1.66 = ( A2
% 2.83/1.66 = ( zero_zero @ A ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(fact_249_abs__eq__0,axiom,
% 2.83/1.66 ! [A: $tType] :
% 2.83/1.66 ( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
% 2.83/1.66 => ! [A2: A] :
% 2.83/1.66 ( ( ( abs_abs @ A @ A2 )
% 2.83/1.66 = ( zero_zero @ A ) )
% 2.83/1.66 = ( A2
% 2.83/1.66 = ( zero_zero @ A ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 2.83/1.66 ! [A6: $tType,A7: $tType] :
% 2.83/1.66 ( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
% 2.83/1.66 => ( comple187826305attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.83/1.66 ! [A6: $tType,A7: $tType] :
% 2.83/1.66 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 2.83/1.66 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.83/1.66 ! [A6: $tType,A7: $tType] :
% 2.83/1.66 ( ( order @ A7 @ ( type2 @ A7 ) )
% 2.83/1.66 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_fun___Orderings_Oord,axiom,
% 2.83/1.66 ! [A6: $tType,A7: $tType] :
% 2.83/1.66 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 2.83/1.66 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_fun___Groups_Ominus,axiom,
% 2.83/1.66 ! [A6: $tType,A7: $tType] :
% 2.83/1.66 ( ( minus @ A7 @ ( type2 @ A7 ) )
% 2.83/1.66 => ( minus @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
% 2.83/1.66 ordere142940540dd_abs @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.83/1.66 cancel146912293up_add @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 2.83/1.66 cancel1352612707id_add @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 2.83/1.66 ordered_ab_group_add @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 2.83/1.66 linordered_semidom @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 2.83/1.66 linordered_idom @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 2.83/1.66 semiring_char_0 @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 2.83/1.66 preorder @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 2.83/1.66 linorder @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 2.83/1.66 semiring_1 @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 2.83/1.66 group_add @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 2.83/1.66 order @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
% 2.83/1.66 neg_numeral @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 2.83/1.66 ring_char_0 @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 2.83/1.66 ord @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Rings_Oring__1,axiom,
% 2.83/1.66 ring_1 @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Groups_Ominus_4,axiom,
% 2.83/1.66 minus @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Num_Onumeral,axiom,
% 2.83/1.66 numeral @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 2.83/1.66 zero @ int @ ( type2 @ int ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.83/1.66 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 2.83/1.66
% 2.83/1.66 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_5,axiom,
% 2.83/1.67 cancel146912293up_add @ nat @ ( type2 @ nat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_6,axiom,
% 2.83/1.67 cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_7,axiom,
% 2.83/1.67 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 2.83/1.67 comm_monoid_diff @ nat @ ( type2 @ nat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_8,axiom,
% 2.83/1.67 semiring_char_0 @ nat @ ( type2 @ nat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Nat_Onat___Orderings_Opreorder_9,axiom,
% 2.83/1.67 preorder @ nat @ ( type2 @ nat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Nat_Onat___Orderings_Olinorder_10,axiom,
% 2.83/1.67 linorder @ nat @ ( type2 @ nat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Nat_Onat___Rings_Osemiring__1_11,axiom,
% 2.83/1.67 semiring_1 @ nat @ ( type2 @ nat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Nat_Onat___Orderings_Oorder_12,axiom,
% 2.83/1.67 order @ nat @ ( type2 @ nat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Nat_Onat___Orderings_Oord_13,axiom,
% 2.83/1.67 ord @ nat @ ( type2 @ nat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Nat_Onat___Groups_Ominus_14,axiom,
% 2.83/1.67 minus @ nat @ ( type2 @ nat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Nat_Onat___Num_Onumeral_15,axiom,
% 2.83/1.67 numeral @ nat @ ( type2 @ nat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Nat_Onat___Groups_Ozero_16,axiom,
% 2.83/1.67 zero @ nat @ ( type2 @ nat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Num_Onum___Orderings_Opreorder_17,axiom,
% 2.83/1.67 preorder @ num @ ( type2 @ num ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Num_Onum___Orderings_Olinorder_18,axiom,
% 2.83/1.67 linorder @ num @ ( type2 @ num ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Num_Onum___Orderings_Oorder_19,axiom,
% 2.83/1.67 order @ num @ ( type2 @ num ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Num_Onum___Orderings_Oord_20,axiom,
% 2.83/1.67 ord @ num @ ( type2 @ num ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_21,axiom,
% 2.83/1.67 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_HOL_Obool___Orderings_Opreorder_22,axiom,
% 2.83/1.67 preorder @ $o @ ( type2 @ $o ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_HOL_Obool___Orderings_Olinorder_23,axiom,
% 2.83/1.67 linorder @ $o @ ( type2 @ $o ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_HOL_Obool___Orderings_Oorder_24,axiom,
% 2.83/1.67 order @ $o @ ( type2 @ $o ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_HOL_Obool___Orderings_Oord_25,axiom,
% 2.83/1.67 ord @ $o @ ( type2 @ $o ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_HOL_Obool___Groups_Ominus_26,axiom,
% 2.83/1.67 minus @ $o @ ( type2 @ $o ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 2.83/1.67 archim1804426504_field @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_27,axiom,
% 2.83/1.67 ordere142940540dd_abs @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling,axiom,
% 2.83/1.67 archim1727834104eiling @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_28,axiom,
% 2.83/1.67 cancel146912293up_add @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_29,axiom,
% 2.83/1.67 cancel1352612707id_add @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_30,axiom,
% 2.83/1.67 ordered_ab_group_add @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_31,axiom,
% 2.83/1.67 linordered_semidom @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Rings_Olinordered__idom_32,axiom,
% 2.83/1.67 linordered_idom @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_33,axiom,
% 2.83/1.67 semiring_char_0 @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Orderings_Opreorder_34,axiom,
% 2.83/1.67 preorder @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Orderings_Olinorder_35,axiom,
% 2.83/1.67 linorder @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Rings_Osemiring__1_36,axiom,
% 2.83/1.67 semiring_1 @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Groups_Ogroup__add_37,axiom,
% 2.83/1.67 group_add @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Orderings_Oorder_38,axiom,
% 2.83/1.67 order @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Num_Oneg__numeral_39,axiom,
% 2.83/1.67 neg_numeral @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Nat_Oring__char__0_40,axiom,
% 2.83/1.67 ring_char_0 @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Orderings_Oord_41,axiom,
% 2.83/1.67 ord @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Rings_Oring__1_42,axiom,
% 2.83/1.67 ring_1 @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Groups_Ominus_43,axiom,
% 2.83/1.67 minus @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Num_Onumeral_44,axiom,
% 2.83/1.67 numeral @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Real_Oreal___Groups_Ozero_45,axiom,
% 2.83/1.67 zero @ real @ ( type2 @ real ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_46,axiom,
% 2.83/1.67 canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_47,axiom,
% 2.83/1.67 comple187826305attice @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_48,axiom,
% 2.83/1.67 semiring_char_0 @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_49,axiom,
% 2.83/1.67 preorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_50,axiom,
% 2.83/1.67 linorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_51,axiom,
% 2.83/1.67 semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_52,axiom,
% 2.83/1.67 order @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_53,axiom,
% 2.83/1.67 ord @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Extended__Nat_Oenat___Groups_Ominus_54,axiom,
% 2.83/1.67 minus @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Extended__Nat_Oenat___Num_Onumeral_55,axiom,
% 2.83/1.67 numeral @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_56,axiom,
% 2.83/1.67 zero @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_57,axiom,
% 2.83/1.67 cancel146912293up_add @ code_natural @ ( type2 @ code_natural ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_58,axiom,
% 2.83/1.67 cancel1352612707id_add @ code_natural @ ( type2 @ code_natural ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_59,axiom,
% 2.83/1.67 comm_monoid_diff @ code_natural @ ( type2 @ code_natural ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_60,axiom,
% 2.83/1.67 preorder @ code_natural @ ( type2 @ code_natural ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_61,axiom,
% 2.83/1.67 linorder @ code_natural @ ( type2 @ code_natural ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_62,axiom,
% 2.83/1.67 semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_63,axiom,
% 2.83/1.67 order @ code_natural @ ( type2 @ code_natural ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Code__Numeral_Onatural___Orderings_Oord_64,axiom,
% 2.83/1.67 ord @ code_natural @ ( type2 @ code_natural ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Code__Numeral_Onatural___Groups_Ominus_65,axiom,
% 2.83/1.67 minus @ code_natural @ ( type2 @ code_natural ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Code__Numeral_Onatural___Num_Onumeral_66,axiom,
% 2.83/1.67 numeral @ code_natural @ ( type2 @ code_natural ) ).
% 2.83/1.67
% 2.83/1.67 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_67,axiom,
% 2.83/1.67 zero @ code_natural @ ( type2 @ code_natural ) ).
% 2.83/1.67
% 2.83/1.67 thf(help_If_3_1_T,axiom,
% 2.83/1.67 ! [P: $o] :
% 2.83/1.67 ( ( P = $true )
% 2.83/1.67 | ( P = $false ) ) ).
% 2.83/1.67
% 2.83/1.67 thf(help_If_2_1_T,axiom,
% 2.83/1.67 ! [A: $tType,X: A,Y2: A] :
% 2.83/1.67 ( ( if @ A @ $false @ X @ Y2 )
% 2.83/1.67 = Y2 ) ).
% 2.83/1.67
% 2.83/1.67 thf(help_If_1_1_T,axiom,
% 2.83/1.67 ! [A: $tType,X: A,Y2: A] :
% 2.83/1.67 ( ( if @ A @ $true @ X @ Y2 )
% 2.83/1.67 = X ) ).
% 2.83/1.67
% 2.83/1.67 thf(conj_0,hypothesis,
% 2.83/1.67 ord_less_eq @ extended_enat @ ( coindu1018505716length @ a @ xs ) @ m ).
% 2.83/1.67
% 2.83/1.67 thf(conj_1,conjecture,
% 2.83/1.67 ( ( coindu191418589_ldrop @ a @ m @ xs )
% 2.83/1.67 = ( coindu1598213697e_LNil @ a ) ) ).
% 2.83/1.67 ------------------------
% 2.88/1.70 % (17445)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_17343 for (2999ds/27Mi)
% 2.88/1.70 % (17447)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_17343 for (2999ds/2Mi)
% 2.88/1.70 % (17448)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_17343 for (2999ds/275Mi)
% 2.88/1.70 % (17446)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_17343 for (2999ds/2Mi)
% 2.88/1.70 % (17443)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_17343 for (2999ds/183Mi)
% 2.88/1.70 % (17444)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_17343 for (2999ds/4Mi)
% 2.88/1.70 % (17449)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_17343 for (2999ds/18Mi)
% 2.88/1.70 % (17446)Instruction limit reached!
% 2.88/1.70 % (17446)------------------------------
% 2.88/1.70 % (17446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.70 % (17446)Termination reason: Unknown
% 2.88/1.70 % (17447)Instruction limit reached!
% 2.88/1.70 % (17447)------------------------------
% 2.88/1.70 % (17447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.70 % (17446)Termination phase: shuffling
% 2.88/1.70
% 2.88/1.70 % (17446)Memory used [KB]: 1279
% 2.88/1.70 % (17446)Time elapsed: 0.003 s
% 2.88/1.70 % (17446)Instructions burned: 2 (million)
% 2.88/1.70 % (17446)------------------------------
% 2.88/1.70 % (17446)------------------------------
% 2.88/1.70 % (17447)Termination reason: Unknown
% 2.88/1.70 % (17447)Termination phase: shuffling
% 2.88/1.70
% 2.88/1.70 % (17447)Memory used [KB]: 1279
% 2.88/1.70 % (17447)Time elapsed: 0.003 s
% 2.88/1.70 % (17447)Instructions burned: 2 (million)
% 2.88/1.70 % (17447)------------------------------
% 2.88/1.70 % (17447)------------------------------
% 2.88/1.70 % (17444)Instruction limit reached!
% 2.88/1.70 % (17444)------------------------------
% 2.88/1.70 % (17444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.70 % (17444)Termination reason: Unknown
% 2.88/1.70 % (17444)Termination phase: shuffling
% 2.88/1.70
% 2.88/1.70 % (17444)Memory used [KB]: 1407
% 2.88/1.70 % (17444)Time elapsed: 0.005 s
% 2.88/1.70 % (17444)Instructions burned: 6 (million)
% 2.88/1.70 % (17444)------------------------------
% 2.88/1.70 % (17444)------------------------------
% 2.88/1.71 % (17449)Instruction limit reached!
% 2.88/1.71 % (17449)------------------------------
% 2.88/1.71 % (17449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.71 % (17449)Termination reason: Unknown
% 2.88/1.71 % (17449)Termination phase: Property scanning
% 2.88/1.71
% 2.88/1.71 % (17449)Memory used [KB]: 1663
% 2.88/1.71 % (17449)Time elapsed: 0.012 s
% 2.88/1.71 % (17449)Instructions burned: 19 (million)
% 2.88/1.71 % (17449)------------------------------
% 2.88/1.71 % (17449)------------------------------
% 2.88/1.71 % (17445)Instruction limit reached!
% 2.88/1.71 % (17445)------------------------------
% 2.88/1.71 % (17445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.71 % (17445)Termination reason: Unknown
% 2.88/1.71 % (17445)Termination phase: Preprocessing 1
% 2.88/1.71
% 2.88/1.71 % (17445)Memory used [KB]: 1663
% 2.88/1.71 % (17445)Time elapsed: 0.016 s
% 2.88/1.71 % (17445)Instructions burned: 28 (million)
% 2.88/1.71 % (17445)------------------------------
% 2.88/1.71 % (17445)------------------------------
% 2.88/1.71 % (17450)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17343 for (2999ds/3Mi)
% 2.88/1.71 % (17451)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_17343 for (2999ds/37Mi)
% 2.88/1.72 % (17450)Instruction limit reached!
% 2.88/1.72 % (17450)------------------------------
% 2.88/1.72 % (17450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.72 % (17450)Termination reason: Unknown
% 2.88/1.72 % (17450)Termination phase: shuffling
% 2.88/1.72
% 2.88/1.72 % (17450)Memory used [KB]: 1407
% 2.88/1.72 % (17450)Time elapsed: 0.003 s
% 2.88/1.72 % (17450)Instructions burned: 3 (million)
% 2.88/1.72 % (17450)------------------------------
% 2.88/1.72 % (17450)------------------------------
% 2.88/1.72 % (17452)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_17343 for (2999ds/15Mi)
% 2.88/1.72 % (17448)First to succeed.
% 2.88/1.72 % (17448)Refutation found. Thanks to Tanya!
% 2.88/1.72 % SZS status Theorem for DTF2THF_17343
% 2.88/1.72 % SZS output start Proof for DTF2THF_17343
% 2.88/1.72 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.88/1.72 thf(type_def_7, type, int: $tType).
% 2.88/1.72 thf(type_def_8, type, code_natural: $tType).
% 2.88/1.72 thf(type_def_9, type, nat: $tType).
% 2.88/1.72 thf(type_def_10, type, coindu1593790203_llist: ($tType) > $tType).
% 2.88/1.72 thf(type_def_11, type, extended_enat: $tType).
% 2.88/1.72 thf(type_def_12, type, num: $tType).
% 2.88/1.72 thf(type_def_13, type, a: $tType).
% 2.88/1.72 thf(type_def_14, type, real: $tType).
% 2.88/1.72 thf(func_def_0, type, coindu1593790203_llist: ($tType) > $tType).
% 2.88/1.72 thf(func_def_1, type, code_natural: $tType).
% 2.88/1.72 thf(func_def_2, type, extended_enat: $tType).
% 2.88/1.72 thf(func_def_3, type, real: $tType).
% 2.88/1.72 thf(func_def_4, type, num: $tType).
% 2.88/1.72 thf(func_def_5, type, nat: $tType).
% 2.88/1.72 thf(func_def_6, type, int: $tType).
% 2.88/1.72 thf(func_def_7, type, itself: ($tType) > $tType).
% 2.88/1.72 thf(func_def_8, type, a: $tType).
% 2.88/1.72 thf(func_def_9, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_10, type, zero: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_11, type, numeral: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_12, type, minus: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_13, type, ring_1: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_14, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_15, type, ring_char_0: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_16, type, neg_numeral: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_17, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_18, type, group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_19, type, semiring_1: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_20, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_21, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_22, type, semiring_char_0: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_23, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_24, type, comm_monoid_diff: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_25, type, linordered_semidom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_26, type, ordered_ab_group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_27, type, cancel1352612707id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_28, type, cancel146912293up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_29, type, archim1727834104eiling: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_30, type, ordere142940540dd_abs: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_31, type, comple187826305attice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_32, type, archim1804426504_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_33, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.72 thf(func_def_34, type, archimedean_ceiling: !>[X0: $tType]:(X0 > int)).
% 2.88/1.72 thf(func_def_35, type, archimedean_round: !>[X0: $tType]:(X0 > int)).
% 2.88/1.72 thf(func_def_36, type, code_Suc: code_natural > code_natural).
% 2.88/1.72 thf(func_def_37, type, code_case_natural: !>[X0: $tType]:(X0 > (code_natural > X0) > code_natural > X0)).
% 2.88/1.72 thf(func_def_38, type, code_rec_natural: !>[X0: $tType]:(X0 > (code_natural > X0 > X0) > code_natural > X0)).
% 2.88/1.72 thf(func_def_39, type, code_size_natural: code_natural > nat).
% 2.88/1.72 thf(func_def_40, type, coindu493225792length: !>[X0: $tType]:(nat > coindu1593790203_llist @ X0 > extended_enat)).
% 2.88/1.72 thf(func_def_41, type, coindu268472904append: !>[X0: $tType]:(coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 2.88/1.72 thf(func_def_42, type, coindu191418589_ldrop: !>[X0: $tType]:(extended_enat > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 2.88/1.72 thf(func_def_43, type, coindu1018505716length: !>[X0: $tType]:(coindu1593790203_llist @ X0 > extended_enat)).
% 2.88/1.72 thf(func_def_44, type, coindu1598213697e_LNil: !>[X0: $tType]:(coindu1593790203_llist @ X0)).
% 2.88/1.72 thf(func_def_45, type, coindu1046438764le_lhd: !>[X0: $tType]:(coindu1593790203_llist @ X0 > X0)).
% 2.88/1.72 thf(func_def_46, type, coindu335574135_lnull: !>[X0: $tType]:(coindu1593790203_llist @ X0 > $o)).
% 2.88/1.72 thf(func_def_47, type, coindu567634248member: !>[X0: $tType]:(X0 > coindu1593790203_llist @ X0 > $o)).
% 2.88/1.72 thf(func_def_48, type, coindu1696667936prefix: !>[X0: $tType]:(coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0 > $o)).
% 2.88/1.72 thf(func_def_49, type, coindu574146665prefix: !>[X0: $tType]:(coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0 > $o)).
% 2.88/1.72 thf(func_def_50, type, coindu1802687541_ltake: !>[X0: $tType]:(extended_enat > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 2.88/1.72 thf(func_def_51, type, coindu721411036eWhile: !>[X0: $tType]:((X0 > $o) > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 2.88/1.72 thf(func_def_52, type, extended_case_enat: !>[X0: $tType]:((nat > X0) > X0 > extended_enat > X0)).
% 2.88/1.72 thf(func_def_53, type, extended_the_enat: extended_enat > nat).
% 2.88/1.72 thf(func_def_54, type, abs_abs: !>[X0: $tType]:(X0 > X0)).
% 2.88/1.72 thf(func_def_55, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.88/1.72 thf(func_def_56, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.88/1.72 thf(func_def_57, type, hilbert_LeastM: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0)).
% 2.88/1.72 thf(func_def_58, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.88/1.72 thf(func_def_59, type, nat2: int > nat).
% 2.88/1.72 thf(func_def_60, type, ring_1_of_int: !>[X0: $tType]:(int > X0)).
% 2.88/1.72 thf(func_def_61, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 2.88/1.72 thf(func_def_62, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.88/1.72 thf(func_def_63, type, nat_tsub: int > int > int).
% 2.88/1.72 thf(func_def_64, type, neg_numeral_dbl: !>[X0: $tType]:(X0 > X0)).
% 2.88/1.72 thf(func_def_65, type, numeral_numeral: !>[X0: $tType]:(num > X0)).
% 2.88/1.72 thf(func_def_66, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.88/1.72 thf(func_def_67, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.88/1.72 thf(func_def_68, type, m: extended_enat).
% 2.88/1.72 thf(func_def_69, type, xs: coindu1593790203_llist @ a).
% 2.88/1.72 thf(func_def_81, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.88/1.72 thf(f1214,plain,(
% 2.88/1.72 $false),
% 2.88/1.72 inference(subsumption_resolution,[status(thm)],[f1213,f1079])).
% 2.88/1.72 thf(f1079,plain,(
% 2.88/1.72 ((ord_less_eq @ extended_enat @ (coindu1018505716length @ a @ xs) @ m) = $true)),
% 2.88/1.72 inference(cnf_transformation,[status(thm)],[f900])).
% 2.88/1.72 thf(f900,plain,(
% 2.88/1.72 ((ord_less_eq @ extended_enat @ (coindu1018505716length @ a @ xs) @ m) = $true)),
% 2.88/1.72 inference(fool_elimination,[status(thm)],[f899])).
% 2.88/1.72 thf(f899,plain,(
% 2.88/1.72 (ord_less_eq @ extended_enat @ (coindu1018505716length @ a @ xs) @ m)),
% 2.88/1.72 inference(rectify,[status(thm)],[f345])).
% 2.88/1.72 thf(f345,axiom,(
% 2.88/1.72 (ord_less_eq @ extended_enat @ (coindu1018505716length @ a @ xs) @ m)),
% 2.88/1.72 file('/export/starexec/sandbox/tmp/tmp.bImcdsmqd4/DTF2THF_17343.p',conj_0)).
% 2.88/1.72 thf(f1213,plain,(
% 2.88/1.72 ((ord_less_eq @ extended_enat @ (coindu1018505716length @ a @ xs) @ m) != $true)),
% 2.88/1.72 inference(trivial_inequality_removal,[status(thm)],[f1211])).
% 2.88/1.72 thf(f1211,plain,(
% 2.88/1.72 ((ord_less_eq @ extended_enat @ (coindu1018505716length @ a @ xs) @ m) != $true) | (coindu1598213697e_LNil @ a != coindu1598213697e_LNil @ a)),
% 2.88/1.72 inference(superposition,[status(thm)],[f1128,f1181])).
% 2.88/1.72 thf(f1181,plain,(
% 2.88/1.72 ( ! [X0 : $tType,X2 : coindu1593790203_llist @ X0,X1 : extended_enat] : ((coindu1598213697e_LNil @ X0 = (coindu191418589_ldrop @ X0 @ X1 @ X2)) | ((ord_less_eq @ extended_enat @ (coindu1018505716length @ X0 @ X2) @ X1) != $true)) )),
% 2.88/1.72 inference(cnf_transformation,[status(thm)],[f1066])).
% 2.88/1.72 thf(f1066,plain,(
% 2.88/1.72 ! [X0 : $tType,X1 : extended_enat,X2 : coindu1593790203_llist @ X0] : ((((ord_less_eq @ extended_enat @ (coindu1018505716length @ X0 @ X2) @ X1) = $true) | (coindu1598213697e_LNil @ X0 != (coindu191418589_ldrop @ X0 @ X1 @ X2))) & ((coindu1598213697e_LNil @ X0 = (coindu191418589_ldrop @ X0 @ X1 @ X2)) | ((ord_less_eq @ extended_enat @ (coindu1018505716length @ X0 @ X2) @ X1) != $true)))),
% 2.88/1.72 inference(nnf_transformation,[status(thm)],[f692])).
% 2.88/1.72 thf(f692,plain,(
% 2.88/1.72 ! [X0 : $tType,X1 : extended_enat,X2 : coindu1593790203_llist @ X0] : (((ord_less_eq @ extended_enat @ (coindu1018505716length @ X0 @ X2) @ X1) = $true) <=> (coindu1598213697e_LNil @ X0 = (coindu191418589_ldrop @ X0 @ X1 @ X2)))),
% 2.88/1.72 inference(fool_elimination,[status(thm)],[f691])).
% 2.88/1.72 thf(f691,plain,(
% 2.88/1.72 ! [X0 : $tType,X1 : extended_enat,X2 : coindu1593790203_llist @ X0] : ((coindu1598213697e_LNil @ X0 = (coindu191418589_ldrop @ X0 @ X1 @ X2)) = (ord_less_eq @ extended_enat @ (coindu1018505716length @ X0 @ X2) @ X1))),
% 2.88/1.72 inference(rectify,[status(thm)],[f2])).
% 2.88/1.72 thf(f2,axiom,(
% 2.88/1.72 ! [X0 : $tType,X3 : extended_enat,X4 : coindu1593790203_llist @ X0] : ((coindu1598213697e_LNil @ X0 = (coindu191418589_ldrop @ X0 @ X3 @ X4)) = (ord_less_eq @ extended_enat @ (coindu1018505716length @ X0 @ X4) @ X3))),
% 2.88/1.72 file('/export/starexec/sandbox/tmp/tmp.bImcdsmqd4/DTF2THF_17343.p',fact_1_ldrop__eq__LNil)).
% 2.88/1.72 thf(f1128,plain,(
% 2.88/1.72 ((coindu191418589_ldrop @ a @ m @ xs) != coindu1598213697e_LNil @ a)),
% 2.88/1.72 inference(cnf_transformation,[status(thm)],[f949])).
% 2.88/1.72 thf(f949,plain,(
% 2.88/1.72 ((coindu191418589_ldrop @ a @ m @ xs) != coindu1598213697e_LNil @ a)),
% 2.88/1.72 inference(flattening,[status(thm)],[f347])).
% 2.88/1.72 thf(f347,negated_conjecture,(
% 2.88/1.72 ~((coindu191418589_ldrop @ a @ m @ xs) = coindu1598213697e_LNil @ a)),
% 2.88/1.72 inference(negated_conjecture,[status(cth)],[f346])).
% 2.88/1.72 thf(f346,conjecture,(
% 2.88/1.72 ((coindu191418589_ldrop @ a @ m @ xs) = coindu1598213697e_LNil @ a)),
% 2.88/1.72 file('/export/starexec/sandbox/tmp/tmp.bImcdsmqd4/DTF2THF_17343.p',conj_1)).
% 2.88/1.72 % SZS output end Proof for DTF2THF_17343
% 2.88/1.72 % (17448)------------------------------
% 2.88/1.72 % (17448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.72 % (17448)Termination reason: Refutation
% 2.88/1.72
% 2.88/1.72 % (17448)Memory used [KB]: 6396
% 2.88/1.72 % (17448)Time elapsed: 0.026 s
% 2.88/1.72 % (17448)Instructions burned: 46 (million)
% 2.88/1.72 % (17448)------------------------------
% 2.88/1.72 % (17448)------------------------------
% 2.88/1.72 % (17442)Success in time 0.054 s
%------------------------------------------------------------------------------